Subgroups — C₃₁×#9755 #21466514

Order 7936.

394 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1846	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1518	C₂	1	4	yes	maximal
4	C₃₁×#1450	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#533	C₂²	1	2	yes	—
6	C₃₁×#500	C₄	1	2	yes	complemented
7	C₃₁×#499	—	2	1	no	—
8	C₃₁×#499	—	2	1	no	—
9	C₃₁×#498	C₄	1	2	yes	complemented
10	C₃₁×#431	C₂²	1	2	yes	—
11	C₃₁×#427	—	2	1	no	—
12	C₃₁×#427	—	2	1	no	—
13	C₂×C₃₁×#107	—	2	1	no	—
14	C₂×C₃₁×#106	—	2	1	no	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#382	C₂²	1	2	yes	—
17	C₃₁×#328	—	2	8	no	—
18	C₃₁×#322	—	2	1	no	—
19	C₈²×C₃₁	—	2	1	no	—
20	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
22	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
23	C₃₁×#125	—	2	2	no	—
24	C₃₁×C₂²⋊Q₈	—	2	4	no	—
25	C₄×Q₈×C₃₁	—	2	4	no	—
26	C₃₁×#117	—	2	2	no	—
27	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
28	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
29	C₃₁×#107	—	2	2	no	—
30	C₃₁×#107	—	2	2	no	—
31	C₃₁×#107	—	4	2	no	—
32	C₃₁×#106	—	2	2	no	—
33	C₃₁×#106	—	2	2	no	—
34	C₃₁×#106	—	4	2	no	—
35	C₃₁×C₄⋊C₈	—	2	2	no	—
36	C₃₁×C₂²⋊C₈	D₄	1	4	yes	—
37	C₄×C₈×C₃₁	—	2	2	no	—
38	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
41	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
42	C₃₁×#95	—	2	4	no	—
43	C₂×Q₈×C₃₁	—	2	4	no	—
44	C₂²×C₄×C₃₁	—	4	2	no	—
45	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₈×C₃₁	—	2	4	no	—
48	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
53	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
54	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
55	C₂×C₈×C₃₁	Q₁₆	1	2	yes	—
56	C₄⋊C₄×C₃₁	—	2	4	no	—
57	C₄⋊C₄×C₃₁	—	2	4	no	—
58	C₄⋊C₄×C₃₁	—	4	4	no	—
59	C₄⋊C₄×C₃₁	—	4	2	no	—
60	C₄⋊C₄×C₃₁	—	4	2	no	—
61	C₂²⋊C₄×C₃₁	—	4	2	no	—
62	C₄²×C₃₁	—	4	2	no	—
63	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₃₁	#41	1	2	yes	—
66	#9755	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂³×C₃₁	#127	1	1	yes	characteristic
68	Q₈×C₃₁	—	8	4	no	—
69	C₂×C₄×C₃₁	—	4	2	no	—
70	C₂×C₄×C₃₁	—	4	4	no	—
71	C₂×C₄×C₃₁	—	8	4	no	—
72	C₂×C₄×C₃₁	—	8	2	no	—
73	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
75	C₂×C₄×C₃₁	#121	1	2	yes	—
76	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
77	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
78	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
79	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
81	C₈×C₃₁	—	4	4	no	—
82	C₈×C₃₁	#103	1	2	yes	—
83	C₈×C₃₁	#103	1	2	yes	—
84	C₈×C₃₁	#103	1	2	yes	—
85	C₈×C₃₁	#103	1	2	yes	—
86	#3160	C₂×C₃₁	1	1	yes	characteristic
87	#1846	C₂×C₃₁	1	1	yes	characteristic
88	#1518	C₂×C₃₁	1	4	yes	—
89	#1450	C₂×C₃₁	1	1	yes	characteristic
90	C₂²×C₃₁	—	2	2	no	—
91	C₂²×C₃₁	#390	1	1	yes	characteristic
92	C₂²×C₃₁	#492	1	1	yes	characteristic
93	C₂²×C₃₁	#494	1	1	yes	characteristic
94	C₄×C₃₁	—	2	2	no	—
95	C₄×C₃₁	—	2	2	no	—
96	C₄×C₃₁	—	8	4	no	—
97	C₄×C₃₁	—	16	2	no	—
98	C₄×C₃₁	C₂×#103	1	1	yes	characteristic
99	C₄×C₃₁	C₂×#103	1	1	yes	characteristic
100	C₄×C₃₁	#487	1	1	yes	characteristic
101	C₄×C₃₁	#487	1	1	yes	characteristic
102	#533	C₂²×C₃₁	1	2	yes	—
103	#500	C₄×C₃₁	1	2	yes	complemented
104	#499	—	2	1	no	—
105	#499	—	2	1	no	—
106	#498	C₄×C₃₁	1	2	yes	complemented
107	#431	C₂²×C₃₁	1	2	yes	—
108	#427	—	2	1	no	—
109	#427	—	2	1	no	—
110	C₂×#107	—	2	1	no	—
111	C₂×#106	—	2	1	no	—
112	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
113	#382	C₂²×C₃₁	1	2	yes	—
114	#328	—	2	8	no	—
115	#322	—	2	1	no	—
116	C₈²	—	2	1	no	—
117	C₂×C₃₁	—	2	1	no	—
118	C₂×C₃₁	—	2	1	no	—
119	C₂×C₃₁	#1956	1	1	yes	characteristic
