Group #22625280 — subgroups

Subgroups — #22625280

Order 9344.

232 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 74 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	C₄×#11924036	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#10596833	C₂	1	4	yes	maximal, complemented
4	#17989142	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924038	—	2	1	no	—
6	C₂×#11924036	—	2	1	no	—
7	C₄⋊#10596833	—	2	1	no	—
8	C₄⋊#10596833	C₂²	1	2	yes	—
9	C₄×#10596833	—	2	1	no	—
10	C₄×#10596833	C₂²	1	2	yes	—
11	#14556591	C₄	1	1	yes	characteristic
12	#14556582	—	2	1	no	—
13	#14556571	—	2	1	no	—
14	#14556567	C₄	1	2	yes	—
15	Q₈×Dic₇₃	C₂²	1	2	yes	—
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
20	#11924038	—	4	2	no	—
21	#11924036	—	2	4	no	—
22	C₂×#10596833	—	2	2	no	—
23	C₂×#10596833	—	2	2	no	—
24	C₂×Dic₁₄₆	—	2	1	no	—
25	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
27	C₂×C₄×D₇₃	D₄	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	D₄	1	1	yes	characteristic
35	C₄×Dic₇₃	D₄	1	1	yes	characteristic
36	C₄×Dic₇₃	C₂³	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	C₂²⋊C₄	1	1	yes	characteristic
42	Q₈×C₇₃	—	2	4	no	—
43	C₂×C₄×C₇₃	—	2	2	no	—
44	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×Dic₇₃	—	2	2	no	—
48	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
51	C₄×D₇₃	—	2	2	no	—
52	C₄×D₇₃	—	2	2	no	—
53	C₄×D₇₃	SD₁₆	1	2	yes	—
54	C₄×D₇₃	Q₁₆	1	2	yes	—
55	Dic₁₄₆	—	4	2	no	—
56	#10596833	—	2	4	no	—
57	#10596833	—	4	2	no	—
58	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×D₇₃	—	2	1	no	—
60	C₂×D₇₃	—	2	1	no	—
61	C₂×D₇₃	#103	1	2	yes	—
62	C₄×C₇₃	—	4	2	no	—
63	C₄×C₇₃	#103	1	2	yes	—
64	C₄×C₇₃	#104	1	2	yes	—
65	C₄×C₇₃	#131	1	2	yes	—
66	Dic₇₃	—	2	1	no	—
67	Dic₇₃	—	2	1	no	—
68	Dic₇₃	—	4	2	no	—
69	Dic₇₃	C₂×SD₁₆	1	1	yes	characteristic
70	Dic₇₃	C₂×Q₁₆	1	1	yes	characteristic
71	C₂×C₇₃	#408	1	1	yes	characteristic
72	C₂×C₇₃	C₂×#103	1	1	yes	characteristic
73	C₂×C₇₃	C₂×#104	1	1	yes	characteristic
74	D₇₃	—	2	2	no	—
75	#1493	—	73	1	no	maximal, Sylow
76	C₇₃	#1493	1	1	yes	characteristic, complemented, Sylow
77	#533	—	73	1	no	—
78	#423	—	73	1	no	—
79	C₂×C₄×C₈	—	73	1	no	—
80	Q₈⋊C₈	—	73	4	no	—
81	C₂×M₁₆	—	146	1	no	—
82	C₂²×C₈	—	146	1	no	—
83	C₄⋊Q₈	—	73	1	no	—
84	C₄⋊Q₈	#64408	1	1	yes	characteristic
85	#125	—	146	1	no	—
86	C₂²⋊Q₈	—	73	2	no	—
87	C₂²⋊Q₈	—	146	1	no	—
88	C₄×Q₈	—	73	2	no	—
89	C₄×Q₈	—	146	1	no	—
90	#117	—	146	1	no	—
91	C₂×C₄²	—	73	1	no	—
92	C₄⋊C₈	—	73	2	no	—
93	C₄⋊C₈	—	146	1	no	—
94	C₄×C₈	—	73	2	no	—
95	C₄×C₈	—	146	1	no	—
96	C₂×Q₈	—	146	1	no	—
97	C₂×Q₈	C₂×#64408	1	2	yes	—
98	C₂²×C₄	—	73	1	no	—
99	C₂²×C₄	—	73	1	no	—
100	C₂²×C₄	—	73	1	no	—
101	M₁₆	—	292	2	no	—
102	C₂×C₈	—	146	2	no	—
103	C₂×C₈	—	146	2	no	—
104	C₂×C₈	—	146	4	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	73	2	no	—
108	C₄⋊C₄	—	146	2	no	—
109	C₄⋊C₄	—	146	2	no	—
110	C₄⋊C₄	—	146	1	no	—
111	C₂²⋊C₄	—	146	2	no	—
112	C₄²	—	73	1	no	—
113	C₄²	—	73	1	no	—
114	C₄²	—	73	1	no	—
115	C₄²	—	146	2	no	—
116	C₄²	C₂×#64408	1	1	yes	characteristic
117	C₂³	—	73	1	no	—
118	Q₈	—	2	4	no	—
119	Q₈	—	292	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	73	1	no	—
122	C₂×C₄	—	73	2	no	—
123	C₂×C₄	—	73	2	no	—
124	C₂×C₄	—	73	1	no	—
125	C₂×C₄	—	73	1	no	—
126	C₂×C₄	—	146	2	no	—
127	C₂×C₄	—	146	2	no	—
128	C₂×C₄	—	146	2	no	—
129	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
130	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
131	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
132	C₈	—	292	2	no	—
133	C₈	—	146	4	no	—
134	C₂²	—	73	2	no	—
135	C₂²	—	146	1	no	—
136	C₂²	—	146	1	no	—
137	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
138	C₄	—	4	2	no	—
139	C₄	—	292	2	no	—
140	C₄	—	73	1	no	—
141	C₄	—	73	1	no	—
142	C₄	—	146	1	no	—
143	C₄	—	146	1	no	—
144	C₄	#14556654	1	2	yes	—
145	C₄	#14556656	1	2	yes	—
146	C₄	#14556678	1	2	yes	—
147	C₂	—	146	2	no	—
148	C₂	#17989561	1	1	yes	characteristic
149	C₂	C₂×#14556654	1	1	yes	characteristic
150	C₂	C₂×#14556656	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989591

