Group #22626116 — subgroups

Subgroups — #22626116

Order 9344.

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

normal subgroups only

This group has 284 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	#17989591	C₂	1	1	yes	maximal, characteristic
2	C₄²×#64408	C₂	1	1	yes	maximal, characteristic
3	#17989457	C₂	1	4	yes	maximal
4	#17989142	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924038	C₂²	1	2	yes	—
6	C₂×C₄×#64408	—	2	1	no	—
7	C₂×C₄×#64408	—	2	1	no	—
8	C₂×C₄×#64408	C₂²	1	2	yes	—
9	C₄⋊#10596833	—	2	1	no	—
10	C₄⋊#10596833	—	2	1	no	—
11	#14556591	C₄	1	1	yes	characteristic, complemented
12	#14556582	C₄	1	2	yes	complemented
13	#14556571	—	2	1	no	—
14	#14556567	—	2	1	no	—
15	Q₈×Dic₇₃	—	2	1	no	—
16	#14556556	—	2	1	no	—
17	#14556549	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	#11924038	—	2	4	no	—
22	C₂×#10596833	—	2	2	no	—
23	C₄×#64408	—	2	4	no	—
24	C₄×#64408	—	2	4	no	—
25	C₄×#64408	—	2	2	no	—
26	C₄×#64408	Q₈	1	4	yes	complemented
27	C₂×Dic₁₄₆	—	2	1	no	—
28	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
29	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
30	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₇₃	—	2	1	no	—
32	C₄⋊Dic₇₃	—	2	2	no	—
33	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
34	#11924007	—	2	2	no	—
35	#11924005	—	2	2	no	—
36	C₄×Dic₇₃	—	2	2	no	—
37	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₇₃	D₄	1	1	yes	characteristic
39	C₄×Dic₇₃	D₄	1	1	yes	characteristic
40	C₂×Q₈×C₇₃	—	2	1	no	—
41	C₄⋊C₄×C₇₃	—	2	1	no	—
42	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
43	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂×#64408	—	2	4	no	—
46	C₂×#64408	—	2	4	no	—
47	C₂×#64408	—	2	4	no	—
48	Q₈×C₇₃	—	4	2	no	—
49	C₂×C₄×C₇₃	—	2	2	no	—
50	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×Dic₇₃	—	2	2	no	—
54	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
57	C₄×D₇₃	—	2	2	no	—
58	C₄×D₇₃	—	2	2	no	—
59	C₄×D₇₃	C₂×Q₈	1	2	yes	—
60	C₄×D₇₃	#41	1	2	yes	—
61	Dic₁₄₆	—	4	2	no	—
62	#10596833	—	4	2	no	—
63	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×D₇₃	—	2	1	no	—
65	C₂×D₇₃	—	2	1	no	—
66	C₂×D₇₃	C₂²⋊Q₈	1	2	yes	—
67	#64408	—	2	8	no	—
68	C₄×C₇₃	—	4	2	no	—
69	C₄×C₇₃	#104	1	2	yes	—
70	C₄×C₇₃	#104	1	2	yes	—
71	C₄×C₇₃	C₄×Q₈	1	2	yes	—
72	Dic₇₃	—	2	1	no	—
73	Dic₇₃	—	2	1	no	—
74	Dic₇₃	—	4	2	no	—
75	Dic₇₃	#124	1	1	yes	characteristic
76	Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₇₃	#391	1	1	yes	characteristic
78	C₂×C₇₃	C₂×#104	1	1	yes	characteristic
79	C₂×C₇₃	C₂×#104	1	1	yes	characteristic
80	D₇₃	—	2	2	no	—
81	#1994	—	73	1	no	maximal, Sylow
82	C₇₃	#1994	1	1	yes	characteristic, complemented, Sylow
83	#533	—	73	1	no	—
84	#423	—	73	1	no	—
85	C₄³	—	73	1	no	—
86	#339	—	73	4	no	—
87	C₂×M₁₆	—	73	2	no	—
88	C₄⋊Q₈	—	73	1	no	—
89	C₄⋊Q₈	#64408	1	1	yes	characteristic, complemented
90	#125	—	73	2	no	—
91	C₂²⋊Q₈	—	146	1	no	—
92	C₂²⋊Q₈	—	146	1	no	—
93	C₄×Q₈	—	146	1	no	—
94	C₄×Q₈	—	146	1	no	—
95	#117	—	73	2	no	—
96	C₂×C₄²	—	73	1	no	—
97	C₂×C₄²	—	73	2	no	—
98	C₂×C₄²	—	146	1	no	—
99	C₂×C₄²	—	146	1	no	—
100	C₄⋊C₈	—	146	1	no	—
101	C₄⋊C₈	—	146	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	146	1	no	—
104	C₂²×C₄	—	73	1	no	—
105	C₂²×C₄	—	73	1	no	—
106	C₂²×C₄	—	73	1	no	—
107	C₂²×C₄	—	146	2	no	—
108	M₁₆	—	146	4	no	—
109	C₂×C₈	—	146	2	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	73	2	no	—
112	C₄⋊C₄	—	146	2	no	—
113	C₄⋊C₄	—	146	2	no	—
114	C₄⋊C₄	—	146	1	no	—
115	C₄⋊C₄	C₂×#64408	1	2	yes	—
116	C₂²⋊C₄	—	146	2	no	—
117	C₄²	—	73	1	no	—
118	C₄²	—	73	1	no	—
119	C₄²	—	73	1	no	—
120	C₄²	—	73	4	no	—
121	C₄²	—	146	2	no	—
122	C₄²	—	146	2	no	—
123	C₄²	—	146	4	no	—
124	C₄²	—	146	4	no	—
125	C₄²	C₂×#64408	1	1	yes	characteristic
126	C₂³	—	73	1	no	—
127	Q₈	—	4	2	no	—
128	Q₈	—	292	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	73	1	no	—
131	C₂×C₄	—	73	2	no	—
132	C₂×C₄	—	73	2	no	—
133	C₂×C₄	—	73	1	no	—
134	C₂×C₄	—	73	1	no	—
135	C₂×C₄	—	146	2	no	—
136	C₂×C₄	—	146	4	no	—
137	C₂×C₄	—	146	4	no	—
138	C₂×C₄	—	146	4	no	—
139	C₂×C₄	—	146	2	no	—
140	C₂×C₄	—	146	2	no	—
141	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
143	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
144	C₈	—	292	2	no	—
145	C₂²	—	73	2	no	—
146	C₂²	—	146	1	no	—
147	C₂²	—	146	1	no	—
148	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
149	C₄	—	4	2	no	—
150	C₄	—	292	2	no	—
151	C₄	—	73	1	no	—
152	C₄	—	73	1	no	—
153	C₄	—	146	8	no	—
154	C₄	—	146	1	no	—
155	C₄	—	146	1	no	—
156	C₄	#14556655	1	2	yes	—
157	C₄	#14556656	1	2	yes	—
158	C₄	Q₈×#64408	1	2	yes	—
159	C₂	—	146	2	no	—
160	C₂	(C₂×#64408)⋊Q₈	1	1	yes	characteristic
161	C₂	C₂×#14556655	1	1	yes	characteristic
162	C₂	C₂×#14556656	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989591

