Subgroups — C₃₁×#35150 #21491909

Order 7936.

3150 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#537	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#525	C₂	1	2	yes	maximal, complemented
3	C₂²×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2854	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2617	C₂	1	4	yes	maximal, complemented
6	C₃₁×#2415	C₂	1	8	yes	maximal, complemented
7	C₃₁×#2414	C₂	1	8	yes	maximal, complemented
8	C₃₁×#2316	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2311	C₂	1	2	yes	maximal, complemented
10	C₂²×#41×C₃₁	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#537	—	2	8	no	—
13	C₃₁×#525	—	2	16	no	—
14	C₂×C₃₁×#123	C₂²	1	4	yes	—
15	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
16	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
17	C₂×C₃₁×#121	C₂²	1	4	yes	—
18	C₂×C₃₁×#121	C₂²	1	4	yes	—
19	C₂×C₃₁×#121	C₂²	1	4	yes	complemented
20	C₂×C₃₁×#120	C₂²	1	4	yes	—
21	C₂×C₄×D₄×C₃₁	C₂²	1	8	yes	—
22	C₂×C₃₁×#117	—	2	8	no	—
23	C₂×C₃₁×#117	C₂²	1	2	yes	—
24	C₂×C₃₁×#117	C₂²	1	8	yes	—
25	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#397	C₂²	1	8	yes	—
30	C₃₁×#394	C₂²	1	16	yes	—
31	C₃₁×#392	C₂²	1	8	yes	—
32	C₃₁×#391	C₂²	1	4	yes	complemented
33	C₃₁×#390	C₂²	1	4	yes	complemented
34	C₃₁×#390	C₂²	1	8	yes	complemented
35	C₃₁×#389	C₂²	1	8	yes	complemented
36	C₃₁×#389	C₂²	1	8	yes	complemented
37	C₃₁×#388	C₂²	1	16	yes	complemented
38	C₃₁×#386	C₂²	1	8	yes	—
39	C₃₁×#380	C₂²	1	4	yes	—
40	C₃₁×#380	C₂²	1	8	yes	complemented
41	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×#41×C₃₁	—	2	8	no	—
43	C₂×#41×C₃₁	—	2	8	no	—
44	C₂×#41×C₃₁	—	2	8	no	—
45	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
46	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
47	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
48	C₂²×D₄×C₃₁	C₂³	1	8	yes	—
49	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
50	C₂³×C₄×C₃₁	—	2	4	no	—
51	C₂³×C₄×C₃₁	—	2	4	no	—
52	C₂³×C₄×C₃₁	D₄	1	8	yes	complemented
53	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
54	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
56	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
57	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
58	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×#123	—	2	16	no	—
60	C₃₁×C₂²⋊Q₈	—	2	16	no	—
61	C₃₁×C₂²⋊Q₈	—	2	16	no	—
62	C₃₁×#121	—	2	16	no	—
63	C₃₁×#121	—	2	16	no	—
64	C₃₁×#121	—	2	16	no	—
65	C₃₁×#120	—	2	16	no	—
66	C₄×D₄×C₃₁	—	2	32	no	—
67	C₃₁×#117	—	2	8	no	—
68	C₃₁×#117	—	2	32	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
70	C₂×C₄⋊C₄×C₃₁	D₄	1	8	yes	complemented
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	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	8	yes	—
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
78	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
79	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	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	8	yes	—
83	C₂×C₂²⋊C₄×C₃₁	C₂³	1	16	yes	—
84	C₂×C₄²×C₃₁	—	2	8	no	—
85	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
86	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
87	C₃₁×#95	C₂³	1	8	yes	—
88	C₃₁×#95	C₂³	1	16	yes	—
89	C₂⁴×C₃₁	—	2	2	no	—
90	C₂⁴×C₃₁	—	2	2	no	—
91	C₂⁴×C₃₁	—	4	2	no	—
92	C₂⁴×C₃₁	—	4	2	no	—
93	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
94	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
