Group #22611254 — subgroups

Subgroups — #22611254

Order 9344.

288 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 89 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	C₂²×Dic₂₉₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556516	C₂	1	2	yes	maximal
3	#17988493	C₂	1	2	yes	maximal
4	#17988350	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₄₆	C₂²	1	2	yes	—
7	C₂×Dic₂₉₂	—	2	4	no	—
8	C₂×Dic₂₉₂	—	2	2	no	—
9	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
10	#14556516	—	2	4	no	—
11	#14556498	—	2	2	no	—
12	#14556498	—	2	2	no	—
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	C₂×Dic₁₄₆	—	2	2	no	—
18	C₂×Dic₁₄₆	—	2	2	no	—
19	C₂×Dic₁₄₆	—	2	2	no	—
20	C₂×Dic₁₄₆	—	2	2	no	—
21	C₂×Dic₁₄₆	—	2	2	no	—
22	C₂×Dic₁₄₆	D₄	1	4	yes	—
23	C₂²×Dic₇₃	—	2	2	no	—
24	C₂²×Dic₇₃	—	2	2	no	—
25	Dic₂₉₂	—	2	8	no	—
26	C₄⋊Dic₇₃	—	2	2	no	—
27	C₄⋊Dic₇₃	—	2	2	no	—
28	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₇₃	—	2	2	no	—
32	C₂×C₈×C₇₃	—	2	1	no	—
33	C₂×C₈×C₇₃	—	2	2	no	—
34	C₂×C₈×C₇₃	D₄	1	2	yes	—
35	C₄²×C₇₃	—	2	1	no	—
36	C₄²×C₇₃	—	2	1	no	—
37	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₃	—	2	1	no	—
39	C₂×C₄×C₇₃	—	2	1	no	—
40	C₂×C₄×C₇₃	—	2	1	no	—
41	C₂×C₄×C₇₃	—	2	2	no	—
42	C₂×C₄×C₇₃	Q₁₆	1	2	yes	complemented
43	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
46	C₂×C₄×C₇₃	#41	1	2	yes	—
47	C₂×Dic₇₃	—	2	4	no	—
48	C₂×Dic₇₃	—	4	2	no	—
49	C₂×Dic₇₃	—	4	2	no	—
50	C₂×Dic₇₃	—	4	2	no	—
51	C₂×Dic₇₃	—	4	2	no	—
52	C₂×Dic₇₃	—	4	2	no	—
53	Dic₁₄₆	—	2	4	no	—
54	Dic₁₄₆	—	2	4	no	—
55	Dic₁₄₆	—	4	2	no	—
56	Dic₁₄₆	—	4	2	no	—
57	C₈×C₇₃	—	2	2	no	—
58	C₈×C₇₃	—	4	1	no	—
59	C₂²×C₇₃	#121	1	1	yes	characteristic
60	C₂²×C₇₃	#121	1	2	yes	—
61	C₂²×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₇₃	#135	1	1	yes	characteristic
63	C₂²×C₇₃	#137	1	2	yes	—
64	C₄×C₇₃	—	2	2	no	—
65	C₄×C₇₃	—	4	1	no	—
66	C₄×C₇₃	#120	1	1	yes	characteristic
67	C₄×C₇₃	#123	1	2	yes	—
68	C₄×C₇₃	#124	1	1	yes	characteristic
69	Dic₇₃	—	4	4	no	—
70	Dic₇₃	—	8	2	no	—
71	C₂×C₇₃	#394	1	1	yes	characteristic
72	C₂×C₇₃	#462	1	2	yes	—
73	C₂×C₇₃	#464	1	2	yes	—
74	C₂×C₇₃	#467	1	1	yes	characteristic
75	C₂×C₇₃	#470	1	1	yes	characteristic
76	#2056	—	73	1	no	maximal, Sylow
77	C₇₃	#2056	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₁₆	—	73	1	no	—
79	C₂×C₄⋊C₈	D₇₃	1	1	yes	characteristic
80	C₂×#103	—	73	2	no	—
81	#391	—	73	2	no	—
82	#340	—	73	1	no	—
83	C₂²×Q₈	—	73	2	no	—
84	C₂×Q₁₆	—	146	4	no	—
85	C₂×Q₁₆	—	146	2	no	—
86	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
87	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
88	C₂×C₄⋊C₄	—	73	2	no	—
89	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
90	C₄⋊C₈	—	2	2	no	—
91	#103	—	146	4	no	—
92	#95	—	146	2	no	—
93	#95	—	146	2	no	—
94	C₂×Q₈	—	73	4	no	—
95	C₂×Q₈	—	146	2	no	—
96	C₂×Q₈	—	146	2	no	—
97	C₂×Q₈	—	146	2	no	—
98	C₂×Q₈	—	146	2	no	—
99	C₂×Q₈	—	146	2	no	—
100	C₂²×C₄	—	146	2	no	—
101	C₂²×C₄	—	146	2	no	—
102	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
103	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
104	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
105	Q₁₆	—	146	8	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	1	no	—
109	C₂×C₈	C₂²⋊D₇₃	1	2	yes	—
110	C₄⋊C₄	—	146	2	no	—
111	C₄⋊C₄	—	146	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	—	2	1	no	—
114	C₂³	C₂×D₂₉₂	1	1	yes	characteristic
115	Q₈	—	292	2	no	—
116	Q₈	—	292	2	no	—
117	Q₈	—	146	4	no	—
118	Q₈	—	146	4	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	292	2	no	—
124	C₂×C₄	—	292	2	no	—
125	C₂×C₄	—	292	2	no	—
126	C₂×C₄	—	292	2	no	—
127	C₂×C₄	—	292	2	no	—
128	C₂×C₄	—	146	4	no	—
129	C₂×C₄	Dic₂₉₂	1	2	yes	complemented
130	C₂×C₄	D₄×D₇₃	1	2	yes	—
131	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
132	C₂×C₄	#11924028	1	1	yes	characteristic
133	C₂×C₄	#11924029	1	2	yes	—
134	C₈	—	2	2	no	—
135	C₈	—	4	1	no	—
136	C₂²	#14556565	1	1	yes	characteristic
137	C₂²	#14556566	1	2	yes	—
138	C₂²	C₂×Dic₂₉₂	1	1	yes	characteristic
139	C₂²	#14556613	1	1	yes	characteristic
140	C₂²	#14556624	1	2	yes	—
141	C₄	—	2	2	no	—
142	C₄	—	4	1	no	—
143	C₄	—	292	4	no	—
144	C₄	—	584	2	no	—
145	C₄	#14556559	1	1	yes	characteristic
146	C₄	#14556576	1	2	yes	—
147	C₄	#14556580	1	1	yes	characteristic
148	C₂	#17988504	1	1	yes	characteristic
149	C₂	#17988798	1	2	yes	—
150	C₂	#17988810	1	2	yes	—
151	C₂	#17988828	1	1	yes	characteristic
152	C₂	#17988845	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₉₂

