Group #36353484 — subgroups

Subgroups — #36353484

Order 22144.

820 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 820 conjugacy classes of subgroups — too many to draw. Only its 133 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×#19149596	C₂	1	4	yes	maximal, complemented
2	#26011455	C₂	1	2	yes	maximal, complemented
3	C₂³×D₄×C₁₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14583810	C₂²	1	4	yes	complemented
5	#19149596	—	2	8	no	—
6	C₂²⋊#13063436	—	2	4	no	—
7	C₂⁵×C₁₇₃	C₄	1	2	yes	—
8	C₂²×D₄×C₁₇₃	—	2	8	no	—
9	C₂²×D₄×C₁₇₃	—	2	4	no	—
10	C₂²×D₄×C₁₇₃	C₄	1	2	yes	—
11	C₂²×D₄×C₁₇₃	C₂²	1	2	yes	—
12	C₂³×C₄×C₁₇₃	C₂²	1	1	yes	characteristic
13	#14583810	—	2	8	no	—
14	C₂×#13063436	—	2	4	no	—
15	C₂⁴×C₁₇₃	—	2	8	no	—
16	C₂⁴×C₁₇₃	—	2	4	no	—
17	C₂⁴×C₁₇₃	—	4	8	no	—
18	C₂⁴×C₁₇₃	C₂×C₄	1	1	yes	characteristic
19	C₂⁴×C₁₇₃	C₂×C₄	1	4	yes	—
20	C₂×D₄×C₁₇₃	—	2	32	no	—
21	C₂×D₄×C₁₇₃	—	2	8	no	—
22	C₂×D₄×C₁₇₃	—	2	4	no	—
23	C₂×D₄×C₁₇₃	—	2	4	no	—
24	C₂×D₄×C₁₇₃	—	2	4	no	—
25	C₂×D₄×C₁₇₃	D₄	1	8	yes	—
26	C₂²×C₄×C₁₇₃	—	2	4	no	—
27	C₂²×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₇₃	D₄	1	4	yes	—
29	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
30	#13063436	—	4	4	no	—
31	C₂³×C₁₇₃	—	2	4	no	—
32	C₂³×C₁₇₃	—	2	2	no	—
33	C₂³×C₁₇₃	—	2	16	no	—
34	C₂³×C₁₇₃	—	2	4	no	—
35	C₂³×C₁₇₃	—	2	4	no	—
36	C₂³×C₁₇₃	—	4	8	no	—
37	C₂³×C₁₇₃	—	4	32	no	—
38	C₂³×C₁₇₃	—	4	8	no	—
39	C₂³×C₁₇₃	—	4	8	no	—
40	C₂³×C₁₇₃	C₂²⋊C₄	1	2	yes	—
41	C₂³×C₁₇₃	C₂²⋊C₄	1	8	yes	—
42	C₂³×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
43	D₄×C₁₇₃	—	2	16	no	—
44	D₄×C₁₇₃	—	2	16	no	—
45	C₂×C₄×C₁₇₃	—	2	2	no	—
46	C₂×C₄×C₁₇₃	—	2	8	no	—
47	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₁₇₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₇₃	C₂×D₄	1	4	yes	—
50	C₂²×C₁₇₃	—	2	8	no	—
51	C₂²×C₁₇₃	—	2	2	no	—
52	C₂²×C₁₇₃	—	2	2	no	—
53	C₂²×C₁₇₃	—	2	2	no	—
54	C₂²×C₁₇₃	—	2	8	no	—
55	C₂²×C₁₇₃	—	4	32	no	—
56	C₂²×C₁₇₃	—	4	8	no	—
57	C₂²×C₁₇₃	—	4	8	no	—
58	C₂²×C₁₇₃	—	4	8	no	—
59	C₂²×C₁₇₃	#100	1	4	yes	complemented
60	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	2	yes	—
61	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₁₇₃	—	2	2	no	—
