Group #22612161 — subgroups

Subgroups — #22612161

Order 9344.

138 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 47 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	#17988988	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988970	C₂	1	1	yes	maximal, characteristic
3	#17988392	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988390	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988388	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₉₂	C₂²	1	1	yes	characteristic
9	C₂×#11924016	—	2	1	no	—
10	#14556582	—	2	1	no	—
11	#14556578	—	2	1	no	—
12	C₂×#11924003	C₂²	1	1	yes	characteristic
13	#14556525	C₂²	1	1	yes	characteristic
14	#14556522	—	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	—	2	1	no	—
20	Dic₂₉₂	—	2	1	no	—
21	Dic₂₉₂	—	2	1	no	—
22	#11924016	—	4	2	no	—
23	C₂×#10596831	D₄	1	1	yes	characteristic
24	C₄⋊Dic₇₃	—	2	1	no	—
25	C₄⋊Dic₇₃	—	4	1	no	—
26	#11924007	—	4	1	no	—
27	C₂²⋊Dic₇₃	—	4	1	no	—
28	C₂²⋊Dic₇₃	—	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	D₄×C₇₃	—	4	1	no	—
39	D₄×C₇₃	—	4	1	no	—
40	C₂×C₄×C₇₃	—	4	1	no	—
41	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₇₃	—	4	1	no	—
43	C₂×Dic₇₃	—	4	1	no	—
44	C₂×Dic₇₃	D₈	1	1	yes	characteristic
45	C₂×Dic₇₃	D₈	1	1	yes	characteristic
46	Dic₁₄₆	—	4	1	no	—
47	Dic₁₄₆	—	4	1	no	—
48	C₈×C₇₃	#41	1	1	yes	characteristic
49	C₈×C₇₃	#41	1	1	yes	characteristic
50	#10596831	SD₁₆	1	2	yes	—
51	C₂²×C₇₃	—	8	1	no	—
52	C₂²×C₇₃	—	8	1	no	—
53	C₂²×C₇₃	—	8	1	no	—
54	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
55	C₄×C₇₃	—	8	1	no	—
56	C₄×C₇₃	#124	1	1	yes	characteristic
57	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
58	Dic₇₃	—	2	1	no	—
59	Dic₇₃	—	2	1	no	—
60	Dic₇₃	—	8	1	no	—
61	Dic₇₃	—	8	1	no	—
62	C₂×C₇₃	—	16	1	no	—
63	C₂×C₇₃	#486	1	1	yes	characteristic
64	C₂×C₇₃	#508	1	1	yes	characteristic
65	C₂×C₇₃	#508	1	1	yes	characteristic
66	#2242	—	73	1	no	maximal, Sylow
67	C₇₃	#2242	1	1	yes	characteristic, complemented, Sylow
68	#499	—	73	1	no	—
69	#495	—	73	1	no	—
70	#358	—	73	1	no	—
71	#358	—	73	1	no	—
72	#357	—	73	1	no	—
73	#357	D₇₃	1	1	yes	characteristic
74	C₄×C₁₆	—	73	1	no	—
75	C₂×Q₁₆	—	73	1	no	—
76	C₂×SD₁₆	—	146	1	no	—
77	C₂×D₈	C₂×D₇₃	1	1	yes	characteristic
78	#125	—	146	1	no	—
79	#124	—	146	1	no	—
80	C₂×C₁₆	—	73	1	no	—
81	C₂×C₁₆	C₂×D₇₃	1	1	yes	characteristic
82	#107	—	73	1	no	—
83	#107	C₂×D₇₃	1	1	yes	characteristic
84	#106	—	146	1	no	—
85	C₄×C₈	—	73	1	no	—
86	C₂×Q₈	—	146	1	no	—
87	C₂×D₄	—	2	1	no	—
88	Q₁₆	—	146	1	no	—
89	Q₁₆	—	146	1	no	—
90	SD₁₆	—	292	2	no	—
91	D₈	—	2	1	no	—
92	D₈	—	2	1	no	—
93	C₂×C₈	—	73	1	no	—
94	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	292	1	no	—
97	C₄⋊C₄	—	292	1	no	—
98	C₄⋊C₄	—	146	1	no	—
99	C₂²⋊C₄	—	292	1	no	—
100	C₂²⋊C₄	—	292	1	no	—
101	C₄²	—	73	1	no	—
102	C₁₆	—	2	1	no	—
103	C₁₆	—	146	1	no	—
104	C₂³	—	4	1	no	—
105	Q₈	—	292	1	no	—
106	Q₈	—	292	1	no	—
107	D₄	—	4	1	no	—
108	D₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	292	1	no	—
111	C₂×C₄	—	292	1	no	—
112	C₂×C₄	—	73	1	no	—
113	C₂×C₄	—	73	1	no	—
114	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
115	C₈	—	73	2	no	—
116	C₈	#11924027	1	1	yes	characteristic
117	C₈	#11924028	1	1	yes	characteristic
118	C₂²	—	8	1	no	—
119	C₂²	—	8	1	no	—
120	C₂²	—	8	1	no	—
121	C₂²	D₈×D₇₃	1	1	yes	characteristic
122	C₄	—	8	1	no	—
123	C₄	—	584	1	no	—
124	C₄	—	584	1	no	—
125	C₄	—	146	1	no	—
126	C₄	—	146	1	no	—
127	C₄	#14556577	1	1	yes	characteristic
128	C₄	SD₁₆×D₇₃	1	1	yes	characteristic
129	C₂	—	16	1	no	—
130	C₂	#17988932	1	1	yes	characteristic
131	C₂	#17989016	1	1	yes	characteristic
132	C₂	#17989018	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988988

