Subgroups — C₃₁×#19979 #21476738

Order 7936.

672 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3298	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3283	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1707	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1679	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1673	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1670	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1654	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1651	C₂	1	2	yes	maximal, complemented
10	C₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	—	2	1	no	—
13	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#547	C₂²	1	2	yes	—
15	C₃₁×#500	—	2	1	no	—
16	C₃₁×#500	C₂²	1	1	yes	characteristic
17	C₃₁×#498	C₂²	1	2	yes	—
18	C₃₁×#494	C₂²	1	1	yes	characteristic
19	C₃₁×#487	C₂²	1	2	yes	—
20	C₃₁×#483	—	2	2	no	—
21	C₃₁×#475	C₂²	1	2	yes	—
22	C₃₁×#474	—	2	2	no	—
23	C₃₁×#474	C₂²	1	2	yes	—
24	C₃₁×#467	—	2	1	no	—
25	C₃₁×#465	—	2	2	no	—
26	C₃₁×#461	C₂²	1	2	yes	—
27	C₃₁×#461	C₂²	1	2	yes	—
28	C₃₁×#452	—	2	2	no	—
29	C₃₁×#448	—	2	2	no	—
30	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
32	C₄×D₈×C₃₁	—	2	1	no	—
33	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₃₁	C₂²	1	2	yes	—
35	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#103	—	2	2	no	—
37	C₃₁×#335	C₂²	1	1	yes	characteristic
38	C₃₁×#334	C₂²	1	1	yes	characteristic
39	C₃₁×#332	C₂²	1	2	yes	—
40	C₃₁×#332	C₂²	1	2	yes	—
41	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
43	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₃₁	C₂²	1	1	yes	characteristic
45	C₂×#41×C₃₁	—	2	2	no	—
46	C₂²×Q₈×C₃₁	—	2	2	no	—
47	C₃₁×#135	—	4	4	no	—
48	C₂×Q₁₆×C₃₁	—	2	1	no	—
49	C₂×Q₁₆×C₃₁	—	2	1	no	—
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
53	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	—	2	1	no	—
55	C₂²×C₈×C₃₁	—	2	2	no	—
56	C₃₁×C₄⋊Q₈	—	2	2	no	—
57	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
58	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
59	C₃₁×#123	—	4	2	no	—
60	C₃₁×C₂²⋊Q₈	—	2	2	no	—
61	C₃₁×C₂²⋊Q₈	—	2	2	no	—
62	C₃₁×C₂²⋊Q₈	—	4	2	no	—
63	C₃₁×#121	—	2	2	no	—
64	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₄×D₄×C₃₁	—	2	2	no	—
66	C₄×D₄×C₃₁	C₂³	1	2	yes	—
67	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₃₁×#107	—	2	1	no	—
70	C₃₁×#107	—	2	1	no	—
71	C₃₁×#107	—	2	2	no	—
72	C₃₁×#107	D₄	1	1	yes	characteristic
73	C₃₁×#106	—	2	2	no	—
74	C₃₁×#106	—	4	2	no	—
75	C₃₁×#106	D₄	1	1	yes	characteristic
76	C₃₁×C₄⋊C₈	—	2	1	no	—
77	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
79	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
80	C₃₁×#103	—	2	2	no	—
81	C₃₁×#103	—	2	2	no	—
82	C₃₁×#103	—	2	2	no	—
83	C₃₁×#103	—	2	2	no	—
84	C₃₁×#103	—	2	2	no	—
85	C₃₁×#103	D₄	1	1	yes	characteristic
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×#102	D₄	1	2	yes	—
88	C₃₁×#102	D₄	1	1	yes	characteristic
89	C₃₁×C₂²⋊C₈	—	2	2	no	—
90	C₃₁×C₂²⋊C₈	—	2	1	no	—
91	C₄×C₈×C₃₁	—	2	1	no	—
92	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
95	#41×C₃₁	—	4	4	no	—
96	#41×C₃₁	—	8	2	no	—
97	C₂×Q₈×C₃₁	—	2	2	no	—
98	C₂×Q₈×C₃₁	—	2	2	no	—
99	C₂×Q₈×C₃₁	—	4	2	no	—
100	C₂×Q₈×C₃₁	—	4	4	no	—
101	C₂×Q₈×C₃₁	—	8	2	no	—
