Subgroups — C₃₁×#19976 #21476735

Order 7936.

726 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×#3380	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3295	C₂	1	2	yes	maximal, complemented
3	D₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1677	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1675	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1668	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1629	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1622	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	C₂²×D₈×C₃₁	—	2	1	no	—
13	C₃₁×#554	C₂²	1	2	yes	complemented
14	D₄²×C₃₁	C₂²	1	2	yes	—
15	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
16	C₃₁×#499	—	2	1	no	—
17	C₃₁×#493	C₂²	1	1	yes	characteristic
18	C₃₁×#492	C₂²	1	2	yes	complemented
19	C₃₁×#486	C₂²	1	2	yes	complemented
20	C₃₁×#481	—	2	2	no	—
21	C₃₁×#478	—	2	2	no	—
22	C₃₁×#476	C₂²	1	2	yes	—
23	C₃₁×#476	C₂²	1	2	yes	complemented
24	C₃₁×#466	—	2	1	no	—
25	C₃₁×#465	—	2	2	no	—
26	C₃₁×#460	C₂²	1	2	yes	complemented
27	C₃₁×#459	C₂²	1	2	yes	—
28	C₃₁×#450	—	2	2	no	—
29	C₃₁×#447	—	2	2	no	—
30	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
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	1	yes	characteristic
35	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
36	C₂×C₃₁×#102	—	2	2	no	—
37	C₃₁×#335	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#334	C₂²	1	1	yes	characteristic, complemented
39	C₃₁×#333	C₂²	1	2	yes	complemented
40	C₃₁×#331	C₂²	1	2	yes	complemented
41	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
44	C₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
45	C₂²×D₄×C₃₁	—	2	2	no	—
46	C₂²×D₄×C₃₁	—	2	2	no	—
47	C₂×SD₁₆×C₃₁	—	2	2	no	—
48	C₂×SD₁₆×C₃₁	—	4	2	no	—
49	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
50	C₂×D₈×C₃₁	—	2	2	no	—
51	C₂×D₈×C₃₁	—	2	1	no	—
52	C₂×D₈×C₃₁	—	2	1	no	—
53	C₂×D₈×C₃₁	—	4	4	no	—
54	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	—	2	1	no	—
56	C₂²×C₈×C₃₁	—	2	2	no	—
57	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
58	C₃₁×#127	C₂³	1	2	yes	—
59	C₃₁×#125	—	2	2	no	—
60	C₃₁×C₂²⋊Q₈	—	2	2	no	—
61	C₃₁×C₂²⋊Q₈	—	4	2	no	—
62	C₃₁×#121	—	2	2	no	—
63	C₃₁×#121	—	2	2	no	—
64	C₃₁×#120	—	4	2	no	—
65	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₄×D₄×C₃₁	—	2	2	no	—
67	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₄×D₄×C₃₁	C₂³	1	2	yes	—
69	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₃₁×#107	—	2	1	no	—
71	C₃₁×#107	—	2	1	no	—
72	C₃₁×#107	D₄	1	1	yes	characteristic
73	C₃₁×#106	—	2	2	no	—
74	C₃₁×#106	—	2	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	D₄	1	1	yes	characteristic
81	C₃₁×#102	—	2	2	no	—
82	C₃₁×#102	—	2	2	no	—
83	C₃₁×#102	—	2	2	no	—
84	C₃₁×#102	—	2	2	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×#102	D₄	1	1	yes	characteristic
88	C₃₁×#102	D₄	1	2	yes	—
