Subgroups — C₃₁×#12983 #21469742

Order 7936.

508 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2929	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#350	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2112	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1379	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#525	C₄	1	1	yes	characteristic
9	C₂×C₃₁×M₃₂	C₂²	1	2	yes	—
10	C₂²×C₁₆×C₃₁	—	2	2	no	—
11	C₂²×C₁₆×C₃₁	—	2	4	no	—
12	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#435	—	2	2	no	—
14	C₈×D₄×C₃₁	—	2	2	no	—
15	C₃₁×#424	C₄	1	1	yes	characteristic
16	C₃₁×#408	C₄	1	1	yes	characteristic
17	C₃₁×#407	C₄	1	1	yes	characteristic
18	C₃₁×#405	C₂²	1	1	yes	characteristic
19	C₃₁×#350	—	2	4	no	—
20	C₃₁×C₂²⋊C₁₆	—	2	2	no	—
21	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₃₁×#131	—	2	2	no	—
24	C₃₁×#131	—	2	2	no	—
25	C₂×M₁₆×C₃₁	—	2	1	no	—
26	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂×M₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	—	2	2	no	—
29	C₂²×C₈×C₃₁	—	2	2	no	—
30	C₂²×C₈×C₃₁	—	2	1	no	—
31	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	D₄	1	2	yes	—
33	C₂²×C₈×C₃₁	D₄	1	2	yes	—
34	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
35	C₃₁×#123	—	2	1	no	—
36	C₃₁×C₂²⋊Q₈	C₈	1	2	yes	—
37	C₃₁×#121	C₈	1	2	yes	—
38	C₃₁×#120	—	2	1	no	—
39	C₄×D₄×C₃₁	—	2	2	no	—
40	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
41	C₃₁×M₃₂	—	2	2	no	—
42	C₃₁×M₃₂	—	2	2	no	—
43	C₂×C₁₆×C₃₁	—	2	8	no	—
44	C₂×C₁₆×C₃₁	—	2	2	no	—
45	C₂×C₁₆×C₃₁	—	2	2	no	—
46	C₂×C₁₆×C₃₁	—	2	2	no	—
47	C₂×C₁₆×C₃₁	D₄	1	4	yes	complemented
48	C₃₁×C₄⋊C₈	—	2	2	no	—
49	C₃₁×C₂²⋊C₈	—	2	2	no	—
50	C₃₁×C₂²⋊C₈	—	2	2	no	—
51	C₃₁×#97	—	2	1	no	—
52	C₄×C₈×C₃₁	—	2	1	no	—
53	C₂⁴×C₃₁	M₁₆	1	1	yes	characteristic
54	#41×C₃₁	—	2	2	no	—
55	#41×C₃₁	—	2	2	no	—
56	C₂×Q₈×C₃₁	C₂×C₈	1	1	yes	characteristic
57	C₂×D₄×C₃₁	—	2	2	no	—
58	C₂×D₄×C₃₁	—	2	1	no	—
59	C₂×D₄×C₃₁	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	—	2	2	no	—
61	C₂²×C₄×C₃₁	—	2	2	no	—
62	C₂²×C₄×C₃₁	—	2	1	no	—
63	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₃₁	M₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
67	M₁₆×C₃₁	—	2	2	no	—
68	M₁₆×C₃₁	—	2	2	no	—
69	M₁₆×C₃₁	—	4	1	no	—
70	M₁₆×C₃₁	—	4	1	no	—
71	C₂×C₈×C₃₁	—	2	2	no	—
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	—	2	1	no	—
76	C₂×C₈×C₃₁	—	2	1	no	—
77	C₂×C₈×C₃₁	—	4	2	no	—
78	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₃₁	#41	1	2	yes	—
83	C₄⋊C₄×C₃₁	—	2	2	no	—
84	C₄⋊C₄×C₃₁	C₂×C₈	1	2	yes	—
85	C₂²⋊C₄×C₃₁	—	2	2	no	—
86	C₂²⋊C₄×C₃₁	—	2	1	no	—
87	C₂²⋊C₄×C₃₁	—	2	2	no	—
88	C₄²×C₃₁	—	2	1	no	—
89	C₁₆×C₃₁	—	2	4	no	—
90	C₁₆×C₃₁	—	4	2	no	—
91	#12983	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₃₁	—	2	4	no	—
93	C₂³×C₃₁	—	2	2	no	—
94	C₂³×C₃₁	—	2	1	no	—
95	C₂³×C₃₁	C₂²⋊C₈	1	2	yes	—
96	C₂³×C₃₁	C₂×M₁₆	1	1	yes	characteristic
97	Q₈×C₃₁	—	2	2	no	—
98	D₄×C₃₁	—	2	2	no	—
99	D₄×C₃₁	—	4	4	no	—
