Group #36353462 — subgroups

Subgroups — #36353462

Order 22144.

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

normal subgroups only

This group has 272 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×#19149611	C₂	1	1	yes	maximal, characteristic
2	C₂×#19149589	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×Dic₁₇₃	C₂	1	2	yes	maximal
4	#26011253	C₂	1	2	yes	maximal
5	C₂×#107×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13063436	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₇₃	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₇₃	—	2	2	no	—
9	C₂×C₄×Dic₁₇₃	C₂²	1	1	yes	characteristic
10	#19149611	C₄	1	4	yes	complemented
11	#19149589	C₄	1	4	yes	complemented
12	#19149586	—	2	2	no	—
13	C₂²×C₈×C₁₇₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₇₃	C₂²	1	2	yes	—
15	#107×C₁₇₃	C₄	1	4	yes	complemented
16	C₂²×Dic₁₇₃	—	2	2	no	—
17	C₂²×Dic₁₇₃	D₄	1	2	yes	—
18	C₂×#13063436	—	2	1	no	—
19	C₂×#13063436	—	2	1	no	—
20	C₂×#13063436	C₂×C₄	1	2	yes	—
21	C₄⋊Dic₁₇₃	—	2	2	no	—
22	C₄⋊Dic₁₇₃	C₂×C₄	1	4	yes	—
23	C₄×Dic₁₇₃	—	4	4	no	—
24	C₄×Dic₁₇₃	C₂×C₄	1	2	yes	—
25	C₄×Dic₁₇₃	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₇₃	—	2	2	no	—
27	C₂²×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
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	C₄⋊C₄×C₁₇₃	—	2	2	no	—
32	C₄⋊C₄×C₁₇₃	C₂×C₄	1	4	yes	—
33	C₂×Dic₁₇₃	—	2	2	no	—
34	C₂×Dic₁₇₃	—	2	2	no	—
35	C₂×Dic₁₇₃	—	2	4	no	—
36	C₂×Dic₁₇₃	—	4	2	no	—
37	C₂×Dic₁₇₃	—	4	2	no	—
38	C₂×Dic₁₇₃	C₄⋊C₄	1	4	yes	—
39	#13063436	—	2	2	no	—
40	C₂³×C₁₇₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇₃	—	2	4	no	—
42	C₂×C₄×C₁₇₃	—	4	2	no	—
43	C₂×C₄×C₁₇₃	—	4	2	no	—
44	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₇₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₁₇₃	#41	1	1	yes	characteristic
49	C₂×C₄×C₁₇₃	#41	1	1	yes	characteristic
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	—	4	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	—	2	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	D₁₇₃	1	1	yes	characteristic
71	C₂×#106	—	173	1	no	—
72	C₂×#97	—	173	1	no	—
73	C₄×C₄⋊C₄	—	173	2	no	—
74	#340	—	173	2	no	—
75	C₂²×C₈	—	173	1	no	—
76	C₂²×C₈	C₂×D₁₇₃	1	1	yes	characteristic
77	C₂×C₄⋊C₄	—	173	2	no	—
78	C₂×C₄⋊C₄	C₂×D₁₇₃	1	2	yes	—
79	C₂×C₄²	—	346	2	no	—
80	C₂×C₄²	—	173	1	no	—
81	#107	Dic₁₇₃	1	4	yes	complemented
82	#106	—	173	4	no	—
83	#97	—	173	4	no	—
84	#95	—	346	2	no	—
85	C₂²×C₄	—	2	2	no	—
86	C₂²×C₄	—	346	2	no	—
87	C₂²×C₄	—	173	2	no	—
88	C₂²×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
89	C₂×C₈	—	346	1	no	—
90	C₂×C₈	—	346	1	no	—
91	C₂×C₈	—	173	2	no	—
92	C₂×C₈	C₄×D₁₇₃	1	2	yes	—
93	C₂×C₈	C₄×D₁₇₃	1	2	yes	—
94	C₂×C₈	C₂×Dic₁₇₃	1	2	yes	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	346	2	no	—
97	C₄⋊C₄	—	173	4	no	—
98	C₄⋊C₄	C₂×Dic₁₇₃	1	4	yes	—
99	C₄²	—	173	2	no	—
100	C₄²	—	173	2	no	—
101	C₄²	—	692	4	no	—
102	C₂³	D₄×D₁₇₃	1	1	yes	characteristic
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	346	2	no	—
107	C₂×C₄	—	346	2	no	—
108	C₂×C₄	—	346	4	no	—
109	C₂×C₄	—	173	4	no	—
110	C₂×C₄	—	692	2	no	—
111	C₂×C₄	—	692	2	no	—
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	Q₈×D₁₇₃	1	1	yes	characteristic
116	C₂×C₄	#14583825	1	1	yes	characteristic
117	C₂×C₄	#14583827	1	1	yes	characteristic
118	C₈	—	346	2	no	—
119	C₈	C₄×Dic₁₇₃	1	4	yes	complemented
120	C₂²	C₄⋊C₄×D₁₇₃	1	1	yes	characteristic
121	C₂²	#19149641	1	1	yes	characteristic
122	C₂²	D₄×Dic₁₇₃	1	1	yes	characteristic
123	C₂²	#19149703	1	2	yes	—
124	C₂²	#19149711	1	2	yes	—
125	C₄	—	4	4	no	—
126	C₄	—	346	4	no	—
127	C₄	—	692	4	no	—
128	C₄	C₂×C₄×Dic₁₇₃	1	1	yes	characteristic
129	C₄	#19149637	1	1	yes	characteristic
130	C₄	#19149644	1	1	yes	characteristic
131	C₄	Q₈×Dic₁₇₃	1	1	yes	characteristic
132	C₂	C₄⋊C₄×Dic₁₇₃	1	1	yes	characteristic
133	C₂	#26011532	1	2	yes	—
134	C₂	#26011590	1	2	yes	—
135	C₂	#26011598	1	1	yes	characteristic
136	C₂	#26011603	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19149611

