Group #22612153 — subgroups

Subgroups — #22612153

Order 9344.

160 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 51 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	#17988991	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988962	C₂	1	1	yes	maximal, characteristic
3	#17988389	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988388	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988387	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₅₈₄	C₂²	1	1	yes	characteristic
9	C₂×#11924013	—	2	1	no	—
10	#14556590	—	2	1	no	—
11	#14556587	—	2	1	no	—
12	C₂×#11924003	C₂²	1	1	yes	characteristic
13	#14556525	C₂²	1	1	yes	characteristic
14	#14556524	—	2	1	no	—
15	C₈×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂×D₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
18	#107×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₄₆	—	4	1	no	—
20	C₂×D₂₉₂	—	2	1	no	—
21	C₂×C₂²⋊D₇₃	—	4	1	no	—
22	C₂×C₂²⋊D₇₃	—	4	1	no	—
23	D₅₈₄	—	2	1	no	—
24	D₅₈₄	—	2	1	no	—
25	#11924013	—	4	2	no	—
26	C₂×#10596831	D₄	1	1	yes	characteristic
27	C₄⋊Dic₇₃	—	2	1	no	—
28	#11924007	—	4	1	no	—
29	C₄×Dic₇₃	D₄	1	1	yes	characteristic
30	#11924003	—	2	1	no	—
31	C₂×D₄×C₇₃	—	2	1	no	—
32	D₈×C₇₃	—	2	1	no	—
33	D₈×C₇₃	—	2	1	no	—
34	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₇₃	—	2	1	no	—
36	C₁₆×C₇₃	—	2	1	no	—
37	C₂³×C₇₃	—	4	1	no	—
38	C₂²×D₇₃	—	4	1	no	—
39	D₄×C₇₃	—	4	1	no	—
40	D₄×C₇₃	—	4	1	no	—
41	C₂×C₄×C₇₃	—	4	1	no	—
42	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₂²⋊D₇₃	—	8	2	no	—
44	C₂²⋊D₇₃	—	8	2	no	—
45	C₂×Dic₇₃	—	4	1	no	—
46	C₂×Dic₇₃	D₈	1	1	yes	characteristic
47	C₂×Dic₇₃	D₈	1	1	yes	characteristic
48	D₂₉₂	—	4	1	no	—
49	D₂₉₂	—	4	1	no	—
50	Dic₁₄₆	—	8	2	no	—
51	C₈×C₇₃	#41	1	1	yes	characteristic
52	C₈×C₇₃	#41	1	1	yes	characteristic
53	#10596831	SD₁₆	1	2	yes	—
54	C₂²×C₇₃	—	8	1	no	—
55	C₂²×C₇₃	—	8	1	no	—
56	C₂²×C₇₃	—	8	1	no	—
57	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
58	C₂×D₇₃	—	8	1	no	—
59	C₂×D₇₃	—	8	1	no	—
60	C₂×D₇₃	—	8	1	no	—
61	C₄×C₇₃	—	8	1	no	—
62	C₄×C₇₃	#124	1	1	yes	characteristic
63	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
64	Dic₇₃	—	8	1	no	—
65	Dic₇₃	D₁₆	1	2	yes	—
66	Dic₇₃	SD₃₂	1	2	yes	—
67	C₂×C₇₃	—	16	1	no	—
68	C₂×C₇₃	#486	1	1	yes	characteristic
69	C₂×C₇₃	C₂×D₁₆	1	1	yes	characteristic
70	C₂×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
71	D₇₃	—	16	1	no	—
72	#2240	—	73	1	no	maximal, Sylow
73	C₇₃	#2240	1	1	yes	characteristic, complemented, Sylow
74	#500	—	73	1	no	—
75	#493	—	73	1	no	—
76	#357	—	73	1	no	—
77	#357	—	73	1	no	—
78	#357	—	73	1	no	—
79	#357	D₇₃	1	1	yes	characteristic, complemented
80	C₄×C₁₆	—	73	1	no	—
81	C₂×D₈	—	73	1	no	—
82	C₂×D₈	—	146	1	no	—
83	C₂×D₈	C₂×D₇₃	1	1	yes	characteristic
84	C₄⋊Q₈	—	146	1	no	—
85	#127	—	146	1	no	—
86	C₂×C₁₆	—	73	1	no	—
87	C₂×C₁₆	C₂×D₇₃	1	1	yes	characteristic
88	#107	—	73	1	no	—
89	#107	—	146	1	no	—
90	#107	C₂×D₇₃	1	1	yes	characteristic
91	C₄×C₈	—	73	1	no	—
92	C₂×Q₈	—	292	1	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	—	292	1	no	—
95	C₂×D₄	—	292	1	no	—
96	C₂×D₄	—	146	1	no	—
97	D₈	—	2	1	no	—
98	D₈	—	2	1	no	—
99	D₈	—	292	2	no	—
100	D₈	—	146	1	no	—
101	D₈	—	146	1	no	—
102	C₂×C₈	—	73	1	no	—
103	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	292	1	no	—
106	C₄⋊C₄	—	146	1	no	—
107	C₄²	—	73	1	no	—
108	C₁₆	—	2	1	no	—
109	C₁₆	—	146	1	no	—
110	C₂³	—	4	1	no	—
111	C₂³	—	292	1	no	—
112	Q₈	—	584	2	no	—
113	D₄	—	4	1	no	—
114	D₄	—	4	1	no	—
115	D₄	—	292	1	no	—
116	D₄	—	292	1	no	—
117	D₄	—	584	2	no	—
118	D₄	—	584	2	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	292	1	no	—
121	C₂×C₄	—	73	1	no	—
122	C₂×C₄	—	73	1	no	—
123	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
124	C₈	—	73	2	no	—
125	C₈	#11924027	1	1	yes	characteristic
126	C₈	#11924028	1	1	yes	characteristic
127	C₂²	—	8	1	no	—
128	C₂²	—	8	1	no	—
129	C₂²	—	8	1	no	—
130	C₂²	—	584	1	no	—
131	C₂²	—	584	1	no	—
132	C₂²	—	584	1	no	—
133	C₂²	D₈×D₇₃	1	1	yes	characteristic
134	C₄	—	8	1	no	—
135	C₄	—	584	1	no	—
136	C₄	—	73	2	no	—
137	C₄	—	73	2	no	—
138	C₄	#14556577	1	1	yes	characteristic
139	C₄	SD₁₆×D₇₃	1	1	yes	characteristic
140	C₂	—	16	1	no	—
141	C₂	—	1168	1	no	—
142	C₂	#17988932	1	1	yes	characteristic
143	C₂	D₁₆×D₇₃	1	1	yes	characteristic
144	C₂	SD₃₂×D₇₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988991

