Group #36353460 — subgroups

Subgroups — #36353460

Order 22144.

272 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 141 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×#19149613	C₂	1	1	yes	maximal, characteristic
2	C₂×#19149611	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊Dic₁₇₃	C₂	1	2	yes	maximal
4	#26011254	C₂	1	2	yes	maximal
5	C₂×#97×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₁₇₃	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₇₃	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₇₃	—	2	2	no	—
9	#19149613	C₄	1	4	yes	complemented
10	#19149611	C₄	1	4	yes	complemented
11	#19149586	—	2	2	no	—
12	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
15	#97×C₁₇₃	C₄	1	4	yes	complemented
16	C₂²×Dic₁₇₃	—	2	2	no	—
17	C₂²×Dic₁₇₃	—	2	2	no	—
18	C₄⋊Dic₁₇₃	—	2	2	no	—
19	C₄⋊Dic₁₇₃	—	2	2	no	—
20	C₄⋊Dic₁₇₃	C₂×C₄	1	4	yes	—
21	C₄⋊Dic₁₇₃	C₂×C₄	1	4	yes	—
22	C₄×Dic₁₇₃	—	4	4	no	—
23	C₂²×C₄×C₁₇₃	D₄	1	2	yes	—
24	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₇₃	—	2	1	no	—
26	C₂×C₈×C₁₇₃	—	2	1	no	—
27	C₂×C₈×C₁₇₃	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₁₇₃	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₇₃	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₇₃	C₂×C₄	1	2	yes	—
31	C₄²×C₁₇₃	C₂×C₄	1	2	yes	—
32	C₄²×C₁₇₃	Q₈	1	2	yes	—
33	C₂×Dic₁₇₃	—	2	4	no	—
34	C₂×Dic₁₇₃	—	2	4	no	—
35	C₂×Dic₁₇₃	—	4	2	no	—
36	C₂×Dic₁₇₃	—	4	2	no	—
37	C₂×Dic₁₇₃	—	4	2	no	—
38	C₂×Dic₁₇₃	—	4	2	no	—
39	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₇₃	—	2	2	no	—
41	C₂×C₄×C₁₇₃	—	2	2	no	—
42	C₂×C₄×C₁₇₃	C₄⋊C₄	1	4	yes	—
43	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₇₃	#41	1	1	yes	characteristic
48	C₂×C₄×C₁₇₃	#41	1	1	yes	characteristic
49	C₈×C₁₇₃	—	2	2	no	—
50	C₈×C₁₇₃	C₄²	1	4	yes	complemented
51	C₂²×C₁₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₇₃	C₄×D₄	1	1	yes	characteristic
53	C₂²×C₁₇₃	C₄×D₄	1	1	yes	characteristic
54	C₂²×C₁₇₃	AGL(1,ℤ/8)	1	2	yes	—
55	C₂²×C₁₇₃	#137	1	2	yes	—
56	C₄×C₁₇₃	—	2	4	no	—
57	C₄×C₁₇₃	C₂×C₄²	1	1	yes	characteristic
58	C₄×C₁₇₃	#117	1	1	yes	characteristic
59	C₄×C₁₇₃	C₄×Q₈	1	1	yes	characteristic
60	C₄×C₁₇₃	C₄×Q₈	1	1	yes	characteristic
61	Dic₁₇₃	—	4	4	no	—
62	Dic₁₇₃	—	4	4	no	—
63	C₂×C₁₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₁₇₃	#428	1	2	yes	—
65	C₂×C₁₇₃	#440	1	2	yes	—
66	C₂×C₁₇₃	#441	1	1	yes	characteristic
67	C₂×C₁₇₃	#442	1	1	yes	characteristic
68	C₁₇₃	#1776	1	1	yes	characteristic, complemented, Sylow
69	#1776	—	173	1	no	maximal, Sylow
70	C₂×#107	—	173	1	no	—
71	C₂×#106	—	173	1	no	—
72	C₂×#97	D₁₇₃	1	1	yes	characteristic
73	C₄×C₄⋊C₄	—	173	2	no	—
74	#340	—	173	2	no	—
75	C₂²×C₈	C₂×D₁₇₃	1	1	yes	characteristic
76	C₂²×C₈	C₂×D₁₇₃	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	173	2	no	—
78	C₂×C₄⋊C₄	—	173	2	no	—
79	C₂×C₄²	—	346	2	no	—
80	C₂×C₄²	C₂×D₁₇₃	1	1	yes	characteristic
81	#107	—	173	4	no	—
82	#106	—	173	4	no	—
83	#97	Dic₁₇₃	1	4	yes	complemented
84	#95	—	346	2	no	—
85	C₂²×C₄	—	346	2	no	—
86	C₂²×C₄	—	346	2	no	—
87	C₂²×C₄	D₆₉₂	1	2	yes	—
88	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	C₄×D₁₇₃	1	2	yes	—
92	C₂×C₈	C₄×D₁₇₃	1	2	yes	—
93	C₂×C₈	C₂×Dic₁₇₃	1	2	yes	—
94	C₂×C₈	C₂×Dic₁₇₃	1	2	yes	—
95	C₄⋊C₄	—	346	2	no	—
96	C₄⋊C₄	—	346	2	no	—
97	C₄⋊C₄	—	173	4	no	—
98	C₄⋊C₄	—	173	4	no	—
99	C₄²	—	692	4	no	—
100	C₄²	C₂×Dic₁₇₃	1	2	yes	—
101	C₄²	Dic₃₄₆	1	2	yes	—
102	C₂³	C₂×D₆₉₂	1	1	yes	characteristic
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	346	4	no	—
106	C₂×C₄	—	346	4	no	—
107	C₂×C₄	—	692	2	no	—
108	C₂×C₄	—	692	2	no	—
109	C₂×C₄	—	692	2	no	—
110	C₂×C₄	—	692	2	no	—
111	C₂×C₄	C₄⋊Dic₁₇₃	1	4	yes	—
112	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
113	C₂×C₄	C₂×C₄×D₁₇₃	1	1	yes	characteristic
114	C₂×C₄	C₂²×Dic₁₇₃	1	1	yes	characteristic
115	C₂×C₄	C₂×Dic₃₄₆	1	1	yes	characteristic
116	C₂×C₄	#14583826	1	1	yes	characteristic
117	C₂×C₄	#14583826	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	C₄×Dic₁₇₃	1	4	yes	complemented
120	C₂²	C₂×C₄⋊Dic₁₇₃	1	1	yes	characteristic
121	C₂²	C₄×D₆₉₂	1	1	yes	characteristic
122	C₂²	C₄×D₆₉₂	1	1	yes	characteristic
123	C₂²	#19149707	1	2	yes	—
124	C₂²	#19149712	1	2	yes	—
125	C₄	—	2	4	no	—
126	C₄	—	692	4	no	—
127	C₄	—	692	4	no	—
128	C₄	C₂×C₄×Dic₁₇₃	1	1	yes	characteristic
129	C₄	#19149638	1	1	yes	characteristic
130	C₄	C₄×Dic₃₄₆	1	1	yes	characteristic
131	C₄	C₄×Dic₃₄₆	1	1	yes	characteristic
132	C₂	C₄×C₄⋊Dic₁₇₃	1	1	yes	characteristic
133	C₂	#26011533	1	2	yes	—
134	C₂	#26011593	1	2	yes	—
135	C₂	#26011599	1	1	yes	characteristic
136	C₂	#26011604	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19149613

