Subgroups — C₁₃₁×#34972 #45720856

Order 33536.

2474 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#529×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	#2604×C₁₃₁	C₂	1	4	yes	maximal, complemented
4	#2565×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2545×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#2503×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#2501×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	#2436×C₁₃₁	C₂	1	1	yes	maximal, characteristic
9	#2434×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	#2413×C₁₃₁	C₂	1	4	yes	maximal, complemented
11	#2396×C₁₃₁	C₂	1	4	yes	maximal, complemented
12	C₄×C₄⋊Q₈×C₁₃₁	C₂	1	2	yes	maximal, complemented
13	C₄×#124×C₁₃₁	C₂	1	2	yes	maximal, complemented
14	C₄×#123×C₁₃₁	C₂	1	4	yes	maximal, complemented
15	C₄×#117×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂³×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₁₃₁	C₂²	1	2	yes	—
18	#529×C₁₃₁	—	2	8	no	—
19	C₂×#126×C₁₃₁	C₂²	1	2	yes	—
20	C₂×#125×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₂×#124×C₁₃₁	C₂²	1	1	yes	characteristic
22	C₂×#124×C₁₃₁	C₂²	1	2	yes	—
23	C₂×#123×C₁₃₁	C₂²	1	4	yes	—
24	C₂×#123×C₁₃₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
26	C₂×#121×C₁₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₁₃₁	—	2	8	no	—
28	C₂×C₄×Q₈×C₁₃₁	C₂²	1	2	yes	—
29	C₂×C₄×Q₈×C₁₃₁	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₁₃₁	C₂²	1	2	yes	—
32	C₂×C₄×Q₈×C₁₃₁	C₂²	1	2	yes	—
33	C₂×C₄×D₄×C₁₃₁	C₂²	1	2	yes	—
34	C₂×C₄×D₄×C₁₃₁	C₂²	1	1	yes	characteristic
35	C₂×#117×C₁₃₁	C₂²	1	2	yes	—
36	C₂×#117×C₁₃₁	C₂²	1	2	yes	—
37	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
38	C₂²×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
39	C₂²×C₄²×C₁₃₁	C₂²	1	2	yes	—
40	C₂²×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
41	#400×C₁₃₁	C₂²	1	4	yes	—
42	#399×C₁₃₁	C₂²	1	4	yes	—
43	#398×C₁₃₁	C₂²	1	8	yes	—
44	#396×C₁₃₁	C₂²	1	8	yes	—
45	#395×C₁₃₁	C₂²	1	4	yes	—
46	#393×C₁₃₁	C₂²	1	4	yes	—
47	#391×C₁₃₁	C₂²	1	2	yes	—
48	#391×C₁₃₁	C₂²	1	2	yes	—
49	#391×C₁₃₁	C₂²	1	2	yes	—
50	#390×C₁₃₁	C₂²	1	2	yes	—
51	#389×C₁₃₁	C₂²	1	8	yes	—
52	#388×C₁₃₁	C₂²	1	8	yes	—
53	#388×C₁₃₁	C₂²	1	8	yes	—
54	#387×C₁₃₁	C₂²	1	8	yes	—
55	#386×C₁₃₁	C₂²	1	4	yes	—
56	#385×C₁₃₁	C₂²	1	4	yes	—
57	#384×C₁₃₁	C₂²	1	2	yes	—
58	#382×C₁₃₁	C₂²	1	2	yes	—
59	#381×C₁₃₁	C₂²	1	4	yes	—
60	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
61	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
62	C₄×C₄⋊C₄×C₁₃₁	C₂²	1	8	yes	—
63	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
64	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	1	yes	characteristic
65	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	2	yes	—
