Subgroups — C₁₃₁×#33380 #45719264

Order 33536.

1170 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#135×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×M₃₂×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#427×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#2141×C₁₃₁	C₂	1	2	yes	maximal, complemented
5	#2140×C₁₃₁	C₂	1	2	yes	maximal, complemented
6	#2139×C₁₃₁	C₂	1	4	yes	maximal, complemented
7	#2138×C₁₃₁	C₂	1	4	yes	maximal, complemented
8	C₂²×#41×C₁₃₁	—	2	1	no	—
9	C₂×#135×C₁₃₁	—	2	1	no	—
10	C₂×#135×C₁₃₁	—	2	1	no	—
11	C₂×#135×C₁₃₁	—	2	4	no	—
12	C₂×#135×C₁₃₁	C₄	1	8	yes	complemented
13	C₂×#135×C₁₃₁	C₂²	1	2	yes	—
14	C₂×#135×C₁₃₁	C₂²	1	2	yes	—
15	C₂²×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
16	C₂²×SD₁₆×C₁₃₁	—	2	1	no	—
17	C₂²×D₈×C₁₃₁	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂×#117×C₁₃₁	—	2	1	no	—
20	C₂×M₃₂×C₁₃₁	—	2	4	no	—
21	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	—
22	C₂×M₃₂×C₁₃₁	C₂²	1	2	yes	complemented
23	C₂²×C₁₆×C₁₃₁	C₂²	1	2	yes	complemented
24	#427×C₁₃₁	—	2	1	no	—
25	#427×C₁₃₁	—	2	1	no	—
26	#427×C₁₃₁	C₂²	1	2	yes	—
27	#427×C₁₃₁	C₂²	1	2	yes	complemented
28	C₂×#107×C₁₃₁	C₂²	1	2	yes	complemented
29	C₂×#106×C₁₃₁	—	2	1	no	—
30	#358×C₁₃₁	C₂²	1	8	yes	complemented
31	#357×C₁₃₁	C₂²	1	8	yes	complemented
32	C₂×#41×C₁₃₁	—	2	8	no	—
33	C₂×#41×C₁₃₁	—	2	2	no	—
34	C₂×#41×C₁₃₁	—	2	2	no	—
35	C₂×#41×C₁₃₁	—	4	1	no	—
36	C₂×#41×C₁₃₁	—	4	1	no	—
37	C₂×#41×C₁₃₁	—	4	4	no	—
38	C₂²×Q₈×C₁₃₁	—	2	1	no	—
39	C₂²×D₄×C₁₃₁	—	2	1	no	—
40	C₂²×D₄×C₁₃₁	—	4	1	no	—
41	C₂³×C₄×C₁₃₁	—	4	1	no	—
42	C₂³×C₄×C₁₃₁	D₄	1	1	yes	characteristic
43	#135×C₁₃₁	—	2	8	no	—
44	#135×C₁₃₁	—	2	8	no	—
45	#135×C₁₃₁	—	2	4	no	—
46	#135×C₁₃₁	—	2	4	no	—
47	#135×C₁₃₁	C₂×C₄	1	8	yes	—
48	#135×C₁₃₁	C₂×C₄	1	8	yes	complemented
49	C₂×Q₁₆×C₁₃₁	—	2	1	no	—
50	C₂×Q₁₆×C₁₃₁	—	2	2	no	—
51	C₂×Q₁₆×C₁₃₁	C₂×C₄	1	4	yes	complemented
52	C₂×Q₁₆×C₁₃₁	C₂³	1	2	yes	—
53	C₂×SD₁₆×C₁₃₁	—	2	8	no	—
54	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₁₃₁	—	2	2	no	—
56	C₂×D₈×C₁₃₁	—	2	1	no	—
57	C₂×D₈×C₁₃₁	—	2	2	no	—
58	C₂×D₈×C₁₃₁	C₂×C₄	1	4	yes	complemented
59	C₂×D₈×C₁₃₁	C₂³	1	2	yes	—
60	C₂²×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
61	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₁₃₁	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₁₃₁	D₄	1	4	yes	—
65	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
68	#117×C₁₃₁	—	2	2	no	—
69	#117×C₁₃₁	—	2	2	no	—
70	#117×C₁₃₁	—	4	1	no	—
71	#117×C₁₃₁	—	4	1	no	—
72	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₁₃₁	—	4	1	no	—
