Group #22609697 — subgroups

Subgroups — #22609697

Order 9344.

336 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂×C₂²⋊#10596831	C₂	1	2	yes	maximal
2	C₂×#14556502	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988342	C₂	1	2	yes	maximal, complemented
5	C₂×#117×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10596831	C₂²	1	2	yes	—
7	C₂²×#10596831	C₂²	1	2	yes	—
8	C₂²⋊#10596831	C₄	1	8	yes	complemented
9	#14556502	—	2	2	no	—
10	C₄×#10596831	—	2	2	no	—
11	C₂³×C₄×C₇₃	C₂²	1	1	yes	characteristic
12	#117×C₇₃	—	2	2	no	—
13	#117×C₇₃	—	2	2	no	—
14	C₂×C₄⋊C₄×C₇₃	C₄	1	2	yes	—
15	C₂×C₂²⋊C₄×C₇₃	C₄	1	2	yes	—
16	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×#10596831	—	2	2	no	—
19	C₂×#10596831	—	2	2	no	—
20	C₂×#10596831	—	2	2	no	—
21	C₂×#10596831	—	2	2	no	—
22	C₂×#10596831	C₂×C₄	1	4	yes	—
23	C₂×#10596831	C₂×C₄	1	4	yes	—
24	C₂⁴×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₇₃	—	2	1	no	—
26	C₂²×C₄×C₇₃	—	2	1	no	—
27	C₂²×C₄×C₇₃	—	2	2	no	—
28	C₂²×C₄×C₇₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₇₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₇₃	—	2	4	no	—
34	C₂²⋊C₄×C₇₃	—	2	4	no	—
35	C₄²×C₇₃	—	2	1	no	—
36	C₄²×C₇₃	—	2	1	no	—
37	C₄²×C₇₃	D₄	1	2	yes	—
38	C₄²×C₇₃	D₄	1	2	yes	—
39	C₂³×C₇₃	—	2	1	no	—
40	C₂³×C₇₃	—	2	1	no	—
41	C₂³×C₇₃	—	2	4	no	—
42	C₂³×C₇₃	C₄²	1	2	yes	—
43	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₃	—	2	1	no	—
45	C₂×C₄×C₇₃	—	2	1	no	—
46	C₂×C₄×C₇₃	—	2	4	no	—
47	C₂×C₄×C₇₃	—	2	4	no	—
48	C₂×C₄×C₇₃	—	2	4	no	—
49	C₂×C₄×C₇₃	—	2	2	no	—
50	C₂×C₄×C₇₃	—	2	4	no	—
51	C₂×C₄×C₇₃	C₄²	1	2	yes	—
52	C₂×C₄×C₇₃	C₂²⋊C₄	1	4	yes	—
53	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₇₃	#41	1	2	yes	—
57	#10596831	—	2	4	no	—
58	#10596831	—	2	4	no	—
59	C₂²×C₇₃	—	2	2	no	—
60	C₂²×C₇₃	—	2	4	no	—
61	C₂²×C₇₃	—	4	2	no	—
62	C₂²×C₇₃	—	4	2	no	—
63	C₂²×C₇₃	C₂×C₄²	1	1	yes	characteristic
64	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₇₃	#117	1	1	yes	characteristic
66	C₂²×C₇₃	#131	1	4	yes	—
67	C₄×C₇₃	—	2	2	no	—
68	C₄×C₇₃	—	2	4	no	—
69	C₄×C₇₃	—	4	2	no	—
70	C₄×C₇₃	C₄×D₄	1	2	yes	—
71	C₄×C₇₃	C₄×D₄	1	2	yes	—
72	C₂×C₇₃	—	4	2	no	—
