Subgroups — C₃₁×#36626 #21493385

Order 7936.

3250 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#525	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₄²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2855	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2854	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2853	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2849	C₂	1	4	yes	maximal, complemented
7	C₄×C₃₁×#123	C₂	1	8	yes	maximal, complemented
8	C₃₁×#2297	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2296	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2280	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
13	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#525	—	2	8	no	—
15	C₂×C₃₁×#123	C₂²	1	8	yes	—
16	C₂×C₃₁×#123	C₂²	1	2	yes	—
17	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
18	C₂×C₃₁×#121	C₂²	1	2	yes	—
19	C₂×C₃₁×#120	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
21	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#117	C₂²	1	2	yes	—
23	C₂²×C₄²×C₃₁	—	2	8	no	—
24	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
25	C₂²×C₄²×C₃₁	C₂²	1	4	yes	—
26	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
27	C₂²×C₄²×C₃₁	C₂²	1	4	yes	—
28	C₃₁×#391	C₂²	1	4	yes	complemented
29	C₃₁×#390	C₂²	1	4	yes	complemented
30	C₃₁×#389	C₂²	1	8	yes	complemented
31	C₃₁×#388	C₂²	1	16	yes	complemented
32	C₃₁×#387	C₂²	1	16	yes	complemented
33	C₃₁×#386	C₂²	1	8	yes	—
34	C₃₁×#386	C₂²	1	8	yes	—
35	C₃₁×#385	C₂²	1	8	yes	—
36	C₃₁×#385	C₂²	1	8	yes	—
37	C₃₁×#381	C₂²	1	8	yes	—
38	C₃₁×#381	C₂²	1	2	yes	—
39	C₃₁×#380	C₂²	1	4	yes	—
40	C₃₁×#379	C₂²	1	2	yes	—
41	C₄×C₄⋊C₄×C₃₁	C₂²	1	8	yes	complemented
42	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
43	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
44	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×#41×C₃₁	—	2	4	no	—
46	C₂×#41×C₃₁	—	2	8	no	—
47	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
49	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
50	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₂³×C₄×C₃₁	—	2	8	no	—
52	C₂³×C₄×C₃₁	—	2	8	no	—
53	C₂³×C₄×C₃₁	—	2	16	no	—
54	C₂³×C₄×C₃₁	D₄	1	8	yes	—
55	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
56	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
57	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
58	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
59	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
61	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
62	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₃₁×#123	—	2	8	no	—
64	C₃₁×#123	—	2	32	no	—
65	C₃₁×C₂²⋊Q₈	—	2	8	no	—
66	C₃₁×#121	—	2	8	no	—
67	C₃₁×#120	—	2	8	no	—
68	C₄×D₄×C₃₁	—	2	16	no	—
69	C₃₁×#117	—	2	4	no	—
70	C₃₁×#117	—	2	8	no	—
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	16	yes	—
75	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
76	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
77	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
78	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
79	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
80	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
81	C₂×C₄²×C₃₁	—	2	16	no	—
82	C₂×C₄²×C₃₁	—	2	16	no	—
83	C₂×C₄²×C₃₁	—	2	8	no	—
84	C₂×C₄²×C₃₁	—	2	8	no	—
85	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
86	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
