Subgroups — C₃₁×#26462 #21483221

Order 7936.

810 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×#475	C₂	1	2	yes	maximal
2	C₃₁×#2551	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2365	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2066	C₂	1	2	yes	maximal
5	C₃₁×#2024	C₂	1	2	yes	maximal
6	C₃₁×#1981	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#1956	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#1865	C₂	1	2	yes	maximal
11	C₄×C₃₁×#106	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#475	—	2	8	no	—
17	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#106	C₂²	1	2	yes	—
19	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#103	C₂²	1	2	yes	—
23	C₂×C₃₁×#103	C₂²	1	2	yes	—
24	C₂×C₃₁×#103	C₂²	1	2	yes	—
25	C₂×C₃₁×#103	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#395	—	2	2	no	—
28	C₃₁×#391	C₂²	1	2	yes	—
29	C₃₁×#389	—	2	1	no	—
30	C₃₁×#389	—	2	1	no	—
31	C₃₁×#389	C₂²	1	2	yes	—
32	C₃₁×#387	—	2	2	no	—
33	C₃₁×#384	C₂²	1	1	yes	characteristic
34	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	1	yes	characteristic
40	C₃₁×#340	C₂²	1	1	yes	characteristic
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₃₁	—	2	2	no	—
43	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×C₄⋊Q₈	—	2	4	no	—
48	C₃₁×C₄⋊Q₈	—	2	2	no	—
49	C₃₁×C₄⋊Q₈	—	4	2	no	—
50	C₃₁×C₄⋊Q₈	—	4	4	no	—
51	C₃₁×C₄⋊Q₈	Q₈	1	4	yes	complemented
52	C₄×Q₈×C₃₁	—	2	4	no	—
53	C₄×Q₈×C₃₁	—	4	2	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₃₁	—	2	2	no	—
69	C₂×C₄²×C₃₁	—	2	1	no	—
70	C₂×C₄²×C₃₁	—	2	1	no	—
71	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×#106	—	2	4	no	—
74	C₃₁×#106	—	2	2	no	—
75	C₃₁×#106	D₄	1	4	yes	—
76	C₃₁×C₄⋊C₈	—	2	4	no	—
77	C₃₁×C₄⋊C₈	—	2	2	no	—
78	C₃₁×#103	—	2	4	no	—
79	C₃₁×#103	—	2	4	no	—
80	C₃₁×#103	—	2	4	no	—
81	C₃₁×#103	—	4	2	no	—
82	C₄×C₈×C₃₁	—	2	2	no	—
83	C₃₁×#95	—	2	1	no	—
84	C₃₁×#95	—	2	2	no	—
85	C₃₁×#95	—	2	2	no	—
86	C₃₁×#95	—	2	1	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	D₄	1	2	yes	—
89	C₂×Q₈×C₃₁	—	2	2	no	—
90	C₂×Q₈×C₃₁	—	2	2	no	—
91	C₂×Q₈×C₃₁	—	2	2	no	—
92	C₂×Q₈×C₃₁	—	2	2	no	—
93	C₂×Q₈×C₃₁	—	2	2	no	—
94	C₂×Q₈×C₃₁	—	4	4	no	—
95	C₂×Q₈×C₃₁	—	4	4	no	—
96	C₂×Q₈×C₃₁	—	8	2	no	—
97	C₂×Q₈×C₃₁	C₂×Q₈	1	4	yes	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	2	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
109	C₂×C₈×C₃₁	—	2	1	no	—
110	C₂×C₈×C₃₁	—	2	1	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	1	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
118	C₄⋊C₄×C₃₁	—	2	4	no	—
119	C₄⋊C₄×C₃₁	—	2	4	no	—
120	C₄⋊C₄×C₃₁	—	2	2	no	—
121	C₄⋊C₄×C₃₁	—	2	1	no	—
122	C₄⋊C₄×C₃₁	—	2	1	no	—
123	C₄⋊C₄×C₃₁	—	2	1	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	1	no	—
128	C₄⋊C₄×C₃₁	—	4	4	no	—
129	C₄⋊C₄×C₃₁	—	4	4	no	—
130	C₄⋊C₄×C₃₁	—	4	4	no	—
131	C₄⋊C₄×C₃₁	—	4	2	no	—
132	C₄⋊C₄×C₃₁	—	8	1	no	—
133	C₄⋊C₄×C₃₁	SD₁₆	1	4	yes	—
134	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
135	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
136	C₄⋊C₄×C₃₁	#41	1	2	yes	—
