Group #22608619 — subgroups

Subgroups — #22608619

Order 9344.

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

normal subgroups only

This group has 216 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	#17989144	C₂	1	1	yes	maximal, characteristic
2	#17988611	C₂	1	1	yes	maximal, characteristic
3	#17988335	C₂	1	2	yes	maximal, complemented
4	#17988334	C₂	1	2	yes	maximal, complemented
5	C₂×C₄⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924012	—	2	1	no	—
7	#14556591	C₂²	1	1	yes	characteristic
8	#14556591	C₂²	1	1	yes	characteristic
9	#14556583	—	2	1	no	—
10	#14556570	—	2	1	no	—
11	#14556570	C₄	1	2	yes	—
12	C₄×Dic₁₄₆	—	2	1	no	—
13	C₄×Dic₁₄₆	C₄	1	2	yes	—
14	#14556550	—	2	1	no	—
15	C₄⋊#10596831	—	2	1	no	—
16	C₄⋊#10596831	C₂²	1	2	yes	—
17	C₂²×C₈×C₇₃	—	2	1	no	—
18	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₇₃	—	2	1	no	—
20	C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
21	C₂×Dic₁₄₆	—	2	1	no	—
22	C₂×Dic₁₄₆	C₂×C₄	1	2	yes	—
23	#11924012	—	4	2	no	—
24	C₂×#10596831	—	2	2	no	—
25	C₄⋊Dic₇₃	—	2	1	no	—
26	C₄⋊Dic₇₃	—	2	1	no	—
27	C₄⋊Dic₇₃	—	2	1	no	—
28	C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
29	#11924007	—	2	2	no	—
30	#11924007	—	2	2	no	—
31	C₂²⋊Dic₇₃	—	2	2	no	—
32	C₄×Dic₇₃	—	2	2	no	—
33	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
36	C₂×C₈×C₇₃	—	2	2	no	—
37	C₂×C₈×C₇₃	—	4	2	no	—
38	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₇₃	D₄	1	1	yes	characteristic
40	C₄²×C₇₃	D₄	1	1	yes	characteristic
41	C₄²×C₇₃	C₂³	1	1	yes	characteristic
42	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₃	—	2	1	no	—
44	C₂×C₄×C₇₃	—	2	1	no	—
45	C₂×C₄×C₇₃	—	4	1	no	—
46	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₇₃	SD₁₆	1	2	yes	—
50	C₂×C₄×C₇₃	Q₁₆	1	2	yes	—
51	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₇₃	—	2	2	no	—
55	C₂×Dic₇₃	—	2	2	no	—
56	Dic₁₄₆	—	4	1	no	—
57	Dic₁₄₆	—	4	1	no	—
58	Dic₁₄₆	—	4	2	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	#103	1	2	yes	—
64	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₇₃	—	2	1	no	—
66	C₄×C₇₃	—	2	1	no	—
67	C₄×C₇₃	—	2	1	no	—
68	C₄×C₇₃	#103	1	2	yes	—
69	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
70	C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
71	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₄×C₇₃	#137	1	1	yes	characteristic
73	Dic₇₃	—	4	2	no	—
74	Dic₇₃	—	4	2	no	—
75	C₂×C₇₃	—	2	2	no	—
76	C₂×C₇₃	#413	1	1	yes	characteristic
77	C₂×C₇₃	C₂×#103	1	1	yes	characteristic
78	C₂×C₇₃	#417	1	1	yes	characteristic
79	#1550	—	73	1	no	maximal, Sylow
80	C₇₃	#1550	1	1	yes	characteristic, complemented, Sylow
81	#533	—	73	1	no	—
82	#423	—	73	1	no	—
83	C₂×C₄⋊C₈	D₇₃	1	1	yes	characteristic
84	#333	—	73	2	no	—
85	#332	—	73	2	no	—
86	C₂×M₁₆	—	146	1	no	—
87	C₂²×C₈	—	2	1	no	—
88	C₄⋊Q₈	—	73	1	no	—
89	C₄⋊Q₈	—	73	1	no	—
90	#125	—	146	1	no	—
91	C₂²⋊Q₈	—	73	2	no	—
92	C₂²⋊Q₈	—	146	1	no	—
93	C₄×Q₈	—	73	2	no	—
94	C₄×Q₈	—	146	1	no	—
95	#117	—	146	1	no	—
96	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
97	C₄⋊C₈	—	2	1	no	—
98	C₄⋊C₈	—	73	2	no	—
99	C₄⋊C₈	—	146	1	no	—
100	C₄⋊C₈	C₂×D₇₃	1	2	yes	—
101	C₂×Q₈	—	73	2	no	—
102	C₂×Q₈	—	146	1	no	—
103	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
104	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
106	M₁₆	—	292	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	146	2	no	—
110	C₄⋊C₄	—	73	2	no	—
111	C₄⋊C₄	—	146	2	no	—
112	C₄⋊C₄	—	146	2	no	—
113	C₄⋊C₄	—	146	1	no	—
114	C₄⋊C₄	—	146	1	no	—
115	C₄⋊C₄	—	146	1	no	—
116	C₂²⋊C₄	—	146	2	no	—
117	C₄²	—	146	2	no	—
118	C₄²	C₄×D₇₃	1	1	yes	characteristic
119	C₄²	D₂₉₂	1	1	yes	characteristic
120	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
121	C₄²	C₂²×D₇₃	1	1	yes	characteristic
122	C₂³	#11924005	1	1	yes	characteristic
123	Q₈	—	292	2	no	—
124	Q₈	—	292	1	no	—
125	Q₈	—	292	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	4	1	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	#11924017	1	2	yes	—
135	C₂×C₄	Dic₂₉₂	1	2	yes	—
136	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
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	#14556516	1	2	yes	—
144	C₂²	C₂×#11924005	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	292	2	no	—
149	C₄	—	292	2	no	—
150	C₄	#14556516	1	2	yes	—
151	C₄	C₂×#11924017	1	1	yes	characteristic
152	C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
153	C₄	#14556617	1	1	yes	characteristic
154	C₄	#14556622	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	#17988569	1	1	yes	characteristic
157	C₂	C₂×#14556516	1	1	yes	characteristic
158	C₂	#17988586	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989144