C₂

yes

maximal, characteristic, complemented

#17988962

C₂

yes

maximal, characteristic

#17988389

C₂

yes

maximal, characteristic, complemented

#17988388

C₂

yes

maximal, characteristic, complemented

#17988387

C₂

yes

maximal, characteristic, complemented

C₁₆×Dic₇₃

C₂

yes

maximal, characteristic, complemented

#357×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×D₅₈₄

C₂²

yes

—

—

—

C₂²

yes

characteristic

#14556525

C₂²

yes

characteristic

#14556524

—

C₈×Dic₇₃

C₂²

yes

characteristic

C₂×D₈×C₇₃

C₂²

yes

characteristic

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

#107×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×C₂²⋊D₇₃

—

C₂×C₂²⋊D₇₃

—

D₅₈₄

—

D₅₈₄

—

#11924013

—

C₂×#10596831

D₄

yes

characteristic

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

D₄

yes

characteristic

#11924003

—

C₂×D₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₁₆×C₇₃

—

C₂³×C₇₃

—

C₂²×D₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²⋊D₇₃

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

D₈

yes

characteristic

C₂×Dic₇₃

D₈

yes

characteristic

D₂₉₂

—

D₂₉₂

—

Dic₁₄₆

—

C₈×C₇₃

yes

C₈×C₇₃

yes

SD₁₆

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#124

yes

characteristic

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

D₁₆

yes

—

Dic₇₃

SD₃₂

yes

—

C₂×C₇₃

—

C₂×C₇₃

#486

yes

characteristic

C₂×C₇₃

C₂×D₁₆

yes

characteristic

C₂×C₇₃

C₂×SD₃₂

yes

characteristic

D₇₃

—

#2240

—

maximal, Sylow

C₇₃

#2240

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₃

yes

characteristic, complemented

C₄×C₁₆

—

C₂×D₈

—

C₂×D₈

—

146

—

C₂×D₈

C₂×D₇₃

yes

characteristic

C₄⋊Q₈

—

146

—

#127

—

146

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₃

yes

characteristic

#107

—

#107

—

146

—

#107

C₂×D₇₃

yes

characteristic

C₄×C₈

—

C₂×Q₈

—

292

—

C₂×D₄

—

C₂×D₄

—

292

—

C₂×D₄

—

292

—

C₂×D₄

—

146

—

D₈

—

D₈

—

D₈

—

292

—

100

D₈

—

146

—

101

D₈

—

146

—

102

C₂×C₈

—

103

C₂×C₈

C₂²×D₇₃

yes

characteristic

104

C₄⋊C₄

—

105

C₄⋊C₄

—

292

—

106

C₄⋊C₄

—

146

—

107

C₄²

—

108

C₁₆

—

109

C₁₆

—

146

—

110

C₂³

—

111

C₂³

—

292

—

112

Q₈

—

584

—

113

D₄

—

114

D₄

—

115

D₄

—

292

—

116

D₄

—

292

—

117

D₄

—

584

—

118

D₄

—

584

—

119

C₂×C₄

—

120

C₂×C₄

—

292

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

D₄×D₇₃

yes

characteristic

124

C₈

—

125

C₈

#11924027

yes

characteristic

126

C₈

#11924028

yes

characteristic

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

584

—

131

C₂²

—

584

—

132

C₂²

—

584

—

133

C₂²

D₈×D₇₃

yes

characteristic

134

C₄

—

135

C₄

—

584

—

136

C₄

—

137

C₄

—

138

C₄

#14556577

yes

characteristic

139

C₄

SD₁₆×D₇₃

yes

characteristic

140

C₂

—

141

C₂

—

1168

—

142

C₂

#17988932

yes

characteristic

143

C₂

D₁₆×D₇₃

yes

characteristic

144

C₂

SD₃₂×D₇₃

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₃
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	#357×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	#357	D₇₃
hypocenter	C₇₃	#2240
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₇₃

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

#357×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. #17988991	C₂	semidirect
3. #17988389	C₂	semidirect
4. #17988388	C₂	semidirect
5. #17988387	C₂	semidirect
6. C₁₆×Dic₇₃	C₂	semidirect
7. #357×C₇₃	C₂	semidirect
73. C₇₃	#2240	semidirect
79. #357	D₇₃	semidirect

H ≅ G/N

type

1. #17988991

C₂

semidirect

3. #17988389

C₂

semidirect

4. #17988388

C₂

semidirect

5. #17988387

C₂

semidirect

6. C₁₆×Dic₇₃

C₂

semidirect

7. #357×C₇₃

C₂

semidirect

73. C₇₃

#2240

semidirect

79. #357

D₇₃

semidirect

Subgroups — #22612153

Conjugacy classes of subgroups

Complements