137	C₄²×C₃₁	—	2	2	no	—
138	C₄²×C₃₁	—	2	2	no	—
139	C₄²×C₃₁	—	4	4	no	—
140	C₄²×C₃₁	—	4	2	no	—
141	C₄²×C₃₁	—	4	2	no	—
142	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
143	C₄²×C₃₁	#41	1	2	yes	—
144	#26462	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
146	Q₈×C₃₁	—	2	4	no	—
147	Q₈×C₃₁	—	2	4	no	—
148	Q₈×C₃₁	—	4	2	no	—
149	Q₈×C₃₁	—	4	2	no	—
150	Q₈×C₃₁	—	8	8	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	4	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	4	2	no	—
164	C₂×C₄×C₃₁	—	4	1	no	—
165	C₂×C₄×C₃₁	—	4	1	no	—
166	C₂×C₄×C₃₁	—	4	1	no	—
167	C₂×C₄×C₃₁	—	4	1	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	4	1	no	—
176	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
177	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
178	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
179	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
180	C₂×C₄×C₃₁	#137	1	2	yes	—
181	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
183	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
185	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
186	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
187	C₈×C₃₁	—	2	2	no	—
188	C₈×C₃₁	—	4	2	no	—
189	C₈×C₃₁	—	4	1	no	—
190	C₂×#475	C₂×C₃₁	1	2	yes	—
191	#2551	C₂×C₃₁	1	1	yes	characteristic
192	#2365	C₂×C₃₁	1	1	yes	characteristic
193	#2066	C₂×C₃₁	1	2	yes	—
194	#2024	C₂×C₃₁	1	2	yes	—
195	#1981	C₂×C₃₁	1	1	yes	characteristic
196	#1956	C₂×C₃₁	1	1	yes	characteristic
197	#1948	C₂×C₃₁	1	1	yes	characteristic
198	#1939	C₂×C₃₁	1	1	yes	characteristic
199	#1865	C₂×C₃₁	1	2	yes	—
200	C₄×#106	C₂×C₃₁	1	1	yes	characteristic
201	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
202	C₂²×C₃₁	D₄²	1	1	yes	characteristic
203	C₂²×C₃₁	#548	1	1	yes	characteristic
204	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
205	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
206	C₂²×C₃₁	#577	1	1	yes	characteristic
207	C₂²×C₃₁	#578	1	1	yes	characteristic
208	C₄×C₃₁	—	2	2	no	—
209	C₄×C₃₁	—	2	4	no	—
210	C₄×C₃₁	—	4	2	no	—
211	C₄×C₃₁	—	4	4	no	—
212	C₄×C₃₁	—	4	2	no	—
213	C₄×C₃₁	—	4	2	no	—
214	C₄×C₃₁	—	4	2	no	—
215	C₄×C₃₁	—	8	1	no	—
216	C₄×C₃₁	—	8	2	no	—
217	C₄×C₃₁	#475	1	4	yes	complemented
218	C₄×C₃₁	#532	1	1	yes	characteristic
219	C₄×C₃₁	#542	1	1	yes	characteristic
220	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
221	C₄×C₃₁	#554	1	1	yes	characteristic
222	C₂×C₄⋊Q₈	—	2	1	no	—
223	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
224	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
225	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
226	#475	—	2	8	no	—
227	C₂×#106	C₂²×C₃₁	1	2	yes	—
228	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
229	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
231	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
232	C₂×#103	C₂²×C₃₁	1	2	yes	—
233	C₂×#103	C₂²×C₃₁	1	2	yes	—
234	C₂×#103	C₂²×C₃₁	1	2	yes	—
235	C₂×#103	C₂²×C₃₁	1	2	yes	—
236	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
237	#395	—	2	2	no	—
