Group #16846512 — subgroups

Subgroups — #16846512

Order 3968.

332 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 137 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	C₂×#11758446	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758398	C₂	1	4	yes	maximal
3	C₂×C₄×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#384	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₆₂	C₂²	1	2	yes	—
6	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
7	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃₁	—	2	2	no	—
10	C₂²×C₃₁⋊C₈	C₂²	1	2	yes	—
11	#11758446	—	2	2	no	—
12	#11758446	C₄	1	4	yes	complemented
13	#11758398	—	2	8	no	—
14	C₄×C₃₁⋊C₈	—	2	2	no	—
15	C₂²×C₄×C₃₁	—	2	2	no	—
16	C₂²×C₄×C₃₁	D₄	1	2	yes	—
17	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₃₁	—	2	2	no	—
19	C₂×Dic₆₂	—	2	4	no	—
20	C₂×Dic₆₂	—	2	4	no	—
21	C₂×Dic₆₂	—	2	2	no	—
22	C₂×Dic₆₂	C₂×C₄	1	4	yes	—
23	C₄⋊C₄×C₃₁	—	2	4	no	—
24	C₄⋊C₄×C₃₁	—	4	4	no	—
25	C₄²×C₃₁	C₂×C₄	1	2	yes	—
26	C₄²×C₃₁	D₄	1	2	yes	—
27	C₄⋊Dic₃₁	—	2	8	no	—
28	C₂×C₃₁⋊C₈	—	2	2	no	—
29	C₂×C₃₁⋊C₈	—	2	2	no	—
30	C₂×C₃₁⋊C₈	D₄	1	4	yes	—
31	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₃₁	—	4	2	no	—
33	C₂×C₄×C₃₁	—	4	4	no	—
34	C₂×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₃₁	SD₁₆	1	4	yes	—
36	C₂×C₄×C₃₁	Q₁₆	1	4	yes	—
37	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₃₁	#41	1	2	yes	—
41	C₂×Dic₃₁	—	2	4	no	—
42	C₂×Dic₃₁	—	4	2	no	—
43	C₂×Dic₃₁	—	4	2	no	—
44	Dic₆₂	—	2	8	no	—
45	Dic₆₂	—	4	4	no	—
46	C₃₁⋊C₈	—	2	4	no	—
47	#1956	—	31	1	no	maximal, Sylow
48	C₂²×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₃₁	#124	1	1	yes	characteristic
50	C₂²×C₃₁	#127	1	1	yes	characteristic
51	C₂²×C₃₁	C₂×SD₁₆	1	2	yes	—
52	C₂²×C₃₁	C₂×Q₁₆	1	2	yes	—
53	C₄×C₃₁	—	8	2	no	—
54	C₄×C₃₁	#103	1	8	yes	complemented
55	C₄×C₃₁	C₄×D₄	1	2	yes	—
56	C₄×C₃₁	#121	1	2	yes	—
57	Dic₃₁	—	4	4	no	—
58	C₂×C₄⋊Q₈	—	31	1	no	—
59	C₂×#103	—	31	4	no	—
60	C₂×C₄×C₈	—	31	1	no	—
61	#384	D₃₁	1	1	yes	characteristic
62	C₂×C₃₁	#390	1	1	yes	characteristic
63	C₂×C₃₁	C₂×#103	1	2	yes	—
64	C₂×C₃₁	#487	1	2	yes	—
65	C₂×C₃₁	#492	1	1	yes	characteristic
66	C₂×C₃₁	#494	1	1	yes	characteristic
67	C₂²×Q₈	—	31	2	no	—
68	C₂²×C₈	—	31	2	no	—
69	C₄⋊Q₈	—	31	4	no	—
70	C₄⋊Q₈	—	62	2	no	—
71	C₂×C₄⋊C₄	—	2	2	no	—
72	C₂×C₄⋊C₄	—	62	2	no	—
73	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
74	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
75	#103	—	62	8	no	—
76	C₄×C₈	—	62	2	no	—
77	C₃₁	#1956	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₈	—	31	4	no	—
79	C₂×Q₈	—	62	2	no	—
80	C₂×Q₈	—	62	4	no	—
81	C₂×Q₈	—	62	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	62	2	no	—
84	C₂²×C₄	C₂²⋊D₃₁	1	2	yes	—
85	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
86	C₂×C₈	—	31	4	no	—
87	C₂×C₈	—	62	2	no	—
88	C₂×C₈	—	62	2	no	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	4	4	no	—
91	C₄⋊C₄	—	62	8	no	—
92	C₄²	C₄×D₃₁	1	2	yes	—
93	C₄²	D₁₂₄	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
95	Q₈	—	62	8	no	—
96	Q₈	—	124	4	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	62	4	no	—
100	C₂×C₄	—	124	2	no	—
101	C₂×C₄	—	124	2	no	—
102	C₂×C₄	(C₂×D₃₁)⋊C₄	1	4	yes	—
103	C₂×C₄	#92480	1	4	yes	—
104	C₂×C₄	#92482	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
106	C₂×C₄	C₂×D₁₂₄	1	1	yes	characteristic
107	C₂×C₄	D₄×D₃₁	1	2	yes	—
108	C₂×C₄	#92499	1	2	yes	—
109	C₈	—	62	4	no	—
110	C₂²	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
111	C₂²	C₂×#92480	1	2	yes	—
112	C₂²	#11758517	1	1	yes	characteristic
113	C₂²	C₂×#92482	1	2	yes	—
114	C₂²	#11758524	1	1	yes	characteristic
115	C₄	—	8	2	no	—
116	C₄	—	124	4	no	—
117	C₄	#11758398	1	8	yes	complemented
118	C₄	#11758470	1	2	yes	—
119	C₄	#11758472	1	2	yes	—
120	C₂	#14520950	1	1	yes	characteristic
121	C₂	C₂×#11758398	1	2	yes	—
122	C₂	#14521401	1	2	yes	—
123	C₂	#14521421	1	1	yes	characteristic
124	C₂	#14521428	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758446

