Group #22620670 — subgroups

Subgroups — #22620670

Order 9344.

446 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 125 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	#17989138	C₂	1	2	yes	maximal, complemented
2	#17988964	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988935	C₂	1	2	yes	maximal, complemented
4	#17988841	C₂	1	4	yes	maximal, complemented
5	#17988698	C₂	1	4	yes	maximal, complemented
6	C₄×#11924011	C₂	1	1	yes	maximal, characteristic, complemented
7	#498×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924029	—	2	2	no	—
9	C₂×D₅₈₄	C₂²	1	4	yes	—
10	C₂×#11924011	C₂²	1	2	yes	—
11	#14556588	C₂²	1	2	yes	—
12	#14556580	—	2	2	no	—
13	#14556566	—	2	2	no	—
14	#14556566	C₂²	1	4	yes	—
15	#14556553	—	2	2	no	—
16	#14556553	C₂²	1	4	yes	—
17	C₄²×D₇₃	C₂²	1	1	yes	characteristic
18	#14556511	C₂²	1	8	yes	—
19	#14556501	C₂²	1	2	yes	—
20	C₄×#10596831	C₂²	1	1	yes	characteristic
21	C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
22	#106×C₇₃	C₂²	1	4	yes	—
23	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	#11924029	—	4	8	no	—
25	C₂×D₂₉₂	—	2	4	no	—
26	C₂×D₂₉₂	—	2	2	no	—
27	C₂×D₂₉₂	—	2	2	no	—
28	C₂×D₂₉₂	C₂³	1	4	yes	—
29	C₂×C₄×D₇₃	—	2	4	no	—
30	C₂×C₄×D₇₃	D₄	1	2	yes	—
31	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
32	D₅₈₄	—	2	8	no	—
33	#11924011	—	2	4	no	—
34	C₂×#10596831	C₂³	1	2	yes	—
35	#11924005	—	2	4	no	—
36	#11924005	—	2	4	no	—
37	C₄×Dic₇₃	D₄	1	2	yes	—
38	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
39	C₂×Q₈×C₇₃	—	2	2	no	—
40	C₂×C₈×C₇₃	C₂³	1	2	yes	—
41	C₄⋊C₄×C₇₃	—	2	2	no	—
42	C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
43	C₄²×C₇₃	C₂³	1	1	yes	characteristic
44	C₂²×D₇₃	—	2	4	no	—
45	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₇₃	—	4	4	no	—
47	C₂×C₄×C₇₃	—	2	4	no	—
48	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₇₃	C₂×D₄	1	2	yes	—
52	D₂₉₂	—	2	8	no	—
53	D₂₉₂	—	4	8	no	—
54	D₂₉₂	—	4	4	no	—
55	D₂₉₂	—	4	4	no	—
56	C₄×D₇₃	—	2	4	no	—
57	C₄×D₇₃	—	2	2	no	—
58	C₄×D₇₃	—	4	8	no	—
59	C₈×C₇₃	#41	1	4	yes	complemented
60	#10596831	—	2	2	no	—
61	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₇₃	—	2	1	no	—
63	C₂×D₇₃	—	2	2	no	—
64	C₂×D₇₃	—	4	4	no	—
65	C₂×D₇₃	—	4	4	no	—
66	C₂×D₇₃	—	4	4	no	—
67	C₄×C₇₃	—	4	4	no	—
68	C₄×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
69	C₄×C₇₃	C₂×#41	1	2	yes	—
70	Dic₇₃	—	2	1	no	—
71	Dic₇₃	—	2	2	no	—
72	C₂×C₇₃	#532	1	1	yes	characteristic
73	C₂×C₇₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
74	D₇₃	—	4	1	no	—
75	D₇₃	—	8	4	no	—
76	#3184	—	73	1	no	maximal, Sylow
77	C₇₃	#3184	1	1	yes	characteristic, complemented, Sylow
78	#532	—	73	2	no	—
79	#498	D₇₃	1	1	yes	characteristic, complemented
80	#493	—	73	1	no	—
81	#486	—	73	2	no	—
82	#469	—	73	4	no	—
83	#442	—	73	4	no	—
84	C₄×M₁₆	—	73	1	no	—
85	C₂×#41	—	146	2	no	—
86	C₂×D₈	—	73	4	no	—
87	C₂×M₁₆	—	73	2	no	—
88	C₄⋊Q₈	C₂×D₇₃	1	2	yes	—
89	#127	—	73	2	no	—
90	#124	—	146	2	no	—
91	#121	—	73	4	no	—
92	#121	—	146	2	no	—
93	C₄×D₄	—	73	4	no	—
94	C₄×D₄	—	146	2	no	—
95	C₂×C₄²	—	73	1	no	—
96	#106	C₂×D₇₃	1	4	yes	—
97	#102	—	73	8	no	—
98	#97	—	73	2	no	—
99	C₄×C₈	—	73	1	no	—
100	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
101	#41	—	292	8	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×D₄	—	73	4	no	—
104	C₂×D₄	—	146	4	no	—
105	C₂×D₄	—	146	2	no	—
106	C₂×D₄	—	146	2	no	—
107	C₂²×C₄	—	73	2	no	—
108	C₂²×C₄	—	73	1	no	—
109	C₂²×C₄	—	146	4	no	—
110	D₈	—	146	8	no	—
111	M₁₆	—	146	4	no	—
112	C₂×C₈	—	73	2	no	—
113	C₂×C₈	C₂²×D₇₃	1	2	yes	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	C₂²×D₇₃	1	4	yes	—
116	C₂²⋊C₄	—	146	4	no	—
117	C₂²⋊C₄	—	146	4	no	—
118	C₄²	—	73	2	no	—
119	C₄²	—	73	1	no	—
120	C₄²	C₂²×D₇₃	1	1	yes	characteristic
121	C₂³	—	73	1	no	—
122	C₂³	—	146	4	no	—
123	Q₈	—	4	4	no	—
124	D₄	—	292	8	no	—
125	D₄	—	292	4	no	—
126	D₄	—	292	4	no	—
127	D₄	—	146	8	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	—	292	8	no	—
130	C₂×C₄	—	73	1	no	—
131	C₂×C₄	—	73	2	no	—
132	C₂×C₄	—	146	4	no	—
133	C₂×C₄	—	146	2	no	—
134	C₂×C₄	D₄×D₇₃	1	2	yes	—
135	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
136	C₈	—	146	2	no	—
137	C₈	#11924029	1	4	yes	complemented
138	C₂²	—	292	4	no	—
139	C₂²	—	292	4	no	—
140	C₂²	—	292	4	no	—
141	C₂²	—	146	1	no	—
142	C₂²	—	146	2	no	—
143	C₂²	C₂×D₄×D₇₃	1	1	yes	characteristic
144	C₄	—	4	4	no	—
145	C₄	—	146	1	no	—
146	C₄	—	146	2	no	—
147	C₄	#14556618	1	4	yes	—
148	C₄	C₂×#11924029	1	2	yes	—
149	C₂	—	292	1	no	—
150	C₂	—	584	4	no	—
151	C₂	#17989138	1	1	yes	characteristic
152	C₂	C₂×#14556618	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17989138