100	D₄×C₃₁	—	4	2	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	1	no	—
107	C₂×C₄×C₃₁	—	4	1	no	—
108	C₂×C₄×C₃₁	—	4	1	no	—
109	C₂×C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
110	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₃₁	#131	1	2	yes	—
114	C₈×C₃₁	—	2	2	no	—
115	C₈×C₃₁	—	4	1	no	—
116	C₈×C₃₁	—	4	1	no	—
117	C₈×C₃₁	#120	1	1	yes	characteristic
118	C₈×C₃₁	#121	1	2	yes	—
119	C₈×C₃₁	#123	1	1	yes	characteristic
120	C₂³×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
121	#2929	C₂×C₃₁	1	1	yes	characteristic
122	C₂×#350	C₂×C₃₁	1	2	yes	complemented
123	#2112	C₂×C₃₁	1	1	yes	characteristic, complemented
124	#1379	C₂×C₃₁	1	2	yes	complemented
125	C₂²×C₃₁	—	2	8	no	—
126	C₂²×C₃₁	—	2	2	no	—
127	C₂²×C₃₁	—	2	2	no	—
128	C₂²×C₃₁	—	2	2	no	—
129	C₂²×C₃₁	—	4	2	no	—
130	C₂²×C₃₁	—	4	1	no	—
131	C₂²×C₃₁	#350	1	4	yes	complemented
132	C₂²×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
133	C₂²×C₃₁	#435	1	2	yes	—
134	C₄×C₃₁	—	2	2	no	—
135	C₄×C₃₁	—	4	2	no	—
136	C₄×C₃₁	—	4	1	no	—
137	C₄×C₃₁	#386	1	1	yes	characteristic
138	C₄×C₃₁	#408	1	1	yes	characteristic
139	C₄×C₃₁	C₈×D₄	1	2	yes	—
140	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
141	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
142	#525	C₄×C₃₁	1	1	yes	characteristic
143	C₂×M₃₂	C₂²×C₃₁	1	2	yes	—
144	C₂²×C₁₆	—	2	2	no	—
145	C₂²×C₁₆	—	2	4	no	—
146	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	—
147	#435	—	2	2	no	—
148	C₈×D₄	—	2	2	no	—
149	#424	C₄×C₃₁	1	1	yes	characteristic
150	#408	C₄×C₃₁	1	1	yes	characteristic
151	#407	C₄×C₃₁	1	1	yes	characteristic
152	#405	C₂²×C₃₁	1	1	yes	characteristic
153	#350	—	2	4	no	—
154	C₂²⋊C₁₆	—	2	2	no	—
155	C₂×C₃₁	—	2	4	no	—
156	C₂×C₃₁	—	2	2	no	—
157	C₂×C₃₁	—	8	1	no	—
158	C₂×C₃₁	#1869	1	1	yes	characteristic
159	C₂×C₃₁	C₂×#350	1	2	yes	—
160	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
161	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
162	#131	—	2	2	no	—
163	#131	—	2	2	no	—
164	C₂×M₁₆	—	2	1	no	—
165	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
166	C₂×M₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
167	C₂²×C₈	—	2	2	no	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
171	C₂²×C₈	D₄×C₃₁	1	2	yes	—
172	C₂²×C₈	D₄×C₃₁	1	2	yes	—
173	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
174	#123	—	2	1	no	—
175	C₂²⋊Q₈	C₈×C₃₁	1	2	yes	—
176	#121	C₈×C₃₁	1	2	yes	—
177	#120	—	2	1	no	—
178	C₄×D₄	—	2	2	no	—
179	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
180	M₃₂	—	2	2	no	—
181	M₃₂	—	2	2	no	—
182	C₂×C₁₆	—	2	8	no	—
183	C₂×C₁₆	—	2	2	no	—
184	C₂×C₁₆	—	2	2	no	—
185	C₂×C₁₆	—	2	2	no	—
186	C₂×C₁₆	D₄×C₃₁	1	4	yes	complemented
187	C₄⋊C₈	—	2	2	no	—
188	C₂²⋊C₈	—	2	2	no	—
189	C₂²⋊C₈	—	2	2	no	—
190	#97	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₃₁	#12983	1	1	yes	characteristic, direct product factor, Sylow
193	C₂⁴	M₁₆×C₃₁	1	1	yes	characteristic
194	#41	—	2	2	no	—
195	#41	—	2	2	no	—
196	C₂×Q₈	C₂×C₈×C₃₁	1	1	yes	characteristic
