Group #22612026 — subgroups

Subgroups — #22612026

Order 9344.

164 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 49 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×#14556535	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988829	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988820	C₂	1	1	yes	maximal, characteristic
5	#17988414	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988392	C₂	1	1	yes	maximal, characteristic, complemented
7	#357×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₉₂	C₂²	1	1	yes	characteristic
9	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
10	#14556571	—	2	1	no	—
11	#14556562	—	2	1	no	—
12	#14556535	—	2	2	no	—
13	C₂×#11924003	C₂²	1	1	yes	characteristic
14	C₁₆×D₇₃	—	2	2	no	—
15	#14556525	C₂²	1	1	yes	characteristic
16	#14556514	—	2	1	no	—
17	#14556512	—	2	1	no	—
18	C₂×D₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
20	#107×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₄₆	—	2	1	no	—
22	C₂×C₂²⋊D₇₃	—	4	1	no	—
23	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
24	Dic₂₉₂	—	2	1	no	—
25	Dic₂₉₂	—	2	1	no	—
26	C₂×#10596831	D₄	1	1	yes	characteristic
27	C₈×D₇₃	—	2	1	no	—
28	C₈×D₇₃	—	2	1	no	—
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₄⋊Dic₇₃	—	4	1	no	—
31	C₂²⋊Dic₇₃	—	4	1	no	—
32	#11924005	—	4	1	no	—
33	#11924003	D₄	1	2	yes	complemented
34	C₂×D₄×C₇₃	—	2	1	no	—
35	D₈×C₇₃	—	2	1	no	—
36	D₈×C₇₃	—	2	1	no	—
37	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₇₃	—	2	1	no	—
39	C₁₆×C₇₃	—	2	1	no	—
40	C₂³×C₇₃	—	4	1	no	—
41	C₂²×D₇₃	D₈	1	1	yes	characteristic
42	D₄×C₇₃	—	4	1	no	—
43	D₄×C₇₃	—	4	1	no	—
44	C₂×C₄×C₇₃	—	4	1	no	—
45	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂²⋊D₇₃	—	8	2	no	—
47	C₂×Dic₇₃	—	4	1	no	—
48	C₂×Dic₇₃	—	4	1	no	—
49	C₂×Dic₇₃	D₈	1	1	yes	characteristic
50	C₄×D₇₃	—	2	1	no	—
51	C₄×D₇₃	—	2	1	no	—
52	Dic₁₄₆	—	4	1	no	—
53	Dic₁₄₆	—	4	1	no	—
54	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₈×C₇₃	#41	1	1	yes	characteristic
56	#10596831	—	2	1	no	—
57	C₂²×C₇₃	—	8	1	no	—
58	C₂²×C₇₃	—	8	1	no	—
59	C₂²×C₇₃	—	8	1	no	—
60	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
61	C₂×D₇₃	—	2	1	no	—
62	C₂×D₇₃	—	2	1	no	—
63	C₂×D₇₃	SD₃₂	1	2	yes	complemented
64	C₄×C₇₃	—	8	1	no	—
65	C₄×C₇₃	#121	1	1	yes	characteristic
