Group #36324919 — subgroups

Subgroups — #36324919

Order 21376.

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

normal subgroups only

This group has 196 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	#24923311	C₂	1	2	yes	maximal
2	C₂×#19128764	C₂	1	1	yes	maximal, characteristic
3	#24923224	C₂	1	1	yes	maximal, characteristic
4	#24923037	C₂	1	2	yes	maximal
5	C₂×#108×C₁₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581934	C₂²	1	1	yes	characteristic
7	#19128791	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₆₇	C₂²	1	2	yes	—
9	#19128766	—	2	2	no	—
10	#19128764	—	2	2	no	—
11	#19128764	—	2	1	no	—
12	#19128764	—	2	1	no	—
13	#19128742	—	2	1	no	—
14	C₈×Dic₁₆₇	—	2	1	no	—
15	C₂×M₁₆×C₁₆₇	C₂²	1	2	yes	—
16	C₂²×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
17	#108×C₁₆₇	C₄	1	4	yes	complemented
18	C₂²×Dic₁₆₇	—	2	2	no	—
19	#14581934	—	2	1	no	—
20	#14581934	Q₈	1	2	yes	—
21	C₂×C₁₆₇⋊C₈	—	2	1	no	—
22	C₄⋊Dic₁₆₇	—	2	2	no	—
23	C₄⋊Dic₁₆₇	—	2	2	no	—
24	C₄⋊Dic₁₆₇	D₄	1	2	yes	—
25	C₂²⋊Dic₁₆₇	—	2	1	no	—
26	C₄×Dic₁₆₇	—	2	1	no	—
27	C₂²×C₄×C₁₆₇	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₆₇	—	2	2	no	—
29	M₁₆×C₁₆₇	C₂×C₄	1	4	yes	—
30	C₂×C₈×C₁₆₇	—	2	2	no	—
31	C₂×C₈×C₁₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
33	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
34	C₂×C₈×C₁₆₇	Q₈	1	2	yes	—
35	C₂×Dic₁₆₇	—	2	2	no	—
36	C₂×Dic₁₆₇	—	4	2	no	—
37	C₂×Dic₁₆₇	—	8	2	no	—
38	C₁₆₇⋊C₈	—	2	2	no	—
39	C₂³×C₁₆₇	#41	1	1	yes	characteristic
40	C₂×C₄×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₆₇	#41	1	1	yes	characteristic
46	C₈×C₁₆₇	—	2	4	no	—
47	C₈×C₁₆₇	C₄⋊C₄	1	2	yes	—
48	C₈×C₁₆₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₁₆₇	—	2	2	no	—
50	C₂²×C₁₆₇	C₄×Q₈	1	1	yes	characteristic
51	C₂²×C₁₆₇	#121	1	1	yes	characteristic
52	C₂²×C₁₆₇	#125	1	1	yes	characteristic
53	C₄×C₁₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₁₆₇	C₄×D₄	1	1	yes	characteristic
55	C₄×C₁₆₇	C₂²⋊Q₈	1	1	yes	characteristic
56	C₄×C₁₆₇	C₄⋊Q₈	1	1	yes	characteristic
57	Dic₁₆₇	—	4	2	no	—
58	Dic₁₆₇	—	8	2	no	—
59	C₂×C₁₆₇	—	2	1	no	—
