Group #36324176 — subgroups

Subgroups — #36324176

Order 21376.

292 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 117 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×#19128768	C₂	1	2	yes	maximal, complemented
2	#24923233	C₂	1	2	yes	maximal
3	#24923034	C₂	1	2	yes	maximal, complemented
4	C₂³×C₈×C₁₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581934	C₂²	1	4	yes	—
6	#19128768	—	2	4	no	—
7	C₂²⋊(C₁₆₇⋊C₈)	—	2	4	no	—
8	C₂³×C₄×C₁₆₇	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₁₆₇	—	2	4	no	—
10	C₂²×C₈×C₁₆₇	C₄	1	4	yes	—
11	C₂²×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
13	#14581934	—	2	4	no	—
14	#14581934	—	2	4	no	—
15	C₂×C₁₆₇⋊C₈	—	2	4	no	—
16	C₂⁴×C₁₆₇	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₁₆₇	—	2	2	no	—
18	C₂²×C₄×C₁₆₇	—	2	2	no	—
19	C₂²×C₄×C₁₆₇	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₁₆₇	D₄	1	2	yes	—
21	C₂²×C₄×C₁₆₇	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₁₆₇	—	2	2	no	—
24	C₂×C₈×C₁₆₇	—	2	8	no	—
25	C₂×C₈×C₁₆₇	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₁₆₇	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₁₆₇	D₄	1	4	yes	—
28	C₁₆₇⋊C₈	—	4	4	no	—
29	C₂³×C₁₆₇	—	2	4	no	—
30	C₂³×C₁₆₇	—	2	2	no	—
31	C₂³×C₁₆₇	C₄⋊C₄	1	2	yes	—
32	C₂³×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₁₆₇	—	2	2	no	—
34	C₂×C₄×C₁₆₇	—	2	2	no	—
35	C₂×C₄×C₁₆₇	—	2	2	no	—
36	C₂×C₄×C₁₆₇	—	2	4	no	—
37	C₂×C₄×C₁₆₇	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₆₇	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₆₇	#41	1	2	yes	—
42	C₈×C₁₆₇	—	2	2	no	—
43	C₈×C₁₆₇	C₂²⋊C₄	1	4	yes	—
44	C₂²×C₁₆₇	—	2	8	no	—
45	C₂²×C₁₆₇	—	2	2	no	—
46	C₂²×C₁₆₇	—	2	2	no	—
47	C₂²×C₁₆₇	—	2	2	no	—
48	C₂²×C₁₆₇	#108	1	4	yes	complemented
49	C₂²×C₁₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₁₆₇	C₂²⋊Q₈	1	2	yes	—
51	C₄×C₁₆₇	—	2	2	no	—
52	C₄×C₁₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₁₆₇	#117	1	1	yes	characteristic
