Group #22609306 — subgroups

Subgroups — #22609306

Order 9344.

166 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 61 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	#17988985	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988877	C₂	1	1	yes	maximal, characteristic
3	#17988792	C₂	1	1	yes	maximal, characteristic
4	#17988338	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988335	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988332	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×D₄×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924016	—	2	1	no	—
9	#14556591	C₂²	1	1	yes	characteristic
10	#14556591	C₂²	1	1	yes	characteristic
11	#14556523	—	2	1	no	—
12	#14556523	—	2	1	no	—
13	#14556522	—	2	1	no	—
14	C₄⋊#10596831	C₂²	1	1	yes	characteristic
15	C₄⋊#10596831	C₂²	1	1	yes	characteristic
16	#14556514	—	2	1	no	—
17	#14556512	—	2	1	no	—
18	C₂²×C₈×C₇₃	—	2	1	no	—
19	C₄×D₄×C₇₃	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₂²⋊C₈×C₇₃	—	2	1	no	—
22	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₄₆	—	2	1	no	—
24	C₂×Dic₁₄₆	—	2	1	no	—
25	#11924016	—	4	2	no	—
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	—	2	1	no	—
28	C₄⋊Dic₇₃	—	2	1	no	—
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₄⋊Dic₇₃	—	2	1	no	—
32	C₂×D₄×C₇₃	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	—	2	1	no	—
34	C₂×C₈×C₇₃	—	2	1	no	—
35	C₂×C₈×C₇₃	—	4	2	no	—
36	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
37	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₇₃	—	2	1	no	—
40	C₄²×C₇₃	C₂³	1	1	yes	characteristic
41	C₂³×C₇₃	—	2	1	no	—
42	D₄×C₇₃	—	2	1	no	—
43	D₄×C₇₃	SD₁₆	1	2	yes	—
44	C₂×C₄×C₇₃	—	2	1	no	—
45	C₂×C₄×C₇₃	—	4	1	no	—
46	C₂×C₄×C₇₃	—	4	1	no	—
47	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₃	#41	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	Dic₁₄₆	—	8	1	no	—
55	C₈×C₇₃	—	2	1	no	—
56	C₈×C₇₃	—	4	1	no	—
57	C₈×C₇₃	SD₁₆	1	2	yes	complemented
58	#10596831	—	4	1	no	—
59	#10596831	—	4	1	no	—
60	C₂²×C₇₃	—	2	2	no	—
61	C₂²×C₇₃	—	4	1	no	—
62	C₂²×C₇₃	—	4	1	no	—
63	C₂²×C₇₃	#121	1	1	yes	characteristic
64	C₄×C₇₃	—	2	1	no	—
65	C₄×C₇₃	—	4	1	no	—
66	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
67	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
68	C₄×C₇₃	#135	1	1	yes	characteristic
69	C₄×C₇₃	#137	1	1	yes	characteristic
70	Dic₇₃	—	8	1	no	—
71	Dic₇₃	—	8	1	no	—
72	C₂×C₇₃	—	4	2	no	—
73	C₂×C₇₃	#461	1	1	yes	characteristic
74	C₂×C₇₃	#465	1	1	yes	characteristic
75	C₂×C₇₃	#471	1	1	yes	characteristic
76	#1671	—	73	1	no	maximal, Sylow
77	C₇₃	#1671	1	1	yes	characteristic, complemented, Sylow
78	#498	—	73	1	no	—
79	#476	—	73	1	no	—
80	#461	—	73	1	no	—
81	C₈×D₄	D₇₃	1	1	yes	characteristic
82	#334	—	73	1	no	—
83	#333	—	73	1	no	—
84	#332	—	73	1	no	—
85	C₂×SD₁₆	—	146	1	no	—
86	C₂²×C₈	—	2	1	no	—
87	C₄⋊Q₈	—	73	1	no	—
88	C₄⋊Q₈	—	73	1	no	—
89	C₄×D₄	C₂×D₇₃	1	1	yes	characteristic
90	#106	—	146	1	no	—
91	#106	—	146	1	no	—
92	#106	—	146	1	no	—
93	C₄⋊C₈	—	73	1	no	—
94	C₄⋊C₈	—	73	1	no	—
95	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
96	#103	—	146	1	no	—
97	#102	—	146	1	no	—
98	C₂²⋊C₈	—	2	1	no	—
99	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
100	C₂×Q₈	—	146	1	no	—
101	C₂×Q₈	—	146	1	no	—
102	C₂×D₄	D₂₉₂	1	1	yes	characteristic
103	C₂²×C₄	—	2	1	no	—
104	SD₁₆	—	292	2	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	146	1	no	—
108	C₂×C₈	—	146	1	no	—
109	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
110	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
111	C₄⋊C₄	—	146	1	no	—
112	C₄⋊C₄	—	146	1	no	—
113	C₄⋊C₄	—	146	1	no	—
114	C₄⋊C₄	—	146	1	no	—
115	C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
116	C₂²⋊C₄	—	2	1	no	—
117	C₄²	C₂²×D₇₃	1	1	yes	characteristic
118	C₂³	—	2	1	no	—
119	Q₈	—	292	1	no	—
120	Q₈	—	292	1	no	—
121	Q₈	—	584	1	no	—
122	D₄	—	2	1	no	—
123	D₄	#11924017	1	2	yes	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	292	1	no	—
128	C₂×C₄	—	292	1	no	—
129	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
130	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
131	C₂×C₄	#11924028	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	4	1	no	—
134	C₈	—	292	1	no	—
135	C₈	—	292	1	no	—
136	C₈	#11924016	1	2	yes	complemented
137	C₂²	—	2	2	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	4	1	no	—
140	C₂²	#14556565	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	4	1	no	—
143	C₄	—	584	1	no	—
144	C₄	—	584	1	no	—
145	C₄	C₂×#11924016	1	1	yes	characteristic
146	C₄	C₂×#11924017	1	1	yes	characteristic
147	C₄	#14556613	1	1	yes	characteristic
148	C₄	#14556621	1	1	yes	characteristic
149	C₂	—	4	2	no	—
150	C₂	#17988792	1	1	yes	characteristic
151	C₂	#17988816	1	1	yes	characteristic
152	C₂	#17988850	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988985

