Subgroups — C₃₁×#33613 #21490372

Order 7936.

662 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#510	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3061	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3059	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2220	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2219	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2217	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2216	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2159	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2138	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#358	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
13	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
15	C₃₁×#537	—	2	1	no	—
16	C₃₁×#536	—	2	1	no	—
17	C₃₁×#510	—	2	4	no	—
18	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
19	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
20	C₃₁×#504	—	2	4	no	—
21	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
22	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#470	—	2	2	no	—
26	C₃₁×#468	—	2	2	no	—
27	C₃₁×#467	C₂²	1	2	yes	—
28	C₃₁×#467	C₂²	1	2	yes	—
29	C₃₁×#467	C₂²	1	2	yes	—
30	C₃₁×#466	C₂²	1	2	yes	—
31	C₃₁×#427	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
33	C₃₁×#419	—	2	1	no	—
34	C₃₁×#417	—	2	1	no	—
35	C₃₁×#367	C₂²	1	2	yes	—
36	C₃₁×#366	C₂²	1	2	yes	—
37	C₃₁×#358	C₂²	1	2	yes	—
38	C₃₁×#358	C₂²	1	2	yes	—
39	C₃₁×#358	C₂²	1	1	yes	characteristic
40	C₃₁×#358	C₂²	1	1	yes	characteristic
41	C₃₁×#357	C₂²	1	1	yes	characteristic
42	C₃₁×#357	C₂²	1	1	yes	characteristic
43	C₂×#41×C₃₁	—	2	1	no	—
44	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
45	C₂²×Q₈×C₃₁	—	2	1	no	—
46	C₃₁×#135	—	2	2	no	—
47	C₃₁×#135	—	2	2	no	—
48	C₂×Q₁₆×C₃₁	—	2	2	no	—
49	C₂×Q₁₆×C₃₁	—	2	1	no	—
50	C₂×Q₁₆×C₃₁	—	2	2	no	—
51	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
52	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×SD₁₆×C₃₁	—	2	1	no	—
54	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₃₁×#131	—	2	2	no	—
56	C₃₁×#131	—	2	2	no	—
57	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
58	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
59	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
60	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×C₄⋊Q₈	—	4	1	no	—
62	C₃₁×#124	—	4	1	no	—
63	C₃₁×#123	—	4	2	no	—
64	C₃₁×C₂²⋊Q₈	—	2	2	no	—
65	C₃₁×C₂²⋊Q₈	—	2	2	no	—
66	C₃₁×C₂²⋊Q₈	—	2	2	no	—
67	C₃₁×C₂²⋊Q₈	—	4	1	no	—
68	C₃₁×#121	—	2	2	no	—
69	C₃₁×#121	—	4	1	no	—
70	C₃₁×#121	—	4	2	no	—
71	C₃₁×#117	—	2	1	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₃₁×Q₃₂	—	2	4	no	—
74	C₃₁×SD₃₂	—	2	4	no	—
75	C₃₁×M₃₂	—	2	4	no	—
76	C₃₁×M₃₂	D₄	1	4	yes	complemented
77	C₂×C₁₆×C₃₁	—	2	2	no	—
78	C₂×C₁₆×C₃₁	—	2	4	no	—
79	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
80	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
82	C₃₁×#107	—	2	1	no	—
83	C₃₁×#107	C₂³	1	2	yes	—
84	C₃₁×#107	C₂³	1	1	yes	characteristic
85	C₃₁×#107	C₂³	1	1	yes	characteristic
86	C₃₁×#106	—	2	1	no	—
87	C₃₁×#103	—	2	2	no	—
88	C₃₁×#103	—	2	2	no	—
89	C₃₁×#103	—	2	2	no	—
90	C₃₁×#102	—	2	2	no	—
91	#41×C₃₁	—	2	2	no	—
92	#41×C₃₁	—	2	2	no	—
93	#41×C₃₁	—	4	2	no	—
94	#41×C₃₁	—	4	2	no	—
95	C₂×Q₈×C₃₁	—	2	1	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	—	4	1	no	—
98	C₂×Q₈×C₃₁	—	4	2	no	—
99	C₂×Q₈×C₃₁	—	4	2	no	—