54	C₄×C₁₆₇	#121	1	2	yes	—
55	C₂×C₁₆₇	—	2	4	no	—
56	C₂×C₁₆₇	—	2	2	no	—
57	C₂×C₁₆₇	#380	1	1	yes	characteristic
58	C₂×C₁₆₇	C₂×#108	1	2	yes	—
59	C₁₆₇	#1810	1	1	yes	characteristic, complemented, Sylow
60	#1810	—	167	1	no	maximal, Sylow
61	C₂³×C₈	D₁₆₇	1	1	yes	characteristic
62	C₂×#108	—	167	2	no	—
63	#407	—	167	2	no	—
64	#341	—	167	2	no	—
65	C₂³×C₄	C₂×D₁₆₇	1	1	yes	characteristic
66	C₂×M₁₆	—	167	4	no	—
67	C₂²×C₈	—	2	4	no	—
68	C₂²×C₈	Dic₁₆₇	1	4	yes	—
69	C₂²×C₈	C₂×D₁₆₇	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₁₆₇	1	1	yes	characteristic
71	#108	—	334	4	no	—
72	C₂²⋊C₈	—	334	4	no	—
73	C₂⁴	Dic₃₃₄	1	1	yes	characteristic
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂×Dic₁₆₇	1	2	yes	—
77	C₂²×C₄	D₆₆₈	1	2	yes	—
78	C₂²×C₄	Dic₃₃₄	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
80	M₁₆	—	334	4	no	—
81	M₁₆	—	334	4	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	8	no	—
84	C₂×C₈	—	334	4	no	—
85	C₂×C₈	C₂×Dic₁₆₇	1	2	yes	—
86	C₂×C₈	C₂×Dic₁₆₇	1	2	yes	—
87	C₂×C₈	C₂²⋊D₁₆₇	1	4	yes	—
88	C₂³	—	2	4	no	—
89	C₂³	—	2	2	no	—
90	C₂³	C₄⋊Dic₁₆₇	1	2	yes	—
91	C₂³	C₂×Dic₃₃₄	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	C₄⋊Dic₁₆₇	1	2	yes	—
97	C₂×C₄	C₂²×Dic₁₆₇	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₁₆₇	1	2	yes	—
99	C₂×C₄	C₂×D₆₆₈	1	1	yes	characteristic
100	C₂×C₄	#14581950	1	2	yes	—
101	C₈	—	2	2	no	—
102	C₈	—	668	4	no	—
103	C₈	C₂²⋊Dic₁₆₇	1	4	yes	—
104	C₂²	—	2	8	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	2	no	—
108	C₂²	#19128768	1	4	yes	complemented
109	C₂²	C₂×C₄⋊Dic₁₆₇	1	1	yes	characteristic
110	C₂²	#19128811	1	2	yes	—
111	C₄	—	2	2	no	—
112	C₄	C₂×C₂²⋊Dic₁₆₇	1	1	yes	characteristic
113	C₄	#19128791	1	1	yes	characteristic
114	C₄	#19128806	1	2	yes	—
115	C₂	—	2	4	no	—
116	C₂	—	2	2	no	—
117	C₂	#24923129	1	1	yes	characteristic
118	C₂	C₂×#19128768	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128768

