Subgroups — C₃₁×#19384 #21476143

Order 7936.

738 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	C₃₁×#3208	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3077	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#459	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2972	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1677	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1665	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1570	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1542	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₃₁	—	2	1	no	—
11	C₃₁×#533	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#492	C₂²	1	2	yes	complemented
15	C₃₁×#486	C₂²	1	2	yes	complemented
16	C₃₁×#478	—	2	2	no	—
17	C₃₁×#476	C₂²	1	2	yes	complemented
18	C₃₁×#474	C₂²	1	2	yes	complemented
19	C₃₁×#460	—	2	2	no	—
20	C₃₁×#459	—	2	2	no	—
21	C₃₁×#459	C₂²	1	4	yes	—
22	C₃₁×#450	—	2	2	no	—
23	C₈×D₄×C₃₁	—	2	2	no	—
24	C₈×D₄×C₃₁	C₂²	1	4	yes	complemented
25	C₃₁×#431	C₂²	1	1	yes	characteristic
26	C₃₁×#431	C₂²	1	1	yes	characteristic
27	C₃₁×#428	—	2	1	no	—
28	C₃₁×#423	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₃₁×#418	—	2	1	no	—
31	C₂×C₃₁×#102	—	2	1	no	—
32	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
33	C₃₁×#334	C₂²	1	4	yes	complemented
34	C₃₁×#333	C₂²	1	2	yes	complemented
35	C₃₁×#332	C₂²	1	2	yes	complemented
36	C₃₁×#331	C₂²	1	4	yes	complemented
37	C₂²×D₄×C₃₁	—	2	1	no	—
38	C₂²×D₄×C₃₁	—	2	1	no	—
39	C₂²×D₄×C₃₁	—	2	1	no	—
40	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₃₁	—	2	1	no	—
42	C₂×SD₁₆×C₃₁	—	2	4	no	—
43	C₂×D₈×C₃₁	—	2	4	no	—
44	C₂×D₈×C₃₁	—	4	4	no	—
45	C₂×M₁₆×C₃₁	—	2	1	no	—
46	C₂²×C₈×C₃₁	—	2	1	no	—
47	C₂²×C₈×C₃₁	—	2	4	no	—
48	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
49	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
50	C₃₁×#127	—	2	1	no	—
51	C₃₁×#127	—	2	1	no	—
52	C₃₁×#127	—	2	1	no	—
53	C₃₁×#127	C₂³	1	2	yes	—
54	C₃₁×#125	—	2	1	no	—
55	C₃₁×C₂²⋊Q₈	—	2	1	no	—
56	C₃₁×C₂²⋊Q₈	—	2	1	no	—
57	C₄×Q₈×C₃₁	—	2	1	no	—
58	C₄×Q₈×C₃₁	—	2	1	no	—
59	C₄×D₄×C₃₁	—	2	1	no	—
60	C₄×D₄×C₃₁	—	2	1	no	—
61	C₄×D₄×C₃₁	D₄	1	2	yes	—
62	C₄×D₄×C₃₁	C₂³	1	2	yes	—
63	C₃₁×#117	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#107	—	2	2	no	—
68	C₃₁×#106	—	2	2	no	—
69	C₃₁×C₄⋊C₈	—	2	1	no	—
70	C₃₁×C₄⋊C₈	—	2	1	no	—
71	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
72	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
73	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
74	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
75	C₃₁×#102	—	2	2	no	—
76	C₃₁×#102	—	2	2	no	—
77	C₃₁×#102	—	2	4	no	—
78	C₃₁×#102	—	2	4	no	—
79	C₃₁×C₂²⋊C₈	—	2	1	no	—
80	C₃₁×C₂²⋊C₈	—	2	4	no	—
81	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
82	C₄×C₈×C₃₁	—	2	1	no	—
