Group #36366205 — subgroups

Subgroups — #36366205

Order 22144.

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

normal subgroups only

This group has 370 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	#26012220	C₂	1	1	yes	maximal, characteristic, complemented
2	#26011843	C₂	1	3	yes	maximal, complemented
3	#26011773	C₂	1	3	yes	maximal, complemented
4	#26011696	C₂	1	3	yes	maximal
5	Q₈×#13063436	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14583814	C₂	1	3	yes	maximal
7	D₄×Q₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#14583814	C₂²	1	3	yes	—
9	#19149679	C₂²	1	3	yes	—
10	#19149670	—	2	3	no	—
11	Q₈×Dic₁₇₃	—	2	1	no	—
12	C₄×Dic₃₄₆	C₂²	1	3	yes	—
13	#19149610	C₂²	1	3	yes	—
14	C₄⋊#13063436	C₂²	1	3	yes	—
15	#19149602	C₂²	1	6	yes	—
16	#19149602	C₂²	1	3	yes	—
17	#19149600	C₂²	1	1	yes	characteristic
18	C₄×#13063436	C₂²	1	3	yes	—
19	C₂²×Q₈×C₁₇₃	—	2	1	no	—
20	C₄⋊Q₈×C₁₇₃	C₂²	1	3	yes	—
21	C₂²⋊Q₈×C₁₇₃	—	2	3	no	—
22	C₄×Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₁₇₃	C₂²	1	3	yes	—
24	C₂×Dic₃₄₆	C₂³	1	3	yes	—
25	#14583814	—	2	6	no	—
26	C₂×#13063436	C₂³	1	1	yes	characteristic
27	C₂×#13063436	C₂³	1	3	yes	—
28	C₄⋊Dic₁₇₃	—	2	3	no	—
29	C₄⋊Dic₁₇₃	—	2	3	no	—
30	C₄⋊Dic₁₇₃	C₂³	1	1	yes	characteristic
31	#14583805	—	2	3	no	—
32	C₄×Dic₁₇₃	—	2	3	no	—
33	C₂×Q₈×C₁₇₃	—	2	3	no	—
34	C₂×Q₈×C₁₇₃	—	4	4	no	—
35	C₂×Q₈×C₁₇₃	D₄	1	1	yes	characteristic
36	C₂×D₄×C₁₇₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₁₇₃	—	2	3	no	—
38	C₄⋊C₄×C₁₇₃	—	2	3	no	—
39	C₄⋊C₄×C₁₇₃	D₄	1	3	yes	—
40	C₄⋊C₄×C₁₇₃	C₂³	1	3	yes	—
41	C₂²⋊C₄×C₁₇₃	—	2	3	no	—
42	C₄²×C₁₇₃	C₂³	1	3	yes	—
43	Dic₃₄₆	—	2	3	no	—
44	Dic₃₄₆	—	2	3	no	—
45	C₂×Dic₁₇₃	—	2	3	no	—
46	C₂×Dic₁₇₃	—	2	1	no	—
47	#13063436	—	2	3	no	—
48	#13063436	—	2	1	no	—
49	C₂³×C₁₇₃	—	2	1	no	—
50	Q₈×C₁₇₃	—	4	6	no	—
51	Q₈×C₁₇₃	SD₁₆	1	4	yes	complemented
52	D₄×C₁₇₃	—	2	1	no	—
53	D₄×C₁₇₃	#41	1	2	yes	—
54	C₂×C₄×C₁₇₃	—	2	3	no	—
55	C₂×C₄×C₁₇₃	—	4	6	no	—
56	C₂×C₄×C₁₇₃	C₂×D₄	1	3	yes	—
57	C₂×C₄×C₁₇₃	C₂×D₄	1	3	yes	—
58	C₂×C₄×C₁₇₃	C₂⁴	1	1	yes	characteristic
59	C₂²×C₁₇₃	—	2	2	no	—
60	C₂²×C₁₇₃	—	4	1	no	—
61	C₂²×C₁₇₃	—	4	1	no	—
62	C₂²×C₁₇₃	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₁₇₃	—	2	3	no	—
64	C₄×C₁₇₃	—	4	3	no	—
65	C₄×C₁₇₃	C₂×SD₁₆	1	6	yes	—
66	C₄×C₁₇₃	C₂×#41	1	1	yes	characteristic
