Group #22612020 — subgroups

Subgroups — #22612020

Order 9344.

154 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 55 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×#14556537	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#11924003	C₂	1	1	yes	maximal, characteristic
3	#17988828	C₂	1	1	yes	maximal, characteristic
4	#17988412	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988391	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988390	C₂	1	1	yes	maximal, characteristic, complemented
7	#467×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₉₂	C₂²	1	1	yes	characteristic
9	#14556570	—	2	1	no	—
10	#14556537	—	2	2	no	—
11	C₂×#11924003	—	2	2	no	—
12	C₂×#11924003	C₂²	1	1	yes	characteristic
13	C₂×#11924003	C₂²	1	1	yes	characteristic
14	#14556525	C₂²	1	1	yes	characteristic
15	#14556516	—	2	1	no	—
16	C₂×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₂²⋊Q₈×C₇₃	—	2	1	no	—
19	#107×C₇₃	C₂²	1	1	yes	characteristic
20	#103×C₇₃	—	2	1	no	—
21	C₂×Dic₁₄₆	—	2	1	no	—
22	Dic₂₉₂	—	2	1	no	—
23	Dic₂₉₂	—	2	1	no	—
24	C₄⋊Dic₇₃	—	2	1	no	—
25	#11924007	—	4	1	no	—
26	C₂²⋊Dic₇₃	—	4	1	no	—
27	#11924003	—	2	1	no	—
28	#11924003	D₄	1	2	yes	—
29	C₂×Q₈×C₇₃	—	2	1	no	—
30	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
31	Q₁₆×C₇₃	—	2	1	no	—
32	Q₁₆×C₇₃	—	2	1	no	—
33	C₂×C₈×C₇₃	—	2	1	no	—
34	C₂×C₈×C₇₃	—	2	1	no	—
35	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
36	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₇₃	—	2	1	no	—
38	C₄⋊C₄×C₇₃	—	4	1	no	—
39	C₂²⋊C₄×C₇₃	—	4	1	no	—
40	C₂³×C₇₃	D₈	1	1	yes	characteristic
41	Q₈×C₇₃	—	4	1	no	—
42	Q₈×C₇₃	—	4	1	no	—
43	C₂×C₄×C₇₃	—	2	1	no	—
44	C₂×C₄×C₇₃	—	2	1	no	—
45	C₂×C₄×C₇₃	—	4	1	no	—
46	C₂×C₄×C₇₃	—	4	1	no	—
47	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₇₃	—	4	1	no	—
50	C₂×Dic₇₃	—	4	1	no	—
51	Dic₁₄₆	—	4	1	no	—
52	Dic₁₄₆	—	4	1	no	—
53	C₈×C₇₃	—	2	1	no	—
54	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₈×C₇₃	#41	1	1	yes	characteristic
56	C₂²×C₇₃	—	2	1	no	—
57	C₂²×C₇₃	—	2	1	no	—
58	C₂²×C₇₃	Q₃₂	1	2	yes	complemented
59	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
60	C₄×C₇₃	—	2	1	no	—
61	C₄×C₇₃	—	8	1	no	—
62	C₄×C₇₃	—	8	1	no	—
63	C₄×C₇₃	#121	1	1	yes	characteristic
64	C₄×C₇₃	#135	1	1	yes	characteristic
65	Dic₇₃	—	8	1	no	—
66	Dic₇₃	—	8	1	no	—
67	C₂×C₇₃	—	2	2	no	—
68	C₂×C₇₃	#466	1	1	yes	characteristic
69	C₂×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
70	C₂×C₇₃	#508	1	1	yes	characteristic
71	#2216	—	73	1	no	maximal, Sylow
72	C₇₃	#2216	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₃₂	—	73	1	no	—
74	C₂²×C₁₆	—	73	1	no	—
75	#467	—	73	1	no	—
76	#467	D₇₃	1	1	yes	characteristic
77	#366	—	73	1	no	—
78	#358	—	73	1	no	—
79	#358	—	73	1	no	—
80	C₂×Q₁₆	—	73	1	no	—
81	C₂×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
83	C₂²⋊Q₈	—	2	1	no	—
84	C₂²⋊Q₈	—	146	1	no	—
85	Q₃₂	—	146	2	no	—
86	C₂×C₁₆	—	73	1	no	—
87	C₂×C₁₆	—	73	1	no	—
88	C₂×C₁₆	—	146	2	no	—
89	#107	—	73	1	no	—
90	#107	C₂×D₇₃	1	1	yes	characteristic
91	#103	—	2	1	no	—
92	#103	—	146	1	no	—
93	C₂×Q₈	—	2	1	no	—
94	C₂×Q₈	—	146	1	no	—
95	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
96	Q₁₆	—	2	1	no	—
97	Q₁₆	—	2	1	no	—
98	Q₁₆	—	146	1	no	—
99	Q₁₆	—	146	1	no	—
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
103	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	4	1	no	—
106	C₄⋊C₄	—	292	1	no	—
107	C₄⋊C₄	—	146	1	no	—
108	C₂²⋊C₄	—	4	1	no	—
109	C₂²⋊C₄	—	292	1	no	—
110	C₁₆	—	73	2	no	—
111	C₁₆	—	146	1	no	—
112	C₂³	#11924013	1	1	yes	characteristic
113	Q₈	—	4	1	no	—
114	Q₈	—	4	1	no	—
115	Q₈	—	292	1	no	—
116	Q₈	—	292	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	292	1	no	—
122	C₂×C₄	—	292	1	no	—
123	C₂×C₄	#11924013	1	1	yes	characteristic
124	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	D₄×D₇₃	1	1	yes	characteristic
127	C₈	#11924027	1	1	yes	characteristic
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	#14556537	1	2	yes	complemented
131	C₂²	C₂×#11924013	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	8	1	no	—
134	C₄	—	8	1	no	—
135	C₄	—	584	1	no	—
136	C₄	—	584	1	no	—
137	C₄	#14556563	1	1	yes	characteristic
138	C₄	#14556611	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	#17988821	1	1	yes	characteristic
141	C₂	C₂×#14556537	1	1	yes	characteristic
142	C₂	#17989017	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556537

