Group #22625460 — subgroups

Subgroups — #22625460

Order 9344.

228 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 62 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	#17989607	C₂	1	1	yes	maximal, characteristic
2	#17989607	C₂	1	1	yes	maximal, characteristic
3	#17989446	C₂	1	4	yes	maximal, complemented
4	C₂×#14556590	C₂	1	1	yes	maximal, characteristic
5	#14556658	C₂²	1	2	yes	—
6	#14556658	C₂²	1	2	yes	—
7	C₂²⋊#10596833	—	2	1	no	—
8	C₂²⋊#10596833	—	2	1	no	—
9	#14556646	—	2	1	no	—
10	#14556646	—	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	C₂×#10596833	—	2	2	no	—
21	C₂×#10596833	—	2	2	no	—
22	C₂×Dic₁₄₆	—	2	2	no	—
23	C₂×Dic₁₄₆	—	2	2	no	—
24	C₂×Dic₁₄₆	—	4	4	no	—
25	C₂²×Dic₇₃	—	2	1	no	—
26	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
27	C₂²×Dic₇₃	D₄	1	1	yes	characteristic
28	C₄⋊Dic₇₃	—	2	1	no	—
29	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
30	#11924007	—	2	2	no	—
31	#11924007	—	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	C₂²×C₄×C₇₃	—	2	1	no	—
37	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₄⋊C₄×C₇₃	—	2	1	no	—
39	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
40	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₃	—	2	2	no	—
42	C₂×C₄×C₇₃	—	2	2	no	—
43	C₂×C₄×C₇₃	—	4	2	no	—
44	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₇₃	—	2	2	no	—
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	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
54	Dic₁₄₆	—	4	4	no	—
55	Dic₁₄₆	—	4	4	no	—
56	#10596833	—	4	2	no	—
57	#10596833	—	4	2	no	—
58	C₂²×C₇₃	—	2	1	no	—
59	C₂²×C₇₃	—	2	1	no	—
60	C₂²×C₇₃	#101	1	2	yes	—
61	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₇₃	—	2	1	no	—
63	C₄×C₇₃	—	2	1	no	—
64	C₄×C₇₃	—	4	2	no	—
65	Dic₇₃	—	2	1	no	—
66	Dic₇₃	—	2	1	no	—
67	Dic₇₃	—	4	2	no	—
68	Dic₇₃	AGL(1,ℤ/8)	1	2	yes	—
69	Dic₇₃	#137	1	2	yes	—
70	C₂×C₇₃	—	2	2	no	—
71	C₂×C₇₃	C₂×#101	1	1	yes	characteristic
72	C₂×C₇₃	#418	1	1	yes	characteristic
73	C₂×C₇₃	#419	1	1	yes	characteristic
74	#1556	—	73	1	no	maximal, Sylow
75	C₇₃	#1556	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₄⋊Q₈	—	73	1	no	—
77	#432	—	73	1	no	—
78	#432	—	73	1	no	—
79	#332	—	73	4	no	—
80	C₂²×Q₈	—	146	1	no	—
81	C₄⋊Q₈	—	73	2	no	—
82	C₄⋊Q₈	—	73	2	no	—
83	C₄⋊Q₈	—	146	1	no	—
84	C₄⋊Q₈	—	146	1	no	—
85	C₂×C₄⋊C₄	—	73	1	no	—
86	C₂×C₄⋊C₄	—	146	1	no	—
87	C₂×C₄⋊C₄	#64408	1	1	yes	characteristic
88	C₂×C₄²	—	73	1	no	—
89	C₄⋊C₈	—	73	2	no	—
90	C₄⋊C₈	—	73	2	no	—
91	C₂²⋊C₈	—	146	1	no	—
92	C₂²⋊C₈	—	146	1	no	—
93	#97	—	146	1	no	—
94	#97	—	146	1	no	—
95	C₂×Q₈	—	292	4	no	—
96	C₂×Q₈	—	146	2	no	—
97	C₂×Q₈	—	146	2	no	—
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	—	73	1	no	—
100	C₂²×C₄	—	73	1	no	—
101	C₂²×C₄	—	146	1	no	—
102	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
103	C₂×C₈	—	146	2	no	—
104	C₂×C₈	—	146	2	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	73	2	no	—
107	C₄⋊C₄	—	146	2	no	—
108	C₄⋊C₄	—	146	2	no	—
109	C₄⋊C₄	—	146	1	no	—
110	C₄⋊C₄	C₂×#64408	1	2	yes	—
111	C₄²	—	73	1	no	—
112	C₄²	—	73	1	no	—
113	C₄²	—	73	1	no	—
114	C₄²	—	73	1	no	—
115	C₂³	C₂²⋊#64408	1	1	yes	characteristic
116	Q₈	—	292	4	no	—
117	Q₈	—	292	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	292	2	no	—
122	C₂×C₄	—	73	1	no	—
123	C₂×C₄	—	73	1	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	C₈	—	292	2	no	—
132	C₈	—	292	2	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	#14556652	1	2	yes	—
136	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	4	2	no	—
140	C₄	—	292	2	no	—
141	C₄	—	73	2	no	—
142	C₄	—	73	2	no	—
143	C₄	—	146	1	no	—
144	C₄	—	146	1	no	—
145	C₂	—	2	2	no	—
146	C₂	C₂×#14556652	1	1	yes	characteristic
147	C₂	#17989579	1	1	yes	characteristic
148	C₂	#17989580	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989607

C₂

yes

maximal, characteristic

#17989607

C₂

yes

maximal, characteristic

#17989446

C₂

yes

maximal, complemented

C₂×#14556590

C₂

yes

maximal, characteristic

#14556658

C₂²

yes

—

#14556658

C₂²

yes

—

—

—

—

—

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

characteristic

C₂×#10596833

—

C₂×#10596833

—

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

—

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

C₂×D₄

yes

characteristic

Dic₁₄₆

—

Dic₁₄₆

—

#10596833

—

#10596833

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#101

yes

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

AGL(1,ℤ/8)

yes

—

Dic₇₃

#137

yes

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#432

—

#432

—

#332

—

C₂²×Q₈

—

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

—

C₂²⋊C₈

—

146

—

C₂²⋊C₈

—

146

—

#97

—

146

—

#97

—

146

—

C₂×Q₈

—

292

—

C₂×Q₈

—

146

—

C₂×Q₈

—

146

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

146

—

102

C₂²×C₄

C₂×#64408

yes

characteristic

103

C₂×C₈

—

146

—

104

C₂×C₈

—

146

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

146

—

108

C₄⋊C₄

—

146

—

109

C₄⋊C₄

—

146

—

110

C₄⋊C₄

C₂×#64408

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₂³

C₂²⋊#64408

yes

characteristic

116

Q₈

—

292

—

117

Q₈

—

292

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

292

—

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

C₈

—

292

—

132

C₈

—

292

—

133

C₂²

—

134

C₂²

—

135

C₂²

#14556652

yes

—

136

C₂²

C₂×C₂²⋊#64408

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

292

—

141

C₄

—

142

C₄

—

143

C₄

—

146

—

144

C₄

—

146

—

145

C₂

—

146

C₂

C₂×#14556652

yes

characteristic

147

C₂

#17989579

yes

characteristic

148

C₂

#17989580

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	#1556
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
75. C₇₃	#1556	semidirect

H ≅ G/N

type

3. #17989446

C₂

semidirect

75. C₇₃

#1556

semidirect

Subgroups — #22625460

Conjugacy classes of subgroups

Complements