Group #36323655 — subgroups

Subgroups — #36323655

Order 21376.

152 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 51 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	#24923672	C₂	1	1	yes	maximal, characteristic
2	#24923480	C₂	1	1	yes	maximal, characteristic
3	#24923476	C₂	1	1	yes	maximal, characteristic
4	#24923396	C₂	1	1	yes	maximal, characteristic
5	#24923019	C₂	1	1	yes	maximal, characteristic
6	#24923014	C₂	1	1	yes	maximal, characteristic
7	#332×C₁₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₆₆₈	—	2	1	no	—
9	C₂×#14581940	—	2	1	no	—
10	#19128832	C₂²	1	1	yes	characteristic
11	C₄×Dic₃₃₄	C₂²	1	1	yes	characteristic
12	#19128765	—	2	1	no	—
13	#19128765	—	2	1	no	—
14	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	1	yes	characteristic
15	Dic₁₆₇⋊C₈	—	2	1	no	—
16	#19128757	—	2	1	no	—
17	#19128756	—	2	1	no	—
18	#19128742	—	2	1	no	—
19	C₄×C₁₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₆₇	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₃₃₄	—	2	1	no	—
24	C₂×Dic₃₃₄	D₄	1	1	yes	characteristic
25	Dic₆₆₈	—	4	2	no	—
26	#14581940	—	4	2	no	—
27	C₂×C₁₆₇⋊C₈	—	2	1	no	—
28	C₂×C₁₆₇⋊C₈	D₄	1	1	yes	characteristic
29	C₂×C₁₆₇⋊C₈	D₄	1	1	yes	characteristic
30	C₄⋊Dic₁₆₇	—	2	1	no	—
31	C₄⋊Dic₁₆₇	—	2	1	no	—
32	C₄⋊Dic₁₆₇	D₄	1	1	yes	characteristic
33	Dic₁₆₇⋊C₄	—	2	1	no	—
34	C₄×Dic₁₆₇	—	2	1	no	—
35	C₂×Q₈×C₁₆₇	—	2	1	no	—
36	C₂×C₈×C₁₆₇	—	2	1	no	—
37	C₂×C₈×C₁₆₇	—	2	1	no	—
38	C₄⋊C₄×C₁₆₇	—	2	1	no	—
39	C₄⋊C₄×C₁₆₇	—	2	1	no	—
40	C₄²×C₁₆₇	C₂³	1	1	yes	characteristic
41	Dic₃₃₄	—	2	1	no	—
42	Dic₃₃₄	—	2	1	no	—
43	Dic₃₃₄	—	4	1	no	—
44	Dic₃₃₄	—	4	1	no	—
45	C₂×Dic₁₆₇	—	2	1	no	—
46	C₂×Dic₁₆₇	—	2	1	no	—
47	C₂×Dic₁₆₇	—	4	1	no	—
48	C₁₆₇⋊C₈	—	2	1	no	—
49	C₁₆₇⋊C₈	—	4	1	no	—
50	C₁₆₇⋊C₈	Q₁₆	1	2	yes	complemented
51	Q₈×C₁₆₇	—	4	1	no	—
52	Q₈×C₁₆₇	—	4	1	no	—
53	C₂×C₄×C₁₆₇	—	4	1	no	—
54	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₆₇	#41	1	1	yes	characteristic
57	C₈×C₁₆₇	—	4	1	no	—
58	C₈×C₁₆₇	—	4	1	no	—
59	C₂²×C₁₆₇	#121	1	1	yes	characteristic
60	C₄×C₁₆₇	—	2	1	no	—
61	C₄×C₁₆₇	—	8	1	no	—
62	C₄×C₁₆₇	C₂×Q₁₆	1	1	yes	characteristic
63	C₄×C₁₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
64	C₄×C₁₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₄×C₁₆₇	#137	1	1	yes	characteristic
