Group #22609294 — subgroups

Subgroups — #22609294

Order 9344.

172 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 57 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	#17988966	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988867	C₂	1	1	yes	maximal, characteristic
3	D₄×#10596831	C₂	1	1	yes	maximal, characteristic
4	#17988339	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988333	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988332	C₂	1	1	yes	maximal, characteristic, complemented
7	#474×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924018	—	2	1	no	—
9	C₂×#11924018	—	2	1	no	—
10	C₂²×#10596831	—	2	1	no	—
11	#14556591	C₂²	1	1	yes	characteristic
12	#14556524	—	2	1	no	—
13	C₄⋊#10596831	C₂²	1	1	yes	characteristic
14	C₄⋊#10596831	C₂²	1	1	yes	characteristic
15	#14556512	—	2	1	no	—
16	C₂²⋊#10596831	—	2	1	no	—
17	C₄×#10596831	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₇₃	C₂²	1	1	yes	characteristic
20	#107×C₇₃	—	2	1	no	—
21	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
22	#102×C₇₃	—	2	1	no	—
23	C₂×Dic₁₄₆	—	2	1	no	—
24	#11924018	—	4	2	no	—
25	#11924018	—	4	2	no	—
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	—	2	1	no	—
28	C₂×#10596831	—	4	2	no	—
29	C₂×#10596831	D₄	1	1	yes	characteristic
30	C₂×#10596831	D₄	1	1	yes	characteristic
31	C₄⋊Dic₇₃	—	2	1	no	—
32	C₄⋊Dic₇₃	—	2	1	no	—
33	C₂×Q₈×C₇₃	—	2	1	no	—
34	C₂×D₄×C₇₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	—	2	1	no	—
36	C₂×C₈×C₇₃	—	2	1	no	—
37	C₄⋊C₄×C₇₃	—	2	1	no	—
38	C₄⋊C₄×C₇₃	—	2	1	no	—
39	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
40	C₂²⋊C₄×C₇₃	—	2	1	no	—
41	C₄²×C₇₃	C₂³	1	1	yes	characteristic
42	C₂³×C₇₃	—	2	1	no	—
43	Q₈×C₇₃	—	4	1	no	—
44	Q₈×C₇₃	—	4	1	no	—
45	D₄×C₇₃	—	2	1	no	—
46	D₄×C₇₃	Q₁₆	1	2	yes	complemented
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	4	1	no	—
49	C₂×C₄×C₇₃	—	4	1	no	—
50	C₂×C₄×C₇₃	—	4	1	no	—
51	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
54	C₂×Dic₇₃	—	4	1	no	—
55	Dic₁₄₆	—	4	1	no	—
56	Dic₁₄₆	—	4	1	no	—
57	C₈×C₇₃	—	4	1	no	—
58	#10596831	—	2	1	no	—
59	#10596831	—	4	1	no	—
60	#10596831	—	4	1	no	—
61	#10596831	D₈	1	2	yes	—
62	C₂²×C₇₃	—	2	2	no	—
63	C₂²×C₇₃	—	4	1	no	—
64	C₂²×C₇₃	—	4	1	no	—
65	C₂²×C₇₃	#121	1	1	yes	characteristic
66	C₄×C₇₃	—	2	1	no	—
67	C₄×C₇₃	—	4	1	no	—
68	C₄×C₇₃	—	8	1	no	—
69	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
70	C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
71	C₄×C₇₃	#135	1	1	yes	characteristic
72	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
73	Dic₇₃	—	8	1	no	—
74	C₂×C₇₃	—	4	2	no	—
75	C₂×C₇₃	#459	1	1	yes	characteristic
76	C₂×C₇₃	#467	1	1	yes	characteristic
77	C₂×C₇₃	#473	1	1	yes	characteristic
78	#1669	—	73	1	no	maximal, Sylow
79	C₇₃	#1669	1	1	yes	characteristic, complemented, Sylow
80	#494	—	73	1	no	—
81	#474	—	73	1	no	—
82	#474	D₇₃	1	1	yes	characteristic
83	C₈×D₄	—	73	1	no	—
84	#335	—	73	1	no	—
85	#332	—	73	1	no	—
86	#332	—	73	1	no	—
87	C₂×Q₁₆	—	146	1	no	—
88	C₂×Q₁₆	—	146	1	no	—
89	C₂²×C₈	—	146	1	no	—
90	C₄⋊Q₈	—	73	1	no	—
91	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
92	C₄×D₄	C₂×D₇₃	1	1	yes	characteristic
93	#107	—	2	1	no	—
94	#107	—	146	1	no	—
95	C₄⋊C₈	—	73	1	no	—
96	C₄⋊C₈	—	73	1	no	—
97	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
98	#102	—	2	1	no	—
99	#102	—	146	1	no	—
100	C₂²⋊C₈	—	146	1	no	—
101	C₄×C₈	—	73	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	146	1	no	—
104	C₂×D₄	C₂²⋊D₇₃	1	1	yes	characteristic
105	C₂²×C₄	—	2	1	no	—
106	Q₁₆	—	292	2	no	—
107	Q₁₆	—	292	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	—	292	2	no	—
110	C₂×C₈	—	73	1	no	—
111	C₂×C₈	—	73	1	no	—
112	C₂×C₈	—	146	1	no	—
113	C₂×C₈	—	146	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	2	1	no	—
116	C₄⋊C₄	—	146	1	no	—
117	C₄⋊C₄	—	146	1	no	—
118	C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
119	C₂²⋊C₄	—	2	1	no	—
120	C₄²	C₂²×D₇₃	1	1	yes	characteristic
121	C₂³	—	2	1	no	—
122	Q₈	—	4	1	no	—
123	Q₈	—	4	1	no	—
124	Q₈	—	292	1	no	—
125	Q₈	—	292	1	no	—
126	D₄	—	2	1	no	—
127	D₄	#11924018	1	2	yes	complemented
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	292	1	no	—
133	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
134	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
135	C₂×C₄	#11924027	1	1	yes	characteristic
136	C₈	—	4	1	no	—
137	C₈	—	292	1	no	—
138	C₈	—	292	1	no	—
139	C₈	—	73	2	no	—
140	C₈	—	146	1	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	4	1	no	—
143	C₂²	—	4	1	no	—
144	C₂²	#14556563	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	4	1	no	—
147	C₄	—	8	1	no	—
148	C₄	—	584	1	no	—
149	C₄	D₈×D₇₃	1	1	yes	characteristic
150	C₄	C₂×#11924018	1	1	yes	characteristic
151	C₄	#14556611	1	1	yes	characteristic
152	C₄	#14556615	1	1	yes	characteristic
153	C₂	—	4	2	no	—
154	C₂	#17988780	1	1	yes	characteristic
155	C₂	#17988826	1	1	yes	characteristic
156	C₂	#17988860	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988966