74	C₂×C₄²×C₁₃₁	—	4	1	no	—
75	M₃₂×C₁₃₁	—	2	4	no	—
76	M₃₂×C₁₃₁	—	2	4	no	—
77	C₂×C₁₆×C₁₃₁	—	2	4	no	—
78	C₂×C₁₆×C₁₃₁	C₂³	1	4	yes	—
79	#107×C₁₃₁	—	2	2	no	—
80	#107×C₁₃₁	C₂³	1	4	yes	—
81	#106×C₁₃₁	—	2	2	no	—
82	#106×C₁₃₁	—	2	2	no	—
83	C₂⁴×C₁₃₁	—	4	1	no	—
84	C₂⁴×C₁₃₁	D₈	1	1	yes	characteristic
85	#41×C₁₃₁	—	2	8	no	—
86	#41×C₁₃₁	—	2	8	no	—
87	#41×C₁₃₁	—	4	8	no	—
88	#41×C₁₃₁	—	4	8	no	—
89	#41×C₁₃₁	—	4	4	no	—
90	#41×C₁₃₁	—	4	4	no	—
91	C₂×Q₈×C₁₃₁	—	2	4	no	—
92	C₂×Q₈×C₁₃₁	—	2	2	no	—
93	C₂×Q₈×C₁₃₁	—	4	1	no	—
94	C₂×Q₈×C₁₃₁	—	4	2	no	—
95	C₂×D₄×C₁₃₁	—	2	4	no	—
96	C₂×D₄×C₁₃₁	—	2	2	no	—
97	C₂×D₄×C₁₃₁	—	4	8	no	—
98	C₂×D₄×C₁₃₁	—	4	2	no	—
99	C₂×D₄×C₁₃₁	—	4	2	no	—
100	C₂×D₄×C₁₃₁	—	4	1	no	—
101	C₂×D₄×C₁₃₁	—	4	2	no	—
102	C₂²×C₄×C₁₃₁	—	4	1	no	—
103	C₂²×C₄×C₁₃₁	—	4	8	no	—
104	C₂²×C₄×C₁₃₁	—	4	2	no	—
105	C₂²×C₄×C₁₃₁	—	4	2	no	—
106	C₂²×C₄×C₁₃₁	—	4	2	no	—
107	C₂²×C₄×C₁₃₁	C₂²⋊C₄	1	4	yes	—
108	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
109	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
110	C₂²×C₄×C₁₃₁	D₈	1	1	yes	characteristic
111	C₂²×C₄×C₁₃₁	SD₁₆	1	4	yes	—
112	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
115	Q₁₆×C₁₃₁	—	2	4	no	—
116	Q₁₆×C₁₃₁	—	2	2	no	—
117	Q₁₆×C₁₃₁	C₂²×C₄	1	4	yes	—
118	SD₁₆×C₁₃₁	—	2	8	no	—
119	SD₁₆×C₁₃₁	—	2	8	no	—
120	D₈×C₁₃₁	—	2	4	no	—
121	D₈×C₁₃₁	—	2	2	no	—
122	D₈×C₁₃₁	C₂²×C₄	1	4	yes	—
123	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	4	yes	—
124	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	4	yes	—
125	C₂×C₈×C₁₃₁	C₂²⋊C₄	1	4	yes	—
126	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
127	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₁₃₁	C₂²×C₄	1	2	yes	—
129	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₁₃₁	C₂×D₄	1	2	yes	—
135	C₂×C₈×C₁₃₁	C₂⁴	1	1	yes	characteristic
136	C₄⋊C₄×C₁₃₁	—	2	4	no	—
137	C₄⋊C₄×C₁₃₁	—	4	2	no	—
138	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
139	C₂²⋊C₄×C₁₃₁	—	4	2	no	—
140	C₄²×C₁₃₁	—	4	2	no	—
141	C₄²×C₁₃₁	—	4	2	no	—
142	C₁₆×C₁₃₁	—	2	4	no	—
143	C₂³×C₁₃₁	—	4	2	no	—
144	C₂³×C₁₃₁	—	4	2	no	—
145	C₂³×C₁₃₁	—	4	4	no	—
146	C₂³×C₁₃₁	—	8	4	no	—
147	C₂³×C₁₃₁	#102	1	4	yes	—
148	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
149	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
150	C₂³×C₁₃₁	C₂×D₈	1	1	yes	characteristic
151	Q₈×C₁₃₁	—	2	4	no	—
152	Q₈×C₁₃₁	—	4	2	no	—
153	Q₈×C₁₃₁	—	4	4	no	—
154	D₄×C₁₃₁	—	2	4	no	—
155	D₄×C₁₃₁	—	4	8	no	—
156	D₄×C₁₃₁	—	4	8	no	—
157	D₄×C₁₃₁	—	4	2	no	—
158	D₄×C₁₃₁	—	4	4	no	—
159	C₂×C₄×C₁₃₁	—	4	2	no	—
160	C₂×C₄×C₁₃₁	—	4	4	no	—
