Group #36336805 — subgroups

Subgroups — #36336805

Order 21376.

392 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×Dic₃₃₄	C₂	1	1	yes	maximal, characteristic, complemented
2	#24923615	C₂	1	3	yes	maximal, complemented
3	#24923559	C₂	1	3	yes	maximal, complemented
4	#24923466	C₂	1	3	yes	maximal, complemented
5	Q₈×C₁₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14581938	C₂	1	3	yes	maximal, complemented
7	#550×C₁₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#14581938	C₂²	1	3	yes	—
9	#19128832	C₂²	1	3	yes	—
10	#19128831	—	2	3	no	—
11	Q₈×Dic₁₆₇	—	2	1	no	—
12	C₄×Dic₃₃₄	C₂²	1	3	yes	—
13	#19128763	C₂²	1	3	yes	—
14	C₄⋊(C₁₆₇⋊C₈)	C₂²	1	3	yes	—
15	#19128755	C₂²	1	1	yes	characteristic
16	#19128753	C₂²	1	6	yes	—
17	#19128753	C₂²	1	3	yes	—
18	C₄×C₁₆₇⋊C₈	C₂²	1	3	yes	—
19	C₂×#41×C₁₆₇	—	2	1	no	—
20	#127×C₁₆₇	C₂²	1	3	yes	—
21	#121×C₁₆₇	—	2	3	no	—
22	C₄×Q₈×C₁₆₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₆₇	C₂²	1	3	yes	—
24	C₂×Dic₃₃₄	—	2	3	no	—
25	C₂×Dic₃₃₄	C₂³	1	1	yes	characteristic
26	#14581938	—	2	6	no	—
27	C₂×C₁₆₇⋊C₈	C₂³	1	3	yes	—
28	C₂×C₁₆₇⋊C₈	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₁₆₇	—	2	3	no	—
30	C₄⋊Dic₁₆₇	C₂³	1	3	yes	—
31	Dic₁₆₇⋊C₄	—	2	3	no	—
32	C₄×Dic₁₆₇	—	2	3	no	—
33	#41×C₁₆₇	—	4	4	no	—
34	C₂×Q₈×C₁₆₇	D₄	1	1	yes	characteristic
35	C₂×D₄×C₁₆₇	—	2	3	no	—
36	C₂×D₄×C₁₆₇	—	2	3	no	—
37	C₂×D₄×C₁₆₇	C₂³	1	3	yes	—
38	C₂²×C₄×C₁₆₇	—	2	3	no	—
39	C₄⋊C₄×C₁₆₇	D₄	1	3	yes	—
40	C₄⋊C₄×C₁₆₇	C₂³	1	1	yes	characteristic
41	C₂²⋊C₄×C₁₆₇	—	2	3	no	—
42	C₄²×C₁₆₇	C₂³	1	3	yes	—
43	Dic₃₃₄	—	2	1	no	—
44	Dic₃₃₄	—	8	3	no	—
45	Dic₃₃₄	#41	1	2	yes	complemented
46	C₂×Dic₁₆₇	—	2	1	no	—
47	C₂×Dic₁₆₇	—	2	3	no	—
48	C₁₆₇⋊C₈	—	2	3	no	—
49	C₁₆₇⋊C₈	—	2	1	no	—
50	C₂³×C₁₆₇	—	2	3	no	—
51	Q₈×C₁₆₇	SD₁₆	1	4	yes	complemented
52	D₄×C₁₆₇	—	2	3	no	—
53	D₄×C₁₆₇	—	2	3	no	—
54	D₄×C₁₆₇	—	4	6	no	—
55	D₄×C₁₆₇	—	8	3	no	—
56	C₂×C₄×C₁₆₇	—	2	1	no	—
57	C₂×C₄×C₁₆₇	—	4	6	no	—
58	C₂×C₄×C₁₆₇	C₂×D₄	1	3	yes	—
59	C₂×C₄×C₁₆₇	C₂×D₄	1	3	yes	—
60	C₂×C₄×C₁₆₇	C₂⁴	1	1	yes	characteristic
61	C₂²×C₁₆₇	—	4	3	no	—
62	C₂²×C₁₆₇	—	4	3	no	—
63	C₂²×C₁₆₇	—	4	3	no	—
64	C₂²×C₁₆₇	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₁₆₇	—	2	3	no	—
66	C₄×C₁₆₇	—	4	1	no	—
67	C₄×C₁₆₇	C₂×SD₁₆	1	6	yes	—
68	C₄×C₁₆₇	C₂×#41	1	1	yes	characteristic