C₂

yes

maximal, characteristic

C₂×#19149589

C₂

yes

maximal, characteristic

C₄⋊C₄×Dic₁₇₃

C₂

yes

maximal

#26011253

C₂

yes

maximal

C₂×#107×C₁₇₃

C₂

yes

maximal, characteristic

C₂²×#13063436

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇₃

C₂²

yes

—

C₂×C₄×Dic₁₇₃

—

C₂×C₄×Dic₁₇₃

C₂²

yes

characteristic

#19149611

C₄

yes

complemented

#19149589

C₄

yes

complemented

#19149586

—

C₂²×C₈×C₁₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇₃

C₂²

yes

—

#107×C₁₇₃

C₄

yes

complemented

C₂²×Dic₁₇₃

—

C₂²×Dic₁₇₃

D₄

yes

—

C₂×#13063436

—

C₂×#13063436

—

C₂×#13063436

C₂×C₄

yes

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

C₂×C₄

yes

—

C₄×Dic₁₇₃

—

C₄×Dic₁₇₃

C₂×C₄

yes

—

C₄×Dic₁₇₃

Q₈

yes

—

C₂²×C₄×C₁₇₃

—

C₂²×C₄×C₁₇₃

C₂³

yes

characteristic

C₂×C₈×C₁₇₃

C₂×C₄

yes

—

C₂×C₈×C₁₇₃

C₂×C₄

yes

—

C₂×C₈×C₁₇₃

C₂×C₄

yes

—

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

C₂×C₄

yes

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

C₄⋊C₄

yes

—

#13063436

—

C₂³×C₁₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

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

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

D₁₇₃

yes

characteristic

C₂×#106

—

173

—

C₂×#97

—

173

—

C₄×C₄⋊C₄

—

173

—

#340

—

173

—

C₂²×C₈

—

173

—

C₂²×C₈

C₂×D₁₇₃

yes

characteristic

C₂×C₄⋊C₄

—

173

—

C₂×C₄⋊C₄

C₂×D₁₇₃

yes

—

C₂×C₄²

—

346

—

C₂×C₄²

—

173

—

#107

Dic₁₇₃

yes

complemented

#106

—

173

—

#97

—

173

—

#95

—

346

—

C₂²×C₄

—

C₂²×C₄

—

346

—

C₂²×C₄

—

173

—

C₂²×C₄

C₂²×D₁₇₃

yes

characteristic

C₂×C₈

—

346

—

C₂×C₈

—

346

—

C₂×C₈

—

173

—

C₂×C₈

C₄×D₁₇₃

yes

—

C₂×C₈

C₄×D₁₇₃

yes

—

C₂×C₈

C₂×Dic₁₇₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

346

—

C₄⋊C₄

—

173

—

C₄⋊C₄

C₂×Dic₁₇₃

yes

—

C₄²

—

173

—

100

C₄²

—

173

—

101

C₄²

—

692

—

102

C₂³

D₄×D₁₇₃

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

346

—

107

C₂×C₄

—

346

—

108

C₂×C₄

—

346

—

109

C₂×C₄

—

173

—

110

C₂×C₄

—