95	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
96	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
97	#41×C₃₁	—	4	32	no	—
98	C₂×Q₈×C₃₁	—	2	4	no	—
99	C₂×Q₈×C₃₁	—	2	4	no	—
100	C₂×Q₈×C₃₁	—	2	4	no	—
101	C₂×D₄×C₃₁	—	2	16	no	—
102	C₂×D₄×C₃₁	—	2	8	no	—
103	C₂×D₄×C₃₁	—	2	8	no	—
104	C₂×D₄×C₃₁	—	2	8	no	—
105	C₂×D₄×C₃₁	—	2	4	no	—
106	C₂×D₄×C₃₁	—	2	4	no	—
107	C₂×D₄×C₃₁	—	2	8	no	—
108	C₂×D₄×C₃₁	—	2	4	no	—
109	C₂×D₄×C₃₁	—	4	8	no	—
110	C₂×D₄×C₃₁	—	4	8	no	—
111	C₂×D₄×C₃₁	—	4	8	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	2	32	no	—
115	C₂²×C₄×C₃₁	—	2	8	no	—
116	C₂²×C₄×C₃₁	—	2	8	no	—
117	C₂²×C₄×C₃₁	—	2	4	no	—
118	C₂²×C₄×C₃₁	—	2	8	no	—
119	C₂²×C₄×C₃₁	—	2	8	no	—
120	C₂²×C₄×C₃₁	—	2	8	no	—
121	C₂²×C₄×C₃₁	—	4	4	no	—
122	C₂²×C₄×C₃₁	—	4	4	no	—
123	C₂²×C₄×C₃₁	C₂×D₄	1	16	yes	—
124	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	8	yes	—
126	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
127	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
128	C₂²×C₄×C₃₁	#41	1	4	yes	—
129	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
130	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
131	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
132	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
133	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
134	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
135	C₄⋊C₄×C₃₁	—	2	8	no	—
136	C₄⋊C₄×C₃₁	—	2	8	no	—
137	C₄⋊C₄×C₃₁	—	2	16	no	—
138	C₄⋊C₄×C₃₁	—	2	8	no	—
139	C₄⋊C₄×C₃₁	—	4	8	no	—
140	C₄⋊C₄×C₃₁	C₂×D₄	1	16	yes	—
141	C₂²⋊C₄×C₃₁	—	2	8	no	—
142	C₂²⋊C₄×C₃₁	—	2	16	no	—
143	C₂²⋊C₄×C₃₁	—	2	16	no	—
144	C₂²⋊C₄×C₃₁	—	2	16	no	—
145	C₂²⋊C₄×C₃₁	—	2	16	no	—
146	C₂²⋊C₄×C₃₁	—	4	16	no	—
147	C₄²×C₃₁	—	2	16	no	—
148	C₄²×C₃₁	—	2	8	no	—
149	#35150	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂³×C₃₁	—	2	2	no	—
151	C₂³×C₃₁	—	2	2	no	—
152	C₂³×C₃₁	—	2	8	no	—
153	C₂³×C₃₁	—	2	2	no	—
154	C₂³×C₃₁	—	2	2	no	—
155	C₂³×C₃₁	—	2	4	no	—
156	C₂³×C₃₁	—	2	4	no	—
157	C₂³×C₃₁	—	2	4	no	—
158	C₂³×C₃₁	—	2	4	no	—
159	C₂³×C₃₁	—	2	4	no	—
160	C₂³×C₃₁	—	2	8	no	—
161	C₂³×C₃₁	—	2	8	no	—
162	C₂³×C₃₁	—	4	8	no	—
163	C₂³×C₃₁	—	4	4	no	—
164	C₂³×C₃₁	—	4	4	no	—
165	C₂³×C₃₁	—	4	8	no	—
166	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
167	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
168	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
169	C₂³×C₃₁	C₂×#41	1	2	yes	—
170	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
171	Q₈×C₃₁	—	4	8	no	—
172	D₄×C₃₁	—	4	32	no	—
173	D₄×C₃₁	—	4	16	no	—
174	D₄×C₃₁	—	4	8	no	—
175	D₄×C₃₁	—	8	8	no	—
176	C₂×C₄×C₃₁	—	2	8	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	8	no	—
180	C₂×C₄×C₃₁	—	2	32	no	—
181	C₂×C₄×C₃₁	—	2	4	no	—
182	C₂×C₄×C₃₁	—	2	8	no	—
183	C₂×C₄×C₃₁	—	2	4	no	—
184	C₂×C₄×C₃₁	—	2	4	no	—
185	C₂×C₄×C₃₁	—	2	4	no	—
186	C₂×C₄×C₃₁	—	2	8	no	—
187	C₂×C₄×C₃₁	—	2	8	no	—
188	C₂×C₄×C₃₁	—	2	8	no	—
189	C₂×C₄×C₃₁	—	4	8	no	—
190	C₂×C₄×C₃₁	—	4	8	no	—
191	C₂×C₄×C₃₁	—	4	16	no	—
192	C₂×C₄×C₃₁	#121	1	16	yes	—
193	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
194	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
195	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
196	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
197	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
198	C₂×#537	C₂×C₃₁	1	1	yes	characteristic, complemented
199	C₂×#525	C₂×C₃₁	1	2	yes	complemented
200	C₂²×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#2854	C₂×C₃₁	1	4	yes	complemented