63	C₄×C₁₇₃	#120	1	4	yes	—
64	C₂×C₁₇₃	—	2	4	no	—
65	C₂×C₁₇₃	—	2	2	no	—
66	C₂×C₁₇₃	—	4	8	no	—
67	C₂×C₁₇₃	#379	1	1	yes	characteristic
68	C₂×C₁₇₃	C₂×#100	1	2	yes	—
69	C₁₇₃	#1783	1	1	yes	characteristic, complemented, Sylow
70	#1783	—	173	1	no	maximal, Sylow
71	C₂³×D₄	D₁₇₃	1	1	yes	characteristic
72	C₂×#100	—	173	4	no	—
73	#407	—	173	2	no	—
74	C₂⁵	Dic₁₇₃	1	2	yes	—
75	C₂²×D₄	—	2	8	no	—
76	C₂²×D₄	—	2	4	no	—
77	C₂²×D₄	Dic₁₇₃	1	2	yes	—
78	C₂²×D₄	C₂×D₁₇₃	1	2	yes	—
79	C₂³×C₄	C₂×D₁₇₃	1	1	yes	characteristic
80	C₂×M₁₆	—	173	4	no	—
81	#100	—	346	8	no	—
82	C₂²⋊C₈	—	346	4	no	—
83	C₂⁴	—	2	8	no	—
84	C₂⁴	—	2	4	no	—
85	C₂⁴	—	4	8	no	—
86	C₂⁴	C₂×Dic₁₇₃	1	1	yes	characteristic
87	C₂⁴	C₂×Dic₁₇₃	1	4	yes	—
88	C₂×D₄	—	2	32	no	—
89	C₂×D₄	—	2	8	no	—
90	C₂×D₄	—	2	4	no	—
91	C₂×D₄	—	2	4	no	—
92	C₂×D₄	—	2	4	no	—
93	C₂×D₄	C₂²⋊D₁₇₃	1	8	yes	—
94	C₂²×C₄	—	2	4	no	—
95	C₂²×C₄	C₂×Dic₁₇₃	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₁₇₃	1	4	yes	—
97	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
98	M₁₆	—	346	8	no	—
99	C₂×C₈	—	346	4	no	—
100	C₂³	—	2	4	no	—
101	C₂³	—	2	2	no	—
102	C₂³	—	2	16	no	—
103	C₂³	—	2	4	no	—
104	C₂³	—	2	4	no	—
105	C₂³	—	4	8	no	—
106	C₂³	—	4	32	no	—
107	C₂³	—	4	8	no	—
108	C₂³	—	4	8	no	—
109	C₂³	C₂²⋊Dic₁₇₃	1	2	yes	—
110	C₂³	C₂²⋊Dic₁₇₃	1	8	yes	—
111	C₂³	C₂²×Dic₁₇₃	1	1	yes	characteristic
112	D₄	—	2	16	no	—
113	D₄	—	2	16	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	8	no	—
116	C₂×C₄	C₂²⋊Dic₁₇₃	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	4	yes	—
118	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	2	yes	—
119	C₈	—	692	4	no	—
120	C₂²	—	2	8	no	—
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	8	no	—
125	C₂²	—	4	8	no	—
126	C₂²	—	4	32	no	—
127	C₂²	—	4	8	no	—
128	C₂²	—	4	8	no	—
129	C₂²	#19149596	1	4	yes	complemented
130	C₂²	C₂×C₂²⋊Dic₁₇₃	1	2	yes	—
131	C₂²	C₂×C₂²⋊Dic₁₇₃	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	#19149648	1	4	yes	—
134	C₂	—	2	4	no	—
135	C₂	—	2	2	no	—
136	C₂	—	4	8	no	—
137	C₂	#26011347	1	1	yes	characteristic
138	C₂	C₂×#19149596	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19149596