69	C₄×C₁₆₇	2₋¹⁺⁴	1	1	yes	characteristic
70	Dic₁₆₇	—	2	2	no	—
71	Dic₁₆₇	—	4	3	no	—
72	C₂×C₁₆₇	—	8	3	no	—
73	C₂×C₁₆₇	#548	1	1	yes	characteristic
74	C₂×C₁₆₇	C₂²×SD₁₆	1	1	yes	characteristic
75	C₂×C₁₆₇	#576	1	1	yes	characteristic
76	C₁₆₇	#3362	1	1	yes	characteristic, complemented, Sylow
77	#3362	—	167	1	no	maximal, Sylow
78	Q₈²	—	167	1	no	—
79	#550	D₁₆₇	1	1	yes	characteristic
80	#486	—	167	3	no	—
81	#476	—	167	3	no	—
82	#460	—	167	3	no	—
83	C₈×Q₈	—	167	1	no	—
84	C₄×SD₁₆	—	167	3	no	—
85	C₂×#41	—	2	1	no	—
86	C₂×SD₁₆	—	167	3	no	—
87	C₄⋊Q₈	—	334	3	no	—
88	C₄⋊Q₈	—	167	3	no	—
89	#127	C₂×D₁₆₇	1	3	yes	—
90	#121	—	2	3	no	—
91	C₄×Q₈	—	334	1	no	—
92	C₄×Q₈	—	167	3	no	—
93	C₄×Q₈	C₂×D₁₆₇	1	1	yes	characteristic
94	C₄×D₄	C₂×D₁₆₇	1	3	yes	—
95	#106	—	167	3	no	—
96	C₄⋊C₈	—	167	3	no	—
97	#103	—	167	1	no	—
98	#102	—	167	3	no	—
99	#102	—	167	6	no	—
100	C₄×C₈	—	167	3	no	—
101	#41	—	4	4	no	—
102	C₂×Q₈	—	334	3	no	—
103	C₂×Q₈	—	167	1	no	—
104	C₂×Q₈	C₂²⋊D₁₆₇	1	1	yes	characteristic
105	C₂×D₄	—	2	3	no	—
106	C₂×D₄	—	2	3	no	—
107	C₂×D₄	C₂²×D₁₆₇	1	3	yes	—
108	C₂²×C₄	—	2	3	no	—
109	SD₁₆	—	334	6	no	—
110	C₂×C₈	—	167	1	no	—
111	C₂×C₈	—	167	3	no	—
112	C₄⋊C₄	—	334	3	no	—
113	C₄⋊C₄	—	334	3	no	—
114	C₄⋊C₄	—	167	3	no	—
115	C₄⋊C₄	C₂²⋊D₁₆₇	1	3	yes	—
116	C₄⋊C₄	C₂²×D₁₆₇	1	1	yes	characteristic
117	C₂²⋊C₄	—	2	3	no	—
118	C₄²	—	334	3	no	—
119	C₄²	C₂²×D₁₆₇	1	3	yes	—
120	C₂³	—	2	3	no	—
121	Q₈	—	1336	3	no	—
122	Q₈	—	334	1	no	—
123	Q₈	—	167	2	no	—
124	Q₈	#14581938	1	4	yes	complemented
125	D₄	—	2	3	no	—
126	D₄	—	2	3	no	—
127	D₄	—	4	6	no	—
128	D₄	—	8	3	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	4	6	no	—
131	C₂×C₄	—	334	1	no	—
132	C₂×C₄	—	334	3	no	—
133	C₂×C₄	C₂×C₂²⋊D₁₆₇	1	3	yes	—
134	C₂×C₄	C₂×C₂²⋊D₁₆₇	1	3	yes	—
135	C₂×C₄	C₂³×D₁₆₇	1	1	yes	characteristic
136	C₈	—	334	3	no	—
137	C₈	—	334	1	no	—
138	C₂²	—	4	3	no	—
139	C₂²	—	4	3	no	—
140	C₂²	—	4	3	no	—
141	C₂²	C₂²×C₂²⋊D₁₆₇	1	1	yes	characteristic
142	C₄	—	2	3	no	—
143	C₄	—	4	1	no	—
144	C₄	—	334	2	no	—
145	C₄	—	668	3	no	—
146	C₄	C₂×#14581938	1	6	yes	—
147	C₄	#41×D₁₆₇	1	1	yes	characteristic
148	C₄	#19128879	1	1	yes	characteristic
149	C₂	—	8	3	no	—
150	C₂	#24923926	1	1	yes	characteristic
151	C₂	C₂²×#14581938	1	1	yes	characteristic
152	C₂	#24924072	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₃₃₄

