Group #36324892 — subgroups

Subgroups — #36324892

Order 21376.

200 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 97 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	1	yes	maximal, characteristic
2	#24923296	C₂	1	2	yes	maximal
3	C₂×#19128742	C₂	1	1	yes	maximal, characteristic
4	#24923028	C₂	1	2	yes	maximal
5	#427×C₁₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581934	C₂²	1	2	yes	—
7	C₂²×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₆₇	C₂²	1	1	yes	characteristic
9	#19128768	—	2	2	no	—
10	C₄⋊(C₁₆₇⋊C₈)	—	2	4	no	—
11	#19128742	—	2	2	no	—
12	C₂²×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
13	#117×C₁₆₇	C₂²	1	2	yes	—
14	#107×C₁₆₇	C₄	1	2	yes	complemented
15	#106×C₁₆₇	C₄	1	2	yes	complemented
16	C₂²×Dic₁₆₇	—	2	1	no	—
17	#14581934	—	2	2	no	—
18	#14581934	—	2	2	no	—
19	C₂×C₁₆₇⋊C₈	—	2	1	no	—
20	C₂×C₁₆₇⋊C₈	—	2	2	no	—
21	C₂×C₁₆₇⋊C₈	—	2	1	no	—
22	C₂×C₁₆₇⋊C₈	D₄	1	2	yes	—
23	C₄×Dic₁₆₇	—	2	1	no	—
24	C₄×Dic₁₆₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₁₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₁₆₇	D₄	1	2	yes	—
29	C₂×C₈×C₁₆₇	Q₈	1	2	yes	—
30	C₄⋊C₄×C₁₆₇	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₁₆₇	—	2	2	no	—
32	C₄²×C₁₆₇	—	2	2	no	—
33	C₂×Dic₁₆₇	—	2	2	no	—
34	C₂×Dic₁₆₇	—	4	2	no	—
35	C₁₆₇⋊C₈	—	2	2	no	—
36	C₁₆₇⋊C₈	—	4	2	no	—
37	C₂³×C₁₆₇	#41	1	1	yes	characteristic
38	C₂×C₄×C₁₆₇	—	2	4	no	—
39	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₆₇	#41	1	1	yes	characteristic
45	C₈×C₁₆₇	C₄⋊C₄	1	2	yes	—
46	C₈×C₁₆₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₁₆₇	—	2	2	no	—
48	C₂²×C₁₆₇	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₁₆₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₆₇	#125	1	1	yes	characteristic
51	C₄×C₁₆₇	—	4	4	no	—
52	C₄×C₁₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₁₆₇	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₁₆₇	#121	1	1	yes	characteristic
55	C₄×C₁₆₇	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₁₆₇	—	4	2	no	—
57	C₂×C₁₆₇	—	2	1	no	—
58	C₂×C₁₆₇	—	2	1	no	—
59	C₂×C₁₆₇	#389	1	1	yes	characteristic
60	C₂×C₁₆₇	#444	1	2	yes	—
61	C₁₆₇	#1946	1	1	yes	characteristic, complemented, Sylow
62	#1946	—	167	1	no	maximal, Sylow
63	C₂×#108	—	167	1	no	—
64	#427	D₁₆₇	1	1	yes	characteristic
65	#424	—	167	2	no	—
66	C₂×#97	—	167	1	no	—
67	#339	—	167	2	no	—
68	C₂×M₁₆	—	167	2	no	—
69	C₂²×C₈	—	167	1	no	—
70	C₂²×C₈	C₂×D₁₆₇	1	1	yes	characteristic
71	#117	C₂×D₁₆₇	1	2	yes	—
72	C₂×C₄²	—	167	1	no	—
73	#108	—	334	2	no	—
74	#107	Dic₁₆₇	1	2	yes	complemented
75	#106	Dic₁₆₇	1	2	yes	complemented
76	C₄⋊C₈	—	334	4	no	—
77	#97	—	334	2	no	—
78	C₂²×C₄	—	334	1	no	—
79	C₂²×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
80	M₁₆	—	334	2	no	—
81	M₁₆	—	334	2	no	—
82	C₂×C₈	—	334	1	no	—
83	C₂×C₈	—	334	1	no	—
84	C₂×C₈	—	334	2	no	—
85	C₂×C₈	—	167	2	no	—
86	C₂×C₈	C₂×Dic₁₆₇	1	1	yes	characteristic
87	C₂×C₈	C₂×Dic₁₆₇	1	1	yes	characteristic
88	C₂×C₈	D₆₆₈	1	2	yes	—
89	C₂×C₈	Dic₃₃₄	1	2	yes	—
90	C₄⋊C₄	C₂×Dic₁₆₇	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	334	1	no	—
94	C₄²	—	167	2	no	—
95	C₂³	#14581949	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	334	2	no	—
98	C₂×C₄	—	668	2	no	—
99	C₂×C₄	C₂²×Dic₁₆₇	1	1	yes	characteristic
100	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
101	C₂×C₄	C₂×D₆₆₈	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₁₆₇	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₃₃₄	1	1	yes	characteristic
104	C₂×C₄	#14581951	1	1	yes	characteristic
105	C₈	—	334	2	no	—
106	C₈	—	668	2	no	—
107	C₈	C₄⋊Dic₁₆₇	1	2	yes	—
108	C₈	C₄⋊Dic₁₆₇	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₁₆₇	1	1	yes	characteristic
111	C₂²	#19128812	1	1	yes	characteristic
112	C₂²	#19128823	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	668	2	no	—
115	C₄	C₂×C₄⋊Dic₁₆₇	1	1	yes	characteristic
116	C₄	Q₈×Dic₁₆₇	1	1	yes	characteristic
117	C₄	#19128807	1	1	yes	characteristic
118	C₄	#19128831	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#24923168	1	1	yes	characteristic
122	C₂	#24923391	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19128768

