Subgroups — C₃₁×#62653 #21519412

Order 7936.

10930 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×2₋¹⁺⁴	C₂	1	32	yes	maximal, complemented
3	C₃₁×#3428	C₂	1	20	yes	maximal, complemented
4	C₂×C₃₁×#519	C₂	1	10	yes	maximal, complemented
5	C₂×C₃₁×2₋¹⁺⁴	C₄	1	32	yes	complemented
6	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	16	yes	—
7	C₂²×#41×C₃₁	C₂²	1	10	yes	—
8	C₂³×Q₈×C₃₁	C₂²	1	5	yes	—
9	C₃₁×#520	C₂²	1	160	yes	complemented
10	C₃₁×#519	C₂²	1	40	yes	—
11	C₃₁×#519	C₂²	1	80	yes	complemented
12	C₃₁×#518	C₂²	1	40	yes	—
13	C₄×#41×C₃₁	C₂²	1	160	yes	complemented
14	C₂×C₄×Q₈×C₃₁	C₂²	1	80	yes	complemented
15	C₂×C₃₁×#117	C₂²	1	60	yes	—
16	C₃₁×2₋¹⁺⁴	C₂×C₄	1	256	yes	complemented
17	C₂×#41×C₃₁	C₂×C₄	1	160	yes	complemented
18	C₂×#41×C₃₁	C₂³	1	80	yes	—
19	C₂²×Q₈×C₃₁	C₂×C₄	1	80	yes	complemented
20	C₂²×Q₈×C₃₁	C₂³	1	40	yes	—
21	C₂²×Q₈×C₃₁	C₂³	1	10	yes	—
22	C₂²×D₄×C₃₁	C₂³	1	10	yes	—
23	C₂³×C₄×C₃₁	C₂³	1	15	yes	—
24	C₄×Q₈×C₃₁	C₂³	1	320	yes	—
25	C₄×D₄×C₃₁	C₂³	1	320	yes	—
26	C₃₁×#117	C₂³	1	80	yes	—
27	C₃₁×#117	C₂³	1	240	yes	—
28	C₂×C₄⋊C₄×C₃₁	C₂³	1	120	yes	—
29	C₂×C₂²⋊C₄×C₃₁	C₂³	1	40	yes	—
30	C₂×C₄²×C₃₁	C₂³	1	120	yes	—
31	C₂⁴×C₃₁	C₂⁴	1	5	yes	—
32	#41×C₃₁	C₂²×C₄	1	640	yes	—
33	C₂×Q₈×C₃₁	C₂²×C₄	1	80	yes	—
34	C₂×Q₈×C₃₁	C₂²×C₄	1	320	yes	—
35	C₂×Q₈×C₃₁	C₂⁴	1	40	yes	—
36	C₂×D₄×C₃₁	C₂²×C₄	1	80	yes	—
37	C₂×D₄×C₃₁	C₂⁴	1	40	yes	—
38	C₂²×C₄×C₃₁	C₂²×C₄	1	120	yes	—
39	C₂²×C₄×C₃₁	C₂⁴	1	80	yes	—
40	C₂²×C₄×C₃₁	C₂⁴	1	60	yes	—
41	C₂²×C₄×C₃₁	C₂⁴	1	10	yes	—
42	C₄⋊C₄×C₃₁	C₂⁴	1	160	yes	—
43	C₂²⋊C₄×C₃₁	C₂⁴	1	16	yes	—
44	C₂²⋊C₄×C₃₁	C₂⁴	1	80	yes	—
45	C₄²×C₃₁	C₂⁴	1	160	yes	—
46	#62653	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂³×C₃₁	—	4	10	no	—
48	C₂³×C₃₁	C₂³×C₄	1	20	yes	—
49	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
50	C₂³×C₃₁	C₂⁵	1	10	yes	—
51	Q₈×C₃₁	C₂³×C₄	1	160	yes	—
52	D₄×C₃₁	C₂³×C₄	1	160	yes	—
53	C₂×C₄×C₃₁	—	4	80	no	—
54	C₂×C₄×C₃₁	C₂³×C₄	1	40	yes	—
55	C₂×C₄×C₃₁	C₂³×C₄	1	240	yes	—