87	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
88	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
89	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
90	C₃₁×#95	C₂³	1	8	yes	—
91	C₃₁×#95	C₂³	1	16	yes	—
92	C₃₁×#95	C₂³	1	16	yes	—
93	C₂⁴×C₃₁	—	2	4	no	—
94	C₂⁴×C₃₁	—	2	8	no	—
95	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
96	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
97	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
98	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
99	#41×C₃₁	—	4	16	no	—
100	C₂×Q₈×C₃₁	—	2	2	no	—
101	C₂×Q₈×C₃₁	—	2	4	no	—
102	C₂×D₄×C₃₁	—	2	8	no	—
103	C₂×D₄×C₃₁	—	2	4	no	—
104	C₂×D₄×C₃₁	—	2	16	no	—
105	C₂×D₄×C₃₁	—	2	8	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	4	no	—
108	C₂²×C₄×C₃₁	—	2	64	no	—
109	C₂²×C₄×C₃₁	—	2	8	no	—
110	C₂²×C₄×C₃₁	—	2	8	no	—
111	C₂²×C₄×C₃₁	—	2	8	no	—
112	C₂²×C₄×C₃₁	—	2	4	no	—
113	C₂²×C₄×C₃₁	—	2	8	no	—
114	C₂²×C₄×C₃₁	—	2	32	no	—
115	C₂²×C₄×C₃₁	—	2	16	no	—
116	C₂²×C₄×C₃₁	—	2	16	no	—
117	C₂²×C₄×C₃₁	—	2	8	no	—
118	C₂²×C₄×C₃₁	—	2	8	no	—
119	C₂²×C₄×C₃₁	—	2	4	no	—
120	C₂²×C₄×C₃₁	—	2	8	no	—
121	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
122	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
123	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
124	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
125	C₂²×C₄×C₃₁	#41	1	8	yes	—
126	C₂²×C₄×C₃₁	#41	1	16	yes	complemented
127	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
128	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
129	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
131	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
132	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
134	C₄⋊C₄×C₃₁	—	2	4	no	—
135	C₄⋊C₄×C₃₁	—	2	8	no	—
136	C₄⋊C₄×C₃₁	—	2	8	no	—
137	C₄⋊C₄×C₃₁	—	2	32	no	—
138	C₂²⋊C₄×C₃₁	—	2	4	no	—
139	C₂²⋊C₄×C₃₁	—	2	8	no	—
140	C₂²⋊C₄×C₃₁	—	2	8	no	—
141	C₂²⋊C₄×C₃₁	—	2	16	no	—
142	C₂²⋊C₄×C₃₁	—	2	8	no	—
143	C₂²⋊C₄×C₃₁	—	2	32	no	—
144	C₄²×C₃₁	—	2	16	no	—
145	C₄²×C₃₁	—	2	8	no	—
146	C₄²×C₃₁	—	2	8	no	—
147	C₄²×C₃₁	—	2	4	no	—
148	C₄²×C₃₁	—	2	8	no	—
149	#36626	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂³×C₃₁	—	2	32	no	—
151	C₂³×C₃₁	—	2	8	no	—
152	C₂³×C₃₁	—	2	16	no	—
153	C₂³×C₃₁	—	2	4	no	—
154	C₂³×C₃₁	—	2	4	no	—
155	C₂³×C₃₁	—	2	4	no	—
156	C₂³×C₃₁	—	2	2	no	—
157	C₂³×C₃₁	—	2	8	no	—
158	C₂³×C₃₁	—	2	4	no	—
159	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
160	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
161	C₂³×C₃₁	C₂×#41	1	4	yes	—
162	C₂³×C₃₁	C₂×#41	1	8	yes	—
163	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
164	Q₈×C₃₁	—	4	4	no	—
165	D₄×C₃₁	—	4	8	no	—
166	D₄×C₃₁	—	4	16	no	—
167	D₄×C₃₁	—	4	4	no	—
168	C₂×C₄×C₃₁	—	2	32	no	—
169	C₂×C₄×C₃₁	—	2	32	no	—
170	C₂×C₄×C₃₁	—	2	8	no	—
171	C₂×C₄×C₃₁	—	2	32	no	—
172	C₂×C₄×C₃₁	—	2	8	no	—
173	C₂×C₄×C₃₁	—	2	4	no	—
174	C₂×C₄×C₃₁	—	2	8	no	—
175	C₂×C₄×C₃₁	—	2	16	no	—
176	C₂×C₄×C₃₁	—	2	16	no	—
177	C₂×C₄×C₃₁	—	2	4	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	8	no	—
180	C₂×C₄×C₃₁	—	2	16	no	—
181	C₂×C₄×C₃₁	—	2	8	no	—
182	C₂×C₄×C₃₁	—	2	4	no	—
183	C₂×C₄×C₃₁	—	4	8	no	—
184	C₂×C₄×C₃₁	#123	1	16	yes	complemented
185	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
186	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
187	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
188	C₂×C₄×C₃₁	C₂×#41	1	16	yes	—