C₂

yes

maximal, characteristic

#17988611

C₂

yes

maximal, characteristic

#17988335

C₂

yes

maximal, complemented

#17988334

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924012

—

#14556591

C₂²

yes

characteristic

#14556591

C₂²

yes

—

—

C₄

yes

—

C₄×Dic₁₄₆

—

C₄×Dic₁₄₆

C₄

yes

—

#14556550

—

C₄⋊#10596831

—

C₄⋊#10596831

C₂²

yes

—

C₂²×C₈×C₇₃

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

C₂²

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924007

—

#11924007

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

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

SD₁₆

yes

—

C₂×C₄×C₇₃

Q₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#103

yes

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#103

yes

—

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₄×C₇₃

C₂×Q₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#533

—

#423

—

C₂×C₄⋊C₈

D₇₃

yes

characteristic

#333

—

#332

—

C₂×M₁₆

—

146

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

146

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

146

—

C₄×Q₈

—

C₄×Q₈

—

146

—

#117

—

146

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

100

C₄⋊C₈

C₂×D₇₃

yes

—

101

C₂×Q₈

—

102

C₂×Q₈

—

146

—

103

C₂²×C₄

C₄×D₇₃

yes

characteristic

104

C₂²×C₄

D₂₉₂

yes

characteristic

105

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

106

M₁₆

—

292

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

146

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

146

—

112

C₄⋊C₄

—

146

—

113

C₄⋊C₄

—

146

—

114

C₄⋊C₄

—

146

—

115

C₄⋊C₄

—

146

—

116

C₂²⋊C₄

—

146

—

117

C₄²

—

146

—

118

C₄²

C₄×D₇₃

yes

characteristic

119

C₄²

D₂₉₂

yes

characteristic

120

C₄²

C₂²⋊D₇₃

yes

characteristic

121

C₄²

C₂²×D₇₃

yes

characteristic

122

C₂³

#11924005

yes

characteristic

123

Q₈

—

292

—

124

Q₈

—

292

—

125

Q₈

—

292

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

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

#11924017

yes

—

135

C₂×C₄

Dic₂₉₂

yes

—

136

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

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

#14556516

yes

—

144

C₂²

C₂×#11924005

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

292

—

149

C₄

—

292

—

150

C₄

#14556516

yes

—

151

C₄

C₂×#11924017

yes

characteristic

152

C₄

C₂×Dic₂₉₂

yes

characteristic

153

C₄

#14556617

yes

characteristic

154

C₄

#14556622

yes

characteristic

155

C₂

—

156

C₂

#17988569

yes

characteristic

157

C₂

C₂×#14556516

yes

characteristic

158

C₂

#17988586

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11924005
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×D₇₃
Fitting	C₂×C₄⋊C₈×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	C₂×C₄⋊C₈	D₇₃
hypocenter	C₇₃	#1550
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂²

C₂×#11924005

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×D₇₃

Fitting

C₂×C₄⋊C₈×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

hypercenter

C₂×C₄⋊C₈

D₇₃

hypocenter

C₇₃

#1550

LCS

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #17988335	C₂	semidirect
4. #17988334	C₂	semidirect
80. C₇₃	#1550	semidirect

H ≅ G/N

type

3. #17988335

C₂

semidirect

4. #17988334

C₂

semidirect

80. C₇₃

#1550

semidirect

Subgroups — #22608619

Conjugacy classes of subgroups

Complements