66	C₄×C₂²⋊C₄×C₁₃₁	C₂²	1	8	yes	—
67	#376×C₁₃₁	C₂²	1	2	yes	—
68	C₄³×C₁₃₁	C₂²	1	2	yes	—
69	C₂²×Q₈×C₁₃₁	—	2	8	no	—
70	C₂²×Q₈×C₁₃₁	—	2	4	no	—
71	C₂²×Q₈×C₁₃₁	C₂³	1	1	yes	characteristic
72	C₂²×Q₈×C₁₃₁	C₂³	1	2	yes	—
73	C₂²×Q₈×C₁₃₁	C₂³	1	2	yes	—
74	C₂²×D₄×C₁₃₁	C₂³	1	1	yes	characteristic
75	C₂²×D₄×C₁₃₁	C₂³	1	2	yes	—
76	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
77	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
78	C₂³×C₄×C₁₃₁	C₂³	1	2	yes	—
79	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
82	C₄⋊Q₈×C₁₃₁	—	2	8	no	—
83	#126×C₁₃₁	—	2	8	no	—
84	#125×C₁₃₁	—	2	4	no	—
85	#124×C₁₃₁	—	2	8	no	—
86	#124×C₁₃₁	—	2	4	no	—
87	#123×C₁₃₁	—	2	8	no	—
88	#123×C₁₃₁	—	2	16	no	—
89	C₂²⋊Q₈×C₁₃₁	—	2	4	no	—
90	#121×C₁₃₁	—	2	4	no	—
91	C₄×Q₈×C₁₃₁	—	2	8	no	—
92	C₄×Q₈×C₁₃₁	—	2	8	no	—
93	C₄×Q₈×C₁₃₁	—	2	8	no	—
94	C₄×Q₈×C₁₃₁	—	2	8	no	—
95	C₄×Q₈×C₁₃₁	—	2	4	no	—
96	C₄×D₄×C₁₃₁	—	2	8	no	—
97	C₄×D₄×C₁₃₁	—	2	4	no	—
98	#117×C₁₃₁	—	2	8	no	—
99	#117×C₁₃₁	—	2	8	no	—
100	C₂×C₄⋊C₄×C₁₃₁	—	2	8	no	—
101	C₂×C₄⋊C₄×C₁₃₁	—	2	4	no	—
102	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	8	yes	—
104	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	8	yes	—
106	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
111	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
112	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
113	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₁₃₁	C₂³	1	2	yes	—
115	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	2	yes	—
116	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	2	yes	—
117	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	8	yes	—
118	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	8	yes	—
119	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	2	yes	—
120	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	2	yes	—
121	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	2	yes	—
122	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	2	yes	—
123	C₂×C₂²⋊C₄×C₁₃₁	C₂³	1	2	yes	—
124	C₂×C₄²×C₁₃₁	—	2	8	no	—
125	C₂×C₄²×C₁₃₁	D₄	1	8	yes	—
126	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
127	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
128	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
129	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
130	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
131	C₂×C₄²×C₁₃₁	C₂³	1	2	yes	—
132	C₂×C₄²×C₁₃₁	C₂³	1	4	yes	—
133	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
135	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
136	C₂×C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
137	#95×C₁₃₁	C₂³	1	4	yes	—
138	#95×C₁₃₁	C₂³	1	8	yes	—
