Group #22626723 — subgroups

Subgroups — #22626723

Order 9344.

348 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 162 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	#17989639	C₂	1	4	yes	maximal, complemented
2	#17989637	C₂	1	4	yes	maximal, complemented
3	#17989594	C₂	1	2	yes	maximal
4	#17989591	C₂	1	1	yes	maximal, characteristic
5	#17989552	C₂	1	2	yes	maximal
6	#17989546	C₂	1	1	yes	maximal, characteristic
7	#17989143	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924038	C₂²	1	2	yes	—
9	C₂×#11924036	C₂²	1	1	yes	characteristic
10	C₂×#11924036	C₂²	1	1	yes	characteristic
11	#14556658	C₂²	1	4	yes	—
12	C₄⋊#10596833	C₂²	1	4	yes	—
13	C₄⋊#10596833	C₂²	1	4	yes	—
14	#14556646	C₂²	1	2	yes	—
15	#14556646	C₂²	1	4	yes	—
16	#14556646	C₂²	1	2	yes	—
17	C₄×#10596833	C₂²	1	4	yes	—
18	#14556590	C₄	1	2	yes	—
19	#14556583	C₄	1	1	yes	characteristic
20	#14556569	C₄	1	4	yes	—
21	#14556556	C₂²	1	4	yes	—
22	#14556549	C₂²	1	2	yes	—
23	C₄²×D₇₃	C₂²	1	1	yes	characteristic
24	#125×C₇₃	C₄	1	1	yes	characteristic
25	#11924038	—	2	4	no	—
26	#11924036	—	2	2	no	—
27	#11924036	Q₈	1	4	yes	complemented
28	C₂×#10596833	C₂³	1	1	yes	characteristic
29	C₂×#10596833	C₂³	1	1	yes	characteristic
30	C₂×#10596833	C₂³	1	4	yes	—
31	C₂×#10596833	C₂³	1	2	yes	—
32	C₂×Dic₁₄₆	C₂×C₄	1	4	yes	—
33	C₂×C₄×D₇₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
35	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
36	#11924007	C₂×C₄	1	4	yes	—
37	#11924007	C₂×C₄	1	4	yes	—
38	#11924005	C₂×C₄	1	4	yes	—
39	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₇₃	C₂³	1	2	yes	—
41	C₄×Dic₇₃	C₂³	1	4	yes	—
42	C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
43	C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
44	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×D₇₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₇₃	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₇₃	C₂²×C₄	1	2	yes	—
50	C₂×Dic₇₃	C₂²×C₄	1	4	yes	—
51	C₂×Dic₇₃	C₂⁴	1	1	yes	characteristic
52	C₄×D₇₃	—	4	2	no	—
53	C₄×D₇₃	C₂×Q₈	1	2	yes	—
54	C₄×D₇₃	#41	1	2	yes	—
55	Dic₁₄₆	—	4	4	no	—
56	#10596833	—	2	1	no	—
57	#10596833	—	2	1	no	—
58	#10596833	—	2	4	no	—
59	#10596833	C₂×Q₈	1	4	yes	—
60	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
61	C₂×D₇₃	—	2	2	no	—
62	C₂×D₇₃	C₄×Q₈	1	2	yes	—
63	C₄×C₇₃	—	2	4	no	—
64	C₄×C₇₃	—	2	2	no	—
65	C₄×C₇₃	C₄×Q₈	1	2	yes	—
66	Dic₇₃	—	2	2	no	—
67	Dic₇₃	—	2	4	no	—
68	Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
69	Dic₇₃	C₂×#41	1	1	yes	characteristic
70	C₂×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
71	C₂×C₇₃	#568	1	2	yes	—
72	D₇₃	—	2	2	no	—
73	#2961	—	73	1	no	maximal, Sylow
74	C₇₃	#2961	1	1	yes	characteristic, complemented, Sylow
75	#533	—	73	1	no	—
76	#446	—	73	4	no	—
77	#446	—	73	4	no	—
78	#424	—	73	2	no	—
79	#423	—	73	1	no	—
80	#405	—	73	2	no	—
81	C₂×#97	—	73	1	no	—
82	C₂×M₁₆	—	73	2	no	—
83	C₂²×C₈	—	73	1	no	—
84	C₂²×C₈	—	73	1	no	—
85	C₄⋊Q₈	—	73	2	no	—
86	#125	—	73	1	no	—
87	#125	#64408	1	1	yes	characteristic
88	C₂²⋊Q₈	—	73	4	no	—
89	C₄×Q₈	—	73	4	no	—
90	#117	—	73	2	no	—
91	C₂×C₄²	—	73	1	no	—
92	C₄⋊C₈	—	73	4	no	—
93	C₄⋊C₈	—	73	4	no	—
94	C₄⋊C₈	—	73	4	no	—
95	#97	—	73	2	no	—
96	#97	—	73	2	no	—
97	#97	—	73	4	no	—
98	C₄×C₈	—	73	4	no	—
99	C₂×Q₈	—	73	4	no	—
100	C₂²×C₄	—	73	2	no	—
101	C₂²×C₄	—	73	1	no	—
102	M₁₆	—	146	4	no	—
103	C₂×C₈	—	73	2	no	—
104	C₂×C₈	—	73	4	no	—
105	C₂×C₈	—	73	4	no	—
106	C₂×C₈	—	73	1	no	—
107	C₂×C₈	—	73	1	no	—
108	C₂×C₈	—	146	2	no	—
109	C₄⋊C₄	—	73	4	no	—
110	C₄⋊C₄	—	73	4	no	—
111	C₄⋊C₄	—	73	2	no	—
112	C₄⋊C₄	C₂×#64408	1	2	yes	—
113	C₄⋊C₄	C₂×#64408	1	4	yes	—
114	C₂²⋊C₄	—	73	4	no	—
115	C₄²	—	73	1	no	—
116	C₄²	—	73	4	no	—
117	C₄²	—	73	2	no	—
118	C₄²	C₂×#64408	1	1	yes	characteristic
119	C₂³	—	73	1	no	—
120	Q₈	—	292	4	no	—
121	C₂×C₄	—	292	2	no	—
122	C₂×C₄	—	73	1	no	—
123	C₂×C₄	—	73	2	no	—
124	C₂×C₄	—	73	2	no	—
125	C₂×C₄	—	73	2	no	—
126	C₂×C₄	—	73	4	no	—
127	C₂×C₄	C₂²×#64408	1	2	yes	—
128	C₂×C₄	C₂²×#64408	1	4	yes	—
129	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
130	C₈	—	73	4	no	—
131	C₈	—	146	1	no	—
132	C₈	—	146	1	no	—
133	C₈	—	146	4	no	—
134	C₂²	—	73	2	no	—
135	C₂²	—	146	2	no	—
136	C₂²	C₂³×#64408	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	4	no	—
139	C₄	—	73	1	no	—
140	C₄	—	73	1	no	—
141	C₄	—	146	2	no	—
142	C₄	—	146	4	no	—
143	C₄	Q₈×#64408	1	2	yes	—
144	C₂	—	146	2	no	—
145	C₂	C₂×Q₈×#64408	1	1	yes	characteristic
146	C₂	#17989666	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17989639