692

—

111

C₂×C₄

—

692

—

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

Q₈×D₁₇₃

yes

characteristic

116

C₂×C₄

#14583825

yes

characteristic

117

C₂×C₄

#14583827

yes

characteristic

118

C₈

—

346

—

119

C₈

C₄×Dic₁₇₃

yes

complemented

120

C₂²

C₄⋊C₄×D₁₇₃

yes

characteristic

121

C₂²

#19149641

yes

characteristic

122

C₂²

D₄×Dic₁₇₃

yes

characteristic

123

C₂²

#19149703

yes

—

124

C₂²

#19149711

yes

—

125

C₄

—

126

C₄

—

346

—

127

C₄

—

692

—

128

C₄

C₂×C₄×Dic₁₇₃

yes

characteristic

129

C₄

#19149637

yes

characteristic

130

C₄

#19149644

yes

characteristic

131

C₄

Q₈×Dic₁₇₃

yes

characteristic

132

C₂

C₄⋊C₄×Dic₁₇₃

yes

characteristic

133

C₂

#26011532

yes

—

134

C₂

#26011590

yes

—

135

C₂

#26011598

yes

characteristic

136

C₂

#26011603

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₇₃
commutator	C₄×C₁₇₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₇₃
Fitting	C₂×#107×C₁₇₃	C₂
socle, prime radical	C₂³×C₁₇₃	C₂×D₄
cosocle	C₂²×C₄×C₁₇₃	C₂³
hypercenter	C₂×#107	D₁₇₃
hypocenter	C₁₇₃	#1776
LCS	G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

subgroup

G/H

center

C₂³

D₄×D₁₇₃

commutator

C₄×C₁₇₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₇₃

Fitting

C₂×#107×C₁₇₃

C₂

socle, prime radical

C₂³×C₁₇₃

C₂×D₄

cosocle

C₂²×C₄×C₁₇₃

C₂³

D₁₇₃

C₁₇₃

G ▷ C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#107

N	H ≅ G/N	type
10. #19149611	C₄	semidirect
11. #19149589	C₄	semidirect
15. #107×C₁₇₃	C₄	semidirect
50. C₈×C₁₇₃	C₄²	semidirect
68. C₁₇₃	#1776	semidirect
81. #107	Dic₁₇₃	semidirect
119. C₈	C₄×Dic₁₇₃	semidirect

H ≅ G/N

type

10. #19149611

C₄

semidirect

11. #19149589

C₄

semidirect

15. #107×C₁₇₃

C₄

semidirect

50. C₈×C₁₇₃

C₄²

semidirect

68. C₁₇₃

#1776

semidirect

81. #107

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×#19149611	2
C₂×#19149589	2
C₄⋊C₄×Dic₁₇₃	2
#26011253	2
C₂×#107×C₁₇₃	2
#1776	173

subgroup

index

C₄⋊C₄×Dic₁₇₃

C₂×#107×C₁₇₃

173

Subgroups — #36353462

Conjugacy classes of subgroups

Complements

Maximal subgroups