Subgroups — #30680527

Order 16768.

1032 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22554061	C₂	1	2	yes	maximal, complemented
2	#22554058	C₂	1	2	yes	maximal, complemented
3	#22554036	C₂	1	1	yes	maximal, characteristic, complemented
4	#22554034	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553991	C₂	1	2	yes	maximal, complemented
6	#22553989	C₂	1	2	yes	maximal, complemented
7	Q₈×C₂²⋊D₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#22553973	C₂	1	1	yes	maximal, characteristic, complemented
9	#22553972	C₂	1	1	yes	maximal, characteristic, complemented
10	#22553958	C₂	1	1	yes	maximal, characteristic, complemented
11	#22553953	C₂	1	1	yes	maximal, characteristic, complemented
12	#22553937	C₂	1	2	yes	maximal, complemented
13	#22553936	C₂	1	2	yes	maximal, complemented
14	#22553935	C₂	1	1	yes	maximal, characteristic, complemented
15	#22553934	C₂	1	1	yes	maximal, characteristic, complemented
16	#22553932	C₂	1	2	yes	maximal, complemented
17	#22553895	C₂	1	2	yes	maximal, complemented
18	#22553887	C₂	1	1	yes	maximal, characteristic, complemented
19	#22553886	C₂	1	1	yes	maximal, characteristic, complemented
20	#22553864	C₂	1	1	yes	maximal, characteristic, complemented
21	#22553842	C₂	1	1	yes	maximal, characteristic, complemented
22	C₂×#16862235	C₂	1	1	yes	maximal, characteristic, complemented
23	D₄×Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#14523514	C₂²	1	1	yes	characteristic
25	C₂×#14523513	C₂²	1	1	yes	characteristic
26	C₂×Q₈×D₁₃₁	C₂²	1	1	yes	characteristic, complemented
27	C₂²×D₅₂₄	C₂²	1	1	yes	characteristic
28	#16862245	C₂²	1	1	yes	characteristic, complemented
29	#16862244	C₂²	1	1	yes	characteristic, complemented
30	#16862244	C₂²	1	1	yes	characteristic, complemented
31	#16862244	C₂²	1	2	yes	complemented
32	#16862241	C₂²	1	2	yes	complemented
33	#16862241	C₂²	1	2	yes	complemented
34	#16862241	C₂²	1	2	yes	complemented
35	#16862241	C₂²	1	1	yes	characteristic, complemented
36	#16862241	C₂²	1	2	yes	complemented
37	#16862240	C₂²	1	2	yes	complemented
38	#16862240	C₂²	1	2	yes	complemented
39	#16862239	C₂²	1	1	yes	characteristic, complemented
40	#16862236	C₂²	1	2	yes	complemented
41	#16862236	C₂²	1	1	yes	characteristic, complemented
42	#16862235	C₂²	1	1	yes	characteristic
43	#16862235	C₂²	1	1	yes	characteristic
44	#16862235	C₂²	1	1	yes	characteristic
45	#16862235	C₂²	1	2	yes	—
46	#16862235	C₂²	1	1	yes	characteristic
47	#16862235	C₂²	1	1	yes	characteristic
48	#16862235	C₂²	1	2	yes	—
49	#16862235	C₂²	1	2	yes	complemented
50	#16862234	C₂²	1	1	yes	characteristic
51	#16862234	C₂²	1	2	yes	complemented
52	#16862232	C₂²	1	2	yes	complemented
53	#16862232	C₂²	1	2	yes	complemented
54	#16862232	C₂²	1	1	yes	characteristic, complemented
55	#16862231	C₂²	1	2	yes	—
56	#16862231	C₂²	1	2	yes	—
57	#16862231	C₂²	1	2	yes	complemented
58	#16862231	C₂²	1	2	yes	complemented
59	#16862231	C₂²	1	2	yes	complemented
60	#16862230	C₂²	1	1	yes	characteristic
61	#16862229	C₂²	1	2	yes	complemented
62	#16862229	C₂²	1	2	yes	complemented
63	#16862228	C₂²	1	2	yes	complemented
64	#16862227	C₂²	1	1	yes	characteristic