66	Dic₁₆₇	—	4	1	no	—
67	Dic₁₆₇	—	4	1	no	—
68	Dic₁₆₇	—	8	1	no	—
69	C₂×C₁₆₇	#462	1	1	yes	characteristic
70	C₂×C₁₆₇	#469	1	1	yes	characteristic
71	C₂×C₁₆₇	#473	1	1	yes	characteristic
72	C₁₆₇	#1694	1	1	yes	characteristic, complemented, Sylow
73	#1694	—	167	1	no	maximal, Sylow
74	#500	—	167	1	no	—
75	#462	—	167	1	no	—
76	#462	—	167	1	no	—
77	#446	—	167	1	no	—
78	#334	—	167	1	no	—
79	#332	—	167	1	no	—
80	#332	D₁₆₇	1	1	yes	characteristic
81	C₂×Q₁₆	—	334	1	no	—
82	C₂×Q₁₆	—	334	1	no	—
83	C₄⋊Q₈	—	167	1	no	—
84	C₄⋊Q₈	C₂×D₁₆₇	1	1	yes	characteristic
85	C₄×Q₈	—	167	1	no	—
86	#107	—	334	1	no	—
87	#107	—	334	1	no	—
88	C₄⋊C₈	—	334	1	no	—
89	C₄⋊C₈	—	167	1	no	—
90	C₄⋊C₈	C₂×D₁₆₇	1	1	yes	characteristic
91	C₄⋊C₈	C₂×D₁₆₇	1	1	yes	characteristic
92	#103	—	334	1	no	—
93	#103	—	334	1	no	—
94	#97	—	334	1	no	—
95	C₄×C₈	—	167	1	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	334	1	no	—
98	C₂×Q₈	—	167	1	no	—
99	Q₁₆	—	668	2	no	—
100	Q₁₆	—	668	2	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	334	1	no	—
104	C₂×C₈	—	167	1	no	—
105	C₂×C₈	—	167	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	334	1	no	—
109	C₄⋊C₄	—	334	1	no	—
110	C₄⋊C₄	—	334	1	no	—
111	C₄⋊C₄	—	167	1	no	—
112	C₄²	—	334	1	no	—
113	C₄²	C₂²×D₁₆₇	1	1	yes	characteristic
114	Q₈	—	4	1	no	—
115	Q₈	—	4	1	no	—
116	Q₈	—	334	1	no	—
117	Q₈	—	334	1	no	—
118	Q₈	—	668	1	no	—
119	Q₈	—	668	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	334	1	no	—
122	C₂×C₄	—	334	1	no	—
123	C₂×C₄	—	668	1	no	—
124	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
125	C₂×C₄	D₄×D₁₆₇	1	1	yes	characteristic
126	C₂×C₄	#14581950	1	1	yes	characteristic
127	C₈	—	4	1	no	—
128	C₈	—	4	1	no	—
129	C₈	—	334	1	no	—
130	C₈	—	668	1	no	—
131	C₈	—	167	2	no	—
132	C₂²	#19128805	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	1336	1	no	—
136	C₄	—	668	1	no	—
137	C₄	—	668	1	no	—
138	C₄	Q₁₆×D₁₆₇	1	1	yes	characteristic
139	C₄	#19128856	1	1	yes	characteristic
140	C₄	#19128859	1	1	yes	characteristic
141	C₄	#19128861	1	1	yes	characteristic
142	C₂	#24923478	1	1	yes	characteristic
143	C₂	#24923521	1	1	yes	characteristic
144	C₂	#24923543	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24923672