202	#2617	C₂×C₃₁	1	4	yes	complemented
203	#2415	C₂×C₃₁	1	8	yes	complemented
204	#2414	C₂×C₃₁	1	8	yes	complemented
205	#2316	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#2311	C₂×C₃₁	1	2	yes	complemented
207	C₂²×C₃₁	—	2	4	no	—
208	C₂²×C₃₁	—	2	2	no	—
209	C₂²×C₃₁	—	2	2	no	—
210	C₂²×C₃₁	—	2	4	no	—
211	C₂²×C₃₁	—	2	4	no	—
212	C₂²×C₃₁	—	2	8	no	—
213	C₂²×C₃₁	—	4	16	no	—
214	C₂²×C₃₁	—	4	2	no	—
215	C₂²×C₃₁	—	4	2	no	—
216	C₂²×C₃₁	—	4	4	no	—
217	C₂²×C₃₁	—	4	4	no	—
218	C₂²×C₃₁	—	4	8	no	—
219	C₂²×C₃₁	—	4	4	no	—
220	C₂²×C₃₁	—	4	4	no	—
221	C₂²×C₃₁	—	4	8	no	—
222	C₂²×C₃₁	C₂×#121	1	4	yes	—
223	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
224	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
225	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
226	C₂²×C₃₁	#585	1	2	yes	—
227	C₂²×C₃₁	#585	1	2	yes	—
228	C₄×C₃₁	—	2	4	no	—
229	C₄×C₃₁	—	2	8	no	—
230	C₄×C₃₁	—	4	4	no	—
231	C₄×C₃₁	—	4	8	no	—
232	C₄×C₃₁	—	4	4	no	—
233	C₄×C₃₁	C₂×#121	1	8	yes	—
234	C₂²×#41	C₂²×C₃₁	1	2	yes	—
235	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
236	#537	—	2	8	no	—
237	#525	—	2	16	no	—
238	C₂×#123	C₂²×C₃₁	1	4	yes	—
239	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
240	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
241	C₂×#121	C₂²×C₃₁	1	4	yes	—
242	C₂×#121	C₂²×C₃₁	1	4	yes	—
243	C₂×#121	C₂²×C₃₁	1	4	yes	complemented
244	C₂×#120	C₂²×C₃₁	1	4	yes	—
245	C₂×C₄×D₄	C₂²×C₃₁	1	8	yes	—
246	C₂×#117	—	2	8	no	—
247	C₂×#117	C₂²×C₃₁	1	2	yes	—
248	C₂×#117	C₂²×C₃₁	1	8	yes	—
249	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
250	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
251	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
252	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
253	#397	C₂²×C₃₁	1	8	yes	—
254	#394	C₂²×C₃₁	1	16	yes	—
255	#392	C₂²×C₃₁	1	8	yes	—
256	#391	C₂²×C₃₁	1	4	yes	complemented
257	#390	C₂²×C₃₁	1	4	yes	complemented
258	#390	C₂²×C₃₁	1	8	yes	complemented
259	#389	C₂²×C₃₁	1	8	yes	complemented
260	#389	C₂²×C₃₁	1	8	yes	complemented
261	#388	C₂²×C₃₁	1	16	yes	complemented
262	#386	C₂²×C₃₁	1	8	yes	—
263	#380	C₂²×C₃₁	1	4	yes	—
264	#380	C₂²×C₃₁	1	8	yes	complemented
265	C₂×C₃₁	—	2	4	no	—
266	C₂×C₃₁	—	4	4	no	—
267	C₂×C₃₁	—	8	4	no	—
268	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
269	C₂×C₃₁	#3449	1	2	yes	—
270	C₂×C₃₁	#3485	1	1	yes	characteristic
271	C₂×C₃₁	#3487	1	1	yes	characteristic
272	C₂×C₃₁	#3493	1	2	yes	—
273	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
274	C₂×#41	—	2	8	no	—
275	C₂×#41	—	2	8	no	—
276	C₂×#41	—	2	8	no	—
277	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
278	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
279	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
280	C₂²×D₄	C₂³×C₃₁	1	8	yes	—
281	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
282	C₂³×C₄	—	2	4	no	—
283	C₂³×C₄	—	2	4	no	—
284	C₂³×C₄	D₄×C₃₁	1	8	yes	complemented
285	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
286	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
287	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
288	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
289	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
290	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
291	#123	—	2	16	no	—
292	C₂²⋊Q₈	—	2	16	no	—
293	C₂²⋊Q₈	—	2	16	no	—
294	#121	—	2	16	no	—
295	#121	—	2	16	no	—
