Group #22612025 — subgroups

Subgroups — #22612025

Order 9344.

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

normal subgroups only

This group has 164 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	C₂×#14556535	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#11924003	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988828	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988414	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988390	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988388	C₂	1	1	yes	maximal, characteristic, complemented
7	#466×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₉₂	C₂²	1	1	yes	characteristic
9	#14556570	—	2	1	no	—
10	#14556535	—	2	2	no	—
11	C₂×#11924003	—	2	2	no	—
12	C₂×#11924003	C₂²	1	1	yes	characteristic
13	C₂×#11924003	C₂²	1	1	yes	characteristic
14	#14556525	C₂²	1	1	yes	characteristic
15	#14556516	—	2	1	no	—
16	C₂×D₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	#121×C₇₃	—	2	1	no	—
19	#107×C₇₃	C₂²	1	1	yes	characteristic
20	#102×C₇₃	—	2	1	no	—
21	C₂×Dic₁₄₆	—	2	1	no	—
22	Dic₂₉₂	—	2	1	no	—
23	Dic₂₉₂	—	2	1	no	—
24	C₄⋊Dic₇₃	—	2	1	no	—
25	#11924007	—	4	1	no	—
26	C₂²⋊Dic₇₃	—	4	1	no	—
27	#11924003	—	2	1	no	—
28	#11924003	D₄	1	2	yes	complemented
29	C₂×D₄×C₇₃	—	2	1	no	—
30	C₂×D₄×C₇₃	—	4	1	no	—
31	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
32	D₈×C₇₃	—	2	1	no	—
33	D₈×C₇₃	—	2	1	no	—
34	C₂×C₈×C₇₃	—	2	1	no	—
35	C₂×C₈×C₇₃	—	2	1	no	—
36	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₇₃	—	2	1	no	—
39	C₂²⋊C₄×C₇₃	—	4	1	no	—
40	C₂³×C₇₃	—	4	1	no	—
41	C₂³×C₇₃	D₈	1	1	yes	characteristic
42	D₄×C₇₃	—	4	1	no	—
43	D₄×C₇₃	—	4	1	no	—
44	D₄×C₇₃	—	8	2	no	—
45	C₂×C₄×C₇₃	—	2	1	no	—
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	—	4	1	no	—
48	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₇₃	—	4	1	no	—
51	C₂×Dic₇₃	—	4	1	no	—
52	Dic₁₄₆	—	4	1	no	—
53	Dic₁₄₆	—	4	1	no	—
54	C₈×C₇₃	—	2	1	no	—
55	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₈×C₇₃	#41	1	1	yes	characteristic
57	C₂²×C₇₃	—	2	1	no	—
58	C₂²×C₇₃	—	2	1	no	—
59	C₂²×C₇₃	—	8	1	no	—
60	C₂²×C₇₃	—	8	1	no	—
61	C₂²×C₇₃	—	8	1	no	—
62	C₂²×C₇₃	SD₃₂	1	2	yes	complemented
63	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
64	C₄×C₇₃	—	2	1	no	—
65	C₄×C₇₃	—	8	1	no	—
66	C₄×C₇₃	#121	1	1	yes	characteristic
67	C₄×C₇₃	#135	1	1	yes	characteristic
68	Dic₇₃	—	8	1	no	—
69	Dic₇₃	—	8	1	no	—
70	C₂×C₇₃	—	2	2	no	—
71	C₂×C₇₃	—	16	1	no	—
72	C₂×C₇₃	#466	1	1	yes	characteristic
73	C₂×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
74	C₂×C₇₃	#508	1	1	yes	characteristic
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	D₇₃	1	1	yes	characteristic
81	#367	—	73	1	no	—
82	#358	—	73	1	no	—
83	#357	—	73	1	no	—
84	C₂×Q₁₆	—	73	1	no	—
85	C₂×D₈	C₂×D₇₃	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
87	C₂²⋊Q₈	—	146	1	no	—
88	#121	—	2	1	no	—
89	SD₃₂	—	146	2	no	—
90	C₂×C₁₆	—	73	1	no	—
91	C₂×C₁₆	—	73	1	no	—
92	C₂×C₁₆	—	146	2	no	—
93	#107	—	73	1	no	—
94	#107	C₂×D₇₃	1	1	yes	characteristic
95	#103	—	146	1	no	—
96	#102	—	2	1	no	—
97	C₂×Q₈	—	146	1	no	—
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	—	4	1	no	—
100	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
101	Q₁₆	—	146	1	no	—
102	Q₁₆	—	146	1	no	—
103	D₈	—	2	1	no	—
104	D₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
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	—	4	1	no	—
113	C₂²⋊C₄	—	292	1	no	—
114	C₁₆	—	73	2	no	—
115	C₁₆	—	146	1	no	—
116	C₂³	—	4	1	no	—
117	C₂³	#11924013	1	1	yes	characteristic
118	Q₈	—	292	1	no	—
119	Q₈	—	292	1	no	—
120	D₄	—	4	1	no	—
121	D₄	—	4	1	no	—
122	D₄	—	8	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	292	1	no	—
127	C₂×C₄	—	292	1	no	—
128	C₂×C₄	#11924013	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
130	C₈	—	2	1	no	—
131	C₈	D₄×D₇₃	1	1	yes	characteristic
132	C₈	#11924027	1	1	yes	characteristic
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	8	1	no	—
138	C₂²	#14556535	1	2	yes	complemented
139	C₂²	C₂×#11924013	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	8	1	no	—
142	C₄	—	584	1	no	—
143	C₄	—	584	1	no	—
144	C₄	#14556563	1	1	yes	characteristic
145	C₄	#14556611	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	—	16	1	no	—
148	C₂	#17988821	1	1	yes	characteristic
149	C₂	C₂×#14556535	1	1	yes	characteristic
150	C₂	#17989017	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556535

