Subgroups — C₃₁×#34635 #21491394

Order 7936.

2514 conjugacy classes of subgroups.

The subgroup lattice has 420 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	C₂²×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2625	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2624	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2623	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2622	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2395	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	1	yes	characteristic
13	C₃₁×#525	—	2	8	no	—
14	C₂×C₃₁×#123	C₂²	1	2	yes	—
15	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
16	C₂×C₃₁×#121	C₂²	1	2	yes	—
17	C₂×C₃₁×#120	C₂²	1	2	yes	—
18	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
19	C₂×C₃₁×#117	—	2	8	no	—
20	C₂×C₃₁×#117	C₂²	1	4	yes	—
21	C₂×C₃₁×#117	C₂²	1	4	yes	—
22	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#117	C₂²	1	2	yes	—
24	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
27	C₃₁×#400	C₂²	1	8	yes	complemented
28	C₃₁×#399	C₂²	1	8	yes	complemented
29	C₃₁×#398	C₂²	1	8	yes	complemented
30	C₃₁×#397	C₂²	1	16	yes	—
31	C₃₁×#397	C₂²	1	8	yes	complemented
32	C₃₁×#394	C₂²	1	16	yes	—
33	C₃₁×#391	C₂²	1	4	yes	complemented
34	C₃₁×#390	C₂²	1	4	yes	complemented
35	C₃₁×#389	C₂²	1	8	yes	complemented
36	C₃₁×#386	C₂²	1	8	yes	—
37	C₃₁×#385	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	2	yes	—
49	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
51	C₂³×C₄×C₃₁	—	2	8	no	—
52	C₂³×C₄×C₃₁	D₄	1	8	yes	—
53	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
54	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
56	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
57	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
58	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×#123	—	2	8	no	—
60	C₃₁×C₂²⋊Q₈	—	2	8	no	—
61	C₃₁×#121	—	2	8	no	—
62	C₃₁×#120	—	2	8	no	—
63	C₄×D₄×C₃₁	—	2	16	no	—
64	C₃₁×#117	—	2	16	no	—
65	C₃₁×#117	—	2	16	no	—
66	C₃₁×#117	—	2	4	no	—
67	C₃₁×#117	—	2	8	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
74	C₂×C₄⋊C₄×C₃₁	C₂³	1	16	yes	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
76	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
77	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
78	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
79	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
80	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
81	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
82	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
83	C₂×C₄²×C₃₁	—	2	8	no	—
84	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
85	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
86	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
87	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
88	C₃₁×#95	C₂³	1	16	yes	—
89	C₃₁×#95	C₂³	1	16	yes	—
90	C₃₁×#95	C₂³	1	8	yes	—
91	C₂⁴×C₃₁	—	2	4	no	—
92	C₂⁴×C₃₁	—	4	4	no	—
93	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
94	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
95	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
96	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
97	#41×C₃₁	—	4	16	no	—
98	C₂×Q₈×C₃₁	—	2	2	no	—
99	C₂×Q₈×C₃₁	—	2	4	no	—
100	C₂×D₄×C₃₁	—	2	8	no	—
101	C₂×D₄×C₃₁	—	2	4	no	—
102	C₂×D₄×C₃₁	—	2	8	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	4	no	—
105	C₂×D₄×C₃₁	—	4	8	no	—
106	C₂²×C₄×C₃₁	—	2	4	no	—
107	C₂²×C₄×C₃₁	—	2	32	no	—
108	C₂²×C₄×C₃₁	—	2	8	no	—
109	C₂²×C₄×C₃₁	—	2	8	no	—
110	C₂²×C₄×C₃₁	—	2	8	no	—
111	C₂²×C₄×C₃₁	—	2	4	no	—
112	C₂²×C₄×C₃₁	—	2	8	no	—
113	C₂²×C₄×C₃₁	—	4	8	no	—
114	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
115	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
116	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
117	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
118	C₂²×C₄×C₃₁	#41	1	8	yes	—
