Subgroups — C₃₁×#33846 #21490605

Order 7936.

718 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3280	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2249	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2216	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2215	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2210	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2187	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2186	C₂	1	2	yes	maximal, complemented
10	C₄×C₃₁×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	C₂²	1	2	yes	complemented
13	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#546	—	2	2	no	—
15	C₃₁×#508	—	2	4	no	—
16	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic, complemented
17	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	complemented
18	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	complemented
19	C₂×C₃₁×D₁₆	—	2	4	no	—
20	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
21	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×D₁₆	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
25	C₃₁×#495	C₂²	1	2	yes	complemented
26	C₃₁×#467	C₂²	1	2	yes	—
27	C₃₁×#466	C₂²	1	2	yes	complemented
28	C₃₁×#463	—	2	2	no	—
29	C₃₁×#450	—	2	2	no	—
30	C₃₁×#449	—	2	2	no	—
31	C₄×D₈×C₃₁	C₂²	1	2	yes	complemented
32	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#366	C₂²	1	1	yes	characteristic, complemented
34	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
35	C₃₁×#358	C₂²	1	2	yes	complemented
36	C₃₁×#357	C₂²	1	2	yes	complemented
37	C₃₁×#357	C₂²	1	2	yes	complemented
38	C₃₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
39	C₃₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
40	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
41	C₂×#41×C₃₁	—	2	2	no	—
42	C₂²×D₄×C₃₁	—	2	2	no	—
43	C₃₁×#135	—	2	4	no	—
44	C₃₁×#135	—	4	2	no	—
45	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
46	C₂×SD₁₆×C₃₁	—	2	2	no	—
47	C₂×SD₁₆×C₃₁	—	2	2	no	—
48	C₂×D₈×C₃₁	—	2	4	no	—
49	C₂×D₈×C₃₁	—	2	2	no	—
50	C₂×D₈×C₃₁	—	4	2	no	—
51	C₂×D₈×C₃₁	C₂³	1	2	yes	—
52	C₂×D₈×C₃₁	C₂³	1	2	yes	—
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×#124	—	2	2	no	—
56	C₃₁×#123	—	4	2	no	—
57	C₃₁×C₂²⋊Q₈	—	2	2	no	—
58	C₃₁×#121	—	2	2	no	—
59	C₃₁×#121	—	4	2	no	—
60	C₃₁×#120	—	4	2	no	—
61	C₄×D₄×C₃₁	—	2	2	no	—
62	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
63	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
64	C₃₁×Q₃₂	—	2	2	no	—
65	C₃₁×SD₃₂	—	2	4	no	—
66	C₃₁×SD₃₂	—	4	2	no	—
67	C₃₁×D₁₆	—	2	4	no	—
68	C₃₁×D₁₆	—	2	2	no	—
69	C₃₁×D₁₆	D₄	1	4	yes	complemented
70	C₂×C₁₆×C₃₁	—	2	2	no	—
71	C₂×C₁₆×C₃₁	—	2	2	no	—
72	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#107	C₂³	1	2	yes	—
77	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
78	C₃₁×#103	—	2	2	no	—
79	C₃₁×#102	—	2	2	no	—
80	C₃₁×#102	—	2	2	no	—