83	C₄×C₈×C₃₁	C₂³	1	2	yes	—
84	C₂⁴×C₃₁	—	2	1	no	—
85	C₂⁴×C₃₁	—	4	1	no	—
86	C₂×Q₈×C₃₁	—	2	1	no	—
87	C₂×Q₈×C₃₁	—	2	1	no	—
88	C₂×D₄×C₃₁	—	2	1	no	—
89	C₂×D₄×C₃₁	—	2	2	no	—
90	C₂×D₄×C₃₁	—	2	2	no	—
91	C₂×D₄×C₃₁	—	2	2	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	4	1	no	—
94	C₂×D₄×C₃₁	—	4	1	no	—
95	C₂×D₄×C₃₁	—	4	1	no	—
96	C₂×D₄×C₃₁	—	4	2	no	—
97	C₂×D₄×C₃₁	—	4	2	no	—
98	C₂×D₄×C₃₁	—	8	2	no	—
99	C₂×D₄×C₃₁	—	8	2	no	—
100	C₂×D₄×C₃₁	SD₁₆	1	4	yes	complemented
101	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	8	2	no	—
106	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
109	SD₁₆×C₃₁	—	8	4	no	—
110	D₈×C₃₁	—	4	8	no	—
111	M₁₆×C₃₁	—	4	2	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	4	2	no	—
116	C₂×C₈×C₃₁	—	8	4	no	—
117	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
118	C₄⋊C₄×C₃₁	—	2	1	no	—
119	C₄⋊C₄×C₃₁	—	2	1	no	—
120	C₄⋊C₄×C₃₁	—	2	1	no	—
121	C₄⋊C₄×C₃₁	—	2	1	no	—
122	C₄⋊C₄×C₃₁	—	2	1	no	—
123	C₄⋊C₄×C₃₁	—	4	1	no	—
124	C₄⋊C₄×C₃₁	—	4	1	no	—
125	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
126	C₂²⋊C₄×C₃₁	—	2	2	no	—
127	C₂²⋊C₄×C₃₁	—	2	2	no	—
128	C₂²⋊C₄×C₃₁	—	4	1	no	—
129	C₄²×C₃₁	—	4	1	no	—
130	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₄²×C₃₁	#41	1	1	yes	characteristic
133	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
134	#19384	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
135	C₂³×C₃₁	—	2	2	no	—
136	C₂³×C₃₁	—	2	4	no	—
137	C₂³×C₃₁	—	4	2	no	—
138	C₂³×C₃₁	—	8	1	no	—
139	C₂³×C₃₁	—	8	1	no	—
140	C₂³×C₃₁	—	8	2	no	—
141	C₂³×C₃₁	—	8	2	no	—
142	C₂³×C₃₁	—	8	2	no	—
143	C₂³×C₃₁	#121	1	1	yes	characteristic
144	Q₈×C₃₁	—	4	2	no	—
145	Q₈×C₃₁	—	8	1	no	—
146	D₄×C₃₁	—	2	2	no	—
147	D₄×C₃₁	—	2	4	no	—
148	D₄×C₃₁	—	4	2	no	—
149	D₄×C₃₁	—	4	2	no	—
150	D₄×C₃₁	—	4	4	no	—
151	D₄×C₃₁	—	8	2	no	—
152	D₄×C₃₁	—	8	2	no	—
153	D₄×C₃₁	—	8	2	no	—
154	D₄×C₃₁	—	8	2	no	—
155	D₄×C₃₁	C₂×SD₁₆	1	4	yes	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	4	1	no	—
159	C₂×C₄×C₃₁	—	4	4	no	—
160	C₂×C₄×C₃₁	—	4	1	no	—
161	C₂×C₄×C₃₁	—	4	1	no	—
162	C₂×C₄×C₃₁	—	8	2	no	—
163	C₂×C₄×C₃₁	—	8	1	no	—
164	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
165	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
166	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
167	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
168	C₂×C₄×C₃₁	#135	1	2	yes	—
169	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
170	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
171	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
172	C₈×C₃₁	—	2	4	no	—
173	C₈×C₃₁	—	4	2	no	—
174	C₈×C₃₁	—	4	2	no	—
175	C₈×C₃₁	—	4	2	no	—
176	#3208	C₂×C₃₁	1	1	yes	characteristic, complemented
177	#3077	C₂×C₃₁	1	1	yes	characteristic, complemented
178	C₂×#459	C₂×C₃₁	1	1	yes	characteristic