C₂

yes

maximal, complemented

#17989637

C₂

yes

maximal, complemented

#17989594

C₂

yes

maximal

#17989591

C₂

yes

maximal, characteristic

#17989552

C₂

yes

maximal

#17989546

C₂

yes

maximal, characteristic

#17989143

C₂

yes

maximal, characteristic

C₂×#11924038

C₂²

yes

—

C₂×#11924036

C₂²

yes

characteristic

C₂×#11924036

C₂²

yes

characteristic

#14556658

C₂²

yes

—

C₄⋊#10596833

C₂²

yes

—

C₄⋊#10596833

C₂²

yes

—

#14556646

C₂²

yes

—

#14556646

C₂²

yes

—

#14556646

C₂²

yes

—

C₄×#10596833

C₂²

yes

—

#14556590

C₄

yes

—

#14556583

C₄

yes

characteristic

#14556569

C₄

yes

—

#14556556

C₂²

yes

—

#14556549

C₂²

yes

—

C₄²×D₇₃

C₂²

yes

characteristic

#125×C₇₃

C₄

yes

—

—

Q₈

yes

complemented

C₂×#10596833

C₂³

yes

characteristic

C₂×#10596833

C₂³

yes

characteristic

C₂×#10596833

C₂³

yes

—

C₂×#10596833

C₂³

yes

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

C₂×C₄×D₇₃

C₂×C₄

yes

—

C₂×C₄×D₇₃

C₂³

yes

characteristic

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924007

C₂×C₄

yes

—

#11924007

C₂×C₄

yes

—

#11924005

C₂×C₄

yes

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

C₂³

yes

—

C₄×Dic₇₃

C₂³

yes

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

C₂⁴

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

C₂×Q₈

yes

—

C₄×D₇₃

#41

yes

—

Dic₁₄₆

—

—

—

—

C₂×Q₈

yes

—

C₂²×C₇₃

C₂³×C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

C₄×Q₈

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×Q₈

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₂²×Q₈

yes

characteristic

Dic₇₃

C₂×#41

yes

characteristic

C₂×C₇₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₇₃

#568

yes

—

D₇₃

—

#2961

—

maximal, Sylow

C₇₃

#2961

yes

characteristic, complemented, Sylow

—

—

—

—

—

—

C₂×#97

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

—

yes

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₂×Q₈

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

M₁₆

—

146

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

146

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

C₂×#64408

yes

—

113

C₄⋊C₄

C₂×#64408

yes

—

114

C₂²⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

C₂×#64408

yes

characteristic

119

C₂³

—

120

Q₈

—

292

—

121

C₂×C₄

—

292

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₂²×#64408

yes

—

128

C₂×C₄

C₂²×#64408

yes

—

129

C₂×C₄

C₂²×#64408

yes

characteristic

130

C₈

—

131

C₈

—

146

—

132

C₈

—

146

—

133

C₈

—

146

—

134

C₂²

—

135

C₂²

—

146

—

136

C₂²

C₂³×#64408

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

146

—

142

C₄

—

146

—

143

C₄

Q₈×#64408

yes

—

144

C₂

—

146

—

145

C₂

C₂×Q₈×#64408

yes

characteristic

146

C₂

#17989666

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₄
Fitting	#125×C₇₃	C₄
hypercenter	#125	#64408
hypocenter	C₇₃	#2961
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×#64408

commutator, socle

C₂²×C₇₃

C₂³×C₄

Fitting

#125×C₇₃

C₄

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #17989639	C₂	semidirect
2. #17989637	C₂	semidirect
27. #11924036	Q₈	semidirect
74. C₇₃	#2961	semidirect

H ≅ G/N

type

1. #17989639

C₂

semidirect

2. #17989637

C₂

semidirect

27. #11924036

Q₈

semidirect

74. C₇₃

#2961

semidirect

Subgroups — #22626723

Conjugacy classes of subgroups

Complements