65	#16862226	C₂²	1	2	yes	complemented
66	#16862226	C₂²	1	2	yes	complemented
67	#16862225	C₂²	1	1	yes	characteristic
68	#16862225	C₂²	1	2	yes	—
69	#16862224	C₂²	1	2	yes	complemented
70	#16862224	C₂²	1	2	yes	complemented
71	#16862224	C₂²	1	2	yes	complemented
72	#16862224	C₂²	1	2	yes	complemented
73	#16862224	C₂²	1	2	yes	complemented
74	#16862223	C₂²	1	2	yes	—
75	#16862223	C₂²	1	1	yes	characteristic
76	#16862222	C₂²	1	2	yes	complemented
77	#16862222	C₂²	1	2	yes	complemented
78	#16862222	C₂²	1	1	yes	characteristic, complemented
79	#16862222	C₂²	1	2	yes	complemented
80	#16862222	C₂²	1	2	yes	complemented
81	#16862221	C₂²	1	1	yes	characteristic
82	#16862221	C₂²	1	2	yes	—
83	#16862220	C₂²	1	1	yes	characteristic
84	#16862220	C₂²	1	2	yes	complemented
85	#16862219	C₂²	1	2	yes	—
86	#16862219	C₂²	1	2	yes	complemented
87	#16862218	C₂²	1	1	yes	characteristic
88	#16862216	C₂²	1	2	yes	—
89	#16862214	C₂²	1	2	yes	—
90	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic, complemented
91	Q₈×Dic₁₃₁	C₂²	1	1	yes	characteristic, complemented
92	C₄×Dic₂₆₂	C₂²	1	1	yes	characteristic, complemented
93	#16862211	C₂²	1	2	yes	complemented
94	#16862211	C₂²	1	1	yes	characteristic, complemented
95	C₄×D₅₂₄	C₂²	1	1	yes	characteristic
96	#16862208	C₂²	1	1	yes	characteristic
97	#16862208	C₂²	1	2	yes	complemented
98	C₄×C₂²⋊D₁₃₁	C₂²	1	2	yes	—
99	C₄×C₂²⋊D₁₃₁	C₂²	1	1	yes	characteristic
100	C₄×C₂²⋊D₁₃₁	C₂²	1	2	yes	complemented
101	C₄×C₂²⋊D₁₃₁	C₂²	1	1	yes	characteristic, complemented
102	#16862205	C₂²	1	1	yes	characteristic, complemented
103	#16862204	C₂²	1	2	yes	complemented
104	#16862203	C₂²	1	1	yes	characteristic, complemented
105	#16862203	C₂²	1	2	yes	complemented
106	#16862202	C₂²	1	2	yes	—
107	C₂×(C₂×D₁₃₁)⋊C₄	C₂²	1	2	yes	—
108	C₂×(C₂×D₁₃₁)⋊C₄	C₂²	1	2	yes	—
109	C₂²⋊C₄×D₁₃₁	C₂²	1	2	yes	—
110	C₂²⋊C₄×D₁₃₁	C₂²	1	2	yes	—
111	C₂×C₄×Dic₁₃₁	C₂²	1	1	yes	characteristic
112	C₂²×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
113	C₂²×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
114	C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
115	C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	complemented
116	C₂²⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
117	C₂²⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
118	C₂²⋊Q₈×C₁₃₁	C₂²	1	2	yes	complemented
119	C₂²⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
120	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic, complemented
121	C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
122	C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
123	C₂³×D₁₃₁	C₂³	1	2	yes	—
124	#14523514	—	4	2	no	—
125	#14523513	—	4	2	no	—
126	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
127	C₂×Dic₂₆₂	C₂³	1	2	yes	—
128	C₂×Dic₂₆₂	C₂³	1	1	yes	characteristic
129	C₂×Dic₂₆₂	C₂³	1	2	yes	—
130	Q₈×D₁₃₁	—	4	2	no	—
131	C₂×D₅₂₄	—	4	2	no	—
132	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
133	C₂×D₅₂₄	C₂³	1	1	yes	characteristic
134	C₂×D₅₂₄	C₂³	1	2	yes	—
135	C₂×D₅₂₄	C₂³	1	2	yes	—
136	C₂×D₅₂₄	C₂³	1	2	yes	—
137	C₂×C₂²⋊D₁₃₁	C₂³	1	1	yes	characteristic
