Group #22612088 — subgroups

Subgroups — #22612088

Order 9344.

136 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 53 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	#17988989	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988414	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988412	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊#11924003	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988388	C₂	1	1	yes	maximal, characteristic
6	#17988388	C₂	1	1	yes	maximal, characteristic
7	C₄×D₈×C₇₃	C₂	1	1	yes	maximal, characteristic
8	#14556583	—	2	1	no	—
9	C₂×#11924003	C₂²	1	1	yes	characteristic
10	C₂×#11924003	C₂²	1	1	yes	characteristic
11	#14556525	C₂²	1	1	yes	characteristic
12	#14556525	C₂²	1	1	yes	characteristic
13	#14556523	—	2	1	no	—
14	C₂×D₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₇₃	—	2	1	no	—
16	#107×C₇₃	C₂²	1	1	yes	characteristic
17	#102×C₇₃	—	2	1	no	—
18	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₇₃	—	2	1	no	—
20	C₄⋊Dic₇₃	—	2	1	no	—
21	#11924007	—	4	1	no	—
22	#11924007	—	4	1	no	—
23	#11924003	—	2	1	no	—
24	#11924003	—	2	1	no	—
25	C₂×D₄×C₇₃	—	2	1	no	—
26	C₂²×C₄×C₇₃	—	4	1	no	—
27	D₈×C₇₃	—	2	1	no	—
28	D₈×C₇₃	Q₈	1	2	yes	—
29	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₇₃	—	2	1	no	—
32	C₂²⋊C₄×C₇₃	—	4	1	no	—
33	C₄²×C₇₃	D₄	1	1	yes	characteristic
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	—	8	2	no	—
39	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
40	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
41	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₇₃	—	4	1	no	—
43	C₂×Dic₇₃	—	4	1	no	—
44	C₈×C₇₃	—	2	1	no	—
45	C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₇₃	#41	1	1	yes	characteristic
47	C₂²×C₇₃	—	4	2	no	—
48	C₂²×C₇₃	—	8	1	no	—
49	C₂²×C₇₃	—	8	1	no	—
50	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
51	C₄×C₇₃	—	2	1	no	—
52	C₄×C₇₃	—	2	1	no	—
53	C₄×C₇₃	—	8	1	no	—
54	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
55	C₄×C₇₃	#137	1	1	yes	characteristic
56	Dic₇₃	—	8	1	no	—
57	Dic₇₃	—	8	1	no	—
58	C₂×C₇₃	—	8	2	no	—
59	C₂×C₇₃	#474	1	1	yes	characteristic
60	C₂×C₇₃	#508	1	1	yes	characteristic
61	C₂×C₇₃	#509	1	1	yes	characteristic
62	#2228	—	73	1	no	maximal, Sylow
63	C₇₃	#2228	1	1	yes	characteristic, complemented, Sylow
64	#499	—	73	1	no	—
65	C₄×D₈	D₇₃	1	1	yes	characteristic
66	#367	—	73	1	no	—
67	#366	—	73	1	no	—
68	C₄⋊C₁₆	—	73	1	no	—
69	#357	—	73	1	no	—
70	#357	—	73	1	no	—
71	C₂×D₈	C₂×D₇₃	1	1	yes	characteristic
72	#125	—	146	1	no	—
73	C₄×D₄	—	2	1	no	—
74	C₂×C₁₆	—	73	1	no	—
75	C₂×C₁₆	—	73	1	no	—
76	#107	—	73	1	no	—
77	#107	—	73	1	no	—
78	#107	C₂×D₇₃	1	1	yes	characteristic
79	#106	—	146	1	no	—
80	#102	—	2	1	no	—
81	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
82	C₂×D₄	—	2	1	no	—
83	C₂²×C₄	—	4	1	no	—
84	D₈	—	2	1	no	—
85	D₈	Dic₁₄₆	1	2	yes	—
86	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
87	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
88	C₄⋊C₄	—	2	1	no	—
89	C₄⋊C₄	—	292	1	no	—
90	C₄⋊C₄	—	292	1	no	—
91	C₄⋊C₄	—	146	1	no	—
92	C₄⋊C₄	—	146	1	no	—
93	C₂²⋊C₄	—	4	1	no	—
94	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
95	C₁₆	—	146	1	no	—
96	C₁₆	—	146	1	no	—
97	C₂³	—	4	1	no	—
98	D₄	—	2	2	no	—
99	D₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	8	2	no	—
102	C₂×C₄	—	292	1	no	—
103	C₂×C₄	—	292	1	no	—
104	C₂×C₄	#11924013	1	1	yes	characteristic
105	C₂×C₄	#11924013	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
109	C₈	#11924028	1	1	yes	characteristic
110	C₂²	—	4	2	no	—
111	C₂²	—	8	1	no	—
112	C₂²	—	8	1	no	—
113	C₂²	C₂×#11924013	1	1	yes	characteristic
114	C₄	—	2	1	no	—
115	C₄	—	2	1	no	—
116	C₄	—	8	1	no	—
117	C₄	—	584	1	no	—
118	C₄	—	584	1	no	—
119	C₄	#14556570	1	1	yes	characteristic
120	C₄	#14556621	1	1	yes	characteristic
121	C₂	—	8	2	no	—
122	C₂	#17988867	1	1	yes	characteristic
123	C₂	#17989017	1	1	yes	characteristic
124	C₂	#17989023	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988989