C₂

yes

maximal, characteristic, complemented

C₂²×#11924003

C₂

yes

maximal, characteristic, complemented

#17988828

C₂

yes

maximal, characteristic, complemented

#17988414

C₂

yes

maximal, characteristic, complemented

#17988390

C₂

yes

maximal, characteristic, complemented

#17988388

C₂

yes

maximal, characteristic, complemented

#466×C₇₃

C₂

yes

maximal, characteristic

C₂×Dic₂₉₂

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₂×#11924003

C₂²

yes

characteristic

#14556525

C₂²

yes

characteristic

#14556516

—

C₂×D₈×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

#121×C₇₃

—

#107×C₇₃

C₂²

yes

characteristic

#102×C₇₃

—

C₂×Dic₁₄₆

—

Dic₂₉₂

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²⋊Dic₇₃

—

#11924003

—

#11924003

D₄

yes

complemented

C₂×D₄×C₇₃

—

C₂×D₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

D₈

yes

characteristic

D₄×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂×D₄

yes

characteristic

C₈×C₇₃

#41

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

SD₃₂

yes

complemented

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#466

yes

characteristic

C₂×C₇₃

C₂×SD₃₂

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

C₂²×C₁₆

—

#467

—

#466

D₇₃

yes

—

—

—

C₂×Q₁₆

—

C₂×D₈

C₂×D₇₃

yes

characteristic

C₂²×C₈

C₂×D₇₃

yes

characteristic

C₂²⋊Q₈

—

146

—

#121

—

SD₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

#107

—

#107

C₂×D₇₃

yes

characteristic

#103

—

146

—

#102

—

C₂×Q₈

—

146

—

C₂×D₄

—

C₂×D₄

—

100

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

101

Q₁₆

—

146

—

102

Q₁₆

—

146

—

103

D₈

—

104

D₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

108

C₂×C₈

C₂²×D₇₃

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

292

—

111

C₄⋊C₄

—

146

—

112

C₂²⋊C₄

—

113

C₂²⋊C₄

—

292

—

114

C₁₆

—

115

C₁₆

—

146

—

116

C₂³

—

117

C₂³

#11924013

yes

characteristic

118

Q₈

—

292

—

119

Q₈

—

292

—

120

D₄

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

292

—

127

C₂×C₄

—

292

—

128

C₂×C₄

#11924013

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

130

C₈

—

131

C₈

D₄×D₇₃

yes

characteristic

132

C₈

#11924027

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#14556535

yes

complemented

139

C₂²

C₂×#11924013

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

584

—

143

C₄

—

584

—

144

C₄

#14556563

yes

characteristic

145

C₄

#14556611

yes

characteristic

146

C₂

—

147

C₂

—

148

C₂

#17988821

yes

characteristic

149

C₂

C₂×#14556535

yes

characteristic

150

C₂

#17989017

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11924013
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	#466×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	#466	D₇₃
hypocenter	C₇₃	#2217
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #466

subgroup

G/H

center

C₂²

C₂×#11924013

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

#466×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #466

N	H ≅ G/N	type
1. C₂×#14556535	C₂	semidirect
2. C₂²×#11924003	C₂	semidirect
3. #17988828	C₂	semidirect
4. #17988414	C₂	semidirect
5. #17988390	C₂	semidirect
6. #17988388	C₂	semidirect
28. #11924003	D₄	semidirect
62. C₂²×C₇₃	SD₃₂	semidirect
76. C₇₃	#2217	semidirect
138. C₂²	#14556535	semidirect

H ≅ G/N

type

1. C₂×#14556535

C₂

semidirect

2. C₂²×#11924003

C₂

semidirect

3. #17988828

C₂

semidirect

4. #17988414

C₂

semidirect

5. #17988390

C₂

semidirect

6. #17988388

C₂

semidirect

28. #11924003

D₄

semidirect

62. C₂²×C₇₃

SD₃₂

semidirect

76. C₇₃

#2217

semidirect

138. C₂²

#14556535

semidirect

Subgroups — #22612025

Conjugacy classes of subgroups

Complements