Group #22610649 — subgroups

Subgroups — #22610649

Order 9344.

260 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₄⋊#10596831	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556520	C₂	1	2	yes	maximal
3	C₄×C₄⋊Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	#17988341	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10596831	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₃	C₄	1	1	yes	characteristic
8	C₂×C₄⋊Dic₇₃	C₄	1	1	yes	characteristic
9	C₂×C₄×Dic₇₃	C₂²	1	2	yes	—
10	C₄⋊#10596831	—	2	2	no	—
11	#14556520	—	2	4	no	—
12	#14556498	C₄	1	2	yes	—
13	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₇₃	—	2	2	no	—
17	C₂²×Dic₇₃	C₂×C₄	1	2	yes	—
18	C₂²×Dic₇₃	C₂×C₄	1	2	yes	—
19	C₂×#10596831	—	2	2	no	—
20	C₂×#10596831	—	2	2	no	—
21	C₂×#10596831	—	2	2	no	—
22	C₄⋊Dic₇₃	—	2	2	no	—
23	C₄⋊Dic₇₃	C₈	1	4	yes	complemented
24	C₄×Dic₇₃	—	2	2	no	—
25	C₄×Dic₇₃	—	2	2	no	—
26	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
29	C₂×C₈×C₇₃	—	2	1	no	—
30	C₂×C₈×C₇₃	—	2	1	no	—
31	C₂×C₈×C₇₃	—	2	1	no	—
32	C₂×C₈×C₇₃	—	2	1	no	—
33	C₂×C₈×C₇₃	D₄	1	2	yes	—
34	C₂×C₈×C₇₃	Q₈	1	2	yes	—
35	C₄²×C₇₃	D₄	1	2	yes	—
36	C₄²×C₇₃	Q₈	1	2	yes	—
37	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₃	—	2	1	no	—
39	C₂×C₄×C₇₃	—	2	1	no	—
40	C₂×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₇₃	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₇₃	M₁₆	1	2	yes	—
44	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
49	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
50	C₂×Dic₇₃	—	2	2	no	—
51	C₂×Dic₇₃	—	2	2	no	—
52	C₂×Dic₇₃	—	2	2	no	—
53	C₂×Dic₇₃	—	2	2	no	—
54	C₂×Dic₇₃	—	2	2	no	—
55	C₂×Dic₇₃	C₂×C₈	1	4	yes	—
56	C₈×C₇₃	—	2	2	no	—
57	C₈×C₇₃	—	2	2	no	—
58	#10596831	—	4	2	no	—
59	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₇₃	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₇₃	C₄×Q₈	1	1	yes	characteristic
62	C₂²×C₇₃	C₂²×C₈	1	1	yes	characteristic
63	C₂²×C₇₃	C₂×M₁₆	1	1	yes	characteristic
64	C₂²×C₇₃	#131	1	1	yes	characteristic
65	C₂²×C₇₃	#131	1	1	yes	characteristic
66	C₄×C₇₃	—	2	2	no	—
67	C₄×C₇₃	C₄⋊C₈	1	4	yes	complemented
68	C₄×C₇₃	#121	1	1	yes	characteristic
69	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
70	C₄×C₇₃	#125	1	1	yes	characteristic
71	C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
72	Dic₇₃	—	2	4	no	—
73	Dic₇₃	—	4	2	no	—
74	C₂×C₇₃	#389	1	1	yes	characteristic
75	C₂×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
76	C₂×C₇₃	#424	1	1	yes	characteristic
77	C₂×C₇₃	C₈×D₄	1	1	yes	characteristic
78	C₂×C₇₃	#436	1	1	yes	characteristic
79	C₂×C₇₃	C₈×Q₈	1	1	yes	characteristic
80	C₂×C₇₃	#446	1	1	yes	characteristic