C₂

yes

maximal, characteristic, complemented

#17988414

C₂

yes

maximal, characteristic, complemented

#17988412

C₂

yes

maximal, characteristic, complemented

C₄⋊#11924003

C₂

yes

maximal, characteristic, complemented

#17988388

C₂

yes

maximal, characteristic

#17988388

C₂

yes

maximal, characteristic

C₄×D₈×C₇₃

C₂

yes

maximal, characteristic

#14556583

—

C₂×#11924003

C₂²

yes

characteristic

C₂×#11924003

C₂²

yes

characteristic

#14556525

C₂²

yes

characteristic

#14556525

C₂²

yes

characteristic

#14556523

—

C₂×D₈×C₇₃

C₂²

yes

characteristic

C₄×D₄×C₇₃

—

#107×C₇₃

C₂²

yes

characteristic

#102×C₇₃

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

—

—

—

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

Q₈

yes

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

D₄

yes

characteristic

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂×Q₈

yes

characteristic

C₈×C₇₃

#41

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

#137

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#499

—

C₄×D₈

D₇₃

yes

characteristic

#367

—

#366

—

C₄⋊C₁₆

—

#357

—

#357

—

C₂×D₈

C₂×D₇₃

yes

characteristic

#125

—

146

—

C₄×D₄

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

#107

—

#107

C₂×D₇₃

yes

characteristic

#106

—

146

—

#102

—

C₄×C₈

C₂×D₇₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

D₈

—

D₈

Dic₁₄₆

yes

—

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₇₃

yes

characteristic

C₁₆

—

146

—

C₁₆

—

146

—

C₂³

—

D₄

—

D₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

292

—

103

C₂×C₄

—

292

—

104

C₂×C₄

#11924013

yes

characteristic

105

C₂×C₄

#11924013

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

107

C₈

—

108

C₈

C₂×Dic₁₄₆

yes

characteristic

109

C₈

#11924028

yes

characteristic

110

C₂²

—

111

C₂²

—

112

C₂²

—

113

C₂²

C₂×#11924013

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

584

—

118

C₄

—

584

—

119

C₄

#14556570

yes

characteristic

120

C₄

#14556621

yes

characteristic

121

C₂

—

122

C₂

#17988867

yes

characteristic

123

C₂

#17989017

yes

characteristic

124

C₂

#17989023

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11924013
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	C₄×D₈×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	C₄×D₈	D₇₃
hypocenter	C₇₃	#2228
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₈

subgroup

G/H

center

C₂²

C₂×#11924013

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

C₄×D₈×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

hypercenter

C₄×D₈

D₇₃

hypocenter

C₇₃

#2228

LCS

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₈

N	H ≅ G/N	type
1. #17988989	C₂	semidirect
2. #17988414	C₂	semidirect
3. #17988412	C₂	semidirect
4. C₄⋊#11924003	C₂	semidirect
63. C₇₃	#2228	semidirect

H ≅ G/N

type

1. #17988989

C₂

semidirect

2. #17988414

C₂

semidirect

3. #17988412

C₂

semidirect

4. C₄⋊#11924003

C₂

semidirect

63. C₇₃

#2228

semidirect

Subgroups — #22612088

Conjugacy classes of subgroups

Complements