Group #22625315 — subgroups

Subgroups — #22625315

Order 9344.

240 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 70 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	#17989606	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#64408	C₂	1	1	yes	maximal, characteristic
3	#17989441	C₂	1	4	yes	maximal
4	#17989142	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#64408	—	2	1	no	—
6	C₂×C₄×#64408	—	2	1	no	—
7	C₄⋊#10596833	—	2	1	no	—
8	#14556648	C₂²	1	2	yes	—
9	C₄×#10596833	—	2	1	no	—
10	(C₂×#64408)⋊C₄	C₂²	1	2	yes	—
11	#14556591	C₄	1	1	yes	characteristic, complemented
12	#14556582	—	2	1	no	—
13	#14556571	—	2	1	no	—
14	#14556567	C₂²	1	2	yes	—
15	Q₈×Dic₇₃	C₄	1	2	yes	complemented
16	#14556556	—	2	1	no	—
17	#14556549	—	2	1	no	—
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	C₄×#64408	—	4	2	no	—
24	C₂×Dic₁₄₆	—	2	1	no	—
25	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
26	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
27	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
28	C₄⋊Dic₇₃	—	2	1	no	—
29	C₄⋊Dic₇₃	—	2	2	no	—
30	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
31	#11924007	—	2	2	no	—
32	#11924005	—	2	2	no	—
33	C₄×Dic₇₃	—	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	C₂×C₄	1	2	yes	—
38	C₄⋊C₄×C₇₃	—	2	1	no	—
39	C₄⋊C₄×C₇₃	—	2	1	no	—
40	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
42	C₂×#64408	—	2	4	no	—
43	C₂×#64408	—	4	2	no	—
44	C₂×#64408	—	4	2	no	—
45	Q₈×C₇₃	—	4	2	no	—
46	C₂×C₄×C₇₃	—	2	2	no	—
47	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×Dic₇₃	—	2	2	no	—
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	C₄×D₇₃	—	2	2	no	—
55	C₄×D₇₃	—	4	1	no	—
56	C₄×D₇₃	SD₁₆	1	2	yes	—
57	C₄×D₇₃	Q₁₆	1	2	yes	—
58	Dic₁₄₆	—	4	2	no	—
59	#10596833	—	4	2	no	—
60	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×D₇₃	—	2	1	no	—
62	C₂×D₇₃	—	2	1	no	—
63	C₂×D₇₃	C₂×SD₁₆	1	1	yes	characteristic
64	C₂×D₇₃	C₂×Q₁₆	1	1	yes	characteristic
65	#64408	—	4	4	no	—
66	C₄×C₇₃	—	2	1	no	—
67	C₄×C₇₃	—	4	2	no	—
68	C₄×C₇₃	#103	1	2	yes	—
69	C₄×C₇₃	#104	1	2	yes	—
70	Dic₇₃	—	2	1	no	—
71	Dic₇₃	—	2	1	no	—
72	Dic₇₃	—	4	2	no	—
73	Dic₇₃	#103	1	2	yes	—
74	C₂×C₇₃	#410	1	1	yes	characteristic
75	C₂×C₇₃	C₂×#103	1	1	yes	characteristic
76	C₂×C₇₃	C₂×#104	1	1	yes	characteristic
77	D₇₃	—	2	1	no	—
78	D₇₃	—	2	1	no	—
79	#1503	—	73	1	no	maximal, Sylow
80	C₇₃	#1503	1	1	yes	characteristic, complemented, Sylow
81	#533	—	73	1	no	—
82	#431	—	73	1	no	—
83	C₄×C₄⋊C₄	—	73	1	no	—
84	#328	—	73	4	no	—
85	C₄⋊Q₈	—	73	1	no	—
86	C₄⋊Q₈	#64408	1	1	yes	characteristic, complemented
87	#125	—	146	1	no	—
88	C₂²⋊Q₈	—	73	2	no	—
89	C₂²⋊Q₈	—	146	1	no	—
90	C₄×Q₈	—	73	2	no	—
91	C₄×Q₈	—	146	1	no	—
92	#117	—	146	1	no	—
93	C₂×C₄⋊C₄	—	146	1	no	—
94	C₂×C₄²	—	73	1	no	—
95	C₂×C₄²	—	146	1	no	—
96	C₄⋊C₈	—	146	1	no	—
97	C₂²⋊C₈	—	73	2	no	—
98	C₄×C₈	—	146	1	no	—
99	#95	—	73	2	no	—
100	C₂×Q₈	—	146	1	no	—
101	C₂×Q₈	C₂×#64408	1	2	yes	—
102	C₂²×C₄	—	73	1	no	—
103	C₂²×C₄	—	73	1	no	—
104	C₂²×C₄	—	73	1	no	—
105	C₂²×C₄	—	146	2	no	—
106	C₂×C₈	—	146	2	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	1	no	—
109	C₄⋊C₄	—	73	2	no	—
110	C₄⋊C₄	—	146	2	no	—
111	C₄⋊C₄	—	146	2	no	—
112	C₄⋊C₄	—	146	4	no	—
113	C₄⋊C₄	—	146	1	no	—
114	C₂²⋊C₄	—	146	2	no	—
115	C₄²	—	292	2	no	—
116	C₄²	—	73	1	no	—
117	C₄²	—	73	1	no	—
118	C₄²	—	73	1	no	—
119	C₄²	—	146	2	no	—
120	C₄²	C₂×#64408	1	1	yes	characteristic
121	C₂³	—	73	1	no	—
122	Q₈	—	4	2	no	—
123	Q₈	—	292	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	292	1	no	—
126	C₂×C₄	—	292	2	no	—
127	C₂×C₄	—	292	2	no	—
128	C₂×C₄	—	73	1	no	—
129	C₂×C₄	—	73	2	no	—
130	C₂×C₄	—	73	2	no	—
131	C₂×C₄	—	73	1	no	—
132	C₂×C₄	—	73	1	no	—
133	C₂×C₄	—	146	4	no	—
134	C₂×C₄	—	146	2	no	—
135	C₂×C₄	—	146	2	no	—
136	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
137	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
138	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
139	C₈	—	292	2	no	—
140	C₂²	—	73	1	no	—
141	C₂²	—	73	1	no	—
142	C₂²	—	146	1	no	—
143	C₂²	—	146	1	no	—
144	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	4	2	no	—
147	C₄	—	292	4	no	—
148	C₄	—	292	2	no	—
149	C₄	—	73	2	no	—
150	C₄	—	146	1	no	—
151	C₄	—	146	1	no	—
152	C₄	#14556654	1	2	yes	—
153	C₄	#14556656	1	2	yes	—
154	C₂	—	146	1	no	—
155	C₂	—	146	1	no	—
156	C₂	#17989567	1	1	yes	characteristic
157	C₂	C₂×#14556654	1	1	yes	characteristic
158	C₂	C₂×#14556656	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989606