67	C₄×C₁₇₃	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₁₇₃	—	4	3	no	—
69	Dic₁₇₃	—	4	1	no	—
70	C₂×C₁₇₃	—	4	2	no	—
71	C₂×C₁₇₃	#548	1	1	yes	characteristic
72	C₂×C₁₇₃	C₂²×SD₁₆	1	1	yes	characteristic
73	C₂×C₁₇₃	#578	1	1	yes	characteristic
74	C₁₇₃	#3359	1	1	yes	characteristic, complemented, Sylow
75	#3359	—	173	1	no	maximal, Sylow
76	#557	—	173	1	no	—
77	D₄×Q₈	D₁₇₃	1	1	yes	characteristic
78	#487	—	173	3	no	—
79	#475	—	173	3	no	—
80	#461	—	173	3	no	—
81	C₈×Q₈	—	173	1	no	—
82	C₄×SD₁₆	—	173	3	no	—
83	C₂²×Q₈	—	2	1	no	—
84	C₂×SD₁₆	—	173	3	no	—
85	C₄⋊Q₈	—	173	3	no	—
86	C₄⋊Q₈	C₂×D₁₇₃	1	3	yes	—
87	#125	—	346	3	no	—
88	C₂²⋊Q₈	—	2	3	no	—
89	C₄×Q₈	—	346	1	no	—
90	C₄×Q₈	—	173	3	no	—
91	C₄×Q₈	C₂×D₁₇₃	1	1	yes	characteristic
92	C₄×D₄	C₂×D₁₇₃	1	3	yes	—
93	#106	—	173	3	no	—
94	C₄⋊C₈	—	173	3	no	—
95	#103	—	173	3	no	—
96	#103	—	173	6	no	—
97	#102	—	173	1	no	—
98	C₄×C₈	—	173	3	no	—
99	C₂×Q₈	—	2	3	no	—
100	C₂×Q₈	—	4	4	no	—
101	C₂×Q₈	—	173	3	no	—
102	C₂×Q₈	C₂²⋊D₁₇₃	1	1	yes	characteristic
103	C₂×D₄	C₂²×D₁₇₃	1	1	yes	characteristic
104	C₂²×C₄	—	2	3	no	—
105	SD₁₆	—	346	6	no	—
106	C₂×C₈	—	173	1	no	—
107	C₂×C₈	—	173	3	no	—
108	C₄⋊C₄	—	2	3	no	—
109	C₄⋊C₄	—	346	3	no	—
110	C₄⋊C₄	—	346	3	no	—
111	C₄⋊C₄	—	346	3	no	—
112	C₄⋊C₄	—	173	1	no	—
113	C₄⋊C₄	C₂²⋊D₁₇₃	1	3	yes	—
114	C₄⋊C₄	C₂²×D₁₇₃	1	3	yes	—
115	C₂²⋊C₄	—	2	3	no	—
116	C₄²	—	346	3	no	—
117	C₄²	C₂²×D₁₇₃	1	3	yes	—
118	C₂³	—	2	1	no	—
119	Q₈	—	4	6	no	—
120	Q₈	—	346	3	no	—
121	Q₈	—	346	3	no	—
122	Q₈	#14583814	1	4	yes	complemented
123	D₄	—	2	1	no	—
124	D₄	#14583827	1	2	yes	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	4	6	no	—
127	C₂×C₄	—	346	1	no	—
128	C₂×C₄	—	346	3	no	—
129	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	3	yes	—
130	C₂×C₄	C₂×C₂²⋊D₁₇₃	1	3	yes	—
131	C₂×C₄	C₂³×D₁₇₃	1	1	yes	characteristic
132	C₈	—	346	3	no	—
133	C₈	—	346	1	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	1	no	—
137	C₂²	C₂²×C₂²⋊D₁₇₃	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	4	3	no	—
140	C₄	—	692	1	no	—
141	C₄	—	692	3	no	—
142	C₄	C₂×#14583814	1	6	yes	—
143	C₄	C₂×#14583827	1	1	yes	characteristic
144	C₄	#19149727	1	1	yes	characteristic
145	C₂	—	4	2	no	—
146	C₂	#26012152	1	1	yes	characteristic
147	C₂	C₂²×#14583814	1	1	yes	characteristic
148	C₂	#26012306	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26012220