161	C₂×C₄×C₁₃₁	—	4	8	no	—
162	C₂×C₄×C₁₃₁	—	4	8	no	—
163	C₂×C₄×C₁₃₁	—	4	4	no	—
164	C₂×C₄×C₁₃₁	—	8	4	no	—
165	C₂×C₄×C₁₃₁	#102	1	4	yes	—
166	C₂×C₄×C₁₃₁	#102	1	4	yes	—
167	C₂×C₄×C₁₃₁	#102	1	4	yes	—
168	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	2	yes	—
169	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	2	yes	—
170	C₂×C₄×C₁₃₁	C₂×C₂²⋊C₄	1	2	yes	—
171	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
172	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₁₃₁	C₂×D₈	1	1	yes	characteristic
174	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	2	yes	—
175	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	2	yes	—
176	C₂×C₄×C₁₃₁	C₂×SD₁₆	1	2	yes	—
177	C₂×C₄×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
178	C₈×C₁₃₁	C₂×C₂²⋊C₄	1	2	yes	—
179	C₈×C₁₃₁	C₂×C₂²⋊C₄	1	2	yes	—
180	C₈×C₁₃₁	C₂×C₂²⋊C₄	1	2	yes	—
181	C₈×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
182	C₈×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
183	C₂²×C₁₃₁	—	2	2	no	—
184	C₂²×C₁₃₁	—	2	2	no	—
185	C₂²×C₁₃₁	—	2	2	no	—
186	C₂²×C₁₃₁	—	4	4	no	—
187	C₂²×C₁₃₁	—	4	4	no	—
188	C₂²×C₁₃₁	—	8	8	no	—
189	C₂²×C₁₃₁	—	8	4	no	—
190	C₂²×C₁₃₁	C₂×#102	1	2	yes	—
191	C₂²×C₁₃₁	C₂×#102	1	2	yes	—
192	C₂²×C₁₃₁	C₂×#102	1	2	yes	—
193	C₂²×C₁₃₁	C₂²×D₈	1	1	yes	characteristic
194	C₄×C₁₃₁	—	4	4	no	—
195	C₄×C₁₃₁	—	8	4	no	—
196	C₄×C₁₃₁	C₂×#102	1	2	yes	—
197	C₄×C₁₃₁	C₂×#102	1	2	yes	—
198	C₄×C₁₃₁	C₂×#102	1	2	yes	—
199	C₄×C₁₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
200	C₄×C₁₃₁	C₂²×SD₁₆	1	1	yes	characteristic
201	C₂×C₁₃₁	—	2	2	no	—
202	C₂×C₁₃₁	—	2	2	no	—
203	C₂×C₁₃₁	—	2	2	no	—
204	C₂×C₁₃₁	—	8	4	no	—
205	C₂×C₁₃₁	C₂²×#102	1	1	yes	characteristic
206	C₂×C₁₃₁	#3414	1	2	yes	—
207	#33380	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₁₃₁	#33380	1	1	yes	characteristic, direct product factor, Sylow
209	C₂²×#135	C₂×C₁₃₁	1	1	yes	characteristic
210	C₂²×M₃₂	C₂×C₁₃₁	1	1	yes	characteristic, complemented
211	C₂×#427	C₂×C₁₃₁	1	1	yes	characteristic, complemented
212	#2141	C₂×C₁₃₁	1	2	yes	complemented
213	#2140	C₂×C₁₃₁	1	2	yes	complemented
214	#2139	C₂×C₁₃₁	1	4	yes	complemented
215	#2138	C₂×C₁₃₁	1	4	yes	complemented
216	C₂²×#41	—	2	1	no	—
217	C₂×#135	—	2	1	no	—
218	C₂×#135	—	2	1	no	—
219	C₂×#135	—	2	4	no	—
220	C₂×#135	C₄×C₁₃₁	1	8	yes	complemented
221	C₂×#135	C₂²×C₁₃₁	1	2	yes	—
222	C₂×#135	C₂²×C₁₃₁	1	2	yes	—
223	C₂²×Q₁₆	C₂²×C₁₃₁	1	1	yes	characteristic
224	C₂²×SD₁₆	—	2	1	no	—
225	C₂²×D₈	C₂²×C₁₃₁	1	1	yes	characteristic
226	C₂³×C₈	C₂²×C₁₃₁	1	1	yes	characteristic
