Subgroups — C₃₁×#11270 #21468029

Order 7936.

634 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#107	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#343	C₂	1	2	yes	maximal
3	C₂×C₃₁×#341	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1736	C₂	1	2	yes	maximal
5	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
7	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
11	C₂×C₃₁×#107	—	2	2	no	—
12	C₂×C₃₁×#107	—	2	4	no	—
13	C₃₁×#343	—	2	4	no	—
14	C₃₁×#341	C₄	1	4	yes	complemented
15	C₃₁×#340	—	2	4	no	—
16	C₃₁×#336	—	2	2	no	—
17	C₂³×C₄×C₃₁	—	2	2	no	—
18	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₃₁	—	2	2	no	—
20	C₂×M₁₆×C₃₁	—	2	1	no	—
21	C₂×M₁₆×C₃₁	—	2	2	no	—
22	C₂×M₁₆×C₃₁	—	2	2	no	—
23	C₂×M₁₆×C₃₁	—	2	2	no	—
24	C₂×M₁₆×C₃₁	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	2	no	—
26	C₂×M₁₆×C₃₁	—	2	2	no	—
27	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
28	C₂×M₁₆×C₃₁	Q₈	1	2	yes	—
29	C₂²×C₈×C₃₁	—	2	2	no	—
30	C₂²×C₈×C₃₁	—	2	2	no	—
31	C₂²×C₈×C₃₁	—	2	1	no	—
32	C₂²×C₈×C₃₁	—	2	2	no	—
33	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
34	C₂²×C₈×C₃₁	D₄	1	2	yes	—
35	C₂²×C₈×C₃₁	Q₈	1	2	yes	—
36	C₃₁×#117	—	2	2	no	—
37	C₃₁×#117	—	2	2	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	D₄	1	4	yes	—
44	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
45	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
46	C₂×C₄²×C₃₁	—	2	1	no	—
47	C₃₁×#107	—	2	8	no	—
48	C₂⁴×C₃₁	#41	1	1	yes	characteristic
49	C₂²×C₄×C₃₁	—	2	2	no	—
50	C₂²×C₄×C₃₁	—	2	4	no	—
51	C₂²×C₄×C₃₁	—	4	2	no	—
52	C₂²×C₄×C₃₁	—	4	2	no	—
53	C₂²×C₄×C₃₁	—	8	4	no	—
54	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	#41	1	2	yes	—
61	C₂²×C₄×C₃₁	#41	1	2	yes	—
62	C₂²×C₄×C₃₁	#41	1	2	yes	—
63	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
64	M₁₆×C₃₁	—	2	4	no	—
65	M₁₆×C₃₁	—	2	4	no	—
66	M₁₆×C₃₁	—	2	4	no	—
67	M₁₆×C₃₁	—	4	2	no	—
68	M₁₆×C₃₁	—	4	2	no	—
69	M₁₆×C₃₁	—	4	2	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	2	4	no	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	2	4	no	—
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	—	4	1	no	—
77	C₂×C₈×C₃₁	—	4	2	no	—
78	C₂×C₈×C₃₁	—	4	2	no	—
79	C₂×C₈×C₃₁	—	4	1	no	—
80	C₂×C₈×C₃₁	C₄⋊C₄	1	4	yes	—
81	C₄⋊C₄×C₃₁	—	2	4	no	—
82	C₄⋊C₄×C₃₁	—	2	4	no	—
83	C₄⋊C₄×C₃₁	—	2	4	no	—
84	C₄⋊C₄×C₃₁	—	4	2	no	—
85	C₄⋊C₄×C₃₁	—	4	2	no	—
86	C₂²⋊C₄×C₃₁	—	4	2	no	—
87	C₄²×C₃₁	—	4	1	no	—
88	C₄²×C₃₁	—	4	1	no	—
89	#11270	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	2	2	no	—
91	C₂³×C₃₁	—	2	2	no	—
92	C₂³×C₃₁	C₄×Q₈	1	1	yes	characteristic
93	C₂³×C₃₁	#121	1	1	yes	characteristic
94	C₂³×C₃₁	#123	1	2	yes	—
95	C₂³×C₃₁	#125	1	1	yes	characteristic
96	C₂³×C₃₁	#126	1	2	yes	—
97	C₂×C₄×C₃₁	—	4	4	no	—
98	C₂×C₄×C₃₁	—	4	2	no	—
99	C₂×C₄×C₃₁	—	4	2	no	—
100	C₂×C₄×C₃₁	—	4	2	no	—
101	C₂×C₄×C₃₁	—	8	4	no	—
102	C₂×C₄×C₃₁	—	8	4	no	—
103	C₂×C₄×C₃₁	—	8	4	no	—
104	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	#117	1	2	yes	—