C₂

yes

maximal, characteristic

#24923296

C₂

yes

maximal

C₂×#19128742

C₂

yes

maximal, characteristic

#24923028

C₂

yes

maximal

#427×C₁₆₇

C₂

yes

maximal, characteristic

C₂×#14581934

C₂²

yes

—

C₂²×C₁₆₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₁₆₇

C₂²

yes

characteristic

#19128768

—

C₄⋊(C₁₆₇⋊C₈)

—

#19128742

—

C₂²×C₈×C₁₆₇

C₂²

yes

characteristic

#117×C₁₆₇

C₂²

yes

—

#107×C₁₆₇

C₄

yes

complemented

#106×C₁₆₇

C₄

yes

complemented

C₂²×Dic₁₆₇

—

#14581934

—

#14581934

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

D₄

yes

—

C₄×Dic₁₆₇

—

C₄×Dic₁₆₇

Q₈

yes

—

C₂²×C₄×C₁₆₇

C₂³

yes

characteristic

C₂×C₈×C₁₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₆₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₆₇

D₄

yes

—

C₂×C₈×C₁₆₇

Q₈

yes

—

C₄⋊C₄×C₁₆₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₆₇

—

C₄²×C₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₁₆₇⋊C₈

—

C₁₆₇⋊C₈

—

C₂³×C₁₆₇

#41

yes

characteristic

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₆₇

#41

yes

characteristic

C₈×C₁₆₇

C₄⋊C₄

yes

—

C₈×C₁₆₇

C₄⋊C₄

yes

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

C₄×D₄

yes

characteristic

C₂²×C₁₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₆₇

#125

yes

characteristic

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₆₇

C₄×Q₈

yes

characteristic

C₄×C₁₆₇

#121

yes

characteristic

C₄×C₁₆₇

C₄⋊Q₈

yes

characteristic

Dic₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

#389

yes

characteristic

C₂×C₁₆₇

#444

yes

—

C₁₆₇

#1946

yes

characteristic, complemented, Sylow

#1946

—

167

maximal, Sylow

C₂×#108

—

167

—

#427

D₁₆₇

yes

characteristic

#424

—

167

—

C₂×#97

—

167

—

#339

—

167

—

C₂×M₁₆

—

167

—

C₂²×C₈

—

167

—

C₂²×C₈

C₂×D₁₆₇

yes

characteristic

#117

C₂×D₁₆₇

yes

—

C₂×C₄²

—

167

—

#108

—

334

—

#107

Dic₁₆₇

yes

complemented

#106

Dic₁₆₇

yes

complemented

C₄⋊C₈

—

334

—

#97

—

334

—

C₂²×C₄

—

334

—

C₂²×C₄

C₂²×D₁₆₇

yes

characteristic

M₁₆

—

334

—

M₁₆

—

334

—

C₂×C₈

—

334

—

C₂×C₈

—

334

—

C₂×C₈

—

334

—

C₂×C₈

—

167

—

C₂×C₈

C₂×Dic₁₆₇

yes

characteristic

