Group #22609407 — subgroups

Subgroups — #22609407

Order 9344.

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

normal subgroups only

This group has 198 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	#17988779	C₂	1	2	yes	maximal, complemented
2	#17988666	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988338	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988331	C₂	1	2	yes	maximal, complemented
5	#493×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11924013	—	2	2	no	—
7	C₂×#11924011	—	2	1	no	—
8	C₄×D₂₉₂	C₂²	1	1	yes	characteristic
9	C₄⋊#10596831	C₂²	1	1	yes	characteristic
10	C₄⋊#10596831	C₂²	1	2	yes	—
11	#14556512	—	2	2	no	—
12	#14556503	—	2	1	no	—
13	C₂×D₈×C₇₃	—	2	1	no	—
14	C₂×D₈×C₇₃	—	2	1	no	—
15	#127×C₇₃	C₂²	1	2	yes	—
16	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×D₂₉₂	D₄	1	1	yes	characteristic
18	C₂×C₄×D₇₃	—	2	1	no	—
19	#11924013	—	4	4	no	—
20	#11924011	—	4	2	no	—
21	C₂×#10596831	—	2	1	no	—
22	C₂×#10596831	—	2	2	no	—
23	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
24	#11924005	—	2	1	no	—
25	C₂×D₄×C₇₃	—	2	2	no	—
26	C₂×D₄×C₇₃	—	2	2	no	—
27	C₂×D₄×C₇₃	—	2	2	no	—
28	D₈×C₇₃	—	4	2	no	—
29	D₈×C₇₃	—	4	2	no	—
30	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
32	C₄²×C₇₃	C₂³	1	1	yes	characteristic
33	C₂³×C₇₃	—	4	2	no	—
34	C₂²×D₇₃	—	2	1	no	—
35	D₄×C₇₃	—	4	2	no	—
36	D₄×C₇₃	—	4	2	no	—
37	D₄×C₇₃	—	4	2	no	—
38	D₄×C₇₃	—	4	2	no	—
39	D₄×C₇₃	—	8	2	no	—
40	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
43	C₂×Dic₇₃	—	2	1	no	—
44	D₂₉₂	—	2	2	no	—
45	C₄×D₇₃	—	4	1	no	—
46	C₄×D₇₃	—	4	1	no	—
47	C₈×C₇₃	—	2	1	no	—
48	C₈×C₇₃	D₈	1	2	yes	complemented
49	#10596831	—	4	1	no	—
50	#10596831	—	4	2	no	—
51	C₂²×C₇₃	—	8	2	no	—
52	C₂²×C₇₃	—	8	2	no	—
53	C₂²×C₇₃	—	8	2	no	—
54	C₂²×C₇₃	#121	1	1	yes	characteristic
55	C₂×D₇₃	—	4	1	no	—
56	C₂×D₇₃	—	4	1	no	—
57	C₂×D₇₃	—	4	1	no	—
58	C₄×C₇₃	—	2	1	no	—
59	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
60	C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
61	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₇₃	—	4	1	no	—
63	C₂×C₇₃	—	16	2	no	—
64	C₂×C₇₃	#459	1	1	yes	characteristic
65	C₂×C₇₃	#469	1	1	yes	characteristic
66	C₂×C₇₃	#472	1	1	yes	characteristic
67	D₇₃	—	8	1	no	—
68	#1687	—	73	1	no	maximal, Sylow
69	C₇₃	#1687	1	1	yes	characteristic, complemented, Sylow
70	#493	D₇₃	1	1	yes	characteristic, complemented
71	#459	—	73	2	no	—
72	#436	—	73	1	no	—
73	#334	—	73	1	no	—
74	#331	—	73	2	no	—
75	C₂×D₈	—	2	1	no	—
76	C₂×D₈	—	2	1	no	—
77	C₂×D₈	—	146	2	no	—
78	C₂×M₁₆	—	146	1	no	—
79	#127	C₂×D₇₃	1	2	yes	—
80	C₄×D₄	—	73	1	no	—
81	C₄⋊C₈	—	73	1	no	—
82	C₄⋊C₈	—	73	2	no	—
83	#102	—	146	2	no	—
84	C₂²⋊C₈	—	146	1	no	—
85	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	—	73	1	no	—
90	C₂²×C₄	—	146	1	no	—
91	D₈	—	4	2	no	—
92	D₈	—	4	2	no	—
93	D₈	—	292	4	no	—
94	M₁₆	—	292	2	no	—
95	C₂×C₈	—	146	2	no	—
96	C₂×C₈	—	146	1	no	—
97	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
98	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
99	C₄⋊C₄	—	73	1	no	—
100	C₂²⋊C₄	—	146	1	no	—
101	C₄²	C₂²×D₇₃	1	1	yes	characteristic
102	C₂³	—	4	2	no	—
103	C₂³	—	146	1	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	2	no	—
107	D₄	—	4	2	no	—
108	D₄	—	8	2	no	—
109	D₄	—	146	2	no	—
110	C₂×C₄	—	292	1	no	—
111	C₂×C₄	—	292	1	no	—
112	C₂×C₄	—	146	1	no	—
113	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
115	C₂×C₄	#11924027	1	1	yes	characteristic
116	C₈	—	2	1	no	—
117	C₈	—	292	1	no	—
118	C₈	—	292	2	no	—
119	C₈	#11924013	1	2	yes	complemented
120	C₂²	—	8	2	no	—
121	C₂²	—	8	2	no	—
122	C₂²	—	8	2	no	—
123	C₂²	—	292	1	no	—
124	C₂²	—	292	1	no	—
125	C₂²	—	292	1	no	—
126	C₂²	#14556562	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	292	1	no	—
129	C₄	C₂×#11924013	1	1	yes	characteristic
130	C₄	#14556615	1	2	yes	—
131	C₄	#14556617	1	1	yes	characteristic
132	C₂	—	16	2	no	—
133	C₂	—	584	1	no	—
134	C₂	#17988779	1	1	yes	characteristic
135	C₂	#17988837	1	1	yes	characteristic
136	C₂	#17988854	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988779

