Group #22617839 — subgroups

Subgroups — #22617839

Order 9344.

420 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 119 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	#17988511	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988502	C₂	1	3	yes	maximal
4	#17988482	C₂	1	3	yes	maximal, complemented
5	#17988477	C₂	1	3	yes	maximal, complemented
6	#17988475	C₂	1	3	yes	maximal, complemented
7	#399×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₂²⋊D₇₃	C₂²	1	1	yes	characteristic
9	C₂×#11924007	C₂²	1	3	yes	—
10	C₂×#11924007	C₂²	1	3	yes	—
11	C₂×C₂²⋊Dic₇₃	C₂²	1	3	yes	—
12	C₂×#11924005	C₂²	1	3	yes	—
13	C₂×#11924005	C₂²	1	3	yes	—
14	C₂×C₄×Dic₇₃	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	C₂³×D₇₃	C₂³	1	1	yes	characteristic
22	C₂×C₂²⋊D₇₃	—	4	3	no	—
23	C₂²×Dic₇₃	C₂³	1	3	yes	—
24	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₇₃	C₂³	1	3	yes	—
26	#11924007	—	2	6	no	—
27	#11924007	—	4	3	no	—
28	C₂²⋊Dic₇₃	—	2	6	no	—
29	#11924005	—	2	6	no	—
30	#11924005	—	4	3	no	—
31	C₄×Dic₇₃	—	4	3	no	—
32	C₂⁴×C₇₃	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₇₃	C₂³	1	3	yes	—
34	C₂²×C₄×C₇₃	C₂³	1	3	yes	—
35	C₄⋊C₄×C₇₃	—	4	3	no	—
36	C₂²⋊C₄×C₇₃	—	4	3	no	—
37	C₂³×C₇₃	—	2	3	no	—
38	C₂³×C₇₃	—	2	3	no	—
39	C₂³×C₇₃	—	2	1	no	—
40	C₂³×C₇₃	C₂⁴	1	1	yes	characteristic
41	C₂²×D₇₃	—	2	3	no	—
42	C₂²×D₇₃	—	2	3	no	—
43	C₂²×D₇₃	—	2	1	no	—
44	C₂×C₄×C₇₃	—	2	3	no	—
45	C₂×C₄×C₇₃	—	2	3	no	—
46	C₂×C₄×C₇₃	—	2	3	no	—
47	C₂×C₄×C₇₃	—	2	3	no	—
48	C₂×C₄×C₇₃	—	2	3	no	—
49	C₂×C₄×C₇₃	—	2	3	no	—
50	C₂²⋊D₇₃	—	4	4	no	—
51	C₂×Dic₇₃	—	2	3	no	—
52	C₂×Dic₇₃	—	2	3	no	—
53	C₂×Dic₇₃	—	2	3	no	—
54	C₂×Dic₇₃	—	2	3	no	—
55	C₂×Dic₇₃	—	2	3	no	—
56	C₂×Dic₇₃	—	2	3	no	—
57	C₂×Dic₇₃	#41	1	6	yes	—
58	C₂²×C₇₃	—	4	3	no	—
59	C₂²×C₇₃	—	4	3	no	—
60	C₂²×C₇₃	—	4	1	no	—
61	C₂²×C₇₃	C₂×#41	1	3	yes	—
62	C₂²×C₇₃	H(5,𝔽₂)	1	3	yes	—
63	C₂²×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
64	C₂×D₇₃	—	4	3	no	—
65	C₂×D₇₃	—	4	3	no	—
66	C₂×D₇₃	—	4	1	no	—
67	C₄×C₇₃	—	4	3	no	—
68	C₄×C₇₃	—	4	3	no	—
69	Dic₇₃	—	2	6	no	—
70	Dic₇₃	—	4	3	no	—
71	Dic₇₃	—	4	1	no	—
72	C₂×C₇₃	—	8	1	no	—
