Group #22612066 — subgroups

Subgroups — #22612066

Order 9344.

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

normal subgroups only

This group has 142 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	D₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
3	#17988413	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988412	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988405	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988388	C₂	1	1	yes	maximal, characteristic
7	#357×C₇₃	C₂	1	1	yes	maximal, characteristic
8	#14556590	—	2	1	no	—
9	D₄×Dic₇₃	—	2	1	no	—
10	C₂×#11924003	C₂²	1	1	yes	characteristic
11	#14556525	C₂²	1	1	yes	characteristic
12	#14556525	C₂²	1	1	yes	characteristic
13	#14556524	—	2	1	no	—
14	#14556512	—	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	C₂²×Dic₇₃	—	4	1	no	—
21	C₂×#10596831	D₄	1	1	yes	characteristic
22	C₄⋊Dic₇₃	—	2	1	no	—
23	C₄⋊Dic₇₃	—	2	1	no	—
24	#11924007	—	4	1	no	—
25	C₂²⋊Dic₇₃	—	4	1	no	—
26	C₄×Dic₇₃	D₄	1	1	yes	characteristic
27	#11924003	—	2	1	no	—
28	C₂×D₄×C₇₃	—	2	1	no	—
29	D₈×C₇₃	—	2	1	no	—
30	D₈×C₇₃	Q₈	1	2	yes	complemented
31	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₇₃	—	2	1	no	—
33	C₁₆×C₇₃	—	2	1	no	—
34	C₂³×C₇₃	—	4	1	no	—
35	D₄×C₇₃	—	2	2	no	—
36	D₄×C₇₃	—	4	1	no	—
37	C₂×C₄×C₇₃	—	4	1	no	—
38	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₇₃	—	4	1	no	—
40	C₂×Dic₇₃	—	4	1	no	—
41	C₂×Dic₇₃	—	8	2	no	—
42	C₂×Dic₇₃	D₈	1	1	yes	characteristic
43	C₂×Dic₇₃	D₈	1	1	yes	characteristic
44	Dic₁₄₆	—	8	2	no	—
45	C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₇₃	#41	1	1	yes	characteristic
47	#10596831	—	2	1	no	—
48	C₂²×C₇₃	—	4	2	no	—
49	C₂²×C₇₃	—	8	1	no	—
50	C₂²×C₇₃	—	8	1	no	—
51	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
52	C₄×C₇₃	—	8	1	no	—
53	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
54	C₄×C₇₃	#137	1	1	yes	characteristic
55	Dic₇₃	—	2	1	no	—
56	Dic₇₃	—	8	1	no	—
57	Dic₇₃	—	8	1	no	—
58	Dic₇₃	D₁₆	1	2	yes	—
59	C₂×C₇₃	—	8	2	no	—
60	C₂×C₇₃	#474	1	1	yes	characteristic
61	C₂×C₇₃	C₂×D₁₆	1	1	yes	characteristic
62	C₂×C₇₃	#510	1	1	yes	characteristic
63	#2224	—	73	1	no	maximal, Sylow
64	C₇₃	#2224	1	1	yes	characteristic, complemented, Sylow
65	#500	—	73	1	no	—
66	C₄×D₈	—	73	1	no	—
67	#366	—	73	1	no	—
68	#366	—	73	1	no	—
69	C₄⋊C₁₆	—	73	1	no	—
70	#357	—	73	1	no	—
71	#357	D₇₃	1	1	yes	characteristic
72	C₂×D₈	C₂×D₇₃	1	1	yes	characteristic
73	C₄⋊Q₈	—	146	1	no	—
74	C₄×D₄	—	146	1	no	—
75	C₂×C₁₆	—	73	1	no	—
76	C₂×C₁₆	C₂×D₇₃	1	1	yes	characteristic
77	#107	—	73	1	no	—
78	#107	—	73	1	no	—
79	#107	—	146	1	no	—
80	#107	C₂×D₇₃	1	1	yes	characteristic
81	#102	—	146	1	no	—
82	C₄×C₈	—	73	1	no	—
83	C₂×Q₈	—	292	1	no	—
84	C₂×D₄	—	2	1	no	—
85	C₂²×C₄	—	292	1	no	—
86	D₈	—	2	1	no	—
87	D₈	Dic₁₄₆	1	2	yes	complemented
88	C₂×C₈	—	73	1	no	—
89	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	292	1	no	—
92	C₄⋊C₄	—	146	1	no	—
93	C₄⋊C₄	—	146	1	no	—
94	C₂²⋊C₄	—	292	1	no	—
95	C₄²	—	73	1	no	—
96	C₁₆	—	2	1	no	—
97	C₁₆	—	146	1	no	—
98	C₂³	—	4	1	no	—
99	Q₈	—	584	2	no	—
100	D₄	—	2	2	no	—
101	D₄	—	4	1	no	—
102	C₂×C₄	—	4	1	no	—
103	C₂×C₄	—	292	1	no	—
104	C₂×C₄	—	292	1	no	—
105	C₂×C₄	—	584	2	no	—
106	C₂×C₄	—	73	1	no	—
107	C₂×C₄	—	73	1	no	—
108	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
109	C₈	—	146	1	no	—
110	C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
111	C₈	#11924027	1	1	yes	characteristic
112	C₂²	—	4	2	no	—
113	C₂²	—	8	1	no	—
114	C₂²	—	8	1	no	—
115	C₂²	D₈×D₇₃	1	1	yes	characteristic
116	C₄	—	8	1	no	—
117	C₄	—	584	1	no	—
118	C₄	—	584	1	no	—
119	C₄	—	73	2	no	—
120	C₄	—	146	1	no	—
121	C₄	#14556568	1	1	yes	characteristic
122	C₄	#14556620	1	1	yes	characteristic
123	C₂	—	8	2	no	—
124	C₂	#17988865	1	1	yes	characteristic
125	C₂	D₁₆×D₇₃	1	1	yes	characteristic
126	C₂	#17989026	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988991