C₂

yes

maximal, characteristic

C₄×#11924036

C₂

yes

maximal, characteristic

Q₈⋊#10596833

C₂

yes

maximal, complemented

#17989142

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₄×#10596833

—

C₄×#10596833

C₂²

yes

—

#14556591

C₄

yes

—

—

C₄

yes

—

Q₈×Dic₇₃

C₂²

yes

—

#14556556

—

#14556549

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₄

yes

—

—

—

—

C₂×Dic₁₄₆

—

C₂×C₄×D₇₃

C₂×C₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924007

—

#11924005

—

C₄×Dic₇₃

—

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

C₂³

yes

characteristic

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

C₂²⋊C₄

yes

characteristic

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

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

SD₁₆

yes

—

C₄×D₇₃

Q₁₆

yes

—

Dic₁₄₆

—

#10596833

—

#10596833

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

#103

yes

—

C₄×C₇₃

—

C₄×C₇₃

#103

yes

—

C₄×C₇₃

#104

yes

—

C₄×C₇₃

#131

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₂×SD₁₆

yes

characteristic

Dic₇₃

C₂×Q₁₆

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1493

—

maximal, Sylow

C₇₃

#1493

yes

characteristic, complemented, Sylow

#533

—

#423

—

C₂×C₄×C₈

—

Q₈⋊C₈

—

C₂×M₁₆

—

146

—

C₂²×C₈

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

#64408

yes

characteristic

#125

—

146

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

146

—

C₄×Q₈

—

C₄×Q₈

—

146

—

#117

—

146

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

C₄×C₈

—

C₄×C₈

—

146

—

C₂×Q₈

—

146

—

C₂×Q₈

C₂×#64408

yes

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

M₁₆

—

292

—

102

C₂×C₈

—

146

—

103

C₂×C₈

—

146

—

104

C₂×C₈

—

146

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

146

—

109

C₄⋊C₄

—

146

—

110

C₄⋊C₄

—

146

—

111

C₂²⋊C₄

—

146

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

146

—

116

C₄²

C₂×#64408

yes

characteristic

117

C₂³

—

118

Q₈

—

119

Q₈

—

292

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

146

—

127

C₂×C₄

—

146

—

128

C₂×C₄

—

146

—

129

C₂×C₄

C₂²⋊#64408

yes

characteristic

130

C₂×C₄

C₂²⋊#64408

yes

characteristic

131

C₂×C₄

C₂²×#64408

yes

characteristic

132

C₈

—

292

—

133

C₈

—

146

—

134

C₂²

—

135

C₂²

—

146

—

136

C₂²

—

146

—

137

C₂²

C₂×C₂²⋊#64408

yes

characteristic

138

C₄

—

139

C₄

—

292

—

140

C₄

—

141

C₄

—

142

C₄

—

146

—

143

C₄

—

146

—

144

C₄

#14556654

yes

—

145

C₄

#14556656

yes

—

146

C₄

#14556678

yes

—

147

C₂

—

146

—

148

C₂

#17989561

yes

characteristic

149

C₂

C₂×#14556654

yes

characteristic

150

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	#1493
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
3. Q₈⋊#10596833	C₂	semidirect
76. C₇₃	#1493	semidirect

H ≅ G/N

type

3. Q₈⋊#10596833

C₂

semidirect

76. C₇₃

#1493

semidirect

Subgroups — #22625280

Conjugacy classes of subgroups

Complements