139	#95×C₁₃₁	C₂³	1	8	yes	—
140	#95×C₁₃₁	C₂³	1	8	yes	—
141	#95×C₁₃₁	C₂³	1	8	yes	—
142	#95×C₁₃₁	C₂³	1	2	yes	—
143	#95×C₁₃₁	C₂³	1	2	yes	—
144	C₂⁴×C₁₃₁	C₂⁴	1	2	yes	—
145	C₂⁴×C₁₃₁	C₂⁴	1	1	yes	characteristic
146	C₂×Q₈×C₁₃₁	—	2	32	no	—
147	C₂×Q₈×C₁₃₁	—	2	4	no	—
148	C₂×Q₈×C₁₃₁	—	2	4	no	—
149	C₂×Q₈×C₁₃₁	—	2	2	no	—
150	C₂×Q₈×C₁₃₁	—	2	8	no	—
151	C₂×Q₈×C₁₃₁	—	2	4	no	—
152	C₂×Q₈×C₁₃₁	—	2	4	no	—
153	C₂×Q₈×C₁₃₁	#41	1	8	yes	—
154	C₂×D₄×C₁₃₁	—	2	4	no	—
155	C₂×D₄×C₁₃₁	—	2	8	no	—
156	C₂×D₄×C₁₃₁	—	2	2	no	—
157	C₂×D₄×C₁₃₁	—	2	4	no	—
158	C₂²×C₄×C₁₃₁	—	2	4	no	—
159	C₂²×C₄×C₁₃₁	—	2	4	no	—
160	C₂²×C₄×C₁₃₁	—	2	8	no	—
161	C₂²×C₄×C₁₃₁	—	2	4	no	—
162	C₂²×C₄×C₁₃₁	—	2	4	no	—
163	C₂²×C₄×C₁₃₁	—	2	4	no	—
164	C₂²×C₄×C₁₃₁	—	2	8	no	—
165	C₂²×C₄×C₁₃₁	—	2	4	no	—
166	C₂²×C₄×C₁₃₁	—	2	8	no	—
167	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
168	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
169	C₂²×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
170	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₁₃₁	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
176	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
177	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
178	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
179	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
180	C₂²×C₄×C₁₃₁	C₂⁴	1	4	yes	—
181	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
182	C₂²×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
183	C₄⋊C₄×C₁₃₁	—	2	8	no	—
184	C₄⋊C₄×C₁₃₁	—	2	8	no	—
185	C₄⋊C₄×C₁₃₁	—	2	4	no	—
186	C₄⋊C₄×C₁₃₁	—	2	4	no	—
187	C₄⋊C₄×C₁₃₁	—	2	4	no	—
188	C₄⋊C₄×C₁₃₁	—	2	4	no	—
189	C₄⋊C₄×C₁₃₁	—	2	2	no	—
190	C₄⋊C₄×C₁₃₁	—	2	4	no	—
191	C₄⋊C₄×C₁₃₁	—	2	2	no	—
192	C₄⋊C₄×C₁₃₁	—	2	4	no	—
193	C₄⋊C₄×C₁₃₁	—	2	8	no	—
194	C₄⋊C₄×C₁₃₁	—	2	16	no	—
195	C₄⋊C₄×C₁₃₁	—	2	16	no	—
196	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
197	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
198	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
199	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
200	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
201	C₂²⋊C₄×C₁₃₁	—	2	4	no	—
202	C₂²⋊C₄×C₁₃₁	—	2	16	no	—
203	C₂²⋊C₄×C₁₃₁	—	2	16	no	—
204	C₂²⋊C₄×C₁₃₁	#41	1	8	yes	—
205	C₄²×C₁₃₁	—	2	8	no	—
206	C₄²×C₁₃₁	—	2	8	no	—
207	C₄²×C₁₃₁	—	2	4	no	—
208	C₄²×C₁₃₁	—	2	8	no	—
209	C₄²×C₁₃₁	—	2	4	no	—
210	C₄²×C₁₃₁	—	2	4	no	—
211	C₄²×C₁₃₁	—	2	8	no	—
212	C₄²×C₁₃₁	C₂×D₄	1	4	yes	—
213	C₄²×C₁₃₁	C₂×D₄	1	4	yes	—
214	C₄²×C₁₃₁	C₂×D₄	1	4	yes	—
215	C₄²×C₁₃₁	C₂×D₄	1	4	yes	—
216	C₂³×C₁₃₁	—	2	2	no	—
217	C₂³×C₁₃₁	—	2	2	no	—
218	C₂³×C₁₃₁	—	2	2	no	—
219	C₂³×C₁₃₁	—	2	4	no	—
220	C₂³×C₁₃₁	—	2	4	no	—
221	C₂³×C₁₃₁	—	2	4	no	—
222	C₂³×C₁₃₁	C₂²×D₄	1	2	yes	—