138	C₂×C₂²⋊D₁₃₁	C₂³	1	2	yes	—
139	C₂×C₂²⋊D₁₃₁	C₂³	1	2	yes	—
140	C₂×C₂²⋊D₁₃₁	C₂³	1	1	yes	characteristic
141	C₂²×Dic₁₃₁	C₂³	1	1	yes	characteristic
142	C₂²×Dic₁₃₁	C₂³	1	1	yes	characteristic
143	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
144	C₂×C₄×D₁₃₁	C₂³	1	2	yes	—
145	C₂×C₄×D₁₃₁	C₂³	1	2	yes	—
146	C₂×C₄×D₁₃₁	C₂³	1	2	yes	—
147	C₂×C₄×D₁₃₁	C₂³	1	2	yes	—
148	C₂×C₄×D₁₃₁	C₂³	1	2	yes	—
149	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
150	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
151	C₄⋊Dic₁₃₁	C₂³	1	2	yes	—
152	C₄⋊Dic₁₃₁	C₂³	1	2	yes	—
153	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
154	C₄⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
155	Dic₁₃₁⋊C₄	C₂³	1	2	yes	—
156	Dic₁₃₁⋊C₄	C₂³	1	2	yes	—
157	Dic₁₃₁⋊C₄	C₂³	1	2	yes	—
158	Dic₁₃₁⋊C₄	C₂³	1	1	yes	characteristic
159	Dic₁₃₁⋊C₄	C₂³	1	1	yes	characteristic
160	Dic₁₃₁⋊C₄	C₂³	1	1	yes	characteristic
161	Dic₁₃₁⋊C₄	C₂³	1	2	yes	—
162	Dic₁₃₁⋊C₄	C₂³	1	2	yes	—
163	Dic₁₃₁⋊C₄	C₂³	1	2	yes	—
164	Dic₁₃₁⋊C₄	C₂³	1	2	yes	—
165	Dic₁₃₁⋊C₄	C₂³	1	1	yes	characteristic
166	Dic₁₃₁⋊C₄	C₂³	1	1	yes	characteristic
167	Dic₁₃₁⋊C₄	C₂³	1	2	yes	—
168	Dic₁₃₁⋊C₄	C₂³	1	1	yes	characteristic
169	C₂²⋊Dic₁₃₁	C₂³	1	2	yes	—
170	C₂²⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
171	C₂²⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
172	C₂²⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
173	C₂²⋊Dic₁₃₁	C₂³	1	1	yes	characteristic
174	(C₂×D₁₃₁)⋊C₄	C₂³	1	2	yes	—
175	(C₂×D₁₃₁)⋊C₄	C₂³	1	1	yes	characteristic
176	(C₂×D₁₃₁)⋊C₄	C₂³	1	2	yes	—
177	(C₂×D₁₃₁)⋊C₄	C₂³	1	2	yes	—
178	(C₂×D₁₃₁)⋊C₄	C₂³	1	2	yes	—
179	(C₂×D₁₃₁)⋊C₄	C₂³	1	2	yes	—
180	(C₂×D₁₃₁)⋊C₄	C₂³	1	1	yes	characteristic
181	(C₂×D₁₃₁)⋊C₄	C₂³	1	2	yes	—
182	(C₂×D₁₃₁)⋊C₄	C₂³	1	2	yes	—
183	(C₂×D₁₃₁)⋊C₄	C₂³	1	2	yes	—
184	(C₂×D₁₃₁)⋊C₄	C₂³	1	2	yes	—
185	(C₂×D₁₃₁)⋊C₄	C₂³	1	2	yes	—
186	(C₂×D₁₃₁)⋊C₄	C₂³	1	1	yes	characteristic
187	(C₂×D₁₃₁)⋊C₄	C₂³	1	2	yes	—
188	(C₂×D₁₃₁)⋊C₄	C₂³	1	2	yes	—
189	(C₂×D₁₃₁)⋊C₄	C₂³	1	2	yes	—
190	(C₂×D₁₃₁)⋊C₄	C₂³	1	2	yes	—
191	(C₂×D₁₃₁)⋊C₄	C₂³	1	2	yes	—
192	(C₂×D₁₃₁)⋊C₄	C₂³	1	2	yes	—
193	(C₂×D₁₃₁)⋊C₄	C₂³	1	1	yes	characteristic
194	(C₂×D₁₃₁)⋊C₄	C₂³	1	2	yes	—
195	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
196	C₄×Dic₁₃₁	C₂³	1	2	yes	—
197	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
198	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
199	C₄×Dic₁₃₁	C₂³	1	2	yes	—
200	C₄×Dic₁₃₁	C₂³	1	2	yes	—
201	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
202	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
203	C₂×Q₈×C₁₃₁	—	4	2	no	—
204	C₂×Q₈×C₁₃₁	—	4	2	no	—
205	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
206	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
207	C₂×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
208	C₂×Q₈×C₁₃₁	C₂³	1	2	yes	—
209	C₂×Q₈×C₁₃₁	C₂³	1	2	yes	—
210	C₂×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
211	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
212	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
213	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
214	C₂²×C₄×C₁₃₁	C₂³	1	2	yes	—
