Group #22620307 — subgroups

Subgroups — #22620307

Order 9344.

484 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂×#14556588	C₂	1	1	yes	maximal, characteristic
2	#17988953	C₂	1	4	yes	maximal, complemented
3	#17988935	C₂	1	4	yes	maximal, complemented
4	#17988591	C₂	1	4	yes	maximal, complemented
5	C₄×#11924012	C₂	1	1	yes	maximal, characteristic, complemented
6	#533×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₂₉₂	—	2	2	no	—
8	C₂²×D₂₉₂	C₂²	1	2	yes	—
9	C₂×#11924012	C₂²	1	2	yes	—
10	#14556588	—	2	2	no	—
11	#14556588	C₂²	1	2	yes	—
12	#14556588	C₂²	1	2	yes	—
13	#14556511	C₂²	1	16	yes	complemented
14	#14556502	C₂²	1	2	yes	complemented
15	C₄×#10596831	C₂²	1	2	yes	complemented
16	C₄⋊Q₈×C₇₃	C₂²	1	2	yes	complemented
17	#125×C₇₃	C₂²	1	2	yes	complemented
18	C₂²⋊Q₈×C₇₃	—	2	2	no	—
19	C₄×Q₈×C₇₃	—	2	2	no	—
20	#117×C₇₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
22	C₂³×D₇₃	—	2	2	no	—
23	C₂×D₂₉₂	—	2	8	no	—
24	C₂×D₂₉₂	—	2	2	no	—
25	C₂×D₂₉₂	—	2	4	no	—
26	C₂×D₂₉₂	—	2	4	no	—
27	C₂×D₂₉₂	—	4	8	no	—
28	C₂×D₂₉₂	C₂³	1	4	yes	—
29	#11924012	—	2	4	no	—
30	C₂×#10596831	C₂³	1	4	yes	—
31	C₂×Q₈×C₇₃	—	2	2	no	—
32	C₂²×C₄×C₇₃	D₄	1	2	yes	—
33	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₇₃	—	2	4	no	—
35	C₄⋊C₄×C₇₃	—	2	2	no	—
36	C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
37	C₂²⋊C₄×C₇₃	—	2	2	no	—
38	C₄²×C₇₃	—	2	2	no	—
39	C₄²×C₇₃	D₄	1	2	yes	—
40	C₄²×C₇₃	C₂³	1	1	yes	characteristic
41	C₄²×C₇₃	C₂³	1	1	yes	characteristic
42	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₂²×D₇₃	—	2	4	no	—
44	C₂²×D₇₃	—	4	8	no	—
45	C₂²×D₇₃	—	4	4	no	—
46	Q₈×C₇₃	—	4	4	no	—
47	C₂×C₄×C₇₃	—	2	4	no	—
48	C₂×C₄×C₇₃	—	2	4	no	—
49	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₇₃	#41	1	4	yes	—
53	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
54	D₂₉₂	—	2	8	no	—
55	D₂₉₂	—	4	8	no	—
56	D₂₉₂	—	4	4	no	—
57	D₂₉₂	—	4	8	no	—
58	#10596831	—	2	4	no	—
59	C₂²×C₇₃	—	2	2	no	—
60	C₂²×C₇₃	#124	1	2	yes	—
61	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₇₃	—	4	4	no	—
63	C₂×D₇₃	—	4	8	no	—
64	C₂×D₇₃	—	8	8	no	—
65	C₄×C₇₃	—	2	2	no	—
66	C₄×C₇₃	—	4	4	no	—
67	C₄×C₇₃	#124	1	2	yes	—
68	C₄×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
69	C₄×C₇₃	C₂×#41	1	2	yes	—