227	C₂×#117	—	2	1	no	—
228	C₂×M₃₂	—	2	4	no	—
229	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	—
230	C₂×M₃₂	C₂²×C₁₃₁	1	2	yes	complemented
231	C₂²×C₁₆	C₂²×C₁₃₁	1	2	yes	complemented
232	#427	—	2	1	no	—
233	#427	—	2	1	no	—
234	#427	C₂²×C₁₃₁	1	2	yes	—
235	#427	C₂²×C₁₃₁	1	2	yes	complemented
236	C₂×#107	C₂²×C₁₃₁	1	2	yes	complemented
237	C₂×#106	—	2	1	no	—
238	#358	C₂²×C₁₃₁	1	8	yes	complemented
239	#357	C₂²×C₁₃₁	1	8	yes	complemented
240	C₂×#41	—	2	8	no	—
241	C₂×#41	—	2	2	no	—
242	C₂×#41	—	2	2	no	—
243	C₂×#41	—	4	1	no	—
244	C₂×#41	—	4	1	no	—
245	C₂×#41	—	4	4	no	—
246	C₂²×Q₈	—	2	1	no	—
247	C₂²×D₄	—	2	1	no	—
248	C₂²×D₄	—	4	1	no	—
249	C₂³×C₄	—	4	1	no	—
250	C₂³×C₄	D₄×C₁₃₁	1	1	yes	characteristic
251	#135	—	2	8	no	—
252	#135	—	2	8	no	—
253	#135	—	2	4	no	—
254	#135	—	2	4	no	—
255	#135	C₂×C₄×C₁₃₁	1	8	yes	—
256	#135	C₂×C₄×C₁₃₁	1	8	yes	complemented
257	C₂×Q₁₆	—	2	1	no	—
258	C₂×Q₁₆	—	2	2	no	—
259	C₂×Q₁₆	C₂×C₄×C₁₃₁	1	4	yes	complemented
260	C₂×Q₁₆	C₂³×C₁₃₁	1	2	yes	—
261	C₂×SD₁₆	—	2	8	no	—
262	C₂×SD₁₆	—	2	2	no	—
263	C₂×SD₁₆	—	2	2	no	—
264	C₂×D₈	—	2	1	no	—
265	C₂×D₈	—	2	2	no	—
266	C₂×D₈	C₂×C₄×C₁₃₁	1	4	yes	complemented
267	C₂×D₈	C₂³×C₁₃₁	1	2	yes	—
268	C₂²×C₈	C₂×C₄×C₁₃₁	1	4	yes	—
269	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
270	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
271	C₂²×C₈	D₄×C₁₃₁	1	1	yes	characteristic
272	C₂²×C₈	D₄×C₁₃₁	1	4	yes	—
273	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
274	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
275	C₂²×C₈	C₂³×C₁₃₁	1	1	yes	characteristic
276	#117	—	2	2	no	—
277	#117	—	2	2	no	—
278	#117	—	4	1	no	—
279	#117	—	4	1	no	—
280	C₂×C₄⋊C₄	—	2	2	no	—
281	C₂×C₂²⋊C₄	—	4	1	no	—
282	C₂×C₄²	—	4	1	no	—
283	M₃₂	—	2	4	no	—
284	M₃₂	—	2	4	no	—
285	C₂×C₁₆	—	2	4	no	—
286	C₂×C₁₆	C₂³×C₁₃₁	1	4	yes	—
287	#107	—	2	2	no	—
288	#107	C₂³×C₁₃₁	1	4	yes	—
289	#106	—	2	2	no	—
290	#106	—	2	2	no	—
291	C₂⁴	—	4	1	no	—
292	C₂⁴	D₈×C₁₃₁	1	1	yes	characteristic
293	#41	—	2	8	no	—
294	#41	—	2	8	no	—
295	#41	—	4	8	no	—
296	#41	—	4	8	no	—
297	#41	—	4	4	no	—
298	#41	—	4	4	no	—
299	C₂×Q₈	—	2	4	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	4	1	no	—
302	C₂×Q₈	—	4	2	no	—
303	C₂×D₄	—	2	4	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	4	8	no	—
306	C₂×D₄	—	4	2	no	—
307	C₂×D₄	—	4	2	no	—
308	C₂×D₄	—	4	1	no	—
309	C₂×D₄	—	4	2	no	—
310	C₂²×C₄	—	4	1	no	—
311	C₂²×C₄	—	4	8	no	—
312	C₂²×C₄	—	4	2	no	—
313	C₂²×C₄	—	4	2	no	—
314	C₂²×C₄	—	4	2	no	—
315	C₂²×C₄	C₂²⋊C₄×C₁₃₁	1	4	yes	—
316	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
