Group #36354510 — subgroups

Subgroups — #36354510

Order 22144.

292 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 121 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×#19149679	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₁₇₃	C₂	1	1	yes	maximal, characteristic
3	#26011254	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₄₆	—	2	1	no	—
6	#19149679	—	2	2	no	—
7	#19149679	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₇₃	—	2	1	no	—
9	C₂×C₄⋊Dic₁₇₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₇₃	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₇₃	—	2	1	no	—
12	#19149586	—	2	1	no	—
13	C₂²×C₈×C₁₇₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₇₃	—	2	2	no	—
16	C₂×Dic₃₄₆	—	2	4	no	—
17	C₂×Dic₃₄₆	—	4	4	no	—
18	C₂²×Dic₁₇₃	—	2	2	no	—
19	C₂²×Dic₁₇₃	—	2	2	no	—
20	C₄⋊Dic₁₇₃	—	2	4	no	—
21	C₄⋊Dic₁₇₃	—	2	2	no	—
22	C₄⋊Dic₁₇₃	—	2	2	no	—
23	C₄⋊Dic₁₇₃	C₂×C₄	1	4	yes	—
24	C₄⋊Dic₁₇₃	Q₈	1	4	yes	complemented
25	C₄×Dic₁₇₃	—	4	2	no	—
26	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×C₁₇₃	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₁₇₃	—	2	2	no	—
30	C₂×C₈×C₁₇₃	—	2	4	no	—
31	C₄²×C₁₇₃	C₂×C₄	1	2	yes	—
32	C₄²×C₁₇₃	D₄	1	2	yes	—
33	Dic₃₄₆	—	4	8	no	—
34	C₂×Dic₁₇₃	—	2	4	no	—
35	C₂×Dic₁₇₃	—	2	4	no	—
36	C₂×Dic₁₇₃	—	4	2	no	—
37	C₂×Dic₁₇₃	—	4	2	no	—
38	C₂×Dic₁₇₃	—	4	4	no	—
39	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₇₃	—	2	2	no	—
41	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₇₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₇₃	SD₁₆	1	2	yes	—
44	C₂×C₄×C₁₇₃	Q₁₆	1	2	yes	—
45	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇₃	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₁₇₃	#41	1	2	yes	—
49	C₈×C₁₇₃	—	4	2	no	—
50	C₂²×C₁₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₇₃	C₂²⋊Q₈	1	2	yes	—
52	C₂²×C₁₇₃	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₁₇₃	C₂×Q₁₆	1	1	yes	characteristic
54	C₂²×C₁₇₃	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₁₇₃	—	2	2	no	—
56	C₄×C₁₇₃	#103	1	4	yes	—
57	C₄×C₁₇₃	C₄×Q₈	1	2	yes	—
58	C₄×C₁₇₃	#124	1	1	yes	characteristic
59	C₄×C₁₇₃	C₄⋊Q₈	1	1	yes	characteristic
60	Dic₁₇₃	—	4	4	no	—
61	Dic₁₇₃	—	4	4	no	—
62	C₂×C₁₇₃	#391	1	1	yes	characteristic
63	C₂×C₁₇₃	C₂×#103	1	1	yes	characteristic
64	C₂×C₁₇₃	#418	1	1	yes	characteristic
65	C₂×C₁₇₃	#476	1	2	yes	—
66	C₂×C₁₇₃	#477	1	2	yes	—
67	C₁₇₃	#1987	1	1	yes	characteristic, complemented, Sylow
68	#1987	—	173	1	no	maximal, Sylow
69	C₂×C₄⋊Q₈	—	173	1	no	—
70	C₂×C₄⋊C₈	D₁₇₃	1	1	yes	characteristic
71	C₄×C₄⋊C₄	—	173	1	no	—
72	#340	—	173	4	no	—
73	C₂²×Q₈	—	346	1	no	—
74	C₂²×C₈	C₂×D₁₇₃	1	2	yes	—
75	C₄⋊Q₈	—	346	2	no	—
76	C₄⋊Q₈	—	173	4	no	—
77	C₂×C₄⋊C₄	—	346	1	no	—
78	C₂×C₄⋊C₄	—	173	2	no	—
79	C₂×C₄⋊C₄	—	173	2	no	—
80	C₂×C₄²	—	346	1	no	—
81	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
82	C₄⋊C₈	—	2	2	no	—
83	#95	—	346	1	no	—
84	C₂×Q₈	—	346	4	no	—
85	C₂×Q₈	—	692	4	no	—
86	C₂²×C₄	—	346	2	no	—
87	C₂²×C₄	—	346	2	no	—
88	C₂²×C₄	D₆₉₂	1	1	yes	characteristic
89	C₂²×C₄	D₆₉₂	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	2	4	no	—
93	C₄⋊C₄	—	346	2	no	—
94	C₄⋊C₄	—	346	2	no	—
95	C₄⋊C₄	—	346	4	no	—
96	C₄⋊C₄	—	173	4	no	—
97	C₄⋊C₄	—	173	4	no	—
98	C₄²	—	692	2	no	—
99	C₄²	C₄×D₁₇₃	1	2	yes	—
100	C₄²	C₂²⋊D₁₇₃	1	2	yes	—
101	C₂³	C₂×D₆₉₂	1	1	yes	characteristic
102	Q₈	—	692	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	346	4	no	—
105	C₂×C₄	—	346	4	no	—
106	C₂×C₄	—	692	2	no	—
107	C₂×C₄	—	692	2	no	—
108	C₂×C₄	—	692	4	no	—
109	C₂×C₄	#14583803	1	2	yes	—
110	C₂×C₄	#14583803	1	2	yes	—
111	C₂×C₄	#14583815	1	2	yes	—
112	C₂×C₄	Dic₆₉₂	1	2	yes	—
113	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
115	C₂×C₄	Q₈×D₁₇₃	1	2	yes	—
116	C₂×C₄	#14583825	1	2	yes	—
117	C₈	—	4	2	no	—
118	C₂²	C₂×#14583803	1	1	yes	characteristic
119	C₂²	#19149659	1	2	yes	—
120	C₂²	C₂×#14583815	1	1	yes	characteristic
121	C₂²	C₂×Dic₆₉₂	1	1	yes	characteristic
122	C₂²	#19149707	1	2	yes	—
123	C₄	—	2	2	no	—
124	C₄	—	692	4	no	—
125	C₄	—	692	4	no	—
126	C₄	#19149604	1	4	yes	—
127	C₄	#19149644	1	2	yes	—
128	C₄	#19149666	1	1	yes	characteristic
129	C₄	#19149677	1	1	yes	characteristic
130	C₂	#26011396	1	1	yes	characteristic
131	C₂	C₂×#19149604	1	1	yes	characteristic
132	C₂	#26011497	1	1	yes	characteristic
133	C₂	#26011782	1	2	yes	—
134	C₂	#26011787	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19149679

