Subgroups — C₃₁×#24313 #21481072

Order 7936.

998 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3502	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3077	C₂	1	2	yes	maximal
3	C₄×C₃₁×#124	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2061	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2027	C₂	1	2	yes	maximal
6	C₃₁×#1994	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1888	C₂	1	2	yes	maximal
8	C₃₁×#1857	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1801	C₂	1	1	yes	maximal, characteristic, complemented
10	Q₈²×C₃₁	—	2	2	no	—
11	C₃₁×#557	—	2	2	no	—
12	C₃₁×#557	—	2	4	no	—
13	C₃₁×#544	—	2	2	no	—
14	C₃₁×#544	C₂²	1	2	yes	—
15	C₃₁×#533	—	2	2	no	—
16	C₃₁×#533	C₂²	1	2	yes	—
17	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
18	C₃₁×#519	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#478	—	2	4	no	—
22	C₃₁×#476	—	2	2	no	—
23	C₃₁×#474	—	2	2	no	—
24	C₃₁×#428	C₂²	1	4	yes	—
25	C₃₁×#423	C₂²	1	2	yes	—
26	C₃₁×#418	C₂²	1	4	yes	—
27	C₂×C₃₁×#101	C₂²	1	2	yes	—
28	C₃₁×#391	C₂²	1	1	yes	characteristic
29	C₃₁×#390	C₂²	1	1	yes	characteristic
30	C₃₁×#388	—	2	2	no	—
31	C₃₁×#382	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
33	C₄³×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#343	C₂²	1	2	yes	—
35	C₃₁×#339	C₂²	1	4	yes	—
36	C₃₁×#339	C₂²	1	4	yes	—
37	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
38	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
39	C₂³×C₄×C₃₁	—	2	1	no	—
40	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
41	C₃₁×C₄⋊Q₈	—	2	2	no	—
42	C₃₁×C₄⋊Q₈	—	2	2	no	—
43	C₃₁×C₄⋊Q₈	—	2	8	no	—
44	C₃₁×C₄⋊Q₈	—	2	4	no	—
45	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
46	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
47	C₃₁×#125	—	2	4	no	—
48	C₃₁×#125	—	2	8	no	—
49	C₃₁×#125	—	2	4	no	—
50	C₃₁×#125	D₄	1	4	yes	—
51	C₃₁×#124	—	2	2	no	—
52	C₃₁×#124	Q₈	1	4	yes	complemented
53	C₃₁×C₂²⋊Q₈	—	2	2	no	—
54	C₃₁×C₂²⋊Q₈	—	2	4	no	—
55	C₃₁×C₂²⋊Q₈	—	2	2	no	—
56	C₄×Q₈×C₃₁	—	2	4	no	—
57	C₄×Q₈×C₃₁	—	2	4	no	—
58	C₄×Q₈×C₃₁	—	2	4	no	—
59	C₄×Q₈×C₃₁	—	2	4	no	—
60	C₄×Q₈×C₃₁	—	2	2	no	—
61	C₄×Q₈×C₃₁	—	2	2	no	—
62	C₄×Q₈×C₃₁	—	2	2	no	—
63	C₄×Q₈×C₃₁	—	2	2	no	—
64	C₄×D₄×C₃₁	—	2	2	no	—
65	C₄×D₄×C₃₁	—	2	2	no	—
66	C₃₁×#117	—	2	4	no	—
67	C₃₁×#117	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₄²×C₃₁	—	2	1	no	—
75	C₂×C₄²×C₃₁	—	2	1	no	—
76	C₂×C₄²×C₃₁	D₄	1	2	yes	—
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×#107	—	2	4	no	—
81	C₃₁×#106	—	2	4	no	—
82	C₃₁×C₄⋊C₈	—	2	2	no	—
83	C₃₁×C₄⋊C₈	—	2	2	no	—
84	C₃₁×#102	—	2	4	no	—
85	C₃₁×#102	—	2	4	no	—
86	C₃₁×#101	—	2	4	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	—	2	2	no	—
89	C₂⁴×C₃₁	—	2	1	no	—
90	C₂×Q₈×C₃₁	—	2	4	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	—	2	2	no	—
95	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
96	C₂×Q₈×C₃₁	#41	1	2	yes	—
97	C₂×D₄×C₃₁	—	2	2	no	—
98	C₂×D₄×C₃₁	—	2	1	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	4	4	no	—
105	C₂²×C₄×C₃₁	—	4	1	no	—
106	C₂²×C₄×C₃₁	—	4	1	no	—