100	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
101	C₂×D₄×C₃₁	—	2	1	no	—
102	C₂×D₄×C₃₁	—	4	1	no	—
103	C₂×D₄×C₃₁	—	4	2	no	—
104	C₂×D₄×C₃₁	D₈	1	2	yes	—
105	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	—	4	1	no	—
107	C₂²×C₄×C₃₁	—	4	1	no	—
108	C₂²×C₄×C₃₁	—	4	1	no	—
109	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
111	Q₁₆×C₃₁	—	2	2	no	—
112	Q₁₆×C₃₁	—	2	1	no	—
113	Q₁₆×C₃₁	—	2	2	no	—
114	Q₁₆×C₃₁	—	2	1	no	—
115	Q₁₆×C₃₁	—	4	2	no	—
116	SD₁₆×C₃₁	—	4	2	no	—
117	D₈×C₃₁	—	2	1	no	—
118	D₈×C₃₁	—	2	1	no	—
119	M₁₆×C₃₁	—	2	2	no	—
120	M₁₆×C₃₁	—	2	4	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₃₁	#41	1	2	yes	—
127	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	1	no	—
130	C₄⋊C₄×C₃₁	—	2	1	no	—
131	C₄⋊C₄×C₃₁	—	4	1	no	—
132	C₄⋊C₄×C₃₁	—	4	2	no	—
133	C₄⋊C₄×C₃₁	—	4	2	no	—
134	C₄⋊C₄×C₃₁	—	4	2	no	—
135	C₂²⋊C₄×C₃₁	—	4	1	no	—
136	C₂²⋊C₄×C₃₁	—	4	2	no	—
137	C₂²⋊C₄×C₃₁	—	4	2	no	—
138	C₂²⋊C₄×C₃₁	—	4	2	no	—
139	C₂²⋊C₄×C₃₁	—	4	2	no	—
140	C₄²×C₃₁	—	4	1	no	—
141	C₁₆×C₃₁	—	2	1	no	—
142	C₁₆×C₃₁	—	2	1	no	—
143	C₁₆×C₃₁	C₂×D₄	1	4	yes	—
144	#33613	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂³×C₃₁	—	4	1	no	—
146	C₂³×C₃₁	C₂×D₈	1	2	yes	—
147	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
148	Q₈×C₃₁	—	2	2	no	—
149	Q₈×C₃₁	—	4	1	no	—
150	Q₈×C₃₁	—	4	2	no	—
151	Q₈×C₃₁	—	4	2	no	—
152	Q₈×C₃₁	—	4	1	no	—
153	Q₈×C₃₁	—	8	2	no	—
154	D₄×C₃₁	—	2	4	no	—
155	D₄×C₃₁	—	2	2	no	—
156	D₄×C₃₁	—	4	1	no	—
157	D₄×C₃₁	—	4	1	no	—
158	D₄×C₃₁	—	8	2	no	—
159	D₄×C₃₁	—	16	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	4	1	no	—
163	C₂×C₄×C₃₁	—	4	1	no	—
164	C₂×C₄×C₃₁	—	4	2	no	—
165	C₂×C₄×C₃₁	—	4	1	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	8	2	no	—
168	C₂×C₄×C₃₁	—	8	2	no	—
169	C₂×C₄×C₃₁	—	16	1	no	—
170	C₂×C₄×C₃₁	#121	1	2	yes	—
171	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
172	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₃₁	#135	1	2	yes	—
174	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
175	C₈×C₃₁	—	2	2	no	—
176	C₈×C₃₁	#121	1	2	yes	—
177	C₈×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
179	C₂×#510	C₂×C₃₁	1	1	yes	characteristic, complemented
180	C₂×#504	C₂×C₃₁	1	1	yes	characteristic, complemented
181	#3061	C₂×C₃₁	1	1	yes	characteristic
182	#3059	C₂×C₃₁	1	1	yes	characteristic, complemented
183	#2220	C₂×C₃₁	1	2	yes	complemented
184	#2219	C₂×C₃₁	1	2	yes	complemented
185	#2217	C₂×C₃₁	1	2	yes	complemented
186	#2216	C₂×C₃₁	1	2	yes	complemented
187	#2159	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#2138	C₂×C₃₁	1	1	yes	characteristic, complemented
189	C₂×#358	C₂×C₃₁	1	1	yes	characteristic, complemented
190	C₂²×C₃₁	—	2	1	no	—
191	C₂²×C₃₁	—	2	1	no	—
192	C₂²×C₃₁	—	2	2	no	—
193	C₂²×C₃₁	—	2	2	no	—
194	C₂²×C₃₁	—	2	2	no	—
195	C₂²×C₃₁	—	8	1	no	—
196	C₂²×C₃₁	—	8	1	no	—
197	C₂²×C₃₁	—	8	1	no	—
198	C₂²×C₃₁	#466	1	2	yes	—
199	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
200	C₄×C₃₁	—	2	2	no	—
201	C₄×C₃₁	—	8	1	no	—
202	C₄×C₃₁	—	8	2	no	—
203	C₄×C₃₁	—	8	1	no	—
204	C₄×C₃₁	—	8	1	no	—
205	C₄×C₃₁	—	8	2	no	—
206	C₄×C₃₁	#466	1	2	yes	—
207	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
208	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
209	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
210	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
211	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
212	#537	—	2	1	no	—
213	#536	—	2	1	no	—