102	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×D₄×C₃₁	—	4	2	no	—
104	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	1	no	—
108	C₂²×C₄×C₃₁	—	4	2	no	—
109	C₂²×C₄×C₃₁	—	4	2	no	—
110	Q₁₆×C₃₁	—	4	2	no	—
111	Q₁₆×C₃₁	—	8	1	no	—
112	SD₁₆×C₃₁	—	2	2	no	—
113	SD₁₆×C₃₁	—	4	4	no	—
114	SD₁₆×C₃₁	—	8	2	no	—
115	D₈×C₃₁	SD₁₆	1	4	yes	complemented
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	4	2	no	—
121	C₂×C₈×C₃₁	—	8	2	no	—
122	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₄⋊C₄×C₃₁	—	2	1	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	4	2	no	—
132	C₄⋊C₄×C₃₁	—	4	2	no	—
133	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
134	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²⋊C₄×C₃₁	—	2	2	no	—
137	C₂²⋊C₄×C₃₁	—	2	1	no	—
138	C₂²⋊C₄×C₃₁	—	4	2	no	—
139	C₂²⋊C₄×C₃₁	—	4	2	no	—
140	C₄²×C₃₁	—	2	1	no	—
141	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
142	#19979	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂³×C₃₁	—	2	2	no	—
144	C₂³×C₃₁	—	2	1	no	—
145	Q₈×C₃₁	—	2	1	no	—
146	Q₈×C₃₁	—	4	2	no	—
147	Q₈×C₃₁	—	4	2	no	—
148	Q₈×C₃₁	—	4	2	no	—
149	Q₈×C₃₁	—	4	2	no	—
150	Q₈×C₃₁	—	8	4	no	—
151	Q₈×C₃₁	#135	1	2	yes	complemented
152	D₄×C₃₁	—	2	1	no	—
153	D₄×C₃₁	—	2	1	no	—
154	D₄×C₃₁	—	2	2	no	—
155	D₄×C₃₁	—	8	4	no	—
156	D₄×C₃₁	C₂×SD₁₆	1	4	yes	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	4	1	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	2	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	8	4	no	—
168	C₂×C₄×C₃₁	—	8	4	no	—
169	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
172	C₈×C₃₁	—	2	2	no	—
173	C₈×C₃₁	—	2	1	no	—
174	C₈×C₃₁	—	2	1	no	—
175	C₈×C₃₁	—	2	1	no	—
176	C₈×C₃₁	—	4	1	no	—
177	C₈×C₃₁	—	4	2	no	—
178	C₈×C₃₁	—	4	1	no	—
179	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
180	Q₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
181	#3298	C₂×C₃₁	1	2	yes	complemented
182	#3283	C₂×C₃₁	1	1	yes	characteristic
183	#1707	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#1679	C₂×C₃₁	1	2	yes	complemented
185	#1673	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#1670	C₂×C₃₁	1	1	yes	characteristic, complemented
187	#1654	C₂×C₃₁	1	2	yes	complemented
188	#1651	C₂×C₃₁	1	2	yes	complemented
189	C₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
190	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
191	C₂²×C₃₁	—	2	4	no	—
192	C₂²×C₃₁	—	4	2	no	—
193	C₂²×C₃₁	—	4	2	no	—
194	C₂²×C₃₁	—	4	1	no	—
195	C₂²×C₃₁	—	4	1	no	—
196	C₂²×C₃₁	—	4	1	no	—
197	C₂²×C₃₁	D₄²	1	1	yes	characteristic
198	C₄×C₃₁	—	2	2	no	—
199	C₄×C₃₁	—	2	1	no	—
200	C₄×C₃₁	—	4	2	no	—
201	C₄×C₃₁	—	4	1	no	—
202	C₄×C₃₁	—	4	1	no	—
203	C₄×C₃₁	—	8	2	no	—
204	C₄×C₃₁	—	8	2	no	—
205	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
206	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
207	C₄×C₃₁	#576	1	1	yes	characteristic
208	C₄×C₃₁	#577	1	1	yes	characteristic
209	C₂×#135	—	2	1	no	—
210	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
211	#547	C₂²×C₃₁	1	2	yes	—
212	#500	—	2	1	no	—
213	#500	C₂²×C₃₁	1	1	yes	characteristic
214	#498	C₂²×C₃₁	1	2	yes	—
215	#494	C₂²×C₃₁	1	1	yes	characteristic
216	#487	C₂²×C₃₁	1	2	yes	—
217	#483	—	2	2	no	—
218	#475	C₂²×C₃₁	1	2	yes	—
219	#474	—	2	2	no	—
220	#474	C₂²×C₃₁	1	2	yes	—