119	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
120	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
121	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
122	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
123	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
124	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
126	C₄⋊C₄×C₃₁	—	2	8	no	—
127	C₄⋊C₄×C₃₁	—	2	8	no	—
128	C₄⋊C₄×C₃₁	—	2	4	no	—
129	C₄⋊C₄×C₃₁	—	2	8	no	—
130	C₄⋊C₄×C₃₁	—	2	8	no	—
131	C₄⋊C₄×C₃₁	—	4	16	no	—
132	C₂²⋊C₄×C₃₁	—	2	16	no	—
133	C₂²⋊C₄×C₃₁	—	2	4	no	—
134	C₂²⋊C₄×C₃₁	—	2	8	no	—
135	C₂²⋊C₄×C₃₁	—	2	8	no	—
136	C₂²⋊C₄×C₃₁	—	2	8	no	—
137	C₂²⋊C₄×C₃₁	—	4	8	no	—
138	C₂²⋊C₄×C₃₁	—	4	16	no	—
139	C₄²×C₃₁	—	2	8	no	—
140	C₄²×C₃₁	—	2	8	no	—
141	C₄²×C₃₁	—	2	4	no	—
142	C₄²×C₃₁	—	2	8	no	—
143	#34635	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	2	4	no	—
145	C₂³×C₃₁	—	2	8	no	—
146	C₂³×C₃₁	—	2	4	no	—
147	C₂³×C₃₁	—	2	4	no	—
148	C₂³×C₃₁	—	2	4	no	—
149	C₂³×C₃₁	—	2	2	no	—
150	C₂³×C₃₁	—	2	4	no	—
151	C₂³×C₃₁	—	2	8	no	—
152	C₂³×C₃₁	—	4	8	no	—
153	C₂³×C₃₁	—	4	16	no	—
154	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
155	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
156	C₂³×C₃₁	C₂×#41	1	4	yes	—
157	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
158	Q₈×C₃₁	—	4	4	no	—
159	D₄×C₃₁	—	4	16	no	—
160	D₄×C₃₁	—	4	8	no	—
161	D₄×C₃₁	—	4	4	no	—
162	C₂×C₄×C₃₁	—	2	8	no	—
163	C₂×C₄×C₃₁	—	2	4	no	—
164	C₂×C₄×C₃₁	—	2	16	no	—
165	C₂×C₄×C₃₁	—	2	16	no	—
166	C₂×C₄×C₃₁	—	2	8	no	—
167	C₂×C₄×C₃₁	—	2	4	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	8	no	—
170	C₂×C₄×C₃₁	—	2	4	no	—
171	C₂×C₄×C₃₁	—	2	16	no	—
172	C₂×C₄×C₃₁	—	2	8	no	—
173	C₂×C₄×C₃₁	—	2	8	no	—
174	C₂×C₄×C₃₁	—	2	8	no	—
175	C₂×C₄×C₃₁	—	4	16	no	—
176	C₂×C₄×C₃₁	—	4	8	no	—
177	C₂×C₄×C₃₁	#123	1	16	yes	—
178	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
179	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
180	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
181	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
182	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
183	C₂×#525	C₂×C₃₁	1	1	yes	characteristic
184	C₂²×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#2625	C₂×C₃₁	1	4	yes	complemented
186	#2624	C₂×C₃₁	1	4	yes	complemented
187	#2623	C₂×C₃₁	1	4	yes	complemented
188	#2622	C₂×C₃₁	1	4	yes	complemented
189	#2395	C₂×C₃₁	1	8	yes	complemented
190	#2297	C₂×C₃₁	1	2	yes	complemented
191	#2296	C₂×C₃₁	1	2	yes	complemented
192	#2280	C₂×C₃₁	1	1	yes	characteristic, complemented
193	C₂²×C₃₁	—	2	4	no	—
194	C₂²×C₃₁	—	2	8	no	—
195	C₂²×C₃₁	—	2	4	no	—
196	C₂²×C₃₁	—	2	4	no	—
197	C₂²×C₃₁	—	2	4	no	—
198	C₂²×C₃₁	—	4	16	no	—
199	C₂²×C₃₁	—	4	8	no	—
200	C₂²×C₃₁	—	4	2	no	—
201	C₂²×C₃₁	—	4	4	no	—
202	C₂²×C₃₁	—	4	8	no	—
203	C₂²×C₃₁	C₂×#123	1	4	yes	—
204	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
205	C₂²×C₃₁	C₂²×#41	1	2	yes	—
206	C₂²×C₃₁	#585	1	4	yes	—
207	C₄×C₃₁	—	2	4	no	—
208	C₄×C₃₁	—	4	2	no	—
209	C₄×C₃₁	—	4	8	no	—
210	C₄×C₃₁	—	4	4	no	—
211	C₄×C₃₁	—	4	8	no	—
212	C₄×C₃₁	C₂×#123	1	4	yes	—
213	C₄×C₃₁	C₂×#123	1	4	yes	—
214	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
215	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
216	#525	—	2	8	no	—
217	C₂×#123	C₂²×C₃₁	1	2	yes	—
218	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
219	C₂×#121	C₂²×C₃₁	1	2	yes	—
220	C₂×#120	C₂²×C₃₁	1	2	yes	—
221	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
222	C₂×#117	—	2	8	no	—
223	C₂×#117	C₂²×C₃₁	1	4	yes	—
224	C₂×#117	C₂²×C₃₁	1	4	yes	—
225	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
226	C₂×#117	C₂²×C₃₁	1	2	yes	—
227	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
228	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
229	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
230	#400	C₂²×C₃₁	1	8	yes	complemented
231	#399	C₂²×C₃₁	1	8	yes	complemented
232	#398	C₂²×C₃₁	1	8	yes	complemented
233	#397	C₂²×C₃₁	1	16	yes	—
234	#397	C₂²×C₃₁	1	8	yes	complemented