C₂

yes

maximal, characteristic, complemented

#24923615

C₂

yes

maximal, complemented

#24923559

C₂

yes

maximal, complemented

#24923466

C₂

yes

maximal, complemented

Q₈×C₁₆₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#14581938

C₂

yes

maximal, complemented

#550×C₁₆₇

C₂

yes

maximal, characteristic

C₂×#14581938

C₂²

yes

—

#19128832

C₂²

yes

—

#19128831

—

Q₈×Dic₁₆₇

—

C₄×Dic₃₃₄

C₂²

yes

—

#19128763

C₂²

yes

—

C₄⋊(C₁₆₇⋊C₈)

C₂²

yes

—

#19128755

C₂²

yes

characteristic

#19128753

C₂²

yes

—

#19128753

C₂²

yes

—

C₄×C₁₆₇⋊C₈

C₂²

yes

—

C₂×#41×C₁₆₇

—

#127×C₁₆₇

C₂²

yes

—

#121×C₁₆₇

—

C₄×Q₈×C₁₆₇

C₂²

yes

characteristic

C₄×D₄×C₁₆₇

C₂²

yes

—

C₂×Dic₃₃₄

—

C₂×Dic₃₃₄

C₂³

yes

characteristic

#14581938

—

C₂×C₁₆₇⋊C₈

C₂³

yes

—

C₂×C₁₆₇⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₁₆₇

—

C₄⋊Dic₁₆₇

C₂³

yes

—

Dic₁₆₇⋊C₄

—

C₄×Dic₁₆₇

—

#41×C₁₆₇

—

C₂×Q₈×C₁₆₇

D₄

yes

characteristic

C₂×D₄×C₁₆₇

—

C₂×D₄×C₁₆₇

—

C₂×D₄×C₁₆₇

C₂³

yes

—

C₂²×C₄×C₁₆₇

—

C₄⋊C₄×C₁₆₇

D₄

yes

—

C₄⋊C₄×C₁₆₇

C₂³

yes

characteristic

C₂²⋊C₄×C₁₆₇

—

C₄²×C₁₆₇

C₂³

yes

—

Dic₃₃₄

—

Dic₃₃₄

—

Dic₃₃₄

#41

yes

complemented

C₂×Dic₁₆₇

—

C₂×Dic₁₆₇

—

C₁₆₇⋊C₈

—

C₁₆₇⋊C₈

—

C₂³×C₁₆₇

—

Q₈×C₁₆₇

SD₁₆

yes

complemented

D₄×C₁₆₇

—

D₄×C₁₆₇

—

D₄×C₁₆₇

—

D₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

—

C₂×C₄×C₁₆₇

C₂×D₄

yes

—

C₂×C₄×C₁₆₇

C₂⁴

yes

characteristic

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

—

C₂²×C₁₆₇

C₂²×D₄

yes

characteristic

C₄×C₁₆₇

—

C₄×C₁₆₇

—

C₄×C₁₆₇

C₂×SD₁₆

yes

—

C₄×C₁₆₇

C₂×#41

yes

characteristic

C₄×C₁₆₇

2₋¹⁺⁴

yes

characteristic

Dic₁₆₇

—

Dic₁₆₇

—

C₂×C₁₆₇

—

C₂×C₁₆₇

#548

yes

characteristic

C₂×C₁₆₇

C₂²×SD₁₆

yes

C₂×C₁₆₇

yes

C₁₆₇

yes

characteristic, complemented, Sylow

#3362

—

167

maximal, Sylow

Q₈²

—

167

—

#550

D₁₆₇

yes

characteristic

#486

—

167

—

#476

—

167

—

#460

—

167

—

C₈×Q₈

—

167

—

C₄×SD₁₆

—

167

—

C₂×#41

—

C₂×SD₁₆

—

167

—

C₄⋊Q₈

—

334

—

C₄⋊Q₈

—

167

—

#127

C₂×D₁₆₇

yes

—

#121

—

C₄×Q₈

—

334

—

C₄×Q₈

—

167

—

C₄×Q₈

C₂×D₁₆₇

yes

characteristic

C₄×D₄

C₂×D₁₆₇

yes

—

#106

—

167

—

C₄⋊C₈

—

167

—

#103

—

167

—

#102

—

167

—

#102

—

167

—

100

C₄×C₈

—

167

—

101

#41

—

102

C₂×Q₈

—

334

—

103

C₂×Q₈

—

167

—

104

C₂×Q₈

C₂²⋊D₁₆₇

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

C₂²×D₁₆₇

yes

—

108

C₂²×C₄

—

109

