Group #22613233 — subgroups

Subgroups — #22613233

Order 9344.

432 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 139 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	C₇₃⋊#401	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988526	C₂	1	3	yes	maximal, complemented
3	#17988512	C₂	1	3	yes	maximal
4	C₄⋊C₄×Dic₇₃	C₂	1	3	yes	maximal, complemented
5	C₂²⋊C₄×Dic₇₃	C₂	1	3	yes	maximal
6	#17988432	C₂	1	1	yes	maximal, characteristic, complemented
7	#396×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂³×Dic₇₃	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₇₃	C₂²	1	3	yes	—
10	C₂×C₂²⋊Dic₇₃	C₂²	1	3	yes	—
11	C₂×C₄×Dic₇₃	C₂²	1	3	yes	—
12	C₂×C₄×Dic₇₃	C₂²	1	3	yes	—
13	#14556498	C₂²	1	3	yes	—
14	#14556498	C₂²	1	3	yes	—
15	#14556498	C₂²	1	3	yes	—
16	#14556498	C₂²	1	3	yes	—
17	#14556498	C₂²	1	3	yes	—
18	C₂×C₄⋊C₄×C₇₃	C₂²	1	3	yes	—
19	C₂×C₂²⋊C₄×C₇₃	C₂²	1	3	yes	—
20	#95×C₇₃	C₂²	1	1	yes	characteristic
21	C₂²×Dic₇₃	—	2	6	no	—
22	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
23	C₂²×Dic₇₃	C₂³	1	3	yes	—
24	C₂²×Dic₇₃	C₂³	1	3	yes	—
25	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₇₃	—	2	6	no	—
27	C₂²⋊Dic₇₃	—	2	6	no	—
28	C₄×Dic₇₃	—	2	6	no	—
29	C₄×Dic₇₃	—	2	6	no	—
30	C₂⁴×C₇₃	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	C₂³	1	3	yes	—
32	C₂²×C₄×C₇₃	C₂³	1	3	yes	—
33	C₄⋊C₄×C₇₃	—	2	6	no	—
34	C₂²⋊C₄×C₇₃	—	2	6	no	—
35	C₂³×C₇₃	—	2	3	no	—
36	C₂³×C₇₃	—	2	3	no	—
37	C₂³×C₇₃	#41	1	2	yes	—
38	C₂³×C₇₃	C₂⁴	1	1	yes	characteristic
39	C₂×C₄×C₇₃	—	2	3	no	—
40	C₂×C₄×C₇₃	—	2	3	no	—
41	C₂×C₄×C₇₃	—	2	3	no	—
42	C₂×C₄×C₇₃	—	2	3	no	—
43	C₂×C₄×C₇₃	—	2	3	no	—
44	C₂×C₄×C₇₃	#41	1	6	yes	—
45	C₂×Dic₇₃	—	2	3	no	—
46	C₂×Dic₇₃	—	2	3	no	—
47	C₂×Dic₇₃	—	2	3	no	—
48	C₂×Dic₇₃	—	2	3	no	—
49	C₂×Dic₇₃	—	2	3	no	—
50	C₂×Dic₇₃	—	2	3	no	—
51	C₂×Dic₇₃	—	2	3	no	—
52	C₂×Dic₇₃	—	4	4	no	—
53	C₂×Dic₇₃	#41	1	6	yes	—
54	C₂²×C₇₃	—	2	6	no	—
55	C₂²×C₇₃	—	4	1	no	—
56	C₂²×C₇₃	—	4	3	no	—
57	C₂²×C₇₃	C₂×#41	1	3	yes	—
58	C₂²×C₇₃	C₂×#41	1	3	yes	—
59	C₂²×C₇₃	C₂×#41	1	1	yes	characteristic
60	C₄×C₇₃	—	2	6	no	—