56	C₂×C₄×C₃₁	C₂⁵	1	32	yes	—
57	C₂×C₄×C₃₁	C₂⁵	1	20	yes	—
58	C₂²×2₋¹⁺⁴	C₂×C₃₁	1	1	yes	characteristic
59	C₄×2₋¹⁺⁴	C₂×C₃₁	1	32	yes	complemented
60	#3428	C₂×C₃₁	1	20	yes	complemented
61	C₂×#519	C₂×C₃₁	1	10	yes	complemented
62	C₂²×C₃₁	—	2	20	no	—
63	C₂²×C₃₁	—	2	2	no	—
64	C₂²×C₃₁	—	4	20	no	—
65	C₂²×C₃₁	C₂⁴×C₄	1	2	yes	—
66	C₂²×C₃₁	C₂⁴×C₄	1	20	yes	—
67	C₂²×C₃₁	C₂⁶	1	1	yes	characteristic
68	C₄×C₃₁	—	2	32	no	—
69	C₄×C₃₁	C₂⁴×C₄	1	40	yes	—
70	C₂×2₋¹⁺⁴	C₄×C₃₁	1	32	yes	complemented
71	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	16	yes	—
72	C₂²×#41	C₂²×C₃₁	1	10	yes	—
73	C₂³×Q₈	C₂²×C₃₁	1	5	yes	—
74	#520	C₂²×C₃₁	1	160	yes	complemented
75	#519	C₂²×C₃₁	1	40	yes	—
76	#519	C₂²×C₃₁	1	80	yes	complemented
77	#518	C₂²×C₃₁	1	40	yes	—
78	C₄×#41	C₂²×C₃₁	1	160	yes	complemented
79	C₂×C₄×Q₈	C₂²×C₃₁	1	80	yes	complemented
80	C₂×#117	C₂²×C₃₁	1	60	yes	—
81	C₂×C₃₁	—	2	2	no	—
82	C₂×C₃₁	—	2	20	no	—
83	C₂×C₃₁	C₂⁵×C₄	1	1	yes	characteristic
84	C₂×C₃₁	2₋¹⁺⁶	1	2	yes	—
85	2₋¹⁺⁴	C₂×C₄×C₃₁	1	256	yes	complemented
86	C₂×#41	C₂×C₄×C₃₁	1	160	yes	complemented
87	C₂×#41	C₂³×C₃₁	1	80	yes	—
88	C₂²×Q₈	C₂×C₄×C₃₁	1	80	yes	complemented
89	C₂²×Q₈	C₂³×C₃₁	1	40	yes	—
90	C₂²×Q₈	C₂³×C₃₁	1	10	yes	—
91	C₂²×D₄	C₂³×C₃₁	1	10	yes	—
92	C₂³×C₄	C₂³×C₃₁	1	15	yes	—
93	C₄×Q₈	C₂³×C₃₁	1	320	yes	—
94	C₄×D₄	C₂³×C₃₁	1	320	yes	—
95	#117	C₂³×C₃₁	1	80	yes	—
96	#117	C₂³×C₃₁	1	240	yes	—
97	C₂×C₄⋊C₄	C₂³×C₃₁	1	120	yes	—
98	C₂×C₂²⋊C₄	C₂³×C₃₁	1	40	yes	—
99	C₂×C₄²	C₂³×C₃₁	1	120	yes	—
100	C₃₁	#62653	1	1	yes	characteristic, direct product factor, Sylow
101	C₂⁴	C₂⁴×C₃₁	1	5	yes	—
102	#41	C₂²×C₄×C₃₁	1	640	yes	—
103	C₂×Q₈	C₂²×C₄×C₃₁	1	80	yes	—
104	C₂×Q₈	C₂²×C₄×C₃₁	1	320	yes	—
105	C₂×Q₈	C₂⁴×C₃₁	1	40	yes	—
106	C₂×D₄	C₂²×C₄×C₃₁	1	80	yes	—
107	C₂×D₄	C₂⁴×C₃₁	1	40	yes	—
108	C₂²×C₄	C₂²×C₄×C₃₁	1	120	yes	—
109	C₂²×C₄	C₂⁴×C₃₁	1	80	yes	—
110	C₂²×C₄	C₂⁴×C₃₁	1	60	yes	—
111	C₂²×C₄	C₂⁴×C₃₁	1	10	yes	—
112	C₄⋊C₄	C₂⁴×C₃₁	1	160	yes	—
113	C₂²⋊C₄	C₂⁴×C₃₁	1	16	yes	—
114	C₂²⋊C₄	C₂⁴×C₃₁	1	80	yes	—