C₂

yes

maximal, complemented

#17988964

C₂

yes

maximal, characteristic, complemented

#17988935

C₂

yes

maximal, complemented

#17988841

C₂

yes

maximal, complemented

#17988698

C₂

yes

maximal, complemented

C₄×#11924011

C₂

yes

maximal, characteristic, complemented

#498×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924029

—

C₂×D₅₈₄

C₂²

yes

—

C₂×#11924011

C₂²

yes

—

#14556588

C₂²

yes

—

#14556580

—

#14556566

—

#14556566

C₂²

yes

—

#14556553

—

#14556553

C₂²

yes

—

C₄²×D₇₃

C₂²

yes

characteristic

#14556511

C₂²

yes

—

#14556501

C₂²

yes

—

C₄×#10596831

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

—

#106×C₇₃

C₂²

yes

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

#11924029

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

C₂³

yes

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

D₄

yes

—

C₂×C₄×D₇₃

C₂³

yes

characteristic

D₅₈₄

—

#11924011

—

C₂×#10596831

C₂³

yes

—

#11924005

—

#11924005

—

C₄×Dic₇₃

D₄

yes

—

C₄×Dic₇₃

C₂³

yes

characteristic

C₂×Q₈×C₇₃

—

C₂×C₈×C₇₃

C₂³

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂³

yes

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂⁴

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

#41

yes

complemented

#10596831

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₄×C₇₃

C₂×#41

yes

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

#532

yes

characteristic

C₂×C₇₃

C₂×AGL(1,ℤ/8)

