Group #21470419 — subgroups

Subgroups — C₃₁×#13660 #21470419

Order 7936.

158 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₃₁×C₄⋊C₃₂	C₂	1	2	yes	maximal
4	C₃₁×#1397	C₂	1	2	yes	maximal
5	C₄×C₃₁×C₃₂	C₂	1	1	yes	maximal, characteristic
6	C₈×Q₈×C₃₁	C₄	1	1	yes	characteristic
7	C₂×C₃₁×C₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₃₁×C₃₂	C₂²	1	2	yes	—
9	C₂×C₃₁×C₃₂	C₂²	1	1	yes	characteristic
10	C₃₁×C₄⋊C₁₆	C₄	1	1	yes	characteristic
11	C₃₁×C₄⋊C₁₆	C₄	1	2	yes	—
12	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
14	C₄×Q₈×C₃₁	C₈	1	1	yes	characteristic
15	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆×C₃₁	C₂×C₄	1	2	yes	—
17	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
18	C₃₁×C₄⋊C₈	C₈	1	2	yes	—
19	C₃₁×C₄⋊C₈	C₈	1	1	yes	characteristic
20	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
22	C₃₁×C₃₂	—	2	1	no	—
23	C₃₁×C₃₂	—	2	2	no	—
24	C₃₁×C₃₂	Q₈	1	2	yes	complemented
25	C₂×Q₈×C₃₁	C₁₆	1	1	yes	characteristic
26	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₃₁	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
29	C₄⋊C₄×C₃₁	C₁₆	1	2	yes	—
30	C₄⋊C₄×C₃₁	C₁₆	1	1	yes	characteristic
31	C₄²×C₃₁	C₂×C₈	1	2	yes	—
32	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
33	C₁₆×C₃₁	—	2	1	no	—
34	C₁₆×C₃₁	—	2	2	no	—
35	C₁₆×C₃₁	C₂×Q₈	1	1	yes	characteristic
36	C₁₆×C₃₁	#41	1	1	yes	characteristic
37	#13660	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	Q₈×C₃₁	—	2	2	no	—
39	C₂×C₄×C₃₁	C₂×C₁₆	1	2	yes	—
40	C₂×C₄×C₃₁	C₂×C₁₆	1	1	yes	characteristic
41	C₂×C₄×C₃₁	C₂×C₁₆	1	2	yes	—
42	C₂×C₄×C₃₁	C₂×C₁₆	1	1	yes	characteristic
43	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
44	C₈×C₃₁	—	2	1	no	—
45	C₈×C₃₁	—	2	2	no	—
46	C₈×C₃₁	C₄×Q₈	1	1	yes	characteristic
47	C₈×C₃₁	#131	1	1	yes	characteristic
48	Q₈×C₁₆	C₂×C₃₁	1	1	yes	characteristic
49	C₄⋊C₃₂	C₂×C₃₁	1	1	yes	characteristic
50	C₄⋊C₃₂	C₂×C₃₁	1	2	yes	—
51	#1397	C₂×C₃₁	1	2	yes	—
52	C₄×C₃₂	C₂×C₃₁	1	1	yes	characteristic
53	C₂²×C₃₁	C₂²×C₁₆	1	1	yes	characteristic
54	C₄×C₃₁	—	2	2	no	—
55	C₄×C₃₁	—	2	2	no	—
56	C₄×C₃₁	—	2	1	no	—
57	C₄×C₃₁	M₆₄	1	2	yes	—
58	C₄×C₃₁	C₈×Q₈	1	1	yes	characteristic
59	C₄×C₃₁	#504	1	1	yes	characteristic
60	C₈×Q₈	C₄×C₃₁	1	1	yes	characteristic
61	C₂×C₃₂	C₂²×C₃₁	1	1	yes	characteristic
62	C₂×C₃₂	C₂²×C₃₁	1	2	yes	—
63	C₂×C₃₂	C₂²×C₃₁	1	1	yes	characteristic
64	C₄⋊C₁₆	C₄×C₃₁	1	1	yes	characteristic
65	C₄⋊C₁₆	C₄×C₃₁	1	2	yes	—
66	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
67	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
68	C₂×C₃₁	Q₈×C₁₆	1	1	yes	characteristic
69	C₂×C₃₁	C₂×M₆₄	1	1	yes	characteristic
70	C₂×C₃₁	#2258	1	1	yes	characteristic
71	C₄×Q₈	C₈×C₃₁	1	1	yes	characteristic
72	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
73	C₂×C₁₆	C₂×C₄×C₃₁	1	2	yes	—
74	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
75	C₄⋊C₈	C₈×C₃₁	1	2	yes	—
76	C₄⋊C₈	C₈×C₃₁	1	1	yes	characteristic
77	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
78	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
79	C₃₂	—	2	1	no	—
80	C₃₂	—	2	2	no	—
81	C₃₂	Q₈×C₃₁	1	2	yes	complemented
82	C₃₁	#13660	1	1	yes	characteristic, direct product factor, Sylow
83	C₂×Q₈	C₁₆×C₃₁	1	1	yes	characteristic
84	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
85	C₂×C₈	C₂×C₈×C₃₁	1	2	yes	—
86	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
87	C₄⋊C₄	C₁₆×C₃₁	1	2	yes	—
88	C₄⋊C₄	C₁₆×C₃₁	1	1	yes	characteristic
89	C₄²	C₂×C₈×C₃₁	1	2	yes	—
90	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
91	C₁₆	—	2	1	no	—
92	C₁₆	—	2	2	no	—
93	C₁₆	C₂×Q₈×C₃₁	1	1	yes	characteristic
94	C₁₆	#41×C₃₁	1	1	yes	characteristic
95	Q₈	—	2	2	no	—
96	C₂×C₄	C₂×C₁₆×C₃₁	1	2	yes	—
97	C₂×C₄	C₂×C₁₆×C₃₁	1	1	yes	characteristic
98	C₂×C₄	C₂×C₁₆×C₃₁	1	2	yes	—
99	C₂×C₄	C₂×C₁₆×C₃₁	1	1	yes	characteristic
100	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	2	2	no	—
103	C₈	C₄×Q₈×C₃₁	1	1	yes	characteristic
104	C₈	C₃₁×#131	1	1	yes	characteristic
105	C₂²	C₂²×C₁₆×C₃₁	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	—	2	2	no	—
108	C₄	—	2	1	no	—
109	C₄	C₃₁×M₆₄	1	2	yes	—
110	C₄	C₈×Q₈×C₃₁	1	1	yes	characteristic
111	C₄	C₃₁×#504	1	1	yes	characteristic
112	C₂	Q₈×C₁₆×C₃₁	1	1	yes	characteristic
113	C₂	C₂×C₃₁×M₆₄	1	1	yes	characteristic
114	C₂	C₃₁×#2258	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×C₁₆×C₃₁