81	#1939	—	73	1	no	maximal, Sylow
82	C₇₃	#1939	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₄⋊C₈	—	73	2	no	—
84	C₂×C₄⋊C₈	—	73	1	no	—
85	C₂×C₄×C₈	D₇₃	1	1	yes	characteristic
86	C₄×C₄⋊C₄	—	73	1	no	—
87	#336	—	73	2	no	—
88	C₂²×C₈	—	73	2	no	—
89	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
90	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
91	C₂×C₄⋊C₄	—	73	1	no	—
92	C₂×C₄⋊C₄	—	73	1	no	—
93	C₂×C₄²	—	73	2	no	—
94	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
95	C₄⋊C₈	—	146	4	no	—
96	C₄⋊C₈	—	146	2	no	—
97	C₄×C₈	—	2	2	no	—
98	#95	—	73	2	no	—
99	C₂²×C₄	—	73	2	no	—
100	C₂²×C₄	—	73	2	no	—
101	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
102	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	146	2	no	—
109	C₂×C₈	—	146	2	no	—
110	C₂×C₈	—	146	2	no	—
111	C₂×C₈	Dic₁₄₆	1	2	yes	—
112	C₂×C₈	D₂₉₂	1	2	yes	—
113	C₄⋊C₄	—	73	4	no	—
114	C₄⋊C₄	—	146	2	no	—
115	C₄²	—	146	2	no	—
116	C₄²	—	146	2	no	—
117	C₄²	Dic₁₄₆	1	2	yes	—
118	C₄²	C₂²⋊D₇₃	1	2	yes	—
119	C₂³	C₂×C₄×D₇₃	1	1	yes	characteristic
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	73	4	no	—
123	C₂×C₄	—	146	2	no	—
124	C₂×C₄	—	146	2	no	—
125	C₂×C₄	—	146	2	no	—
126	C₂×C₄	—	146	2	no	—
127	C₂×C₄	—	146	2	no	—
128	C₂×C₄	#11924007	1	2	yes	—
129	C₂×C₄	#11924007	1	2	yes	—
130	C₂×C₄	C₈×D₇₃	1	2	yes	—
131	C₂×C₄	#11924011	1	2	yes	—
132	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
133	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
134	C₂×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
135	C₂×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
136	C₂×C₄	#11924028	1	1	yes	characteristic
137	C₂×C₄	#11924028	1	1	yes	characteristic
138	C₈	—	2	2	no	—
139	C₈	—	2	2	no	—
140	C₈	—	292	2	no	—
141	C₂²	C₂×#11924007	1	1	yes	characteristic
142	C₂²	C₄×D₂₉₂	1	1	yes	characteristic
143	C₂²	C₄×Dic₁₄₆	1	1	yes	characteristic
144	C₂²	C₂×C₈×D₇₃	1	1	yes	characteristic
145	C₂²	C₂×#11924011	1	1	yes	characteristic
146	C₂²	#14556597	1	1	yes	characteristic
147	C₂²	#14556597	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	292	2	no	—
150	C₄	—	146	4	no	—
151	C₄	#14556520	1	4	yes	complemented
152	C₄	#14556565	1	1	yes	characteristic
153	C₄	#14556570	1	1	yes	characteristic
154	C₄	#14556583	1	1	yes	characteristic
155	C₄	#14556591	1	1	yes	characteristic
156	C₂	#17988485	1	1	yes	characteristic
157	C₂	C₂×#14556520	1	1	yes	characteristic
158	C₂	#17988613	1	1	yes	characteristic
159	C₂	C₈×D₂₉₂	1	1	yes	characteristic
160	C₂	#17988666	1	1	yes	characteristic
161	C₂	C₈×Dic₁₄₆	1	1	yes	characteristic
162	C₂	#17988717	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#10596831