214	#510	—	2	4	no	—
215	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
216	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
217	#504	—	2	4	no	—
218	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
219	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
220	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
221	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
222	#470	—	2	2	no	—
223	#468	—	2	2	no	—
224	#467	C₂²×C₃₁	1	2	yes	—
225	#467	C₂²×C₃₁	1	2	yes	—
226	#467	C₂²×C₃₁	1	2	yes	—
227	#466	C₂²×C₃₁	1	2	yes	—
228	#427	C₂²×C₃₁	1	1	yes	characteristic
229	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
230	#419	—	2	1	no	—
231	#417	—	2	1	no	—
232	#367	C₂²×C₃₁	1	2	yes	—
233	#366	C₂²×C₃₁	1	2	yes	—
234	#358	C₂²×C₃₁	1	2	yes	—
235	#358	C₂²×C₃₁	1	2	yes	—
236	#358	C₂²×C₃₁	1	1	yes	characteristic
237	#358	C₂²×C₃₁	1	1	yes	characteristic
238	#357	C₂²×C₃₁	1	1	yes	characteristic
239	#357	C₂²×C₃₁	1	1	yes	characteristic
240	C₂×C₃₁	—	2	2	no	—
241	C₂×C₃₁	—	4	2	no	—
242	C₂×C₃₁	—	16	1	no	—
243	C₂×C₃₁	C₂×#466	1	1	yes	characteristic
244	C₂×C₃₁	#3416	1	1	yes	characteristic
245	C₂×C₃₁	#3417	1	1	yes	characteristic
246	C₂×#41	—	2	1	no	—
247	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
248	C₂²×Q₈	—	2	1	no	—
249	#135	—	2	2	no	—
250	#135	—	2	2	no	—
251	C₂×Q₁₆	—	2	2	no	—
252	C₂×Q₁₆	—	2	1	no	—
253	C₂×Q₁₆	—	2	2	no	—
254	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
255	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
256	C₂×SD₁₆	—	2	1	no	—
257	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
258	#131	—	2	2	no	—
259	#131	—	2	2	no	—
260	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
261	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
262	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
263	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
264	C₄⋊Q₈	—	4	1	no	—
265	#124	—	4	1	no	—
266	#123	—	4	2	no	—
267	C₂²⋊Q₈	—	2	2	no	—
268	C₂²⋊Q₈	—	2	2	no	—
269	C₂²⋊Q₈	—	2	2	no	—
270	C₂²⋊Q₈	—	4	1	no	—
271	#121	—	2	2	no	—
272	#121	—	4	1	no	—
273	#121	—	4	2	no	—
274	#117	—	2	1	no	—
275	C₂×C₄⋊C₄	—	2	1	no	—
276	Q₃₂	—	2	4	no	—
277	SD₃₂	—	2	4	no	—
278	M₃₂	—	2	4	no	—
279	M₃₂	D₄×C₃₁	1	4	yes	complemented
280	C₂×C₁₆	—	2	2	no	—
281	C₂×C₁₆	—	2	4	no	—
282	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
283	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
284	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
285	#107	—	2	1	no	—
286	#107	C₂³×C₃₁	1	1	yes	characteristic
287	#107	C₂³×C₃₁	1	1	yes	characteristic
288	#107	C₂³×C₃₁	1	2	yes	—
289	#106	—	2	1	no	—
290	#103	—	2	2	no	—
291	#103	—	2	2	no	—
292	#103	—	2	2	no	—
293	#102	—	2	2	no	—
294	C₃₁	#33613	1	1	yes	characteristic, direct product factor, Sylow
295	#41	—	2	2	no	—
296	#41	—	2	2	no	—
297	#41	—	4	2	no	—
298	#41	—	4	2	no	—
299	C₂×Q₈	—	2	1	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	4	1	no	—
302	C₂×Q₈	—	4	2	no	—
303	C₂×Q₈	—	4	2	no	—
304	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
305	C₂×D₄	—	2	1	no	—
306	C₂×D₄	—	4	1	no	—
307	C₂×D₄	—	4	2	no	—
308	C₂×D₄	D₈×C₃₁	1	2	yes	—
309	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
310	C₂²×C₄	—	4	1	no	—
311	C₂²×C₄	—	4	1	no	—
312	C₂²×C₄	—	4	1	no	—
313	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
314	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
315	Q₁₆	—	2	2	no	—
316	Q₁₆	—	2	1	no	—
317	Q₁₆	—	2	2	no	—
318	Q₁₆	—	2	1	no	—