C₂

yes

maximal, complemented

#24923233

C₂

yes

maximal

#24923034

C₂

yes

maximal, complemented

C₂³×C₈×C₁₆₇

C₂

yes

maximal, characteristic

C₂×#14581934

C₂²

yes

—

#19128768

—

C₂²⋊(C₁₆₇⋊C₈)

—

C₂³×C₄×C₁₆₇

C₂²

yes

characteristic

C₂²×C₈×C₁₆₇

—

C₂²×C₈×C₁₆₇

C₄

yes

—

C₂²×C₈×C₁₆₇

C₂²

yes

characteristic

C₂²×C₈×C₁₆₇

C₂²

yes

characteristic

#14581934

—

#14581934

—

C₂×C₁₆₇⋊C₈

—

C₂⁴×C₁₆₇

Q₈

yes

characteristic

C₂²×C₄×C₁₆₇

—

C₂²×C₄×C₁₆₇

—

C₂²×C₄×C₁₆₇

C₂×C₄

yes

—

C₂²×C₄×C₁₆₇

D₄

yes

—

C₂²×C₄×C₁₆₇

Q₈

yes

characteristic

C₂²×C₄×C₁₆₇

C₂³

yes

characteristic

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

C₂×C₄

yes

—

C₂×C₈×C₁₆₇

C₂×C₄

yes

—

C₂×C₈×C₁₆₇

D₄

yes

—

C₁₆₇⋊C₈

—

C₂³×C₁₆₇

—

C₂³×C₁₆₇

—

C₂³×C₁₆₇

C₄⋊C₄

yes

—

C₂³×C₁₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

#41

yes

—

C₈×C₁₆₇

—

C₈×C₁₆₇

C₂²⋊C₄

yes

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

#108

yes

complemented

C₂²×C₁₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₆₇

C₂²⋊Q₈

yes

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×C₂²⋊C₄

yes

C₄×C₁₆₇

yes

C₄×C₁₆₇

yes

—

C₂×C₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

#380

yes

characteristic

C₂×C₁₆₇

C₂×#108

yes

—

C₁₆₇

#1810

yes

characteristic, complemented, Sylow

#1810

—

167

maximal, Sylow

C₂³×C₈

D₁₆₇

yes

characteristic

C₂×#108

—

167

—

#407

—

167

—

#341

—

167

—

C₂³×C₄

C₂×D₁₆₇

yes

characteristic

C₂×M₁₆

—

167

—

C₂²×C₈

—

C₂²×C₈

Dic₁₆₇

yes

—

C₂²×C₈

C₂×D₁₆₇

yes

characteristic

C₂²×C₈

C₂×D₁₆₇

yes

characteristic

#108

—

334

—

C₂²⋊C₈

—

334

—

C₂⁴

Dic₃₃₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₆₇

yes

—

C₂²×C₄

D₆₆₈

yes

—

C₂²×C₄

Dic₃₃₄

yes

characteristic

C₂²×C₄

C₂²×D₁₆₇

yes

characteristic

M₁₆

—

334

—

M₁₆

—

334

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

334

—

C₂×C₈

C₂×Dic₁₆₇

yes

—

C₂×C₈

C₂×Dic₁₆₇

yes

—

C₂×C₈

C₂²⋊D₁₆₇

yes

—

C₂³

—

C₂³

—

C₂³

C₄⋊Dic₁₆₇

yes

—

C₂³

C₂×Dic₃₃₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₁₆₇

yes

—

C₂×C₄

C₂²×Dic₁₆₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₆₇

yes

—

C₂×C₄

C₂×D₆₆₈

yes

characteristic

100

C₂×C₄

#14581950

yes

—

101

C₈

—

102

C₈

—

668

—

103

C₈

C₂²⋊Dic₁₆₇

yes

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

#19128768

yes

complemented

109

C₂²

C₂×C₄⋊Dic₁₆₇

yes

characteristic

110

C₂²

#19128811

yes

—

111

C₄

—

112

C₄

C₂×C₂²⋊Dic₁₆₇

yes

characteristic

113

C₄

#19128791

yes

characteristic

114

C₄

#19128806

yes

—

115

C₂

—

116

C₂

—

117

C₂

#24923129

yes

characteristic

118

C₂

C₂×#19128768

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₆₆₈
commutator	C₂×C₄×C₁₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₇
Fitting	C₂³×C₈×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₆₇	Q₈
cosocle	C₂²×C₄×C₁₆₇	C₂³
hypercenter	C₂³×C₈	D₁₆₇
hypocenter	C₁₆₇	#1810
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₆₆₈

commutator

C₂×C₄×C₁₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₇

Fitting

C₂³×C₈×C₁₆₇

C₂

socle

C₂²×C₁₆₇

C₂×C₄⋊C₄

prime radical

C₂⁴×C₁₆₇

Q₈

cosocle

C₂²×C₄×C₁₆₇

C₂³

hypercenter

C₂³×C₈

D₁₆₇

hypocenter

C₁₆₇

#1810

LCS

G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
1. C₂×#19128768	C₂	semidirect
3. #24923034	C₂	semidirect
48. C₂²×C₁₆₇	#108	semidirect
59. C₁₆₇	#1810	semidirect
108. C₂²	#19128768	semidirect

H ≅ G/N

type

1. C₂×#19128768

C₂

semidirect

3. #24923034

C₂

semidirect

48. C₂²×C₁₆₇

#108

semidirect

59. C₁₆₇

#1810

semidirect

108. C₂²

#19128768

semidirect

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

subgroup	index
C₂×#19128768	2
#24923233	2
#24923034	2
C₂³×C₈×C₁₆₇	2
#1810	167

subgroup

index

C₂³×C₈×C₁₆₇

167

Subgroups — #36324176

Conjugacy classes of subgroups

Complements

Maximal subgroups