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
112	M₁₆×C₃₁	—	2	4	no	—
113	M₁₆×C₃₁	—	2	4	no	—
114	C₂×C₈×C₃₁	—	2	4	no	—
115	C₄⋊C₄×C₃₁	—	2	4	no	—
116	C₄⋊C₄×C₃₁	—	2	4	no	—
117	C₄⋊C₄×C₃₁	—	2	4	no	—
118	C₄⋊C₄×C₃₁	—	2	8	no	—
119	C₄⋊C₄×C₃₁	—	2	4	no	—
120	C₄⋊C₄×C₃₁	—	2	4	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	4	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
127	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
128	C₂²⋊C₄×C₃₁	—	2	4	no	—
129	C₂²⋊C₄×C₃₁	—	2	8	no	—
130	C₂²⋊C₄×C₃₁	—	4	4	no	—
131	C₂²⋊C₄×C₃₁	—	4	2	no	—
132	C₄²×C₃₁	—	2	8	no	—
133	C₄²×C₃₁	—	2	2	no	—
134	C₄²×C₃₁	—	2	2	no	—
135	C₄²×C₃₁	—	2	4	no	—
136	C₄²×C₃₁	—	2	1	no	—
137	C₄²×C₃₁	—	2	2	no	—
138	C₄²×C₃₁	—	4	2	no	—
139	C₄²×C₃₁	—	4	2	no	—
140	C₄²×C₃₁	C₂×D₄	1	2	yes	—
141	C₄²×C₃₁	C₂×D₄	1	2	yes	—
142	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
143	#24313	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	2	2	no	—
145	C₂³×C₃₁	—	4	2	no	—
146	C₂³×C₃₁	—	4	2	no	—
147	C₂³×C₃₁	—	4	1	no	—
148	C₂³×C₃₁	—	4	1	no	—
149	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
150	Q₈×C₃₁	—	2	4	no	—
151	Q₈×C₃₁	—	2	4	no	—
152	Q₈×C₃₁	—	4	4	no	—
153	Q₈×C₃₁	—	8	2	no	—
154	Q₈×C₃₁	—	8	4	no	—
155	D₄×C₃₁	—	2	4	no	—
156	D₄×C₃₁	—	4	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	4	no	—
162	C₂×C₄×C₃₁	—	2	4	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	2	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	2	no	—
172	C₂×C₄×C₃₁	—	4	1	no	—
173	C₂×C₄×C₃₁	—	8	2	no	—
174	C₂×C₄×C₃₁	—	8	2	no	—
175	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
176	C₂×C₄×C₃₁	#120	1	2	yes	—
177	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
179	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
180	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
182	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
183	C₈×C₃₁	—	4	4	no	—
184	#3502	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#3077	C₂×C₃₁	1	2	yes	—
186	C₄×#124	C₂×C₃₁	1	1	yes	characteristic
187	#2061	C₂×C₃₁	1	2	yes	complemented
188	#2027	C₂×C₃₁	1	2	yes	—
189	#1994	C₂×C₃₁	1	2	yes	complemented
190	#1888	C₂×C₃₁	1	2	yes	—
191	#1857	C₂×C₃₁	1	2	yes	complemented
192	#1801	C₂×C₃₁	1	1	yes	characteristic, complemented
193	C₂²×C₃₁	—	2	1	no	—
194	C₂²×C₃₁	—	2	1	no	—
195	C₂²×C₃₁	—	4	4	no	—
196	C₂²×C₃₁	—	4	2	no	—
197	C₂²×C₃₁	—	8	2	no	—
198	C₂²×C₃₁	—	8	1	no	—
199	C₂²×C₃₁	—	8	1	no	—
200	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
201	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
202	C₂²×C₃₁	#536	1	1	yes	characteristic
203	C₄×C₃₁	—	2	4	no	—
204	C₄×C₃₁	—	2	1	no	—
205	C₄×C₃₁	—	2	2	no	—
206	C₄×C₃₁	—	4	2	no	—
207	C₄×C₃₁	—	4	2	no	—
208	C₄×C₃₁	—	4	4	no	—
209	C₄×C₃₁	—	4	2	no	—
210	C₄×C₃₁	—	4	4	no	—
211	C₄×C₃₁	—	8	1	no	—
212	C₄×C₃₁	#454	1	2	yes	—
213	C₄×C₃₁	#548	1	2	yes	—
214	C₄×C₃₁	D₄×Q₈	1	2	yes	—
215	Q₈²	—	2	2	no	—
216	#557	—	2	2	no	—
217	#557	—	2	4	no	—
218	#544	—	2	2	no	—
219	#544	C₂²×C₃₁	1	2	yes	—
220	#533	—	2	2	no	—
221	#533	C₂²×C₃₁	1	2	yes	—
222	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
