Group #22625322 — subgroups

Subgroups — #22625322

Order 9344.

272 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 60 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	#17989609	C₂	1	1	yes	maximal, characteristic, complemented
2	#17989523	C₂	1	1	yes	maximal, characteristic, complemented
3	#17989439	C₂	1	4	yes	maximal, complemented
4	#17989138	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×#64408	—	2	1	no	—
6	C₄⋊#10596833	—	2	1	no	—
7	#14556648	C₂²	1	2	yes	—
8	#14556646	—	2	1	no	—
9	(C₂×#64408)⋊C₄	—	2	1	no	—
10	(C₂×#64408)⋊C₄	C₂²	1	2	yes	—
11	C₂×#11924029	—	2	1	no	—
12	#14556588	C₄	1	1	yes	characteristic, complemented
13	#14556580	—	2	1	no	—
14	#14556566	—	2	1	no	—
15	#14556566	C₂²	1	2	yes	—
16	#14556553	—	2	1	no	—
17	#14556553	C₄	1	2	yes	complemented
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	C₂×#10596833	—	2	2	no	—
22	C₄×#64408	—	2	4	no	—
23	#11924029	—	4	4	no	—
24	C₂×D₂₉₂	—	2	2	no	—
25	C₂×D₂₉₂	—	2	1	no	—
26	C₂×D₂₉₂	—	2	1	no	—
27	C₂×D₂₉₂	C₂×C₄	1	2	yes	—
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	#11924005	—	2	2	no	—
33	#11924005	—	2	2	no	—
34	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄×Dic₇₃	D₄	1	1	yes	characteristic
36	C₄×Dic₇₃	D₄	1	1	yes	characteristic
37	C₂×Q₈×C₇₃	—	2	1	no	—
38	C₄⋊C₄×C₇₃	—	2	1	no	—
39	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
40	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×D₇₃	—	2	2	no	—
42	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
43	C₂×#64408	—	2	4	no	—
44	C₂×#64408	—	4	2	no	—
45	C₂×#64408	—	4	2	no	—
46	Q₈×C₇₃	—	4	2	no	—
47	C₂×C₄×C₇₃	—	2	2	no	—
48	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
54	D₂₉₂	—	4	2	no	—
55	D₂₉₂	—	4	2	no	—
56	D₂₉₂	—	4	4	no	—
57	D₂₉₂	—	4	2	no	—
58	C₄×D₇₃	—	2	2	no	—
59	C₄×D₇₃	—	2	2	no	—
60	C₄×D₇₃	—	4	1	no	—
61	C₄×D₇₃	—	4	4	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	—	4	2	no	—
67	C₂×D₇₃	—	4	2	no	—
68	C₂×D₇₃	—	4	2	no	—
69	C₂×D₇₃	AGL(1,ℤ/8)	1	2	yes	—
70	#64408	—	4	4	no	—
71	C₄×C₇₃	—	2	1	no	—
72	C₄×C₇₃	—	2	1	no	—
73	C₄×C₇₃	—	4	2	no	—
74	C₄×C₇₃	#104	1	2	yes	—
75	Dic₇₃	—	2	1	no	—
76	Dic₇₃	—	2	1	no	—
77	Dic₇₃	—	2	1	no	—
78	C₂×C₇₃	#410	1	1	yes	characteristic
79	C₂×C₇₃	#418	1	1	yes	characteristic
80	C₂×C₇₃	C₂×#104	1	1	yes	characteristic
81	D₇₃	—	2	2	no	—
82	D₇₃	—	8	2	no	—
83	#1505	—	73	1	no	maximal, Sylow
84	C₇₃	#1505	1	1	yes	characteristic, complemented, Sylow
85	#532	—	73	1	no	—
86	#432	—	73	1	no	—
87	#376	—	73	1	no	—
88	#327	—	73	4	no	—
89	C₂×#41	—	146	1	no	—
90	C₄⋊Q₈	#64408	1	1	yes	characteristic, complemented
91	#127	—	73	1	no	—
92	#124	—	146	1	no	—
93	#121	—	73	2	no	—
94	#121	—	146	1	no	—
95	C₄×D₄	—	73	2	no	—
96	C₄×D₄	—	146	1	no	—
97	C₂×C₄²	—	73	1	no	—
98	C₂×C₄²	—	146	1	no	—
99	C₄⋊C₈	—	146	1	no	—
100	C₂²⋊C₈	—	73	2	no	—
101	#97	—	146	1	no	—
102	#95	—	73	2	no	—
103	#95	—	146	1	no	—
104	#41	—	292	4	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×D₄	—	73	2	no	—
107	C₂×D₄	—	146	2	no	—
108	C₂×D₄	—	146	1	no	—
109	C₂×D₄	—	146	1	no	—
110	C₂²×C₄	—	73	1	no	—
111	C₂²×C₄	—	73	1	no	—
112	C₂²×C₄	—	73	1	no	—
113	C₂²×C₄	—	146	2	no	—
114	C₂²×C₄	—	146	2	no	—
115	C₂×C₈	—	146	2	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	C₂×#64408	1	2	yes	—
118	C₂²⋊C₄	—	146	2	no	—
119	C₂²⋊C₄	—	146	2	no	—
120	C₄²	—	73	1	no	—
121	C₄²	—	73	1	no	—
122	C₄²	—	73	1	no	—
123	C₄²	—	146	4	no	—
124	C₄²	C₂×#64408	1	1	yes	characteristic
125	C₂³	—	73	1	no	—
126	C₂³	—	146	2	no	—
127	Q₈	—	4	2	no	—
128	D₄	—	292	2	no	—
129	D₄	—	292	2	no	—
130	D₄	—	292	4	no	—
131	D₄	—	292	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	292	1	no	—
134	C₂×C₄	—	292	4	no	—
135	C₂×C₄	—	292	2	no	—
136	C₂×C₄	—	292	2	no	—
137	C₂×C₄	—	73	1	no	—
138	C₂×C₄	—	73	1	no	—
139	C₂×C₄	—	73	1	no	—
140	C₂×C₄	—	146	2	no	—
141	C₂×C₄	—	146	4	no	—
142	C₂×C₄	—	146	2	no	—
143	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
144	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
145	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
146	C₈	—	292	2	no	—
147	C₂²	—	292	2	no	—
148	C₂²	—	292	2	no	—
149	C₂²	—	292	2	no	—
150	C₂²	—	73	2	no	—
151	C₂²	—	146	1	no	—
152	C₂²	—	146	1	no	—
153	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	4	2	no	—
157	C₄	—	292	4	no	—
158	C₄	—	146	1	no	—
159	C₄	—	146	1	no	—
160	C₄	—	146	1	no	—
161	C₄	#14556656	1	2	yes	—
162	C₂	—	584	2	no	—
163	C₂	—	146	2	no	—
164	C₂	#17989568	1	1	yes	characteristic
165	C₂	#17989579	1	1	yes	characteristic
166	C₂	C₂×#14556656	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989609