223	C₂³×C₁₃₁	C₂×#41	1	4	yes	—
224	C₂³×C₁₃₁	C₂⁵	1	1	yes	characteristic
225	Q₈×C₁₃₁	—	2	16	no	—
226	Q₈×C₁₃₁	—	2	16	no	—
227	Q₈×C₁₃₁	—	4	4	no	—
228	D₄×C₁₃₁	—	4	8	no	—
229	D₄×C₁₃₁	—	4	4	no	—
230	C₂×C₄×C₁₃₁	—	2	16	no	—
231	C₂×C₄×C₁₃₁	—	2	8	no	—
232	C₂×C₄×C₁₃₁	—	2	4	no	—
233	C₂×C₄×C₁₃₁	—	2	2	no	—
234	C₂×C₄×C₁₃₁	—	2	4	no	—
235	C₂×C₄×C₁₃₁	—	2	2	no	—
236	C₂×C₄×C₁₃₁	—	2	2	no	—
237	C₂×C₄×C₁₃₁	—	2	4	no	—
238	C₂×C₄×C₁₃₁	—	2	4	no	—
239	C₂×C₄×C₁₃₁	—	2	2	no	—
240	C₂×C₄×C₁₃₁	—	2	8	no	—
241	C₂×C₄×C₁₃₁	—	2	8	no	—
242	C₂×C₄×C₁₃₁	—	2	4	no	—
243	C₂×C₄×C₁₃₁	—	2	2	no	—
244	C₂×C₄×C₁₃₁	—	2	2	no	—
245	C₂×C₄×C₁₃₁	—	2	2	no	—
246	C₂×C₄×C₁₃₁	—	2	4	no	—
247	C₂×C₄×C₁₃₁	—	2	4	no	—
248	C₂×C₄×C₁₃₁	—	2	8	no	—
249	C₂×C₄×C₁₃₁	—	2	4	no	—
250	C₂×C₄×C₁₃₁	—	2	4	no	—
251	C₂×C₄×C₁₃₁	—	2	4	no	—
252	C₂×C₄×C₁₃₁	—	2	4	no	—
253	C₂×C₄×C₁₃₁	—	4	8	no	—
254	C₂×C₄×C₁₃₁	—	4	8	no	—
255	C₂×C₄×C₁₃₁	C₂²×D₄	1	2	yes	—
256	C₂×C₄×C₁₃₁	C₂²×D₄	1	2	yes	—
257	C₂×C₄×C₁₃₁	C₂²×D₄	1	2	yes	—
258	C₂×C₄×C₁₃₁	C₂²×D₄	1	2	yes	—
259	C₂×C₄×C₁₃₁	C₂²×D₄	1	2	yes	—
260	C₂×C₄×C₁₃₁	C₂²×D₄	1	2	yes	—
261	C₂×C₄×C₁₃₁	C₂×#41	1	8	yes	—
262	C₂×C₄×C₁₃₁	C₂×#41	1	8	yes	—
263	C₂×C₄×C₁₃₁	C₂×#41	1	4	yes	—
264	C₂×C₄×C₁₃₁	2₋¹⁺⁴	1	4	yes	—
265	C₂²×C₁₃₁	—	2	2	no	—
266	C₂²×C₁₃₁	—	2	2	no	—
267	C₂²×C₁₃₁	—	2	8	no	—
268	C₂²×C₁₃₁	—	4	2	no	—
269	C₂²×C₁₃₁	—	4	2	no	—
270	C₂²×C₁₃₁	—	4	2	no	—
271	C₂²×C₁₃₁	—	4	8	no	—
272	C₂²×C₁₃₁	#532	1	4	yes	—
273	C₂²×C₁₃₁	C₂³×D₄	1	1	yes	characteristic
274	C₂²×C₁₃₁	C₂²×#41	1	2	yes	—
275	C₂²×C₁₃₁	C₂²×#41	1	2	yes	—
276	C₂²×C₁₃₁	C₂×2₋¹⁺⁴	1	2	yes	—
277	C₄×C₁₃₁	—	2	2	no	—
278	C₄×C₁₃₁	—	2	8	no	—
279	C₄×C₁₃₁	—	2	2	no	—
280	C₄×C₁₃₁	—	2	2	no	—
281	C₄×C₁₃₁	—	2	2	no	—
282	C₄×C₁₃₁	—	2	8	no	—
283	C₄×C₁₃₁	—	4	2	no	—
284	C₄×C₁₃₁	—	4	4	no	—
285	C₄×C₁₃₁	—	4	2	no	—
286	C₄×C₁₃₁	—	4	2	no	—
287	C₄×C₁₃₁	—	4	4	no	—
288	C₄×C₁₃₁	#525	1	4	yes	—
289	C₄×C₁₃₁	#536	1	4	yes	—
290	C₂×C₁₃₁	—	2	4	no	—
291	C₂×C₁₃₁	—	8	2	no	—
292	C₂×C₁₃₁	C₂×#525	1	1	yes	characteristic
293	C₂×C₁₃₁	C₂×#532	1	1	yes	characteristic
294	C₂×C₁₃₁	C₂×#536	1	1	yes	characteristic
295	C₂×C₁₃₁	#3482	1	4	yes	—
296	#34972	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
297	C₁₃₁	#34972	1	1	yes	characteristic, direct product factor, Sylow
298	C₂×#529	C₂×C₁₃₁	1	1	yes	characteristic
299	C₂²×C₄×Q₈	C₂×C₁₃₁	1	1	yes	characteristic, complemented
300	#2604	C₂×C₁₃₁	1	4	yes	complemented
301	#2565	C₂×C₁₃₁	1	2	yes	complemented
302	#2545	C₂×C₁₃₁	1	2	yes	complemented
303	#2503	C₂×C₁₃₁	1	1	yes	characteristic, complemented
304	#2501	C₂×C₁₃₁	1	1	yes	characteristic
305	#2436	C₂×C₁₃₁	1	1	yes	characteristic
306	#2434	C₂×C₁₃₁	1	1	yes	characteristic, complemented
307	#2413	C₂×C₁₃₁	1	4	yes	complemented
308	#2396	C₂×C₁₃₁	1	4	yes	complemented