317	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
318	C₂²×C₄	D₈×C₁₃₁	1	1	yes	characteristic
319	C₂²×C₄	SD₁₆×C₁₃₁	1	4	yes	—
320	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
321	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
322	C₂²×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
323	Q₁₆	—	2	4	no	—
324	Q₁₆	—	2	2	no	—
325	Q₁₆	C₂²×C₄×C₁₃₁	1	4	yes	—
326	SD₁₆	—	2	8	no	—
327	SD₁₆	—	2	8	no	—
328	D₈	—	2	4	no	—
329	D₈	—	2	2	no	—
330	D₈	C₂²×C₄×C₁₃₁	1	4	yes	—
331	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	4	yes	—
332	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	4	yes	—
333	C₂×C₈	C₂²⋊C₄×C₁₃₁	1	4	yes	—
334	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
335	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
336	C₂×C₈	C₂²×C₄×C₁₃₁	1	2	yes	—
337	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
338	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
339	C₂×C₈	C₂×D₄×C₁₃₁	1	1	yes	characteristic
340	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
341	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
342	C₂×C₈	C₂×D₄×C₁₃₁	1	2	yes	—
343	C₂×C₈	C₂⁴×C₁₃₁	1	1	yes	characteristic
344	C₄⋊C₄	—	2	4	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₂²⋊C₄	—	4	2	no	—
347	C₂²⋊C₄	—	4	2	no	—
348	C₄²	—	4	2	no	—
349	C₄²	—	4	2	no	—
350	C₁₆	—	2	4	no	—
351	C₂³	—	4	2	no	—
352	C₂³	—	4	4	no	—
353	C₂³	—	4	2	no	—
354	C₂³	—	8	4	no	—
355	C₂³	#102×C₁₃₁	1	4	yes	—
356	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
357	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
358	C₂³	C₂×D₈×C₁₃₁	1	1	yes	characteristic
359	Q₈	—	2	4	no	—
360	Q₈	—	4	4	no	—
361	Q₈	—	4	2	no	—
362	D₄	—	2	4	no	—
363	D₄	—	4	8	no	—
364	D₄	—	4	8	no	—
365	D₄	—	4	4	no	—
366	D₄	—	4	2	no	—
367	C₂×C₄	—	4	4	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	8	no	—
370	C₂×C₄	—	4	8	no	—
371	C₂×C₄	—	4	4	no	—
372	C₂×C₄	—	8	4	no	—
373	C₂×C₄	#102×C₁₃₁	1	4	yes	—
374	C₂×C₄	#102×C₁₃₁	1	4	yes	—
375	C₂×C₄	#102×C₁₃₁	1	4	yes	—
376	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	2	yes	—
377	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	2	yes	—
378	C₂×C₄	C₂×C₂²⋊C₄×C₁₃₁	1	2	yes	—
379	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
380	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
381	C₂×C₄	C₂×D₈×C₁₃₁	1	1	yes	characteristic
382	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	2	yes	—
383	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	2	yes	—
384	C₂×C₄	C₂×SD₁₆×C₁₃₁	1	2	yes	—