73	C₂×C₇₃	#542	1	3	yes	—
74	C₂×C₇₃	#553	1	3	yes	—
75	C₂×C₇₃	#560	1	1	yes	characteristic
76	D₇₃	—	8	1	no	—
77	#2801	—	73	1	no	maximal, Sylow
78	C₇₃	#2801	1	1	yes	characteristic, complemented, Sylow
79	#401	—	73	1	no	—
80	#399	D₇₃	1	1	yes	characteristic, complemented
81	#396	—	73	1	no	—
82	#394	—	73	3	no	—
83	#389	—	73	3	no	—
84	#388	—	73	3	no	—
85	#388	—	73	3	no	—
86	C₂²×D₄	—	73	1	no	—
87	C₂×C₄⋊C₄	—	73	3	no	—
88	C₂×C₄⋊C₄	—	73	3	no	—
89	C₂×C₄⋊C₄	C₂×D₇₃	1	3	yes	—
90	C₂×C₂²⋊C₄	—	73	3	no	—
91	C₂×C₂²⋊C₄	—	73	3	no	—
92	C₂×C₂²⋊C₄	—	73	3	no	—
93	C₂×C₂²⋊C₄	C₂×D₇₃	1	3	yes	—
94	C₂×C₄²	—	73	3	no	—
95	#95	—	73	3	no	—
96	#95	—	73	3	no	—
97	#95	—	73	3	no	—
98	#95	C₂×D₇₃	1	1	yes	characteristic
99	C₂⁴	—	73	1	no	—
100	C₂⁴	C₂²×D₇₃	1	1	yes	characteristic
101	C₂×D₄	—	292	3	no	—
102	C₂²×C₄	—	73	3	no	—
103	C₂²×C₄	—	73	3	no	—
104	C₂²×C₄	—	73	1	no	—
105	C₂²×C₄	C₂²×D₇₃	1	3	yes	—
106	C₂²×C₄	C₂²×D₇₃	1	3	yes	—
107	C₄⋊C₄	—	4	3	no	—
108	C₄⋊C₄	—	292	3	no	—
109	C₄⋊C₄	—	146	6	no	—
110	C₂²⋊C₄	—	4	3	no	—
111	C₂²⋊C₄	—	292	3	no	—
112	C₂²⋊C₄	—	146	6	no	—
113	C₂²⋊C₄	—	146	6	no	—
114	C₄²	—	292	3	no	—
115	C₂³	—	2	3	no	—
116	C₂³	—	2	3	no	—
117	C₂³	—	2	1	no	—
118	C₂³	—	146	3	no	—
119	C₂³	—	146	3	no	—
120	C₂³	—	146	1	no	—
121	C₂³	C₂³×D₇₃	1	1	yes	characteristic
122	D₄	—	292	4	no	—
123	C₂×C₄	—	2	3	no	—
124	C₂×C₄	—	2	3	no	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	2	3	no	—
127	C₂×C₄	—	2	3	no	—
128	C₂×C₄	—	2	3	no	—
129	C₂×C₄	—	73	6	no	—
130	C₂×C₄	—	146	3	no	—
131	C₂×C₄	—	146	3	no	—
132	C₂×C₄	—	146	3	no	—
133	C₂×C₄	—	146	3	no	—
134	C₂×C₄	—	146	3	no	—
135	C₂×C₄	—	146	3	no	—
136	C₂²	—	4	3	no	—
137	C₂²	—	4	3	no	—
138	C₂²	—	4	1	no	—
139	C₂²	—	292	3	no	—
140	C₂²	—	292	3	no	—
141	C₂²	—	292	1	no	—
142	C₂²	#41×D₇₃	1	3	yes	—
143	C₂²	#14556636	1	3	yes	—
144	C₂²	#14556639	1	1	yes	characteristic
145	C₄	—	4	3	no	—
146	C₄	—	4	3	no	—
147	C₄	—	292	3	no	—
148	C₄	—	292	1	no	—
149	C₄	—	146	6	no	—
150	C₂	—	8	1	no	—
151	C₂	—	584	1	no	—
152	C₂	#17989207	1	3	yes	—
153	C₂	#17989282	1	3	yes	—
154	C₂	#17989324	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₇₃⋊#401

