Group #22625205 — subgroups

Subgroups — #22625205

Order 9344.

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

normal subgroups only

This group has 204 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	#17989608	C₂	1	1	yes	maximal, characteristic
2	#17989604	C₂	1	1	yes	maximal, characteristic
3	#17989433	C₂	1	4	yes	maximal
4	C₂×#14556582	C₂	1	1	yes	maximal, characteristic
5	#14556658	—	2	1	no	—
6	C₄⋊#10596833	—	2	1	no	—
7	C₂²⋊#10596833	C₂²	1	2	yes	—
8	C₂²⋊#10596833	C₂²	1	2	yes	—
9	#14556646	—	2	1	no	—
10	C₄×#10596833	—	2	1	no	—
11	#14556582	—	2	1	no	—
12	#14556582	—	2	1	no	—
13	#14556582	—	2	1	no	—
14	#14556582	—	2	1	no	—
15	C₂×C₄⋊Dic₇₃	C₄	1	2	yes	—
16	C₂×C₄⋊Dic₇₃	C₄	1	1	yes	characteristic
17	C₂×#11924007	C₂²	1	2	yes	—
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	C₂²×Dic₇₃	C₂×C₄	1	2	yes	—
23	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
25	C₄⋊Dic₇₃	—	2	1	no	—
26	C₄⋊Dic₇₃	—	2	1	no	—
27	C₄⋊Dic₇₃	—	2	2	no	—
28	C₄⋊Dic₇₃	—	2	2	no	—
29	#11924007	—	2	2	no	—
30	#11924007	—	2	2	no	—
31	C₄×Dic₇₃	D₄	1	1	yes	characteristic
32	C₄×Dic₇₃	D₄	1	1	yes	characteristic
33	C₄×Dic₇₃	D₄	1	1	yes	characteristic
34	C₄×Dic₇₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₇₃	—	2	1	no	—
38	C₄⋊C₄×C₇₃	—	2	1	no	—
39	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₇₃	—	2	2	no	—
41	C₂×C₄×C₇₃	—	2	2	no	—
42	C₂×C₄×C₇₃	—	4	2	no	—
43	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×Dic₇₃	—	2	2	no	—
46	C₂×Dic₇₃	—	2	2	no	—
47	C₂×Dic₇₃	—	2	4	no	—
48	C₂×Dic₇₃	—	4	1	no	—
49	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×Dic₇₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₇₃	C₂×D₄	1	1	yes	characteristic
53	#10596833	—	4	2	no	—
54	#10596833	—	4	2	no	—
55	C₂²×C₇₃	—	2	1	no	—
56	C₂²×C₇₃	—	2	1	no	—
57	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₇₃	#131	1	2	yes	—
59	C₄×C₇₃	—	2	1	no	—
60	C₄×C₇₃	—	4	2	no	—
61	C₄×C₇₃	#101	1	2	yes	—
62	Dic₇₃	—	2	1	no	—
63	Dic₇₃	—	2	1	no	—
64	Dic₇₃	—	2	1	no	—
65	Dic₇₃	—	2	1	no	—
66	Dic₇₃	—	4	2	no	—
67	C₂×C₇₃	—	2	2	no	—
68	C₂×C₇₃	#408	1	1	yes	characteristic
69	C₂×C₇₃	#410	1	1	yes	characteristic
70	C₂×C₇₃	C₂×#101	1	1	yes	characteristic
71	#1470	—	73	1	no	maximal, Sylow
72	C₇₃	#1470	1	1	yes	characteristic, complemented, Sylow
73	C₂×#125	—	73	1	no	—
74	#432	—	73	1	no	—
75	#431	—	73	1	no	—
76	#324	—	73	4	no	—
77	#125	—	146	1	no	—
78	#125	—	146	1	no	—
79	#125	—	146	1	no	—
80	#125	—	146	1	no	—
81	C₂×C₄⋊C₄	—	73	2	no	—
82	C₂×C₄⋊C₄	—	73	2	no	—
83	C₂×C₄⋊C₄	—	73	1	no	—
84	C₂×C₄⋊C₄	#64408	1	1	yes	characteristic
85	C₂×C₄²	—	73	1	no	—
86	C₄⋊C₈	—	146	1	no	—
87	C₄⋊C₈	—	146	1	no	—
88	C₂²⋊C₈	—	73	2	no	—
89	C₂²⋊C₈	—	73	2	no	—
90	#97	—	146	1	no	—
91	C₄×C₈	—	146	1	no	—
92	C₂²×C₄	—	73	1	no	—
93	C₂²×C₄	—	73	1	no	—
94	C₂²×C₄	—	73	2	no	—
95	C₂²×C₄	C₂×#64408	1	2	yes	—
96	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
97	C₂×C₈	—	146	2	no	—
98	C₂×C₈	—	146	2	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	146	2	no	—
102	C₄⋊C₄	—	146	2	no	—
103	C₄⋊C₄	—	146	2	no	—
104	C₄⋊C₄	—	146	2	no	—
105	C₄⋊C₄	—	146	1	no	—
106	C₄⋊C₄	—	146	1	no	—
107	C₄²	—	73	1	no	—
108	C₄²	—	73	1	no	—
109	C₄²	—	73	1	no	—
110	C₄²	—	73	1	no	—
111	C₂³	C₂²×#64408	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	—	292	1	no	—
116	C₂×C₄	—	73	1	no	—
117	C₂×C₄	—	73	1	no	—
118	C₂×C₄	—	73	1	no	—
119	C₂×C₄	—	73	1	no	—
120	C₂×C₄	—	146	4	no	—
121	C₂×C₄	—	146	2	no	—
122	C₂×C₄	—	146	2	no	—
123	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
124	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
125	C₈	—	292	2	no	—
126	C₈	—	292	2	no	—
127	C₂²	—	2	1	no	—
128	C₂²	—	2	1	no	—
129	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
130	C₂²	#14556679	1	2	yes	—
131	C₄	—	2	1	no	—
132	C₄	—	4	2	no	—
133	C₄	—	292	2	no	—
134	C₄	—	146	1	no	—
135	C₄	—	146	1	no	—
136	C₄	—	146	1	no	—
137	C₄	—	146	1	no	—
138	C₄	#14556652	1	2	yes	—
139	C₂	—	2	2	no	—
140	C₂	#17989562	1	1	yes	characteristic
141	C₂	#17989567	1	1	yes	characteristic
142	C₂	C₂×#14556652	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989608