106	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
109	C₂×C₄×C₃₁	#121	1	2	yes	—
110	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
114	C₂×C₄×C₃₁	#123	1	2	yes	—
115	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
116	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
117	C₂×C₄×C₃₁	#124	1	2	yes	—
118	C₂×C₄×C₃₁	#125	1	2	yes	—
119	C₂×C₄×C₃₁	#125	1	2	yes	—
120	C₂×C₄×C₃₁	#126	1	2	yes	—
121	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
122	C₈×C₃₁	—	2	4	no	—
123	C₈×C₃₁	—	4	4	no	—
124	C₈×C₃₁	—	4	2	no	—
125	C₂²×#107	C₂×C₃₁	1	1	yes	characteristic
126	C₂×#343	C₂×C₃₁	1	2	yes	—
127	C₂×#341	C₂×C₃₁	1	1	yes	characteristic
128	#1736	C₂×C₃₁	1	2	yes	—
129	#1729	C₂×C₃₁	1	1	yes	characteristic
130	C₂²×C₃₁	—	2	4	no	—
131	C₂²×C₃₁	—	2	2	no	—
132	C₂²×C₃₁	—	2	2	no	—
133	C₂²×C₃₁	—	2	2	no	—
134	C₂²×C₃₁	—	2	2	no	—
135	C₂²×C₃₁	#387	1	2	yes	—
136	C₂²×C₃₁	#389	1	1	yes	characteristic
137	C₂²×C₃₁	#394	1	1	yes	characteristic
138	C₂²×C₃₁	#397	1	1	yes	characteristic
139	C₂²×C₃₁	#400	1	2	yes	—
140	C₂²×C₃₁	#472	1	2	yes	—
141	C₂²×C₃₁	#473	1	2	yes	—
142	C₄×C₃₁	—	8	2	no	—
143	C₄×C₃₁	—	8	4	no	—
144	C₄×C₃₁	#382	1	1	yes	characteristic
145	C₄×C₃₁	#385	1	1	yes	characteristic
146	C₄×C₃₁	#388	1	2	yes	—
147	C₄×C₃₁	#393	1	1	yes	characteristic
148	C₄×C₃₁	#398	1	2	yes	—
149	C₄×C₃₁	#400	1	1	yes	characteristic
150	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
151	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
152	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
153	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
154	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
155	C₂×#107	—	2	2	no	—
156	C₂×#107	—	2	4	no	—
157	#343	—	2	4	no	—
158	#341	C₄×C₃₁	1	4	yes	complemented
159	#340	—	2	4	no	—
160	#336	—	2	2	no	—
161	C₂×C₃₁	—	2	2	no	—
162	C₂×C₃₁	—	2	2	no	—
163	C₂×C₃₁	#1442	1	1	yes	characteristic
164	C₂×C₃₁	#1952	1	2	yes	—
165	C₂×C₃₁	#2045	1	1	yes	characteristic
166	C₂×C₃₁	#2047	1	1	yes	characteristic
167	C₂×C₃₁	#2078	1	2	yes	—
168	C₂³×C₄	—	2	2	no	—
169	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
179	C₂×M₁₆	Q₈×C₃₁	1	2	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
185	C₂²×C₈	D₄×C₃₁	1	2	yes	—
186	C₂²×C₈	Q₈×C₃₁	1	2	yes	—
187	#117	—	2	2	no	—
188	#117	—	2	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	D₄×C₃₁	1	4	yes	—
195	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
196	C₂×C₂²⋊C₄	—	2	1	no	—
197	C₂×C₄²	—	2	1	no	—
198	#107	—	2	8	no	—
199	C₃₁	#11270	1	1	yes	characteristic, direct product factor, Sylow
200	C₂⁴	#41×C₃₁	1	1	yes	characteristic
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	4	2	no	—
205	C₂²×C₄	—	8	4	no	—
206	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
210	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
211	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	#41×C₃₁	1	2	yes	—
213	C₂²×C₄	#41×C₃₁	1	2	yes	—
214	C₂²×C₄	#41×C₃₁	1	2	yes	—
215	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
