Group #22626115 — subgroups

Subgroups — #22626115

Order 9344.

276 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 78 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	#17989546	C₂	1	1	yes	maximal, characteristic
2	#17989531	C₂	1	1	yes	maximal, characteristic
3	#17989460	C₂	1	4	yes	maximal
4	C₄⋊Q₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924036	C₂²	1	2	yes	—
6	C₂×C₄⋊#64408	C₂²	1	2	yes	—
7	#14556646	—	2	1	no	—
8	#14556646	—	2	1	no	—
9	(C₂×#64408)⋊C₄	—	2	1	no	—
10	(C₂×#64408)⋊C₄	—	2	1	no	—
11	C₂×Q₈×D₇₃	—	2	1	no	—
12	#14556591	C₄	1	1	yes	characteristic, complemented
13	#14556590	—	2	1	no	—
14	#14556590	C₄	1	2	yes	complemented
15	#14556589	—	2	1	no	—
16	C₄⋊C₄×D₇₃	—	2	1	no	—
17	C₄⋊C₄×D₇₃	C₂²	1	2	yes	—
18	C₄²×D₇₃	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₇₃	C₄	1	1	yes	characteristic, complemented
20	C₂²×#64408	—	2	2	no	—
21	#11924036	—	2	2	no	—
22	#11924036	Q₈	1	4	yes	complemented
23	C₂×#10596833	—	2	2	no	—
24	C₄⋊#64408	—	2	4	no	—
25	C₂×Dic₁₄₆	—	2	1	no	—
26	C₂×Dic₁₄₆	—	2	2	no	—
27	Q₈×D₇₃	—	4	4	no	—
28	C₂×C₄×D₇₃	—	2	2	no	—
29	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
30	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
31	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
32	C₄⋊Dic₇₃	—	2	1	no	—
33	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
34	#11924007	—	2	2	no	—
35	#11924007	—	2	2	no	—
36	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₇₃	D₄	1	1	yes	characteristic
38	C₄×Dic₇₃	D₄	1	1	yes	characteristic
39	C₂×Q₈×C₇₃	—	2	1	no	—
40	C₄⋊C₄×C₇₃	—	2	1	no	—
41	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
42	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
44	C₂×#64408	—	4	2	no	—
45	C₂×#64408	—	4	2	no	—
46	C₂×#64408	—	4	2	no	—
47	Q₈×C₇₃	—	4	2	no	—
48	C₂×C₄×C₇₃	—	2	2	no	—
49	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×Dic₇₃	—	2	2	no	—
53	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
56	C₄×D₇₃	—	2	2	no	—
57	C₄×D₇₃	—	2	2	no	—
58	C₄×D₇₃	—	4	4	no	—
59	C₄×D₇₃	C₂×Q₈	1	2	yes	—
60	C₄×D₇₃	#41	1	2	yes	—
61	Dic₁₄₆	—	4	4	no	—
62	Dic₁₄₆	—	4	2	no	—
63	#10596833	—	2	4	no	—
64	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×D₇₃	#124	1	1	yes	characteristic
66	C₂×D₇₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×D₇₃	AGL(1,ℤ/8)	1	2	yes	—
68	C₂×D₇₃	#137	1	2	yes	—
69	#64408	—	8	2	no	—
70	C₄×C₇₃	—	2	1	no	—
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	4	2	no	—
73	C₄×C₇₃	C₄×Q₈	1	2	yes	—
74	Dic₇₃	—	2	1	no	—
75	Dic₇₃	—	2	1	no	—
76	Dic₇₃	—	4	2	no	—
77	Dic₇₃	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₇₃	#391	1	1	yes	characteristic
79	C₂×C₇₃	#418	1	1	yes	characteristic
80	C₂×C₇₃	#419	1	1	yes	characteristic
81	D₇₃	#489	1	2	yes	—
82	D₇₃	#501	1	2	yes	—
83	#1993	—	73	1	no	maximal, Sylow
84	C₇₃	#1993	1	1	yes	characteristic, complemented, Sylow
85	C₂×C₄⋊Q₈	—	73	1	no	—
86	C₂×#97	—	73	1	no	—
87	#382	—	73	1	no	—
88	#340	—	73	4	no	—
89	C₂²×Q₈	—	146	1	no	—
90	C₂²×C₈	—	73	2	no	—
91	C₄⋊Q₈	—	73	2	no	—
92	C₄⋊Q₈	—	73	1	no	—
93	C₄⋊Q₈	—	146	1	no	—
94	C₄⋊Q₈	—	146	1	no	—
95	C₄⋊Q₈	#64408	1	1	yes	characteristic, complemented
96	C₂×C₄⋊C₄	—	73	2	no	—
97	C₂×C₄⋊C₄	—	73	2	no	—
98	C₂×C₄⋊C₄	—	146	1	no	—
99	C₂×C₄²	—	73	1	no	—
100	#97	—	146	1	no	—
101	#97	—	146	1	no	—
102	#95	—	146	1	no	—
103	#95	—	146	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	292	4	no	—
106	C₂×Q₈	—	146	2	no	—
107	C₂×Q₈	—	146	1	no	—
108	C₂²×C₄	—	73	1	no	—
109	C₂²×C₄	—	73	1	no	—
110	C₂²×C₄	—	73	1	no	—
111	C₂²×C₄	—	146	2	no	—
112	C₂²×C₄	—	146	2	no	—
113	C₂×C₈	—	73	4	no	—
114	C₂×C₈	—	146	2	no	—
115	C₂×C₈	—	146	2	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	73	2	no	—
118	C₄⋊C₄	—	146	2	no	—
119	C₄⋊C₄	—	146	2	no	—
120	C₄⋊C₄	—	146	4	no	—
121	C₄⋊C₄	—	146	1	no	—
122	C₄⋊C₄	C₂×#64408	1	2	yes	—
123	C₄²	—	73	1	no	—
124	C₄²	—	73	1	no	—
125	C₄²	—	73	1	no	—
126	C₄²	C₂×#64408	1	1	yes	characteristic
127	C₂³	—	73	1	no	—
128	Q₈	—	4	2	no	—
129	Q₈	—	292	2	no	—
130	Q₈	—	292	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	292	4	no	—
133	C₂×C₄	—	292	2	no	—
134	C₂×C₄	—	292	2	no	—
135	C₂×C₄	—	292	2	no	—
136	C₂×C₄	—	73	1	no	—
137	C₂×C₄	—	73	2	no	—
138	C₂×C₄	—	73	2	no	—
139	C₂×C₄	—	73	1	no	—
140	C₂×C₄	—	73	1	no	—
141	C₂×C₄	—	146	2	no	—
142	C₂×C₄	—	146	2	no	—
143	C₂×C₄	—	146	2	no	—
144	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
145	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
146	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
147	C₈	—	146	4	no	—
148	C₂²	—	73	1	no	—
149	C₂²	—	73	1	no	—
150	C₂²	—	73	2	no	—
151	C₂²	—	73	2	no	—
152	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	2	no	—
156	C₄	—	292	2	no	—
157	C₄	—	584	2	no	—
158	C₄	—	73	2	no	—
159	C₄	—	146	1	no	—
160	C₄	—	146	1	no	—
161	C₄	Q₈×#64408	1	2	yes	—
162	C₂	—	73	2	no	—
163	C₂	—	73	2	no	—
164	C₂	(C₂×#64408)⋊Q₈	1	1	yes	characteristic
165	C₂	#17989579	1	1	yes	characteristic
166	C₂	#17989580	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989546

