Group #22609348 — subgroups

Subgroups — #22609348

Order 9344.

160 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 53 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	#17988934	C₂	1	1	yes	maximal, characteristic
2	#17988883	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988788	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988335	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988330	C₂	1	1	yes	maximal, characteristic
7	#334×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924017	—	2	1	no	—
9	#14556591	C₂²	1	1	yes	characteristic
10	#14556588	C₂²	1	1	yes	characteristic
11	C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
12	#14556525	—	2	1	no	—
13	C₄⋊#10596831	C₂²	1	1	yes	characteristic
14	#14556520	—	2	1	no	—
15	#14556516	—	2	1	no	—
16	#14556513	—	2	1	no	—
17	#14556513	—	2	1	no	—
18	#14556513	—	2	1	no	—
19	C₈×Dic₇₃	—	2	1	no	—
20	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₄₆	—	2	1	no	—
24	C₂×Dic₁₄₆	D₄	1	1	yes	characteristic
25	C₂×D₂₉₂	—	2	1	no	—
26	C₂×D₂₉₂	—	2	1	no	—
27	C₂×D₂₉₂	—	2	1	no	—
28	#11924017	—	4	2	no	—
29	C₂×#10596831	—	2	1	no	—
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₄⋊Dic₇₃	—	2	1	no	—
32	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
33	#11924007	—	2	1	no	—
34	C₄×Dic₇₃	—	2	1	no	—
35	C₂×C₈×C₇₃	—	2	1	no	—
36	C₂×C₈×C₇₃	—	2	1	no	—
37	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
38	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
39	C₄²×C₇₃	C₂³	1	1	yes	characteristic
40	C₂²×D₇₃	—	4	1	no	—
41	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
44	C₂×Dic₇₃	—	2	1	no	—
45	C₂×Dic₇₃	—	2	1	no	—
46	C₂×Dic₇₃	—	4	1	no	—
47	D₂₉₂	—	4	1	no	—
48	D₂₉₂	—	4	1	no	—
49	D₂₉₂	—	4	1	no	—
50	D₂₉₂	—	4	1	no	—
51	D₂₉₂	—	8	1	no	—
52	Dic₁₄₆	—	2	1	no	—
53	Dic₁₄₆	—	8	1	no	—
54	Dic₁₄₆	SD₁₆	1	2	yes	—
55	C₈×C₇₃	—	2	1	no	—
56	C₈×C₇₃	—	2	1	no	—
57	C₈×C₇₃	—	4	1	no	—
58	C₈×C₇₃	—	4	1	no	—
59	#10596831	—	4	1	no	—
60	C₂²×C₇₃	#121	1	1	yes	characteristic
61	C₂×D₇₃	—	8	1	no	—
62	C₂×D₇₃	—	8	1	no	—
63	C₂×D₇₃	—	8	1	no	—
64	C₄×C₇₃	—	2	1	no	—
65	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
66	C₄×C₇₃	#135	1	1	yes	characteristic
67	C₄×C₇₃	#135	1	1	yes	characteristic
68	C₄×C₇₃	#137	1	1	yes	characteristic
69	Dic₇₃	—	2	2	no	—
70	Dic₇₃	—	4	1	no	—
71	Dic₇₃	—	8	1	no	—
72	C₂×C₇₃	#464	1	1	yes	characteristic
73	C₂×C₇₃	#465	1	1	yes	characteristic
74	C₂×C₇₃	#472	1	1	yes	characteristic
75	D₇₃	—	16	1	no	—
76	#1678	—	73	1	no	maximal, Sylow
77	C₇₃	#1678	1	1	yes	characteristic, complemented, Sylow
78	#486	—	73	1	no	—
79	#477	—	73	1	no	—
80	#460	—	73	1	no	—
81	C₈×Q₈	—	73	1	no	—
82	#334	D₇₃	1	1	yes	characteristic, complemented
83	#333	—	73	1	no	—
84	#331	—	73	1	no	—
85	C₂×SD₁₆	—	146	1	no	—
86	C₄⋊Q₈	—	73	1	no	—
87	#127	—	73	1	no	—
88	C₄×Q₈	—	73	1	no	—
89	#107	—	146	1	no	—
90	C₄⋊C₈	—	73	1	no	—
91	C₄⋊C₈	—	146	1	no	—
92	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
93	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
94	#103	—	146	1	no	—
95	#102	—	146	1	no	—
96	#102	—	146	1	no	—
97	#102	—	146	1	no	—
98	C₄×C₈	—	146	1	no	—
99	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
100	C₂×Q₈	—	73	1	no	—
101	C₂×Q₈	—	146	1	no	—
102	C₂×D₄	—	146	1	no	—
103	C₂×D₄	—	146	1	no	—
104	C₂×D₄	—	146	1	no	—
105	SD₁₆	—	292	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	146	1	no	—
109	C₂×C₈	D₂₉₂	1	1	yes	characteristic
110	C₂×C₈	D₂₉₂	1	1	yes	characteristic
111	C₄⋊C₄	—	73	1	no	—
112	C₄⋊C₄	—	146	1	no	—
113	C₄⋊C₄	—	146	1	no	—
114	C₄⋊C₄	—	146	1	no	—
115	C₄²	—	146	1	no	—
116	C₄²	C₂²×D₇₃	1	1	yes	characteristic
117	C₂³	—	292	1	no	—
118	Q₈	—	584	1	no	—
119	Q₈	—	73	2	no	—
120	Q₈	—	146	1	no	—
121	D₄	—	292	1	no	—
122	D₄	—	292	1	no	—
123	D₄	—	292	1	no	—
124	D₄	—	292	1	no	—
125	D₄	—	584	1	no	—
126	C₂×C₄	—	292	1	no	—
127	C₂×C₄	—	146	1	no	—
128	C₂×C₄	—	146	1	no	—
129	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
130	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
131	C₂×C₄	#11924029	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	4	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	292	1	no	—
137	C₂²	—	584	1	no	—
138	C₂²	—	584	1	no	—
139	C₂²	—	584	1	no	—
140	C₂²	#14556566	1	1	yes	characteristic
141	C₄	—	2	1	no	—
142	C₄	—	292	1	no	—
143	C₄	—	584	1	no	—
144	C₄	—	146	2	no	—
145	C₄	SD₁₆×D₇₃	1	1	yes	characteristic
146	C₄	#14556612	1	1	yes	characteristic
147	C₄	#14556613	1	1	yes	characteristic
148	C₄	#14556624	1	1	yes	characteristic
149	C₂	—	1168	1	no	—
150	C₂	#17988810	1	1	yes	characteristic
151	C₂	#17988817	1	1	yes	characteristic
152	C₂	#17988858	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988934