C₂

yes

maximal, complemented

#17988666

C₂

yes

maximal, characteristic, complemented

#17988338

C₂

yes

maximal, characteristic, complemented

#17988331

C₂

yes

maximal, complemented

#493×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924013

—

C₂×#11924011

—

C₄×D₂₉₂

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

#14556512

—

#14556503

—

C₂×D₈×C₇₃

—

C₂×D₈×C₇₃

—

#127×C₇₃

C₂²

yes

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×D₂₉₂

D₄

yes

characteristic

C₂×C₄×D₇₃

—

—

—

—

—

C₄⋊Dic₇₃

D₄

yes

characteristic

#11924005

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

C₂²×D₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

C₈×C₇₃

D₈

yes

complemented

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#121

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₄×C₇₃

AGL(1,ℤ/8)

yes

characteristic

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1687

—

maximal, Sylow

C₇₃

#1687

yes

characteristic, complemented, Sylow

#493

D₇₃

yes

characteristic, complemented

—

—

—

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

146

—

C₂×M₁₆

—

146

—

#127

C₂×D₇₃

yes

—

C₄×D₄

—

C₄⋊C₈

—

C₄⋊C₈

—

#102

—

146

—

C₂²⋊C₈

—

146

—

C₄×C₈

C₂×D₇₃

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

146

—

D₈

—

D₈

—

D₈

—

292

—

M₁₆

—

292

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

C₄⋊C₄

—

100

C₂²⋊C₄

—

146

—

101

C₄²

C₂²×D₇₃

yes

characteristic

102

C₂³

—

103

C₂³

—

146

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

108

D₄

—

109

D₄

—

146

—

110

C₂×C₄

—

292

—

111

C₂×C₄

—

292

—

112

C₂×C₄

—

146

—

113

C₂×C₄

D₄×D₇₃

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

115

C₂×C₄

#11924027

yes

characteristic

116

C₈

—

117

C₈

—

292

—

118

C₈

—

292

—

119

C₈

#11924013

yes

complemented

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

292

—

124

C₂²

—

292

—

125

C₂²

—

292

—

126

C₂²

#14556562

yes

characteristic

127

C₄

—

128

C₄

—

292

—

129

C₄

C₂×#11924013

yes

characteristic

130

C₄

#14556615

yes

—

131

C₄

#14556617

yes

characteristic

132

C₂

—

133

C₂

—

584

—

134

C₂

#17988779

yes

characteristic

135

C₂

#17988837

yes

characteristic

136

C₂

#17988854

yes

characteristic

	subgroup	G/H
center	C₂²	#14556562
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#493×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#493	D₇₃
hypocenter	C₇₃	#1687
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#14556562

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#493×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #17988779	C₂	semidirect
2. #17988666	C₂	semidirect
3. #17988338	C₂	semidirect
4. #17988331	C₂	semidirect
5. #493×C₇₃	C₂	semidirect
48. C₈×C₇₃	D₈	semidirect
69. C₇₃	#1687	semidirect
70. #493	D₇₃	semidirect
119. C₈	#11924013	semidirect

H ≅ G/N

type

1. #17988779

C₂

semidirect

2. #17988666

C₂

semidirect

3. #17988338

C₂

semidirect

4. #17988331

C₂

semidirect

5. #493×C₇₃

C₂

semidirect

48. C₈×C₇₃

D₈

semidirect

69. C₇₃

#1687

semidirect

70. #493

D₇₃

semidirect

119. C₈

#11924013

semidirect

Subgroups — #22609407

Conjugacy classes of subgroups

Complements