221	#467	—	2	1	no	—
222	#465	—	2	2	no	—
223	#461	C₂²×C₃₁	1	2	yes	—
224	#461	C₂²×C₃₁	1	2	yes	—
225	#452	—	2	2	no	—
226	#448	—	2	2	no	—
227	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
228	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
229	C₄×D₈	—	2	1	no	—
230	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
231	C₈×D₄	C₂²×C₃₁	1	2	yes	—
232	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×#103	—	2	2	no	—
234	#335	C₂²×C₃₁	1	1	yes	characteristic
235	#334	C₂²×C₃₁	1	1	yes	characteristic
236	#332	C₂²×C₃₁	1	2	yes	—
237	#332	C₂²×C₃₁	1	2	yes	—
238	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
239	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
240	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
241	C₈²	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×C₃₁	—	4	4	no	—
243	C₂×C₃₁	—	8	1	no	—
244	C₂×C₃₁	#3280	1	1	yes	characteristic
245	C₂×C₃₁	D₄×SD₁₆	1	1	yes	characteristic
246	C₂×C₃₁	#3293	1	1	yes	characteristic
247	C₂×#41	—	2	2	no	—
248	C₂²×Q₈	—	2	2	no	—
249	#135	—	4	4	no	—
250	C₂×Q₁₆	—	2	1	no	—
251	C₂×Q₁₆	—	2	1	no	—
252	C₂×SD₁₆	—	2	2	no	—
253	C₂×SD₁₆	—	2	2	no	—
254	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
255	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
256	C₂²×C₈	—	2	1	no	—
257	C₂²×C₈	—	2	2	no	—
258	C₄⋊Q₈	—	2	2	no	—
259	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
260	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
261	#123	—	4	2	no	—
262	C₂²⋊Q₈	—	2	2	no	—
263	C₂²⋊Q₈	—	2	2	no	—
264	C₂²⋊Q₈	—	4	2	no	—
265	#121	—	2	2	no	—
266	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
267	C₄×D₄	—	2	2	no	—
268	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
269	C₄×D₄	C₂³×C₃₁	1	2	yes	—
270	C₂×C₄⋊C₄	—	2	2	no	—
271	#107	—	2	1	no	—
272	#107	—	2	1	no	—
273	#107	—	2	2	no	—
274	#107	D₄×C₃₁	1	1	yes	characteristic
275	#106	—	2	2	no	—
276	#106	—	4	2	no	—
277	#106	D₄×C₃₁	1	1	yes	characteristic
278	C₄⋊C₈	—	2	1	no	—
279	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
280	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
281	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
282	#103	—	2	2	no	—
283	#103	—	2	2	no	—
284	#103	—	2	2	no	—
285	#103	—	2	2	no	—
286	#103	—	2	2	no	—
287	#103	D₄×C₃₁	1	1	yes	characteristic
288	#102	—	2	2	no	—
289	#102	D₄×C₃₁	1	2	yes	—
290	#102	D₄×C₃₁	1	1	yes	characteristic
291	C₂²⋊C₈	—	2	2	no	—
292	C₂²⋊C₈	—	2	1	no	—
293	C₄×C₈	—	2	1	no	—
294	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
295	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
296	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
297	C₃₁	#19979	1	1	yes	characteristic, direct product factor, Sylow
298	#41	—	4	4	no	—
299	#41	—	8	2	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	4	2	no	—
303	C₂×Q₈	—	4	4	no	—
304	C₂×Q₈	—	8	2	no	—
305	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
306	C₂×D₄	—	4	2	no	—
307	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
308	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	4	2	no	—
312	C₂²×C₄	—	4	2	no	—
313	Q₁₆	—	4	2	no	—
314	Q₁₆	—	8	1	no	—
315	SD₁₆	—	2	2	no	—
316	SD₁₆	—	4	4	no	—
317	SD₁₆	—	8	2	no	—
318	D₈	SD₁₆×C₃₁	1	4	yes	complemented
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	4	2	no	—