189	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
190	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
191	C₂×#525	C₂×C₃₁	1	1	yes	characteristic
192	C₂³×C₄²	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#2855	C₂×C₃₁	1	4	yes	complemented
194	#2854	C₂×C₃₁	1	4	yes	complemented
195	#2853	C₂×C₃₁	1	4	yes	complemented
196	#2849	C₂×C₃₁	1	4	yes	complemented
197	C₄×#123	C₂×C₃₁	1	8	yes	complemented
198	#2297	C₂×C₃₁	1	2	yes	complemented
199	#2296	C₂×C₃₁	1	2	yes	complemented
200	#2280	C₂×C₃₁	1	1	yes	characteristic, complemented
201	C₂²×C₃₁	—	2	32	no	—
202	C₂²×C₃₁	—	2	4	no	—
203	C₂²×C₃₁	—	2	4	no	—
204	C₂²×C₃₁	—	2	4	no	—
205	C₂²×C₃₁	—	2	8	no	—
206	C₂²×C₃₁	—	2	16	no	—
207	C₂²×C₃₁	—	4	2	no	—
208	C₂²×C₃₁	—	4	4	no	—
209	C₂²×C₃₁	—	4	8	no	—
210	C₂²×C₃₁	C₂×#123	1	4	yes	—
211	C₂²×C₃₁	#525	1	8	yes	—
212	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
213	C₂²×C₃₁	C₂²×#41	1	2	yes	—
214	C₂²×C₃₁	C₂²×#41	1	4	yes	—
215	C₄×C₃₁	—	2	4	no	—
216	C₄×C₃₁	—	2	16	no	—
217	C₄×C₃₁	—	4	2	no	—
218	C₄×C₃₁	—	4	4	no	—
219	C₄×C₃₁	—	4	8	no	—
220	C₄×C₃₁	C₂×#123	1	4	yes	—
221	C₄×C₃₁	C₂×#123	1	4	yes	—
222	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
223	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
224	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
225	#525	—	2	8	no	—
226	C₂×#123	C₂²×C₃₁	1	8	yes	—
227	C₂×#123	C₂²×C₃₁	1	2	yes	—
228	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
229	C₂×#121	C₂²×C₃₁	1	2	yes	—
230	C₂×#120	C₂²×C₃₁	1	2	yes	—
231	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
232	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×#117	C₂²×C₃₁	1	2	yes	—
234	C₂²×C₄²	—	2	8	no	—
235	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
236	C₂²×C₄²	C₂²×C₃₁	1	4	yes	—
237	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
238	C₂²×C₄²	C₂²×C₃₁	1	4	yes	—
239	#391	C₂²×C₃₁	1	4	yes	complemented
240	#390	C₂²×C₃₁	1	4	yes	complemented
241	#389	C₂²×C₃₁	1	8	yes	complemented
242	#388	C₂²×C₃₁	1	16	yes	complemented
243	#387	C₂²×C₃₁	1	16	yes	complemented
244	#386	C₂²×C₃₁	1	8	yes	—
245	#386	C₂²×C₃₁	1	8	yes	—
246	#385	C₂²×C₃₁	1	8	yes	—
247	#385	C₂²×C₃₁	1	8	yes	—
248	#381	C₂²×C₃₁	1	8	yes	—
249	#381	C₂²×C₃₁	1	2	yes	—
250	#380	C₂²×C₃₁	1	4	yes	—
251	#379	C₂²×C₃₁	1	2	yes	—
252	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	complemented
253	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
254	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
255	C₂×C₃₁	—	2	4	no	—
256	C₂×C₃₁	—	2	8	no	—
257	C₂×C₃₁	—	8	2	no	—
258	C₂×C₃₁	C₂²×#123	1	1	yes	characteristic
259	C₂×C₃₁	C₂×#525	1	2	yes	—
260	C₂×C₃₁	#3444	1	4	yes	—
261	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
262	C₂×#41	—	2	4	no	—
263	C₂×#41	—	2	8	no	—
264	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
265	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
266	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
267	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
268	C₂³×C₄	—	2	8	no	—
269	C₂³×C₄	—	2	8	no	—
270	C₂³×C₄	—	2	16	no	—
271	C₂³×C₄	D₄×C₃₁	1	8	yes	—
272	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
273	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
274	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
275	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
276	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
277	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