C₂

yes

maximal, characteristic

C₃₁×C₄⋊C₃₂

C₂

yes

maximal, characteristic

C₃₁×C₄⋊C₃₂

C₂

yes

maximal

C₃₁×#1397

C₂

yes

maximal

C₄×C₃₁×C₃₂

C₂

yes

maximal, characteristic

C₈×Q₈×C₃₁

C₄

yes

characteristic

C₂×C₃₁×C₃₂

C₂²

yes

characteristic

C₂×C₃₁×C₃₂

C₂²

yes

—

C₂×C₃₁×C₃₂

C₂²

yes

characteristic

C₃₁×C₄⋊C₁₆

C₄

yes

characteristic

C₃₁×C₄⋊C₁₆

C₄

yes

—

C₄×C₁₆×C₃₁

C₂²

yes

characteristic

C₄×C₁₆×C₃₁

C₂²

yes

—

C₄×Q₈×C₃₁

C₈

yes

characteristic

C₂×C₁₆×C₃₁

C₂×C₄

yes

characteristic

C₂×C₁₆×C₃₁

C₂×C₄

yes

—

C₂×C₁₆×C₃₁

C₂³

yes

characteristic

C₃₁×C₄⋊C₈

C₈

yes

—

C₃₁×C₄⋊C₈

C₈

yes

characteristic

C₄×C₈×C₃₁

C₂×C₄

yes

characteristic

C₄×C₈×C₃₁

C₂×C₄

yes

—

C₃₁×C₃₂

—

C₃₁×C₃₂

—

C₃₁×C₃₂

Q₈

yes

complemented

C₂×Q₈×C₃₁

C₁₆

yes

characteristic

C₂×C₈×C₃₁

C₂×C₈

yes

characteristic

C₂×C₈×C₃₁

C₂×C₈

yes

—

C₂×C₈×C₃₁

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₃₁

C₁₆

yes

—

C₄⋊C₄×C₃₁

C₁₆

yes

characteristic

C₄²×C₃₁

C₂×C₈

yes

—

C₄²×C₃₁

C₂×C₈

yes

characteristic

C₁₆×C₃₁

—

C₁₆×C₃₁

—

C₁₆×C₃₁

C₂×Q₈

yes

C₁₆×C₃₁

yes

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

Q₈×C₃₁

—

C₂×C₄×C₃₁

C₂×C₁₆

yes

—

C₂×C₄×C₃₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₃₁

C₂×C₁₆

yes

—

C₂×C₄×C₃₁

C₂×C₁₆

yes

characteristic

C₂×C₄×C₃₁

C₂²×C₈

yes

characteristic

C₈×C₃₁

—

C₈×C₃₁

—

C₈×C₃₁

C₄×Q₈

yes

characteristic

C₈×C₃₁

#131

yes

characteristic

Q₈×C₁₆

C₂×C₃₁

yes

characteristic

C₄⋊C₃₂

C₂×C₃₁

yes

characteristic

C₄⋊C₃₂

C₂×C₃₁

yes

—

#1397

C₂×C₃₁

yes

—

C₄×C₃₂

C₂×C₃₁

yes

characteristic

C₂²×C₃₁

C₂²×C₁₆

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

M₆₄

yes

—

C₄×C₃₁

C₈×Q₈

yes

characteristic

C₄×C₃₁

#504

yes

characteristic

C₈×Q₈

C₄×C₃₁

yes

characteristic

C₂×C₃₂

C₂²×C₃₁

yes

characteristic

C₂×C₃₂

C₂²×C₃₁

yes

—

C₂×C₃₂

C₂²×C₃₁

yes

characteristic

C₄⋊C₁₆

C₄×C₃₁

yes

characteristic

C₄⋊C₁₆

C₄×C₃₁

yes

—

C₄×C₁₆

C₂²×C₃₁

yes

characteristic

C₄×C₁₆

C₂²×C₃₁

yes

—

C₂×C₃₁

Q₈×C₁₆

yes

characteristic

C₂×C₃₁

C₂×M₆₄

yes

characteristic

C₂×C₃₁

#2258

yes

characteristic

C₄×Q₈

C₈×C₃₁

yes

characteristic

C₂×C₁₆

C₂×C₄×C₃₁

yes

characteristic

C₂×C₁₆

C₂×C₄×C₃₁

yes

—

C₂×C₁₆

C₂³×C₃₁