C₂

yes

maximal, characteristic, complemented

#26011843

C₂

yes

maximal, complemented

#26011773

C₂

yes

maximal, complemented

#26011696

C₂

yes

maximal

Q₈×#13063436

C₂

yes

maximal, characteristic, complemented

C₄×#14583814

C₂

yes

maximal

D₄×Q₈×C₁₇₃

C₂

yes

maximal, characteristic

C₂×#14583814

C₂²

yes

—

#19149679

C₂²

yes

—

#19149670

—

Q₈×Dic₁₇₃

—

C₄×Dic₃₄₆

C₂²

yes

—

#19149610

C₂²

yes

—

C₄⋊#13063436

C₂²

yes

—

#19149602

C₂²

yes

—

#19149602

C₂²

yes

—

#19149600

C₂²

yes

characteristic

C₄×#13063436

C₂²

yes

—

C₂²×Q₈×C₁₇₃

—

C₄⋊Q₈×C₁₇₃

C₂²

yes

—

C₂²⋊Q₈×C₁₇₃

—

C₄×Q₈×C₁₇₃

C₂²

yes

characteristic

C₄×D₄×C₁₇₃

C₂²

yes

—

C₂×Dic₃₄₆

C₂³

yes

—

#14583814

—

C₂×#13063436

C₂³

yes

characteristic

C₂×#13063436

C₂³

yes

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

—

C₄⋊Dic₁₇₃

C₂³

yes

characteristic

#14583805

—

C₄×Dic₁₇₃

—

C₂×Q₈×C₁₇₃

—

C₂×Q₈×C₁₇₃

—

C₂×Q₈×C₁₇₃

D₄

yes

characteristic

C₂×D₄×C₁₇₃

C₂³

yes

characteristic

C₂²×C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

—

C₄⋊C₄×C₁₇₃

D₄

yes

—

C₄⋊C₄×C₁₇₃

C₂³

yes

—

C₂²⋊C₄×C₁₇₃

—

C₄²×C₁₇₃

C₂³

yes

—

Dic₃₄₆

—

Dic₃₄₆

—

C₂×Dic₁₇₃

—

C₂×Dic₁₇₃

—

#13063436

—

#13063436

—

C₂³×C₁₇₃

—

Q₈×C₁₇₃

—

Q₈×C₁₇₃

SD₁₆

yes

complemented

D₄×C₁₇₃

—

D₄×C₁₇₃

#41

yes

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂×D₄

yes

—

C₂×C₄×C₁₇₃

C₂⁴

yes

characteristic

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

—

C₂²×C₁₇₃

C₂²×D₄

yes

characteristic

C₄×C₁₇₃

—

C₄×C₁₇₃

—

C₄×C₁₇₃

C₂×SD₁₆

yes

—

C₄×C₁₇₃

C₂×#41

yes

characteristic

C₄×C₁₇₃

2₋¹⁺⁴

yes

characteristic

Dic₁₇₃

—

Dic₁₇₃

—

C₂×C₁₇₃

—

C₂×C₁₇₃

#548

yes

characteristic

C₂×C₁₇₃

C₂²×SD₁₆

yes

C₂×C₁₇₃

yes

C₁₇₃

yes

characteristic, complemented, Sylow

#3359

—

173

maximal, Sylow

#557

—

173

—

D₄×Q₈

D₁₇₃

yes

characteristic

#487

—

173

—

#475

—

173

—

#461

—

173

—

C₈×Q₈

—

173

—

C₄×SD₁₆

—

173

—

C₂²×Q₈

—

C₂×SD₁₆

—

173

—

C₄⋊Q₈

—

173

—

C₄⋊Q₈

C₂×D₁₇₃

yes

—

#125

—

346

—

C₂²⋊Q₈

—

C₄×Q₈

—

346

—

C₄×Q₈

—

173

—

C₄×Q₈

C₂×D₁₇₃

yes

characteristic

C₄×D₄

C₂×D₁₇₃

yes

—

#106

—

173

—

C₄⋊C₈

—

173

—

#103

—

173

—

#103

—

173

—

#102

—

173

—

C₄×C₈

—

173

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

173

—

102

C₂×Q₈

C₂²⋊D₁₇₃

yes

characteristic

103

C₂×D₄

C₂²×D₁₇₃

yes

characteristic

104