81	C₃₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
82	C₃₁×C₂²⋊C₈	D₄	1	1	yes	characteristic
83	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
84	C₂⁴×C₃₁	—	4	2	no	—
85	#41×C₃₁	—	4	4	no	—
86	#41×C₃₁	—	8	2	no	—
87	C₂×Q₈×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	—	2	2	no	—
90	C₂×D₄×C₃₁	—	4	4	no	—
91	C₂×D₄×C₃₁	—	4	2	no	—
92	C₂×D₄×C₃₁	—	4	2	no	—
93	C₂×D₄×C₃₁	—	4	2	no	—
94	C₂×D₄×C₃₁	—	8	2	no	—
95	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	—	4	2	no	—
97	C₂²×C₄×C₃₁	—	4	2	no	—
98	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	Q₁₆×C₃₁	—	2	2	no	—
101	Q₁₆×C₃₁	—	2	2	no	—
102	SD₁₆×C₃₁	—	4	4	no	—
103	SD₁₆×C₃₁	—	8	2	no	—
104	D₈×C₃₁	—	2	2	no	—
105	D₈×C₃₁	—	2	2	no	—
106	D₈×C₃₁	—	2	2	no	—
107	D₈×C₃₁	—	4	4	no	—
108	D₈×C₃₁	C₂×D₄	1	4	yes	—
109	C₂×C₈×C₃₁	—	2	1	no	—
110	C₂×C₈×C₃₁	—	2	1	no	—
111	C₂×C₈×C₃₁	—	2	1	no	—
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₄⋊C₄×C₃₁	—	4	2	no	—
119	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
120	C₂²⋊C₄×C₃₁	—	4	2	no	—
121	C₂²⋊C₄×C₃₁	—	4	2	no	—
122	C₂²⋊C₄×C₃₁	—	4	2	no	—
123	C₂²⋊C₄×C₃₁	—	4	2	no	—
124	C₂²⋊C₄×C₃₁	—	4	2	no	—
125	C₂²⋊C₄×C₃₁	D₈	1	1	yes	characteristic
126	C₂²⋊C₄×C₃₁	D₈	1	1	yes	characteristic
127	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₁₆×C₃₁	—	2	1	no	—
129	C₁₆×C₃₁	—	2	1	no	—
130	C₁₆×C₃₁	—	2	1	no	—
131	C₁₆×C₃₁	C₂×D₄	1	2	yes	—
132	#33846	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₂³×C₃₁	—	4	2	no	—
134	C₂³×C₃₁	—	4	2	no	—
135	C₂³×C₃₁	—	8	4	no	—
136	C₂³×C₃₁	—	8	4	no	—
137	C₂³×C₃₁	—	16	2	no	—
138	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
139	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
140	Q₈×C₃₁	—	4	2	no	—
141	Q₈×C₃₁	—	4	2	no	—
142	D₄×C₃₁	—	2	2	no	—
143	D₄×C₃₁	—	2	4	no	—
144	D₄×C₃₁	—	4	2	no	—
145	D₄×C₃₁	—	4	2	no	—
146	D₄×C₃₁	—	4	2	no	—
147	D₄×C₃₁	—	8	4	no	—
148	D₄×C₃₁	—	8	4	no	—
149	D₄×C₃₁	—	16	2	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	4	2	no	—
155	C₂×C₄×C₃₁	—	4	2	no	—
156	C₂×C₄×C₃₁	—	8	4	no	—
157	C₂×C₄×C₃₁	—	16	2	no	—
158	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
163	C₈×C₃₁	—	2	1	no	—
164	C₈×C₃₁	—	2	1	no	—
165	C₈×C₃₁	—	2	1	no	—
166	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
167	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
168	C₂×#508	C₂×C₃₁	1	1	yes	characteristic, complemented
169	C₂²×D₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
170	#3280	C₂×C₃₁	1	2	yes	complemented
171	#2249	C₂×C₃₁	1	1	yes	characteristic, complemented
172	#2216	C₂×C₃₁	1	1	yes	characteristic, complemented
173	#2215	C₂×C₃₁	1	1	yes	characteristic, complemented
174	#2210	C₂×C₃₁	1	2	yes	complemented
175	#2187	C₂×C₃₁	1	2	yes	complemented
176	#2186	C₂×C₃₁	1	2	yes	complemented
177	C₄×D₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
178	D₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
179	C₂²×C₃₁	—	2	1	no	—