yes

—

D₇₃

—

D₇₃

—

#3184

—

maximal, Sylow

C₇₃

#3184

yes

characteristic, complemented, Sylow

#532

—

#498

D₇₃

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₂×#41

—

146

—

C₂×D₈

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₇₃

yes

—

#127

—

#124

—

146

—

#121

—

#121

—

146

—

C₄×D₄

—

C₄×D₄

—

146

—

C₂×C₄²

—

#106

C₂×D₇₃

yes

—

#102

—

#97

—

C₄×C₈

—

100

C₄×C₈

C₂×D₇₃

yes

characteristic

101

#41

—

292

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

146

—

105

C₂×D₄

—

146

—

106

C₂×D₄

—

146

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

146

—

110

D₈

—

146

—

111

M₁₆

—

146

—

112

C₂×C₈

—

113

C₂×C₈

C₂²×D₇₃

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂²×D₇₃

yes

—

116

C₂²⋊C₄

—

146

—

117

C₂²⋊C₄

—

146

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₂²×D₇₃

yes

characteristic

121

C₂³

—

122

C₂³

—

146

—

123

Q₈

—

124

D₄

—

292

—

125

D₄

—

292

—

126

D₄

—

292

—

127

D₄

—

146

—

128

C₂×C₄

—

129

C₂×C₄

—

292

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

146

—

133

C₂×C₄

—

146

—

134

C₂×C₄

D₄×D₇₃

yes

—

135

C₂×C₄

C₂³×D₇₃

yes

characteristic

136

C₈

—

146

—

137

C₈

#11924029

yes

complemented

138

C₂²

—

292

—

139

C₂²

—

292

—

140

C₂²

—

292

—

141

C₂²

—

146

—

142

C₂²

—

146

—

143

C₂²

C₂×D₄×D₇₃

yes

characteristic

144

C₄

—

145

C₄

—

146

—

146

C₄

—

146

—

147

C₄

#14556618

yes

—

148

C₄

C₂×#11924029

yes

—

149

C₂

—

292

—

150

C₂

—

584

—

151

C₂

#17989138

yes

characteristic

152

C₂

C₂×#14556618

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₃
Fitting	#498×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₄
hypercenter	#498	D₇₃
hypocenter	C₇₃	#3184
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₇₃

commutator

C₂×C₄×C₇₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₃

Fitting

#498×C₇₃

C₂

socle

C₂²×C₇₃

C₂²×D₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
1. #17989138	C₂	semidirect
2. #17988964	C₂	semidirect
3. #17988935	C₂	semidirect
4. #17988841	C₂	semidirect
5. #17988698	C₂	semidirect
6. C₄×#11924011	C₂	semidirect
7. #498×C₇₃	C₂	semidirect
59. C₈×C₇₃	#41	semidirect
77. C₇₃	#3184	semidirect
79. #498	D₇₃	semidirect
137. C₈	#11924029	semidirect

H ≅ G/N

type

1. #17989138

C₂

semidirect

2. #17988964

C₂

semidirect

3. #17988935

C₂

semidirect

4. #17988841

C₂

semidirect

5. #17988698

C₂

semidirect

6. C₄×#11924011

C₂

semidirect

7. #498×C₇₃

C₂

semidirect

59. C₈×C₇₃

#41

semidirect

77. C₇₃

#3184

semidirect

79. #498

D₇₃

semidirect

137. C₈

#11924029

semidirect

Subgroups — #22620670

Conjugacy classes of subgroups

Complements