Group #36370353 — subgroups

Subgroups — #36370353

Order 22144.

364 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 166 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	#26012543	C₂	1	4	yes	maximal, complemented
2	#26012540	C₂	1	4	yes	maximal, complemented
3	#26012498	C₂	1	2	yes	maximal
4	#26012496	C₂	1	1	yes	maximal, characteristic
5	#26012456	C₂	1	2	yes	maximal
6	C₄×D₁₇₃⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#26012047	C₂	1	1	yes	maximal, characteristic
8	C₂×#14583836	C₂²	1	2	yes	—
9	C₂×D₁₇₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×D₁₇₃⋊C₈	C₂²	1	1	yes	characteristic
11	#19149746	C₂²	1	4	yes	—
12	#19149746	C₂²	1	4	yes	—
13	C₄⋊#13063443	C₂²	1	4	yes	—
14	#19149734	C₂²	1	4	yes	—
15	C₄×#13063443	C₂²	1	2	yes	—
16	C₄×#13063443	C₂²	1	2	yes	—
17	C₄×#13063443	C₂²	1	4	yes	—
18	#19149678	C₄	1	2	yes	—
19	#19149671	C₄	1	1	yes	characteristic
20	#19149657	C₄	1	4	yes	—
21	#19149644	C₂²	1	4	yes	—
22	#19149637	C₂²	1	2	yes	—
23	C₄²×D₁₇₃	C₂²	1	1	yes	characteristic
24	#125×C₁₇₃	C₄	1	1	yes	characteristic
25	#14583836	—	2	4	no	—
26	D₁₇₃⋊C₈	—	2	2	no	—
27	D₁₇₃⋊C₈	Q₈	1	4	yes	complemented
28	C₂×#13063443	C₂³	1	2	yes	—
29	C₂×#13063443	C₂³	1	4	yes	—
30	C₂×#13063443	C₂³	1	1	yes	characteristic
31	C₂×#13063443	C₂³	1	1	yes	characteristic
32	C₂×Dic₃₄₆	C₂×C₄	1	4	yes	—
33	C₂×C₄×D₁₇₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₁₇₃	C₂³	1	1	yes	characteristic
35	C₄⋊Dic₁₇₃	C₂×C₄	1	2	yes	—
36	#14583805	C₂×C₄	1	4	yes	—
37	#14583805	C₂×C₄	1	4	yes	—
38	#14583803	C₂×C₄	1	4	yes	—
39	C₄×Dic₁₇₃	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₁₇₃	C₂³	1	2	yes	—
41	C₄×Dic₁₇₃	C₂³	1	4	yes	—
42	C₄⋊C₄×C₁₇₃	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₁₇₃	C₂×C₄	1	4	yes	—
44	C₄²×C₁₇₃	C₂×C₄	1	1	yes	characteristic
45	#13063443	—	2	2	no	—
46	#13063443	—	2	4	no	—
47	#13063443	C₂×Q₈	1	2	yes	—
48	#13063443	C₂×Q₈	1	2	yes	—
49	C₂²×D₁₇₃	C₂²×C₄	1	1	yes	characteristic
50	Dic₃₄₆	—	2	8	no	—
51	C₂×Dic₁₇₃	C₂²×C₄	1	4	yes	—
52	C₂×Dic₁₇₃	C₂²×C₄	1	2	yes	—
53	C₂×Dic₁₇₃	C₂⁴	1	1	yes	characteristic
54	C₄×D₁₇₃	—	2	4	no	—
55	C₄×D₁₇₃	C₂×Q₈	1	2	yes	—
56	C₄×D₁₇₃	#41	1	2	yes	—
57	C₂×C₄×C₁₇₃	C₂²×C₄	1	2	yes	—
58	C₂×C₄×C₁₇₃	C₂²×C₄	1	4	yes	—
59	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₁₇₃	C₂³×C₄	1	1	yes	characteristic