180	C₂²×C₃₁	—	2	1	no	—
181	C₂²×C₃₁	—	2	1	no	—
182	C₂²×C₃₁	—	2	1	no	—
183	C₂²×C₃₁	—	2	1	no	—
184	C₂²×C₃₁	—	4	4	no	—
185	C₂²×C₃₁	—	8	2	no	—
186	C₂²×C₃₁	—	8	2	no	—
187	C₂²×C₃₁	—	8	2	no	—
188	C₂²×C₃₁	—	8	2	no	—
189	C₂²×C₃₁	—	8	2	no	—
190	C₂²×C₃₁	—	16	8	no	—
191	C₂²×C₃₁	#508	1	2	yes	complemented
192	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
193	C₄×C₃₁	—	2	1	no	—
194	C₄×C₃₁	—	2	1	no	—
195	C₄×C₃₁	—	2	1	no	—
196	C₄×C₃₁	—	2	1	no	—
197	C₄×C₃₁	—	8	2	no	—
198	C₄×C₃₁	—	8	2	no	—
199	C₄×C₃₁	D₄²	1	1	yes	characteristic
200	C₄×C₃₁	#576	1	1	yes	characteristic
201	C₂×#135	C₂²×C₃₁	1	2	yes	complemented
202	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
203	#546	—	2	2	no	—
204	#508	—	2	4	no	—
205	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic, complemented
206	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	complemented
207	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	complemented
208	C₂×D₁₆	—	2	4	no	—
209	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
210	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
211	C₂×D₁₆	C₂²×C₃₁	1	1	yes	characteristic
212	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
213	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
214	#495	C₂²×C₃₁	1	2	yes	complemented
215	#467	C₂²×C₃₁	1	2	yes	—
216	#466	C₂²×C₃₁	1	2	yes	complemented
217	#463	—	2	2	no	—
218	#450	—	2	2	no	—
219	#449	—	2	2	no	—
220	C₄×D₈	C₂²×C₃₁	1	2	yes	complemented
221	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
222	#366	C₂²×C₃₁	1	1	yes	characteristic, complemented
223	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
224	#358	C₂²×C₃₁	1	2	yes	complemented
225	#357	C₂²×C₃₁	1	2	yes	complemented
226	#357	C₂²×C₃₁	1	2	yes	complemented
227	C₂²⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
228	C₂²⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
229	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
230	C₂×C₃₁	—	2	2	no	—
231	C₂×C₃₁	—	4	1	no	—
232	C₂×C₃₁	—	8	4	no	—
233	C₂×C₃₁	—	16	2	no	—
234	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
235	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
236	C₂×C₃₁	#3415	1	1	yes	characteristic
237	C₂×#41	—	2	2	no	—
238	C₂²×D₄	—	2	2	no	—
239	#135	—	2	4	no	—
240	#135	—	4	2	no	—
241	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
242	C₂×SD₁₆	—	2	2	no	—
243	C₂×SD₁₆	—	2	2	no	—
244	C₂×D₈	—	2	4	no	—
245	C₂×D₈	—	2	2	no	—
246	C₂×D₈	—	4	2	no	—
247	C₂×D₈	C₂³×C₃₁	1	2	yes	—
248	C₂×D₈	C₂³×C₃₁	1	2	yes	—
249	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
250	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
251	#124	—	2	2	no	—
252	#123	—	4	2	no	—
253	C₂²⋊Q₈	—	2	2	no	—
254	#121	—	2	2	no	—
255	#121	—	4	2	no	—
256	#120	—	4	2	no	—
257	C₄×D₄	—	2	2	no	—
258	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
259	C₂×C₂²⋊C₄	—	4	2	no	—
260	Q₃₂	—	2	2	no	—
261	SD₃₂	—	2	4	no	—