216	M₁₆	—	2	4	no	—
217	M₁₆	—	2	4	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	4	2	no	—
220	M₁₆	—	4	2	no	—
221	M₁₆	—	4	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	4	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	4	1	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	1	no	—
232	C₂×C₈	C₄⋊C₄×C₃₁	1	4	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	4	2	no	—
238	C₂²⋊C₄	—	4	2	no	—
239	C₄²	—	4	1	no	—
240	C₄²	—	4	1	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₄×Q₈×C₃₁	1	1	yes	characteristic
244	C₂³	C₃₁×#121	1	1	yes	characteristic
245	C₂³	C₃₁×#123	1	2	yes	—
246	C₂³	C₃₁×#125	1	1	yes	characteristic
247	C₂³	C₃₁×#126	1	2	yes	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	8	4	no	—
253	C₂×C₄	—	8	4	no	—
254	C₂×C₄	—	8	4	no	—
255	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
256	C₂×C₄	C₃₁×#117	1	2	yes	—
257	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
258	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
259	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
260	C₂×C₄	C₃₁×#121	1	2	yes	—
261	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
262	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
263	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
264	C₂×C₄	C₃₁×#123	1	2	yes	—
265	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
266	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
267	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
268	C₂×C₄	C₃₁×#124	1	2	yes	—
269	C₂×C₄	C₃₁×#125	1	2	yes	—
270	C₂×C₄	C₃₁×#125	1	2	yes	—
271	C₂×C₄	C₃₁×#126	1	2	yes	—
272	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	—	4	4	no	—
275	C₈	—	4	2	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	C₃₁×#387	1	2	yes	—
282	C₂²	C₃₁×#389	1	1	yes	characteristic
283	C₂²	C₃₁×#394	1	1	yes	characteristic
284	C₂²	C₃₁×#397	1	1	yes	characteristic
285	C₂²	C₃₁×#400	1	2	yes	—
286	C₂²	C₃₁×#472	1	2	yes	—
287	C₂²	C₃₁×#473	1	2	yes	—
288	C₄	—	8	2	no	—
289	C₄	—	8	4	no	—
290	C₄	C₃₁×#382	1	1	yes	characteristic
291	C₄	C₃₁×#385	1	1	yes	characteristic
292	C₄	C₃₁×#388	1	2	yes	—
293	C₄	C₃₁×#393	1	1	yes	characteristic
294	C₄	C₃₁×#398	1	2	yes	—
295	C₄	C₃₁×#400	1	1	yes	characteristic
296	C₂	—	2	2	no	—
297	C₂	—	2	2	no	—
298	C₂	C₃₁×#1442	1	1	yes	characteristic
299	C₂	C₃₁×#1952	1	2	yes	—
300	C₂	C₃₁×#2045	1	1	yes	characteristic
301	C₂	C₃₁×#2047	1	1	yes	characteristic
302	C₂	C₃₁×#2078	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	#121
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
prime radical	C₂⁴×C₃₁	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
14. C₃₁×#341	C₄	semidirect
89. #11270	C₃₁	direct (N × H)
158. #341	C₄×C₃₁	semidirect
199. C₃₁	#11270	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#107	2
C₂×C₃₁×#343	2
C₂×C₃₁×#341	2
C₃₁×#1736	2
C₃₁×#1729	2
#11270	31

Subgroups — C31×#11270 #21468029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#11270 #21468029