C₂

yes

maximal, characteristic

C₂×#14556520

C₂

yes

maximal

C₄×C₄⋊Dic₇₃

C₂

yes

maximal, characteristic

#17988341

C₂

yes

maximal

C₂×C₄×C₈×C₇₃

C₂

yes

maximal, characteristic

C₂²×#10596831

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₄

yes

characteristic

C₂×C₄⋊Dic₇₃

C₄

yes

characteristic

C₂×C₄×Dic₇₃

C₂²

yes

—

C₄⋊#10596831

—

#14556520

—

#14556498

C₄

yes

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄×C₈×C₇₃

—

C₂²×Dic₇₃

C₂×C₄

yes

—

C₂²×Dic₇₃

C₂×C₄

yes

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₈

yes

complemented

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

—

C₂×C₈×C₇₃

Q₈

yes

—

C₄²×C₇₃

D₄

yes

—

C₄²×C₇₃

Q₈

yes

—

C₂³×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂×C₈

yes

—

C₂×C₄×C₇₃

M₁₆

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×C₈

yes

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₃

C₄×D₄

yes

characteristic

C₂²×C₇₃

C₄×Q₈

yes

characteristic

C₂²×C₇₃

C₂²×C₈

yes

characteristic

C₂²×C₇₃

C₂×M₁₆

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

C₄⋊C₈

yes

complemented

C₄×C₇₃

#121

yes

characteristic

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

#125

yes

characteristic

C₄×C₇₃

C₄⋊Q₈

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

#389

yes

characteristic

C₂×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₂×C₇₃

#424

yes

characteristic

C₂×C₇₃

C₈×D₄

yes

characteristic

C₂×C₇₃

#436

yes

characteristic

C₂×C₇₃

C₈×Q₈

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₇₃

yes

characteristic

C₄×C₄⋊C₄

—

#336

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₃

yes

characteristic

C₂²×C₈

C₂×D₇₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₄⋊C₈

—

146

—

C₄⋊C₈

—

146

—

C₄×C₈

—

#95

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₄×D₇₃

yes

characteristic

102

C₂²×C₄

C₄×D₇₃

yes

characteristic

103

C₂²×C₄

C₂²×D₇₃

yes

characteristic

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

146

—

109

C₂×C₈

—

146

—

110

C₂×C₈

—

146

—

111

C₂×C₈

Dic₁₄₆

yes

—

112

C₂×C₈

D₂₉₂

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

146

—

115

C₄²

—

146

—

116

C₄²

—

146

—

117

C₄²

Dic₁₄₆

yes

—

118

C₄²

C₂²⋊D₇₃

yes

—

119

C₂³

C₂×C₄×D₇₃

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

146

—

124

C₂×C₄

—

146

—

125

C₂×C₄

—

146

—

126

C₂×C₄

—

146

—

127

C₂×C₄

—

146

—

128

C₂×C₄

#11924007

yes

—

129

C₂×C₄

#11924007

yes

—

130

C₂×C₄

C₈×D₇₃

yes

—

131

C₂×C₄

#11924011

yes

—

132

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

133

C₂×C₄

C₂×D₂₉₂

yes

characteristic

134

C₂×C₄

C₂×Dic₁₄₆

yes

characteristic

135

C₂×C₄

C₂×Dic₁₄₆

yes

characteristic

136

C₂×C₄

#11924028

yes

characteristic

137

C₂×C₄

#11924028

yes

characteristic

138

C₈

—

139

C₈

—

140

C₈

—

292

—

141

C₂²

C₂×#11924007

yes

characteristic

142

C₂²

C₄×D₂₉₂

yes

characteristic

143

C₂²

C₄×Dic₁₄₆

yes

characteristic

144

C₂²

C₂×C₈×D₇₃

yes

characteristic

145

C₂²

C₂×#11924011

yes

characteristic

146

C₂²

#14556597

yes

characteristic

147

C₂²

#14556597

yes

characteristic

148

C₄

—

149

C₄

—

292

—

150

C₄

—

146

—

151

C₄

#14556520

yes

complemented

152

C₄

#14556565

yes

characteristic

153

C₄

#14556570

yes

characteristic

154

C₄

#14556583

yes

characteristic

155

C₄

#14556591

yes

characteristic

156

C₂

#17988485

yes

characteristic

157

C₂

C₂×#14556520

yes

characteristic

158

C₂

#17988613

yes

characteristic

159

C₂

C₈×D₂₉₂

yes

characteristic

160

C₂

#17988666

yes

characteristic

161

C₂

C₈×Dic₁₄₆

yes

characteristic

162

C₂

#17988717

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂²×C₈
Fitting	C₂×C₄×C₈×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×C₄
hypercenter	C₂×C₄×C₈	D₇₃
hypocenter	C₇₃	#1939
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₇₃

commutator

C₂²×C₇₃

C₂²×C₈

Fitting

C₂×C₄×C₈×C₇₃

C₂

socle

C₂³×C₇₃

C₂²×C₄

hypercenter

C₂×C₄×C₈

D₇₃

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
23. C₄⋊Dic₇₃	C₈	semidirect
67. C₄×C₇₃	C₄⋊C₈	semidirect
82. C₇₃	#1939	semidirect
151. C₄	#14556520	semidirect

H ≅ G/N

type

23. C₄⋊Dic₇₃

C₈

semidirect

67. C₄×C₇₃

C₄⋊C₈

semidirect

82. C₇₃

#1939

semidirect

151. C₄

#14556520

semidirect

Subgroups — #22610649

Conjugacy classes of subgroups

Complements