C₂

yes

maximal, characteristic

C₂×#14556516

C₂

yes

maximal

#17988493

C₂

yes

maximal

#17988350

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

#14556516

—

#14556498

—

#14556498

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

D₄

yes

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

Q₁₆

yes

complemented

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₂²×C₇₃

#121

yes

characteristic

C₂²×C₇₃

#121

yes

—

C₂²×C₇₃

C₂×Q₁₆

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#120

yes

characteristic

C₄×C₇₃

#123

yes

—

C₄×C₇₃

#124

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

#394

yes

characteristic

C₂×C₇₃

#462

yes

—

C₂×C₇₃

#464

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊C₈

D₇₃

yes

—

—

—

C₂²×Q₈

—

C₂×Q₁₆

—

146

—

C₂×Q₁₆

—

146

—

C₂²×C₈

C₂×D₇₃

yes

characteristic

C₂²×C₈

C₂×D₇₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₄⋊C₈

—

#103

—

146

—

#95

—

146

—

#95

—

146

—

C₂×Q₈

—

C₂×Q₈

—

146

—

C₂×Q₈

—

146

—

C₂×Q₈

—

146

—

C₂×Q₈

—

146

—

C₂×Q₈

—

146

—

100

C₂²×C₄

—

146

—

101

C₂²×C₄

—

146

—

102

C₂²×C₄

D₂₉₂

yes

characteristic

103

C₂²×C₄

D₂₉₂

yes

characteristic

104

C₂²×C₄

C₂²×D₇₃

yes

characteristic

105

Q₁₆

—

146

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂²⋊D₇₃

yes

—

110

C₄⋊C₄

—

146

—

111

C₄⋊C₄

—

146

—

112

C₄²

—

113

C₄²

—

114

C₂³

C₂×D₂₉₂

yes

characteristic

115

Q₈

—

292

—

116

Q₈

—

292

—

117

Q₈

—

146

—

118

Q₈

—

146

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

292

—

124

C₂×C₄

—

292

—

125

C₂×C₄

—

292

—

126

C₂×C₄

—

292

—

127

C₂×C₄

—

292

—

128

C₂×C₄

—

146

—

129

C₂×C₄

Dic₂₉₂

yes

complemented

130

C₂×C₄

D₄×D₇₃

yes

—

131

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

132

C₂×C₄

#11924028

yes

characteristic

133

C₂×C₄

#11924029

yes

—

134

C₈

—

135

C₈

—

136

C₂²

#14556565

yes

characteristic

137

C₂²

#14556566

yes

—

138

C₂²

C₂×Dic₂₉₂

yes

characteristic

139

C₂²

#14556613

yes

characteristic

140

C₂²

#14556624

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

292

—

144

C₄

—

584

—

145

C₄

#14556559

yes

characteristic

146

C₄

#14556576

yes

—

147

C₄

#14556580

yes

characteristic

148

C₂

#17988504

yes

characteristic

149

C₂

#17988798

yes

—

150

C₂

#17988810

yes

—

151

C₂

#17988828

yes

characteristic

152

C₂

#17988845

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₉₂
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×C₄⋊C₈×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₇₃
hypocenter	C₇₃	#2056
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₉₂

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×C₄⋊C₈×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₇₃

hypocenter

C₇₃

#2056

LCS

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
42. C₂×C₄×C₇₃	Q₁₆	semidirect
77. C₇₃	#2056	semidirect
129. C₂×C₄	Dic₂₉₂	semidirect

H ≅ G/N

type

42. C₂×C₄×C₇₃

Q₁₆

semidirect

77. C₇₃

#2056

semidirect

129. C₂×C₄

Dic₂₉₂

semidirect

Subgroups — #22611254

Conjugacy classes of subgroups

Complements