278	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
279	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
280	#123	—	2	8	no	—
281	#123	—	2	32	no	—
282	C₂²⋊Q₈	—	2	8	no	—
283	#121	—	2	8	no	—
284	#120	—	2	8	no	—
285	C₄×D₄	—	2	16	no	—
286	#117	—	2	4	no	—
287	#117	—	2	8	no	—
288	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
289	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
290	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
291	C₂×C₄⋊C₄	C₂³×C₃₁	1	16	yes	—
292	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
293	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
294	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
295	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
296	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
297	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
298	C₂×C₄²	—	2	16	no	—
299	C₂×C₄²	—	2	16	no	—
300	C₂×C₄²	—	2	8	no	—
301	C₂×C₄²	—	2	8	no	—
302	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
303	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
304	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
305	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
306	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
307	#95	C₂³×C₃₁	1	8	yes	—
308	#95	C₂³×C₃₁	1	16	yes	—
309	#95	C₂³×C₃₁	1	16	yes	—
310	C₃₁	#36626	1	1	yes	characteristic, direct product factor, Sylow
311	C₂⁴	—	2	4	no	—
312	C₂⁴	—	2	8	no	—
313	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
314	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
315	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
316	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
317	#41	—	4	16	no	—
318	C₂×Q₈	—	2	2	no	—
319	C₂×Q₈	—	2	4	no	—
320	C₂×D₄	—	2	8	no	—
321	C₂×D₄	—	2	4	no	—
322	C₂×D₄	—	2	16	no	—
323	C₂×D₄	—	2	8	no	—
324	C₂×D₄	—	2	2	no	—
325	C₂×D₄	—	2	4	no	—
326	C₂²×C₄	—	2	64	no	—
327	C₂²×C₄	—	2	8	no	—
328	C₂²×C₄	—	2	8	no	—
329	C₂²×C₄	—	2	8	no	—
330	C₂²×C₄	—	2	4	no	—
331	C₂²×C₄	—	2	8	no	—
332	C₂²×C₄	—	2	32	no	—
333	C₂²×C₄	—	2	16	no	—
334	C₂²×C₄	—	2	16	no	—
335	C₂²×C₄	—	2	8	no	—
336	C₂²×C₄	—	2	8	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	2	8	no	—
339	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
340	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
341	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
342	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
343	C₂²×C₄	#41×C₃₁	1	8	yes	—
344	C₂²×C₄	#41×C₃₁	1	16	yes	complemented
345	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
346	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
347	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
348	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
349	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
350	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
351	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
352	C₄⋊C₄	—	2	4	no	—
353	C₄⋊C₄	—	2	8	no	—
354	C₄⋊C₄	—	2	8	no	—
355	C₄⋊C₄	—	2	32	no	—
356	C₂²⋊C₄	—	2	4	no	—
357	C₂²⋊C₄	—	2	8	no	—
358	C₂²⋊C₄	—	2	8	no	—
359	C₂²⋊C₄	—	2	16	no	—
360	C₂²⋊C₄	—	2	8	no	—
361	C₂²⋊C₄	—	2	32	no	—
362	C₄²	—	2	16	no	—
363	C₄²	—	2	8	no	—
364	C₄²	—	2	8	no	—
365	C₄²	—	2	4	no	—
366	C₄²	—	2	8	no	—
367	C₂³	—	2	32	no	—
368	C₂³	—	2	8	no	—
369	C₂³	—	2	16	no	—
370	C₂³	—	2	4	no	—
371	C₂³	—	2	4	no	—
372	C₂³	—	2	4	no	—
373	C₂³	—	2	2	no	—
374	C₂³	—	2	8	no	—