70	C₂×C₇₃	—	2	2	no	—
71	C₂×C₇₃	C₂×#124	1	1	yes	characteristic
72	C₂×C₇₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
73	D₇₃	—	8	4	no	—
74	#3138	—	73	1	no	maximal, Sylow
75	C₇₃	#3138	1	1	yes	characteristic, complemented, Sylow
76	#533	D₇₃	1	1	yes	characteristic, complemented
77	C₂×#127	—	73	1	no	—
78	#490	—	73	4	no	—
79	#486	—	73	4	no	—
80	#418	—	73	4	no	—
81	C₄×M₁₆	—	73	1	no	—
82	C₂²×D₄	—	73	2	no	—
83	C₂²×D₄	—	146	2	no	—
84	C₂×M₁₆	—	73	2	no	—
85	C₄⋊Q₈	C₂×D₇₃	1	2	yes	complemented
86	#127	—	73	2	no	—
87	#127	—	73	2	no	—
88	#127	—	146	2	no	—
89	#125	C₂×D₇₃	1	2	yes	complemented
90	C₂²⋊Q₈	—	2	2	no	—
91	C₄×Q₈	—	2	2	no	—
92	#117	C₂×D₇₃	1	2	yes	—
93	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
94	#102	—	73	16	no	—
95	#97	—	73	2	no	—
96	C₄×C₈	—	73	2	no	—
97	C₂⁴	—	146	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×D₄	—	292	8	no	—
100	C₂×D₄	—	73	4	no	—
101	C₂×D₄	—	146	4	no	—
102	C₂×D₄	—	146	4	no	—
103	C₂×D₄	—	146	2	no	—
104	C₂×D₄	—	146	8	no	—
105	C₂²×C₄	C₂²⋊D₇₃	1	2	yes	—
106	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
107	M₁₆	—	146	4	no	—
108	C₂×C₈	—	73	4	no	—
109	C₄⋊C₄	—	2	4	no	—
110	C₄⋊C₄	—	2	2	no	—
111	C₄⋊C₄	C₂²×D₇₃	1	4	yes	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₄²	—	2	2	no	—
114	C₄²	C₂²⋊D₇₃	1	2	yes	—
115	C₄²	C₂²×D₇₃	1	1	yes	characteristic
116	C₄²	C₂²×D₇₃	1	1	yes	characteristic
117	C₂³	—	292	8	no	—
118	C₂³	—	292	4	no	—
119	C₂³	—	146	4	no	—
120	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
121	Q₈	—	4	4	no	—
122	D₄	—	292	8	no	—
123	D₄	—	292	4	no	—
124	D₄	—	292	8	no	—
125	D₄	—	146	8	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
131	C₂×C₄	#11924029	1	4	yes	—
132	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
133	C₈	—	146	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	292	4	no	—
136	C₂²	—	292	8	no	—
137	C₂²	—	584	8	no	—
138	C₂²	#14556580	1	2	yes	—
139	C₂²	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	4	4	no	—
142	C₄	#14556580	1	2	yes	—
143	C₄	#14556616	1	4	yes	—
144	C₄	C₂×#11924029	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	584	4	no	—
147	C₂	C₂×#14556580	1	1	yes	characteristic
148	C₂	C₂×#14556616	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556588