238	#391	C₂²×C₃₁	1	2	yes	—
239	#389	—	2	1	no	—
240	#389	—	2	1	no	—
241	#389	C₂²×C₃₁	1	2	yes	—
242	#387	—	2	2	no	—
243	#384	C₂²×C₃₁	1	1	yes	characteristic
244	C₄×C₄⋊C₄	—	2	1	no	—
245	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
246	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
247	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
248	#340	C₂²×C₃₁	1	2	yes	—
249	#340	C₂²×C₃₁	1	1	yes	characteristic
250	#340	C₂²×C₃₁	1	1	yes	characteristic
251	#336	C₂²×C₃₁	1	2	yes	—
252	C₂×C₃₁	#2452	1	1	yes	characteristic
253	C₂×C₃₁	C₂×#475	1	1	yes	characteristic
254	C₂×C₃₁	#3082	1	1	yes	characteristic
255	C₂×C₃₁	#3276	1	1	yes	characteristic
256	C₂×C₃₁	D₄×SD₁₆	1	1	yes	characteristic
257	C₂×C₃₁	#3359	1	1	yes	characteristic
258	C₂×C₃₁	#3369	1	1	yes	characteristic
259	C₂²×Q₈	—	2	2	no	—
260	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
261	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
263	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
264	C₄⋊Q₈	—	2	4	no	—
265	C₄⋊Q₈	—	2	2	no	—
266	C₄⋊Q₈	—	4	2	no	—
267	C₄⋊Q₈	—	4	4	no	—
268	C₄⋊Q₈	Q₈×C₃₁	1	4	yes	complemented
269	C₄×Q₈	—	2	4	no	—
270	C₄×Q₈	—	4	2	no	—
271	C₂×C₄⋊C₄	—	2	1	no	—
272	C₂×C₄⋊C₄	—	2	1	no	—
273	C₂×C₄⋊C₄	—	2	2	no	—
274	C₂×C₄⋊C₄	—	2	1	no	—
275	C₂×C₄⋊C₄	—	2	2	no	—
276	C₂×C₄⋊C₄	—	2	2	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
280	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
281	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
282	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
283	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
284	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
285	C₂×C₄²	—	2	2	no	—
286	C₂×C₄²	—	2	1	no	—
287	C₂×C₄²	—	2	1	no	—
288	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
289	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
290	#106	—	2	4	no	—
291	#106	—	2	2	no	—
292	#106	D₄×C₃₁	1	4	yes	—
293	C₄⋊C₈	—	2	4	no	—
294	C₄⋊C₈	—	2	2	no	—
295	#103	—	2	4	no	—
296	#103	—	2	4	no	—
297	#103	—	2	4	no	—
298	#103	—	4	2	no	—
299	C₄×C₈	—	2	2	no	—
300	#95	—	2	2	no	—
301	#95	—	2	1	no	—
302	#95	—	2	1	no	—
303	#95	—	2	2	no	—
304	#95	—	2	2	no	—
305	#95	D₄×C₃₁	1	2	yes	—
306	C₃₁	#26462	1	1	yes	characteristic, direct product factor, Sylow
307	C₂×Q₈	—	2	2	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	—	2	2	no	—
310	C₂×Q₈	—	2	2	no	—
311	C₂×Q₈	—	2	2	no	—
312	C₂×Q₈	—	4	4	no	—
313	C₂×Q₈	—	4	4	no	—
314	C₂×Q₈	—	8	2	no	—
315	C₂×Q₈	C₂×Q₈×C₃₁	1	4	yes	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	2	2	no	—
318	C₂²×C₄	—	2	1	no	—
319	C₂²×C₄	—	2	1	no	—
320	C₂²×C₄	—	2	1	no	—
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
323	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
324	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
325	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
326	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
336	C₄⋊C₄	—	2	4	no	—
337	C₄⋊C₄	—	2	4	no	—
338	C₄⋊C₄	—	2	2	no	—
339	C₄⋊C₄	—	2	1	no	—
340	C₄⋊C₄	—	2	1	no	—
