Subgroups — C₃₁×#19663 #21476422

Order 7936.

690 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3230	C₂	1	2	yes	maximal
2	C₃₁×#3209	C₂	1	2	yes	maximal
3	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#498	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1723	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1707	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#533	C₂²	1	1	yes	characteristic
9	C₃₁×#533	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₃₁×#500	C₂²	1	2	yes	—
12	C₃₁×#500	C₂²	1	4	yes	—
13	C₃₁×#499	—	2	2	no	—
14	C₃₁×#498	—	2	2	no	—
15	C₃₁×#498	C₂²	1	2	yes	—
16	C₃₁×#498	C₂²	1	2	yes	—
17	C₃₁×#498	C₂²	1	2	yes	—
18	C₃₁×#494	C₂²	1	4	yes	—
19	C₃₁×#487	C₂²	1	4	yes	—
20	C₃₁×#487	C₂²	1	4	yes	—
21	C₃₁×#484	—	2	4	no	—
22	C₃₁×#477	—	2	4	no	—
23	C₃₁×#462	—	2	4	no	—
24	C₃₁×#451	—	2	4	no	—
25	C₃₁×#431	C₂²	1	2	yes	—
26	C₃₁×#427	—	2	2	no	—
27	C₂×C₃₁×#106	—	2	2	no	—
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#322	C₂²	1	2	yes	—
30	C₈²×C₃₁	C₂²	1	2	yes	—
31	C₂²×Q₈×C₃₁	—	2	2	no	—
32	C₂×Q₁₆×C₃₁	—	2	8	no	—
33	C₂²×C₈×C₃₁	D₄	1	2	yes	—
34	C₃₁×C₄⋊Q₈	—	2	2	no	—
35	C₃₁×C₄⋊Q₈	—	2	2	no	—
36	C₃₁×C₄⋊Q₈	—	2	2	no	—
37	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
38	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
39	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
40	C₃₁×#125	—	2	1	no	—
41	C₃₁×#125	—	2	1	no	—
42	C₃₁×C₂²⋊Q₈	—	2	2	no	—
43	C₃₁×C₂²⋊Q₈	—	2	2	no	—
44	C₄×Q₈×C₃₁	—	2	2	no	—
45	C₄×Q₈×C₃₁	—	2	2	no	—
46	C₃₁×#117	—	2	1	no	—
47	C₃₁×#117	—	2	1	no	—
48	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×#107	—	2	4	no	—
52	C₃₁×#107	—	2	8	no	—
53	C₃₁×#106	—	2	4	no	—
54	C₃₁×#106	—	2	4	no	—
55	C₃₁×#106	—	2	4	no	—
56	C₃₁×C₄⋊C₈	D₄	1	4	yes	—
57	C₃₁×#103	—	2	8	no	—
58	C₃₁×#103	—	2	8	no	—
59	C₃₁×C₂²⋊C₈	D₄	1	4	yes	—
60	C₄×C₈×C₃₁	D₄	1	2	yes	—
61	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₃₁	C₂³	1	4	yes	—
64	C₂×Q₈×C₃₁	—	2	2	no	—
65	C₂×Q₈×C₃₁	—	2	4	no	—
66	C₂×Q₈×C₃₁	—	2	2	no	—
67	C₂×Q₈×C₃₁	—	4	2	no	—
68	C₂×Q₈×C₃₁	—	8	4	no	—
69	C₂²×C₄×C₃₁	—	4	2	no	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
72	Q₁₆×C₃₁	—	8	8	no	—
73	C₂×C₈×C₃₁	Q₁₆	1	8	yes	complemented
74	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
75	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
77	C₄⋊C₄×C₃₁	—	2	2	no	—
78	C₄⋊C₄×C₃₁	—	2	4	no	—
79	C₄⋊C₄×C₃₁	—	2	4	no	—
80	C₄⋊C₄×C₃₁	—	2	2	no	—
81	C₄⋊C₄×C₃₁	—	2	2	no	—
82	C₄⋊C₄×C₃₁	—	2	2	no	—
83	C₄⋊C₄×C₃₁	—	4	2	no	—
84	C₄⋊C₄×C₃₁	—	4	2	no	—
85	C₄⋊C₄×C₃₁	—	4	2	no	—
86	C₄⋊C₄×C₃₁	—	4	2	no	—
87	C₂²⋊C₄×C₃₁	—	4	1	no	—
88	C₂²⋊C₄×C₃₁	—	4	1	no	—
