Group #22625446 — subgroups

Subgroups — #22625446

Order 9344.

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

normal subgroups only

This group has 240 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	#17989604	C₂	1	1	yes	maximal, characteristic
2	#17989590	C₂	1	1	yes	maximal, characteristic
3	#17989446	C₂	1	4	yes	maximal, complemented
4	C₂×#14556590	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924037	—	2	1	no	—
6	#14556658	C₂²	1	2	yes	—
7	#14556658	C₂²	1	2	yes	—
8	C₄⋊#10596833	—	2	1	no	—
9	C₂²⋊#10596833	—	2	1	no	—
10	C₄×#10596833	—	2	1	no	—
11	C₂²×Dic₁₄₆	—	2	1	no	—
12	#14556590	—	2	1	no	—
13	#14556590	—	2	1	no	—
14	#14556590	C₄	1	2	yes	—
15	#14556590	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₇₃	C₄	1	1	yes	characteristic
17	C₂×#11924007	—	2	1	no	—
18	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₇₃	C₄	1	1	yes	characteristic
20	#11924037	—	4	2	no	—
21	C₂×#10596833	—	2	2	no	—
22	C₂×#10596833	—	2	2	no	—
23	C₂×Dic₁₄₆	—	2	2	no	—
24	C₂×Dic₁₄₆	—	2	2	no	—
25	C₂×Dic₁₄₆	—	4	4	no	—
26	C₂²×Dic₇₃	—	2	1	no	—
27	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
28	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
31	#11924007	—	2	2	no	—
32	#11924007	—	2	2	no	—
33	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
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	C₂²×C₄×C₇₃	—	2	1	no	—
38	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄⋊C₄×C₇₃	—	2	1	no	—
40	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
41	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₃	—	2	2	no	—
43	C₂×C₄×C₇₃	—	2	2	no	—
44	C₂×C₄×C₇₃	—	4	2	no	—
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	—	2	2	no	—
49	C₂×Dic₇₃	—	4	2	no	—
50	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Dic₇₃	SD₁₆	1	2	yes	—
53	C₂×Dic₇₃	Q₁₆	1	2	yes	—
54	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
56	Dic₁₄₆	—	4	4	no	—
57	Dic₁₄₆	—	4	4	no	—
58	#10596833	—	4	2	no	—
59	#10596833	—	4	2	no	—
60	C₂²×C₇₃	—	2	1	no	—
61	C₂²×C₇₃	—	2	1	no	—
62	C₂²×C₇₃	#103	1	2	yes	—
63	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₄×C₇₃	—	2	1	no	—
65	C₄×C₇₃	—	4	2	no	—
66	C₄×C₇₃	#101	1	2	yes	—
67	Dic₇₃	—	2	1	no	—
68	Dic₇₃	—	4	2	no	—
69	Dic₇₃	#103	1	2	yes	—
70	Dic₇₃	C₂×SD₁₆	1	1	yes	characteristic
71	Dic₇₃	C₂×Q₁₆	1	1	yes	characteristic
72	Dic₇₃	AGL(1,ℤ/8)	1	2	yes	—
73	C₂×C₇₃	—	2	2	no	—
74	C₂×C₇₃	C₂×#101	1	1	yes	characteristic
75	C₂×C₇₃	C₂×#103	1	1	yes	characteristic
76	C₂×C₇₃	#418	1	1	yes	characteristic
77	#1549	—	73	1	no	maximal, Sylow
78	C₇₃	#1549	1	1	yes	characteristic, complemented, Sylow
79	C₂×C₄⋊Q₈	—	73	1	no	—
80	#431	—	73	1	no	—
81	#423	—	73	1	no	—
82	#332	—	73	4	no	—
83	C₂²×Q₈	—	146	1	no	—
84	C₂×M₁₆	—	146	1	no	—
85	C₄⋊Q₈	—	73	2	no	—
86	C₄⋊Q₈	—	73	2	no	—
87	C₄⋊Q₈	—	146	1	no	—
88	C₄⋊Q₈	—	146	1	no	—
89	C₂×C₄⋊C₄	—	73	1	no	—
90	C₂×C₄⋊C₄	—	146	1	no	—
91	C₂×C₄⋊C₄	#64408	1	1	yes	characteristic
92	C₂×C₄²	—	73	1	no	—
93	C₄⋊C₈	—	73	2	no	—
94	C₄⋊C₈	—	73	2	no	—
95	C₄⋊C₈	—	146	1	no	—
96	C₂²⋊C₈	—	146	1	no	—
97	C₄×C₈	—	146	1	no	—
98	C₂×Q₈	—	292	4	no	—
99	C₂×Q₈	—	146	2	no	—
100	C₂×Q₈	—	146	2	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	73	1	no	—
103	C₂²×C₄	—	73	1	no	—
104	C₂²×C₄	—	146	1	no	—
105	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
106	M₁₆	—	292	2	no	—
107	C₂×C₈	—	146	2	no	—
108	C₂×C₈	—	146	2	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	73	2	no	—
111	C₄⋊C₄	—	146	2	no	—
112	C₄⋊C₄	—	146	2	no	—
113	C₄⋊C₄	—	146	1	no	—
114	C₄⋊C₄	C₂×#64408	1	2	yes	—
115	C₄²	—	73	1	no	—
116	C₄²	—	73	1	no	—
117	C₄²	—	73	1	no	—
118	C₄²	—	73	1	no	—
119	C₂³	C₂²⋊#64408	1	1	yes	characteristic
120	Q₈	—	292	4	no	—
121	Q₈	—	292	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	292	2	no	—
126	C₂×C₄	—	73	2	no	—
127	C₂×C₄	—	73	2	no	—
128	C₂×C₄	—	73	1	no	—
129	C₂×C₄	—	73	1	no	—
130	C₂×C₄	—	73	1	no	—
131	C₂×C₄	—	73	1	no	—
132	C₂×C₄	—	146	2	no	—
133	C₂×C₄	—	146	2	no	—
134	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
135	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
136	C₈	—	292	2	no	—
137	C₈	—	292	2	no	—
138	C₂²	—	2	1	no	—
139	C₂²	—	2	1	no	—
140	C₂²	#14556654	1	2	yes	—
141	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	4	2	no	—
144	C₄	—	292	2	no	—
145	C₄	—	73	1	no	—
146	C₄	—	73	2	no	—
147	C₄	—	73	1	no	—
148	C₄	—	73	2	no	—
149	C₄	—	146	1	no	—
150	C₄	#14556652	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	C₂×#14556652	1	1	yes	characteristic
153	C₂	C₂×#14556654	1	1	yes	characteristic
154	C₂	#17989579	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989604