SD₁₆

—

334

—

110

C₂×C₈

—

167

—

111

C₂×C₈

—

167

—

112

C₄⋊C₄

—

334

—

113

C₄⋊C₄

—

334

—

114

C₄⋊C₄

—

167

—

115

C₄⋊C₄

C₂²⋊D₁₆₇

yes

—

116

C₄⋊C₄

C₂²×D₁₆₇

yes

characteristic

117

C₂²⋊C₄

—

118

C₄²

—

334

—

119

C₄²

C₂²×D₁₆₇

yes

—

120

C₂³

—

121

Q₈

—

1336

—

122

Q₈

—

334

—

123

Q₈

—

167

—

124

Q₈

#14581938

yes

complemented

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

334

—

132

C₂×C₄

—

334

—

133

C₂×C₄

C₂×C₂²⋊D₁₆₇

yes

—

134

C₂×C₄

C₂×C₂²⋊D₁₆₇

yes

—

135

C₂×C₄

C₂³×D₁₆₇

yes

characteristic

136

C₈

—

334

—

137

C₈

—

334

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂²×C₂²⋊D₁₆₇

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

334

—

145

C₄

—

668

—

146

C₄

C₂×#14581938

yes

—

147

C₄

#41×D₁₆₇

yes

characteristic

148

C₄

#19128879

yes

characteristic

149

C₂

—

150

C₂

#24923926

yes

characteristic

151

C₂

C₂²×#14581938

yes

characteristic

152

C₂

#24924072

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₆₇
commutator, cosocle	C₂×C₄×C₁₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₆₇
Fitting, prime radical	#550×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂²×D₄
hypercenter	#550	D₁₆₇
hypocenter	C₁₆₇	#3362
LCS	G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₆₇

commutator, cosocle

C₂×C₄×C₁₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₆₇

Fitting, prime radical

#550×C₁₆₇

C₂

socle

C₂²×C₁₆₇

C₂²×D₄

D₁₆₇

C₁₆₇

G ▷ C₂×C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #550

N	H ≅ G/N	type
1. Q₈×Dic₃₃₄	C₂	semidirect
2. #24923615	C₂	semidirect
3. #24923559	C₂	semidirect
4. #24923466	C₂	semidirect
5. Q₈×C₁₆₇⋊C₈	C₂	semidirect
6. C₄×#14581938	C₂	semidirect
45. Dic₃₃₄	#41	semidirect
51. Q₈×C₁₆₇	SD₁₆	semidirect
76. C₁₆₇	#3362	semidirect
124. Q₈	#14581938	semidirect

H ≅ G/N

type

1. Q₈×Dic₃₃₄

C₂

semidirect

2. #24923615

C₂

semidirect

3. #24923559

C₂

semidirect

4. #24923466

C₂

semidirect

5. Q₈×C₁₆₇⋊C₈

C₂

semidirect

6. C₄×#14581938

C₂

semidirect

45. Dic₃₃₄

#41

semidirect

51. Q₈×C₁₆₇

SD₁₆

semidirect

76. C₁₆₇

#3362

semidirect

124. Q₈

#14581938

semidirect

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

subgroup	index
Q₈×Dic₃₃₄	2
#24923615	2
#24923559	2
#24923466	2
Q₈×C₁₆₇⋊C₈	2
C₄×#14581938	2
#550×C₁₆₇	2
#3362	167

subgroup

index

Q₈×Dic₃₃₄

Q₈×C₁₆₇⋊C₈

#550×C₁₆₇

167

Subgroups — #36336805

Conjugacy classes of subgroups

Complements

Maximal subgroups