215	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
216	C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
217	C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
218	C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
219	C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
220	C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
221	C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
222	C₂²⋊C₄×C₁₃₁	C₂³	1	2	yes	—
223	C₂²⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
224	C₂²⋊C₄×C₁₃₁	C₂³	1	2	yes	—
225	C₂²⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
226	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
227	C₄²×C₁₃₁	C₂³	1	2	yes	—
228	C₂²×D₁₃₁	—	2	4	no	—
229	C₂²×D₁₃₁	—	4	2	no	—
230	C₂²×D₁₃₁	—	4	2	no	—
231	C₂²×D₁₃₁	C₂⁴	1	2	yes	—
232	C₂²×D₁₃₁	C₂⁴	1	2	yes	—
233	C₂²×D₁₃₁	C₂⁴	1	1	yes	characteristic
234	Dic₂₆₂	—	2	2	no	—
235	Dic₂₆₂	—	4	2	no	—
236	Dic₂₆₂	—	4	1	no	—
237	Dic₂₆₂	—	4	2	no	—
238	D₅₂₄	—	2	2	no	—
239	D₅₂₄	—	2	2	no	—
240	D₅₂₄	—	4	2	no	—
241	D₅₂₄	—	4	2	no	—
242	D₅₂₄	—	4	2	no	—
243	C₂²⋊D₁₃₁	—	2	2	no	—
244	C₂²⋊D₁₃₁	—	4	2	no	—
245	C₂²⋊D₁₃₁	—	4	1	no	—
246	C₂²⋊D₁₃₁	—	4	2	no	—
247	C₂×Dic₁₃₁	—	2	2	no	—
248	C₂×Dic₁₃₁	—	4	1	no	—
249	C₂×Dic₁₃₁	C₂⁴	1	1	yes	characteristic
250	C₂×Dic₁₃₁	C₂⁴	1	1	yes	characteristic
251	C₂×Dic₁₃₁	C₂⁴	1	1	yes	characteristic
252	C₂×Dic₁₃₁	C₂⁴	1	1	yes	characteristic
253	C₂×Dic₁₃₁	C₂⁴	1	1	yes	characteristic
254	C₂×Dic₁₃₁	C₂⁴	1	2	yes	—
255	C₂×Dic₁₃₁	C₂⁴	1	1	yes	characteristic
256	C₂×Dic₁₃₁	C₂⁴	1	2	yes	—
257	C₂×Dic₁₃₁	C₂⁴	1	1	yes	characteristic
258	C₄×D₁₃₁	—	2	2	no	—
259	C₄×D₁₃₁	—	4	2	no	—
260	C₄×D₁₃₁	—	4	2	no	—
261	C₄×D₁₃₁	—	4	2	no	—
262	C₄×D₁₃₁	—	4	2	no	—
263	C₄×D₁₃₁	—	4	2	no	—
264	C₂³×C₁₃₁	C₂⁴	1	1	yes	characteristic
265	C₂³×C₁₃₁	C₂⁴	1	1	yes	characteristic
266	Q₈×C₁₃₁	—	2	2	no	—
267	Q₈×C₁₃₁	—	2	2	no	—
268	Q₈×C₁₃₁	—	2	2	no	—
269	Q₈×C₁₃₁	—	4	2	no	—
270	Q₈×C₁₃₁	—	4	2	no	—
271	D₄×C₁₃₁	—	2	2	no	—
272	C₂×C₄×C₁₃₁	—	2	2	no	—
273	C₂×C₄×C₁₃₁	—	2	2	no	—
274	C₂×C₄×C₁₃₁	—	4	2	no	—
275	C₂×C₄×C₁₃₁	—	4	2	no	—
276	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
277	C₂×C₄×C₁₃₁	C₂⁴	1	2	yes	—
278	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
279	C₂×C₄×C₁₃₁	C₂⁴	1	2	yes	—
280	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
281	C₂×C₄×C₁₃₁	C₂⁴	1	2	yes	—
282	C₂×C₄×C₁₃₁	C₂⁴	1	2	yes	—
283	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
284	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
285	C₂²×C₁₃₁	—	2	1	no	—
286	C₂²×C₁₃₁	—	2	1	no	—
287	C₂²×C₁₃₁	—	2	1	no	—
288	C₂²×C₁₃₁	—	2	1	no	—
289	C₂²×C₁₃₁	—	2	1	no	—
290	C₂²×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
291	C₂²×C₁₃₁	C₂⁵	1	1	yes	characteristic
292	C₂×D₁₃₁	—	2	1	no	—
293	C₂×D₁₃₁	—	2	2	no	—
294	C₂×D₁₃₁	—	2	2	no	—
295	C₂×D₁₃₁	—	2	1	no	—
296	C₂×D₁₃₁	—	2	2	no	—
297	C₂×D₁₃₁	—	2	2	no	—
298	C₂×D₁₃₁	—	2	2	no	—
299	C₂×D₁₃₁	—	2	2	no	—
300	C₂×D₁₃₁	—	2	1	no	—
301	C₂×D₁₃₁	—	8	4	no	—
302	C₄×C₁₃₁	—	2	1	no	—
303	C₄×C₁₃₁	—	2	1	no	—
304	C₄×C₁₃₁	—	2	2	no	—
305	C₄×C₁₃₁	—	2	2	no	—
306	C₄×C₁₃₁	—	2	2	no	—
307	C₄×C₁₃₁	—	2	1	no	—
308	C₄×C₁₃₁	—	2	1	no	—
309	C₄×C₁₃₁	—	2	2	no	—
310	C₄×C₁₃₁	H(5,𝔽₂)	1	2	yes	—