C₂

yes

maximal, characteristic

#17989590

C₂

yes

maximal, characteristic

#17989446

C₂

yes

maximal, complemented

C₂×#14556590

C₂

yes

maximal, characteristic

C₂×#11924037

—

#14556658

C₂²

yes

—

#14556658

C₂²

yes

—

C₄⋊#10596833

—

C₂²⋊#10596833

—

C₄×#10596833

—

C₂²×Dic₁₄₆

—

#14556590

—

#14556590

—

#14556590

C₄

yes

—

#14556590

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₄

yes

characteristic

C₂×#11924007

—

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₄

yes

—

—

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

D₄

yes

characteristic

C₂²×Dic₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924007

—

#11924007

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

C₂³

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂³×C₇₃

C₂²⋊C₄

yes

characteristic

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

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

SD₁₆

yes

—

C₂×Dic₇₃

Q₁₆

yes

—

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

#10596833

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#103

yes

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#101

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#103

yes

—

Dic₇₃

C₂×SD₁₆

yes

characteristic

Dic₇₃

C₂×Q₁₆

yes

characteristic

Dic₇₃

AGL(1,ℤ/8)

yes

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#431

—

#423

—

#332

—

C₂²×Q₈

—

146

—

C₂×M₁₆

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄⋊C₄

#64408

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

146

—

C₄×C₈

—

146

—

C₂×Q₈

—

292

—

C₂×Q₈

—

146

—

100

C₂×Q₈

—

146

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

146

—

105

C₂²×C₄

C₂×#64408

yes

characteristic

106

M₁₆

—

292

—

107

C₂×C₈

—

146

—

108

C₂×C₈

—

146

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

146

—

112

C₄⋊C₄

—

146

—

113

C₄⋊C₄

—

146

—

114

C₄⋊C₄

C₂×#64408

yes

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₂³

C₂²⋊#64408

yes

characteristic

120

Q₈

—

292

—

121

Q₈

—

292

—

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

—

131

C₂×C₄

—

132

C₂×C₄

—

146

—

133

C₂×C₄

—

146

—

134

C₂×C₄

C₂²⋊#64408

yes

characteristic

135

C₂×C₄

C₂²×#64408

yes

characteristic

136

C₈

—

292

—

137

C₈

—

292

—

138

C₂²

—

139

C₂²

—

140

C₂²

#14556654

yes

—

141

C₂²

C₂×C₂²⋊#64408

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

292

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

146

—

150

C₄

#14556652

yes

—

151

C₂

—

152

C₂

C₂×#14556652

yes

characteristic

153

C₂

C₂×#14556654

yes

characteristic

154

C₂

#17989579

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₂×C₄⋊C₄×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₂×C₄⋊C₄	#64408
hypocenter	C₇₃	#1549
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₂×C₄⋊C₄×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

C₂×C₄⋊C₄

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
3. #17989446	C₂	semidirect
78. C₇₃	#1549	semidirect

H ≅ G/N

type

3. #17989446

C₂

semidirect

78. C₇₃

#1549

semidirect

Subgroups — #22625446

Conjugacy classes of subgroups

Complements