Group #22612028 — subgroups

Subgroups — #22612028

Order 9344.

164 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 51 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×#14556536	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	#17988824	C₂	1	1	yes	maximal, characteristic
5	#17988390	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988387	C₂	1	1	yes	maximal, characteristic, complemented
7	#367×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Dic₂₉₂	C₂²	1	1	yes	characteristic
9	C₂×D₅₈₄	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
11	#14556571	—	2	1	no	—
12	#14556566	—	2	1	no	—
13	#14556536	—	2	2	no	—
14	C₂×#11924003	C₂²	1	1	yes	characteristic
15	C₁₆×D₇₃	—	2	2	no	—
16	#14556514	—	2	1	no	—
17	#14556511	—	2	1	no	—
18	C₂×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
19	#107×C₇₃	C₂²	1	1	yes	characteristic
20	#107×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₄₆	—	2	1	no	—
22	C₂×D₂₉₂	—	2	1	no	—
23	C₂×D₂₉₂	—	4	1	no	—
24	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
25	Dic₂₉₂	—	2	1	no	—
26	Dic₂₉₂	—	2	1	no	—
27	D₅₈₄	—	2	1	no	—
28	D₅₈₄	—	2	1	no	—
29	C₂×#10596831	D₄	1	1	yes	characteristic
30	C₈×D₇₃	—	2	1	no	—
31	C₈×D₇₃	—	2	1	no	—
32	C₄⋊Dic₇₃	—	4	1	no	—
33	#11924005	—	4	1	no	—
34	#11924005	—	4	1	no	—
35	#11924003	—	2	1	no	—
36	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₇₃	—	2	1	no	—
38	C₄⋊C₄×C₇₃	—	2	1	no	—
39	C₁₆×C₇₃	D₄	1	2	yes	complemented
40	C₂²×D₇₃	—	4	1	no	—
41	C₂²×D₇₃	D₈	1	1	yes	characteristic
42	C₂×C₄×C₇₃	—	4	1	no	—
43	C₂×C₄×C₇₃	—	4	1	no	—
44	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₇₃	—	4	1	no	—
46	C₂×Dic₇₃	D₈	1	1	yes	characteristic
47	D₂₉₂	—	4	1	no	—
48	D₂₉₂	—	4	1	no	—
49	D₂₉₂	—	8	2	no	—
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	C₂×D₈	1	1	yes	characteristic
58	C₂×D₇₃	—	2	1	no	—
59	C₂×D₇₃	—	2	1	no	—
60	C₂×D₇₃	—	8	1	no	—
61	C₂×D₇₃	—	8	1	no	—
62	C₂×D₇₃	—	8	1	no	—
63	C₂×D₇₃	SD₃₂	1	2	yes	complemented
64	C₄×C₇₃	—	8	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	—	2	1	no	—
69	Dic₇₃	—	8	1	no	—
70	C₂×C₇₃	#466	1	1	yes	characteristic
71	C₂×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
72	C₂×C₇₃	#508	1	1	yes	characteristic
73	D₇₃	—	2	2	no	—
74	D₇₃	—	16	1	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	D₇₃	1	1	yes	characteristic, complemented
82	#358	—	73	1	no	—
83	#357	—	73	1	no	—
84	C₂×Q₁₆	—	73	1	no	—
85	C₂×D₈	—	73	1	no	—
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	C₂×D₇₃	1	1	yes	characteristic
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	—	292	1	no	—
99	C₂×D₄	—	146	1	no	—
100	C₂²×C₄	—	73	1	no	—
101	Q₁₆	—	146	1	no	—
102	Q₁₆	—	146	1	no	—
103	D₈	—	146	1	no	—
104	D₈	—	146	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	—	2	1	no	—
111	C₄⋊C₄	—	292	1	no	—
112	C₂²⋊C₄	—	292	1	no	—
113	C₂²⋊C₄	—	292	1	no	—
114	C₁₆	—	146	1	no	—
115	C₁₆	D₂₉₂	1	2	yes	complemented
116	C₂³	—	292	1	no	—
117	C₂³	—	73	1	no	—
118	Q₈	—	292	1	no	—
119	Q₈	—	292	1	no	—
120	D₄	—	292	1	no	—
121	D₄	—	292	1	no	—
122	D₄	—	584	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	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	C₂×D₂₉₂	1	1	yes	characteristic
132	C₈	#11924029	1	1	yes	characteristic
133	C₂²	—	584	1	no	—
134	C₂²	—	584	1	no	—
135	C₂²	—	584	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	—	8	1	no	—
142	C₄	—	584	1	no	—
143	C₄	—	146	1	no	—
144	C₄	#14556566	1	1	yes	characteristic
145	C₄	#14556614	1	1	yes	characteristic
146	C₂	—	1168	1	no	—
147	C₂	—	146	2	no	—
148	C₂	#17988824	1	1	yes	characteristic
149	C₂	SD₃₂×D₇₃	1	1	yes	characteristic
150	C₂	#17989018	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556536