311	Dic₁₃₁	—	2	1	no	—
312	Dic₁₃₁	—	2	2	no	—
313	Dic₁₃₁	—	2	1	no	—
314	Dic₁₃₁	—	2	1	no	—
315	Dic₁₃₁	—	2	1	no	—
316	Dic₁₃₁	—	2	1	no	—
317	Dic₁₃₁	—	2	2	no	—
318	Dic₁₃₁	—	2	1	no	—
319	Dic₁₃₁	—	2	1	no	—
320	C₂×C₁₃₁	—	2	2	no	—
321	C₂×C₁₃₁	—	4	1	no	—
322	C₂×C₁₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
323	C₂×C₁₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
324	C₂×C₁₃₁	#585	1	1	yes	characteristic
325	D₁₃₁	—	4	1	no	—
326	D₁₃₁	—	4	2	no	—
327	D₁₃₁	—	4	2	no	—
328	C₁₃₁	#3561	1	1	yes	characteristic, complemented, Sylow
329	#3561	—	131	1	no	maximal, Sylow
330	#563	—	131	2	no	—
331	#562	—	131	2	no	—
332	#556	—	131	1	no	—
333	#556	—	131	1	no	—
334	#551	—	131	2	no	—
335	#551	—	131	2	no	—
336	D₄×Q₈	—	131	1	no	—
337	D₄×Q₈	D₁₃₁	1	1	yes	characteristic, complemented
338	#548	—	131	1	no	—
339	#548	—	131	1	no	—
340	#546	—	131	1	no	—
341	#546	—	131	1	no	—
342	#543	—	131	2	no	—
343	#542	—	131	2	no	—
344	#542	—	131	2	no	—
345	#542	—	131	1	no	—
346	#542	—	131	1	no	—
347	#538	—	131	2	no	—
348	#536	—	131	1	no	—
349	#536	—	131	1	no	—
350	#533	—	131	1	no	—
351	#529	—	131	1	no	—
352	C₂×#124	—	131	1	no	—
353	C₂×#41	—	131	1	no	—
354	C₂×#41	—	131	1	no	—
355	C₂²×Q₈	—	131	1	no	—
356	C₂²×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
357	C₂²×Q₈	C₂×D₁₃₁	1	1	yes	characteristic, complemented
358	C₂²×D₄	—	131	1	no	—
359	C₄⋊Q₈	—	131	1	no	—
360	C₄⋊Q₈	—	131	1	no	—
361	C₄⋊Q₈	—	131	2	no	—
362	C₄⋊Q₈	—	131	1	no	—
363	C₄⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic, complemented
364	C₄⋊Q₈	C₂×D₁₃₁	1	2	yes	complemented
365	#126	—	131	1	no	—
366	#126	—	131	2	no	—
367	#126	—	131	2	no	—
368	#126	—	131	2	no	—
369	#126	—	131	2	no	—
370	#126	—	131	1	no	—
371	#126	—	131	2	no	—
372	#126	—	131	2	no	—
373	#125	—	131	1	no	—
374	#125	—	131	2	no	—
375	#124	—	131	1	no	—
376	#124	—	131	1	no	—
377	#124	—	131	2	no	—
378	#124	—	131	1	no	—
379	#124	—	131	1	no	—
380	#124	—	131	2	no	—
381	#124	—	131	1	no	—
382	#124	—	131	2	no	—
383	#124	—	131	1	no	—
384	#124	—	131	2	no	—
385	#124	—	131	2	no	—
386	#124	—	131	1	no	—
387	#124	—	131	2	no	—
388	#123	—	131	1	no	—
389	#123	—	131	2	no	—
390	#123	—	131	2	no	—
391	#123	—	131	2	no	—
392	#123	—	131	2	no	—
393	#123	—	131	2	no	—
394	#123	—	131	2	no	—
395	#123	—	131	2	no	—
396	#123	—	131	1	no	—
397	#123	—	131	2	no	—
398	C₂²⋊Q₈	—	131	2	no	—
399	C₂²⋊Q₈	—	131	2	no	—
400	C₂²⋊Q₈	—	131	1	no	—
401	C₂²⋊Q₈	—	131	2	no	—
402	C₂²⋊Q₈	—	131	2	no	—
403	C₂²⋊Q₈	—	131	1	no	—
404	C₂²⋊Q₈	—	131	2	no	—
405	C₂²⋊Q₈	—	131	2	no	—
406	C₂²⋊Q₈	—	131	2	no	—
407	C₂²⋊Q₈	—	131	2	no	—
408	C₂²⋊Q₈	—	131	2	no	—
409	C₂²⋊Q₈	—	131	2	no	—
410	C₂²⋊Q₈	—	131	2	no	—
411	C₂²⋊Q₈	—	131	2	no	—
412	C₂²⋊Q₈	—	131	1	no	—
413	C₂²⋊Q₈	—	131	2	no	—
414	C₂²⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic
415	C₂²⋊Q₈	C₂×D₁₃₁	1	2	yes	—
416	C₂²⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic, complemented
417	C₂²⋊Q₈	C₂×D₁₃₁	1	2	yes	complemented
418	#121	—	131	1	no	—
419	#121	—	131	1	no	—
420	#121	—	131	2	no	—
421	#121	—	131	1	no	—
422	#121	—	131	2	no	—
423	#121	—	131	2	no	—
424	#121	—	131	2	no	—
425	#120	—	131	2	no	—