89	C₄²×C₃₁	—	4	1	no	—
90	C₄²×C₃₁	—	4	1	no	—
91	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₃₁	C₂×D₄	1	2	yes	—
93	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
94	#19663	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₃₁	#127	1	1	yes	characteristic
96	Q₈×C₃₁	—	4	4	no	—
97	Q₈×C₃₁	—	4	8	no	—
98	Q₈×C₃₁	—	4	4	no	—
99	Q₈×C₃₁	—	8	4	no	—
100	C₂×C₄×C₃₁	—	2	4	no	—
101	C₂×C₄×C₃₁	—	4	2	no	—
102	C₂×C₄×C₃₁	—	4	2	no	—
103	C₂×C₄×C₃₁	—	4	4	no	—
104	C₂×C₄×C₃₁	—	8	4	no	—
105	C₂×C₄×C₃₁	#127	1	2	yes	—
106	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
107	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
108	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
110	C₈×C₃₁	—	2	8	no	—
111	C₈×C₃₁	C₂×Q₁₆	1	4	yes	—
112	C₈×C₃₁	C₂×Q₁₆	1	4	yes	—
113	#3230	C₂×C₃₁	1	2	yes	—
114	#3209	C₂×C₃₁	1	2	yes	—
115	#3160	C₂×C₃₁	1	1	yes	characteristic
116	C₂×#498	C₂×C₃₁	1	1	yes	characteristic
117	#1723	C₂×C₃₁	1	4	yes	complemented
118	#1707	C₂×C₃₁	1	4	yes	complemented
119	#1450	C₂×C₃₁	1	1	yes	characteristic
120	C₂²×C₃₁	—	2	2	no	—
121	C₂²×C₃₁	#494	1	2	yes	—
122	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
123	C₄×C₃₁	—	2	2	no	—
124	C₄×C₃₁	—	8	4	no	—
125	C₄×C₃₁	—	8	2	no	—
126	C₄×C₃₁	—	8	2	no	—
127	C₄×C₃₁	#494	1	2	yes	—
128	C₄×C₃₁	C₂²×Q₁₆	1	2	yes	—
129	C₄×C₃₁	#577	1	4	yes	—
130	#533	C₂²×C₃₁	1	1	yes	characteristic
131	#533	C₂²×C₃₁	1	1	yes	characteristic
132	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
133	#500	C₂²×C₃₁	1	2	yes	—
134	#500	C₂²×C₃₁	1	4	yes	—
135	#499	—	2	2	no	—
136	#498	—	2	2	no	—
137	#498	C₂²×C₃₁	1	2	yes	—
138	#498	C₂²×C₃₁	1	2	yes	—
139	#498	C₂²×C₃₁	1	2	yes	—
140	#494	C₂²×C₃₁	1	4	yes	—
141	#487	C₂²×C₃₁	1	4	yes	—
142	#487	C₂²×C₃₁	1	4	yes	—
143	#484	—	2	4	no	—
144	#477	—	2	4	no	—
145	#462	—	2	4	no	—
146	#451	—	2	4	no	—
147	#431	C₂²×C₃₁	1	2	yes	—
148	#427	—	2	2	no	—
149	C₂×#106	—	2	2	no	—
150	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
151	#322	C₂²×C₃₁	1	2	yes	—
152	C₈²	C₂²×C₃₁	1	2	yes	—
153	C₂×C₃₁	—	2	2	no	—
154	C₂×C₃₁	C₂×#494	1	1	yes	characteristic
155	C₂×C₃₁	#3154	1	2	yes	—
156	C₂²×Q₈	—	2	2	no	—
157	C₂×Q₁₆	—	2	8	no	—
158	C₂²×C₈	D₄×C₃₁	1	2	yes	—
159	C₄⋊Q₈	—	2	2	no	—
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	—	2	2	no	—
162	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
163	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
164	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
165	#125	—	2	1	no	—
166	#125	—	2	1	no	—
167	C₂²⋊Q₈	—	2	2	no	—
168	C₂²⋊Q₈	—	2	2	no	—
169	C₄×Q₈	—	2	2	no	—
170	C₄×Q₈	—	2	2	no	—
171	#117	—	2	1	no	—