179	#2972	C₂×C₃₁	1	1	yes	characteristic, complemented
180	#1677	C₂×C₃₁	1	4	yes	complemented
181	#1665	C₂×C₃₁	1	4	yes	complemented
182	#1570	C₂×C₃₁	1	1	yes	characteristic, complemented
183	#1545	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#1542	C₂×C₃₁	1	1	yes	characteristic, complemented
185	C₂²×C₃₁	—	2	1	no	—
186	C₂²×C₃₁	—	2	1	no	—
187	C₂²×C₃₁	—	2	4	no	—
188	C₂²×C₃₁	—	4	2	no	—
189	C₂²×C₃₁	—	4	2	no	—
190	C₂²×C₃₁	—	8	4	no	—
191	C₂²×C₃₁	—	8	2	no	—
192	C₂²×C₃₁	—	8	2	no	—
193	C₂²×C₃₁	—	8	2	no	—
194	C₂²×C₃₁	—	16	4	no	—
195	C₂²×C₃₁	#465	1	2	yes	—
196	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
197	C₄×C₃₁	—	2	1	no	—
198	C₄×C₃₁	—	2	1	no	—
199	C₄×C₃₁	—	2	1	no	—
200	C₄×C₃₁	—	4	2	no	—
201	C₄×C₃₁	—	8	1	no	—
202	C₄×C₃₁	—	8	1	no	—
203	C₄×C₃₁	#465	1	2	yes	—
204	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
205	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
206	C₄×C₃₁	#576	1	2	yes	—
207	C₂²×D₈	—	2	1	no	—
208	#533	C₂²×C₃₁	1	1	yes	characteristic
209	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
210	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
211	#492	C₂²×C₃₁	1	2	yes	complemented
212	#486	C₂²×C₃₁	1	2	yes	complemented
213	#478	—	2	2	no	—
214	#476	C₂²×C₃₁	1	2	yes	complemented
215	#474	C₂²×C₃₁	1	2	yes	complemented
216	#460	—	2	2	no	—
217	#459	—	2	2	no	—
218	#459	C₂²×C₃₁	1	4	yes	—
219	#450	—	2	2	no	—
220	C₈×D₄	—	2	2	no	—
221	C₈×D₄	C₂²×C₃₁	1	4	yes	complemented
222	#431	C₂²×C₃₁	1	1	yes	characteristic
223	#431	C₂²×C₃₁	1	1	yes	characteristic
224	#428	—	2	1	no	—
225	#423	C₂²×C₃₁	1	1	yes	characteristic
226	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
227	#418	—	2	1	no	—
228	C₂×#102	—	2	1	no	—
229	C₂×C₂²⋊C₈	—	2	2	no	—
230	#334	C₂²×C₃₁	1	4	yes	complemented
231	#333	C₂²×C₃₁	1	2	yes	complemented
232	#332	C₂²×C₃₁	1	2	yes	complemented
233	#331	C₂²×C₃₁	1	4	yes	complemented
234	C₂×C₃₁	—	2	2	no	—
235	C₂×C₃₁	—	4	4	no	—
236	C₂×C₃₁	—	16	2	no	—
237	C₂×C₃₁	#3041	1	1	yes	characteristic
238	C₂×C₃₁	C₂×#465	1	1	yes	characteristic
239	C₂×C₃₁	#3068	1	1	yes	characteristic
240	C₂²×D₄	—	2	1	no	—
241	C₂²×D₄	—	2	1	no	—
242	C₂²×D₄	—	2	1	no	—
243	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
244	C₂³×C₄	—	2	1	no	—
245	C₂×SD₁₆	—	2	4	no	—
246	C₂×D₈	—	2	4	no	—
247	C₂×D₈	—	4	4	no	—
248	C₂×M₁₆	—	2	1	no	—
249	C₂²×C₈	—	2	1	no	—
250	C₂²×C₈	—	2	4	no	—
251	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
252	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
253	#127	—	2	1	no	—
254	#127	—	2	1	no	—
255	#127	—	2	1	no	—
256	#127	C₂³×C₃₁	1	2	yes	—
257	#125	—	2	1	no	—
258	C₂²⋊Q₈	—	2	1	no	—
259	C₂²⋊Q₈	—	2	1	no	—
260	C₄×Q₈	—	2	1	no	—
261	C₄×Q₈	—	2	1	no	—
262	C₄×D₄	—	2	1	no	—
263	C₄×D₄	—	2	1	no	—
264	C₄×D₄	D₄×C₃₁	1	2	yes	—
265	C₄×D₄	C₂³×C₃₁	1	2	yes	—
266	#117	—	2	1	no	—