C₂

yes

maximal, characteristic

#17988883

C₂

yes

maximal, characteristic, complemented

#17988788

C₂

yes

maximal, characteristic

C₈×Dic₁₄₆

C₂

yes

maximal, characteristic, complemented

#17988335

C₂

yes

maximal, characteristic, complemented

#17988330

C₂

yes

maximal, characteristic

#334×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924017

—

#14556591

C₂²

yes

characteristic

#14556588

C₂²

yes

characteristic

C₄×Dic₁₄₆

C₂²

yes

characteristic

#14556525

—

C₄⋊#10596831

C₂²

yes

—

—

—

—

—

C₈×Dic₇₃

—

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

D₄

yes

characteristic

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

#11924017

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

characteristic

#11924007

—

C₄×Dic₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

SD₁₆

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#121

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×SD₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1678

—

maximal, Sylow

C₇₃

#1678

yes

characteristic, complemented, Sylow

#486

—

#477

—

#460

—

C₈×Q₈

—

#334

D₇₃

yes

characteristic, complemented

#333

—

#331

—

C₂×SD₁₆

—

146

—

C₄⋊Q₈

—

#127

—

C₄×Q₈

—

#107

—

146

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

C₄⋊C₈

C₂×D₇₃

yes

characteristic

C₄⋊C₈

C₂×D₇₃

yes

characteristic

#103

—

146

—

#102

—

146

—

#102

—

146

—

#102

—

146

—

C₄×C₈

—

146

—

C₄×C₈

C₂×D₇₃

yes

characteristic

100

C₂×Q₈

—

101

C₂×Q₈

—

146

—

102

C₂×D₄

—

146

—

103

C₂×D₄

—

146

—

104

C₂×D₄

—

146

—

105

SD₁₆

—

292

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

146

—

109

C₂×C₈

D₂₉₂

yes

characteristic

110

C₂×C₈

D₂₉₂

yes

characteristic

111

C₄⋊C₄

—

112

C₄⋊C₄

—

146

—

113

C₄⋊C₄

—

146

—

114

C₄⋊C₄

—

146

—

115

C₄²

—

146

—

116

C₄²

C₂²×D₇₃

yes

characteristic

117

C₂³

—

292

—

118

Q₈

—

584

—

119

Q₈

—

120

Q₈

—

146

—

121

D₄

—

292

—

122

D₄

—

292

—

123

D₄

—

292

—

124

D₄

—

292

—

125

D₄

—

584

—

126

C₂×C₄

—

292

—

127

C₂×C₄

—

146

—

128

C₂×C₄

—

146

—

129

C₂×C₄

D₄×D₇₃

yes

characteristic

130

C₂×C₄

C₂×D₂₉₂

yes

characteristic

131

C₂×C₄

#11924029

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

292

—

137

C₂²

—

584

—

138

C₂²

—

584

—

139

C₂²

—

584

—

140

C₂²

#14556566

yes

characteristic

141

C₄

—

142

C₄

—

292

—

143

C₄

—

584

—

144

C₄

—

146

—

145

C₄

SD₁₆×D₇₃

yes

characteristic

146

C₄

#14556612

yes

characteristic

147

C₄

#14556613

yes

characteristic

148

C₄

#14556624

yes

characteristic

149

C₂

—

1168

—

150

C₂

#17988810

yes

characteristic

151

C₂

#17988817

yes

characteristic

152

C₂

#17988858

yes

characteristic

	subgroup	G/H
center	C₂²	#14556566
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#334×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#334	D₇₃
hypocenter	C₇₃	#1678
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #334

subgroup

G/H

center

C₂²

#14556566

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#334×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #334

N	H ≅ G/N	type
2. #17988883	C₂	semidirect
4. C₈×Dic₁₄₆	C₂	semidirect
5. #17988335	C₂	semidirect
7. #334×C₇₃	C₂	semidirect
77. C₇₃	#1678	semidirect
82. #334	D₇₃	semidirect

H ≅ G/N

type

2. #17988883

C₂

semidirect

4. C₈×Dic₁₄₆

C₂

semidirect

5. #17988335

C₂

semidirect

7. #334×C₇₃

C₂

semidirect

77. C₇₃

#1678

semidirect

82. #334

D₇₃

semidirect

Subgroups — #22609348

Conjugacy classes of subgroups

Complements