C₂

yes

maximal, characteristic, complemented

#17988877

C₂

yes

maximal, characteristic

#17988792

C₂

yes

maximal, characteristic

#17988338

C₂

yes

maximal, characteristic, complemented

#17988335

C₂

yes

maximal, characteristic, complemented

#17988332

C₂

yes

maximal, characteristic, complemented

C₈×D₄×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924016

—

#14556591

C₂²

yes

characteristic

#14556591

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

characteristic

#14556514

—

#14556512

—

C₂²×C₈×C₇₃

—

C₄×D₄×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

#11924016

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×D₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

D₄

yes

characteristic

C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

SD₁₆

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

SD₁₆

yes

complemented

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#121

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₄×C₇₃

C₂×SD₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#498

—

#476

—

#461

—

C₈×D₄

D₇₃

yes

—

—

—

C₂×SD₁₆

—

146

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×D₄

C₂×D₇₃

yes

characteristic

#106

—

146

—

#106

—

146

—

#106

—

146

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₃

yes

characteristic

#103

—

146

—

#102

—

146

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₇₃

yes

characteristic

100

C₂×Q₈

—

146

—

101

C₂×Q₈

—

146

—

102

C₂×D₄

D₂₉₂

yes

characteristic

103

C₂²×C₄

—

104

SD₁₆

—

292

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

146

—

108

C₂×C₈

—

146

—

109

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

110

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

111

C₄⋊C₄

—

146

—

112

C₄⋊C₄

—

146

—

113

C₄⋊C₄

—

146

—

114

C₄⋊C₄

—

146

—

115

C₄⋊C₄

D₂₉₂

yes

characteristic

116

C₂²⋊C₄

—

117

C₄²

C₂²×D₇₃

yes

characteristic

118

C₂³

—

119

Q₈

—

292

—

120

Q₈

—

292

—

121

Q₈

—

584

—

122

D₄

—

123

D₄

#11924017

yes

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

292

—

128

C₂×C₄

—

292

—

129

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

130

C₂×C₄

C₂×D₂₉₂

yes

characteristic

131

C₂×C₄

#11924028

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

292

—

135

C₈

—

292

—

136

C₈

#11924016

yes

complemented

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

#14556565

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

584

—

144

C₄

—

584

—

145

C₄

C₂×#11924016

yes

characteristic

146

C₄

C₂×#11924017

yes

characteristic

147

C₄

#14556613

yes

characteristic

148

C₄

#14556621

yes

characteristic

149

C₂

—

150

C₂

#17988792

yes

characteristic

151

C₂

#17988816

yes

characteristic

152

C₂

#17988850

yes

characteristic

	subgroup	G/H
center	C₂²	#14556565
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	C₈×D₄×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	C₈×D₄	D₇₃
hypocenter	C₇₃	#1671
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄

subgroup

G/H

center

C₂²

#14556565

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

C₈×D₄×C₇₃

C₂

socle

C₂²×C₇₃

#121

hypercenter

C₈×D₄

D₇₃

hypocenter

C₇₃

#1671

LCS

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄

N	H ≅ G/N	type
1. #17988985	C₂	semidirect
4. #17988338	C₂	semidirect
5. #17988335	C₂	semidirect
6. #17988332	C₂	semidirect
57. C₈×C₇₃	SD₁₆	semidirect
77. C₇₃	#1671	semidirect
136. C₈	#11924016	semidirect

H ≅ G/N

type

1. #17988985

C₂

semidirect

4. #17988338

C₂

semidirect

5. #17988335

C₂

semidirect

6. #17988332

C₂

semidirect

57. C₈×C₇₃

SD₁₆

semidirect

77. C₇₃

#1671

semidirect

136. C₈

#11924016

semidirect

Subgroups — #22609306

Conjugacy classes of subgroups

Complements