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₇₃

C₂

yes

maximal, characteristic

#26011254

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₃₄₆

—

#19149679

—

#19149679

C₄

yes

complemented

C₂×C₄⋊Dic₁₇₃

—

C₂×C₄⋊Dic₁₇₃

C₂²

yes

—

C₂×C₄⋊Dic₁₇₃

C₂²

yes

—

C₂×C₄×Dic₁₇₃

—

#19149586

—

C₂²×C₈×C₁₇₃

C₂²

yes

—

C₂×C₄²×C₁₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₁₇₃

—

C₂×Dic₃₄₆

—

C₂×Dic₃₄₆

—

C₂²×Dic₁₇₃

—

C₂²×Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

C₂×C₄

yes

—

C₄⋊Dic₁₇₃

Q₈

yes

complemented

C₄×Dic₁₇₃

—

C₂²×C₄×C₁₇₃

D₄

yes

characteristic

C₂²×C₄×C₁₇₃

D₄

yes

characteristic

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

C₂×C₈×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₄²×C₁₇₃

C₂×C₄

yes

—

C₄²×C₁₇₃

D₄

yes

—

Dic₃₄₆

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂³×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇₃

SD₁₆

yes

—

C₂×C₄×C₁₇₃

Q₁₆

yes

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂×Q₈

yes

—

C₂×C₄×C₁₇₃

#41

yes

—

C₈×C₁₇₃

—

C₂²×C₁₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇₃

C₂²⋊Q₈

yes

—

C₂²×C₁₇₃

C₂×SD₁₆

yes

characteristic

C₂²×C₁₇₃

C₂×Q₁₆

yes

characteristic

C₂²×C₁₇₃

AGL(1,ℤ/8)

