Group #22609073 — subgroups

Subgroups — #22609073

Order 9344.

206 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 59 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	Q₈×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic
2	#17988967	C₂	1	1	yes	maximal, characteristic
3	#17988883	C₂	1	1	yes	maximal, characteristic
4	#17988882	C₂	1	1	yes	maximal, characteristic
5	#17988335	C₂	1	1	yes	maximal, characteristic
6	#17988319	C₂	1	1	yes	maximal, characteristic
7	Q₈⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₂₉₂	—	2	2	no	—
9	#14556591	—	2	1	no	—
10	#14556591	C₂²	1	1	yes	characteristic
11	#14556591	C₂²	1	1	yes	characteristic
12	#14556590	—	2	2	no	—
13	#14556590	—	2	1	no	—
14	Q₈×Dic₇₃	—	2	1	no	—
15	C₄×Dic₁₄₆	—	2	1	no	—
16	C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
17	#14556525	—	2	1	no	—
18	#14556524	—	2	1	no	—
19	C₄⋊#10596831	C₂²	1	1	yes	characteristic
20	#14556516	—	2	1	no	—
21	#14556515	—	2	1	no	—
22	C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×Dic₁₄₆	—	2	1	no	—
26	C₂×Dic₁₄₆	—	2	2	no	—
27	C₂×Dic₁₄₆	D₄	1	1	yes	characteristic
28	C₂×Dic₁₄₆	D₄	1	2	yes	—
29	Dic₂₉₂	—	4	4	no	—
30	C₂×#10596831	—	2	1	no	—
31	C₄⋊Dic₇₃	—	2	2	no	—
32	C₄⋊Dic₇₃	—	2	1	no	—
33	C₄⋊Dic₇₃	—	2	1	no	—
34	C₄⋊Dic₇₃	—	2	1	no	—
35	C₄⋊Dic₇₃	—	2	1	no	—
36	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
37	#11924007	—	2	2	no	—
38	#11924007	—	2	1	no	—
39	C₄×Dic₇₃	—	2	2	no	—
40	C₄×Dic₇₃	—	2	1	no	—
41	C₂×Q₈×C₇₃	D₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	—	2	1	no	—
43	C₂×C₈×C₇₃	—	2	1	no	—
44	C₄⋊C₄×C₇₃	—	2	1	no	—
45	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
46	C₄²×C₇₃	—	2	1	no	—
47	C₄²×C₇₃	C₂³	1	1	yes	characteristic
48	Q₈×C₇₃	—	2	1	no	—
49	Q₈×C₇₃	Q₁₆	1	2	yes	complemented
50	C₂×C₄×C₇₃	—	2	1	no	—
51	C₂×C₄×C₇₃	—	2	1	no	—
52	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₇₃	—	2	1	no	—
56	C₂×Dic₇₃	—	2	2	no	—
57	C₂×Dic₇₃	—	2	1	no	—
58	C₂×Dic₇₃	—	4	1	no	—
59	Dic₁₄₆	—	2	1	no	—
60	Dic₁₄₆	—	4	2	no	—
61	Dic₁₄₆	—	4	2	no	—
62	Dic₁₄₆	—	8	2	no	—
63	Dic₁₄₆	Q₁₆	1	2	yes	complemented
64	C₈×C₇₃	—	2	2	no	—
65	C₈×C₇₃	—	4	1	no	—
66	#10596831	—	4	1	no	—
67	C₂²×C₇₃	#120	1	1	yes	characteristic
68	C₄×C₇₃	—	2	2	no	—
69	C₄×C₇₃	—	2	1	no	—
70	C₄×C₇₃	—	4	1	no	—
71	C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
72	C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
73	C₄×C₇₃	#137	1	1	yes	characteristic
74	C₄×C₇₃	#137	1	1	yes	characteristic
75	Dic₇₃	—	2	2	no	—
76	Dic₇₃	—	4	2	no	—
77	Dic₇₃	—	4	1	no	—
78	Dic₇₃	—	8	1	no	—
