Group #22611440 — subgroups

Subgroups — #22611440

Order 9344.

176 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 67 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	#17988629	C₂	1	1	yes	maximal, characteristic
2	#17988627	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556524	C₂	1	1	yes	maximal, characteristic
4	#17988615	C₂	1	1	yes	maximal, characteristic
5	#17988355	C₂	1	1	yes	maximal, characteristic
6	#17988351	C₂	1	1	yes	maximal, characteristic
7	#343×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924012	C₂²	1	1	yes	characteristic
9	C₂²×#10596831	C₂²	1	1	yes	characteristic
10	#14556550	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
12	#14556525	—	2	1	no	—
13	#14556524	—	2	2	no	—
14	#14556524	—	2	1	no	—
15	#14556523	—	2	1	no	—
16	#14556522	—	2	1	no	—
17	#14556520	—	2	2	no	—
18	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
21	C₂²×Dic₇₃	—	2	1	no	—
22	#11924012	—	2	1	no	—
23	#11924012	Q₈	1	2	yes	—
24	C₂×#10596831	—	2	1	no	—
25	C₂×#10596831	—	2	1	no	—
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	D₄	1	2	yes	—
28	C₄⋊Dic₇₃	—	2	1	no	—
29	C₄⋊Dic₇₃	—	2	1	no	—
30	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
31	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
32	C₂²⋊Dic₇₃	—	2	1	no	—
33	C₄×Dic₇₃	—	2	1	no	—
34	C₂²×C₄×C₇₃	—	2	1	no	—
35	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
36	M₁₆×C₇₃	—	2	1	no	—
37	M₁₆×C₇₃	—	2	1	no	—
38	M₁₆×C₇₃	—	2	1	no	—
39	M₁₆×C₇₃	Q₈	1	2	yes	—
40	C₂×C₈×C₇₃	—	2	1	no	—
41	C₂×C₈×C₇₃	—	2	1	no	—
42	C₄⋊C₄×C₇₃	—	2	1	no	—
43	C₄⋊C₄×C₇₃	—	2	1	no	—
44	C₂³×C₇₃	#41	1	1	yes	characteristic
45	C₂×C₄×C₇₃	—	4	1	no	—
46	C₂×C₄×C₇₃	—	8	1	no	—
47	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
53	C₂×Dic₇₃	—	2	1	no	—
54	C₂×Dic₇₃	—	2	1	no	—
55	C₂×Dic₇₃	—	4	1	no	—
56	C₂×Dic₇₃	—	8	1	no	—
57	C₈×C₇₃	—	2	2	no	—
58	C₈×C₇₃	—	4	1	no	—
59	#10596831	—	2	2	no	—
60	#10596831	—	2	2	no	—
61	C₂²×C₇₃	—	2	1	no	—
62	C₂²×C₇₃	—	2	1	no	—
63	C₂²×C₇₃	#121	1	1	yes	characteristic
64	C₂²×C₇₃	#125	1	1	yes	characteristic
65	C₂²×C₇₃	#125	1	1	yes	characteristic
66	C₄×C₇₃	—	8	1	no	—
67	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
68	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
69	C₄×C₇₃	#123	1	1	yes	characteristic
70	C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
71	Dic₇₃	—	4	1	no	—
72	Dic₇₃	—	4	1	no	—
73	Dic₇₃	—	8	1	no	—
74	C₂×C₇₃	—	2	1	no	—
75	C₂×C₇₃	—	2	1	no	—
76	C₂×C₇₃	#398	1	1	yes	characteristic
77	C₂×C₇₃	#472	1	1	yes	characteristic
78	C₂×C₇₃	#473	1	1	yes	characteristic
79	#2089	—	73	1	no	maximal, Sylow
80	C₇₃	#2089	1	1	yes	characteristic, complemented, Sylow
81	#428	—	73	1	no	—
82	#428	—	73	1	no	—
83	C₂×#107	—	73	1	no	—
84	#424	—	73	1	no	—
85	#343	—	73	1	no	—
86	#343	D₇₃	1	1	yes	characteristic
87	#341	—	73	1	no	—
88	C₂×M₁₆	—	73	1	no	—
89	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
90	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
91	C₂²×C₈	—	73	1	no	—
92	#117	—	73	1	no	—
93	C₂×C₄⋊C₄	—	73	1	no	—
94	C₂×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
95	#107	—	146	1	no	—
96	#107	—	146	2	no	—
97	#107	—	146	1	no	—
98	#106	—	146	1	no	—
99	#106	—	146	1	no	—
100	C₄⋊C₈	—	146	2	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	146	1	no	—
103	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
104	M₁₆	—	2	1	no	—
105	M₁₆	—	2	1	no	—
106	M₁₆	—	2	1	no	—
107	M₁₆	—	73	2	no	—
108	M₁₆	—	146	1	no	—
109	M₁₆	Dic₁₄₆	1	2	yes	—
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	73	2	no	—
113	C₂×C₈	—	146	1	no	—
114	C₂×C₈	—	146	1	no	—
115	C₂×C₈	—	146	1	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	2	1	no	—
118	C₄⋊C₄	—	73	1	no	—
119	C₄⋊C₄	—	73	1	no	—
120	C₄⋊C₄	—	146	1	no	—
121	C₄⋊C₄	—	146	1	no	—
122	C₂²⋊C₄	—	146	1	no	—
123	C₄²	—	146	1	no	—
124	C₂³	#11924028	1	1	yes	characteristic
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	8	1	no	—
127	C₂×C₄	—	292	1	no	—
128	C₂×C₄	—	584	1	no	—
129	C₂×C₄	—	146	1	no	—
130	C₂×C₄	—	146	1	no	—
131	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
132	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
133	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
134	C₂×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
135	C₂×C₄	#11924027	1	1	yes	characteristic
136	C₂×C₄	#11924028	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	4	1	no	—
139	C₈	—	146	2	no	—
140	C₈	—	146	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	#14556564	1	1	yes	characteristic
144	C₂²	#14556581	1	1	yes	characteristic
145	C₂²	#14556583	1	1	yes	characteristic
146	C₄	—	8	1	no	—
147	C₄	—	292	1	no	—
148	C₄	—	292	1	no	—
149	C₄	—	584	1	no	—
150	C₄	#14556568	1	1	yes	characteristic
151	C₄	#14556569	1	1	yes	characteristic
152	C₄	#14556573	1	1	yes	characteristic
153	C₄	#14556590	1	1	yes	characteristic
154	C₂	—	2	1	no	—
155	C₂	—	2	1	no	—
156	C₂	#17988521	1	1	yes	characteristic
157	C₂	#17988856	1	1	yes	characteristic
158	C₂	#17988861	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988629