C₂

yes

maximal, characteristic

C₂×#11758398

C₂

yes

maximal

C₂×C₄×C₃₁⋊C₈

C₂

yes

maximal, characteristic

C₃₁×#384

C₂

yes

maximal, characteristic

C₂²×Dic₆₂

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

—

C₂²×C₃₁⋊C₈

C₂²

yes

—

#11758446

—

#11758446

C₄

yes

complemented

#11758398

—

C₄×C₃₁⋊C₈

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

D₄

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₂×Dic₆₂

C₂×C₄

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂×C₄

yes

—

C₄²×C₃₁

D₄

yes

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

—

C₂×C₃₁⋊C₈

D₄

yes

—

C₂³×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

SD₁₆

yes

—

C₂×C₄×C₃₁

Q₁₆

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

C₃₁⋊C₈

—

#1956

—

maximal, Sylow

C₂²×C₃₁

C₂×C₂²⋊C₄

yes

C₂²×C₃₁

yes

C₂²×C₃₁

yes

C₂²×C₃₁

C₂×SD₁₆

yes

—

C₂²×C₃₁

C₂×Q₁₆

yes

—

C₄×C₃₁

—

C₄×C₃₁

#103

yes

complemented

C₄×C₃₁

C₄×D₄

yes

—

C₄×C₃₁

#121

yes

—

Dic₃₁

—

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×C₄×C₈

—

#384

D₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

—

C₂×C₃₁

#487

yes

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#103

—

C₄×C₈

—

C₃₁

#1956

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₃₁

yes

—

C₄²

D₁₂₄

yes

—

C₂³

C₂×C₂²⋊D₃₁

yes

characteristic

Q₈

—

Q₈

—

124

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

124

—

101

C₂×C₄

—

124

—

102

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

103

C₂×C₄

#92480

yes

—

104

C₂×C₄

#92482

yes

—

105

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

106

C₂×C₄

C₂×D₁₂₄

yes

characteristic

107

C₂×C₄

D₄×D₃₁

yes

—

108

C₂×C₄

#92499

yes

—

109

C₈

—

110

C₂²

C₂×(C₂×D₃₁)⋊C₄

yes

characteristic

111

C₂²

C₂×#92480

yes

—

112

C₂²

#11758517

yes

characteristic

113

C₂²

C₂×#92482

yes

—

114

C₂²

#11758524

yes

characteristic

115

C₄

—

116

C₄

—

124

—

117

C₄

#11758398

yes

complemented

118

C₄

#11758470

yes

—

119

C₄

#11758472

yes

—

120

C₂

#14520950

yes

characteristic

121

C₂

C₂×#11758398

yes

—

122

C₂

#14521401

yes

—

123

C₂

#14521421

yes

characteristic

124

C₂

#14521428

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₁
commutator	C₂×C₄×C₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#384	C₂
socle, prime radical	C₂³×C₃₁	C₂×D₄
cosocle	C₂²×C₄×C₃₁	C₂³
hypercenter	#384	D₃₁
hypocenter	C₃₁	#1956
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₁

commutator

C₂×C₄×C₃₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₁

Fitting

C₃₁×#384

C₂

socle, prime radical

C₂³×C₃₁

C₂×D₄

cosocle

C₂²×C₄×C₃₁

C₂³

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
12. #11758446	C₄	semidirect
54. C₄×C₃₁	#103	semidirect
77. C₃₁	#1956	semidirect
117. C₄	#11758398	semidirect

H ≅ G/N

type

12. #11758446

C₄

semidirect

54. C₄×C₃₁

#103

semidirect

77. C₃₁

#1956

semidirect

117. C₄

#11758398

semidirect

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

subgroup	index
C₂×#11758446	2
C₂×#11758398	2
C₂×C₄×C₃₁⋊C₈	2
C₃₁×#384	2
#1956	31

subgroup

index

C₂×#11758446

C₂×#11758398

C₂×C₄×C₃₁⋊C₈

C₃₁×#384

#1956

Subgroups — #16846512

Conjugacy classes of subgroups

Complements

Maximal subgroups