Subgroups — C₃₁×#7621 #21464380

Order 7936.

630 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1839	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1270	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#127	C₄	1	1	yes	characteristic
6	C₂×C₄×D₄×C₃₁	—	2	1	no	—
7	C₂×C₄×D₄×C₃₁	—	2	1	no	—
8	C₂×C₄×D₄×C₃₁	C₄	1	2	yes	—
9	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
10	C₂×C₃₁×#97	—	2	1	no	—
11	C₃₁×#390	—	2	1	no	—
12	C₃₁×#390	—	2	1	no	—
13	C₃₁×#390	C₂²	1	2	yes	—
14	C₃₁×#384	C₄	1	1	yes	characteristic
15	C₄³×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#336	—	2	1	no	—
17	C₃₁×#336	—	2	1	no	—
18	C₃₁×#336	C₂²	1	2	yes	—
19	C₃₁×#336	C₂²	1	2	yes	—
20	C₂²×D₄×C₃₁	—	2	1	no	—
21	C₂²×D₄×C₃₁	—	2	1	no	—
22	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₃₁	—	2	2	no	—
24	C₂²×C₈×C₃₁	—	2	2	no	—
25	C₂²×C₈×C₃₁	—	2	2	no	—
26	C₃₁×#127	—	2	4	no	—
27	C₄×D₄×C₃₁	—	4	4	no	—
28	C₄×D₄×C₃₁	—	4	4	no	—
29	C₄×D₄×C₃₁	—	4	2	no	—
30	C₄×D₄×C₃₁	—	4	2	no	—
31	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
32	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
33	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
34	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
36	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
37	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×C₄⋊C₈	—	2	4	no	—
45	C₃₁×#97	—	4	2	no	—
46	C₂⁴×C₃₁	—	2	2	no	—
47	C₂×D₄×C₃₁	—	2	4	no	—
48	C₂×D₄×C₃₁	—	2	4	no	—
49	C₂×D₄×C₃₁	—	2	2	no	—
50	C₂×D₄×C₃₁	—	4	2	no	—
51	C₂×D₄×C₃₁	—	4	2	no	—
52	C₂×D₄×C₃₁	—	4	2	no	—
53	C₂×D₄×C₃₁	—	4	2	no	—
54	C₂×D₄×C₃₁	—	4	2	no	—
55	C₂×D₄×C₃₁	—	4	2	no	—
56	C₂×D₄×C₃₁	M₁₆	1	4	yes	—
57	C₂²×C₄×C₃₁	—	2	2	no	—
58	C₂²×C₄×C₃₁	—	4	4	no	—
59	C₂²×C₄×C₃₁	—	4	4	no	—
60	C₂²×C₄×C₃₁	—	4	4	no	—
61	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₃₁	—	2	4	no	—
69	C₂×C₈×C₃₁	—	4	2	no	—
70	C₂×C₈×C₃₁	—	4	2	no	—
71	C₂×C₈×C₃₁	—	4	2	no	—
72	C₂×C₈×C₃₁	—	4	2	no	—
73	C₂×C₈×C₃₁	—	4	2	no	—
74	C₄⋊C₄×C₃₁	—	4	2	no	—
75	C₄⋊C₄×C₃₁	—	4	2	no	—
76	C₄⋊C₄×C₃₁	—	4	2	no	—
77	C₂²⋊C₄×C₃₁	—	4	4	no	—
78	C₂²⋊C₄×C₃₁	—	4	2	no	—
79	C₂²⋊C₄×C₃₁	—	4	2	no	—
80	C₂²⋊C₄×C₃₁	—	4	4	no	—
81	C₄²×C₃₁	—	2	2	no	—
82	C₄²×C₃₁	—	2	2	no	—
83	C₄²×C₃₁	—	2	2	no	—
84	C₄²×C₃₁	—	2	2	no	—
85	C₄²×C₃₁	—	4	1	no	—
86	C₄²×C₃₁	—	4	1	no	—
87	C₄²×C₃₁	D₈	1	2	yes	—
88	C₄²×C₃₁	SD₁₆	1	2	yes	—
89	#7621	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	2	4	no	—
91	C₂³×C₃₁	—	4	2	no	—
92	C₂³×C₃₁	—	4	2	no	—
93	C₂³×C₃₁	—	4	2	no	—
94	C₂³×C₃₁	—	4	2	no	—
95	C₂³×C₃₁	—	4	2	no	—
96	C₂³×C₃₁	—	4	2	no	—
97	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	D₄×C₃₁	—	4	4	no	—
99	D₄×C₃₁	—	4	4	no	—
100	D₄×C₃₁	—	4	8	no	—
101	D₄×C₃₁	—	4	8	no	—
102	C₂×C₄×C₃₁	—	2	1	no	—
103	C₂×C₄×C₃₁	—	2	1	no	—
104	C₂×C₄×C₃₁	—	2	1	no	—
105	C₂×C₄×C₃₁	—	2	1	no	—
106	C₂×C₄×C₃₁	—	2	1	no	—
107	C₂×C₄×C₃₁	—	2	1	no	—
108	C₂×C₄×C₃₁	—	2	1	no	—
109	C₂×C₄×C₃₁	—	2	1	no	—
110	C₂×C₄×C₃₁	—	2	1	no	—