61	C₂×D₁₇₃	—	2	2	no	—
62	C₂×D₁₇₃	C₄×Q₈	1	2	yes	—
63	C₄×C₁₇₃	—	2	2	no	—
64	C₄×C₁₇₃	—	2	4	no	—
65	C₄×C₁₇₃	C₄×Q₈	1	2	yes	—
66	Dic₁₇₃	—	2	4	no	—
67	Dic₁₇₃	#131	1	4	yes	—
68	Dic₁₇₃	C₂²×Q₈	1	1	yes	characteristic
69	Dic₁₇₃	C₂×#41	1	1	yes	characteristic
70	C₂×C₁₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
71	C₂×C₁₇₃	C₂×#131	1	2	yes	—
72	D₁₇₃	—	2	2	no	—
73	C₁₇₃	#2960	1	1	yes	characteristic, complemented, Sylow
74	#2960	—	173	1	no	maximal, Sylow
75	#533	—	173	1	no	—
76	#446	—	173	4	no	—
77	C₈×Q₈	—	173	4	no	—
78	#424	—	173	2	no	—
79	#423	—	173	1	no	—
80	#405	—	173	2	no	—
81	C₂×C₄×C₈	—	173	1	no	—
82	C₂×M₁₆	—	173	2	no	—
83	C₂²×C₈	—	173	1	no	—
84	C₂²×C₈	—	173	1	no	—
85	C₄⋊Q₈	—	173	2	no	—
86	#125	—	173	1	no	—
87	#125	#10621924	1	1	yes	characteristic
88	C₂²⋊Q₈	—	173	4	no	—
89	C₄×Q₈	—	173	4	no	—
90	#117	—	173	2	no	—
91	C₂×C₄²	—	173	1	no	—
92	C₄⋊C₈	—	173	4	no	—
93	C₄⋊C₈	—	173	4	no	—
94	C₄⋊C₈	—	173	4	no	—
95	#97	—	173	4	no	—
96	C₄×C₈	—	173	2	no	—
97	C₄×C₈	—	173	2	no	—
98	C₄×C₈	—	173	4	no	—
99	C₂×Q₈	—	173	4	no	—
100	C₂²×C₄	—	173	2	no	—
101	C₂²×C₄	—	173	1	no	—
102	M₁₆	—	346	4	no	—
103	C₂×C₈	—	346	2	no	—
104	C₂×C₈	—	173	2	no	—
105	C₂×C₈	—	173	4	no	—
106	C₂×C₈	—	173	1	no	—
107	C₂×C₈	—	173	1	no	—
108	C₂×C₈	—	173	4	no	—
109	C₄⋊C₄	—	173	2	no	—
110	C₄⋊C₄	—	173	4	no	—
111	C₄⋊C₄	—	173	4	no	—
112	C₄⋊C₄	C₂×#10621924	1	2	yes	—
113	C₄⋊C₄	C₂×#10621924	1	4	yes	—
114	C₂²⋊C₄	—	173	4	no	—
115	C₄²	—	173	2	no	—
116	C₄²	—	173	4	no	—
117	C₄²	—	173	1	no	—
118	C₄²	C₂×#10621924	1	1	yes	characteristic
119	C₂³	—	173	1	no	—
120	Q₈	—	346	8	no	—
121	C₂×C₄	—	346	4	no	—
122	C₂×C₄	—	173	4	no	—
123	C₂×C₄	—	173	2	no	—
124	C₂×C₄	—	173	1	no	—
125	C₂×C₄	—	173	2	no	—
126	C₂×C₄	—	173	2	no	—
127	C₂×C₄	C₂²×#10621924	1	2	yes	—
128	C₂×C₄	C₂²×#10621924	1	4	yes	—
129	C₂×C₄	C₂²×#10621924	1	1	yes	characteristic
130	C₈	—	346	2	no	—
131	C₈	—	346	4	no	—
132	C₈	—	173	2	no	—
133	C₈	—	173	2	no	—
134	C₂²	—	346	2	no	—
135	C₂²	—	173	2	no	—
136	C₂²	C₂³×#10621924	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	4	no	—
139	C₄	—	346	4	no	—
140	C₄	—	173	1	no	—
141	C₄	—	173	1	no	—
142	C₄	—	173	4	no	—
143	C₄	Q₈×#10621924	1	2	yes	—
144	C₂	—	346	2	no	—
145	C₂	C₂×Q₈×#10621924	1	1	yes	characteristic
146	C₂	D₁₇₃⋊#131	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#26012543