C₂

yes

maximal, characteristic

#17989604

C₂

yes

maximal, characteristic

#17989433

C₂

yes

maximal

C₂×#14556582

C₂

yes

maximal, characteristic

#14556658

—

C₄⋊#10596833

—

C₂²⋊#10596833

C₂²

yes

—

C₂²⋊#10596833

C₂²

yes

—

—

—

—

—

—

—

C₂×C₄⋊Dic₇₃

C₄

yes

—

C₂×C₄⋊Dic₇₃

C₄

yes

characteristic

C₂×#11924007

C₂²

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₄

yes

characteristic

C₂×#10596833

—

C₂×#10596833

—

C₂²×Dic₇₃

C₂×C₄

yes

—

C₂²×Dic₇₃

C₂×C₄

yes

characteristic

C₂²×Dic₇₃

C₂³

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

#11924007

—

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₄×Dic₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

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

#10596833

—

#10596833

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

#131

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#101

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

—

—

—

—

—

146

—

#125

—

146

—

#125

—

146

—

#125

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#64408

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

146

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

146

—

C₄×C₈

—

146

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#64408

yes

—

C₂²×C₄

C₂×#64408

yes

characteristic

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

146

—

102

C₄⋊C₄

—

146

—

103

C₄⋊C₄

—

146

—

104

C₄⋊C₄

—

146

—

105

C₄⋊C₄

—

146

—

106

C₄⋊C₄

—

146

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₂³

C₂²×#64408

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

292

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

146

—

121

C₂×C₄

—

146

—

122

C₂×C₄

—

146

—

123

C₂×C₄

C₂²⋊#64408

yes

characteristic

124

C₂×C₄

C₂²⋊#64408

yes

characteristic

125

C₈

—

292

—

126

C₈

—

292

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×C₂²⋊#64408

yes

characteristic

130

C₂²

#14556679

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

292

—

134

C₄

—

146

—

135

C₄

—

146

—

136

C₄

—

146

—

137

C₄

—

146

—

138

C₄

#14556652

yes

—

139

C₂

—

140

C₂

#17989562

yes

characteristic

141

C₂

#17989567

yes

characteristic

142

C₂

C₂×#14556652

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#64408
commutator	C₂³×C₇₃	C₂²×C₄
Frattini	C₂³	C₂²×#64408
Fitting	C₂×C₄⋊C₄×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₂×C₄⋊C₄	#64408
hypocenter	C₇₃	#1470
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊#64408

commutator

C₂³×C₇₃

C₂²×C₄

Frattini

C₂³

C₂²×#64408

Fitting

C₂×C₄⋊C₄×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

C₂×C₄⋊C₄

C₇₃

G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
72. C₇₃	#1470	semidirect

H ≅ G/N

type

72. C₇₃

#1470

semidirect

Subgroups — #22625205

Conjugacy classes of subgroups

Complements