C₂

yes

maximal, characteristic

C₄²×#64408

C₂

yes

maximal, characteristic

#17989457

C₂

yes

maximal

#17989142

C₂

yes

maximal, characteristic

C₂×#11924038

C₂²

yes

—

C₂×C₄×#64408

—

C₂×C₄×#64408

—

C₂×C₄×#64408

C₂²

yes

—

C₄⋊#10596833

—

C₄⋊#10596833

—

#14556591

C₄

yes

characteristic, complemented

#14556582

C₄

yes

complemented

#14556571

—

#14556567

—

Q₈×Dic₇₃

—

#14556556

—

#14556549

C₂²

yes

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

—

—

—

—

—

—

Q₈

yes

complemented

C₂×Dic₁₄₆

—

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

C₂³

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924007

—

#11924005

—

C₄×Dic₇₃

—

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

C₂×Q₈

yes

—

C₄×D₇₃

#41

yes

—

Dic₁₄₆

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂²⋊Q₈

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

#104

yes

—

C₄×C₇₃

#104

yes

—

C₄×C₇₃

C₄×Q₈

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#124

yes

characteristic

Dic₇₃

C₄⋊Q₈

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1994

—

maximal, Sylow

C₇₃

#1994

yes

characteristic, complemented, Sylow

#533

—

#423

—

C₄³

—

#339

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

#64408

yes

characteristic, complemented

#125

—

C₂²⋊Q₈

—

146

—

C₂²⋊Q₈

—

146

—

C₄×Q₈

—

146

—

C₄×Q₈

—

146

—

#117

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

146

—

C₂×C₄²

—

146

—

100

C₄⋊C₈

—

146

—

101

C₄⋊C₈

—

146

—

102

C₂×Q₈

—

103

C₂×Q₈

—

146

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

146

—

108

M₁₆

—

146

—

109

C₂×C₈

—

146

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

146

—

113

C₄⋊C₄

—

146

—

114

C₄⋊C₄

—

146

—

115

C₄⋊C₄

C₂×#64408

yes

—

116

C₂²⋊C₄

—

146

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

146

—

122

C₄²

—

146

—

123

C₄²

—

146

—

124

C₄²

—

146

—

125

C₄²

C₂×#64408

yes

characteristic

126

C₂³

—

127

Q₈

—

128

Q₈

—

292

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

146

—

136

C₂×C₄

—

146

—

137

C₂×C₄

—

146

—

138

C₂×C₄

—

146

—

139

C₂×C₄

—

146

—

140

C₂×C₄

—

146

—

141

C₂×C₄

C₂²⋊#64408

yes

characteristic

142

C₂×C₄

C₂²⋊#64408

yes

characteristic

143

C₂×C₄

C₂²×#64408

yes

characteristic

144

C₈

—

292

—

145

C₂²

—

146

C₂²

—

146

—

147

C₂²

—

146

—

148

C₂²

C₂×C₂²⋊#64408

yes

characteristic

149

C₄

—

150

C₄

—

292

—

151

C₄

—

152

C₄

—

153

C₄

—

146

—

154

C₄

—

146

—

155

C₄

—

146

—

156

C₄

#14556655

yes

—

157

C₄

#14556656

yes

—

158

C₄

Q₈×#64408

yes

—

159

C₂

—

146

—

160

C₂

(C₂×#64408)⋊Q₈

yes

characteristic

161

C₂

C₂×#14556655

yes

characteristic

162

C₂

C₂×#14556656

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	#1994
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
11. #14556591	C₄	semidirect
12. #14556582	C₄	semidirect
19. C₄⋊Q₈×C₇₃	C₄	semidirect
26. C₄×#64408	Q₈	semidirect
82. C₇₃	#1994	semidirect
89. C₄⋊Q₈	#64408	semidirect

H ≅ G/N

type

11. #14556591

C₄

semidirect

12. #14556582

C₄

semidirect

19. C₄⋊Q₈×C₇₃

C₄

semidirect

26. C₄×#64408

Q₈

semidirect

82. C₇₃

#1994

semidirect

89. C₄⋊Q₈

#64408

semidirect

Subgroups — #22626116

Conjugacy classes of subgroups

Complements