111	C₂×C₄×C₃₁	—	2	1	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	2	1	no	—
114	C₂×C₄×C₃₁	—	2	1	no	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	4	2	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	—	4	2	no	—
119	C₂×C₄×C₃₁	—	8	8	no	—
120	C₂×C₄×C₃₁	#100	1	2	yes	—
121	C₂×C₄×C₃₁	#102	1	2	yes	—
122	C₂×C₄×C₃₁	#102	1	2	yes	—
123	C₂×C₄×C₃₁	#104	1	2	yes	—
124	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
125	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
127	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
128	C₈×C₃₁	—	4	4	no	—
129	C₈×C₃₁	—	8	2	no	—
130	C₄×#127	C₂×C₃₁	1	1	yes	characteristic
131	#1839	C₂×C₃₁	1	1	yes	characteristic, complemented
132	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
133	#1270	C₂×C₃₁	1	4	yes	complemented
134	C₂²×C₃₁	—	4	4	no	—
135	C₂²×C₃₁	—	4	4	no	—
136	C₂²×C₃₁	—	4	4	no	—
137	C₂²×C₃₁	—	8	2	no	—
138	C₂²×C₃₁	—	8	2	no	—
139	C₂²×C₃₁	—	8	2	no	—
140	C₂²×C₃₁	—	8	2	no	—
141	C₂²×C₃₁	#407	1	1	yes	characteristic
142	C₂²×C₃₁	#410	1	1	yes	characteristic
143	C₂²×C₃₁	C₂×#100	1	1	yes	characteristic
144	C₂²×C₃₁	C₂×#102	1	1	yes	characteristic
145	C₂²×C₃₁	#418	1	1	yes	characteristic
146	C₂²×C₃₁	C₂×#104	1	1	yes	characteristic
147	C₂²×C₃₁	#421	1	1	yes	characteristic
148	C₄×C₃₁	—	2	2	no	—
149	C₄×C₃₁	—	2	2	no	—
150	C₄×C₃₁	—	2	2	no	—
151	C₄×C₃₁	—	2	2	no	—
152	C₄×C₃₁	—	2	1	no	—
153	C₄×C₃₁	—	2	1	no	—
154	C₄×C₃₁	—	2	2	no	—
155	C₄×C₃₁	—	2	2	no	—
156	C₄×C₃₁	—	8	2	no	—
157	C₂×#127	C₄×C₃₁	1	1	yes	characteristic
158	C₂×C₄×D₄	—	2	1	no	—
159	C₂×C₄×D₄	—	2	1	no	—
160	C₂×C₄×D₄	C₄×C₃₁	1	2	yes	—
161	C₂×C₄⋊C₈	—	2	1	no	—
162	C₂×#97	—	2	1	no	—
163	#390	—	2	1	no	—
164	#390	—	2	1	no	—
165	#390	C₂²×C₃₁	1	2	yes	—
166	#384	C₄×C₃₁	1	1	yes	characteristic
167	C₄³	C₂²×C₃₁	1	1	yes	characteristic
168	#336	—	2	1	no	—
169	#336	—	2	1	no	—
170	#336	C₂²×C₃₁	1	2	yes	—
171	#336	C₂²×C₃₁	1	2	yes	—
172	C₂×C₃₁	—	8	4	no	—
173	C₂×C₃₁	#1466	1	1	yes	characteristic
174	C₂×C₃₁	#1486	1	1	yes	characteristic
175	C₂×C₃₁	#1490	1	1	yes	characteristic
176	C₂×C₃₁	#1502	1	1	yes	characteristic
177	C₂×C₃₁	#1514	1	1	yes	characteristic
178	C₂×C₃₁	#1530	1	1	yes	characteristic
179	C₂×C₃₁	#1545	1	1	yes	characteristic
180	C₂²×D₄	—	2	1	no	—
181	C₂²×D₄	—	2	1	no	—
182	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
183	C₂³×C₄	—	2	2	no	—
184	C₂²×C₈	—	2	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	#127	—	2	4	no	—
187	C₄×D₄	—	4	4	no	—
188	C₄×D₄	—	4	4	no	—
189	C₄×D₄	—	4	2	no	—
190	C₄×D₄	—	4	2	no	—
191	C₂×C₄⋊C₄	—	2	1	no	—
192	C₂×C₄⋊C₄	—	2	1	no	—
193	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
194	C₂×C₂²⋊C₄	—	2	2	no	—
195	C₂×C₂²⋊C₄	—	2	2	no	—
196	C₂×C₂²⋊C₄	—	2	2	no	—
197	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
198	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
199	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