267	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
268	C₂×C₂²⋊C₄	—	2	1	no	—
269	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
270	#107	—	2	2	no	—
271	#106	—	2	2	no	—
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	—	2	1	no	—
274	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
275	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
276	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
277	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
278	#102	—	2	2	no	—
279	#102	—	2	2	no	—
280	#102	—	2	4	no	—
281	#102	—	2	4	no	—
282	C₂²⋊C₈	—	2	1	no	—
283	C₂²⋊C₈	—	2	4	no	—
284	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	C₂³×C₃₁	1	2	yes	—
287	C₃₁	#19384	1	1	yes	characteristic, direct product factor, Sylow
288	C₂⁴	—	2	1	no	—
289	C₂⁴	—	4	1	no	—
290	C₂×Q₈	—	2	1	no	—
291	C₂×Q₈	—	2	1	no	—
292	C₂×D₄	—	2	1	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	2	2	no	—
295	C₂×D₄	—	2	2	no	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	—	4	1	no	—
298	C₂×D₄	—	4	1	no	—
299	C₂×D₄	—	4	1	no	—
300	C₂×D₄	—	4	2	no	—
301	C₂×D₄	—	4	2	no	—
302	C₂×D₄	—	8	2	no	—
303	C₂×D₄	—	8	2	no	—
304	C₂×D₄	SD₁₆×C₃₁	1	4	yes	complemented
305	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	2	2	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	8	2	no	—
310	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
311	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
312	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
313	SD₁₆	—	8	4	no	—
314	D₈	—	4	8	no	—
315	M₁₆	—	4	2	no	—
316	C₂×C₈	—	2	2	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	4	2	no	—
320	C₂×C₈	—	8	4	no	—
321	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	2	1	no	—
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	1	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	—	4	1	no	—
329	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
330	C₂²⋊C₄	—	2	2	no	—
331	C₂²⋊C₄	—	2	2	no	—
332	C₂²⋊C₄	—	4	1	no	—
333	C₄²	—	4	1	no	—
334	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
335	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₄²	#41×C₃₁	1	1	yes	characteristic
337	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
338	C₂³	—	2	2	no	—
339	C₂³	—	2	4	no	—
340	C₂³	—	4	2	no	—
341	C₂³	—	8	1	no	—
342	C₂³	—	8	1	no	—
343	C₂³	—	8	2	no	—
344	C₂³	—	8	2	no	—
345	C₂³	—	8	2	no	—
346	C₂³	C₃₁×#121	1	1	yes	characteristic
347	Q₈	—	4	2	no	—
348	Q₈	—	8	1	no	—
349	D₄	—	2	4	no	—
350	D₄	—	2	2	no	—
351	D₄	—	4	2	no	—
352	D₄	—	4	2	no	—
353	D₄	—	4	4	no	—
354	D₄	—	8	2	no	—
355	D₄	—	8	2	no	—
356	D₄	—	8	2	no	—
357	D₄	—	8	2	no	—
358	D₄	C₂×SD₁₆×C₃₁	1	4	yes	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	4	4	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	8	2	no	—
366	C₂×C₄	—	8	1	no	—
