Group #22608254 — subgroups

Subgroups — #22608254

Order 9344.

248 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 97 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×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic
2	#17988640	C₂	1	1	yes	maximal, characteristic
3	#17988312	C₂	1	4	yes	maximal
4	#431×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₄₆	C₄	1	1	yes	characteristic
6	C₄×Dic₁₄₆	—	2	1	no	—
7	C₄×Dic₁₄₆	—	2	1	no	—
8	C₄×Dic₁₄₆	—	2	1	no	—
9	C₄×Dic₁₄₆	—	2	1	no	—
10	C₂×C₄⋊Dic₇₃	C₄	1	1	yes	characteristic
11	C₂×#11924007	C₂²	1	2	yes	—
12	C₂×C₄×Dic₇₃	C₄	1	2	yes	—
13	C₄⋊#10596831	—	2	1	no	—
14	C₂²⋊#10596831	C₂²	1	2	yes	—
15	C₄×#10596831	—	2	1	no	—
16	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₇₃	—	2	1	no	—
18	C₂²⋊C₈×C₇₃	C₂²	1	2	yes	—
19	C₄×C₈×C₇₃	—	2	1	no	—
20	C₂×Dic₁₄₆	—	2	1	no	—
21	C₂×Dic₁₄₆	—	2	1	no	—
22	C₂×Dic₁₄₆	—	2	2	no	—
23	C₂×Dic₁₄₆	C₈	1	4	yes	complemented
24	C₂²×Dic₇₃	C₂×C₄	1	2	yes	—
25	C₂²×Dic₇₃	C₂×C₄	1	2	yes	—
26	C₂×#10596831	—	2	2	no	—
27	C₄⋊Dic₇₃	—	2	1	no	—
28	C₄⋊Dic₇₃	—	2	1	no	—
29	#11924007	—	2	2	no	—
30	#11924007	—	2	2	no	—
31	C₄×Dic₇₃	—	2	2	no	—
32	C₄×Dic₇₃	—	2	2	no	—
33	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₇₃	—	2	2	no	—
37	C₄²×C₇₃	D₄	1	1	yes	characteristic
38	C₄²×C₇₃	D₄	1	1	yes	characteristic
39	C₄²×C₇₃	D₄	1	1	yes	characteristic
40	C₄²×C₇₃	D₄	1	1	yes	characteristic
41	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₃	—	2	2	no	—
43	C₂×C₄×C₇₃	—	4	1	no	—
44	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂×C₈	1	2	yes	—
49	C₂×C₄×C₇₃	M₁₆	1	2	yes	—
50	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×Dic₇₃	—	2	2	no	—
53	C₂×Dic₇₃	—	2	2	no	—
54	C₂×Dic₇₃	—	2	2	no	—
55	C₂×Dic₇₃	—	4	2	no	—
56	C₂×Dic₇₃	C₂×C₈	1	4	yes	—
57	Dic₁₄₆	—	2	4	no	—
58	Dic₁₄₆	—	2	4	no	—
59	C₈×C₇₃	—	4	2	no	—
60	#10596831	—	4	2	no	—
61	C₂²×C₇₃	—	2	1	no	—
62	C₂²×C₇₃	—	2	1	no	—
63	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₇₃	C₂²×C₈	1	1	yes	characteristic
65	C₂²×C₇₃	C₂×M₁₆	1	1	yes	characteristic
66	C₄×C₇₃	—	2	1	no	—
67	C₄×C₇₃	—	2	1	no	—
68	C₄×C₇₃	—	2	1	no	—
69	C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
70	C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
71	C₄×C₇₃	#101	1	2	yes	—
72	Dic₇₃	—	2	4	no	—
73	Dic₇₃	—	4	2	no	—
74	C₂×C₇₃	—	2	1	no	—
75	C₂×C₇₃	—	2	1	no	—
76	C₂×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
77	C₂×C₇₃	#410	1	1	yes	characteristic
78	C₂×C₇₃	C₂×#101	1	1	yes	characteristic
79	#1461	—	73	1	no	maximal, Sylow
80	C₇₃	#1461	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₄×Q₈	—	73	1	no	—
82	#431	—	73	1	no	—
83	#431	D₇₃	1	1	yes	characteristic
84	#324	—	73	4	no	—
85	C₂²×Q₈	—	73	1	no	—
86	C₄×Q₈	—	146	1	no	—
87	C₄×Q₈	—	146	1	no	—
88	C₄×Q₈	—	146	1	no	—
89	C₄×Q₈	—	146	1	no	—
90	C₂×C₄⋊C₄	—	73	1	no	—
91	C₂×C₄⋊C₄	—	73	2	no	—
92	C₂×C₄²	—	73	2	no	—
93	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
94	C₄⋊C₈	—	2	1	no	—
95	C₄⋊C₈	—	146	1	no	—
96	C₂²⋊C₈	—	73	2	no	—
97	C₂²⋊C₈	C₂×D₇₃	1	2	yes	—
98	C₄×C₈	—	2	1	no	—
99	C₄×C₈	—	146	1	no	—
100	C₂×Q₈	—	73	4	no	—
101	C₂×Q₈	—	146	1	no	—
102	C₂×Q₈	—	146	1	no	—
103	C₂×Q₈	—	146	2	no	—
104	C₂²×C₄	—	73	2	no	—
105	C₂²×C₄	—	73	2	no	—
106	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
107	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
108	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	146	2	no	—
111	C₄⋊C₄	—	146	1	no	—
112	C₄⋊C₄	—	146	1	no	—
113	C₄⋊C₄	—	146	2	no	—
114	C₄⋊C₄	—	146	2	no	—
115	C₄²	—	146	2	no	—
116	C₄²	—	146	2	no	—
117	C₄²	D₂₉₂	1	1	yes	characteristic
118	C₄²	D₂₉₂	1	1	yes	characteristic
119	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
120	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
121	C₂³	C₂×C₄×D₇₃	1	1	yes	characteristic
122	Q₈	—	146	4	no	—
123	Q₈	—	146	4	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	292	2	no	—
127	C₂×C₄	—	73	4	no	—
128	C₂×C₄	—	146	2	no	—
129	C₂×C₄	—	146	2	no	—
130	C₂×C₄	—	146	2	no	—
131	C₂×C₄	#11924005	1	1	yes	characteristic
132	C₂×C₄	#11924005	1	1	yes	characteristic
133	C₂×C₄	#11924005	1	1	yes	characteristic
134	C₂×C₄	#11924005	1	1	yes	characteristic
135	C₂×C₄	C₈×D₇₃	1	2	yes	—
136	C₂×C₄	#11924011	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
138	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
139	C₈	—	4	2	no	—
140	C₈	—	292	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×#11924005	1	1	yes	characteristic
144	C₂²	C₂×C₈×D₇₃	1	1	yes	characteristic
145	C₂²	C₂×#11924011	1	1	yes	characteristic
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	292	2	no	—
150	C₄	—	146	4	no	—
151	C₄	#14556503	1	2	yes	—
152	C₄	#14556503	1	2	yes	—
153	C₄	#14556509	1	2	yes	—
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	C₂×#14556503	1	1	yes	characteristic
157	C₂	#17988560	1	1	yes	characteristic
158	C₂	C₂×#14556509	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₄₆