89	C₃₁×C₂²⋊C₈	—	2	1	no	—
90	C₃₁×C₂²⋊C₈	—	2	2	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	C₂⁴×C₃₁	—	4	2	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	2	2	no	—
101	C₂×D₄×C₃₁	—	4	2	no	—
102	C₂×D₄×C₃₁	—	4	2	no	—
103	C₂×D₄×C₃₁	—	4	4	no	—
104	C₂×D₄×C₃₁	—	4	4	no	—
105	C₂×D₄×C₃₁	—	8	2	no	—
106	C₂×D₄×C₃₁	—	8	2	no	—
107	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
108	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	4	2	no	—
112	SD₁₆×C₃₁	—	2	2	no	—
113	SD₁₆×C₃₁	—	8	4	no	—
114	SD₁₆×C₃₁	—	8	2	no	—
115	D₈×C₃₁	—	4	4	no	—
116	D₈×C₃₁	—	4	2	no	—
117	D₈×C₃₁	—	8	1	no	—
118	D₈×C₃₁	SD₁₆	1	4	yes	complemented
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	1	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	4	2	no	—
124	C₂×C₈×C₃₁	—	8	2	no	—
125	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₄⋊C₄×C₃₁	—	2	1	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	4	2	no	—
133	C₄⋊C₄×C₃₁	—	4	2	no	—
134	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
136	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²⋊C₄×C₃₁	—	2	1	no	—
138	C₂²⋊C₄×C₃₁	—	2	2	no	—
139	C₂²⋊C₄×C₃₁	—	4	2	no	—
140	C₂²⋊C₄×C₃₁	—	4	2	no	—
141	C₄²×C₃₁	—	2	1	no	—
142	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
143	#19976	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	2	2	no	—
145	C₂³×C₃₁	—	2	1	no	—
146	C₂³×C₃₁	—	4	2	no	—
147	C₂³×C₃₁	—	8	4	no	—
148	C₂³×C₃₁	—	8	4	no	—
149	C₂³×C₃₁	—	16	2	no	—
150	Q₈×C₃₁	—	2	1	no	—
151	Q₈×C₃₁	—	2	1	no	—
152	Q₈×C₃₁	—	8	2	no	—
153	D₄×C₃₁	—	2	1	no	—
154	D₄×C₃₁	—	2	2	no	—
155	D₄×C₃₁	—	4	2	no	—
156	D₄×C₃₁	—	4	2	no	—
157	D₄×C₃₁	—	4	2	no	—
158	D₄×C₃₁	—	4	2	no	—
159	D₄×C₃₁	—	8	2	no	—
160	D₄×C₃₁	—	8	4	no	—
161	D₄×C₃₁	—	8	4	no	—
162	D₄×C₃₁	C₂×SD₁₆	1	4	yes	—
163	D₄×C₃₁	#135	1	2	yes	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	1	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	2	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	—	16	2	no	—
174	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
177	C₈×C₃₁	—	2	2	no	—
178	C₈×C₃₁	—	2	1	no	—
179	C₈×C₃₁	—	2	1	no	—
180	C₈×C₃₁	—	2	1	no	—
181	C₈×C₃₁	—	4	1	no	—
182	C₈×C₃₁	—	4	2	no	—
183	C₈×C₃₁	—	4	1	no	—
184	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
185	#3380	C₂×C₃₁	1	1	yes	characteristic, complemented
186	#3295	C₂×C₃₁	1	2	yes	complemented
187	D₄×D₈	C₂×C₃₁	1	1	yes	characteristic
188	#1706	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#1677	C₂×C₃₁	1	2	yes	complemented
190	#1675	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#1668	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#1629	C₂×C₃₁	1	2	yes	complemented
193	#1622	C₂×C₃₁	1	2	yes	complemented
194	C₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
195	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
196	C₂²×C₃₁	—	2	2	no	—
197	C₂²×C₃₁	—	2	4	no	—