C₂

yes

maximal, characteristic, complemented

C₂²×#11924003

C₂

yes

maximal, characteristic

#17988828

C₂

yes

maximal, characteristic

#17988412

C₂

yes

maximal, characteristic, complemented

#17988391

C₂

yes

maximal, characteristic, complemented

#17988390

C₂

yes

maximal, characteristic, complemented

#467×C₇₃

C₂

yes

maximal, characteristic

C₂×Dic₂₉₂

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₂×#11924003

C₂²

yes

characteristic

#14556525

C₂²

yes

characteristic

#14556516

—

C₂×Q₁₆×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂²⋊Q₈×C₇₃

—

#107×C₇₃

C₂²

yes

characteristic

#103×C₇₃

—

C₂×Dic₁₄₆

—

Dic₂₉₂

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²⋊Dic₇₃

—

#11924003

—

#11924003

D₄

yes

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂³×C₇₃

D₈

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂×D₄

yes

characteristic

C₈×C₇₃

#41

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

Q₃₂

yes

complemented

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#466

yes

characteristic

C₂×C₇₃

C₂×Q₃₂

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂²×C₁₆

—

#467

—

#467

D₇₃

yes

—

—

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₇₃

yes

characteristic

C₂²×C₈

C₂×D₇₃

yes

characteristic

C₂²⋊Q₈

—

C₂²⋊Q₈

—

146

—

Q₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

#107

—

#107

C₂×D₇₃

yes

characteristic

#103

—

#103

—

146

—

C₂×Q₈

—

C₂×Q₈

—

146

—

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

146

—

Q₁₆

—

146

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

103

C₂×C₈

C₂²×D₇₃

yes

characteristic

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

292

—

107

C₄⋊C₄

—

146

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

292

—

110

C₁₆

—

111

C₁₆

—

146

—

112

C₂³

#11924013

yes

characteristic

113

Q₈

—

114

Q₈

—

115

Q₈

—

292

—

116

Q₈

—

292

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

292

—

122

C₂×C₄

—

292

—

123

C₂×C₄

#11924013

yes

characteristic

124

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

125

C₈

—

126

C₈

D₄×D₇₃

yes

characteristic

127

C₈

#11924027

yes

characteristic

128

C₂²

—

129

C₂²

—

130

C₂²

#14556537

yes

complemented

131

C₂²

C₂×#11924013

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

584

—

136

C₄

—

584

—

137

C₄

#14556563

yes

characteristic

138

C₄

#14556611

yes

characteristic

139

C₂

—

140

C₂

#17988821

yes

characteristic

141

C₂

C₂×#14556537

yes

characteristic

142

C₂

#17989017

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11924013
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	#467×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	#467	D₇₃
hypocenter	C₇₃	#2216
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #467

subgroup

G/H

center

C₂²

C₂×#11924013

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

#467×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #467

N	H ≅ G/N	type
1. C₂×#14556537	C₂	semidirect
4. #17988412	C₂	semidirect
5. #17988391	C₂	semidirect
6. #17988390	C₂	semidirect
58. C₂²×C₇₃	Q₃₂	semidirect
72. C₇₃	#2216	semidirect
130. C₂²	#14556537	semidirect

H ≅ G/N

type

1. C₂×#14556537

C₂

semidirect

4. #17988412

C₂

semidirect

5. #17988391

C₂

semidirect

6. #17988390

C₂

semidirect

58. C₂²×C₇₃

Q₃₂

semidirect

72. C₇₃

#2216

semidirect

130. C₂²

#14556537

semidirect

Subgroups — #22612020

Conjugacy classes of subgroups

Complements