Group #22625273 — subgroups

Subgroups — #22625273

Order 9344.

252 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 72 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	#17989608	C₂	1	1	yes	maximal, characteristic
2	C₄×#11924037	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#10596833	C₂	1	4	yes	maximal, complemented
4	C₂×Q₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924037	—	2	1	no	—
6	C₄⋊#10596833	C₂²	1	2	yes	—
7	C₂²⋊#10596833	—	2	1	no	—
8	#14556646	—	2	1	no	—
9	#14556646	—	2	1	no	—
10	C₄×#10596833	C₂²	1	2	yes	—
11	Q₈×Dic₇₃	—	2	1	no	—
12	Q₈×Dic₇₃	—	2	1	no	—
13	Q₈×Dic₇₃	C₄	1	2	yes	—
14	Q₈×Dic₇₃	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₇₃	—	2	1	no	—
16	C₂×C₄⋊Dic₇₃	C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₇₃	—	2	1	no	—
18	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
19	C₂²×Q₈×C₇₃	C₄	1	1	yes	characteristic
20	#11924037	—	2	4	no	—
21	C₂×#10596833	—	2	2	no	—
22	C₂×#10596833	—	2	2	no	—
23	C₂²×Dic₇₃	—	2	1	no	—
24	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
25	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
26	C₄⋊Dic₇₃	—	2	1	no	—
27	C₄⋊Dic₇₃	—	2	2	no	—
28	C₄⋊Dic₇₃	—	2	2	no	—
29	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
30	C₄×Dic₇₃	—	2	2	no	—
31	C₄×Dic₇₃	—	2	2	no	—
32	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
33	C₄×Dic₇₃	D₄	1	1	yes	characteristic
34	C₄×Dic₇₃	D₄	1	1	yes	characteristic
35	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
36	C₂×Q₈×C₇₃	—	2	4	no	—
37	C₂×Q₈×C₇₃	—	2	1	no	—
38	C₂×Q₈×C₇₃	C₂×C₄	1	2	yes	—
39	C₂²×C₄×C₇₃	—	2	1	no	—
40	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	Q₈×C₇₃	—	2	4	no	—
43	Q₈×C₇₃	—	2	2	no	—
44	Q₈×C₇₃	M₁₆	1	4	yes	complemented
45	C₂×C₄×C₇₃	—	2	4	no	—
46	C₂×C₄×C₇₃	—	2	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×Dic₇₃	—	2	2	no	—
51	C₂×Dic₇₃	—	2	2	no	—
52	C₂×Dic₇₃	—	2	2	no	—
53	C₂×Dic₇₃	—	4	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	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
58	#10596833	—	2	4	no	—
59	#10596833	—	4	2	no	—
60	C₂²×C₇₃	—	2	1	no	—
61	C₂²×C₇₃	—	2	1	no	—
62	C₂²×C₇₃	#104	1	2	yes	—
63	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₄×C₇₃	—	2	4	no	—
65	C₄×C₇₃	—	2	1	no	—
66	C₄×C₇₃	C₂×M₁₆	1	2	yes	—
67	Dic₇₃	—	2	1	no	—
68	Dic₇₃	—	2	1	no	—
69	Dic₇₃	—	2	1	no	—
70	Dic₇₃	—	4	2	no	—
71	Dic₇₃	#137	1	2	yes	—
72	C₂×C₇₃	—	2	2	no	—
73	C₂×C₇₃	#407	1	1	yes	characteristic
74	C₂×C₇₃	#419	1	1	yes	characteristic
75	C₂×C₇₃	C₂×#104	1	1	yes	characteristic
76	#1491	—	73	1	no	maximal, Sylow
77	C₇₃	#1491	1	1	yes	characteristic, complemented, Sylow
78	C₂×C₄×Q₈	—	73	1	no	—
79	#432	—	73	1	no	—
80	C₄×M₁₆	—	73	1	no	—
81	Q₈⋊C₈	—	73	4	no	—
82	C₂²×Q₈	#64408	1	1	yes	characteristic
83	C₂×M₁₆	—	146	1	no	—
84	C₄×Q₈	—	73	2	no	—
85	C₄×Q₈	—	73	2	no	—
86	C₄×Q₈	—	146	1	no	—
87	C₄×Q₈	—	146	1	no	—
88	C₂×C₄⋊C₄	—	73	1	no	—
89	C₂×C₄⋊C₄	—	146	1	no	—
90	C₂×C₄²	—	73	1	no	—
91	C₂×C₄²	—	146	1	no	—
92	C₄⋊C₈	—	73	2	no	—
93	C₂²⋊C₈	—	146	1	no	—
94	#97	—	146	1	no	—
95	#97	—	146	1	no	—
96	C₄×C₈	—	73	2	no	—
97	C₂×Q₈	—	2	4	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	C₂×#64408	1	2	yes	—
100	C₂²×C₄	—	2	1	no	—
101	C₂²×C₄	—	73	1	no	—
102	C₂²×C₄	—	73	1	no	—
103	C₂²×C₄	—	146	1	no	—
104	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
105	M₁₆	—	146	4	no	—
106	C₂×C₈	—	146	2	no	—
107	C₂×C₈	—	146	2	no	—
108	C₄⋊C₄	—	73	2	no	—
109	C₄⋊C₄	—	146	2	no	—
110	C₄⋊C₄	—	146	2	no	—
111	C₄⋊C₄	—	146	1	no	—
112	C₄²	—	73	1	no	—
113	C₄²	—	73	1	no	—
114	C₄²	—	73	1	no	—
115	C₄²	—	73	1	no	—
116	C₄²	—	146	2	no	—
117	C₄²	—	146	2	no	—
118	C₂³	C₂²⋊#64408	1	1	yes	characteristic
119	Q₈	—	2	4	no	—
120	Q₈	—	2	2	no	—
121	Q₈	#11924037	1	4	yes	complemented
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	292	2	no	—
126	C₂×C₄	—	73	1	no	—
127	C₂×C₄	—	73	1	no	—
128	C₂×C₄	—	73	1	no	—
129	C₂×C₄	—	73	1	no	—
130	C₂×C₄	—	146	2	no	—
131	C₂×C₄	—	146	2	no	—
132	C₂×C₄	—	146	2	no	—
133	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
134	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
135	C₈	—	292	2	no	—
136	C₈	—	146	4	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	1	no	—
139	C₂²	#14556656	1	2	yes	—
140	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	2	1	no	—
143	C₄	—	292	2	no	—
144	C₄	—	73	2	no	—
145	C₄	—	146	1	no	—
146	C₄	—	146	1	no	—
147	C₄	—	146	1	no	—
148	C₄	C₂×#11924037	1	2	yes	—
149	C₂	—	2	2	no	—
150	C₂	#17989557	1	1	yes	characteristic
151	C₂	#17989580	1	1	yes	characteristic
152	C₂	C₂×#14556656	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989608