198	C₂²×C₃₁	—	4	2	no	—
199	C₂²×C₃₁	—	4	1	no	—
200	C₂²×C₃₁	—	4	2	no	—
201	C₂²×C₃₁	—	4	1	no	—
202	C₂²×C₃₁	—	8	2	no	—
203	C₂²×C₃₁	—	8	2	no	—
204	C₂²×C₃₁	—	8	2	no	—
205	C₂²×C₃₁	—	16	8	no	—
206	C₂²×C₃₁	D₄²	1	1	yes	characteristic
207	C₄×C₃₁	—	2	1	no	—
208	C₄×C₃₁	—	4	2	no	—
209	C₄×C₃₁	—	4	1	no	—
210	C₄×C₃₁	—	4	1	no	—
211	C₄×C₃₁	—	4	1	no	—
212	C₄×C₃₁	—	8	2	no	—
213	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
214	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
215	C₄×C₃₁	#576	1	1	yes	characteristic
216	C₄×C₃₁	#577	1	1	yes	characteristic
217	C₂²×D₈	—	2	1	no	—
218	#554	C₂²×C₃₁	1	2	yes	complemented
219	D₄²	C₂²×C₃₁	1	2	yes	—
220	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
221	#499	—	2	1	no	—
222	#493	C₂²×C₃₁	1	1	yes	characteristic
223	#492	C₂²×C₃₁	1	2	yes	complemented
224	#486	C₂²×C₃₁	1	2	yes	complemented
225	#481	—	2	2	no	—
226	#478	—	2	2	no	—
227	#476	C₂²×C₃₁	1	2	yes	—
228	#476	C₂²×C₃₁	1	2	yes	complemented
229	#466	—	2	1	no	—
230	#465	—	2	2	no	—
231	#460	C₂²×C₃₁	1	2	yes	complemented
232	#459	C₂²×C₃₁	1	2	yes	—
233	#450	—	2	2	no	—
234	#447	—	2	2	no	—
235	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
236	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
237	C₄×D₈	—	2	1	no	—
238	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
239	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
240	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
241	C₂×#102	—	2	2	no	—
242	#335	C₂²×C₃₁	1	1	yes	characteristic, complemented
243	#334	C₂²×C₃₁	1	1	yes	characteristic, complemented
244	#333	C₂²×C₃₁	1	2	yes	complemented
245	#331	C₂²×C₃₁	1	2	yes	complemented
246	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
247	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
248	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
249	C₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
250	C₂×C₃₁	—	4	4	no	—
251	C₂×C₃₁	—	4	2	no	—
252	C₂×C₃₁	—	16	2	no	—
253	C₂×C₃₁	#3280	1	1	yes	characteristic
254	C₂×C₃₁	D₄×SD₁₆	1	1	yes	characteristic
255	C₂×C₃₁	#3292	1	1	yes	characteristic
256	C₂²×D₄	—	2	2	no	—
257	C₂²×D₄	—	2	2	no	—
258	C₂×SD₁₆	—	2	2	no	—
259	C₂×SD₁₆	—	4	2	no	—
260	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
261	C₂×D₈	—	2	2	no	—
262	C₂×D₈	—	2	1	no	—
263	C₂×D₈	—	2	1	no	—
264	C₂×D₈	—	4	4	no	—
265	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
266	C₂²×C₈	—	2	1	no	—
267	C₂²×C₈	—	2	2	no	—
268	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
269	#127	C₂³×C₃₁	1	2	yes	—
270	#125	—	2	2	no	—
271	C₂²⋊Q₈	—	2	2	no	—
272	C₂²⋊Q₈	—	4	2	no	—
273	#121	—	2	2	no	—
274	#121	—	2	2	no	—
275	#120	—	4	2	no	—
276	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
277	C₄×D₄	—	2	2	no	—
278	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
279	C₄×D₄	C₂³×C₃₁	1	2	yes	—
280	C₂×C₄⋊C₄	—	2	2	no	—
281	#107	—	2	1	no	—