C₂

yes

maximal, characteristic

#17988640

C₂

yes

maximal, characteristic

#17988312

C₂

yes

maximal

#431×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₄

yes

characteristic

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

—

C₂×C₄⋊Dic₇₃

C₄

yes

characteristic

C₂×#11924007

C₂²

yes

—

C₂×C₄×Dic₇₃

C₄

yes

—

C₄⋊#10596831

—

C₂²⋊#10596831

C₂²

yes

—

C₄×#10596831

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

C₂²

yes

—

C₄×C₈×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₈

yes

complemented

C₂²×Dic₇₃

C₂×C₄

yes

—

C₂²×Dic₇₃

C₂×C₄

yes

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

#11924007

—

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

—

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₂³×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

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×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×C₈

yes

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₃

C₂²×C₈

yes

characteristic

C₂²×C₇₃

C₂×M₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊C₈

yes

—

C₄×C₇₃

C₂²⋊C₈

yes

—

C₄×C₇₃

#101

yes

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₂×C₂²⋊C₈

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

#431

—

#431

D₇₃

yes

characteristic

#324

—

C₂²×Q₈

—

C₄×Q₈

—

146

—

C₄×Q₈

—

146

—

C₄×Q₈

—

146

—

C₄×Q₈

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₇₃

yes

—

C₄×C₈

—

C₄×C₈

—

146

—

100

C₂×Q₈

—

101

C₂×Q₈

—

146

—

102

C₂×Q₈

—

146

—

103

C₂×Q₈

—

146

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₄×D₇₃

yes

characteristic

107

C₂²×C₄

C₄×D₇₃

yes

characteristic

108

C₂²×C₄

C₂²×D₇₃

yes

characteristic

109

C₂×C₈

—

110

C₂×C₈

—

146

—

111

C₄⋊C₄

—

146

—

112

C₄⋊C₄

—

146

—

113

C₄⋊C₄

—

146

—

114

C₄⋊C₄

—

146

—

115

C₄²

—

146

—

116

C₄²

—

146

—

117

C₄²

D₂₉₂

yes

characteristic

118

C₄²

D₂₉₂

yes

characteristic

119

C₄²

C₂²⋊D₇₃

yes

characteristic

120

C₄²

C₂²⋊D₇₃

yes

characteristic

121

C₂³

C₂×C₄×D₇₃

yes

characteristic

122

Q₈

—

146

—

123

Q₈

—

146

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

292

—

127

C₂×C₄

—

128

C₂×C₄

—

146

—

129

C₂×C₄

—

146

—

130

C₂×C₄

—

146

—

131

C₂×C₄

#11924005

yes

characteristic

132

C₂×C₄

#11924005

yes

characteristic

133

C₂×C₄

#11924005

yes

characteristic

134

C₂×C₄

#11924005

yes

characteristic

135

C₂×C₄

C₈×D₇₃

yes

—

136

C₂×C₄

#11924011

yes

—

137

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

138

C₂×C₄

C₂×D₂₉₂

yes

characteristic

139

C₈

—

140

C₈

—

292

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×#11924005

yes

characteristic

144

C₂²

C₂×C₈×D₇₃

yes

characteristic

145

C₂²

C₂×#11924011

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

292

—

150

C₄

—

146

—

151

C₄

#14556503

yes

—

152

C₄

#14556503

yes

—

153

C₄

#14556509

yes

—

154

C₂

—

155

C₂

—

156

C₂

C₂×#14556503

yes

characteristic

157

C₂

#17988560

yes

characteristic

158

C₂

C₂×#14556509

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#11924005
commutator	C₂²×C₇₃	C₂²×C₈
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#431×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	#431	D₇₃
hypocenter	C₇₃	#1461
LCS	G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂×C₄

#11924005

commutator

C₂²×C₇₃

C₂²×C₈

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#431×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

D₇₃

C₇₃

G ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
23. C₂×Dic₁₄₆	C₈	semidirect
80. C₇₃	#1461	semidirect

H ≅ G/N

type

23. C₂×Dic₁₄₆

C₈

semidirect

80. C₇₃

#1461

semidirect

Subgroups — #22608254

Conjugacy classes of subgroups

Complements