C₂²×C₄

—

105

SD₁₆

—

346

—

106

C₂×C₈

—

173

—

107

C₂×C₈

—

173

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

346

—

110

C₄⋊C₄

—

346

—

111

C₄⋊C₄

—

346

—

112

C₄⋊C₄

—

173

—

113

C₄⋊C₄

C₂²⋊D₁₇₃

yes

—

114

C₄⋊C₄

C₂²×D₁₇₃

yes

—

115

C₂²⋊C₄

—

116

C₄²

—

346

—

117

C₄²

C₂²×D₁₇₃

yes

—

118

C₂³

—

119

Q₈

—

120

Q₈

—

346

—

121

Q₈

—

346

—

122

Q₈

#14583814

yes

complemented

123

D₄

—

124

D₄

#14583827

yes

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

346

—

128

C₂×C₄

—

346

—

129

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

—

130

C₂×C₄

C₂×C₂²⋊D₁₇₃

yes

—

131

C₂×C₄

C₂³×D₁₇₃

yes

characteristic

132

C₈

—

346

—

133

C₈

—

346

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂²×C₂²⋊D₁₇₃

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

692

—

141

C₄

—

692

—

142

C₄

C₂×#14583814

yes

—

143

C₄

C₂×#14583827

yes

characteristic

144

C₄

#19149727

yes

characteristic

145

C₂

—

146

C₂

#26012152

yes

characteristic

147

C₂

C₂²×#14583814

yes

characteristic

148

C₂

#26012306

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₇₃
commutator, cosocle	C₂×C₄×C₁₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇₃
Fitting	D₄×Q₈×C₁₇₃	C₂
socle	C₂²×C₁₇₃	C₂²×D₄
prime radical	C₂×D₄×C₁₇₃	C₂³
hypercenter	D₄×Q₈	D₁₇₃
hypocenter	C₁₇₃	#3359
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₇₃

commutator, cosocle

C₂×C₄×C₁₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₇₃

Fitting

D₄×Q₈×C₁₇₃

C₂

socle

C₂²×C₁₇₃

C₂²×D₄

prime radical

C₂×D₄×C₁₇₃

C₂³

hypercenter

D₄×Q₈

D₁₇₃

hypocenter

C₁₇₃

#3359

LCS

G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

N	H ≅ G/N	type
1. #26012220	C₂	semidirect
2. #26011843	C₂	semidirect
3. #26011773	C₂	semidirect
5. Q₈×#13063436	C₂	semidirect
51. Q₈×C₁₇₃	SD₁₆	semidirect
74. C₁₇₃	#3359	semidirect
122. Q₈	#14583814	semidirect

H ≅ G/N

type

1. #26012220

C₂

semidirect

2. #26011843

C₂

semidirect

3. #26011773

C₂

semidirect

5. Q₈×#13063436

C₂

semidirect

51. Q₈×C₁₇₃

SD₁₆

semidirect

74. C₁₇₃

#3359

semidirect

122. Q₈

#14583814

semidirect

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

subgroup	index
#26012220	2
#26011843	2
#26011773	2
#26011696	2
Q₈×#13063436	2
C₄×#14583814	2
D₄×Q₈×C₁₇₃	2
#3359	173

subgroup

index

D₄×Q₈×C₁₇₃

173

Subgroups — #36366205

Conjugacy classes of subgroups

Complements

Maximal subgroups