C₂

yes

maximal, characteristic

#17989531

C₂

yes

maximal, characteristic

#17989460

C₂

yes

maximal

C₄⋊Q₈×D₇₃

C₂

yes

maximal, characteristic

C₂×#11924036

C₂²

yes

—

C₂×C₄⋊#64408

C₂²

yes

—

—

—

—

—

C₂×Q₈×D₇₃

—

#14556591

C₄

yes

characteristic, complemented

#14556590

—

#14556590

C₄

yes

complemented

#14556589

—

C₄⋊C₄×D₇₃

—

C₄⋊C₄×D₇₃

C₂²

yes

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

—

#11924036

—

#11924036

Q₈

yes

complemented

C₂×#10596833

—

C₄⋊#64408

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

Q₈×D₇₃

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

C₂³

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924007

—

#11924007

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₂×Q₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

C₂×D₄

yes

—

—

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×Q₈

yes

—

C₄×D₇₃

#41

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

#124

yes

characteristic

C₂×D₇₃

C₄⋊Q₈

yes

characteristic

C₂×D₇₃

AGL(1,ℤ/8)

yes

—

C₂×D₇₃

#137

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×Q₈

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₂²⋊Q₈

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

yes

—

D₇₃

#501

yes

—

#1993

—

maximal, Sylow

C₇₃

#1993

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×#97

—

#382

—

#340

—

C₂²×Q₈

—

146

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

#64408

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

—

100

#97

—

146

—

101

#97

—

146

—

102

#95

—

146

—

103

#95

—

146

—

104

C₂×Q₈

—

105

C₂×Q₈

—

292

—

106

C₂×Q₈

—

146

—

107

C₂×Q₈

—

146

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

146

—

112

C₂²×C₄

—

146

—

113

C₂×C₈

—

114

C₂×C₈

—

146

—

115

C₂×C₈

—

146

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

146

—

119

C₄⋊C₄

—

146

—

120

C₄⋊C₄

—

146

—

121

C₄⋊C₄

—

146

—

122

C₄⋊C₄

C₂×#64408

yes

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

C₂×#64408

yes

characteristic

127

C₂³

—

128

Q₈

—

129

Q₈

—

292

—

130

Q₈

—

292

—

131

C₂×C₄

—

132

C₂×C₄

—

292

—

133

C₂×C₄

—

292

—

134

C₂×C₄

—

292

—

135

C₂×C₄

—

292

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

146

—

142

C₂×C₄

—

146

—

143

C₂×C₄

—

146

—

144

C₂×C₄

C₂²⋊#64408

yes

characteristic

145

C₂×C₄

C₂²⋊#64408

yes

characteristic

146

C₂×C₄

C₂²×#64408

yes

characteristic

147

C₈

—

146

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×C₂²⋊#64408

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

292

—

157

C₄

—

584

—

158

C₄

—

159

C₄

—

146

—

160

C₄

—

146

—

161

C₄

Q₈×#64408

yes

—

162

C₂

—

163

C₂

—

164

C₂

(C₂×#64408)⋊Q₈

yes

characteristic

165

C₂

#17989579

yes

characteristic

166

C₂

#17989580

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₄⋊Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#64408
hypocenter	C₇₃	#1993
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₄⋊Q₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

C₄⋊Q₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
12. #14556591	C₄	semidirect
14. #14556590	C₄	semidirect
19. C₄⋊Q₈×C₇₃	C₄	semidirect
22. #11924036	Q₈	semidirect
84. C₇₃	#1993	semidirect
95. C₄⋊Q₈	#64408	semidirect

H ≅ G/N

type

12. #14556591

C₄

semidirect

14. #14556590

C₄

semidirect

19. C₄⋊Q₈×C₇₃

C₄

semidirect

22. #11924036

Q₈

semidirect

84. C₇₃

#1993

semidirect

95. C₄⋊Q₈

#64408

semidirect

Subgroups — #22626115

Conjugacy classes of subgroups

Complements