262	SD₃₂	—	4	2	no	—
263	D₁₆	—	2	4	no	—
264	D₁₆	—	2	2	no	—
265	D₁₆	D₄×C₃₁	1	4	yes	complemented
266	C₂×C₁₆	—	2	2	no	—
267	C₂×C₁₆	—	2	2	no	—
268	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
269	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
270	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
271	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
272	#107	C₂³×C₃₁	1	2	yes	—
273	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
274	#103	—	2	2	no	—
275	#102	—	2	2	no	—
276	#102	—	2	2	no	—
277	C₂²⋊C₈	D₄×C₃₁	1	1	yes	characteristic
278	C₂²⋊C₈	D₄×C₃₁	1	1	yes	characteristic
279	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
280	C₃₁	#33846	1	1	yes	characteristic, direct product factor, Sylow
281	C₂⁴	—	4	2	no	—
282	#41	—	4	4	no	—
283	#41	—	8	2	no	—
284	C₂×Q₈	—	2	2	no	—
285	C₂×D₄	—	2	2	no	—
286	C₂×D₄	—	2	2	no	—
287	C₂×D₄	—	4	4	no	—
288	C₂×D₄	—	4	2	no	—
289	C₂×D₄	—	4	2	no	—
290	C₂×D₄	—	4	2	no	—
291	C₂×D₄	—	8	2	no	—
292	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
293	C₂²×C₄	—	4	2	no	—
294	C₂²×C₄	—	4	2	no	—
295	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
297	Q₁₆	—	2	2	no	—
298	Q₁₆	—	2	2	no	—
299	SD₁₆	—	4	4	no	—
300	SD₁₆	—	8	2	no	—
301	D₈	—	2	2	no	—
302	D₈	—	2	2	no	—
303	D₈	—	2	2	no	—
304	D₈	—	4	4	no	—
305	D₈	C₂×D₄×C₃₁	1	4	yes	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	2	1	no	—
310	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
311	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
312	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
313	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	4	2	no	—
316	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
317	C₂²⋊C₄	—	4	2	no	—
318	C₂²⋊C₄	—	4	2	no	—
319	C₂²⋊C₄	—	4	2	no	—
320	C₂²⋊C₄	—	4	2	no	—
321	C₂²⋊C₄	—	4	2	no	—
322	C₂²⋊C₄	D₈×C₃₁	1	1	yes	characteristic
323	C₂²⋊C₄	D₈×C₃₁	1	1	yes	characteristic
324	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
325	C₁₆	—	2	1	no	—
326	C₁₆	—	2	1	no	—
327	C₁₆	—	2	1	no	—
328	C₁₆	C₂×D₄×C₃₁	1	2	yes	—
329	C₂³	—	4	2	no	—
330	C₂³	—	4	2	no	—
331	C₂³	—	8	4	no	—
332	C₂³	—	8	4	no	—
333	C₂³	—	16	2	no	—
334	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
335	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
336	Q₈	—	4	2	no	—
337	Q₈	—	4	2	no	—
338	D₄	—	2	2	no	—
339	D₄	—	2	4	no	—
340	D₄	—	4	2	no	—
341	D₄	—	4	2	no	—
342	D₄	—	4	2	no	—
343	D₄	—	8	4	no	—
344	D₄	—	8	4	no	—
345	D₄	—	16	2	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	2	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	1	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	—	8	4	no	—
353	C₂×C₄	—	16	2	no	—
354	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
355	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
356	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
357	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