319	Q₁₆	—	4	2	no	—
320	SD₁₆	—	4	2	no	—
321	D₈	—	2	1	no	—
322	D₈	—	2	1	no	—
323	M₁₆	—	2	2	no	—
324	M₁₆	—	2	4	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
328	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
330	C₂×C₈	#41×C₃₁	1	2	yes	—
331	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	2	1	no	—
335	C₄⋊C₄	—	4	1	no	—
336	C₄⋊C₄	—	4	2	no	—
337	C₄⋊C₄	—	4	2	no	—
338	C₄⋊C₄	—	4	2	no	—
339	C₂²⋊C₄	—	4	1	no	—
340	C₂²⋊C₄	—	4	2	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₂²⋊C₄	—	4	2	no	—
343	C₂²⋊C₄	—	4	2	no	—
344	C₄²	—	4	1	no	—
345	C₁₆	—	2	1	no	—
346	C₁₆	—	2	1	no	—
347	C₁₆	C₂×D₄×C₃₁	1	4	yes	—
348	C₂³	—	4	1	no	—
349	C₂³	C₂×D₈×C₃₁	1	2	yes	—
350	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
351	Q₈	—	2	2	no	—
352	Q₈	—	4	1	no	—
353	Q₈	—	4	2	no	—
354	Q₈	—	4	2	no	—
355	Q₈	—	4	1	no	—
356	Q₈	—	8	2	no	—
357	D₄	—	2	4	no	—
358	D₄	—	2	2	no	—
359	D₄	—	4	1	no	—
360	D₄	—	4	1	no	—
361	D₄	—	8	2	no	—
362	D₄	—	16	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	1	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	1	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	8	2	no	—
371	C₂×C₄	—	8	2	no	—
372	C₂×C₄	—	16	1	no	—
373	C₂×C₄	C₃₁×#121	1	2	yes	—
374	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
375	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
376	C₂×C₄	C₃₁×#135	1	2	yes	—
377	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
378	C₈	—	2	2	no	—
379	C₈	C₃₁×#121	1	2	yes	—
380	C₈	C₂²×D₄×C₃₁	1	1	yes	characteristic
381	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
382	C₂²	—	2	1	no	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	2	no	—
385	C₂²	—	2	2	no	—
386	C₂²	—	2	2	no	—
387	C₂²	—	8	1	no	—
388	C₂²	—	8	1	no	—
389	C₂²	—	8	1	no	—
390	C₂²	C₃₁×#466	1	2	yes	—
391	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
392	C₄	—	2	2	no	—
393	C₄	—	8	2	no	—
394	C₄	—	8	1	no	—
395	C₄	—	8	1	no	—
396	C₄	—	8	1	no	—
397	C₄	—	8	2	no	—
398	C₄	C₃₁×#466	1	2	yes	—
399	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
400	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
401	C₂	—	2	2	no	—
402	C₂	—	4	2	no	—
403	C₂	—	16	1	no	—
404	C₂	C₂×C₃₁×#466	1	1	yes	characteristic
405	C₂	C₃₁×#3416	1	1	yes	characteristic
406	C₂	C₃₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#3061	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×#41×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#510	C₂	semidirect
2. C₂×C₃₁×#504	C₂	semidirect
4. C₃₁×#3059	C₂	semidirect
5. C₃₁×#2220	C₂	semidirect
6. C₃₁×#2219	C₂	semidirect
7. C₃₁×#2217	C₂	semidirect
8. C₃₁×#2216	C₂	semidirect
9. C₃₁×#2159	C₂	semidirect
10. C₃₁×#2138	C₂	semidirect
11. C₂×C₃₁×#358	C₂	semidirect
76. C₃₁×M₃₂	D₄	semidirect
144. #33613	C₃₁	direct (N × H)
179. C₂×#510	C₂×C₃₁	semidirect
180. C₂×#504	C₂×C₃₁	semidirect
182. #3059	C₂×C₃₁	semidirect
183. #2220	C₂×C₃₁	semidirect
184. #2219	C₂×C₃₁	semidirect
185. #2217	C₂×C₃₁	semidirect
186. #2216	C₂×C₃₁	semidirect
187. #2159	C₂×C₃₁	semidirect
188. #2138	C₂×C₃₁	semidirect
189. C₂×#358	C₂×C₃₁	semidirect
279. M₃₂	D₄×C₃₁	semidirect
294. C₃₁	#33613	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#510	2
C₂×C₃₁×#504	2
C₃₁×#3061	2
C₃₁×#3059	2
C₃₁×#2220	2
C₃₁×#2219	2
C₃₁×#2217	2
C₃₁×#2216	2
C₃₁×#2159	2
C₃₁×#2138	2
C₂×C₃₁×#358	2
#33613	31

Subgroups — C31×#33613 #21490372

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33613 #21490372