324	C₂×C₈	—	8	2	no	—
325	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
326	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
327	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
328	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	4	2	no	—
335	C₄⋊C₄	—	4	2	no	—
336	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
337	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
338	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
339	C₂²⋊C₄	—	2	2	no	—
340	C₂²⋊C₄	—	2	1	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₂²⋊C₄	—	4	2	no	—
343	C₄²	—	2	1	no	—
344	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
345	C₂³	—	2	2	no	—
346	C₂³	—	2	1	no	—
347	Q₈	—	2	1	no	—
348	Q₈	—	4	2	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	4	2	no	—
351	Q₈	—	4	2	no	—
352	Q₈	—	8	4	no	—
353	Q₈	C₃₁×#135	1	2	yes	complemented
354	D₄	—	2	1	no	—
355	D₄	—	2	1	no	—
356	D₄	—	2	2	no	—
357	D₄	—	8	4	no	—
358	D₄	C₂×SD₁₆×C₃₁	1	4	yes	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	8	4	no	—
370	C₂×C₄	—	8	4	no	—
371	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
372	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
373	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
374	C₈	—	2	2	no	—
375	C₈	—	2	1	no	—
376	C₈	—	2	1	no	—
377	C₈	—	2	1	no	—
378	C₈	—	4	1	no	—
379	C₈	—	4	2	no	—
380	C₈	—	4	1	no	—
381	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
382	C₂²	—	2	4	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	4	2	no	—
385	C₂²	—	4	1	no	—
386	C₂²	—	4	1	no	—
387	C₂²	—	4	1	no	—
388	C₂²	D₄²×C₃₁	1	1	yes	characteristic
389	C₄	—	2	2	no	—
390	C₄	—	2	1	no	—
391	C₄	—	4	2	no	—
392	C₄	—	4	1	no	—
393	C₄	—	4	1	no	—
394	C₄	—	8	2	no	—
395	C₄	—	8	2	no	—
396	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
397	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
398	C₄	C₃₁×#576	1	1	yes	characteristic
399	C₄	C₃₁×#577	1	1	yes	characteristic
400	C₂	—	4	4	no	—
401	C₂	—	8	1	no	—
402	C₂	C₃₁×#3280	1	1	yes	characteristic
403	C₂	D₄×SD₁₆×C₃₁	1	1	yes	characteristic
404	C₂	C₃₁×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3283	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×D₈×C₃₁	C₂	semidirect
2. C₃₁×#3298	C₂	semidirect
4. C₃₁×#1707	C₂	semidirect
5. C₃₁×#1679	C₂	semidirect
6. C₃₁×#1673	C₂	semidirect
7. C₃₁×#1670	C₂	semidirect
8. C₃₁×#1654	C₂	semidirect
9. C₃₁×#1651	C₂	semidirect
10. C₈×SD₁₆×C₃₁	C₂	semidirect
11. C₈×D₈×C₃₁	C₂	semidirect
115. D₈×C₃₁	SD₁₆	semidirect
142. #19979	C₃₁	direct (N × H)
151. Q₈×C₃₁	#135	semidirect
180. Q₈×D₈	C₂×C₃₁	semidirect
181. #3298	C₂×C₃₁	semidirect
183. #1707	C₂×C₃₁	semidirect
184. #1679	C₂×C₃₁	semidirect
185. #1673	C₂×C₃₁	semidirect
186. #1670	C₂×C₃₁	semidirect
187. #1654	C₂×C₃₁	semidirect
188. #1651	C₂×C₃₁	semidirect
189. C₈×SD₁₆	C₂×C₃₁	semidirect
190. C₈×D₈	C₂×C₃₁	semidirect
297. C₃₁	#19979	direct (N × H)
318. D₈	SD₁₆×C₃₁	semidirect
353. Q₈	C₃₁×#135	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₈×C₃₁	2
C₃₁×#3298	2
C₃₁×#3283	2
C₃₁×#1707	2
C₃₁×#1679	2
C₃₁×#1673	2
C₃₁×#1670	2
C₃₁×#1654	2
C₃₁×#1651	2
C₈×SD₁₆×C₃₁	2
C₈×D₈×C₃₁	2
#19979	31

Subgroups — C31×#19979 #21476738

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19979 #21476738