115	C₄²	C₂⁴×C₃₁	1	160	yes	—
116	C₂³	—	4	10	no	—
117	C₂³	C₂³×C₄×C₃₁	1	20	yes	—
118	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
119	C₂³	C₂⁵×C₃₁	1	10	yes	—
120	Q₈	C₂³×C₄×C₃₁	1	160	yes	—
121	D₄	C₂³×C₄×C₃₁	1	160	yes	—
122	C₂×C₄	—	4	80	no	—
123	C₂×C₄	C₂³×C₄×C₃₁	1	40	yes	—
124	C₂×C₄	C₂³×C₄×C₃₁	1	240	yes	—
125	C₂×C₄	C₂⁵×C₃₁	1	32	yes	—
126	C₂×C₄	C₂⁵×C₃₁	1	20	yes	—
127	C₂²	—	2	20	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	4	20	no	—
130	C₂²	C₂⁴×C₄×C₃₁	1	2	yes	—
131	C₂²	C₂⁴×C₄×C₃₁	1	20	yes	—
132	C₂²	C₂⁶×C₃₁	1	1	yes	characteristic
133	C₄	—	2	32	no	—
134	C₄	C₂⁴×C₄×C₃₁	1	40	yes	—
135	C₂	—	2	2	no	—
136	C₂	—	2	20	no	—
137	C₂	C₂⁵×C₄×C₃₁	1	1	yes	characteristic
138	C₂	C₃₁×2₋¹⁺⁶	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂⁶
commutator	C₂	C₂⁵×C₄×C₃₁
Frattini, cosocle	C₂²	C₂⁶×C₃₁
prime radical	C₂²×C₃₁×2₋¹⁺⁴	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₃₁×2₋¹⁺⁴	C₂	semidirect
3. C₃₁×#3428	C₂	semidirect
4. C₂×C₃₁×#519	C₂	semidirect
5. C₂×C₃₁×2₋¹⁺⁴	C₄	semidirect
9. C₃₁×#520	C₂²	semidirect
11. C₃₁×#519	C₂²	semidirect
13. C₄×#41×C₃₁	C₂²	semidirect
14. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
16. C₃₁×2₋¹⁺⁴	C₂×C₄	semidirect
17. C₂×#41×C₃₁	C₂×C₄	semidirect
19. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
46. #62653	C₃₁	direct (N × H)
59. C₄×2₋¹⁺⁴	C₂×C₃₁	semidirect
60. #3428	C₂×C₃₁	semidirect
61. C₂×#519	C₂×C₃₁	semidirect
70. C₂×2₋¹⁺⁴	C₄×C₃₁	semidirect
74. #520	C₂²×C₃₁	semidirect
76. #519	C₂²×C₃₁	semidirect
78. C₄×#41	C₂²×C₃₁	semidirect
79. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
85. 2₋¹⁺⁴	C₂×C₄×C₃₁	semidirect
86. C₂×#41	C₂×C₄×C₃₁	semidirect
88. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
100. C₃₁	#62653	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×C₃₁×2₋¹⁺⁴	2
C₄×C₃₁×2₋¹⁺⁴	2
C₃₁×#3428	2
C₂×C₃₁×#519	2
#62653	31

Subgroups — C31×#62653 #21519412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#62653 #21519412