235	#394	C₂²×C₃₁	1	16	yes	—
236	#391	C₂²×C₃₁	1	4	yes	complemented
237	#390	C₂²×C₃₁	1	4	yes	complemented
238	#389	C₂²×C₃₁	1	8	yes	complemented
239	#386	C₂²×C₃₁	1	8	yes	—
240	#385	C₂²×C₃₁	1	8	yes	—
241	#381	C₂²×C₃₁	1	2	yes	—
242	#380	C₂²×C₃₁	1	4	yes	—
243	#379	C₂²×C₃₁	1	2	yes	—
244	C₄×C₄⋊C₄	C₂²×C₃₁	1	8	yes	complemented
245	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
246	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
247	C₂×C₃₁	—	2	4	no	—
248	C₂×C₃₁	—	4	4	no	—
249	C₂×C₃₁	—	8	2	no	—
250	C₂×C₃₁	C₂²×#123	1	1	yes	characteristic
251	C₂×C₃₁	#3449	1	2	yes	—
252	C₂×C₃₁	#3455	1	4	yes	—
253	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
254	C₂×#41	—	2	4	no	—
255	C₂×#41	—	2	8	no	—
256	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
257	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
258	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
259	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
260	C₂³×C₄	—	2	8	no	—
261	C₂³×C₄	D₄×C₃₁	1	8	yes	—
262	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
263	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
264	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
265	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
266	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
267	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
268	#123	—	2	8	no	—
269	C₂²⋊Q₈	—	2	8	no	—
270	#121	—	2	8	no	—
271	#120	—	2	8	no	—
272	C₄×D₄	—	2	16	no	—
273	#117	—	2	16	no	—
274	#117	—	2	16	no	—
275	#117	—	2	4	no	—
276	#117	—	2	8	no	—
277	C₂×C₄⋊C₄	—	2	8	no	—
278	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
279	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
280	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
281	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
282	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
283	C₂×C₄⋊C₄	C₂³×C₃₁	1	16	yes	—
284	C₂×C₂²⋊C₄	—	2	8	no	—
285	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
286	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
287	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
288	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
289	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
290	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
291	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
292	C₂×C₄²	—	2	8	no	—
293	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
294	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
295	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
296	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
297	#95	C₂³×C₃₁	1	8	yes	—
298	#95	C₂³×C₃₁	1	16	yes	—
299	#95	C₂³×C₃₁	1	16	yes	—
300	C₃₁	#34635	1	1	yes	characteristic, direct product factor, Sylow
301	C₂⁴	—	2	4	no	—
302	C₂⁴	—	4	4	no	—
303	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
304	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
305	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
306	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
307	#41	—	4	16	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	2	4	no	—
310	C₂×D₄	—	2	8	no	—
311	C₂×D₄	—	2	4	no	—
312	C₂×D₄	—	2	8	no	—
313	C₂×D₄	—	2	2	no	—
314	C₂×D₄	—	2	4	no	—
315	C₂×D₄	—	4	8	no	—
316	C₂²×C₄	—	2	4	no	—
317	C₂²×C₄	—	2	32	no	—
318	C₂²×C₄	—	2	8	no	—
319	C₂²×C₄	—	2	8	no	—
320	C₂²×C₄	—	2	8	no	—
321	C₂²×C₄	—	2	4	no	—
322	C₂²×C₄	—	2	8	no	—
323	C₂²×C₄	—	4	8	no	—
324	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
325	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
326	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
327	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
328	C₂²×C₄	#41×C₃₁	1	8	yes	—
329	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
330	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
331	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
332	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