426	#120	—	131	2	no	—
427	C₄×Q₈	—	131	1	no	—
428	C₄×Q₈	—	131	1	no	—
429	C₄×Q₈	—	131	2	no	—
430	C₄×Q₈	—	131	1	no	—
431	C₄×Q₈	—	131	1	no	—
432	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic, complemented
433	C₄×D₄	—	131	1	no	—
434	C₄×D₄	—	131	1	no	—
435	C₄×D₄	—	131	2	no	—
436	C₄×D₄	—	131	2	no	—
437	C₄×D₄	—	131	1	no	—
438	C₄×D₄	—	131	2	no	—
439	C₄×D₄	—	131	1	no	—
440	C₄×D₄	C₂×D₁₃₁	1	1	yes	characteristic
441	C₄×D₄	C₂×D₁₃₁	1	2	yes	—
442	#117	—	131	1	no	—
443	#117	—	131	2	no	—
444	#117	—	131	2	no	—
445	#117	—	131	1	no	—
446	#117	—	131	2	no	—
447	C₂×C₂²⋊C₄	—	131	2	no	—
448	C₂×C₂²⋊C₄	—	131	2	no	—
449	C₂×C₂²⋊C₄	—	131	2	no	—
450	C₂×C₂²⋊C₄	—	131	2	no	—
451	C₂×C₄²	—	131	1	no	—
452	C₂⁴	—	131	2	no	—
453	#41	—	524	2	no	—
454	#41	—	524	2	no	—
455	C₂×Q₈	—	4	2	no	—
456	C₂×Q₈	—	4	2	no	—
457	C₂×Q₈	—	524	2	no	—
458	C₂×Q₈	—	131	1	no	—
459	C₂×Q₈	—	131	1	no	—
460	C₂×Q₈	—	131	2	no	—
461	C₂×Q₈	—	131	2	no	—
462	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
463	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
464	C₂×Q₈	C₂²×D₁₃₁	1	1	yes	characteristic
465	C₂×Q₈	C₂²×D₁₃₁	1	2	yes	—
466	C₂×Q₈	C₂²×D₁₃₁	1	2	yes	—
467	C₂×D₄	—	524	2	no	—
468	C₂×D₄	—	131	1	no	—
469	C₂×D₄	—	131	2	no	—
470	C₂×D₄	—	131	1	no	—
471	C₂×D₄	—	131	2	no	—
472	C₂×D₄	—	131	1	no	—
473	C₂×D₄	—	131	1	no	—
474	C₂×D₄	—	131	2	no	—
475	C₂×D₄	—	131	2	no	—
476	C₂×D₄	—	131	2	no	—
477	C₂×D₄	C₂²×D₁₃₁	1	1	yes	characteristic
478	C₂²×C₄	—	131	1	no	—
479	C₂²×C₄	—	131	1	no	—
480	C₂²×C₄	—	131	1	no	—
481	C₂²×C₄	—	131	2	no	—
482	C₂²×C₄	—	131	2	no	—
483	C₂²×C₄	—	131	2	no	—
484	C₂²×C₄	—	131	2	no	—
485	C₂²×C₄	—	131	2	no	—
486	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
487	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
488	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
489	C₂²×C₄	C₂²×D₁₃₁	1	2	yes	—
490	C₄⋊C₄	—	131	1	no	—
491	C₄⋊C₄	—	131	1	no	—
492	C₄⋊C₄	—	131	2	no	—
493	C₄⋊C₄	—	131	2	no	—
494	C₄⋊C₄	—	131	1	no	—
495	C₄⋊C₄	—	131	2	no	—
496	C₄⋊C₄	—	131	1	no	—
497	C₄⋊C₄	—	131	2	no	—
498	C₄⋊C₄	—	131	2	no	—
499	C₄⋊C₄	—	131	1	no	—
500	C₄⋊C₄	—	131	2	no	—
501	C₄⋊C₄	—	131	2	no	—
502	C₄⋊C₄	—	131	1	no	—
503	C₄⋊C₄	—	131	2	no	—
504	C₄⋊C₄	—	131	1	no	—
505	C₄⋊C₄	—	131	1	no	—
506	C₄⋊C₄	—	131	1	no	—
507	C₄⋊C₄	—	131	1	no	—
508	C₄⋊C₄	—	131	2	no	—
509	C₄⋊C₄	—	131	2	no	—
510	C₄⋊C₄	C₂²×D₁₃₁	1	2	yes	—
511	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
512	C₄⋊C₄	C₂²×D₁₃₁	1	2	yes	—
513	C₄⋊C₄	C₂²×D₁₃₁	1	2	yes	—
514	C₄⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
515	C₄⋊C₄	C₂²×D₁₃₁	1	2	yes	—
516	C₄⋊C₄	C₂²×D₁₃₁	1	2	yes	—
517	C₂²⋊C₄	—	131	1	no	—
518	C₂²⋊C₄	—	131	2	no	—
519	C₂²⋊C₄	—	131	1	no	—
520	C₂²⋊C₄	—	131	1	no	—
521	C₂²⋊C₄	—	131	1	no	—
522	C₂²⋊C₄	—	131	2	no	—
523	C₂²⋊C₄	—	131	2	no	—
524	C₂²⋊C₄	—	131	2	no	—
525	C₂²⋊C₄	—	131	2	no	—
526	C₂²⋊C₄	—	131	2	no	—
527	C₂²⋊C₄	—	131	2	no	—
528	C₂²⋊C₄	—	131	2	no	—
529	C₂²⋊C₄	—	131	2	no	—
530	C₂²⋊C₄	—	131	2	no	—
531	C₂²⋊C₄	—	131	2	no	—
532	C₂²⋊C₄	—	131	2	no	—
533	C₂²⋊C₄	—	131	2	no	—
534	C₂²⋊C₄	—	131	2	no	—
535	C₂²⋊C₄	—	131	2	no	—
536	C₂²⋊C₄	—	131	1	no	—
537	C₂²⋊C₄	—	131	1	no	—
538	C₂²⋊C₄	—	131	1	no	—