296	#121	—	2	16	no	—
297	#120	—	2	16	no	—
298	C₄×D₄	—	2	32	no	—
299	#117	—	2	8	no	—
300	#117	—	2	32	no	—
301	C₂×C₄⋊C₄	—	2	4	no	—
302	C₂×C₄⋊C₄	D₄×C₃₁	1	8	yes	complemented
303	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
304	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
305	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
306	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
307	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
308	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
309	C₂×C₂²⋊C₄	—	2	8	no	—
310	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
311	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
312	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
313	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
314	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
315	C₂×C₂²⋊C₄	C₂³×C₃₁	1	16	yes	—
316	C₂×C₄²	—	2	8	no	—
317	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
318	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
319	#95	C₂³×C₃₁	1	8	yes	—
320	#95	C₂³×C₃₁	1	16	yes	—
321	C₃₁	#35150	1	1	yes	characteristic, direct product factor, Sylow
322	C₂⁴	—	2	2	no	—
323	C₂⁴	—	2	2	no	—
324	C₂⁴	—	4	2	no	—
325	C₂⁴	—	4	2	no	—
326	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
327	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
328	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
329	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
330	#41	—	4	32	no	—
331	C₂×Q₈	—	2	4	no	—
332	C₂×Q₈	—	2	4	no	—
333	C₂×Q₈	—	2	4	no	—
334	C₂×D₄	—	2	16	no	—
335	C₂×D₄	—	2	8	no	—
336	C₂×D₄	—	2	8	no	—
337	C₂×D₄	—	2	8	no	—
338	C₂×D₄	—	2	4	no	—
339	C₂×D₄	—	2	4	no	—
340	C₂×D₄	—	2	8	no	—
341	C₂×D₄	—	2	4	no	—
342	C₂×D₄	—	4	8	no	—
343	C₂×D₄	—	4	8	no	—
344	C₂×D₄	—	4	8	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	32	no	—
348	C₂²×C₄	—	2	8	no	—
349	C₂²×C₄	—	2	8	no	—
350	C₂²×C₄	—	2	4	no	—
351	C₂²×C₄	—	2	8	no	—
352	C₂²×C₄	—	2	8	no	—
353	C₂²×C₄	—	2	8	no	—
354	C₂²×C₄	—	4	4	no	—
355	C₂²×C₄	—	4	4	no	—
356	C₂²×C₄	C₂×D₄×C₃₁	1	16	yes	—
357	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
358	C₂²×C₄	C₂×D₄×C₃₁	1	8	yes	—
359	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
360	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
361	C₂²×C₄	#41×C₃₁	1	4	yes	—
362	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
363	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
364	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
365	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
366	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
367	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
368	C₄⋊C₄	—	2	8	no	—
369	C₄⋊C₄	—	2	8	no	—
370	C₄⋊C₄	—	2	16	no	—
371	C₄⋊C₄	—	2	8	no	—
372	C₄⋊C₄	—	4	8	no	—
373	C₄⋊C₄	C₂×D₄×C₃₁	1	16	yes	—
374	C₂²⋊C₄	—	2	8	no	—
375	C₂²⋊C₄	—	2	16	no	—
376	C₂²⋊C₄	—	2	16	no	—
377	C₂²⋊C₄	—	2	16	no	—
378	C₂²⋊C₄	—	2	16	no	—
379	C₂²⋊C₄	—	4	16	no	—
380	C₄²	—	2	16	no	—
381	C₄²	—	2	8	no	—
382	C₂³	—	2	2	no	—
383	C₂³	—	2	2	no	—
384	C₂³	—	2	8	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	2	2	no	—
387	C₂³	—	2	4	no	—
388	C₂³	—	2	4	no	—
389	C₂³	—	2	4	no	—
390	C₂³	—	2	4	no	—
391	C₂³	—	2	4	no	—
392	C₂³	—	2	8	no	—
393	C₂³	—	2	8	no	—
394	C₂³	—	4	8	no	—
395	C₂³	—	4	4	no	—
396	C₂³	—	4	4	no	—
397	C₂³	—	4	8	no	—
398	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
399	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