C₂

yes

maximal, complemented

#26012540

C₂

yes

maximal, complemented

#26012498

C₂

yes

maximal

#26012496

C₂

yes

maximal, characteristic

#26012456

C₂

yes

maximal

C₄×D₁₇₃⋊C₈

C₂

yes

maximal, characteristic

#26012047

C₂

yes

maximal, characteristic

C₂×#14583836

C₂²

yes

—

C₂×D₁₇₃⋊C₈

C₂²

yes

characteristic

C₂×D₁₇₃⋊C₈

C₂²

yes

characteristic

#19149746

C₂²

yes

—

#19149746

C₂²

yes

—

C₄⋊#13063443

C₂²

yes

—

#19149734

C₂²

yes

—

C₄×#13063443

C₂²

yes

—

C₄×#13063443

C₂²

yes

—

C₄×#13063443

C₂²

yes

—

#19149678

C₄

yes

—

#19149671

C₄

yes

characteristic

#19149657

C₄

yes

—

#19149644

C₂²

yes

—

#19149637

C₂²

yes

—

C₄²×D₁₇₃

C₂²

yes

characteristic

#125×C₁₇₃

C₄

yes

characteristic

#14583836

—

D₁₇₃⋊C₈

—

D₁₇₃⋊C₈

Q₈

yes

complemented

C₂×#13063443

C₂³

yes

—

C₂×#13063443

C₂³

yes

—

C₂×#13063443

C₂³

yes

characteristic

C₂×#13063443

C₂³

yes

characteristic

C₂×Dic₃₄₆

C₂×C₄

yes

—

C₂×C₄×D₁₇₃

C₂×C₄

yes

—

C₂×C₄×D₁₇₃

C₂³

yes

characteristic

C₄⋊Dic₁₇₃

C₂×C₄

yes

—

#14583805

C₂×C₄

yes

—

#14583805

C₂×C₄

yes

—

#14583803

C₂×C₄

yes

—

C₄×Dic₁₇₃

C₂×C₄

yes

characteristic

C₄×Dic₁₇₃

C₂³

yes

—

C₄×Dic₁₇₃

C₂³

yes

—

C₄⋊C₄×C₁₇₃

C₂×C₄

yes

—

C₄⋊C₄×C₁₇₃

C₂×C₄

yes

—

C₄²×C₁₇₃

C₂×C₄

yes

—

—

C₂×Q₈

yes

—

#13063443

C₂×Q₈

yes

—

C₂²×D₁₇₃

C₂²×C₄

yes

characteristic

Dic₃₄₆

—

C₂×Dic₁₇₃

C₂²×C₄

yes

—

C₂×Dic₁₇₃

C₂²×C₄

yes

—

C₂×Dic₁₇₃

C₂⁴

yes

characteristic

C₄×D₁₇₃

—

C₄×D₁₇₃

C₂×Q₈

yes

—

C₄×D₁₇₃

#41

yes

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₂²×C₁₇₃

C₂³×C₄

yes

characteristic

C₂×D₁₇₃

—

C₂×D₁₇₃

C₄×Q₈

yes

—

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

C₄×Q₈

yes

—

Dic₁₇₃

—

Dic₁₇₃

#131

yes

—

Dic₁₇₃

C₂²×Q₈

yes

characteristic

Dic₁₇₃

C₂×#41

yes

characteristic

C₂×C₁₇₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₁₇₃

C₂×#131

yes

—

D₁₇₃

—

C₁₇₃

#2960

yes

characteristic, complemented, Sylow

#2960

—

173

maximal, Sylow

#533

—

173

—

#446

—

173

—

C₈×Q₈

—

173

—

#424

—

173

—

#423

—

173

—

#405

—

173

—

C₂×C₄×C₈

—

173

—

C₂×M₁₆

—

173

—

C₂²×C₈

—

173

—

C₂²×C₈

—

173

—

C₄⋊Q₈

—

173

—

#125

—

173

—

#125

#10621924

yes

characteristic

C₂²⋊Q₈

—

173

—

C₄×Q₈

—

173

—

#117

—

173

—