309	C₄×C₄⋊Q₈	C₂×C₁₃₁	1	2	yes	complemented
310	C₄×#124	C₂×C₁₃₁	1	2	yes	complemented
311	C₄×#123	C₂×C₁₃₁	1	4	yes	complemented
312	C₄×#117	C₂×C₁₃₁	1	1	yes	characteristic, complemented
313	C₂³×Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
314	C₂×C₄⋊Q₈	C₂²×C₁₃₁	1	2	yes	—
315	#529	—	2	8	no	—
316	C₂×#126	C₂²×C₁₃₁	1	2	yes	—
317	C₂×#125	C₂²×C₁₃₁	1	1	yes	characteristic
318	C₂×#124	C₂²×C₁₃₁	1	1	yes	characteristic
319	C₂×#124	C₂²×C₁₃₁	1	2	yes	—
320	C₂×#123	C₂²×C₁₃₁	1	2	yes	—
321	C₂×#123	C₂²×C₁₃₁	1	4	yes	—
322	C₂×C₂²⋊Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
323	C₂×#121	C₂²×C₁₃₁	1	1	yes	characteristic
324	C₂×C₄×Q₈	—	2	8	no	—
325	C₂×C₄×Q₈	C₂²×C₁₃₁	1	1	yes	characteristic
326	C₂×C₄×Q₈	C₂²×C₁₃₁	1	2	yes	—
327	C₂×C₄×Q₈	C₂²×C₁₃₁	1	2	yes	—
328	C₂×C₄×Q₈	C₂²×C₁₃₁	1	2	yes	—
329	C₂×C₄×Q₈	C₂²×C₁₃₁	1	2	yes	—
330	C₂×C₄×D₄	C₂²×C₁₃₁	1	2	yes	—
331	C₂×C₄×D₄	C₂²×C₁₃₁	1	1	yes	characteristic
332	C₂×#117	C₂²×C₁₃₁	1	2	yes	—
333	C₂×#117	C₂²×C₁₃₁	1	2	yes	—
334	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
335	C₂²×C₄⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
336	C₂²×C₄²	C₂²×C₁₃₁	1	2	yes	—
337	C₂²×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
338	#400	C₂²×C₁₃₁	1	4	yes	—
339	#399	C₂²×C₁₃₁	1	4	yes	—
340	#398	C₂²×C₁₃₁	1	8	yes	—
341	#396	C₂²×C₁₃₁	1	8	yes	—
342	#395	C₂²×C₁₃₁	1	4	yes	—
343	#393	C₂²×C₁₃₁	1	4	yes	—
344	#391	C₂²×C₁₃₁	1	2	yes	—
345	#391	C₂²×C₁₃₁	1	2	yes	—
346	#391	C₂²×C₁₃₁	1	2	yes	—
347	#390	C₂²×C₁₃₁	1	2	yes	—
348	#389	C₂²×C₁₃₁	1	8	yes	—
349	#388	C₂²×C₁₃₁	1	8	yes	—
350	#388	C₂²×C₁₃₁	1	8	yes	—
351	#387	C₂²×C₁₃₁	1	8	yes	—
352	#386	C₂²×C₁₃₁	1	4	yes	—
353	#385	C₂²×C₁₃₁	1	4	yes	—
354	#384	C₂²×C₁₃₁	1	2	yes	—
355	#382	C₂²×C₁₃₁	1	2	yes	—
356	#381	C₂²×C₁₃₁	1	4	yes	—
357	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	8	yes	—
358	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
359	C₄×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
360	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	8	yes	—
361	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
362	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	1	yes	characteristic
363	C₄×C₂²⋊C₄	C₂²×C₁₃₁	1	2	yes	—
364	#376	C₂²×C₁₃₁	1	2	yes	—
365	C₄³	C₂²×C₁₃₁	1	2	yes	—
366	C₂²×Q₈	—	2	8	no	—
367	C₂²×Q₈	—	2	4	no	—
368	C₂²×Q₈	C₂³×C₁₃₁	1	2	yes	—
369	C₂²×Q₈	C₂³×C₁₃₁	1	1	yes	characteristic
370	C₂²×Q₈	C₂³×C₁₃₁	1	2	yes	—
371	C₂²×D₄	C₂³×C₁₃₁	1	2	yes	—
372	C₂²×D₄	C₂³×C₁₃₁	1	1	yes	characteristic
373	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
374	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
375	C₂³×C₄	C₂³×C₁₃₁	1	2	yes	—
376	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