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₇₃

C₂

yes

maximal, characteristic, complemented

#17988829

C₂

yes

maximal, characteristic, complemented

#17988824

C₂

yes

maximal, characteristic

#17988390

C₂

yes

maximal, characteristic, complemented

#17988387

C₂

yes

maximal, characteristic, complemented

#367×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₉₂

C₂²

yes

characteristic

C₂×D₅₈₄

C₂²

yes

characteristic

C₂×C₈×D₇₃

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₁₆×D₇₃

—

#14556514

—

#14556511

—

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

#107×C₇₃

C₂²

yes

characteristic

#107×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×C₄×D₇₃

D₄

yes

characteristic

Dic₂₉₂

—

Dic₂₉₂

—

D₅₈₄

—

D₅₈₄

—

C₂×#10596831

D₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₄⋊Dic₇₃

—

#11924005

—

#11924005

—

#11924003

—

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₁₆×C₇₃

D₄

yes

complemented

C₂²×D₇₃

—

C₂²×D₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

D₈

yes

characteristic

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

C₂×D₄

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

SD₃₂

yes

complemented

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

#466

yes

characteristic

C₂×C₇₃

C₂×SD₃₂

yes

characteristic

C₂×C₇₃

#508

yes

characteristic

D₇₃

—

D₇₃

—

#2217

—

maximal, Sylow

C₇₃

#2217

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

C₂²×C₁₆

—

#467

—

#466

—

#367

D₇₃

yes

characteristic, complemented

#358

—

#357

—

C₂×Q₁₆

—

C₂×D₈

—

C₂²×C₈

—

C₂²⋊Q₈

—

146

—

#121

—

146

—

SD₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₂×D₇₃

yes

characteristic

#107

C₂×D₇₃

yes

characteristic

#107

C₂×D₇₃

yes

characteristic

#103

—

146

—

#102

—

146

—

C₂×Q₈

—

146

—

C₂×D₄

—

292

—

C₂×D₄

—

146

—

100

C₂²×C₄

—

101

Q₁₆

—

146

—

102

Q₁₆

—

146

—

103

D₈

—

146

—

104

D₈

—

146

—

105

C₂×C₈

—

106

C₂×C₈

—

146

—

107

C₂×C₈

—

146

—

108

C₂×C₈

C₂²×D₇₃

yes

characteristic

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

292

—

112

C₂²⋊C₄

—

292

—

113

C₂²⋊C₄

—

292

—

114

C₁₆

—

146

—

115

C₁₆

D₂₉₂

yes

complemented

116

C₂³

—

292

—

117

C₂³

—

118

Q₈

—

292

—

119

Q₈

—

292

—

120

D₄

—

292

—

121

D₄

—

292

—

122

D₄

—

584

—

123

C₂×C₄

—

124

C₂×C₄

—

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

C₂×D₂₉₂

yes

characteristic

132

C₈

#11924029

yes

characteristic

133

C₂²

—

584

—

134

C₂²

—

584

—

135

C₂²

—

584

—

136

C₂²

—

137

C₂²

—

146

—

138

C₂²

—

146

—

139

C₂²

D₈×D₇₃

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

584

—

143

C₄

—

146

—

144

C₄

#14556566

yes

characteristic

145

C₄

#14556614

yes

characteristic

146

C₂

—

1168

—

147

C₂

—

146

—

148

C₂

#17988824

yes

characteristic

149

C₂

SD₃₂×D₇₃

yes

characteristic

150

C₂

#17989018

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₃
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	#367×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	#367	D₇₃
hypocenter	C₇₃	#2217
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

subgroup

G/H

center

C₂²

D₈×D₇₃

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

#367×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 ◁ #367

N	H ≅ G/N	type
1. C₂×#14556536	C₂	semidirect
2. C₂×C₁₆×D₇₃	C₂	semidirect
3. #17988829	C₂	semidirect
5. #17988390	C₂	semidirect
6. #17988387	C₂	semidirect
7. #367×C₇₃	C₂	semidirect
39. C₁₆×C₇₃	D₄	semidirect
63. C₂×D₇₃	SD₃₂	semidirect
76. C₇₃	#2217	semidirect
81. #367	D₇₃	semidirect
115. C₁₆	D₂₉₂	semidirect

H ≅ G/N

type

1. C₂×#14556536

C₂

semidirect

2. C₂×C₁₆×D₇₃

C₂

semidirect

3. #17988829

C₂

semidirect

5. #17988390

C₂

semidirect

6. #17988387

C₂

semidirect

7. #367×C₇₃

C₂

semidirect

39. C₁₆×C₇₃

D₄

semidirect

63. C₂×D₇₃

SD₃₂

semidirect

76. C₇₃

#2217

semidirect

81. #367

D₇₃

semidirect

115. C₁₆

D₂₉₂

semidirect

Subgroups — #22612028

Conjugacy classes of subgroups

Complements