66	C₄×C₇₃	#135	1	1	yes	characteristic
67	Dic₇₃	—	2	1	no	—
68	Dic₇₃	—	8	1	no	—
69	Dic₇₃	—	8	1	no	—
70	C₂×C₇₃	—	16	1	no	—
71	C₂×C₇₃	#466	1	1	yes	characteristic
72	C₂×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
73	C₂×C₇₃	#508	1	1	yes	characteristic
74	D₇₃	—	2	2	no	—
75	#2217	—	73	1	no	maximal, Sylow
76	C₇₃	#2217	1	1	yes	characteristic, complemented, Sylow
77	C₂×SD₃₂	—	73	1	no	—
78	C₂²×C₁₆	—	73	1	no	—
79	#467	—	73	1	no	—
80	#466	—	73	1	no	—
81	#367	—	73	1	no	—
82	#358	—	73	1	no	—
83	#357	D₇₃	1	1	yes	characteristic, complemented
84	C₂×Q₁₆	—	73	1	no	—
85	C₂×D₈	C₂×D₇₃	1	1	yes	characteristic
86	C₂²×C₈	—	73	1	no	—
87	C₂²⋊Q₈	—	146	1	no	—
88	#121	—	146	1	no	—
89	SD₃₂	—	146	2	no	—
90	C₂×C₁₆	—	73	1	no	—
91	C₂×C₁₆	—	146	2	no	—
92	C₂×C₁₆	C₂×D₇₃	1	1	yes	characteristic
93	#107	—	73	1	no	—
94	#107	C₂×D₇₃	1	1	yes	characteristic
95	#103	—	146	1	no	—
96	#102	—	146	1	no	—
97	C₂×Q₈	—	146	1	no	—
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	—	292	1	no	—
100	C₂²×C₄	—	73	1	no	—
101	Q₁₆	—	146	1	no	—
102	Q₁₆	—	146	1	no	—
103	D₈	—	2	1	no	—
104	D₈	—	2	1	no	—
105	C₂×C₈	—	73	1	no	—
106	C₂×C₈	—	146	1	no	—
107	C₂×C₈	—	146	1	no	—
108	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	292	1	no	—
111	C₄⋊C₄	—	146	1	no	—
112	C₂²⋊C₄	—	292	1	no	—
113	C₂²⋊C₄	—	292	1	no	—
114	C₁₆	—	2	1	no	—
115	C₁₆	—	73	2	no	—
116	C₂³	—	4	1	no	—
117	C₂³	—	73	1	no	—
118	Q₈	—	292	1	no	—
119	Q₈	—	292	1	no	—
120	D₄	—	4	1	no	—
121	D₄	—	4	1	no	—
122	D₄	—	584	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	292	1	no	—
125	C₂×C₄	—	292	1	no	—
126	C₂×C₄	—	73	1	no	—
127	C₂×C₄	—	146	1	no	—
128	C₂×C₄	—	146	1	no	—
129	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
130	C₈	—	146	1	no	—
131	C₈	D₄×D₇₃	1	1	yes	characteristic
132	C₈	#11924028	1	1	yes	characteristic
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	73	2	no	—
137	C₂²	—	146	1	no	—
138	C₂²	—	146	1	no	—
139	C₂²	D₈×D₇₃	1	1	yes	characteristic
140	C₄	—	8	1	no	—
141	C₄	—	584	1	no	—
142	C₄	—	584	1	no	—
143	C₄	—	146	1	no	—
144	C₄	#14556564	1	1	yes	characteristic
145	C₄	#14556612	1	1	yes	characteristic
146	C₂	—	16	1	no	—
147	C₂	—	146	2	no	—
148	C₂	#17988822	1	1	yes	characteristic
149	C₂	SD₃₂×D₇₃	1	1	yes	characteristic
150	C₂	#17989016	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556535