539	C₂²⋊C₄	—	131	2	no	—
540	C₂²⋊C₄	—	131	2	no	—
541	C₂²⋊C₄	—	131	1	no	—
542	C₂²⋊C₄	—	131	2	no	—
543	C₂²⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
544	C₂²⋊C₄	C₂²×D₁₃₁	1	2	yes	—
545	C₂²⋊C₄	C₂²×D₁₃₁	1	1	yes	characteristic
546	C₂²⋊C₄	C₂²×D₁₃₁	1	2	yes	—
547	C₄²	—	131	1	no	—
548	C₄²	—	131	1	no	—
549	C₄²	—	131	1	no	—
550	C₄²	—	131	1	no	—
551	C₄²	—	131	2	no	—
552	C₄²	—	131	2	no	—
553	C₄²	—	131	1	no	—
554	C₄²	—	131	2	no	—
555	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
556	C₄²	C₂²×D₁₃₁	1	2	yes	—
557	C₂³	—	262	4	no	—
558	C₂³	—	524	2	no	—
559	C₂³	—	524	2	no	—
560	C₂³	—	131	2	no	—
561	C₂³	—	131	2	no	—
562	C₂³	—	131	1	no	—
563	C₂³	C₂³×D₁₃₁	1	1	yes	characteristic
564	C₂³	C₂³×D₁₃₁	1	1	yes	characteristic
565	Q₈	—	2	2	no	—
566	Q₈	—	2	2	no	—
567	Q₈	—	2	2	no	—
568	Q₈	—	4	2	no	—
569	Q₈	—	4	2	no	—
570	Q₈	—	262	2	no	—
571	Q₈	—	524	2	no	—
572	Q₈	—	524	1	no	—
573	Q₈	—	524	2	no	—
574	D₄	—	2	2	no	—
575	D₄	—	262	2	no	—
576	D₄	—	262	2	no	—
577	D₄	—	262	2	no	—
578	D₄	—	524	2	no	—
579	D₄	—	524	2	no	—
580	D₄	—	524	2	no	—
581	D₄	—	524	2	no	—
582	D₄	—	524	2	no	—
583	D₄	—	524	1	no	—
584	C₂×C₄	—	2	2	no	—
585	C₂×C₄	—	2	2	no	—
586	C₂×C₄	—	4	2	no	—
587	C₂×C₄	—	4	2	no	—
588	C₂×C₄	—	262	2	no	—
589	C₂×C₄	—	262	2	no	—
590	C₂×C₄	—	524	2	no	—
591	C₂×C₄	—	524	2	no	—
592	C₂×C₄	—	524	2	no	—
593	C₂×C₄	—	524	2	no	—
594	C₂×C₄	—	524	2	no	—
595	C₂×C₄	—	524	1	no	—
596	C₂×C₄	—	131	1	no	—
597	C₂×C₄	—	131	1	no	—
598	C₂×C₄	—	131	1	no	—
599	C₂×C₄	—	131	1	no	—
600	C₂×C₄	—	131	1	no	—
601	C₂×C₄	—	131	2	no	—
602	C₂×C₄	—	131	1	no	—
603	C₂×C₄	—	131	2	no	—
604	C₂×C₄	—	131	1	no	—
605	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
606	C₂×C₄	C₂³×D₁₃₁	1	2	yes	—
607	C₂×C₄	C₂³×D₁₃₁	1	2	yes	—
608	C₂×C₄	C₂³×D₁₃₁	1	2	yes	—
609	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
610	C₂×C₄	C₂³×D₁₃₁	1	2	yes	—
611	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
612	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
613	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
614	C₂²	—	2	1	no	—
615	C₂²	—	2	1	no	—
616	C₂²	—	2	1	no	—
617	C₂²	—	2	1	no	—
618	C₂²	—	2	1	no	—
619	C₂²	—	262	1	no	—
620	C₂²	—	262	2	no	—
621	C₂²	—	262	2	no	—
622	C₂²	—	262	1	no	—
623	C₂²	—	262	2	no	—
624	C₂²	—	262	2	no	—
625	C₂²	—	262	2	no	—
626	C₂²	—	262	2	no	—
627	C₂²	—	262	1	no	—
628	C₂²	—	1048	4	no	—
629	C₂²	#16862294	1	2	yes	—
630	C₂²	C₂⁴×D₁₃₁	1	1	yes	characteristic
631	C₄	—	2	1	no	—
632	C₄	—	2	1	no	—
633	C₄	—	2	2	no	—
634	C₄	—	2	2	no	—
635	C₄	—	2	2	no	—
636	C₄	—	2	1	no	—
637	C₄	—	2	1	no	—
638	C₄	—	2	2	no	—
639	C₄	—	262	1	no	—
640	C₄	—	262	2	no	—
641	C₄	—	262	1	no	—
642	C₄	—	262	1	no	—
643	C₄	—	262	1	no	—
644	C₄	—	262	1	no	—
645	C₄	—	262	2	no	—
646	C₄	—	262	1	no	—
647	C₄	—	262	1	no	—
648	C₄	#16862291	1	2	yes	—
649	C₂	—	2	2	no	—
650	C₂	—	4	1	no	—
651	C₂	—	524	2	no	—
652	C₂	—	524	2	no	—
653	C₂	—	524	1	no	—
654	C₂	C₂×#16862291	1	1	yes	characteristic
655	C₂	C₂×#16862294	1	1	yes	characteristic
656	C₂	#22554149	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃₁
commutator, socle, cosocle	C₂²×C₁₃₁	C₂⁵
Fitting	D₄×Q₈×C₁₃₁	C₂
hypercenter	D₄×Q₈	D₁₃₁