341	C₄⋊C₄	—	2	1	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	1	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	4	4	no	—
347	C₄⋊C₄	—	4	4	no	—
348	C₄⋊C₄	—	4	4	no	—
349	C₄⋊C₄	—	4	2	no	—
350	C₄⋊C₄	—	8	1	no	—
351	C₄⋊C₄	SD₁₆×C₃₁	1	4	yes	—
352	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
353	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
354	C₄⋊C₄	#41×C₃₁	1	2	yes	—
355	C₄²	—	2	2	no	—
356	C₄²	—	2	2	no	—
357	C₄²	—	4	4	no	—
358	C₄²	—	4	2	no	—
359	C₄²	—	4	2	no	—
360	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
361	C₄²	#41×C₃₁	1	2	yes	—
362	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
363	Q₈	—	2	4	no	—
364	Q₈	—	2	4	no	—
365	Q₈	—	4	2	no	—
366	Q₈	—	4	2	no	—
367	Q₈	—	8	8	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	1	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
394	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
395	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
396	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
397	C₂×C₄	C₃₁×#137	1	2	yes	—
398	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
399	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
400	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
401	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
402	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
403	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
404	C₈	—	2	2	no	—
405	C₈	—	4	2	no	—
406	C₈	—	4	1	no	—
407	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
408	C₂²	D₄²×C₃₁	1	1	yes	characteristic
409	C₂²	C₃₁×#548	1	1	yes	characteristic
410	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
411	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
412	C₂²	C₃₁×#577	1	1	yes	characteristic
413	C₂²	C₃₁×#578	1	1	yes	characteristic
414	C₄	—	2	2	no	—
415	C₄	—	2	4	no	—
416	C₄	—	4	2	no	—
417	C₄	—	4	4	no	—
418	C₄	—	4	2	no	—
419	C₄	—	4	2	no	—
420	C₄	—	4	2	no	—
421	C₄	—	8	1	no	—
422	C₄	—	8	2	no	—
423	C₄	C₃₁×#475	1	4	yes	complemented
424	C₄	C₃₁×#532	1	1	yes	characteristic
425	C₄	C₃₁×#542	1	1	yes	characteristic
426	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
427	C₄	C₃₁×#554	1	1	yes	characteristic
428	C₂	C₃₁×#2452	1	1	yes	characteristic
429	C₂	C₂×C₃₁×#475	1	1	yes	characteristic
430	C₂	C₃₁×#3082	1	1	yes	characteristic
431	C₂	C₃₁×#3276	1	1	yes	characteristic
432	C₂	D₄×SD₁₆×C₃₁	1	1	yes	characteristic
433	C₂	C₃₁×#3359	1	1	yes	characteristic
434	C₂	C₃₁×#3369	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₄⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
51. C₃₁×C₄⋊Q₈	Q₈	semidirect
144. #26462	C₃₁	direct (N × H)
217. C₄×C₃₁	#475	semidirect
268. C₄⋊Q₈	Q₈×C₃₁	semidirect
306. C₃₁	#26462	direct (N × H)
423. C₄	C₃₁×#475	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#475	2
C₃₁×#2551	2
C₃₁×#2365	2
C₃₁×#2066	2
C₃₁×#2024	2
C₃₁×#1981	2
C₃₁×#1956	2
C₃₁×#1948	2
C₃₁×#1939	2
C₃₁×#1865	2
C₄×C₃₁×#106	2
#26462	31

Subgroups — C31×#26462 #21483221

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#26462 #21483221