73	C₂×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₇₃	#405	1	2	yes	—
75	C₂×C₇₃	#408	1	2	yes	—
76	C₂×C₇₃	#433	1	2	yes	—
77	#1750	—	73	1	no	maximal, Sylow
78	C₇₃	#1750	1	1	yes	characteristic, complemented, Sylow
79	C₂×#117	D₇₃	1	1	yes	characteristic
80	C₂×C₂²⋊C₈	—	73	2	no	—
81	C₂×#97	—	73	1	no	—
82	C₂×C₄×C₈	—	73	1	no	—
83	#336	—	73	2	no	—
84	C₂³×C₄	C₂×D₇₃	1	1	yes	characteristic
85	C₂²×C₈	—	73	2	no	—
86	C₂²×C₈	—	73	2	no	—
87	#117	—	2	2	no	—
88	#117	—	2	2	no	—
89	C₂×C₄⋊C₄	Dic₇₃	1	2	yes	—
90	C₂×C₂²⋊C₄	Dic₇₃	1	2	yes	—
91	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
93	C₂²⋊C₈	—	73	8	no	—
94	#97	—	146	2	no	—
95	C₄×C₈	—	146	2	no	—
96	C₂⁴	C₂×Dic₇₃	1	1	yes	characteristic
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	C₄×D₇₃	1	4	yes	—
101	C₂²×C₄	C₂×Dic₇₃	1	2	yes	—
102	C₂²×C₄	C₂×Dic₇₃	1	1	yes	characteristic
103	C₂²×C₄	C₂×Dic₇₃	1	2	yes	—
104	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
105	C₂×C₈	—	73	4	no	—
106	C₂×C₈	—	73	4	no	—
107	C₂×C₈	—	146	2	no	—
108	C₂×C₈	—	146	2	no	—
109	C₂×C₈	—	146	2	no	—
110	C₂×C₈	—	146	2	no	—
111	C₄⋊C₄	—	2	4	no	—
112	C₂²⋊C₄	—	2	4	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	2	1	no	—
115	C₄²	C₂²⋊D₇₃	1	2	yes	—
116	C₄²	C₂²⋊D₇₃	1	2	yes	—
117	C₂³	—	2	1	no	—
118	C₂³	—	2	1	no	—
119	C₂³	—	2	4	no	—
120	C₂³	C₄×Dic₇₃	1	2	yes	—
121	C₂³	C₂²×Dic₇₃	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	C₄×Dic₇₃	1	2	yes	—
130	C₂×C₄	C₂²⋊Dic₇₃	1	4	yes	—
131	C₂×C₄	C₂×C₄×D₇₃	1	2	yes	—
132	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
134	C₂×C₄	#11924028	1	2	yes	—
135	C₈	—	146	4	no	—
136	C₈	—	146	4	no	—
137	C₂²	—	2	2	no	—
138	C₂²	—	2	4	no	—
139	C₂²	—	4	2	no	—
140	C₂²	—	4	2	no	—
141	C₂²	C₂×C₄×Dic₇₃	1	1	yes	characteristic
142	C₂²	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
143	C₂²	#14556550	1	1	yes	characteristic
144	C₂²	#14556598	1	4	yes	—
145	C₄	—	2	2	no	—
146	C₄	—	2	4	no	—
147	C₄	—	4	2	no	—
148	C₄	C₄×C₂²⋊D₇₃	1	2	yes	—
149	C₄	C₄×C₂²⋊D₇₃	1	2	yes	—
150	C₂	—	4	2	no	—
151	C₂	C₄×C₂²⋊Dic₇₃	1	1	yes	characteristic
152	C₂	#17988543	1	2	yes	—
153	C₂	#17988554	1	2	yes	—
154	C₂	#17988648	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊#10596831