223	#519	C₂²×C₃₁	1	2	yes	—
224	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
225	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
226	#478	—	2	4	no	—
227	#476	—	2	2	no	—
228	#474	—	2	2	no	—
229	#428	C₂²×C₃₁	1	4	yes	—
230	#423	C₂²×C₃₁	1	2	yes	—
231	#418	C₂²×C₃₁	1	4	yes	—
232	C₂×#101	C₂²×C₃₁	1	2	yes	—
233	#391	C₂²×C₃₁	1	1	yes	characteristic
234	#390	C₂²×C₃₁	1	1	yes	characteristic
235	#388	—	2	2	no	—
236	#382	C₂²×C₃₁	1	1	yes	characteristic
237	C₄×C₂²⋊C₄	—	2	2	no	—
238	C₄³	C₂²×C₃₁	1	1	yes	characteristic
239	#343	C₂²×C₃₁	1	2	yes	—
240	#339	C₂²×C₃₁	1	4	yes	—
241	#339	C₂²×C₃₁	1	4	yes	—
242	C₂×C₃₁	—	2	1	no	—
243	C₂×C₃₁	—	2	1	no	—
244	C₂×C₃₁	—	8	2	no	—
245	C₂×C₃₁	#2409	1	1	yes	characteristic
246	C₂×C₃₁	C₂×#454	1	1	yes	characteristic
247	C₂×C₃₁	#3018	1	1	yes	characteristic
248	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
249	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
250	C₂³×C₄	—	2	1	no	—
251	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
252	C₄⋊Q₈	—	2	2	no	—
253	C₄⋊Q₈	—	2	2	no	—
254	C₄⋊Q₈	—	2	8	no	—
255	C₄⋊Q₈	—	2	4	no	—
256	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
257	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
258	#125	—	2	4	no	—
259	#125	—	2	8	no	—
260	#125	—	2	4	no	—
261	#125	D₄×C₃₁	1	4	yes	—
262	#124	—	2	2	no	—
263	#124	Q₈×C₃₁	1	4	yes	complemented
264	C₂²⋊Q₈	—	2	2	no	—
265	C₂²⋊Q₈	—	2	4	no	—
266	C₂²⋊Q₈	—	2	2	no	—
267	C₄×Q₈	—	2	4	no	—
268	C₄×Q₈	—	2	4	no	—
269	C₄×Q₈	—	2	4	no	—
270	C₄×Q₈	—	2	4	no	—
271	C₄×Q₈	—	2	2	no	—
272	C₄×Q₈	—	2	2	no	—
273	C₄×Q₈	—	2	2	no	—
274	C₄×Q₈	—	2	2	no	—
275	C₄×D₄	—	2	2	no	—
276	C₄×D₄	—	2	2	no	—
277	#117	—	2	4	no	—
278	#117	C₂³×C₃₁	1	4	yes	—
279	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
280	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
281	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
282	C₂×C₂²⋊C₄	—	2	2	no	—
283	C₂×C₂²⋊C₄	—	2	1	no	—
284	C₂×C₂²⋊C₄	—	2	2	no	—
285	C₂×C₄²	—	2	1	no	—
286	C₂×C₄²	—	2	1	no	—
287	C₂×C₄²	D₄×C₃₁	1	2	yes	—
288	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
289	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
290	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
291	#107	—	2	4	no	—
292	#106	—	2	4	no	—
293	C₄⋊C₈	—	2	2	no	—
294	C₄⋊C₈	—	2	2	no	—
295	#102	—	2	4	no	—
296	#102	—	2	4	no	—
297	#101	—	2	4	no	—
298	#95	—	2	2	no	—
299	#95	—	2	2	no	—
300	C₃₁	#24313	1	1	yes	characteristic, direct product factor, Sylow
301	C₂⁴	—	2	1	no	—
302	C₂×Q₈	—	2	4	no	—
303	C₂×Q₈	—	2	2	no	—
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	—	2	2	no	—
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
308	C₂×Q₈	#41×C₃₁	1	2	yes	—
309	C₂×D₄	—	2	2	no	—
310	C₂×D₄	—	2	1	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	4	4	no	—
317	C₂²×C₄	—	4	1	no	—
318	C₂²×C₄	—	4	1	no	—
319	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
321	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
322	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
324	M₁₆	—	2	4	no	—