C₂×C₈

C₂×Dic₁₆₇

yes

characteristic

C₂×C₈

D₆₆₈

yes

—

C₂×C₈

Dic₃₃₄

yes

—

C₄⋊C₄

C₂×Dic₁₆₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

334

—

C₄²

—

167

—

C₂³

#14581949

yes

characteristic

C₂×C₄

—

C₂×C₄

—

334

—

C₂×C₄

—

668

—

C₂×C₄

C₂²×Dic₁₆₇

yes

characteristic

100

C₂×C₄

D₄×D₁₆₇

yes

characteristic

101

C₂×C₄

C₂×D₆₆₈

yes

characteristic

102

C₂×C₄

Q₈×D₁₆₇

yes

characteristic

103

C₂×C₄

C₂×Dic₃₃₄

yes

characteristic

104

C₂×C₄

#14581951

yes

characteristic

105

C₈

—

334

—

106

C₈

—

668

—

107

C₈

C₄⋊Dic₁₆₇

yes

—

108

C₈

C₄⋊Dic₁₆₇

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₁₆₇

yes

characteristic

111

C₂²

#19128812

yes

characteristic

112

C₂²

#19128823

yes

characteristic

113

C₄

—

114

C₄

—

668

—

115

C₄

C₂×C₄⋊Dic₁₆₇

yes

characteristic

116

C₄

Q₈×Dic₁₆₇

yes

characteristic

117

C₄

#19128807

yes

characteristic

118

C₄

#19128831

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#24923168

yes

characteristic

122

C₂

#24923391

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₆₇
commutator	C₂×C₄×C₁₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₇
Fitting	#427×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₄×D₄
prime radical	C₂³×C₁₆₇	#41
cosocle	C₂²×C₄×C₁₆₇	C₂³
hypercenter	#427	D₁₆₇
hypocenter	C₁₆₇	#1946
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₁₆₇

commutator

C₂×C₄×C₁₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₇

Fitting

#427×C₁₆₇

C₂

socle

C₂²×C₁₆₇

C₄×D₄

prime radical

C₂³×C₁₆₇

#41

cosocle

C₂²×C₄×C₁₆₇

C₂³

D₁₆₇

C₁₆₇

G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₁₆₇	C₄	semidirect
15. #106×C₁₆₇	C₄	semidirect
61. C₁₆₇	#1946	semidirect
74. #107	Dic₁₆₇	semidirect
75. #106	Dic₁₆₇	semidirect

H ≅ G/N

type

14. #107×C₁₆₇

C₄

semidirect

15. #106×C₁₆₇

C₄

semidirect

61. C₁₆₇

#1946

semidirect

74. #107

Dic₁₆₇

semidirect

75. #106

Dic₁₆₇

semidirect

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

subgroup	index
C₂×#19128768	2
#24923296	2
C₂×#19128742	2
#24923028	2
#427×C₁₆₇	2
#1946	167

subgroup

index

#427×C₁₆₇

167

Subgroups — #36324892

Conjugacy classes of subgroups

Complements

Maximal subgroups