377	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
378	C₂³×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
379	C₄⋊Q₈	—	2	8	no	—
380	#126	—	2	8	no	—
381	#125	—	2	4	no	—
382	#124	—	2	8	no	—
383	#124	—	2	4	no	—
384	#123	—	2	8	no	—
385	#123	—	2	16	no	—
386	C₂²⋊Q₈	—	2	4	no	—
387	#121	—	2	4	no	—
388	C₄×Q₈	—	2	8	no	—
389	C₄×Q₈	—	2	8	no	—
390	C₄×Q₈	—	2	8	no	—
391	C₄×Q₈	—	2	8	no	—
392	C₄×Q₈	—	2	4	no	—
393	C₄×D₄	—	2	8	no	—
394	C₄×D₄	—	2	4	no	—
395	#117	—	2	8	no	—
396	#117	—	2	8	no	—
397	C₂×C₄⋊C₄	—	2	8	no	—
398	C₂×C₄⋊C₄	—	2	4	no	—
399	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
400	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
401	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
402	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
403	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
404	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
405	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
406	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
407	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
408	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	1	yes	characteristic
409	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	2	yes	—
410	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	8	yes	—
411	C₂×C₄⋊C₄	C₂³×C₁₃₁	1	8	yes	—
412	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	2	yes	—
413	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	2	yes	—
414	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	2	yes	—
415	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	2	yes	—
416	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	2	yes	—
417	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	2	yes	—
418	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	8	yes	—
419	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	8	yes	—
420	C₂×C₂²⋊C₄	C₂³×C₁₃₁	1	2	yes	—
421	C₂×C₄²	—	2	8	no	—
422	C₂×C₄²	D₄×C₁₃₁	1	8	yes	—
423	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
424	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
425	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
426	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
427	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
428	C₂×C₄²	C₂³×C₁₃₁	1	2	yes	—
429	C₂×C₄²	C₂³×C₁₃₁	1	4	yes	—
430	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
431	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
433	C₂×C₄²	C₂³×C₁₃₁	1	1	yes	characteristic
434	#95	C₂³×C₁₃₁	1	4	yes	—
435	#95	C₂³×C₁₃₁	1	2	yes	—
436	#95	C₂³×C₁₃₁	1	2	yes	—
437	#95	C₂³×C₁₃₁	1	8	yes	—
438	#95	C₂³×C₁₃₁	1	8	yes	—
439	#95	C₂³×C₁₃₁	1	8	yes	—
440	#95	C₂³×C₁₃₁	1	8	yes	—
441	C₂⁴	C₂⁴×C₁₃₁	1	2	yes	—