C₂

yes

maximal, characteristic

#24923480

C₂

yes

maximal, characteristic

#24923476

C₂

yes

maximal, characteristic

#24923396

C₂

yes

maximal, characteristic

#24923019

C₂

yes

maximal, characteristic

#24923014

C₂

yes

maximal, characteristic

#332×C₁₆₇

C₂

yes

maximal, characteristic

C₂×Dic₆₆₈

—

C₂×#14581940

—

#19128832

C₂²

yes

characteristic

C₄×Dic₃₃₄

C₂²

yes

characteristic

#19128765

—

#19128765

—

C₄⋊(C₁₆₇⋊C₈)

C₂²

yes

characteristic

Dic₁₆₇⋊C₈

—

#19128757

—

#19128756

—

#19128742

—

C₄×C₁₆₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₆₇

C₂²

yes

characteristic

C₂×Dic₃₃₄

—

C₂×Dic₃₃₄

D₄

yes

characteristic

Dic₆₆₈

—

#14581940

—

C₂×C₁₆₇⋊C₈

—

C₂×C₁₆₇⋊C₈

D₄

yes

characteristic

C₂×C₁₆₇⋊C₈

D₄

yes

characteristic

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

D₄

yes

characteristic

Dic₁₆₇⋊C₄

—

C₄×Dic₁₆₇

—

C₂×Q₈×C₁₆₇

—

C₂×C₈×C₁₆₇

—

C₂×C₈×C₁₆₇

—

C₄⋊C₄×C₁₆₇

—

C₄⋊C₄×C₁₆₇

—

C₄²×C₁₆₇

C₂³

yes

characteristic

Dic₃₃₄

—

Dic₃₃₄

—

Dic₃₃₄

—

Dic₃₃₄

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₁₆₇⋊C₈

—

C₁₆₇⋊C₈

—

C₁₆₇⋊C₈

Q₁₆

yes

complemented

Q₈×C₁₆₇

—

Q₈×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₆₇

#41

yes

characteristic

C₈×C₁₆₇

—

C₈×C₁₆₇

—

C₂²×C₁₆₇

#121

yes

characteristic

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×Q₁₆

yes

C₄×C₁₆₇

yes

C₄×C₁₆₇

yes

C₄×C₁₆₇

yes

Dic₁₆₇

—

Dic₁₆₇

—

Dic₁₆₇

—

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₂×C₁₆₇

yes

C₁₆₇

yes

characteristic, complemented, Sylow

#1694

—

167

maximal, Sylow

#500

—

167

—

#462

—

167

—

#462

—

167

—

#446

—

167

—

#334

—

167

—

#332

—

167

—

#332

D₁₆₇

yes

characteristic

C₂×Q₁₆

—

334

—

C₂×Q₁₆

—

334

—

C₄⋊Q₈

—

167

—

C₄⋊Q₈

C₂×D₁₆₇

yes

characteristic

C₄×Q₈

—

167

—

#107

—

334

—

#107

—

334

—

C₄⋊C₈

—

334

—

C₄⋊C₈

—

167

—

C₄⋊C₈

C₂×D₁₆₇

yes

characteristic

C₄⋊C₈

C₂×D₁₆₇

yes

characteristic

#103

—

334

—

#103

—

334

—

#97

—

334

—

C₄×C₈

—

167

—

C₂×Q₈

—

C₂×Q₈

—

334

—

C₂×Q₈

—

167

—

Q₁₆

—

668

—

100

Q₁₆

—

668

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

334

—

104

C₂×C₈

—

167

—

105

C₂×C₈

—

167

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

334

—

109

C₄⋊C₄

—

334

—

110

C₄⋊C₄

—

334

—

111

C₄⋊C₄

—

167

—

112

C₄²

—

334

—

113

C₄²

C₂²×D₁₆₇

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

334

—

117

Q₈

—

334

—

118

Q₈

—

668

—

119

Q₈

—

668

—

120

C₂×C₄

—

121

C₂×C₄

—

334

—

122

C₂×C₄

—

334

—

123

C₂×C₄

—

668

—

124

C₂×C₄

D₄×D₁₆₇

yes

characteristic

125

C₂×C₄

D₄×D₁₆₇

yes

characteristic

126

C₂×C₄

#14581950

yes

characteristic

127

C₈

—

128

C₈

—

129

C₈

—

334

—

130

C₈

—

668

—

131

C₈

—

167

—

132

C₂²

#19128805

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

1336

—

136

C₄

—

668

—

137

C₄

—

668

—

138

C₄

Q₁₆×D₁₆₇

yes

characteristic

139

C₄

#19128856

yes

characteristic

140

C₄

#19128859

yes

characteristic

141

C₄

#19128861

yes

characteristic

142

C₂

#24923478

yes

characteristic

143

C₂

#24923521

yes

characteristic

144

C₂

#24923543

yes

characteristic

	subgroup	G/H
center	C₂²	#19128805
commutator, cosocle	C₄²×C₁₆₇	C₂³
Frattini	C₄²	C₂²×D₁₆₇
Fitting	#332×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	#121
hypercenter	#332	D₁₆₇
hypocenter	C₁₆₇	#1694
LCS	G ▷ C₄²×C₁₆₇ ▷ C₂²×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#19128805

commutator, cosocle

C₄²×C₁₆₇

C₂³

Frattini

C₄²

C₂²×D₁₆₇

Fitting

#332×C₁₆₇

C₂

C₂²×C₁₆₇

D₁₆₇

C₁₆₇

G ▷ C₄²×C₁₆₇ ▷ C₂²×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
50. C₁₆₇⋊C₈	Q₁₆	semidirect
72. C₁₆₇	#1694	semidirect

H ≅ G/N

type

50. C₁₆₇⋊C₈

Q₁₆

semidirect

72. C₁₆₇

#1694

semidirect

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

subgroup	index
#24923672	2
#24923480	2
#24923476	2
#24923396	2
#24923019	2
#24923014	2
#332×C₁₆₇	2
#1694	167

subgroup

index

#332×C₁₆₇

167

Subgroups — #36323655

Conjugacy classes of subgroups

Complements

Maximal subgroups