367	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
368	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
369	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
370	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
371	C₂×C₄	C₃₁×#135	1	2	yes	—
372	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
373	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
374	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
375	C₈	—	2	4	no	—
376	C₈	—	4	2	no	—
377	C₈	—	4	2	no	—
378	C₈	—	4	2	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	4	no	—
382	C₂²	—	4	2	no	—
383	C₂²	—	4	2	no	—
384	C₂²	—	8	4	no	—
385	C₂²	—	8	2	no	—
386	C₂²	—	8	2	no	—
387	C₂²	—	8	2	no	—
388	C₂²	—	16	4	no	—
389	C₂²	C₃₁×#465	1	2	yes	—
390	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
391	C₄	—	2	1	no	—
392	C₄	—	2	1	no	—
393	C₄	—	2	1	no	—
394	C₄	—	4	2	no	—
395	C₄	—	8	1	no	—
396	C₄	—	8	1	no	—
397	C₄	C₃₁×#465	1	2	yes	—
398	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
399	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
400	C₄	C₃₁×#576	1	2	yes	—
401	C₂	—	2	2	no	—
402	C₂	—	4	4	no	—
403	C₂	—	16	2	no	—
404	C₂	C₃₁×#3041	1	1	yes	characteristic
405	C₂	C₂×C₃₁×#465	1	1	yes	characteristic
406	C₂	C₃₁×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#459	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ 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×#3208	C₂	semidirect
2. C₃₁×#3077	C₂	semidirect
4. C₃₁×#2972	C₂	semidirect
5. C₃₁×#1677	C₂	semidirect
6. C₃₁×#1665	C₂	semidirect
7. C₃₁×#1570	C₂	semidirect
8. C₃₁×#1545	C₂	semidirect
9. C₃₁×#1542	C₂	semidirect
14. C₃₁×#492	C₂²	semidirect
15. C₃₁×#486	C₂²	semidirect
17. C₃₁×#476	C₂²	semidirect
18. C₃₁×#474	C₂²	semidirect
24. C₈×D₄×C₃₁	C₂²	semidirect
33. C₃₁×#334	C₂²	semidirect
34. C₃₁×#333	C₂²	semidirect
35. C₃₁×#332	C₂²	semidirect
36. C₃₁×#331	C₂²	semidirect
100. C₂×D₄×C₃₁	SD₁₆	semidirect
134. #19384	C₃₁	direct (N × H)
176. #3208	C₂×C₃₁	semidirect
177. #3077	C₂×C₃₁	semidirect
179. #2972	C₂×C₃₁	semidirect
180. #1677	C₂×C₃₁	semidirect
181. #1665	C₂×C₃₁	semidirect
182. #1570	C₂×C₃₁	semidirect
183. #1545	C₂×C₃₁	semidirect
184. #1542	C₂×C₃₁	semidirect
211. #492	C₂²×C₃₁	semidirect
212. #486	C₂²×C₃₁	semidirect
214. #476	C₂²×C₃₁	semidirect
215. #474	C₂²×C₃₁	semidirect
221. C₈×D₄	C₂²×C₃₁	semidirect
230. #334	C₂²×C₃₁	semidirect
231. #333	C₂²×C₃₁	semidirect
232. #332	C₂²×C₃₁	semidirect
233. #331	C₂²×C₃₁	semidirect
287. C₃₁	#19384	direct (N × H)
304. C₂×D₄	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×#3208	2
C₃₁×#3077	2
C₂×C₃₁×#459	2
C₃₁×#2972	2
C₃₁×#1677	2
C₃₁×#1665	2
C₃₁×#1570	2
C₃₁×#1545	2
C₃₁×#1542	2
#19384	31

Subgroups — C31×#19384 #21476143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19384 #21476143