C₂

yes

maximal, characteristic, complemented

#17988511

C₂

yes

maximal, characteristic, complemented

#17988502

C₂

yes

maximal

#17988482

C₂

yes

maximal, complemented

#17988477

C₂

yes

maximal, complemented

#17988475

C₂

yes

maximal, complemented

#399×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×C₂²⋊D₇₃

C₂²

yes

characteristic

C₂×#11924007

C₂²

yes

—

C₂×#11924007

C₂²

yes

—

C₂×C₂²⋊Dic₇₃

C₂²

yes

—

C₂×#11924005

C₂²

yes

—

C₂×#11924005

C₂²

yes

—

C₂×C₄×Dic₇₃

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

C₂³×D₇₃

C₂³

yes

characteristic

C₂×C₂²⋊D₇₃

—

C₂²×Dic₇₃

C₂³

yes

—

C₂²×Dic₇₃

C₂³

yes

characteristic

C₂²×Dic₇₃

C₂³

yes

—

#11924007

—

#11924007

—

C₂²⋊Dic₇₃

—

#11924005

—

#11924005

—

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

—

C₂³×C₇₃

C₂⁴

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂²⋊D₇₃

—

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

H(5,𝔽₂)

yes

—

C₂²×C₇₃

2₋¹⁺⁴

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#542

yes

—

C₂×C₇₃

#553

yes

—

C₂×C₇₃

#560

yes

characteristic

D₇₃

—

#2801

—

maximal, Sylow

C₇₃

#2801

yes

characteristic, complemented, Sylow

#401

—

#399

D₇₃

yes

characteristic, complemented

—

—

—

—

—

C₂²×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₇₃

yes

—

C₂×C₄²

—

—

—

—

C₂×D₇₃

yes

characteristic

C₂⁴

—

100

C₂⁴

C₂²×D₇₃

yes

characteristic

101

C₂×D₄

—

292

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²×D₇₃

yes

—

106

C₂²×C₄

C₂²×D₇₃

yes

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

292

—

109

C₄⋊C₄

—

146

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

292

—

112

C₂²⋊C₄

—

146

—

113

C₂²⋊C₄

—

146

—

114

C₄²

—

292

—

115

C₂³

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

146

—

119

C₂³

—

146

—

120

C₂³

—

146

—

121

C₂³

C₂³×D₇₃

yes

characteristic

122

D₄

—

292

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

146

—

131

C₂×C₄

—

146

—

132

C₂×C₄

—

146

—

133

C₂×C₄

—

146

—

134

C₂×C₄

—

146

—

135

C₂×C₄

—

146

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

292

—

140

C₂²

—

292

—

141

C₂²

—

292

—

142

C₂²

#41×D₇₃

yes

—

143

C₂²

#14556636

yes

—

144

C₂²

#14556639

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

292

—

148

C₄

—

292

—

149

C₄

—

146

—

150

C₂

—

151

C₂

—

584

—

152

C₂

#17989207

yes

—

153

C₂

#17989282

yes

—

154

C₂

#17989324

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₃
commutator, socle	C₂³×C₇₃	C₂⁴
Fitting	#399×C₇₃	C₂
hypercenter	#399	D₇₃
hypocenter	C₇₃	#2801
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ #399

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₃

commutator, socle

C₂³×C₇₃

C₂⁴

Fitting

#399×C₇₃

C₂

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ #399

N	H ≅ G/N	type
1. C₇₃⋊#401	C₂	semidirect
2. #17988511	C₂	semidirect
4. #17988482	C₂	semidirect
5. #17988477	C₂	semidirect
6. #17988475	C₂	semidirect
7. #399×C₇₃	C₂	semidirect
78. C₇₃	#2801	semidirect
80. #399	D₇₃	semidirect

H ≅ G/N

type

1. C₇₃⋊#401

C₂

semidirect

2. #17988511

C₂

semidirect

4. #17988482

C₂

semidirect

5. #17988477

C₂

semidirect

6. #17988475

C₂

semidirect

7. #399×C₇₃

C₂

semidirect

78. C₇₃

#2801

semidirect

80. #399

D₇₃

semidirect

Subgroups — #22617839

Conjugacy classes of subgroups

Complements