C₂

yes

maximal, characteristic, complemented

D₈×Dic₇₃

C₂

yes

maximal, characteristic

#17988413

C₂

yes

maximal, characteristic, complemented

#17988412

C₂

yes

maximal, characteristic, complemented

#17988405

C₂

yes

maximal, characteristic, complemented

#17988388

C₂

yes

maximal, characteristic

#357×C₇₃

C₂

yes

maximal, characteristic

#14556590

—

D₄×Dic₇₃

—

C₂×#11924003

C₂²

yes

characteristic

#14556525

C₂²

yes

characteristic

#14556525

C₂²

yes

characteristic

#14556524

—

#14556512

—

C₈×Dic₇₃

C₂²

yes

characteristic

C₂×D₈×C₇₃

C₂²

yes

characteristic

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

#107×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂×#10596831

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

D₄

yes

characteristic

#11924003

—

C₂×D₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

Q₈

yes

complemented

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

—

C₂×Dic₇₃

D₈

yes

characteristic

C₂×Dic₇₃

D₈

yes

characteristic

Dic₁₄₆

—

C₈×C₇₃

C₂×Q₈

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

#137

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

D₁₆

yes

—

C₂×C₇₃

—

C₂×C₇₃

#474

yes

characteristic

C₂×C₇₃

C₂×D₁₆

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#500

—

C₄×D₈

—

#366

—

#366

—

C₄⋊C₁₆

—

#357

—

#357

D₇₃

yes

characteristic

C₂×D₈

C₂×D₇₃

yes

characteristic

C₄⋊Q₈

—

146

—

C₄×D₄

—

146

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₃

yes

—

—

—

146

—

#107

C₂×D₇₃

yes

characteristic

#102

—

146

—

C₄×C₈

—

C₂×Q₈

—

292

—

C₂×D₄

—

C₂²×C₄

—

292

—

D₈

—

D₈

Dic₁₄₆

yes

complemented

C₂×C₈

—

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

292

—

C₄²

—

C₁₆

—

C₁₆

—

146

—

C₂³

—

Q₈

—

584

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

292

—

104

C₂×C₄

—

292

—

105

C₂×C₄

—

584

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

D₄×D₇₃

yes

characteristic

109

C₈

—

146

—

110

C₈

C₂×Dic₁₄₆

yes

characteristic

111

C₈

#11924027

yes

characteristic

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

D₈×D₇₃

yes

characteristic

116

C₄

—

117

C₄

—

584

—

118

C₄

—

584

—

119

C₄

—

120

C₄

—

146

—

121

C₄

#14556568

yes

characteristic

122

C₄

#14556620

yes

characteristic

123

C₂

—

124

C₂

#17988865

yes

characteristic

125

C₂

D₁₆×D₇₃

yes

characteristic

126

C₂

#17989026

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	#2224
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. #17988413	C₂	semidirect
4. #17988412	C₂	semidirect
5. #17988405	C₂	semidirect
30. D₈×C₇₃	Q₈	semidirect
64. C₇₃	#2224	semidirect
87. D₈	Dic₁₄₆	semidirect

H ≅ G/N

type

1. #17988991

C₂

semidirect

3. #17988413

C₂

semidirect

4. #17988412

C₂

semidirect

5. #17988405

C₂

semidirect

30. D₈×C₇₃

Q₈

semidirect

64. C₇₃

#2224

semidirect

87. D₈

Dic₁₄₆

semidirect

Subgroups — #22612066

Conjugacy classes of subgroups

Complements