60	C₂×C₁₆₇	—	2	1	no	—
61	C₂×C₁₆₇	#389	1	1	yes	characteristic
62	C₂×C₁₆₇	#471	1	2	yes	—
63	C₁₆₇	#1951	1	1	yes	characteristic, complemented, Sylow
64	#1951	—	167	1	no	maximal, Sylow
65	C₂×#108	D₁₆₇	1	1	yes	characteristic
66	#428	—	167	2	no	—
67	C₂×#106	—	167	1	no	—
68	#405	—	167	1	no	—
69	#343	—	167	2	no	—
70	C₂×M₁₆	—	167	1	no	—
71	C₂×M₁₆	C₂×D₁₆₇	1	2	yes	—
72	C₂²×C₈	C₂×D₁₆₇	1	1	yes	characteristic
73	#117	—	167	1	no	—
74	C₂×C₄⋊C₄	—	167	2	no	—
75	#108	Dic₁₆₇	1	4	yes	complemented
76	#107	—	334	2	no	—
77	#106	—	334	1	no	—
78	#106	—	334	1	no	—
79	#106	—	334	2	no	—
80	#97	—	334	1	no	—
81	C₄×C₈	—	334	1	no	—
82	C₂²×C₄	—	334	2	no	—
83	C₂²×C₄	C₂²×D₁₆₇	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	334	1	no	—
86	M₁₆	—	167	2	no	—
87	M₁₆	C₂×Dic₁₆₇	1	4	yes	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	334	1	no	—
90	C₂×C₈	C₂×Dic₁₆₇	1	2	yes	—
91	C₂×C₈	D₆₆₈	1	1	yes	characteristic
92	C₂×C₈	D₆₆₈	1	1	yes	characteristic
93	C₂×C₈	Dic₃₃₄	1	2	yes	—
94	C₄⋊C₄	—	334	2	no	—
95	C₄⋊C₄	—	334	2	no	—
96	C₄⋊C₄	—	167	2	no	—
97	C₂²⋊C₄	—	334	1	no	—
98	C₄²	—	334	1	no	—
99	C₂³	#14581951	1	1	yes	characteristic
100	C₂×C₄	—	1336	2	no	—
101	C₂×C₄	—	334	2	no	—
102	C₂×C₄	—	668	2	no	—
103	C₂×C₄	C₂²×Dic₁₆₇	1	1	yes	characteristic
104	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
105	C₂×C₄	C₂×D₆₆₈	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₆₇	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₃₃₄	1	1	yes	characteristic
108	C₂×C₄	#14581949	1	1	yes	characteristic
109	C₈	—	2	4	no	—
110	C₈	—	334	2	no	—
111	C₈	C₄⋊Dic₁₆₇	1	2	yes	—
112	C₈	C₄⋊Dic₁₆₇	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	Q₈×Dic₁₆₇	1	1	yes	characteristic
115	C₂²	#19128807	1	1	yes	characteristic
116	C₂²	#19128823	1	1	yes	characteristic
117	C₄	—	1336	2	no	—
118	C₄	—	668	2	no	—
119	C₄	C₂×C₄⋊Dic₁₆₇	1	1	yes	characteristic
120	C₄	D₄×Dic₁₆₇	1	1	yes	characteristic
121	C₄	#19128812	1	1	yes	characteristic
122	C₄	#19128831	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#24923168	1	1	yes	characteristic
126	C₂	#24923533	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24923311