120	C₂×C₃₁	#1976	1	2	yes	—
121	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
122	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
123	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
124	#125	—	2	2	no	—
125	C₂²⋊Q₈	—	2	4	no	—
126	C₄×Q₈	—	2	4	no	—
127	#117	—	2	2	no	—
128	C₂×C₄⋊C₄	—	2	2	no	—
129	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
130	#107	—	2	2	no	—
131	#107	—	2	2	no	—
132	#107	—	4	2	no	—
133	#106	—	2	2	no	—
134	#106	—	2	2	no	—
135	#106	—	4	2	no	—
136	C₄⋊C₈	—	2	2	no	—
137	C₂²⋊C₈	D₄×C₃₁	1	4	yes	—
138	C₄×C₈	—	2	2	no	—
139	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
140	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
141	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
142	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
143	#95	—	2	4	no	—
144	C₃₁	#9755	1	1	yes	characteristic, direct product factor, Sylow
145	C₂×Q₈	—	2	4	no	—
146	C₂²×C₄	—	4	2	no	—
147	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
148	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
149	C₂×C₈	—	2	4	no	—
150	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
151	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
152	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
153	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
154	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
155	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
156	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
157	C₂×C₈	Q₁₆×C₃₁	1	2	yes	—
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	4	no	—
160	C₄⋊C₄	—	4	4	no	—
161	C₄⋊C₄	—	4	2	no	—
162	C₄⋊C₄	—	4	2	no	—
163	C₂²⋊C₄	—	4	2	no	—
164	C₄²	—	4	2	no	—
165	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
166	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
167	C₄²	#41×C₃₁	1	2	yes	—
168	C₂³	C₃₁×#127	1	1	yes	characteristic
169	Q₈	—	8	4	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	4	4	no	—
172	C₂×C₄	—	8	4	no	—
173	C₂×C₄	—	8	2	no	—
174	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
175	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
176	C₂×C₄	C₃₁×#121	1	2	yes	—
177	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
178	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
179	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
180	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
181	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
182	C₈	—	4	4	no	—
183	C₈	C₃₁×#103	1	2	yes	—
184	C₈	C₃₁×#103	1	2	yes	—
185	C₈	C₃₁×#103	1	2	yes	—
186	C₈	C₃₁×#103	1	2	yes	—
187	C₂²	—	2	2	no	—
188	C₂²	C₃₁×#390	1	1	yes	characteristic
189	C₂²	C₃₁×#492	1	1	yes	characteristic
190	C₂²	C₃₁×#494	1	1	yes	characteristic
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	8	4	no	—
194	C₄	—	16	2	no	—
195	C₄	C₂×C₃₁×#103	1	1	yes	characteristic
196	C₄	C₂×C₃₁×#103	1	1	yes	characteristic
197	C₄	C₃₁×#487	1	1	yes	characteristic
198	C₄	C₃₁×#487	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	C₃₁×#1956	1	1	yes	characteristic
202	C₂	C₃₁×#1976	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	#390
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×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
6. C₃₁×#500	C₄	semidirect
9. C₃₁×#498	C₄	semidirect
66. #9755	C₃₁	direct (N × H)
103. #500	C₄×C₃₁	semidirect
106. #498	C₄×C₃₁	semidirect
144. C₃₁	#9755	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3160	2
C₃₁×#1846	2
C₃₁×#1518	2
C₃₁×#1450	2
#9755	31

Subgroups — C31×#9755 #21466514

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9755 #21466514