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₇₃

C₂

yes

maximal, characteristic, complemented

#17988829

C₂

yes

maximal, characteristic, complemented

#17988820

C₂

yes

maximal, characteristic

#17988414

C₂

yes

maximal, characteristic, complemented

#17988392

C₂

yes

maximal, characteristic, complemented

#357×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₉₂

C₂²

yes

characteristic

C₂×C₈×D₇₃

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₁₆×D₇₃

—

#14556525

C₂²

yes

characteristic

#14556514

—

#14556512

—

C₂×D₈×C₇₃

C₂²

yes

characteristic

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

#107×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×C₂²⋊D₇₃

—

C₂×C₄×D₇₃

D₄

yes

characteristic

Dic₂₉₂

—

Dic₂₉₂

—

C₂×#10596831

D₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

#11924005

—

#11924003

D₄

yes

complemented

C₂×D₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₁₆×C₇₃

—

C₂³×C₇₃

—

C₂²×D₇₃

D₈

yes

characteristic

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

D₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

C₂×D₄

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

SD₃₂

yes

complemented

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#466

yes

characteristic

C₂×C₇₃

C₂×SD₃₂

yes

characteristic

C₂×C₇₃

#508

yes

characteristic

D₇₃

—

#2217

—

maximal, Sylow

C₇₃

#2217

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

C₂²×C₁₆

—

—

—

—

—

D₇₃

yes

characteristic, complemented

C₂×Q₁₆

—

C₂×D₈

C₂×D₇₃

yes

characteristic

C₂²×C₈

—

C₂²⋊Q₈

—

146

—

#121

—

146

—

SD₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₂×D₇₃

yes

characteristic

#107

—

#107

C₂×D₇₃

yes

characteristic

#103

—

146

—

#102

—

146

—

C₂×Q₈

—

146

—

C₂×D₄

—

C₂×D₄

—

292

—

100

C₂²×C₄

—

101

Q₁₆

—

146

—

102

Q₁₆

—

146

—

103

D₈

—

104

D₈

—

105

C₂×C₈

—

106

C₂×C₈

—

146

—

107

C₂×C₈

—

146

—

108

C₂×C₈

C₂²×D₇₃

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

292

—

111

C₄⋊C₄

—

146

—

112

C₂²⋊C₄

—

292

—

113

C₂²⋊C₄

—

292

—

114

C₁₆

—

115

C₁₆

—

116

C₂³

—

117

C₂³

—

118

Q₈

—

292

—

119

Q₈

—

292

—

120

D₄

—

121

D₄

—

122

D₄

—

584

—

123

C₂×C₄

—

124

C₂×C₄

—

292

—

125

C₂×C₄

—

292

—

126

C₂×C₄

—

127

C₂×C₄

—

146

—

128

C₂×C₄

—

146

—

129

C₂×C₄

D₄×D₇₃

yes

characteristic

130

C₈

—

146

—

131

C₈

D₄×D₇₃

yes

characteristic

132

C₈

#11924028

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

146

—

138

C₂²

—

146

—

139

C₂²

D₈×D₇₃

yes

characteristic

140

C₄

—

141

C₄

—

584

—

142

C₄

—

584

—

143

C₄

—

146

—

144

C₄

#14556564

yes

characteristic

145

C₄

#14556612

yes

characteristic

146

C₂

—

147

C₂

—

146

—

148

C₂

#17988822

yes

characteristic

149

C₂

SD₃₂×D₇₃

yes

characteristic

150

C₂

#17989016

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₃
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	#357×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	#357	D₇₃
hypocenter	C₇₃	#2217
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

subgroup

G/H

center

C₂²

D₈×D₇₃

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

#357×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #357

N	H ≅ G/N	type
1. C₂×#14556535	C₂	semidirect
2. C₂×C₁₆×D₇₃	C₂	semidirect
3. #17988829	C₂	semidirect
5. #17988414	C₂	semidirect
6. #17988392	C₂	semidirect
7. #357×C₇₃	C₂	semidirect
33. #11924003	D₄	semidirect
63. C₂×D₇₃	SD₃₂	semidirect
76. C₇₃	#2217	semidirect
83. #357	D₇₃	semidirect

H ≅ G/N

type

1. C₂×#14556535

C₂

semidirect

2. C₂×C₁₆×D₇₃

C₂

semidirect

3. #17988829

C₂

semidirect

5. #17988414

C₂

semidirect

6. #17988392

C₂

semidirect

7. #357×C₇₃

C₂

semidirect

33. #11924003

D₄

semidirect

63. C₂×D₇₃

SD₃₂

semidirect

76. C₇₃

#2217

semidirect

83. #357

D₇₃

semidirect

Subgroups — #22612026

Conjugacy classes of subgroups

Complements