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	1	no	—
199	C₂×D₄	C₂×C₈×C₃₁	1	1	yes	characteristic
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	1	no	—
203	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
204	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
205	C₂²×C₄	M₁₆×C₃₁	1	1	yes	characteristic
206	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
207	M₁₆	—	2	2	no	—
208	M₁₆	—	2	2	no	—
209	M₁₆	—	4	1	no	—
210	M₁₆	—	4	1	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	1	no	—
216	C₂×C₈	—	2	1	no	—
217	C₂×C₈	—	4	2	no	—
218	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
219	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
220	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
221	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
222	C₂×C₈	#41×C₃₁	1	2	yes	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	C₂×C₈×C₃₁	1	2	yes	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	1	no	—
227	C₂²⋊C₄	—	2	2	no	—
228	C₄²	—	2	1	no	—
229	C₁₆	—	2	4	no	—
230	C₁₆	—	4	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	4	no	—
233	C₂³	—	2	1	no	—
234	C₂³	C₃₁×C₂²⋊C₈	1	2	yes	—
235	C₂³	C₂×M₁₆×C₃₁	1	1	yes	characteristic
236	Q₈	—	2	2	no	—
237	D₄	—	2	2	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	C₃₁×C₂²⋊C₈	1	2	yes	—
249	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
250	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
251	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
252	C₂×C₄	C₃₁×#131	1	2	yes	—
253	C₈	—	2	2	no	—
254	C₈	—	4	1	no	—
255	C₈	—	4	1	no	—
256	C₈	C₃₁×#120	1	1	yes	characteristic
257	C₈	C₃₁×#121	1	2	yes	—
258	C₈	C₃₁×#123	1	1	yes	characteristic
259	C₂²	—	2	8	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	C₃₁×#350	1	4	yes	complemented
266	C₂²	C₂×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
267	C₂²	C₃₁×#435	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	1	no	—
271	C₄	C₃₁×#386	1	1	yes	characteristic
272	C₄	C₃₁×#408	1	1	yes	characteristic
273	C₄	C₈×D₄×C₃₁	1	2	yes	—
274	C₂	—	2	2	no	—
275	C₂	—	2	4	no	—
276	C₂	—	8	1	no	—
277	C₂	C₃₁×#1869	1	1	yes	characteristic
278	C₂	C₂×C₃₁×#350	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
socle	C₂²×C₃₁	C₂×C₂²⋊C₈
prime radical	C₃₁×#525	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₁₆×C₃₁	C₂	semidirect
3. C₂×C₃₁×#350	C₂	semidirect
4. C₃₁×#2112	C₂	semidirect
5. C₃₁×#1379	C₂	semidirect
47. C₂×C₁₆×C₃₁	D₄	semidirect
91. #12983	C₃₁	direct (N × H)
120. C₂³×C₁₆	C₂×C₃₁	semidirect
122. C₂×#350	C₂×C₃₁	semidirect
123. #2112	C₂×C₃₁	semidirect
124. #1379	C₂×C₃₁	semidirect
131. C₂²×C₃₁	#350	semidirect
186. C₂×C₁₆	D₄×C₃₁	semidirect
192. C₃₁	#12983	direct (N × H)
265. C₂²	C₃₁×#350	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₆×C₃₁	2
C₃₁×#2929	2
C₂×C₃₁×#350	2
C₃₁×#2112	2
C₃₁×#1379	2
#12983	31

Subgroups — C31×#12983 #21469742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#12983 #21469742