333	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
334	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
335	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
336	C₄⋊C₄	—	2	8	no	—
337	C₄⋊C₄	—	2	8	no	—
338	C₄⋊C₄	—	2	4	no	—
339	C₄⋊C₄	—	2	8	no	—
340	C₄⋊C₄	—	2	8	no	—
341	C₄⋊C₄	—	4	16	no	—
342	C₂²⋊C₄	—	2	16	no	—
343	C₂²⋊C₄	—	2	4	no	—
344	C₂²⋊C₄	—	2	8	no	—
345	C₂²⋊C₄	—	2	8	no	—
346	C₂²⋊C₄	—	2	8	no	—
347	C₂²⋊C₄	—	4	8	no	—
348	C₂²⋊C₄	—	4	16	no	—
349	C₄²	—	2	8	no	—
350	C₄²	—	2	8	no	—
351	C₄²	—	2	4	no	—
352	C₄²	—	2	8	no	—
353	C₂³	—	2	4	no	—
354	C₂³	—	2	8	no	—
355	C₂³	—	2	4	no	—
356	C₂³	—	2	4	no	—
357	C₂³	—	2	4	no	—
358	C₂³	—	2	2	no	—
359	C₂³	—	2	4	no	—
360	C₂³	—	2	8	no	—
361	C₂³	—	4	8	no	—
362	C₂³	—	4	16	no	—
363	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
364	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
365	C₂³	C₂×#41×C₃₁	1	4	yes	—
366	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
367	Q₈	—	4	4	no	—
368	D₄	—	4	16	no	—
369	D₄	—	4	8	no	—
370	D₄	—	4	4	no	—
371	C₂×C₄	—	2	8	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	16	no	—
374	C₂×C₄	—	2	16	no	—
375	C₂×C₄	—	2	8	no	—
376	C₂×C₄	—	2	4	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	8	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	16	no	—
381	C₂×C₄	—	2	8	no	—
382	C₂×C₄	—	2	8	no	—
383	C₂×C₄	—	2	8	no	—
384	C₂×C₄	—	4	16	no	—
385	C₂×C₄	—	4	8	no	—
386	C₂×C₄	C₃₁×#123	1	16	yes	—
387	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
388	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
389	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
390	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
391	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
392	C₂²	—	2	4	no	—
393	C₂²	—	2	8	no	—
394	C₂²	—	2	4	no	—
395	C₂²	—	2	4	no	—
396	C₂²	—	2	4	no	—
397	C₂²	—	4	16	no	—
398	C₂²	—	4	8	no	—
399	C₂²	—	4	2	no	—
400	C₂²	—	4	4	no	—
401	C₂²	—	4	8	no	—
402	C₂²	C₂×C₃₁×#123	1	4	yes	—
403	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
404	C₂²	C₂²×#41×C₃₁	1	2	yes	—
405	C₂²	C₃₁×#585	1	4	yes	—
406	C₄	—	2	4	no	—
407	C₄	—	4	2	no	—
408	C₄	—	4	8	no	—
409	C₄	—	4	4	no	—
410	C₄	—	4	8	no	—
411	C₄	C₂×C₃₁×#123	1	4	yes	—
412	C₄	C₂×C₃₁×#123	1	4	yes	—
413	C₂	—	2	4	no	—
414	C₂	—	4	4	no	—
415	C₂	—	8	2	no	—
416	C₂	C₂²×C₃₁×#123	1	1	yes	characteristic
417	C₂	C₃₁×#3449	1	2	yes	—
418	C₂	C₃₁×#3455	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. C₂²×C₃₁×#117	C₂	semidirect
3. C₃₁×#2625	C₂	semidirect
4. C₃₁×#2624	C₂	semidirect
5. C₃₁×#2623	C₂	semidirect
6. C₃₁×#2622	C₂	semidirect
7. C₃₁×#2395	C₂	semidirect
8. C₃₁×#2297	C₂	semidirect
9. C₃₁×#2296	C₂	semidirect
10. C₃₁×#2280	C₂	semidirect
27. C₃₁×#400	C₂²	semidirect
28. C₃₁×#399	C₂²	semidirect
29. C₃₁×#398	C₂²	semidirect
31. C₃₁×#397	C₂²	semidirect
33. C₃₁×#391	C₂²	semidirect
34. C₃₁×#390	C₂²	semidirect
35. C₃₁×#389	C₂²	semidirect
41. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
43. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
143. #34635	C₃₁	direct (N × H)
184. C₂²×#117	C₂×C₃₁	semidirect
185. #2625	C₂×C₃₁	semidirect
186. #2624	C₂×C₃₁	semidirect
187. #2623	C₂×C₃₁	semidirect
188. #2622	C₂×C₃₁	semidirect
189. #2395	C₂×C₃₁	semidirect
190. #2297	C₂×C₃₁	semidirect
191. #2296	C₂×C₃₁	semidirect
192. #2280	C₂×C₃₁	semidirect
230. #400	C₂²×C₃₁	semidirect
231. #399	C₂²×C₃₁	semidirect
232. #398	C₂²×C₃₁	semidirect
234. #397	C₂²×C₃₁	semidirect
236. #391	C₂²×C₃₁	semidirect
237. #390	C₂²×C₃₁	semidirect
238. #389	C₂²×C₃₁	semidirect
244. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
246. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
300. C₃₁	#34635	direct (N × H)

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
C₂²×C₃₁×#117	2
C₃₁×#2625	2
C₃₁×#2624	2
C₃₁×#2623	2
C₃₁×#2622	2
C₃₁×#2395	2
C₃₁×#2297	2
C₃₁×#2296	2
C₃₁×#2280	2
#34635	31

Subgroups — C31×#34635 #21491394

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34635 #21491394