C₂

yes

maximal, characteristic

C₂×#19149611

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₁₇₃

C₂

yes

maximal

#26011254

C₂

yes

maximal

C₂×#97×C₁₇₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₁₇₃

C₂²

yes

—

C₂×C₄⋊Dic₁₇₃

C₂²

yes

—

C₂×C₄×Dic₁₇₃

—

#19149613

C₄

yes

complemented

#19149611

C₄

yes

complemented

#19149586

—

C₂²×C₈×C₁₇₃

C₂²

yes

characteristic

C₂²×C₈×C₁₇₃

C₂²

yes

characteristic

C₂×C₄²×C₁₇₃

C₂²

yes

characteristic

#97×C₁₇₃

C₄

yes

complemented

C₂²×Dic₁₇₃

—

C₂²×Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

C₂×C₄

yes

—

C₄⋊Dic₁₇₃

C₂×C₄

yes

—

C₄×Dic₁₇₃

—

C₂²×C₄×C₁₇₃

D₄

yes

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

C₂×C₈×C₁₇₃

—

C₂×C₈×C₁₇₃

—

C₂×C₈×C₁₇₃

C₂×C₄

yes

—

C₂×C₈×C₁₇₃

C₂×C₄

yes

—

C₂×C₈×C₁₇₃

C₂×C₄

yes

—

C₂×C₈×C₁₇₃

C₂×C₄

yes

—

C₄²×C₁₇₃

C₂×C₄

yes

—

C₄²×C₁₇₃

Q₈

yes

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂³×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇₃

C₂×Q₈

yes

C₂×C₄×C₁₇₃

yes

C₂×C₄×C₁₇₃

yes

C₈×C₁₇₃

—

C₈×C₁₇₃

C₄²

yes

complemented

C₂²×C₁₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇₃

C₄×D₄

yes

characteristic

C₂²×C₁₇₃

C₄×D₄

yes

characteristic

C₂²×C₁₇₃

AGL(1,ℤ/8)