358	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
363	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	2	1	no	—
368	C₂²	—	2	1	no	—
369	C₂²	—	4	4	no	—
370	C₂²	—	8	2	no	—
371	C₂²	—	8	2	no	—
372	C₂²	—	8	2	no	—
373	C₂²	—	8	2	no	—
374	C₂²	—	8	2	no	—
375	C₂²	—	16	8	no	—
376	C₂²	C₃₁×#508	1	2	yes	complemented
377	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	2	1	no	—
382	C₄	—	8	2	no	—
383	C₄	—	8	2	no	—
384	C₄	D₄²×C₃₁	1	1	yes	characteristic
385	C₄	C₃₁×#576	1	1	yes	characteristic
386	C₂	—	2	2	no	—
387	C₂	—	4	1	no	—
388	C₂	—	8	4	no	—
389	C₂	—	16	2	no	—
390	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
391	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
392	C₂	C₃₁×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×D₄×C₃₁ ◁ C₈×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#508	C₂	semidirect
2. C₂²×C₃₁×D₁₆	C₂	semidirect
3. C₃₁×#3280	C₂	semidirect
4. C₃₁×#2249	C₂	semidirect
5. C₃₁×#2216	C₂	semidirect
6. C₃₁×#2215	C₂	semidirect
7. C₃₁×#2210	C₂	semidirect
8. C₃₁×#2187	C₂	semidirect
9. C₃₁×#2186	C₂	semidirect
10. C₄×C₃₁×D₁₆	C₂	semidirect
11. D₄×C₁₆×C₃₁	C₂	semidirect
12. C₂×C₃₁×#135	C₂²	semidirect
16. C₂×C₃₁×Q₃₂	C₂²	semidirect
17. C₂×C₃₁×SD₃₂	C₂²	semidirect
18. C₂×C₃₁×SD₃₂	C₂²	semidirect
24. C₂²×C₁₆×C₃₁	C₂²	semidirect
25. C₃₁×#495	C₂²	semidirect
27. C₃₁×#466	C₂²	semidirect
31. C₄×D₈×C₃₁	C₂²	semidirect
33. C₃₁×#366	C₂²	semidirect
34. C₃₁×C₄⋊C₁₆	C₂²	semidirect
35. C₃₁×#358	C₂²	semidirect
36. C₃₁×#357	C₂²	semidirect
37. C₃₁×#357	C₂²	semidirect
39. C₃₁×C₂²⋊C₁₆	C₂²	semidirect
40. C₄×C₁₆×C₃₁	C₂²	semidirect
69. C₃₁×D₁₆	D₄	semidirect
132. #33846	C₃₁	direct (N × H)
168. C₂×#508	C₂×C₃₁	semidirect
169. C₂²×D₁₆	C₂×C₃₁	semidirect
170. #3280	C₂×C₃₁	semidirect
171. #2249	C₂×C₃₁	semidirect
172. #2216	C₂×C₃₁	semidirect
173. #2215	C₂×C₃₁	semidirect
174. #2210	C₂×C₃₁	semidirect
175. #2187	C₂×C₃₁	semidirect
176. #2186	C₂×C₃₁	semidirect
177. C₄×D₁₆	C₂×C₃₁	semidirect
178. D₄×C₁₆	C₂×C₃₁	semidirect
191. C₂²×C₃₁	#508	semidirect
201. C₂×#135	C₂²×C₃₁	semidirect
205. C₂×Q₃₂	C₂²×C₃₁	semidirect
206. C₂×SD₃₂	C₂²×C₃₁	semidirect
207. C₂×SD₃₂	C₂²×C₃₁	semidirect
213. C₂²×C₁₆	C₂²×C₃₁	semidirect
214. #495	C₂²×C₃₁	semidirect
216. #466	C₂²×C₃₁	semidirect
220. C₄×D₈	C₂²×C₃₁	semidirect
222. #366	C₂²×C₃₁	semidirect
223. C₄⋊C₁₆	C₂²×C₃₁	semidirect
224. #358	C₂²×C₃₁	semidirect
225. #357	C₂²×C₃₁	semidirect
226. #357	C₂²×C₃₁	semidirect
228. C₂²⋊C₁₆	C₂²×C₃₁	semidirect
229. C₄×C₁₆	C₂²×C₃₁	semidirect
265. D₁₆	D₄×C₃₁	semidirect
280. C₃₁	#33846	direct (N × H)
376. C₂²	C₃₁×#508	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#508	2
C₂²×C₃₁×D₁₆	2
C₃₁×#3280	2
C₃₁×#2249	2
C₃₁×#2216	2
C₃₁×#2215	2
C₃₁×#2210	2
C₃₁×#2187	2
C₃₁×#2186	2
C₄×C₃₁×D₁₆	2
D₄×C₁₆×C₃₁	2
#33846	31

Subgroups — C31×#33846 #21490605

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33846 #21490605