385	C₂×C₄	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
386	C₈	C₂×C₂²⋊C₄×C₁₃₁	1	2	yes	—
387	C₈	C₂×C₂²⋊C₄×C₁₃₁	1	2	yes	—
388	C₈	C₂×C₂²⋊C₄×C₁₃₁	1	2	yes	—
389	C₈	C₂³×C₄×C₁₃₁	1	1	yes	characteristic
390	C₈	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
391	C₂²	—	2	2	no	—
392	C₂²	—	2	2	no	—
393	C₂²	—	2	2	no	—
394	C₂²	—	4	4	no	—
395	C₂²	—	4	4	no	—
396	C₂²	—	8	8	no	—
397	C₂²	—	8	4	no	—
398	C₂²	C₂×#102×C₁₃₁	1	2	yes	—
399	C₂²	C₂×#102×C₁₃₁	1	2	yes	—
400	C₂²	C₂×#102×C₁₃₁	1	2	yes	—
401	C₂²	C₂²×D₈×C₁₃₁	1	1	yes	characteristic
402	C₄	—	4	4	no	—
403	C₄	—	8	4	no	—
404	C₄	C₂×#102×C₁₃₁	1	2	yes	—
405	C₄	C₂×#102×C₁₃₁	1	2	yes	—
406	C₄	C₂×#102×C₁₃₁	1	2	yes	—
407	C₄	C₂²×C₂²⋊C₄×C₁₃₁	1	1	yes	characteristic
408	C₄	C₂²×SD₁₆×C₁₃₁	1	1	yes	characteristic
409	C₂	—	2	2	no	—
410	C₂	—	2	2	no	—
411	C₂	—	2	2	no	—
412	C₂	—	8	4	no	—
413	C₂	C₂²×#102×C₁₃₁	1	1	yes	characteristic
414	C₂	#3414×C₁₃₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃₁	C₂×D₈
commutator	C₈	C₂³×C₄×C₁₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃₁
socle	C₂²×C₁₃₁	C₂²×D₈
prime radical	C₂²×#135×C₁₃₁	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃₁ ◁ C₂³×C₄×C₁₃₁ ◁ C₂³×C₈×C₁₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×M₃₂×C₁₃₁	C₂	semidirect
3. C₂×#427×C₁₃₁	C₂	semidirect
4. #2141×C₁₃₁	C₂	semidirect
5. #2140×C₁₃₁	C₂	semidirect
6. #2139×C₁₃₁	C₂	semidirect
7. #2138×C₁₃₁	C₂	semidirect
12. C₂×#135×C₁₃₁	C₄	semidirect
22. C₂×M₃₂×C₁₃₁	C₂²	semidirect
23. C₂²×C₁₆×C₁₃₁	C₂²	semidirect
27. #427×C₁₃₁	C₂²	semidirect
28. C₂×#107×C₁₃₁	C₂²	semidirect
30. #358×C₁₃₁	C₂²	semidirect
31. #357×C₁₃₁	C₂²	semidirect
48. #135×C₁₃₁	C₂×C₄	semidirect
51. C₂×Q₁₆×C₁₃₁	C₂×C₄	semidirect
58. C₂×D₈×C₁₃₁	C₂×C₄	semidirect
207. #33380	C₁₃₁	direct (N × H)
208. C₁₃₁	#33380	direct (N × H)
210. C₂²×M₃₂	C₂×C₁₃₁	semidirect
211. C₂×#427	C₂×C₁₃₁	semidirect
212. #2141	C₂×C₁₃₁	semidirect
213. #2140	C₂×C₁₃₁	semidirect
214. #2139	C₂×C₁₃₁	semidirect
215. #2138	C₂×C₁₃₁	semidirect
220. C₂×#135	C₄×C₁₃₁	semidirect
230. C₂×M₃₂	C₂²×C₁₃₁	semidirect
231. C₂²×C₁₆	C₂²×C₁₃₁	semidirect
235. #427	C₂²×C₁₃₁	semidirect
236. C₂×#107	C₂²×C₁₃₁	semidirect
238. #358	C₂²×C₁₃₁	semidirect
239. #357	C₂²×C₁₃₁	semidirect
256. #135	C₂×C₄×C₁₃₁	semidirect
259. C₂×Q₁₆	C₂×C₄×C₁₃₁	semidirect
266. C₂×D₈	C₂×C₄×C₁₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#135×C₁₃₁	2
C₂²×M₃₂×C₁₃₁	2
C₂×#427×C₁₃₁	2
#2141×C₁₃₁	2
#2140×C₁₃₁	2
#2139×C₁₃₁	2
#2138×C₁₃₁	2
#33380	131

Subgroups — C131×#33380 #45719264

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₁×#33380 #45719264