79	C₂×C₇₃	#451	1	1	yes	characteristic
80	C₂×C₇₃	#451	1	1	yes	characteristic
81	C₂×C₇₃	#453	1	1	yes	characteristic
82	#1633	—	73	1	no	maximal, Sylow
83	C₇₃	#1633	1	1	yes	characteristic, complemented, Sylow
84	Q₈²	—	73	1	no	—
85	#494	—	73	1	no	—
86	#477	—	73	1	no	—
87	#477	—	73	1	no	—
88	#333	—	73	1	no	—
89	Q₈⋊C₈	—	73	1	no	—
90	Q₈⋊C₈	D₇₃	1	1	yes	characteristic
91	C₂×Q₁₆	—	146	2	no	—
92	C₄⋊Q₈	—	73	1	no	—
93	C₄⋊Q₈	—	73	1	no	—
94	C₄⋊Q₈	—	146	1	no	—
95	C₄⋊Q₈	—	146	2	no	—
96	C₄⋊Q₈	—	146	1	no	—
97	C₄×Q₈	—	73	1	no	—
98	C₄×Q₈	—	146	1	no	—
99	C₄×Q₈	—	146	1	no	—
100	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
101	#107	—	146	1	no	—
102	#107	—	146	1	no	—
103	C₄⋊C₈	—	73	1	no	—
104	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
105	#103	—	146	1	no	—
106	#103	—	146	1	no	—
107	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
108	C₂×Q₈	—	73	2	no	—
109	C₂×Q₈	—	73	1	no	—
110	C₂×Q₈	—	146	2	no	—
111	C₂×Q₈	—	146	1	no	—
112	C₂×Q₈	D₂₉₂	1	1	yes	characteristic
113	Q₁₆	—	292	4	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	146	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	73	1	no	—
119	C₄⋊C₄	—	146	2	no	—
120	C₄⋊C₄	—	146	2	no	—
121	C₄⋊C₄	—	146	1	no	—
122	C₄⋊C₄	—	146	1	no	—
123	C₄⋊C₄	—	146	1	no	—
124	C₄⋊C₄	—	146	1	no	—
125	C₄⋊C₄	—	146	1	no	—
126	C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
127	C₄²	—	2	1	no	—
128	C₄²	—	146	2	no	—
129	C₄²	—	146	1	no	—
130	C₄²	C₂²×D₇₃	1	1	yes	characteristic
131	Q₈	—	2	1	no	—
132	Q₈	—	292	2	no	—
133	Q₈	—	292	2	no	—
134	Q₈	—	584	2	no	—
135	Q₈	—	73	2	no	—
136	Q₈	—	146	1	no	—
137	Q₈	Dic₂₉₂	1	2	yes	complemented
138	C₂×C₄	—	2	1	no	—
139	C₂×C₄	—	2	1	no	—
140	C₂×C₄	—	292	1	no	—
141	C₂×C₄	—	146	1	no	—
142	C₂×C₄	—	146	2	no	—
143	C₂×C₄	—	146	1	no	—
144	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
145	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
146	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
147	C₈	—	2	2	no	—
148	C₈	—	4	1	no	—
149	C₈	—	292	1	no	—
150	C₂²	#14556561	1	1	yes	characteristic
151	C₄	—	2	2	no	—
152	C₄	—	2	1	no	—
153	C₄	—	4	1	no	—
154	C₄	—	292	2	no	—
155	C₄	—	292	1	no	—
156	C₄	—	584	1	no	—
157	C₄	—	146	2	no	—
158	C₄	Q₁₆×D₇₃	1	1	yes	characteristic
159	C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
160	C₄	#14556620	1	1	yes	characteristic
161	C₄	#14556624	1	1	yes	characteristic
162	C₂	#17988744	1	1	yes	characteristic
163	C₂	#17988745	1	1	yes	characteristic
164	C₂	#17988756	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₄₆