325	M₁₆	—	2	4	no	—
326	C₂×C₈	—	2	4	no	—
327	C₄⋊C₄	—	2	4	no	—
328	C₄⋊C₄	—	2	4	no	—
329	C₄⋊C₄	—	2	4	no	—
330	C₄⋊C₄	—	2	8	no	—
331	C₄⋊C₄	—	2	4	no	—
332	C₄⋊C₄	—	2	4	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	4	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
339	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
340	C₂²⋊C₄	—	2	4	no	—
341	C₂²⋊C₄	—	2	8	no	—
342	C₂²⋊C₄	—	4	4	no	—
343	C₂²⋊C₄	—	4	2	no	—
344	C₄²	—	2	8	no	—
345	C₄²	—	2	2	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	2	4	no	—
348	C₄²	—	2	1	no	—
349	C₄²	—	2	2	no	—
350	C₄²	—	4	2	no	—
351	C₄²	—	4	2	no	—
352	C₄²	C₂×D₄×C₃₁	1	2	yes	—
353	C₄²	C₂×D₄×C₃₁	1	2	yes	—
354	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
355	C₂³	—	2	2	no	—
356	C₂³	—	4	2	no	—
357	C₂³	—	4	2	no	—
358	C₂³	—	4	1	no	—
359	C₂³	—	4	1	no	—
360	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
361	Q₈	—	2	4	no	—
362	Q₈	—	2	4	no	—
363	Q₈	—	4	4	no	—
364	Q₈	—	8	2	no	—
365	Q₈	—	8	4	no	—
366	D₄	—	2	4	no	—
367	D₄	—	4	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	4	1	no	—
377	C₂×C₄	—	4	1	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	8	2	no	—
385	C₂×C₄	—	8	2	no	—
386	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
387	C₂×C₄	C₃₁×#120	1	2	yes	—
388	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
389	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
390	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
391	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
392	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
393	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
394	C₈	—	4	4	no	—
395	C₂²	—	2	1	no	—
396	C₂²	—	2	1	no	—
397	C₂²	—	4	4	no	—
398	C₂²	—	4	2	no	—
399	C₂²	—	8	2	no	—
400	C₂²	—	8	1	no	—
401	C₂²	—	8	1	no	—
402	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
403	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
404	C₂²	C₃₁×#536	1	1	yes	characteristic
405	C₄	—	2	4	no	—
406	C₄	—	2	1	no	—
407	C₄	—	2	2	no	—
408	C₄	—	4	2	no	—
409	C₄	—	4	2	no	—
410	C₄	—	4	4	no	—
411	C₄	—	4	2	no	—
412	C₄	—	4	4	no	—
413	C₄	—	8	1	no	—
414	C₄	C₃₁×#454	1	2	yes	—
415	C₄	C₃₁×#548	1	2	yes	—
416	C₄	D₄×Q₈×C₃₁	1	2	yes	—
417	C₂	—	2	1	no	—
418	C₂	—	2	1	no	—
419	C₂	—	8	2	no	—
420	C₂	C₃₁×#2409	1	1	yes	characteristic
421	C₂	C₂×C₃₁×#454	1	1	yes	characteristic
422	C₂	C₃₁×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3502	C₂	semidirect
4. C₃₁×#2061	C₂	semidirect
6. C₃₁×#1994	C₂	semidirect
8. C₃₁×#1857	C₂	semidirect
9. C₃₁×#1801	C₂	semidirect
52. C₃₁×#124	Q₈	semidirect
143. #24313	C₃₁	direct (N × H)
184. #3502	C₂×C₃₁	semidirect
187. #2061	C₂×C₃₁	semidirect
189. #1994	C₂×C₃₁	semidirect
191. #1857	C₂×C₃₁	semidirect
192. #1801	C₂×C₃₁	semidirect
263. #124	Q₈×C₃₁	semidirect
300. C₃₁	#24313	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3502	2
C₃₁×#3077	2
C₄×C₃₁×#124	2
C₃₁×#2061	2
C₃₁×#2027	2
C₃₁×#1994	2
C₃₁×#1888	2
C₃₁×#1857	2
C₃₁×#1801	2
#24313	31

Subgroups — C31×#24313 #21481072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24313 #21481072