hypocenter	C₁₃₁	#3561
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ D₄×Q₈

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #22554061	C₂	semidirect
2. #22554058	C₂	semidirect
3. #22554036	C₂	semidirect
4. #22554034	C₂	semidirect
5. #22553991	C₂	semidirect
6. #22553989	C₂	semidirect
7. Q₈×C₂²⋊D₁₃₁	C₂	semidirect
8. #22553973	C₂	semidirect
9. #22553972	C₂	semidirect
10. #22553958	C₂	semidirect
11. #22553953	C₂	semidirect
12. #22553937	C₂	semidirect
13. #22553936	C₂	semidirect
14. #22553935	C₂	semidirect
15. #22553934	C₂	semidirect
16. #22553932	C₂	semidirect
17. #22553895	C₂	semidirect
18. #22553887	C₂	semidirect
19. #22553886	C₂	semidirect
20. #22553864	C₂	semidirect
21. #22553842	C₂	semidirect
22. C₂×#16862235	C₂	semidirect
23. D₄×Q₈×C₁₃₁	C₂	semidirect
26. C₂×Q₈×D₁₃₁	C₂²	semidirect
28. #16862245	C₂²	semidirect
29. #16862244	C₂²	semidirect
30. #16862244	C₂²	semidirect
31. #16862244	C₂²	semidirect
32. #16862241	C₂²	semidirect
33. #16862241	C₂²	semidirect
34. #16862241	C₂²	semidirect
35. #16862241	C₂²	semidirect
36. #16862241	C₂²	semidirect
37. #16862240	C₂²	semidirect
38. #16862240	C₂²	semidirect
39. #16862239	C₂²	semidirect
40. #16862236	C₂²	semidirect
41. #16862236	C₂²	semidirect
49. #16862235	C₂²	semidirect
51. #16862234	C₂²	semidirect
52. #16862232	C₂²	semidirect
53. #16862232	C₂²	semidirect
54. #16862232	C₂²	semidirect
57. #16862231	C₂²	semidirect
58. #16862231	C₂²	semidirect
59. #16862231	C₂²	semidirect
61. #16862229	C₂²	semidirect
62. #16862229	C₂²	semidirect
63. #16862228	C₂²	semidirect
65. #16862226	C₂²	semidirect
66. #16862226	C₂²	semidirect
69. #16862224	C₂²	semidirect
70. #16862224	C₂²	semidirect
71. #16862224	C₂²	semidirect
72. #16862224	C₂²	semidirect
73. #16862224	C₂²	semidirect
76. #16862222	C₂²	semidirect
77. #16862222	C₂²	semidirect
78. #16862222	C₂²	semidirect
79. #16862222	C₂²	semidirect
80. #16862222	C₂²	semidirect
84. #16862220	C₂²	semidirect
86. #16862219	C₂²	semidirect
90. Q₈×Dic₁₃₁	C₂²	semidirect
91. Q₈×Dic₁₃₁	C₂²	semidirect
92. C₄×Dic₂₆₂	C₂²	semidirect
93. #16862211	C₂²	semidirect
94. #16862211	C₂²	semidirect
97. #16862208	C₂²	semidirect
100. C₄×C₂²⋊D₁₃₁	C₂²	semidirect
101. C₄×C₂²⋊D₁₃₁	C₂²	semidirect
102. #16862205	C₂²	semidirect
103. #16862204	C₂²	semidirect
104. #16862203	C₂²	semidirect
105. #16862203	C₂²	semidirect
113. C₂²×Q₈×C₁₃₁	C₂²	semidirect
114. C₄⋊Q₈×C₁₃₁	C₂²	semidirect
115. C₄⋊Q₈×C₁₃₁	C₂²	semidirect
118. C₂²⋊Q₈×C₁₃₁	C₂²	semidirect
119. C₂²⋊Q₈×C₁₃₁	C₂²	semidirect
120. C₄×Q₈×C₁₃₁	C₂²	semidirect
328. C₁₃₁	#3561	semidirect
337. D₄×Q₈	D₁₃₁	semidirect
357. C₂²×Q₈	C₂×D₁₃₁	semidirect
363. C₄⋊Q₈	C₂×D₁₃₁	semidirect
364. C₄⋊Q₈	C₂×D₁₃₁	semidirect
416. C₂²⋊Q₈	C₂×D₁₃₁	semidirect
417. C₂²⋊Q₈	C₂×D₁₃₁	semidirect
432. C₄×Q₈	C₂×D₁₃₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22554061	2
#22554058	2
#22554036	2
#22554034	2
#22553991	2
#22553989	2
Q₈×C₂²⋊D₁₃₁	2
#22553973	2
#22553972	2
#22553958	2
#22553953	2
#22553937	2
#22553936	2
#22553935	2
#22553934	2
#22553932	2
#22553895	2
#22553887	2
#22553886	2
#22553864	2
#22553842	2
C₂×#16862235	2
D₄×Q₈×C₁₃₁	2
#3561	131