C₂

yes

maximal, complemented

#26011455

C₂

yes

maximal, complemented

C₂³×D₄×C₁₇₃

C₂

yes

maximal, characteristic

C₂×#14583810

C₂²

yes

complemented

#19149596

—

C₂²⋊#13063436

—

C₂⁵×C₁₇₃

C₄

yes

—

C₂²×D₄×C₁₇₃

—

C₂²×D₄×C₁₇₃

—

C₂²×D₄×C₁₇₃

C₄

yes

—

C₂²×D₄×C₁₇₃

C₂²

yes

—

C₂³×C₄×C₁₇₃

C₂²

yes

characteristic

#14583810

—

C₂×#13063436

—

C₂⁴×C₁₇₃

—

C₂⁴×C₁₇₃

—

C₂⁴×C₁₇₃

—

C₂⁴×C₁₇₃

C₂×C₄

yes

characteristic

C₂⁴×C₁₇₃

C₂×C₄

yes

—

C₂×D₄×C₁₇₃

—

C₂×D₄×C₁₇₃

—

C₂×D₄×C₁₇₃

—

C₂×D₄×C₁₇₃

—

C₂×D₄×C₁₇₃

—

C₂×D₄×C₁₇₃

D₄

yes

—

C₂²×C₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇₃

D₄

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

#13063436

—

C₂³×C₁₇₃

—

C₂³×C₁₇₃

—

C₂³×C₁₇₃

—

C₂³×C₁₇₃

—

C₂³×C₁₇₃

—

C₂³×C₁₇₃

—

C₂³×C₁₇₃

—

C₂³×C₁₇₃

—

C₂³×C₁₇₃

—

C₂³×C₁₇₃

C₂²⋊C₄

yes

—

C₂³×C₁₇₃

C₂²⋊C₄

yes

—

C₂³×C₁₇₃

C₂²×C₄

yes

characteristic

D₄×C₁₇₃

—

D₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

#100

yes

complemented

C₂²×C₁₇₃

C₂×C₂²⋊C₄

yes

—

C₂²×C₁₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₇₃

—

C₄×C₁₇₃

#120

yes

—

C₂×C₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

#379

yes

characteristic

C₂×C₁₇₃

C₂×#100

yes

—

C₁₇₃

#1783

yes

characteristic, complemented, Sylow

#1783

—

173

maximal, Sylow

C₂³×D₄

D₁₇₃

yes

characteristic

C₂×#100

—

173

—

#407

—

173

—

C₂⁵

Dic₁₇₃

yes

—

C₂²×D₄

—

C₂²×D₄

—

C₂²×D₄

Dic₁₇₃

yes

—

C₂²×D₄

C₂×D₁₇₃

yes

—

C₂³×C₄

C₂×D₁₇₃

yes

characteristic

C₂×M₁₆

—

173

—

#100

—

346

—

C₂²⋊C₈

—

346

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

C₂×Dic₁₇₃

yes

characteristic

C₂⁴

C₂×Dic₁₇₃

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₇₃

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₇₃

yes

characteristic

C₂²×C₄

C₂²⋊D₁₇₃

yes

—

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

M₁₆

—

346

—

C₂×C₈

—

346

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

C₂²⋊Dic₁₇₃

yes

—

110

C₂³

C₂²⋊Dic₁₇₃

yes

—

111

C₂³

C₂²×Dic₁₇₃

yes

characteristic

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂²⋊Dic₁₇₃

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

—

118

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

—

119

C₈

—

692

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#19149596

yes

complemented

130

C₂²

C₂×C₂²⋊Dic₁₇₃

yes

—

131

C₂²

C₂×C₂²⋊Dic₁₇₃

yes

characteristic

132

C₄

—

133

C₄

#19149648

yes

—

134

C₂

—

135

C₂

—

136

C₂

—

137

C₂

#26011347

yes

characteristic

138

C₂

C₂×#19149596

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₇₃
commutator	C₂³×C₁₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting, prime radical	C₂³×D₄×C₁₇₃	C₂
socle	C₂²×C₁₇₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	C₂³×D₄	D₁₇₃
hypocenter	C₁₇₃	#1783
LCS	G ▷ C₂³×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×D₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₇₃

commutator

C₂³×C₁₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₃

Fitting, prime radical

C₂³×D₄×C₁₇₃

C₂

socle

C₂²×C₁₇₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₇₃

C₂³

hypercenter

C₂³×D₄

D₁₇₃

hypocenter

C₁₇₃

#1783

LCS

G ▷ C₂³×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×D₄

N	H ≅ G/N	type
1. C₂×#19149596	C₂	semidirect
2. #26011455	C₂	semidirect
4. C₂×#14583810	C₂²	semidirect
59. C₂²×C₁₇₃	#100	semidirect
69. C₁₇₃	#1783	semidirect
129. C₂²	#19149596	semidirect

H ≅ G/N

type

1. C₂×#19149596

C₂

semidirect

2. #26011455

C₂

semidirect

4. C₂×#14583810

C₂²

semidirect

59. C₂²×C₁₇₃

#100

semidirect

69. C₁₇₃

#1783

semidirect

129. C₂²

#19149596

semidirect

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

subgroup	index
C₂×#19149596	2
#26011455	2
C₂³×D₄×C₁₇₃	2
#1783	173

subgroup

index

C₂×#19149596

#26011455

C₂³×D₄×C₁₇₃

#1783

173

Subgroups — #36353484

Conjugacy classes of subgroups

Complements

Maximal subgroups