282	#107	—	2	1	no	—
283	#107	D₄×C₃₁	1	1	yes	characteristic
284	#106	—	2	2	no	—
285	#106	—	2	2	no	—
286	#106	D₄×C₃₁	1	1	yes	characteristic
287	C₄⋊C₈	—	2	1	no	—
288	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
289	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
290	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
291	#103	D₄×C₃₁	1	1	yes	characteristic
292	#102	—	2	2	no	—
293	#102	—	2	2	no	—
294	#102	—	2	2	no	—
295	#102	—	2	2	no	—
296	#102	—	2	2	no	—
297	#102	—	2	2	no	—
298	#102	D₄×C₃₁	1	1	yes	characteristic
299	#102	D₄×C₃₁	1	2	yes	—
300	C₂²⋊C₈	—	2	1	no	—
301	C₂²⋊C₈	—	2	2	no	—
302	C₄×C₈	—	2	1	no	—
303	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
304	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
305	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
306	C₃₁	#19976	1	1	yes	characteristic, direct product factor, Sylow
307	C₂⁴	—	4	2	no	—
308	C₂×Q₈	—	2	2	no	—
309	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	4	2	no	—
314	C₂×D₄	—	4	2	no	—
315	C₂×D₄	—	4	4	no	—
316	C₂×D₄	—	4	4	no	—
317	C₂×D₄	—	8	2	no	—
318	C₂×D₄	—	8	2	no	—
319	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
320	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
321	C₂²×C₄	—	2	2	no	—
322	C₂²×C₄	—	2	1	no	—
323	C₂²×C₄	—	4	2	no	—
324	SD₁₆	—	2	2	no	—
325	SD₁₆	—	8	4	no	—
326	SD₁₆	—	8	2	no	—
327	D₈	—	4	4	no	—
328	D₈	—	4	2	no	—
329	D₈	—	8	1	no	—
330	D₈	SD₁₆×C₃₁	1	4	yes	complemented
331	C₂×C₈	—	2	1	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	4	2	no	—
336	C₂×C₈	—	8	2	no	—
337	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
338	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
339	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
340	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₄⋊C₄	—	2	1	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	4	2	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
347	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
348	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
349	C₂²⋊C₄	—	2	1	no	—
350	C₂²⋊C₄	—	2	2	no	—
351	C₂²⋊C₄	—	4	2	no	—
352	C₂²⋊C₄	—	4	2	no	—
353	C₄²	—	2	1	no	—
354	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
355	C₂³	—	2	2	no	—
356	C₂³	—	2	1	no	—
357	C₂³	—	4	2	no	—
358	C₂³	—	8	4	no	—
359	C₂³	—	8	4	no	—
360	C₂³	—	16	2	no	—
361	Q₈	—	2	1	no	—
362	Q₈	—	2	1	no	—
363	Q₈	—	8	2	no	—
364	D₄	—	2	1	no	—
365	D₄	—	2	2	no	—
366	D₄	—	4	2	no	—
367	D₄	—	4	2	no	—
368	D₄	—	4	2	no	—
369	D₄	—	4	2	no	—
370	D₄	—	8	2	no	—
371	D₄	—	8	4	no	—
372	D₄	—	8	4	no	—
373	D₄	C₂×SD₁₆×C₃₁	1	4	yes	—
374	D₄	C₃₁×#135	1	2	yes	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	16	2	no	—
385	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
386	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
387	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
388	C₈	—	2	2	no	—