442	C₂⁴	C₂⁴×C₁₃₁	1	1	yes	characteristic
443	C₂×Q₈	—	2	32	no	—
444	C₂×Q₈	—	2	4	no	—
445	C₂×Q₈	—	2	4	no	—
446	C₂×Q₈	—	2	2	no	—
447	C₂×Q₈	—	2	8	no	—
448	C₂×Q₈	—	2	4	no	—
449	C₂×Q₈	—	2	4	no	—
450	C₂×Q₈	#41×C₁₃₁	1	8	yes	—
451	C₂×D₄	—	2	4	no	—
452	C₂×D₄	—	2	8	no	—
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	4	no	—
455	C₂²×C₄	—	2	4	no	—
456	C₂²×C₄	—	2	4	no	—
457	C₂²×C₄	—	2	8	no	—
458	C₂²×C₄	—	2	4	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	2	4	no	—
461	C₂²×C₄	—	2	8	no	—
462	C₂²×C₄	—	2	4	no	—
463	C₂²×C₄	—	2	8	no	—
464	C₂²×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
465	C₂²×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
466	C₂²×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
467	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
468	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
469	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
470	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
471	C₂²×C₄	C₂⁴×C₁₃₁	1	2	yes	—
472	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
473	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
474	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
475	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
476	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
477	C₂²×C₄	C₂⁴×C₁₃₁	1	4	yes	—
478	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
479	C₂²×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
480	C₄⋊C₄	—	2	8	no	—
481	C₄⋊C₄	—	2	8	no	—
482	C₄⋊C₄	—	2	4	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	2	4	no	—
486	C₄⋊C₄	—	2	2	no	—
487	C₄⋊C₄	—	2	4	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	4	no	—
490	C₄⋊C₄	—	2	8	no	—
491	C₄⋊C₄	—	2	16	no	—
492	C₄⋊C₄	—	2	16	no	—
493	C₂²⋊C₄	—	2	4	no	—
494	C₂²⋊C₄	—	2	4	no	—
495	C₂²⋊C₄	—	2	4	no	—
496	C₂²⋊C₄	—	2	4	no	—
497	C₂²⋊C₄	—	2	4	no	—
498	C₂²⋊C₄	—	2	4	no	—
499	C₂²⋊C₄	—	2	16	no	—
500	C₂²⋊C₄	—	2	16	no	—
501	C₂²⋊C₄	#41×C₁₃₁	1	8	yes	—
502	C₄²	—	2	8	no	—
503	C₄²	—	2	8	no	—
504	C₄²	—	2	4	no	—
505	C₄²	—	2	8	no	—
506	C₄²	—	2	4	no	—
507	C₄²	—	2	4	no	—
508	C₄²	—	2	8	no	—
509	C₄²	C₂×D₄×C₁₃₁	1	4	yes	—
510	C₄²	C₂×D₄×C₁₃₁	1	4	yes	—
511	C₄²	C₂×D₄×C₁₃₁	1	4	yes	—
512	C₄²	C₂×D₄×C₁₃₁	1	4	yes	—
513	C₂³	—	2	2	no	—
514	C₂³	—	2	2	no	—
515	C₂³	—	2	2	no	—
516	C₂³	—	2	4	no	—
517	C₂³	—	2	4	no	—
518	C₂³	—	2	4	no	—
519	C₂³	C₂²×D₄×C₁₃₁	1	2	yes	—
520	C₂³	C₂×#41×C₁₃₁	1	4	yes	—
521	C₂³	C₂⁵×C₁₃₁	1	1	yes	characteristic
522	Q₈	—	2	16	no	—
523	Q₈	—	2	16	no	—
524	Q₈	—	4	4	no	—
525	D₄	—	4	8	no	—
526	D₄	—	4	4	no	—
527	C₂×C₄	—	2	16	no	—
528	C₂×C₄	—	2	8	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	2	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	8	no	—