C₂

yes

maximal

C₂×#14556502

C₂

yes

maximal, characteristic, complemented

C₂×C₄×#10596831

C₂

yes

maximal, characteristic, complemented

#17988342

C₂

yes

maximal, complemented

C₂×#117×C₇₃

C₂

yes

maximal, characteristic

C₂²×#10596831

C₂²

yes

—

C₂²×#10596831

C₂²

yes

—

C₂²⋊#10596831

C₄

yes

complemented

#14556502

—

C₄×#10596831

—

C₂³×C₄×C₇₃

C₂²

yes

characteristic

#117×C₇₃

—

#117×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₄

yes

—

C₂×C₂²⋊C₄×C₇₃

C₄

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂×C₄²×C₇₃

C₂²

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#10596831

C₂×C₄

yes

—

C₂⁴×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

D₄

yes

—

C₄²×C₇₃

D₄

yes

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₄²

yes

—

C₂³×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄²

yes

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₄²

yes

characteristic

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄×D₄

yes

—

C₄×C₇₃

C₄×D₄

yes

—

C₂×C₇₃

—

C₂×C₇₃

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₇₃

#405

yes

—

C₂×C₇₃

#408

yes

—

C₂×C₇₃

#433

yes

—

#1750

—

maximal, Sylow

C₇₃

#1750

yes

characteristic, complemented, Sylow

C₂×#117

D₇₃

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

#336

—

C₂³×C₄

C₂×D₇₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

—

C₂×C₄⋊C₄

Dic₇₃

yes

—

C₂×C₂²⋊C₄

Dic₇₃

yes

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₂²⋊C₈

—

#97

—

146

—

C₄×C₈

—

146

—

C₂⁴

C₂×Dic₇₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

C₄×D₇₃

yes

—

101

C₂²×C₄

C₂×Dic₇₃

yes

—

102

C₂²×C₄

C₂×Dic₇₃

yes

characteristic

103

C₂²×C₄

C₂×Dic₇₃

yes

—

104

C₂²×C₄

C₂²×D₇₃

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

146

—

108

C₂×C₈

—

146

—

109

C₂×C₈

—

146

—

110

C₂×C₈

—

146

—

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₄²

C₂²⋊D₇₃

yes

—

116

C₄²

C₂²⋊D₇₃

yes

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

C₄×Dic₇₃

yes

—

121

C₂³

C₂²×Dic₇₃

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₄×Dic₇₃

yes

—

130

C₂×C₄

C₂²⋊Dic₇₃

yes

—

131

C₂×C₄

C₂×C₄×D₇₃

yes

—

132

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

134

C₂×C₄

#11924028

yes

—

135

C₈

—

146

—

136

C₈

—

146

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×C₄×Dic₇₃

yes

characteristic

142

C₂²

C₂×C₂²⋊Dic₇₃

yes

characteristic

143

C₂²

#14556550

yes

characteristic

144

C₂²

#14556598

yes

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

C₄×C₂²⋊D₇₃

yes

—

149

C₄

C₄×C₂²⋊D₇₃

yes

—

150

C₂

—

151

C₂

C₄×C₂²⋊Dic₇₃

yes

characteristic

152

C₂

#17988543

yes

—

153

C₂

#17988554

yes

—

154

C₂

#17988648

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂×C₄²
Fitting	C₂×#117×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×C₄
hypercenter	C₂×#117	D₇₃
hypocenter	C₇₃	#1750
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₇₃

commutator

C₂²×C₇₃

C₂×C₄²

Fitting

C₂×#117×C₇₃

C₂

socle

C₂³×C₇₃

C₂²×C₄

D₇₃

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂²×C₄ ◁ C₂×#117

N	H ≅ G/N	type
2. C₂×#14556502	C₂	semidirect
3. C₂×C₄×#10596831	C₂	semidirect
4. #17988342	C₂	semidirect
8. C₂²⋊#10596831	C₄	semidirect
78. C₇₃	#1750	semidirect

H ≅ G/N

type

2. C₂×#14556502

C₂

semidirect

3. C₂×C₄×#10596831

C₂

semidirect

4. #17988342

C₂

semidirect

8. C₂²⋊#10596831

C₄

semidirect

78. C₇₃

#1750

semidirect

Subgroups — #22609697

Conjugacy classes of subgroups

Complements