389	C₈	—	2	1	no	—
390	C₈	—	2	1	no	—
391	C₈	—	2	1	no	—
392	C₈	—	4	1	no	—
393	C₈	—	4	2	no	—
394	C₈	—	4	1	no	—
395	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
396	C₂²	—	2	2	no	—
397	C₂²	—	2	4	no	—
398	C₂²	—	4	2	no	—
399	C₂²	—	4	1	no	—
400	C₂²	—	4	2	no	—
401	C₂²	—	4	1	no	—
402	C₂²	—	8	2	no	—
403	C₂²	—	8	2	no	—
404	C₂²	—	8	2	no	—
405	C₂²	—	16	8	no	—
406	C₂²	D₄²×C₃₁	1	1	yes	characteristic
407	C₄	—	2	1	no	—
408	C₄	—	4	2	no	—
409	C₄	—	4	1	no	—
410	C₄	—	4	1	no	—
411	C₄	—	4	1	no	—
412	C₄	—	8	2	no	—
413	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
414	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
415	C₄	C₃₁×#576	1	1	yes	characteristic
416	C₄	C₃₁×#577	1	1	yes	characteristic
417	C₂	—	4	2	no	—
418	C₂	—	4	4	no	—
419	C₂	—	16	2	no	—
420	C₂	C₃₁×#3280	1	1	yes	characteristic
421	C₂	D₄×SD₁₆×C₃₁	1	1	yes	characteristic
422	C₂	C₃₁×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	D₄×D₈×C₃₁	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. C₃₁×#3380	C₂	semidirect
2. C₃₁×#3295	C₂	semidirect
4. C₃₁×#1706	C₂	semidirect
5. C₃₁×#1677	C₂	semidirect
6. C₃₁×#1675	C₂	semidirect
7. C₃₁×#1668	C₂	semidirect
8. C₃₁×#1629	C₂	semidirect
9. C₃₁×#1622	C₂	semidirect
10. C₈×SD₁₆×C₃₁	C₂	semidirect
11. C₈×D₈×C₃₁	C₂	semidirect
13. C₃₁×#554	C₂²	semidirect
15. C₃₁×#500	C₂²	semidirect
18. C₃₁×#492	C₂²	semidirect
19. C₃₁×#486	C₂²	semidirect
23. C₃₁×#476	C₂²	semidirect
26. C₃₁×#460	C₂²	semidirect
30. C₈×Q₈×C₃₁	C₂²	semidirect
35. C₈×D₄×C₃₁	C₂²	semidirect
37. C₃₁×#335	C₂²	semidirect
38. C₃₁×#334	C₂²	semidirect
39. C₃₁×#333	C₂²	semidirect
40. C₃₁×#331	C₂²	semidirect
41. C₃₁×Q₈⋊C₈	C₂²	semidirect
43. C₃₁×D₄⋊C₈	C₂²	semidirect
44. C₈²×C₃₁	C₂²	semidirect
118. D₈×C₃₁	SD₁₆	semidirect
143. #19976	C₃₁	direct (N × H)
185. #3380	C₂×C₃₁	semidirect
186. #3295	C₂×C₃₁	semidirect
188. #1706	C₂×C₃₁	semidirect
189. #1677	C₂×C₃₁	semidirect
190. #1675	C₂×C₃₁	semidirect
191. #1668	C₂×C₃₁	semidirect
192. #1629	C₂×C₃₁	semidirect
193. #1622	C₂×C₃₁	semidirect
194. C₈×SD₁₆	C₂×C₃₁	semidirect
195. C₈×D₈	C₂×C₃₁	semidirect
218. #554	C₂²×C₃₁	semidirect
220. #500	C₂²×C₃₁	semidirect
223. #492	C₂²×C₃₁	semidirect
224. #486	C₂²×C₃₁	semidirect
228. #476	C₂²×C₃₁	semidirect
231. #460	C₂²×C₃₁	semidirect
235. C₈×Q₈	C₂²×C₃₁	semidirect
240. C₈×D₄	C₂²×C₃₁	semidirect
242. #335	C₂²×C₃₁	semidirect
243. #334	C₂²×C₃₁	semidirect
244. #333	C₂²×C₃₁	semidirect
245. #331	C₂²×C₃₁	semidirect
246. Q₈⋊C₈	C₂²×C₃₁	semidirect
248. D₄⋊C₈	C₂²×C₃₁	semidirect
249. C₈²	C₂²×C₃₁	semidirect
306. C₃₁	#19976	direct (N × H)
330. D₈	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3380	2
C₃₁×#3295	2
D₄×D₈×C₃₁	2
C₃₁×#1706	2
C₃₁×#1677	2
C₃₁×#1675	2
C₃₁×#1668	2
C₃₁×#1629	2
C₃₁×#1622	2
C₈×SD₁₆×C₃₁	2
C₈×D₈×C₃₁	2
#19976	31

Subgroups — C31×#19976 #21476735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19976 #21476735