yes

characteristic

C₄⋊C₈

C₈×C₃₁

yes

—

C₄⋊C₈

C₈×C₃₁

yes

characteristic

C₄×C₈

C₂×C₄×C₃₁

yes

characteristic

C₄×C₈

C₂×C₄×C₃₁

yes

—

C₃₂

—

C₃₂

—

C₃₂

Q₈×C₃₁

yes

complemented

C₃₁

#13660

yes

characteristic, direct product factor, Sylow

C₂×Q₈

C₁₆×C₃₁

yes

characteristic

C₂×C₈

C₂×C₈×C₃₁

yes

characteristic

C₂×C₈

C₂×C₈×C₃₁

yes

—

C₂×C₈

C₂²×C₄×C₃₁

yes

characteristic

C₄⋊C₄

C₁₆×C₃₁

yes

—

C₄⋊C₄

C₁₆×C₃₁

yes

characteristic

C₄²

C₂×C₈×C₃₁

yes

—

C₄²

C₂×C₈×C₃₁

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

C₂×Q₈×C₃₁

yes

characteristic

C₁₆

#41×C₃₁

yes

characteristic

Q₈

—

C₂×C₄

C₂×C₁₆×C₃₁

yes

—

C₂×C₄

C₂×C₁₆×C₃₁

yes

characteristic

C₂×C₄

C₂×C₁₆×C₃₁

yes

—

C₂×C₄

C₂×C₁₆×C₃₁

yes

characteristic

100

C₂×C₄

C₂²×C₈×C₃₁

yes

characteristic

101

C₈

—

102

C₈

—

103

C₈

C₄×Q₈×C₃₁

yes

characteristic

104

C₈

C₃₁×#131

yes

characteristic

105

C₂²

C₂²×C₁₆×C₃₁

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

C₃₁×M₆₄

yes

—

110

C₄

C₈×Q₈×C₃₁

yes

characteristic

111

C₄

C₃₁×#504

yes

characteristic

112

C₂

Q₈×C₁₆×C₃₁

yes

characteristic

113

C₂

C₂×C₃₁×M₆₄

yes

characteristic

114

C₂

C₃₁×#2258

yes

characteristic

	subgroup	G/H
center	C₂×C₁₆×C₃₁	C₂³
commutator	C₂²	C₂²×C₁₆×C₃₁
Frattini, cosocle	C₂×C₁₆	C₂³×C₃₁
socle, prime radical	C₂²×C₃₁	C₂²×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₃₁ ◁ G

subgroup

G/H

center

C₂×C₁₆×C₃₁

C₂³

commutator

C₂²

C₂²×C₁₆×C₃₁

Frattini, cosocle

C₂×C₁₆

C₂³×C₃₁

socle, prime radical

C₂²×C₃₁

C₂²×C₁₆

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₁₆×C₃₁ ◁ G

N	H ≅ G/N	type
24. C₃₁×C₃₂	Q₈	semidirect
37. #13660	C₃₁	direct (N × H)
81. C₃₂	Q₈×C₃₁	semidirect
82. C₃₁	#13660	direct (N × H)

H ≅ G/N

type

24. C₃₁×C₃₂

Q₈

semidirect

37. #13660

C₃₁

direct (N × H)

81. C₃₂

Q₈×C₃₁

semidirect

82. C₃₁

#13660

direct (N × H)

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

subgroup	index
Q₈×C₁₆×C₃₁	2
C₃₁×C₄⋊C₃₂	2
C₃₁×#1397	2
C₄×C₃₁×C₃₂	2
#13660	31

subgroup

index

Q₈×C₁₆×C₃₁

C₃₁×C₄⋊C₃₂

C₃₁×#1397

C₄×C₃₁×C₃₂

#13660

Subgroups — C₃₁×#13660 #21470419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C31×#13660 #21470419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13660 #21470419