538	C₂×C₄	—	2	8	no	—
539	C₂×C₄	—	2	4	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	8	no	—
546	C₂×C₄	—	2	4	no	—
547	C₂×C₄	—	2	4	no	—
548	C₂×C₄	—	2	4	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	4	8	no	—
551	C₂×C₄	—	4	8	no	—
552	C₂×C₄	C₂²×D₄×C₁₃₁	1	2	yes	—
553	C₂×C₄	C₂²×D₄×C₁₃₁	1	2	yes	—
554	C₂×C₄	C₂²×D₄×C₁₃₁	1	2	yes	—
555	C₂×C₄	C₂²×D₄×C₁₃₁	1	2	yes	—
556	C₂×C₄	C₂²×D₄×C₁₃₁	1	2	yes	—
557	C₂×C₄	C₂²×D₄×C₁₃₁	1	2	yes	—
558	C₂×C₄	C₂×#41×C₁₃₁	1	8	yes	—
559	C₂×C₄	C₂×#41×C₁₃₁	1	8	yes	—
560	C₂×C₄	C₂×#41×C₁₃₁	1	4	yes	—
561	C₂×C₄	2₋¹⁺⁴×C₁₃₁	1	4	yes	—
562	C₂²	—	2	2	no	—
563	C₂²	—	2	2	no	—
564	C₂²	—	2	8	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	4	2	no	—
567	C₂²	—	4	2	no	—
568	C₂²	—	4	8	no	—
569	C₂²	#532×C₁₃₁	1	4	yes	—
570	C₂²	C₂³×D₄×C₁₃₁	1	1	yes	characteristic
571	C₂²	C₂²×#41×C₁₃₁	1	2	yes	—
572	C₂²	C₂²×#41×C₁₃₁	1	2	yes	—
573	C₂²	C₂×2₋¹⁺⁴×C₁₃₁	1	2	yes	—
574	C₄	—	2	2	no	—
575	C₄	—	2	8	no	—
576	C₄	—	2	2	no	—
577	C₄	—	2	2	no	—
578	C₄	—	2	2	no	—
579	C₄	—	2	8	no	—
580	C₄	—	4	2	no	—
581	C₄	—	4	4	no	—
582	C₄	—	4	2	no	—
583	C₄	—	4	2	no	—
584	C₄	—	4	4	no	—
585	C₄	#525×C₁₃₁	1	4	yes	—
586	C₄	#536×C₁₃₁	1	4	yes	—
587	C₂	—	2	4	no	—
588	C₂	—	8	2	no	—
589	C₂	C₂×#525×C₁₃₁	1	1	yes	characteristic
590	C₂	C₂×#532×C₁₃₁	1	1	yes	characteristic
591	C₂	C₂×#536×C₁₃₁	1	1	yes	characteristic
592	C₂	#3482×C₁₃₁	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃₁
socle	C₂³×C₁₃₁	C₂⁵
prime radical	C₂×#121×C₁₃₁	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×Q₈×C₁₃₁	C₂	semidirect
3. #2604×C₁₃₁	C₂	semidirect
4. #2565×C₁₃₁	C₂	semidirect
5. #2545×C₁₃₁	C₂	semidirect
6. #2503×C₁₃₁	C₂	semidirect
9. #2434×C₁₃₁	C₂	semidirect
10. #2413×C₁₃₁	C₂	semidirect
11. #2396×C₁₃₁	C₂	semidirect
12. C₄×C₄⋊Q₈×C₁₃₁	C₂	semidirect
13. C₄×#124×C₁₃₁	C₂	semidirect
14. C₄×#123×C₁₃₁	C₂	semidirect
15. C₄×#117×C₁₃₁	C₂	semidirect
296. #34972	C₁₃₁	direct (N × H)
297. C₁₃₁	#34972	direct (N × H)
299. C₂²×C₄×Q₈	C₂×C₁₃₁	semidirect
300. #2604	C₂×C₁₃₁	semidirect
301. #2565	C₂×C₁₃₁	semidirect
302. #2545	C₂×C₁₃₁	semidirect
303. #2503	C₂×C₁₃₁	semidirect
306. #2434	C₂×C₁₃₁	semidirect
307. #2413	C₂×C₁₃₁	semidirect
308. #2396	C₂×C₁₃₁	semidirect
309. C₄×C₄⋊Q₈	C₂×C₁₃₁	semidirect
310. C₄×#124	C₂×C₁₃₁	semidirect
311. C₄×#123	C₂×C₁₃₁	semidirect
312. C₄×#117	C₂×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#529×C₁₃₁	2
C₂²×C₄×Q₈×C₁₃₁	2
#2604×C₁₃₁	2
#2565×C₁₃₁	2
#2545×C₁₃₁	2
#2503×C₁₃₁	2
#2501×C₁₃₁	2
#2436×C₁₃₁	2
#2434×C₁₃₁	2
#2413×C₁₃₁	2
#2396×C₁₃₁	2
C₄×C₄⋊Q₈×C₁₃₁	2
C₄×#124×C₁₃₁	2
C₄×#123×C₁₃₁	2
C₄×#117×C₁₃₁	2
#34972	131

Subgroups — C131×#34972 #45720856

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#34972 #45720856