C₂

yes

maximal, characteristic

C₄×C₄⋊#64408

C₂

yes

maximal, characteristic

#17989441

C₂

yes

maximal

#17989142

C₂

yes

maximal, characteristic

C₂×C₄⋊#64408

—

C₂×C₄×#64408

—

C₄⋊#10596833

—

#14556648

C₂²

yes

—

C₄×#10596833

—

(C₂×#64408)⋊C₄

C₂²

yes

—

#14556591

C₄

yes

characteristic, complemented

#14556582

—

#14556571

—

#14556567

C₂²

yes

—

Q₈×Dic₇₃

C₄

yes

complemented

#14556556

—

#14556549

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

—

C₂×#10596833

—

C₄⋊#64408

—

C₄×#64408

—

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

C₂×C₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

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

SD₁₆

yes

—

C₄×D₇₃

Q₁₆

yes

—

Dic₁₄₆

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂×SD₁₆

yes

characteristic

C₂×D₇₃

C₂×Q₁₆

yes

characteristic

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#103

yes

—

C₄×C₇₃

#104

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#103

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

D₇₃

—

#1503

—

maximal, Sylow

C₇₃

#1503

yes

characteristic, complemented, Sylow

#533

—

#431

—

C₄×C₄⋊C₄

—

#328

—

C₄⋊Q₈

—

C₄⋊Q₈

#64408

yes

characteristic, complemented

#125

—

146

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

146

—

C₄×Q₈

—

C₄×Q₈

—

146

—

#117

—

146

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

—

C₂×C₄²

—

146

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

C₄×C₈

—

146

—

#95

—

100

C₂×Q₈

—

146

—

101

C₂×Q₈

C₂×#64408

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

146

—

106

C₂×C₈

—

146

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

146

—

111

C₄⋊C₄

—

146

—

112

C₄⋊C₄

—

146

—

113

C₄⋊C₄

—

146

—

114

C₂²⋊C₄

—

146

—

115

C₄²

—

292

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

146

—

120

C₄²

C₂×#64408

yes

characteristic

121

C₂³

—

122

Q₈

—

123

Q₈

—

292

—

124

C₂×C₄

—

125

C₂×C₄

—

292

—

126

C₂×C₄

—

292

—

127

C₂×C₄

—

292

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

146

—

134

C₂×C₄

—

146

—

135

C₂×C₄

—

146

—

136

C₂×C₄

C₂²⋊#64408

yes

characteristic

137

C₂×C₄

C₂²⋊#64408

yes

characteristic

138

C₂×C₄

C₂²×#64408

yes

characteristic

139

C₈

—

292

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

146

—

143

C₂²

—

146

—

144

C₂²

C₂×C₂²⋊#64408

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

292

—

148

C₄

—

292

—

149

C₄

—

150

C₄

—

146

—

151

C₄

—

146

—

152

C₄

#14556654

yes

—

153

C₄

#14556656

yes

—

154

C₂

—

146

—

155

C₂

—

146

—

156

C₂

#17989567

yes

characteristic

157

C₂

C₂×#14556654

yes

characteristic

158

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	#1503
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
11. #14556591	C₄	semidirect
15. Q₈×Dic₇₃	C₄	semidirect
19. C₄⋊Q₈×C₇₃	C₄	semidirect
80. C₇₃	#1503	semidirect
86. C₄⋊Q₈	#64408	semidirect

H ≅ G/N

type

11. #14556591

C₄

semidirect

15. Q₈×Dic₇₃

C₄

semidirect

19. C₄⋊Q₈×C₇₃

C₄

semidirect

80. C₇₃

#1503

semidirect

86. C₄⋊Q₈

#64408

semidirect

Subgroups — #22625315

Conjugacy classes of subgroups

Complements