C₂

yes

maximal, characteristic, complemented

#17988867

C₂

yes

maximal, characteristic

D₄×#10596831

C₂

yes

maximal, characteristic

#17988339

C₂

yes

maximal, characteristic, complemented

#17988333

C₂

yes

maximal, characteristic, complemented

#17988332

C₂

yes

maximal, characteristic, complemented

#474×C₇₃

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

characteristic

#14556524

—

C₄⋊#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄×D₄×C₇₃

C₂²

yes

characteristic

#107×C₇₃

—

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

#102×C₇₃

—

C₂×Dic₁₄₆

—

—

—

—

—

—

D₄

yes

characteristic

C₂×#10596831

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

Q₁₆

yes

complemented

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

—

—

—

D₈

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#121

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

C₂×Q₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#494

—

#474

—

#474

D₇₃

yes

characteristic

C₈×D₄

—

#335

—

#332

—

#332

—

C₂×Q₁₆

—

146

—

C₂×Q₁₆

—

146

—

C₂²×C₈

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

C₄×D₄

C₂×D₇₃

yes

characteristic

#107

—

#107

—

146

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₃

yes

characteristic

#102

—

#102

—

146

—

100

C₂²⋊C₈

—

146

—

101

C₄×C₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

146

—

104

C₂×D₄

C₂²⋊D₇₃

yes

characteristic

105

C₂²×C₄

—

106

Q₁₆

—

292

—

107

Q₁₆

—

292

—

108

C₂×C₈

—

109

C₂×C₈

—

292

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

146

—

113

C₂×C₈

—

146

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

146

—

117

C₄⋊C₄

—

146

—

118

C₄⋊C₄

C₂²⋊D₇₃

yes

characteristic

119

C₂²⋊C₄

—

120

C₄²

C₂²×D₇₃

yes

characteristic

121

C₂³

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

292

—

125

Q₈

—

292

—

126

D₄

—

127

D₄

#11924018

yes

complemented

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

292

—

133

C₂×C₄

D₄×D₇₃

yes

characteristic

134

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

135

C₂×C₄

#11924027

yes

characteristic

136

C₈

—

137

C₈

—

292

—

138

C₈

—

292

—

139

C₈

—

140

C₈

—

146

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#14556563

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

584

—

149

C₄

D₈×D₇₃

yes

characteristic

150

C₄

C₂×#11924018

yes

characteristic

151

C₄

#14556611

yes

characteristic

152

C₄

#14556615

yes

characteristic

153

C₂

—

154

C₂

#17988780

yes

characteristic

155

C₂

#17988826

yes

characteristic

156

C₂

#17988860

yes

characteristic

	subgroup	G/H
center	C₂²	#14556563
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#474×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#474	D₇₃
hypocenter	C₇₃	#1669
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #474

subgroup

G/H

center

C₂²

#14556563

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#474×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #474

N	H ≅ G/N	type
1. #17988966	C₂	semidirect
4. #17988339	C₂	semidirect
5. #17988333	C₂	semidirect
6. #17988332	C₂	semidirect
46. D₄×C₇₃	Q₁₆	semidirect
79. C₇₃	#1669	semidirect
127. D₄	#11924018	semidirect

H ≅ G/N

type

1. #17988966

C₂

semidirect

4. #17988339

C₂

semidirect

5. #17988333

C₂

semidirect

6. #17988332

C₂

semidirect

46. D₄×C₇₃

Q₁₆

semidirect

79. C₇₃

#1669

semidirect

127. D₄

#11924018

semidirect

Subgroups — #22609294

Conjugacy classes of subgroups

Complements