172	#117	—	2	1	no	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
176	#107	—	2	4	no	—
177	#107	—	2	8	no	—
178	#106	—	2	4	no	—
179	#106	—	2	4	no	—
180	#106	—	2	4	no	—
181	C₄⋊C₈	D₄×C₃₁	1	4	yes	—
182	#103	—	2	8	no	—
183	#103	—	2	8	no	—
184	C₂²⋊C₈	D₄×C₃₁	1	4	yes	—
185	C₄×C₈	D₄×C₃₁	1	2	yes	—
186	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
187	C₄×C₈	C₂³×C₃₁	1	4	yes	—
188	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
189	C₃₁	#19663	1	1	yes	characteristic, direct product factor, Sylow
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	—	2	2	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×Q₈	—	8	4	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
198	Q₁₆	—	8	8	no	—
199	C₂×C₈	Q₁₆×C₃₁	1	8	yes	complemented
200	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
201	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
202	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	2	no	—
213	C₂²⋊C₄	—	4	1	no	—
214	C₂²⋊C₄	—	4	1	no	—
215	C₄²	—	4	1	no	—
216	C₄²	—	4	1	no	—
217	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
218	C₄²	C₂×D₄×C₃₁	1	2	yes	—
219	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
220	C₂³	C₃₁×#127	1	1	yes	characteristic
221	Q₈	—	4	4	no	—
222	Q₈	—	4	8	no	—
223	Q₈	—	4	4	no	—
224	Q₈	—	8	4	no	—
225	C₂×C₄	—	2	4	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	—	8	4	no	—
230	C₂×C₄	C₃₁×#127	1	2	yes	—
231	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
232	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	—
233	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
234	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
235	C₈	—	2	8	no	—
236	C₈	C₂×Q₁₆×C₃₁	1	4	yes	—
237	C₈	C₂×Q₁₆×C₃₁	1	4	yes	—
238	C₂²	—	2	2	no	—
239	C₂²	C₃₁×#494	1	2	yes	—
240	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	8	4	no	—
243	C₄	—	8	2	no	—
244	C₄	—	8	2	no	—
245	C₄	C₃₁×#494	1	2	yes	—
246	C₄	C₂²×Q₁₆×C₃₁	1	2	yes	—
247	C₄	C₃₁×#577	1	4	yes	—
248	C₂	—	2	2	no	—
249	C₂	C₂×C₃₁×#494	1	1	yes	characteristic
250	C₂	C₃₁×#3154	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂³×C₃₁	#127
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
5. C₃₁×#1723	C₂	semidirect
6. C₃₁×#1707	C₂	semidirect
73. C₂×C₈×C₃₁	Q₁₆	semidirect
94. #19663	C₃₁	direct (N × H)
117. #1723	C₂×C₃₁	semidirect
118. #1707	C₂×C₃₁	semidirect
189. C₃₁	#19663	direct (N × H)
199. C₂×C₈	Q₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3230	2
C₃₁×#3209	2
C₃₁×#3160	2
C₂×C₃₁×#498	2
C₃₁×#1723	2
C₃₁×#1707	2
C₃₁×#1450	2
#19663	31

Subgroups — C31×#19663 #21476422

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19663 #21476422