200	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
201	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
202	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
203	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
204	C₄⋊C₈	—	2	4	no	—
205	#97	—	4	2	no	—
206	C₃₁	#7621	1	1	yes	characteristic, direct product factor, Sylow
207	C₂⁴	—	2	2	no	—
208	C₂×D₄	—	2	4	no	—
209	C₂×D₄	—	2	4	no	—
210	C₂×D₄	—	2	2	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂×D₄	—	4	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂×D₄	M₁₆×C₃₁	1	4	yes	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
224	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
225	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	4	2	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	4	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	4	4	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	4	1	no	—
248	C₄²	D₈×C₃₁	1	2	yes	—
249	C₄²	SD₁₆×C₃₁	1	2	yes	—
250	C₂³	—	2	4	no	—
251	C₂³	—	4	2	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	2	no	—
256	C₂³	—	4	2	no	—
257	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
258	D₄	—	4	4	no	—
259	D₄	—	4	4	no	—
260	D₄	—	4	8	no	—
261	D₄	—	4	8	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	8	8	no	—
280	C₂×C₄	C₃₁×#100	1	2	yes	—
281	C₂×C₄	C₃₁×#102	1	2	yes	—
282	C₂×C₄	C₃₁×#102	1	2	yes	—
283	C₂×C₄	C₃₁×#104	1	2	yes	—
284	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
285	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
286	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
287	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
288	C₈	—	4	4	no	—
289	C₈	—	8	2	no	—
290	C₂²	—	4	4	no	—
291	C₂²	—	4	4	no	—
292	C₂²	—	4	4	no	—
293	C₂²	—	8	2	no	—
294	C₂²	—	8	2	no	—
295	C₂²	—	8	2	no	—
296	C₂²	—	8	2	no	—
297	C₂²	C₃₁×#407	1	1	yes	characteristic
298	C₂²	C₃₁×#410	1	1	yes	characteristic
299	C₂²	C₂×C₃₁×#100	1	1	yes	characteristic
300	C₂²	C₂×C₃₁×#102	1	1	yes	characteristic
301	C₂²	C₃₁×#418	1	1	yes	characteristic
302	C₂²	C₂×C₃₁×#104	1	1	yes	characteristic
303	C₂²	C₃₁×#421	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	8	2	no	—
313	C₂	—	8	4	no	—
314	C₂	C₃₁×#1466	1	1	yes	characteristic
315	C₂	C₃₁×#1486	1	1	yes	characteristic
316	C₂	C₃₁×#1490	1	1	yes	characteristic
317	C₂	C₃₁×#1502	1	1	yes	characteristic
318	C₂	C₃₁×#1514	1	1	yes	characteristic
319	C₂	C₃₁×#1530	1	1	yes	characteristic
320	C₂	C₃₁×#1545	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1839	C₂	semidirect
3. C₃₁×#1767	C₂	semidirect
4. C₃₁×#1270	C₂	semidirect
89. #7621	C₃₁	direct (N × H)
131. #1839	C₂×C₃₁	semidirect
132. #1767	C₂×C₃₁	semidirect
133. #1270	C₂×C₃₁	semidirect
206. C₃₁	#7621	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#127	2
C₃₁×#1839	2
C₃₁×#1767	2
C₃₁×#1270	2
#7621	31

Subgroups — C31×#7621 #21464380

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7621 #21464380