C₂

yes

maximal

C₂×#19128764

C₂

yes

maximal, characteristic

#24923224

C₂

yes

maximal, characteristic

#24923037

C₂

yes

maximal

C₂×#108×C₁₆₇

C₂

yes

maximal, characteristic

C₂×#14581934

C₂²

yes

characteristic

#19128791

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₆₇

C₂²

yes

—

—

—

—

—

—

C₈×Dic₁₆₇

—

C₂×M₁₆×C₁₆₇

C₂²

yes

—

C₂²×C₈×C₁₆₇

C₂²

yes

characteristic

#108×C₁₆₇

C₄

yes

complemented

C₂²×Dic₁₆₇

—

#14581934

—

#14581934

Q₈

yes

—

C₂×C₁₆₇⋊C₈

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

D₄

yes

—

C₂²⋊Dic₁₆₇

—

C₄×Dic₁₆₇

—

C₂²×C₄×C₁₆₇

C₂³

yes

characteristic

M₁₆×C₁₆₇

—

M₁₆×C₁₆₇

C₂×C₄

yes

—

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

C₂×C₄

yes

—

C₂×C₈×C₁₆₇

D₄

yes

characteristic

C₂×C₈×C₁₆₇

D₄

yes

characteristic

C₂×C₈×C₁₆₇

Q₈

yes

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₁₆₇⋊C₈

—

C₂³×C₁₆₇

#41

yes

characteristic

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

—

C₈×C₁₆₇

C₄⋊C₄

yes

—

C₈×C₁₆₇

C₄⋊C₄

yes

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

C₄×Q₈

yes

C₂²×C₁₆₇

yes

C₂²×C₁₆₇

yes

C₄×C₁₆₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₆₇

C₄×D₄

yes

characteristic

C₄×C₁₆₇

C₂²⋊Q₈

yes

characteristic

C₄×C₁₆₇

C₄⋊Q₈

yes

characteristic

Dic₁₆₇

—

Dic₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

#389

yes

characteristic

C₂×C₁₆₇

#471

yes

—

C₁₆₇

#1951

yes

characteristic, complemented, Sylow

#1951

—

167

maximal, Sylow

C₂×#108

D₁₆₇

yes

characteristic

#428

—

167

—

C₂×#106

—

167

—

#405

—

167

—

#343

—

167

—

C₂×M₁₆

—

167

—

C₂×M₁₆

C₂×D₁₆₇

yes

—

C₂²×C₈

C₂×D₁₆₇

yes

characteristic

#117

—

167

—

C₂×C₄⋊C₄

—

167

—

#108

Dic₁₆₇

yes

complemented

#107

—

334

—

#106

—

334

—

#106

—

334

—

#106

—

334

—

#97

—

334

—

C₄×C₈

—

334

—

C₂²×C₄

—

334

—

C₂²×C₄

C₂²×D₁₆₇

yes

characteristic

M₁₆

—

M₁₆

—

334

—

M₁₆

—

167

—

M₁₆

C₂×Dic₁₆₇

yes

—

C₂×C₈

—

C₂×C₈

—

334

—

C₂×C₈

C₂×Dic₁₆₇

yes

—

C₂×C₈

D₆₆₈

yes

characteristic

C₂×C₈

D₆₆₈

yes

characteristic

C₂×C₈

Dic₃₃₄

yes

—

C₄⋊C₄

—

334

—

C₄⋊C₄

—

334

—

C₄⋊C₄

—

167

—

C₂²⋊C₄

—

334

—

C₄²

—

334

—

C₂³

#14581951

yes

characteristic

100

C₂×C₄

—

1336

—

101

C₂×C₄

—

334

—

102

C₂×C₄

—

668

—

103

C₂×C₄

C₂²×Dic₁₆₇

yes

characteristic

104

C₂×C₄

D₄×D₁₆₇

yes

characteristic

105

C₂×C₄

C₂×D₆₆₈

yes

characteristic

106

C₂×C₄

Q₈×D₁₆₇

yes

characteristic

107

C₂×C₄

C₂×Dic₃₃₄

yes

characteristic

108

C₂×C₄

#14581949

yes

characteristic

109

C₈

—

110

C₈

—

334

—

111

C₈

C₄⋊Dic₁₆₇

yes

—

112

C₈

C₄⋊Dic₁₆₇

yes

—

113

C₂²

—

114

C₂²

Q₈×Dic₁₆₇

yes

characteristic

115

C₂²

#19128807

yes

characteristic

116

C₂²

#19128823

yes

characteristic

117

C₄

—

1336

—

118

C₄

—

668

—

119

C₄

C₂×C₄⋊Dic₁₆₇

yes

characteristic

120

C₄

D₄×Dic₁₆₇

yes

characteristic

121

C₄

#19128812

yes

characteristic

122

C₄

#19128831

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#24923168

yes

characteristic

126

C₂

#24923533

yes

—

	subgroup	G/H
center	C₂²	#19128807
commutator	C₂×C₄×C₁₆₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₆₇
Fitting	C₂×#108×C₁₆₇	C₂
socle	C₂²×C₁₆₇	#121
prime radical	C₂³×C₁₆₇	#41
cosocle	C₂²×C₄×C₁₆₇	C₂³
hypercenter	C₂×#108	D₁₆₇
hypocenter	C₁₆₇	#1951
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#19128807

commutator

C₂×C₄×C₁₆₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₆₇

Fitting

C₂×#108×C₁₆₇

C₂

C₂²×C₁₆₇

C₂³×C₁₆₇

C₂²×C₄×C₁₆₇

C₂³

D₁₆₇

C₁₆₇

G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
17. #108×C₁₆₇	C₄	semidirect
63. C₁₆₇	#1951	semidirect
75. #108	Dic₁₆₇	semidirect

H ≅ G/N

type

17. #108×C₁₆₇

C₄

semidirect

63. C₁₆₇

#1951

semidirect

75. #108

Dic₁₆₇

semidirect

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

subgroup	index
#24923311	2
C₂×#19128764	2
#24923224	2
#24923037	2
C₂×#108×C₁₆₇	2
#1951	167

subgroup

index

C₂×#108×C₁₆₇

167

Subgroups — #36324919

Conjugacy classes of subgroups

Complements

Maximal subgroups