C₂

yes

maximal, characteristic

C₄×#11924037

C₂

yes

maximal, characteristic

Q₈⋊#10596833

C₂

yes

maximal, complemented

C₂×Q₈×Dic₇₃

C₂

yes

maximal, characteristic

C₂×#11924037

—

C₄⋊#10596833

C₂²

yes

—

—

—

—

C₂²

yes

—

Q₈×Dic₇₃

—

Q₈×Dic₇₃

—

Q₈×Dic₇₃

C₄

yes

—

Q₈×Dic₇₃

C₂²

yes

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

C₄

yes

characteristic

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

C₂²×Q₈×C₇₃

C₄

yes

—

—

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

D₄

yes

characteristic

C₂²×Dic₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

C₂³

yes

characteristic

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂³×C₇₃

C₂²⋊C₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

M₁₆

yes

complemented

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

#10596833

—

#10596833

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#104

yes

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×M₁₆

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#137

yes

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

#432

—

C₄×M₁₆

—

Q₈⋊C₈

—

C₂²×Q₈

#64408

yes

characteristic

C₂×M₁₆

—

146

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

146

—

C₄×Q₈

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

—

C₂×C₄²

—

146

—

C₄⋊C₈

—

C₂²⋊C₈

—

146

—

#97

—

146

—

#97

—

146

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×#64408

yes

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

146

—

104

C₂²×C₄

C₂×#64408

yes

characteristic

105

M₁₆

—

146

—

106

C₂×C₈

—

146

—

107

C₂×C₈

—

146

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

146

—

110

C₄⋊C₄

—

146

—

111

C₄⋊C₄

—

146

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

—

116

C₄²

—

146

—

117

C₄²

—

146

—

118

C₂³

C₂²⋊#64408

yes

characteristic

119

Q₈

—

120

Q₈

—

121

Q₈

#11924037

yes

complemented

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

292

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

146

—

131

C₂×C₄

—

146

—

132

C₂×C₄

—

146

—

133

C₂×C₄

C₂²⋊#64408

yes

characteristic

134

C₂×C₄

C₂²×#64408

yes

characteristic

135

C₈

—

292

—

136

C₈

—

146

—

137

C₂²

—

138

C₂²

—

139

C₂²

#14556656

yes

—

140

C₂²

C₂×C₂²⋊#64408

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

292

—

144

C₄

—

145

C₄

—

146

—

146

C₄

—

146

—

147

C₄

—

146

—

148

C₄

C₂×#11924037

yes

—

149

C₂

—

150

C₂

#17989557

yes

characteristic

151

C₂

#17989580

yes

characteristic

152

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	C₂²×Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₂²×Q₈	#64408
hypocenter	C₇₃	#1491
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₂²×Q₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

C₂²×Q₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₈

N	H ≅ G/N	type
3. Q₈⋊#10596833	C₂	semidirect
44. Q₈×C₇₃	M₁₆	semidirect
77. C₇₃	#1491	semidirect
121. Q₈	#11924037	semidirect

H ≅ G/N

type

3. Q₈⋊#10596833

C₂

semidirect

44. Q₈×C₇₃

M₁₆

semidirect

77. C₇₃

#1491

semidirect

121. Q₈

#11924037

semidirect

Subgroups — #22625273

Conjugacy classes of subgroups

Complements