Group #22611031 — subgroups

Subgroups — #22611031

Order 9344.

370 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 99 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	C₂²×#11924018	C₂	1	2	yes	maximal
2	C₂×#14556515	C₂	1	2	yes	maximal
3	#17988493	C₂	1	1	yes	maximal, characteristic
4	#17988342	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₄₆	C₂²	1	1	yes	characteristic
7	C₂×#11924018	—	2	4	no	—
8	C₂×#11924018	—	2	4	no	—
9	C₂×#11924018	—	2	4	no	—
10	C₂²×#10596831	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
12	#14556515	—	2	4	no	—
13	#14556498	—	2	2	no	—
14	C₂²×Q₈×C₇₃	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₇₃	—	2	4	no	—
16	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
17	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×Dic₁₄₆	—	2	2	no	—
19	C₂×Dic₁₄₆	—	2	2	no	—
20	C₂×Dic₁₄₆	—	2	1	no	—
21	C₂×Dic₁₄₆	D₄	1	2	yes	—
22	C₂²×Dic₇₃	—	2	1	no	—
23	C₂²×Dic₇₃	—	2	1	no	—
24	#11924018	—	2	16	no	—
25	C₂×#10596831	—	2	2	no	—
26	C₂×#10596831	—	2	2	no	—
27	C₂×#10596831	D₄	1	4	yes	—
28	C₄⋊Dic₇₃	—	2	2	no	—
29	C₂×Q₈×C₇₃	—	2	2	no	—
30	C₂×Q₈×C₇₃	—	2	2	no	—
31	C₂×Q₈×C₇₃	—	2	2	no	—
32	C₂×Q₈×C₇₃	—	2	2	no	—
33	C₂×Q₈×C₇₃	—	2	2	no	—
34	C₂×Q₈×C₇₃	D₄	1	4	yes	—
35	C₂²×C₄×C₇₃	—	2	2	no	—
36	C₂²×C₄×C₇₃	D₄	1	2	yes	—
37	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	C₄⋊C₄×C₇₃	—	2	8	no	—
39	C₄²×C₇₃	—	2	2	no	—
40	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
41	Q₈×C₇₃	—	2	4	no	—
42	Q₈×C₇₃	—	2	4	no	—
43	Q₈×C₇₃	—	4	2	no	—
44	Q₈×C₇₃	—	4	2	no	—
45	C₂×C₄×C₇₃	—	2	2	no	—
46	C₂×C₄×C₇₃	—	2	2	no	—
47	C₂×C₄×C₇₃	—	2	4	no	—
48	C₂×C₄×C₇₃	—	4	2	no	—
49	C₂×C₄×C₇₃	—	4	2	no	—
50	C₂×C₄×C₇₃	Q₁₆	1	4	yes	complemented
51	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
55	C₂×Dic₇₃	—	2	2	no	—
56	C₂×Dic₇₃	—	4	1	no	—
57	C₂×Dic₇₃	—	4	1	no	—
58	C₂×Dic₇₃	—	4	1	no	—
59	C₂×Dic₇₃	—	4	2	no	—
60	Dic₁₄₆	—	2	4	no	—
61	Dic₁₄₆	—	4	2	no	—
62	#10596831	—	2	4	no	—
63	C₂²×C₇₃	#120	1	1	yes	characteristic
64	C₂²×C₇₃	#121	1	1	yes	characteristic
65	C₂²×C₇₃	#127	1	1	yes	characteristic
66	C₂²×C₇₃	C₂×Q₁₆	1	2	yes	—
67	C₂²×C₇₃	#137	1	2	yes	—
68	C₄×C₇₃	—	2	4	no	—
69	C₄×C₇₃	—	4	4	no	—
70	C₄×C₇₃	#120	1	2	yes	—
71	C₄×C₇₃	#121	1	2	yes	—
72	Dic₇₃	—	4	2	no	—
73	Dic₇₃	—	8	1	no	—
74	C₂×C₇₃	#392	1	1	yes	characteristic
75	C₂×C₇₃	#451	1	2	yes	—
76	C₂×C₇₃	#462	1	2	yes	—
77	C₂×C₇₃	#494	1	1	yes	characteristic
78	C₂×C₇₃	#497	1	1	yes	characteristic
79	#2016	—	73	1	no	maximal, Sylow
80	C₇₃	#2016	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₁₆	—	73	2	no	—
82	C₂×C₄⋊Q₈	D₇₃	1	1	yes	characteristic
83	C₂×#103	—	73	2	no	—
84	#391	—	73	1	no	—
85	#336	—	73	1	no	—
86	C₂²×Q₈	—	73	1	no	—
87	C₂²×Q₈	C₂×D₇₃	1	2	yes	—
88	C₂×Q₁₆	—	146	4	no	—
89	C₂×Q₁₆	—	146	4	no	—
90	C₂×Q₁₆	—	146	4	no	—
91	C₂²×C₈	—	73	2	no	—
92	C₄⋊Q₈	—	2	4	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	—	73	1	no	—
95	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
96	#103	—	146	4	no	—
97	#95	—	146	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	73	2	no	—
104	C₂×Q₈	—	146	2	no	—
105	C₂×Q₈	—	146	2	no	—
106	C₂×Q₈	—	146	1	no	—
107	C₂×Q₈	C₂²⋊D₇₃	1	4	yes	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	146	1	no	—
110	C₂²×C₄	—	146	1	no	—
111	C₂²×C₄	C₂²⋊D₇₃	1	2	yes	—
112	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
113	Q₁₆	—	146	16	no	—
114	C₂×C₈	—	73	4	no	—
115	C₂×C₈	—	146	2	no	—
116	C₂×C₈	—	146	2	no	—
117	C₄⋊C₄	—	2	8	no	—
118	C₄⋊C₄	—	146	2	no	—
119	C₄²	—	2	2	no	—
120	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
121	Q₈	—	2	4	no	—
122	Q₈	—	2	4	no	—
123	Q₈	—	4	2	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	292	2	no	—
126	Q₈	—	146	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	292	1	no	—
133	C₂×C₄	—	292	1	no	—
134	C₂×C₄	—	292	1	no	—
135	C₂×C₄	—	292	2	no	—
136	C₂×C₄	—	146	2	no	—
137	C₂×C₄	#11924018	1	4	yes	complemented
138	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
139	C₂×C₄	D₄×D₇₃	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
141	C₂×C₄	#11924027	1	1	yes	characteristic
142	C₈	—	146	4	no	—
143	C₂²	#14556560	1	1	yes	characteristic
144	C₂²	#14556562	1	1	yes	characteristic
145	C₂²	#14556587	1	1	yes	characteristic
146	C₂²	C₂×#11924018	1	2	yes	—
147	C₂²	#14556622	1	2	yes	—
148	C₄	—	2	4	no	—
149	C₄	—	4	4	no	—
150	C₄	—	292	2	no	—
151	C₄	—	584	1	no	—
152	C₄	#14556559	1	2	yes	—
153	C₄	#14556563	1	2	yes	—
154	C₂	#17988495	1	1	yes	characteristic
155	C₂	#17988743	1	2	yes	—
156	C₂	#17988794	1	2	yes	—
157	C₂	#17988966	1	1	yes	characteristic
158	C₂	#17988979	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11924018

