Group #22609416 — subgroups

Subgroups — #22609416

Order 9344.

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

normal subgroups only

This group has 184 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	#17988964	C₂	1	1	yes	maximal, characteristic
2	#17988788	C₂	1	2	yes	maximal, complemented
3	#17988717	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988330	C₂	1	2	yes	maximal, complemented
5	#334×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11924017	—	2	2	no	—
7	C₂×D₅₈₄	—	2	1	no	—
8	C₂×D₅₈₄	—	2	1	no	—
9	#14556588	C₂²	1	2	yes	—
10	C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
11	C₄⋊#10596831	C₂²	1	1	yes	characteristic
12	#14556520	—	2	1	no	—
13	#14556513	—	2	2	no	—
14	#14556501	—	2	1	no	—
15	C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
16	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₄₆	D₄	1	1	yes	characteristic
18	C₂×D₂₉₂	—	2	2	no	—
19	C₂×D₂₉₂	—	2	2	no	—
20	C₂×D₂₉₂	—	2	2	no	—
21	#11924017	—	4	4	no	—
22	D₅₈₄	—	4	2	no	—
23	D₅₈₄	—	4	2	no	—
24	C₂×#10596831	—	2	1	no	—
25	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
26	#11924007	—	2	1	no	—
27	C₄×Dic₇₃	—	2	1	no	—
28	C₂×C₈×C₇₃	—	2	2	no	—
29	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
30	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₄²×C₇₃	C₂³	1	1	yes	characteristic
32	C₂²×D₇₃	—	4	2	no	—
33	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
36	C₂×Dic₇₃	—	2	1	no	—
37	C₂×Dic₇₃	—	2	1	no	—
38	D₂₉₂	—	4	2	no	—
39	D₂₉₂	—	4	2	no	—
40	D₂₉₂	—	4	2	no	—
41	D₂₉₂	—	4	2	no	—
42	D₂₉₂	—	8	2	no	—
43	Dic₁₄₆	—	2	2	no	—
44	C₈×C₇₃	—	2	1	no	—
45	C₈×C₇₃	—	4	2	no	—
46	C₈×C₇₃	SD₁₆	1	2	yes	complemented
47	#10596831	—	4	1	no	—
48	C₂²×C₇₃	#121	1	1	yes	characteristic
49	C₂×D₇₃	—	8	2	no	—
50	C₂×D₇₃	—	8	2	no	—
51	C₂×D₇₃	—	8	2	no	—
52	C₄×C₇₃	—	2	1	no	—
53	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
54	C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
56	Dic₇₃	—	4	1	no	—
57	Dic₇₃	—	4	1	no	—
58	C₂×C₇₃	#460	1	1	yes	characteristic
59	C₂×C₇₃	#469	1	1	yes	characteristic
60	C₂×C₇₃	#472	1	1	yes	characteristic
61	D₇₃	—	16	2	no	—
62	#1688	—	73	1	no	maximal, Sylow
63	C₇₃	#1688	1	1	yes	characteristic, complemented, Sylow
64	#493	—	73	1	no	—
65	#460	—	73	2	no	—
66	#446	—	73	1	no	—
67	#334	D₇₃	1	1	yes	characteristic, complemented
68	#331	—	73	2	no	—
69	C₂×SD₁₆	—	146	2	no	—
70	C₂×D₈	—	146	1	no	—
71	C₂×D₈	—	146	1	no	—
72	#127	—	73	2	no	—
73	C₄×Q₈	—	73	1	no	—
74	C₄⋊C₈	—	73	1	no	—
75	C₄⋊C₈	—	146	1	no	—
76	C₄⋊C₈	C₂×D₇₃	1	2	yes	—
77	#102	—	146	2	no	—
78	#97	—	146	1	no	—
79	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
80	C₂×Q₈	—	73	1	no	—
81	C₂×D₄	—	146	2	no	—
82	C₂×D₄	—	146	2	no	—
83	C₂×D₄	—	146	2	no	—
84	SD₁₆	—	292	4	no	—
85	D₈	—	292	2	no	—
86	D₈	—	292	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	146	1	no	—
89	C₂×C₈	D₂₉₂	1	1	yes	characteristic
90	C₂×C₈	D₂₉₂	1	1	yes	characteristic
91	C₄⋊C₄	—	73	1	no	—
92	C₄⋊C₄	—	146	1	no	—
93	C₄²	—	146	1	no	—
94	C₄²	C₂²×D₇₃	1	1	yes	characteristic
95	C₂³	—	292	2	no	—
96	Q₈	—	146	2	no	—
97	D₄	—	292	2	no	—
98	D₄	—	292	2	no	—
99	D₄	—	292	2	no	—
100	D₄	—	292	2	no	—
101	D₄	—	584	2	no	—
102	C₂×C₄	—	146	1	no	—
103	C₂×C₄	—	146	1	no	—
104	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
105	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
106	C₂×C₄	#11924029	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	4	2	no	—
109	C₈	—	292	1	no	—
110	C₈	#11924017	1	2	yes	complemented
111	C₂²	—	584	2	no	—
112	C₂²	—	584	2	no	—
113	C₂²	—	584	2	no	—
114	C₂²	#14556566	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	292	1	no	—
117	C₄	—	292	1	no	—
118	C₄	C₂×#11924017	1	1	yes	characteristic
119	C₄	#14556618	1	2	yes	—
120	C₄	#14556619	1	1	yes	characteristic
121	C₂	—	1168	2	no	—
122	C₂	#17988788	1	1	yes	characteristic
123	C₂	#17988841	1	1	yes	characteristic
124	C₂	#17988858	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988964