yes

—

C₄×C₁₇₃

—

C₄×C₁₇₃

#103

yes

—

C₄×C₁₇₃

C₄×Q₈

yes

—

C₄×C₁₇₃

#124

yes

characteristic

C₄×C₁₇₃

C₄⋊Q₈

yes

characteristic

Dic₁₇₃

—

Dic₁₇₃

—

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

—

C₂×C₁₇₃

#477

yes

—

C₁₇₃

#1987

yes

characteristic, complemented, Sylow

#1987

—

173

maximal, Sylow

C₂×C₄⋊Q₈

—

173

—

C₂×C₄⋊C₈

D₁₇₃

yes

characteristic

C₄×C₄⋊C₄

—

173

—

#340

—

173

—

C₂²×Q₈

—

346

—

C₂²×C₈

C₂×D₁₇₃

yes

—

C₄⋊Q₈

—

346

—

C₄⋊Q₈

—

173

—

C₂×C₄⋊C₄

—

346

—

C₂×C₄⋊C₄

—

173

—

C₂×C₄⋊C₄

—

173

—

C₂×C₄²

—

346

—

C₂×C₄²

C₂×D₁₇₃

yes

characteristic

C₄⋊C₈

—

#95

—

346

—

C₂×Q₈

—

346

—

C₂×Q₈

—

692

—

C₂²×C₄

—

346

—

C₂²×C₄

—

346

—

C₂²×C₄

D₆₉₂

yes

characteristic

C₂²×C₄

D₆₉₂

yes

characteristic

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

346

—

C₄⋊C₄

—

346

—

C₄⋊C₄

—

346

—

C₄⋊C₄

—

173

—

C₄⋊C₄

—

173

—

C₄²

—

692

—

C₄²

C₄×D₁₇₃

yes

—

100

C₄²

C₂²⋊D₁₇₃

yes

—

101

C₂³

C₂×D₆₉₂

yes

characteristic

102

Q₈

—

692

—

103

C₂×C₄

—

104

C₂×C₄

—

346

—

105

C₂×C₄

—

346

—

106

C₂×C₄

—

692

—

107

C₂×C₄

—

692

—

108

C₂×C₄

—

692

—

109

C₂×C₄

#14583803

yes

—

110

C₂×C₄

#14583803

yes

—

111

C₂×C₄

#14583815

yes

—

112

C₂×C₄

Dic₆₉₂

yes

—

113

C₂×C₄

C₂×C₄×D₁₇₃

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

characteristic

115

C₂×C₄

Q₈×D₁₇₃

yes

—

116

C₂×C₄

#14583825

yes

—

117

C₈

—

118

C₂²

C₂×#14583803

yes

characteristic

119

C₂²

#19149659

yes

—

120

C₂²

C₂×#14583815

yes

characteristic

121

C₂²

C₂×Dic₆₉₂

yes

characteristic

122

C₂²

#19149707

yes

—

123

C₄

—

124

C₄

—

692

—

125

C₄

—

692

—

126

C₄

#19149604

yes

—

127

C₄

#19149644

yes

—

128

C₄

#19149666

yes

characteristic

129

C₄

#19149677

yes

characteristic

130

C₂

#26011396

yes

characteristic

131

C₂

C₂×#19149604

yes

characteristic

132

C₂

#26011497

yes

characteristic

133

C₂

#26011782

yes

—

134

C₂

#26011787

yes

—

	subgroup	G/H
center	C₂³	C₂×D₆₉₂
commutator	C₂×C₄×C₁₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	C₂×C₄⋊C₈×C₁₇₃	C₂
socle, prime radical	C₂³×C₁₇₃	C₂×D₄
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₇₃
hypocenter	C₁₇₃	#1987
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₆₉₂

commutator

C₂×C₄×C₁₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇₃

Fitting

C₂×C₄⋊C₈×C₁₇₃

C₂

socle, prime radical

C₂³×C₁₇₃

C₂×D₄

cosocle

C₂²×C₄×C₁₇₃

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₇₃

hypocenter

C₁₇₃

#1987

LCS

G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
7. #19149679	C₄	semidirect
24. C₄⋊Dic₁₇₃	Q₈	semidirect
67. C₁₇₃	#1987	semidirect

H ≅ G/N

type

7. #19149679

C₄

semidirect

24. C₄⋊Dic₁₇₃

Q₈

semidirect

67. C₁₇₃

#1987

semidirect

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

subgroup	index
C₂×#19149679	2
C₄×C₄⋊Dic₁₇₃	2
#26011254	2
C₂×C₄⋊C₈×C₁₇₃	2
#1987	173

subgroup

index

C₂×#19149679

C₄×C₄⋊Dic₁₇₃

#26011254

C₂×C₄⋊C₈×C₁₇₃

#1987

173

Subgroups — #36354510

Conjugacy classes of subgroups

Complements

Maximal subgroups