C₂×C₄²

—

173

—

C₄⋊C₈

—

173

—

C₄⋊C₈

—

173

—

C₄⋊C₈

—

173

—

#97

—

173

—

C₄×C₈

—

173

—

C₄×C₈

—

173

—

C₄×C₈

—

173

—

C₂×Q₈

—

173

—

100

C₂²×C₄

—

173

—

101

C₂²×C₄

—

173

—

102

M₁₆

—

346

—

103

C₂×C₈

—

346

—

104

C₂×C₈

—

173

—

105

C₂×C₈

—

173

—

106

C₂×C₈

—

173

—

107

C₂×C₈

—

173

—

108

C₂×C₈

—

173

—

109

C₄⋊C₄

—

173

—

110

C₄⋊C₄

—

173

—

111

C₄⋊C₄

—

173

—

112

C₄⋊C₄

C₂×#10621924

yes

—

113

C₄⋊C₄

C₂×#10621924

yes

—

114

C₂²⋊C₄

—

173

—

115

C₄²

—

173

—

116

C₄²

—

173

—

117

C₄²

—

173

—

118

C₄²

C₂×#10621924

yes

characteristic

119

C₂³

—

173

—

120

Q₈

—

346

—

121

C₂×C₄

—

346

—

122

C₂×C₄

—

173

—

123

C₂×C₄

—

173

—

124

C₂×C₄

—

173

—

125

C₂×C₄

—

173

—

126

C₂×C₄

—

173

—

127

C₂×C₄

C₂²×#10621924

yes

—

128

C₂×C₄

C₂²×#10621924

yes

—

129

C₂×C₄

C₂²×#10621924

yes

characteristic

130

C₈

—

346

—

131

C₈

—

346

—

132

C₈

—

173

—

133

C₈

—

173

—

134

C₂²

—

346

—

135

C₂²

—

173

—

136

C₂²

C₂³×#10621924

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

346

—

140

C₄

—

173

—

141

C₄

—

173

—

142

C₄

—

173

—

143

C₄

Q₈×#10621924

yes

—

144

C₂

—

346

—

145

C₂

C₂×Q₈×#10621924

yes

characteristic

146

C₂

D₁₇₃⋊#131

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#10621924
commutator, socle	C₂²×C₁₇₃	C₂³×C₄
Fitting	#125×C₁₇₃	C₄
prime radical	C₂²×D₁₇₃	C₂²×C₄
cosocle	C₂×Dic₁₇₃	C₂⁴
hypercenter	#125	#10621924
hypocenter	C₁₇₃	#2960
LCS	G ▷ C₂²×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×#10621924

commutator, socle

C₂²×C₁₇₃

C₂³×C₄

Fitting

#125×C₁₇₃

C₄

prime radical

C₂²×D₁₇₃

C₂²×C₄

cosocle

C₂×Dic₁₇₃

C₂⁴

C₁₇₃

G ▷ C₂²×C₁₇₃ ▷ C₁₇₃

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #26012543	C₂	semidirect
2. #26012540	C₂	semidirect
27. D₁₇₃⋊C₈	Q₈	semidirect
73. C₁₇₃	#2960	semidirect

H ≅ G/N

type

1. #26012543

C₂

semidirect

2. #26012540

C₂

semidirect

27. D₁₇₃⋊C₈

Q₈

semidirect

73. C₁₇₃

#2960

semidirect

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

subgroup	index
#26012543	2
#26012540	2
#26012498	2
#26012496	2
#26012456	2
C₄×D₁₇₃⋊C₈	2
#26012047	2
#2960	173

subgroup

index

C₄×D₁₇₃⋊C₈

173

Subgroups — #36370353

Conjugacy classes of subgroups

Complements

Maximal subgroups