C₂

yes

maximal, characteristic

#17988788

C₂

yes

maximal, complemented

#17988717

C₂

yes

maximal, characteristic, complemented

#17988330

C₂

yes

maximal, complemented

#334×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924017

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

#14556588

C₂²

yes

—

C₄×Dic₁₄₆

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

—

C₄⋊C₈×C₇₃

C₂²

yes

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

D₄

yes

characteristic

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

#11924017

—

D₅₈₄

—

D₅₈₄

—

C₂×#10596831

—

C₄⋊Dic₇₃

D₄

yes

characteristic

#11924007

—

C₄×Dic₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

SD₁₆

yes

complemented

#10596831

—

C₂²×C₇₃

#121

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₄×C₇₃

AGL(1,ℤ/8)

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1688

—

maximal, Sylow

C₇₃

#1688

yes

characteristic, complemented, Sylow

—

—

—

D₇₃

yes

characteristic, complemented

#331

—

C₂×SD₁₆

—

146

—

C₂×D₈

—

146

—

C₂×D₈

—

146

—

#127

—

C₄×Q₈

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

C₄⋊C₈

C₂×D₇₃

yes

—

#102

—

146

—

#97

—

146

—

C₄×C₈

C₂×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

146

—

C₂×D₄

—

146

—

C₂×D₄

—

146

—

SD₁₆

—

292

—

D₈

—

292

—

D₈

—

292

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

D₂₉₂

yes

characteristic

C₂×C₈

D₂₉₂

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

146

—

C₄²

—

146

—

C₄²

C₂²×D₇₃

yes

characteristic

C₂³

—

292

—

Q₈

—

146

—

D₄

—

292

—

D₄

—

292

—

D₄

—

292

—

100

D₄

—

292

—

101

D₄

—

584

—

102

C₂×C₄

—

146

—

103

C₂×C₄

—

146

—

104

C₂×C₄

D₄×D₇₃

yes

characteristic

105

C₂×C₄

C₂×D₂₉₂

yes

characteristic

106

C₂×C₄

#11924029

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

292

—

110

C₈

#11924017

yes

complemented

111

C₂²

—

584

—

112

C₂²

—

584

—

113

C₂²

—

584

—

114

C₂²

#14556566

yes

characteristic

115

C₄

—

116

C₄

—

292

—

117

C₄

—

292

—

118

C₄

C₂×#11924017

yes

characteristic

119

C₄

#14556618

yes

—

120

C₄

#14556619

yes

characteristic

121

C₂

—

1168

—

122

C₂

#17988788

yes

characteristic

123

C₂

#17988841

yes

characteristic

124

C₂

#17988858

yes

characteristic

	subgroup	G/H
center	C₂²	#14556566
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#334×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#334	D₇₃
hypocenter	C₇₃	#1688
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#14556566

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#334×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
2. #17988788	C₂	semidirect
3. #17988717	C₂	semidirect
4. #17988330	C₂	semidirect
5. #334×C₇₃	C₂	semidirect
46. C₈×C₇₃	SD₁₆	semidirect
63. C₇₃	#1688	semidirect
67. #334	D₇₃	semidirect
110. C₈	#11924017	semidirect

H ≅ G/N

type

2. #17988788

C₂

semidirect

3. #17988717

C₂

semidirect

4. #17988330

C₂

semidirect

5. #334×C₇₃

C₂

semidirect

46. C₈×C₇₃

SD₁₆

semidirect

63. C₇₃

#1688

semidirect

67. #334

D₇₃

semidirect

110. C₈

#11924017

semidirect

Subgroups — #22609416

Conjugacy classes of subgroups

Complements