C₂

yes

maximal, characteristic

#17988627

C₂

yes

maximal, characteristic

C₂×#14556524

C₂

yes

maximal, characteristic

#17988615

C₂

yes

maximal, characteristic

#17988355

C₂

yes

maximal, characteristic

#17988351

C₂

yes

maximal, characteristic

#343×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924012

C₂²

yes

characteristic

C₂²×#10596831

C₂²

yes

characteristic

#14556550

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

—

—

—

—

—

—

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₂²

yes

characteristic

C₂²×Dic₇₃

—

#11924012

—

#11924012

Q₈

yes

—

—

—

—

D₄

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

characteristic

C₄⋊Dic₇₃

D₄

yes

characteristic

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

—

M₁₆×C₇₃

Q₈

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₂³×C₇₃

#41

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

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

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

#123

yes

characteristic

C₄×C₇₃

C₄⋊Q₈

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₃

yes

characteristic

#341

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₃

yes

characteristic

C₂×M₁₆

C₂×D₇₃

yes

characteristic

C₂²×C₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₃

yes

characteristic

#107

—

146

—

#107

—

146

—

#107

—

146

—

#106

—

146

—

#106

—

146

—

100

C₄⋊C₈

—

146

—

101

C₂²×C₄

—

102

C₂²×C₄

—

146

—

103

C₂²×C₄

C₂²×D₇₃

yes

characteristic

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

146

—

109

M₁₆

Dic₁₄₆

yes

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

146

—

114

C₂×C₈

—

146

—

115

C₂×C₈

—

146

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

146

—

121

C₄⋊C₄

—

146

—

122

C₂²⋊C₄

—

146

—

123

C₄²

—

146

—

124

C₂³

#11924028

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

292

—

128

C₂×C₄

—

584

—

129

C₂×C₄

—

146

—

130

C₂×C₄

—

146

—

131

C₂×C₄

D₄×D₇₃

yes

characteristic

132

C₂×C₄

D₄×D₇₃

yes

characteristic

133

C₂×C₄

Q₈×D₇₃

yes

characteristic

134

C₂×C₄

C₂×Dic₁₄₆

yes

characteristic

135

C₂×C₄

#11924027

yes

characteristic

136

C₂×C₄

#11924028

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

146

—

140

C₈

—

146

—

141

C₂²

—

142

C₂²

—

143

C₂²

#14556564

yes

characteristic

144

C₂²

#14556581

yes

characteristic

145

C₂²

#14556583

yes

characteristic

146

C₄

—

147

C₄

—

292

—

148

C₄

—

292

—

149

C₄

—

584

—

150

C₄

#14556568

yes

characteristic

151

C₄

#14556569

yes

characteristic

152

C₄

#14556573

yes

characteristic

153

C₄

#14556590

yes

characteristic

154

C₂

—

155

C₂

—

156

C₂

#17988521

yes

characteristic

157

C₂

#17988856

yes

characteristic

158

C₂

#17988861

yes

characteristic

	subgroup	G/H
center	C₂²	#14556564
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#343×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#343	D₇₃
hypocenter	C₇₃	#2089
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

#14556564

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#343×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
80. C₇₃	#2089	semidirect

H ≅ G/N

type

80. C₇₃

#2089

semidirect

Subgroups — #22611440

Conjugacy classes of subgroups

Complements