375	C₂³	—	2	4	no	—
376	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
377	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
378	C₂³	C₂×#41×C₃₁	1	4	yes	—
379	C₂³	C₂×#41×C₃₁	1	8	yes	—
380	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
381	Q₈	—	4	4	no	—
382	D₄	—	4	8	no	—
383	D₄	—	4	16	no	—
384	D₄	—	4	4	no	—
385	C₂×C₄	—	2	32	no	—
386	C₂×C₄	—	2	32	no	—
387	C₂×C₄	—	2	8	no	—
388	C₂×C₄	—	2	32	no	—
389	C₂×C₄	—	2	8	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	8	no	—
392	C₂×C₄	—	2	16	no	—
393	C₂×C₄	—	2	16	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	8	no	—
397	C₂×C₄	—	2	16	no	—
398	C₂×C₄	—	2	8	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	4	8	no	—
401	C₂×C₄	C₃₁×#123	1	16	yes	complemented
402	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
403	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
404	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
405	C₂×C₄	C₂×#41×C₃₁	1	16	yes	—
406	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
407	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
408	C₂²	—	2	32	no	—
409	C₂²	—	2	4	no	—
410	C₂²	—	2	4	no	—
411	C₂²	—	2	4	no	—
412	C₂²	—	2	8	no	—
413	C₂²	—	2	16	no	—
414	C₂²	—	4	2	no	—
415	C₂²	—	4	4	no	—
416	C₂²	—	4	8	no	—
417	C₂²	C₂×C₃₁×#123	1	4	yes	—
418	C₂²	C₃₁×#525	1	8	yes	—
419	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
420	C₂²	C₂²×#41×C₃₁	1	2	yes	—
421	C₂²	C₂²×#41×C₃₁	1	4	yes	—
422	C₄	—	2	4	no	—
423	C₄	—	2	16	no	—
424	C₄	—	4	2	no	—
425	C₄	—	4	4	no	—
426	C₄	—	4	8	no	—
427	C₄	C₂×C₃₁×#123	1	4	yes	—
428	C₄	C₂×C₃₁×#123	1	4	yes	—
429	C₂	—	2	4	no	—
430	C₂	—	2	8	no	—
431	C₂	—	8	2	no	—
432	C₂	C₂²×C₃₁×#123	1	1	yes	characteristic
433	C₂	C₂×C₃₁×#525	1	2	yes	—
434	C₂	C₃₁×#3444	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂³×C₃₁	C₂⁵
prime radical	C₂×C₃₁×#525	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₄²×C₃₁	C₂	semidirect
3. C₃₁×#2855	C₂	semidirect
4. C₃₁×#2854	C₂	semidirect
5. C₃₁×#2853	C₂	semidirect
6. C₃₁×#2849	C₂	semidirect
7. C₄×C₃₁×#123	C₂	semidirect
8. C₃₁×#2297	C₂	semidirect
9. C₃₁×#2296	C₂	semidirect
10. C₃₁×#2280	C₂	semidirect
28. C₃₁×#391	C₂²	semidirect
29. C₃₁×#390	C₂²	semidirect
30. C₃₁×#389	C₂²	semidirect
31. C₃₁×#388	C₂²	semidirect
32. C₃₁×#387	C₂²	semidirect
41. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
43. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
126. C₂²×C₄×C₃₁	#41	semidirect
149. #36626	C₃₁	direct (N × H)
184. C₂×C₄×C₃₁	#123	semidirect
192. C₂³×C₄²	C₂×C₃₁	semidirect
193. #2855	C₂×C₃₁	semidirect
194. #2854	C₂×C₃₁	semidirect
195. #2853	C₂×C₃₁	semidirect
196. #2849	C₂×C₃₁	semidirect
197. C₄×#123	C₂×C₃₁	semidirect
198. #2297	C₂×C₃₁	semidirect
199. #2296	C₂×C₃₁	semidirect
200. #2280	C₂×C₃₁	semidirect
239. #391	C₂²×C₃₁	semidirect
240. #390	C₂²×C₃₁	semidirect
241. #389	C₂²×C₃₁	semidirect
242. #388	C₂²×C₃₁	semidirect
243. #387	C₂²×C₃₁	semidirect
252. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
254. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
310. C₃₁	#36626	direct (N × H)
344. C₂²×C₄	#41×C₃₁	semidirect
401. C₂×C₄	C₃₁×#123	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#525	2
C₂³×C₄²×C₃₁	2
C₃₁×#2855	2
C₃₁×#2854	2
C₃₁×#2853	2
C₃₁×#2849	2
C₄×C₃₁×#123	2
C₃₁×#2297	2
C₃₁×#2296	2
C₃₁×#2280	2
#36626	31

Subgroups — C31×#36626 #21493385

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#36626 #21493385