61	C₄×C₇₃	—	4	3	no	—
62	Dic₇₃	—	4	1	no	—
63	Dic₇₃	—	4	3	no	—
64	Dic₇₃	—	4	3	no	—
65	Dic₇₃	#126	1	4	yes	—
66	C₂×C₇₃	—	4	2	no	—
67	C₂×C₇₃	C₂×#126	1	1	yes	characteristic
68	C₂×C₇₃	#529	1	3	yes	—
69	C₂×C₇₃	#529	1	3	yes	—
70	#2401	—	73	1	no	maximal, Sylow
71	C₇₃	#2401	1	1	yes	characteristic, complemented, Sylow
72	#401	—	73	1	no	—
73	#400	—	73	3	no	—
74	#397	—	73	3	no	—
75	#396	D₇₃	1	1	yes	characteristic
76	C₄×C₄⋊C₄	—	73	3	no	—
77	C₄×C₂²⋊C₄	—	73	3	no	—
78	#376	—	73	1	no	—
79	C₂³×C₄	—	73	1	no	—
80	C₂×C₄⋊C₄	—	73	3	no	—
81	C₂×C₄⋊C₄	C₂×D₇₃	1	3	yes	—
82	C₂×C₂²⋊C₄	—	73	3	no	—
83	C₂×C₂²⋊C₄	C₂×D₇₃	1	3	yes	—
84	C₂×C₄²	—	73	3	no	—
85	C₂×C₄²	—	73	3	no	—
86	#95	—	73	3	no	—
87	#95	—	73	3	no	—
88	#95	—	73	3	no	—
89	#95	—	73	3	no	—
90	#95	—	73	3	no	—
91	#95	C₂×D₇₃	1	1	yes	characteristic
92	C₂⁴	C₂²×D₇₃	1	1	yes	characteristic
93	C₂²×C₄	—	73	1	no	—
94	C₂²×C₄	—	73	1	no	—
95	C₂²×C₄	—	73	3	no	—
96	C₂²×C₄	—	73	3	no	—
97	C₂²×C₄	—	146	6	no	—
98	C₂²×C₄	C₂²×D₇₃	1	3	yes	—
99	C₂²×C₄	C₂²×D₇₃	1	3	yes	—
100	C₄⋊C₄	—	2	6	no	—
101	C₄⋊C₄	—	146	6	no	—
102	C₂²⋊C₄	—	2	6	no	—
103	C₂²⋊C₄	—	146	6	no	—
104	C₄²	—	146	6	no	—
105	C₄²	—	146	6	no	—
106	C₂³	—	2	3	no	—
107	C₂³	—	2	3	no	—
108	C₂³	#11924029	1	2	yes	—
109	C₂³	C₂³×D₇₃	1	1	yes	characteristic
110	C₂×C₄	—	2	3	no	—
111	C₂×C₄	—	2	3	no	—
112	C₂×C₄	—	2	3	no	—
113	C₂×C₄	—	2	3	no	—
114	C₂×C₄	—	2	3	no	—
115	C₂×C₄	—	292	4	no	—
116	C₂×C₄	—	73	6	no	—
117	C₂×C₄	—	146	3	no	—
118	C₂×C₄	—	146	3	no	—
119	C₂×C₄	—	146	3	no	—
120	C₂×C₄	—	146	3	no	—
121	C₂×C₄	—	146	3	no	—
122	C₂×C₄	—	146	3	no	—
123	C₂×C₄	—	146	3	no	—
124	C₂×C₄	#11924027	1	6	yes	—
125	C₂²	—	2	6	no	—
126	C₂²	—	4	1	no	—
127	C₂²	—	4	3	no	—
128	C₂²	#41×D₇₃	1	3	yes	—
129	C₂²	C₂×#11924027	1	3	yes	—
130	C₂²	C₂×#11924029	1	1	yes	characteristic
131	C₄	—	2	6	no	—
132	C₄	—	4	3	no	—
133	C₄	—	292	3	no	—
134	C₄	—	292	3	no	—
135	C₄	—	292	1	no	—
136	C₄	—	73	4	no	—
137	C₂	—	4	2	no	—
138	C₂	#126×D₇₃	1	1	yes	characteristic
139	C₂	#17989118	1	3	yes	—
140	C₂	#17989123	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃⋊#401