C₂

yes

maximal, characteristic, complemented

#17988970

C₂

yes

maximal, characteristic

#17988392

C₂

yes

maximal, characteristic, complemented

#17988390

C₂

yes

maximal, characteristic, complemented

#17988388

C₂

yes

maximal, characteristic

C₁₆×Dic₇₃

C₂

yes

maximal, characteristic, complemented

#357×C₇₃

C₂

yes

maximal, characteristic

C₂×Dic₂₉₂

C₂²

yes

—

—

—

C₂²

yes

characteristic

#14556525

C₂²

yes

characteristic

#14556522

—

C₈×Dic₇₃

C₂²

yes

characteristic

C₂×D₈×C₇₃

C₂²

yes

characteristic

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

#107×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

Dic₂₉₂

—

Dic₂₉₂

—

#11924016

—

C₂×#10596831

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

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

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

D₈

yes

characteristic

C₂×Dic₇₃

D₈

yes

characteristic

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

yes

C₈×C₇₃

yes

SD₁₆

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

#124

yes

characteristic

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₃

yes

characteristic

C₄×C₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

146

—

C₂×D₈

C₂×D₇₃

yes

characteristic

#125

—

146

—

#124

—

146

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₃

yes

characteristic

#107

—

#107

C₂×D₇₃

yes

characteristic

#106

—

146

—

C₄×C₈

—

C₂×Q₈

—

146

—

C₂×D₄

—

Q₁₆

—

146

—

Q₁₆

—

146

—

SD₁₆

—

292

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

292

—

100

C₂²⋊C₄

—

292

—

101

C₄²

—

102

C₁₆

—

103

C₁₆

—

146

—

104

C₂³

—

105

Q₈

—

292

—

106

Q₈

—

292

—

107

D₄

—

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

292

—

111

C₂×C₄

—

292

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

D₄×D₇₃

yes

characteristic

115

C₈

—

116

C₈

#11924027

yes

characteristic

117

C₈

#11924028

yes

characteristic

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

D₈×D₇₃

yes

characteristic

122

C₄

—

123

C₄

—

584

—

124

C₄

—

584

—

125

C₄

—

146

—

126

C₄

—

146

—

127

C₄

#14556577

yes

characteristic

128

C₄

SD₁₆×D₇₃

yes

characteristic

129

C₂

—

130

C₂

#17988932

yes

characteristic

131

C₂

#17989016

yes

characteristic

132

C₂

#17989018

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	#2242
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. #17988988	C₂	semidirect
3. #17988392	C₂	semidirect
4. #17988390	C₂	semidirect
6. C₁₆×Dic₇₃	C₂	semidirect
67. C₇₃	#2242	semidirect

H ≅ G/N

type

1. #17988988

C₂

semidirect

3. #17988392

C₂

semidirect

4. #17988390

C₂

semidirect

6. C₁₆×Dic₇₃

C₂

semidirect

67. C₇₃

#2242

semidirect

Subgroups — #22612161

Conjugacy classes of subgroups

Complements