C₂

yes

maximal

C₂×#14556515

C₂

yes

maximal

#17988493

C₂

yes

maximal, characteristic

#17988342

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

—

—

—

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

#14556515

—

#14556498

—

C₂²×Q₈×C₇₃

C₂²

yes

—

C₄⋊Q₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

D₄

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

—

—

—

D₄

yes

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

Q₁₆

yes

complemented

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596831

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

C₂×Q₁₆

yes

—

C₂²×C₇₃

#137

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#120

yes

—

C₄×C₇₃

#121

yes

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

#392

yes

characteristic

C₂×C₇₃

#451

yes

—

C₂×C₇₃

#462

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

D₇₃

yes

—

—

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₇₃

yes

—

C₂×Q₁₆

—

146

—

C₂×Q₁₆

—

146

—

C₂×Q₁₆

—

146

—

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

#103

—

146

—

#95

—

146

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

146

—

105

C₂×Q₈

—

146

—

106

C₂×Q₈

—

146

—

107

C₂×Q₈

C₂²⋊D₇₃

yes

—

108

C₂²×C₄

—

109

C₂²×C₄

—

146

—

110

C₂²×C₄

—

146

—

111

C₂²×C₄

C₂²⋊D₇₃

yes

—

112

C₂²×C₄

C₂²×D₇₃

yes

characteristic

113

Q₁₆

—

146

—

114

C₂×C₈

—

115

C₂×C₈

—

146

—

116

C₂×C₈

—

146

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

146

—

119

C₄²

—

120

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

121

Q₈

—

122

Q₈

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

292

—

126

Q₈

—

146

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

292

—

133

C₂×C₄

—

292

—

134

C₂×C₄

—

292

—

135

C₂×C₄

—

292

—

136

C₂×C₄

—

146

—

137

C₂×C₄

#11924018

yes

complemented

138

C₂×C₄

D₄×D₇₃

yes

characteristic

139

C₂×C₄

D₄×D₇₃

yes

—

140

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

141

C₂×C₄

#11924027

yes

characteristic

142

C₈

—

146

—

143

C₂²

#14556560

yes

characteristic

144

C₂²

#14556562

yes

characteristic

145

C₂²

#14556587

yes

characteristic

146

C₂²

C₂×#11924018

yes

—

147

C₂²

#14556622

yes

—

148

C₄

—

149

C₄

—

150

C₄

—

292

—

151

C₄

—

584

—

152

C₄

#14556559

yes

—

153

C₄

#14556563

yes

—

154

C₂

#17988495

yes

characteristic

155

C₂

#17988743

yes

—

156

C₂

#17988794

yes

—

157

C₂

#17988966

yes

characteristic

158

C₂

#17988979

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₃
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×C₄⋊Q₈×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₇₃
hypocenter	C₇₃	#2016
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₃

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×C₄⋊Q₈×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₇₃

hypocenter

C₇₃

#2016

LCS

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
50. C₂×C₄×C₇₃	Q₁₆	semidirect
80. C₇₃	#2016	semidirect
137. C₂×C₄	#11924018	semidirect

H ≅ G/N

type

50. C₂×C₄×C₇₃

Q₁₆

semidirect

80. C₇₃

#2016

semidirect

137. C₂×C₄

#11924018

semidirect

Subgroups — #22611031

Conjugacy classes of subgroups

Complements