C₂

yes

maximal, characteristic, complemented

#17989523

C₂

yes

maximal, characteristic, complemented

#17989439

C₂

yes

maximal, complemented

#17989138

C₂

yes

maximal, characteristic

C₂×C₄×#64408

—

C₄⋊#10596833

—

#14556648

C₂²

yes

—

#14556646

—

(C₂×#64408)⋊C₄

—

(C₂×#64408)⋊C₄

C₂²

yes

—

C₂×#11924029

—

#14556588

C₄

yes

characteristic, complemented

#14556580

—

#14556566

—

#14556566

C₂²

yes

—

#14556553

—

#14556553

C₄

yes

complemented

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

—

C₂×#10596833

—

C₄×#64408

—

#11924029

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

C₂×C₄

yes

—

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

C₂³

yes

characteristic

#11924005

—

#11924005

—

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

—

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

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

AGL(1,ℤ/8)

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#104

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

D₇₃

—

#1505

—

maximal, Sylow

C₇₃

#1505

yes

characteristic, complemented, Sylow

—

—

—

—

C₂×#41

—

146

—

C₄⋊Q₈

#64408

yes

characteristic, complemented

#127

—

#124

—

146

—

#121

—

#121

—

146

—

C₄×D₄

—

C₄×D₄

—

146

—

C₂×C₄²

—

C₂×C₄²

—

146

—

C₄⋊C₈

—

146

—

100

C₂²⋊C₈

—

101

#97

—

146

—

102

#95

—

103

#95

—

146

—

104

#41

—

292

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

146

—

108

C₂×D₄

—

146

—

109

C₂×D₄

—

146

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

146

—

114

C₂²×C₄

—

146

—

115

C₂×C₈

—

146

—

116

C₄⋊C₄

—

117

C₄⋊C₄

C₂×#64408

yes

—

118

C₂²⋊C₄

—

146

—

119

C₂²⋊C₄

—

146

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

146

—

124

C₄²

C₂×#64408

yes

characteristic

125

C₂³

—

126

C₂³

—

146

—

127

Q₈

—

128

D₄

—

292

—

129

D₄

—

292

—

130

D₄

—

292

—

131

D₄

—

292

—

132

C₂×C₄

—

133

C₂×C₄

—

292

—

134

C₂×C₄

—

292

—

135

C₂×C₄

—

292

—

136

C₂×C₄

—

292

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

146

—

141

C₂×C₄

—

146

—

142

C₂×C₄

—

146

—

143

C₂×C₄

C₂²⋊#64408

yes

characteristic

144

C₂×C₄

C₂²⋊#64408

yes

characteristic

145

C₂×C₄

C₂²×#64408

yes

characteristic

146

C₈

—

292

—

147

C₂²

—

292

—

148

C₂²

—

292

—

149

C₂²

—

292

—

150

C₂²

—

151

C₂²

—

146

—

152

C₂²

—

146

—

153

C₂²

C₂×C₂²⋊#64408

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

292

—

158

C₄

—

146

—

159

C₄

—

146

—

160

C₄

—

146

—

161

C₄

#14556656

yes

—

162

C₂

—

584

—

163

C₂

—

146

—

164

C₂

#17989568

yes

characteristic

165

C₂

#17989579

yes

characteristic

166

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	#1505
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×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 ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #17989609	C₂	semidirect
2. #17989523	C₂	semidirect
3. #17989439	C₂	semidirect
12. #14556588	C₄	semidirect
17. #14556553	C₄	semidirect
19. C₄⋊Q₈×C₇₃	C₄	semidirect
84. C₇₃	#1505	semidirect
90. C₄⋊Q₈	#64408	semidirect

H ≅ G/N

type

1. #17989609

C₂

semidirect

2. #17989523

C₂

semidirect

3. #17989439

C₂

semidirect

12. #14556588

C₄

semidirect

17. #14556553

C₄

semidirect

19. C₄⋊Q₈×C₇₃

C₄

semidirect

84. C₇₃

#1505

semidirect

90. C₄⋊Q₈

#64408

semidirect

Subgroups — #22625322

Conjugacy classes of subgroups

Complements