C₂

yes

maximal, characteristic

#17988953

C₂

yes

maximal, complemented

#17988935

C₂

yes

maximal, complemented

#17988591

C₂

yes

maximal, complemented

C₄×#11924012

C₂

yes

maximal, characteristic, complemented

#533×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₉₂

—

C₂²×D₂₉₂

C₂²

yes

—

C₂×#11924012

C₂²

yes

—

#14556588

—

#14556588

C₂²

yes

—

#14556588

C₂²

yes

—

#14556511

C₂²

yes

complemented

#14556502

C₂²

yes

complemented

C₄×#10596831

C₂²

yes

complemented

C₄⋊Q₈×C₇₃

C₂²

yes

complemented

#125×C₇₃

C₂²

yes

complemented

C₂²⋊Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

#117×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

C₂³

yes

—

#11924012

—

C₂×#10596831

C₂³

yes

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂³

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

D₄

yes

—

C₄²×C₇₃

C₂³

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

#41

yes

—

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#124

yes

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#124

yes

—

C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₄×C₇₃

C₂×#41

yes

—

C₂×C₇₃

—

C₂×C₇₃

C₂×#124

yes

characteristic

C₂×C₇₃

C₂×AGL(1,ℤ/8)

yes

—

D₇₃

—

#3138

—

maximal, Sylow

C₇₃

#3138

yes

characteristic, complemented, Sylow

#533

D₇₃

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₂²×D₄

—

C₂²×D₄

—

146

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₇₃

yes

—

—

—

146

—

#125

C₂×D₇₃

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₇₃

yes

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

#102

—

#97

—

C₄×C₈

—

C₂⁴

—

146

—

C₂×Q₈

—

C₂×D₄

—

292

—

100

C₂×D₄

—

101

C₂×D₄

—

146

—

102

C₂×D₄

—

146

—

103

C₂×D₄

—

146

—

104

C₂×D₄

—

146

—

105

C₂²×C₄

C₂²⋊D₇₃

yes

—

106

C₂²×C₄

C₂²×D₇₃

yes

characteristic

107

M₁₆

—

146

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

C₂²×D₇₃

yes

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

C₂²⋊D₇₃

yes

—

115

C₄²

C₂²×D₇₃

yes

characteristic

116

C₄²

C₂²×D₇₃

yes

characteristic

117

C₂³

—

292

—

118

C₂³

—

292

—

119

C₂³

—

146

—

120

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

121

Q₈

—

122

D₄

—

292

—

123

D₄

—

292

—

124

D₄

—

292

—

125

D₄

—

146

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

129

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

131

C₂×C₄

#11924029

yes

—

132

C₂×C₄

C₂³×D₇₃

yes

characteristic

133

C₈

—

146

—

134

C₂²

—

135

C₂²

—

292

—

136

C₂²

—

292

—

137

C₂²

—

584

—

138

C₂²

#14556580

yes

—

139

C₂²

C₂²×C₂²⋊D₇₃

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

#14556580

yes

—

143

C₄

#14556616

yes

—

144

C₄

C₂×#11924029

yes

—

145

C₂

—

146

C₂

—

584

—

147

C₂

C₂×#14556580

yes

characteristic

148

C₂

C₂×#14556616

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#533×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#533	D₇₃
hypocenter	C₇₃	#3138
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#533×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #17988953	C₂	semidirect
3. #17988935	C₂	semidirect
4. #17988591	C₂	semidirect
5. C₄×#11924012	C₂	semidirect
6. #533×C₇₃	C₂	semidirect
13. #14556511	C₂²	semidirect
14. #14556502	C₂²	semidirect
15. C₄×#10596831	C₂²	semidirect
16. C₄⋊Q₈×C₇₃	C₂²	semidirect
17. #125×C₇₃	C₂²	semidirect
75. C₇₃	#3138	semidirect
76. #533	D₇₃	semidirect
85. C₄⋊Q₈	C₂×D₇₃	semidirect
89. #125	C₂×D₇₃	semidirect

H ≅ G/N

type

2. #17988953

C₂

semidirect

3. #17988935

C₂

semidirect

4. #17988591

C₂

semidirect

5. C₄×#11924012

C₂

semidirect

6. #533×C₇₃

C₂

semidirect

13. #14556511

C₂²

semidirect

14. #14556502

C₂²

semidirect

15. C₄×#10596831

C₂²

semidirect

16. C₄⋊Q₈×C₇₃

C₂²

semidirect

17. #125×C₇₃

C₂²

semidirect

75. C₇₃

#3138

semidirect

76. #533

D₇₃

semidirect

85. C₄⋊Q₈

C₂×D₇₃

semidirect

89. #125

C₂×D₇₃

semidirect

Subgroups — #22620307

Conjugacy classes of subgroups

Complements