C₂

yes

maximal, characteristic, complemented

#17988526

C₂

yes

maximal, complemented

#17988512

C₂

yes

maximal

C₄⋊C₄×Dic₇₃

C₂

yes

maximal, complemented

C₂²⋊C₄×Dic₇₃

C₂

yes

maximal

#17988432

C₂

yes

maximal, characteristic, complemented

#396×C₇₃

C₂

yes

maximal, characteristic

C₂³×Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×C₂²⋊Dic₇₃

C₂²

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

—

#14556498

C₂²

yes

—

#14556498

C₂²

yes

—

#14556498

C₂²

yes

—

#14556498

C₂²

yes

—

#14556498

C₂²

yes

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₇₃

C₂²

yes

—

#95×C₇₃

C₂²

yes

characteristic

C₂²×Dic₇₃

—

C₂²×Dic₇₃

C₂³

yes

characteristic

C₂²×Dic₇₃

C₂³

yes

—

C₂²×Dic₇₃

C₂³

yes

—

C₂²×Dic₇₃

C₂³

yes

characteristic

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂⁴×C₇₃

C₂³

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

C₂³

yes

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

#41

yes

—

C₂³×C₇₃

C₂⁴

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#41

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

#41

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×#41

yes

—

C₂²×C₇₃

C₂×#41

yes

—

C₂²×C₇₃

C₂×#41

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

#126

yes

—

C₂×C₇₃

—

C₂×C₇₃

C₂×#126

yes

characteristic

C₂×C₇₃

#529

yes

—

C₂×C₇₃

#529

yes

—

#2401

—

maximal, Sylow

C₇₃

#2401

yes

characteristic, complemented, Sylow

—

—

—

D₇₃

yes

characteristic

C₄×C₄⋊C₄

—

C₄×C₂²⋊C₄

—

#376

—

C₂³×C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₇₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

—

—

—

—

—

C₂×D₇₃

yes

characteristic

C₂⁴

C₂²×D₇₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

146

—

C₂²×C₄

C₂²×D₇₃

yes

—

C₂²×C₄

C₂²×D₇₃

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

146

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

146

—

104

C₄²

—

146

—

105

C₄²

—

146

—

106

C₂³

—

107

C₂³

—

108

C₂³

#11924029

yes

—

109

C₂³

C₂³×D₇₃

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

292

—

116

C₂×C₄

—

117

C₂×C₄

—

146

—

118

C₂×C₄

—

146

—

119

C₂×C₄

—

146

—

120

C₂×C₄

—

146

—

121

C₂×C₄

—

146

—

122

C₂×C₄

—

146

—

123

C₂×C₄

—

146

—

124

C₂×C₄

#11924027

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#41×D₇₃

yes

—

129

C₂²

C₂×#11924027

yes

—

130

C₂²

C₂×#11924029

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

292

—

134

C₄

—

292

—

135

C₄

—

292

—

136

C₄

—

137

C₂

—

138

C₂

#126×D₇₃

yes

characteristic

139

C₂

#17989118

yes

—

140

C₂

#17989123

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₃
commutator, socle	C₂³×C₇₃	C₂⁴
Fitting	#396×C₇₃	C₂
hypercenter	#396	D₇₃
hypocenter	C₇₃	#2401
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ #396

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₃

commutator, socle

C₂³×C₇₃

C₂⁴

Fitting

#396×C₇₃

C₂

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ #396

N	H ≅ G/N	type
1. C₇₃⋊#401	C₂	semidirect
2. #17988526	C₂	semidirect
4. C₄⋊C₄×Dic₇₃	C₂	semidirect
6. #17988432	C₂	semidirect
71. C₇₃	#2401	semidirect

H ≅ G/N

type

1. C₇₃⋊#401

C₂

semidirect

2. #17988526

C₂

semidirect

4. C₄⋊C₄×Dic₇₃

C₂

semidirect

6. #17988432

C₂

semidirect

71. C₇₃

#2401

semidirect

Subgroups — #22613233

Conjugacy classes of subgroups

Complements