yes

—

C₂²×C₁₇₃

#137

yes

—

C₄×C₁₇₃

—

C₄×C₁₇₃

C₂×C₄²

yes

characteristic

C₄×C₁₇₃

#117

yes

characteristic

C₄×C₁₇₃

C₄×Q₈

yes

characteristic

C₄×C₁₇₃

C₄×Q₈

yes

characteristic

Dic₁₇₃

—

Dic₁₇₃

—

C₂×C₁₇₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₇₃

#428

yes

—

C₂×C₁₇₃

#440

yes

—

C₂×C₁₇₃

yes

C₂×C₁₇₃

yes

C₁₇₃

yes

characteristic, complemented, Sylow

#1776

—

173

maximal, Sylow

C₂×#107

—

173

—

C₂×#106

—

173

—

C₂×#97

D₁₇₃

yes

characteristic

C₄×C₄⋊C₄

—

173

—

#340

—

173

—

C₂²×C₈

C₂×D₁₇₃

yes

characteristic

C₂²×C₈

C₂×D₁₇₃

yes

characteristic

C₂×C₄⋊C₄

—

173

—

C₂×C₄⋊C₄

—

173

—

C₂×C₄²

—

346

—

C₂×C₄²

C₂×D₁₇₃

yes

characteristic

#107

—

173

—

#106

—

173

—

#97

Dic₁₇₃

yes

complemented

#95

—

346

—

C₂²×C₄

—

346

—

C₂²×C₄

—

346

—

C₂²×C₄

D₆₉₂

yes

—

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₁₇₃

yes

—

C₂×C₈

C₄×D₁₇₃

yes

—

C₂×C₈

C₂×Dic₁₇₃

yes

—

C₂×C₈

C₂×Dic₁₇₃

yes

—

C₄⋊C₄

—

346

—

C₄⋊C₄

—

346

—

C₄⋊C₄

—

173

—

C₄⋊C₄

—

173

—

C₄²

—

692

—

100

C₄²

C₂×Dic₁₇₃

yes

—

101

C₄²

Dic₃₄₆

yes

—

102

C₂³

C₂×D₆₉₂

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

346

—

106

C₂×C₄

—

346

—

107

C₂×C₄

—

692

—

108

C₂×C₄

—

692

—

109

C₂×C₄

—

692

—

110

C₂×C₄

—

692

—

111

C₂×C₄

C₄⋊Dic₁₇₃

yes

—

112

C₂×C₄

C₂×C₄×D₁₇₃

yes

characteristic

113

C₂×C₄

C₂×C₄×D₁₇₃

yes

characteristic

114

C₂×C₄

C₂²×Dic₁₇₃

yes

characteristic

115

C₂×C₄

C₂×Dic₃₄₆

yes

characteristic

116

C₂×C₄

#14583826

yes

characteristic

117

C₂×C₄

#14583826

yes

characteristic

118

C₈

—

119

C₈

C₄×Dic₁₇₃

yes

complemented

120

C₂²

C₂×C₄⋊Dic₁₇₃

yes

characteristic

121

C₂²

C₄×D₆₉₂

yes

characteristic

122

C₂²

C₄×D₆₉₂

yes

characteristic

123

C₂²

#19149707

yes

—

124

C₂²

#19149712

yes

—

125

C₄

—

126

C₄

—

692

—

127

C₄

—

692

—

128

C₄

C₂×C₄×Dic₁₇₃

yes

characteristic

129

C₄

#19149638

yes

characteristic

130

C₄

C₄×Dic₃₄₆

yes

characteristic

131

C₄

C₄×Dic₃₄₆

yes

characteristic

132

C₂

C₄×C₄⋊Dic₁₇₃

yes

characteristic

133

C₂

#26011533

yes

—

134

C₂

#26011593

yes

—

135

C₂

#26011599

yes

characteristic

136

C₂

#26011604

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₉₂
commutator	C₄×C₁₇₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	C₂×#97×C₁₇₃	C₂
socle, prime radical	C₂³×C₁₇₃	C₂×D₄
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	C₂×#97	D₁₇₃
hypocenter	C₁₇₃	#1776
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₆₉₂

commutator

C₄×C₁₇₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₇₃

Fitting

C₂×#97×C₁₇₃

C₂

socle, prime radical

C₂³×C₁₇₃

C₂×D₄

cosocle

C₂²×C₄×C₁₇₃

C₂³

hypercenter

C₂×#97

D₁₇₃

hypocenter

C₁₇₃

#1776

LCS

G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
9. #19149613	C₄	semidirect
10. #19149611	C₄	semidirect
15. #97×C₁₇₃	C₄	semidirect
50. C₈×C₁₇₃	C₄²	semidirect
68. C₁₇₃	#1776	semidirect
83. #97	Dic₁₇₃	semidirect
119. C₈	C₄×Dic₁₇₃	semidirect

H ≅ G/N

type

9. #19149613

C₄

semidirect

10. #19149611

C₄

semidirect

15. #97×C₁₇₃

C₄

semidirect

50. C₈×C₁₇₃

C₄²

semidirect

68. C₁₇₃

#1776

semidirect

83. #97

Dic₁₇₃

semidirect

119. C₈

C₄×Dic₁₇₃

semidirect

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

subgroup	index
C₂×#19149613	2
C₂×#19149611	2
C₄×C₄⋊Dic₁₇₃	2
#26011254	2
C₂×#97×C₁₇₃	2
#1776	173

subgroup

index

C₄×C₄⋊Dic₁₇₃

C₂×#97×C₁₇₃

173

Subgroups — #36353460

Conjugacy classes of subgroups

Complements

Maximal subgroups