400	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
401	C₂³	C₂×#41×C₃₁	1	2	yes	—
402	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
403	Q₈	—	4	8	no	—
404	D₄	—	4	32	no	—
405	D₄	—	4	16	no	—
406	D₄	—	4	8	no	—
407	D₄	—	8	8	no	—
408	C₂×C₄	—	2	8	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	8	no	—
412	C₂×C₄	—	2	32	no	—
413	C₂×C₄	—	2	4	no	—
414	C₂×C₄	—	2	8	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	2	8	no	—
419	C₂×C₄	—	2	8	no	—
420	C₂×C₄	—	2	8	no	—
421	C₂×C₄	—	4	8	no	—
422	C₂×C₄	—	4	8	no	—
423	C₂×C₄	—	4	16	no	—
424	C₂×C₄	C₃₁×#121	1	16	yes	—
425	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
426	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
427	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
428	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
429	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
430	C₂²	—	2	4	no	—
431	C₂²	—	2	2	no	—
432	C₂²	—	2	2	no	—
433	C₂²	—	2	4	no	—
434	C₂²	—	2	4	no	—
435	C₂²	—	2	8	no	—
436	C₂²	—	4	16	no	—
437	C₂²	—	4	2	no	—
438	C₂²	—	4	2	no	—
439	C₂²	—	4	4	no	—
440	C₂²	—	4	4	no	—
441	C₂²	—	4	8	no	—
442	C₂²	—	4	4	no	—
443	C₂²	—	4	4	no	—
444	C₂²	—	4	8	no	—
445	C₂²	C₂×C₃₁×#121	1	4	yes	—
446	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
447	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
448	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
449	C₂²	C₃₁×#585	1	2	yes	—
450	C₂²	C₃₁×#585	1	2	yes	—
451	C₄	—	2	4	no	—
452	C₄	—	2	8	no	—
453	C₄	—	4	4	no	—
454	C₄	—	4	8	no	—
455	C₄	—	4	4	no	—
456	C₄	C₂×C₃₁×#121	1	8	yes	—
457	C₂	—	2	4	no	—
458	C₂	—	4	4	no	—
459	C₂	—	8	4	no	—
460	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic
461	C₂	C₃₁×#3449	1	2	yes	—
462	C₂	C₃₁×#3485	1	1	yes	characteristic
463	C₂	C₃₁×#3487	1	1	yes	characteristic
464	C₂	C₃₁×#3493	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#537	C₂	semidirect
2. C₂×C₃₁×#525	C₂	semidirect
3. C₂²×C₃₁×#117	C₂	semidirect
4. C₃₁×#2854	C₂	semidirect
5. C₃₁×#2617	C₂	semidirect
6. C₃₁×#2415	C₂	semidirect
7. C₃₁×#2414	C₂	semidirect
8. C₃₁×#2316	C₂	semidirect
9. C₃₁×#2311	C₂	semidirect
16. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
19. C₂×C₃₁×#121	C₂²	semidirect
32. C₃₁×#391	C₂²	semidirect
33. C₃₁×#390	C₂²	semidirect
34. C₃₁×#390	C₂²	semidirect
35. C₃₁×#389	C₂²	semidirect
36. C₃₁×#389	C₂²	semidirect
37. C₃₁×#388	C₂²	semidirect
40. C₃₁×#380	C₂²	semidirect
52. C₂³×C₄×C₃₁	D₄	semidirect
70. C₂×C₄⋊C₄×C₃₁	D₄	semidirect
149. #35150	C₃₁	direct (N × H)
198. C₂×#537	C₂×C₃₁	semidirect
199. C₂×#525	C₂×C₃₁	semidirect
200. C₂²×#117	C₂×C₃₁	semidirect
201. #2854	C₂×C₃₁	semidirect
202. #2617	C₂×C₃₁	semidirect
203. #2415	C₂×C₃₁	semidirect
204. #2414	C₂×C₃₁	semidirect
205. #2316	C₂×C₃₁	semidirect
206. #2311	C₂×C₃₁	semidirect
240. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
243. C₂×#121	C₂²×C₃₁	semidirect
256. #391	C₂²×C₃₁	semidirect
257. #390	C₂²×C₃₁	semidirect
258. #390	C₂²×C₃₁	semidirect
259. #389	C₂²×C₃₁	semidirect
260. #389	C₂²×C₃₁	semidirect
261. #388	C₂²×C₃₁	semidirect
264. #380	C₂²×C₃₁	semidirect
284. C₂³×C₄	D₄×C₃₁	semidirect
302. C₂×C₄⋊C₄	D₄×C₃₁	semidirect
321. C₃₁	#35150	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×#537	2
C₂×C₃₁×#525	2
C₂²×C₃₁×#117	2
C₃₁×#2854	2
C₃₁×#2617	2
C₃₁×#2415	2
C₃₁×#2414	2
C₃₁×#2316	2
C₃₁×#2311	2
#35150	31

Subgroups — C31×#35150 #21491909

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#35150 #21491909