C₂

yes

maximal, characteristic

#17988967

C₂

yes

maximal, characteristic

#17988883

C₂

yes

maximal, characteristic

#17988882

C₂

yes

maximal, characteristic

#17988335

C₂

yes

maximal, characteristic

#17988319

C₂

yes

maximal, characteristic

Q₈⋊C₈×C₇₃

C₂

yes

maximal, characteristic

C₂×Dic₂₉₂

—

#14556591

—

#14556591

C₂²

yes

characteristic

#14556591

C₂²

yes

characteristic

#14556590

—

#14556590

—

Q₈×Dic₇₃

—

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

C₂²

yes

—

—

C₂²

yes

characteristic

#14556516

—

#14556515

—

C₄×Q₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

D₄

yes

characteristic

C₂×Dic₁₄₆

D₄

yes

—

Dic₂₉₂

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

characteristic

#11924007

—

#11924007

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂×Q₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

Q₁₆

yes

complemented

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

Q₁₆

yes

complemented

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

#120

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×Q₁₆

yes

characteristic

C₄×C₇₃

C₂×Q₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

Q₈²

—

—

—

—

—

Q₈⋊C₈

—

Q₈⋊C₈

D₇₃

yes

characteristic

C₂×Q₁₆

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

—

146

—

C₄×Q₈

—

C₄×Q₈

—

146

—

C₄×Q₈

—

146

—

100

C₄×Q₈

C₂×D₇₃

yes

characteristic

101

#107

—

146

—

102

#107

—

146

—

103

C₄⋊C₈

—

104

C₄⋊C₈

C₂×D₇₃

yes

characteristic

105

#103

—

146

—

106

#103

—

146

—

107

C₄×C₈

C₂×D₇₃

yes

characteristic

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

146

—

111

C₂×Q₈

—

146

—

112

C₂×Q₈

D₂₉₂

yes

characteristic

113

Q₁₆

—

292

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

146

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

146

—

120

C₄⋊C₄

—

146

—

121

C₄⋊C₄

—

146

—

122

C₄⋊C₄

—

146

—

123

C₄⋊C₄

—

146

—

124

C₄⋊C₄

—

146

—

125

C₄⋊C₄

—

146

—

126

C₄⋊C₄

D₂₉₂

yes

characteristic

127

C₄²

—

128

C₄²

—

146

—

129

C₄²

—

146

—

130

C₄²

C₂²×D₇₃

yes

characteristic

131

Q₈

—

132

Q₈

—

292

—

133

Q₈

—

292

—

134

Q₈

—

584

—

135

Q₈

—

136

Q₈

—

146

—

137

Q₈

Dic₂₉₂

yes

complemented

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

292

—

141

C₂×C₄

—

146

—

142

C₂×C₄

—

146

—

143

C₂×C₄

—

146

—

144

C₂×C₄

D₄×D₇₃

yes

characteristic

145

C₂×C₄

D₄×D₇₃

yes

characteristic

146

C₂×C₄

C₂×D₂₉₂

yes

characteristic

147

C₈

—

148

C₈

—

149

C₈

—

292

—

150

C₂²

#14556561

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

292

—

155

C₄

—

292

—

156

C₄

—

584

—

157

C₄

—

146

—

158

C₄

Q₁₆×D₇₃

yes

characteristic

159

C₄

C₂×Dic₂₉₂

yes

characteristic

160

C₄

#14556620

yes

characteristic

161

C₄

#14556624

yes

characteristic

162

C₂

#17988744

yes

characteristic

163

C₂

#17988745

yes

characteristic

164

C₂

#17988756

yes

characteristic

	subgroup	G/H
center	C₂²	#14556561
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	Q₈⋊C₈×C₇₃	C₂
socle	C₂²×C₇₃	#120
hypercenter	Q₈⋊C₈	D₇₃
hypocenter	C₇₃	#1633
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

subgroup

G/H

center

C₂²

#14556561

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

Q₈⋊C₈×C₇₃

C₂

socle

C₂²×C₇₃

#120

hypercenter

Q₈⋊C₈

D₇₃

hypocenter

C₇₃

#1633

LCS

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈⋊C₈

N	H ≅ G/N	type
49. Q₈×C₇₃	Q₁₆	semidirect
63. Dic₁₄₆	Q₁₆	semidirect
83. C₇₃	#1633	semidirect
137. Q₈	Dic₂₉₂	semidirect

H ≅ G/N

type

49. Q₈×C₇₃

Q₁₆

semidirect

63. Dic₁₄₆

Q₁₆

semidirect

83. C₇₃

#1633

semidirect

137. Q₈

Dic₂₉₂

semidirect

Subgroups — #22609073

Conjugacy classes of subgroups

Complements