Group #22611237 — subgroups

Subgroups — #22611237

Order 9344.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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	#17989387	C₂	1	1	yes	maximal, characteristic
2	#17988584	C₂	1	2	yes	maximal, complemented
3	#17988560	C₂	1	2	yes	maximal, complemented
4	#17988344	C₂	1	1	yes	maximal, characteristic, complemented
5	#424×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11924028	C₂²	1	2	yes	—
7	#14556624	—	2	2	no	—
8	#14556619	—	2	2	no	—
9	#14556613	—	2	2	no	—
10	#14556550	C₂²	1	2	yes	—
11	#14556516	—	2	2	no	—
12	#14556513	—	2	2	no	—
13	#14556505	—	2	4	no	—
14	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	#117×C₇₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₇₃	—	2	2	no	—
18	#11924028	—	2	2	no	—
19	#11924028	—	2	2	no	—
20	#11924028	—	2	2	no	—
21	#11924028	—	2	2	no	—
22	C₂×Dic₁₄₆	D₄	1	2	yes	—
23	C₂×D₂₉₂	D₄	1	2	yes	—
24	C₂×C₂²⋊D₇₃	—	2	2	no	—
25	C₂×C₄×D₇₃	—	2	2	no	—
26	Dic₂₉₂	—	2	2	no	—
27	#11924017	—	2	4	no	—
28	D₅₈₄	—	2	2	no	—
29	C₄⋊Dic₇₃	—	2	2	no	—
30	C₂²⋊Dic₇₃	—	2	2	no	—
31	C₄×Dic₇₃	—	2	2	no	—
32	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
33	M₁₆×C₇₃	—	2	2	no	—
34	C₂×C₈×C₇₃	—	2	1	no	—
35	C₂×C₈×C₇₃	—	2	2	no	—
36	C₂×C₈×C₇₃	D₄	1	2	yes	—
37	C₄⋊C₄×C₇₃	—	2	1	no	—
38	C₂²⋊C₄×C₇₃	D₄	1	2	yes	—
39	C₄²×C₇₃	—	2	1	no	—
40	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₇₃	—	2	2	no	—
42	C₂×C₄×C₇₃	—	2	2	no	—
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	C₂²⋊D₇₃	—	4	2	no	—
48	C₂²⋊D₇₃	—	4	2	no	—
49	C₂×Dic₇₃	—	2	2	no	—
50	C₂×Dic₇₃	—	4	2	no	—
51	D₂₉₂	—	2	2	no	—
52	D₂₉₂	—	2	2	no	—
53	C₄×D₇₃	—	4	2	no	—
54	C₄×D₇₃	—	4	2	no	—
55	Dic₁₄₆	—	2	2	no	—
56	Dic₁₄₆	—	2	2	no	—
57	C₈×C₇₃	—	2	2	no	—
58	C₈×C₇₃	—	4	1	no	—
59	C₂²×C₇₃	—	4	1	no	—
60	C₂²×C₇₃	#121	1	1	yes	characteristic
61	C₂²×C₇₃	#121	1	2	yes	—
62	C₂×D₇₃	—	4	2	no	—
63	C₂×D₇₃	—	8	2	no	—
64	C₄×C₇₃	—	4	2	no	—
65	C₄×C₇₃	#120	1	1	yes	characteristic
66	C₄×C₇₃	#123	1	2	yes	—
67	C₄×C₇₃	#124	1	1	yes	characteristic
68	Dic₇₃	—	4	2	no	—
69	Dic₇₃	—	8	2	no	—
70	C₂×C₇₃	—	2	2	no	—
71	C₂×C₇₃	—	2	1	no	—
72	C₂×C₇₃	#394	1	1	yes	characteristic
73	D₇₃	—	8	2	no	—
74	#2053	—	73	1	no	maximal, Sylow
75	C₇₃	#2053	1	1	yes	characteristic, complemented, Sylow
76	#575	—	73	1	no	—
77	#424	D₇₃	1	1	yes	characteristic, complemented
78	#416	—	73	2	no	—
79	#410	—	73	2	no	—
80	H(3,ℤ/4)	—	73	1	no	—
81	C₂×#41	—	73	2	no	—
82	#137	—	146	2	no	—
83	AGL(1,ℤ/8)	—	146	2	no	—
84	#135	—	146	2	no	—
85	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
86	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
87	#117	—	73	2	no	—
88	#117	C₂×D₇₃	1	1	yes	characteristic
89	C₄⋊C₈	—	2	2	no	—
90	#103	—	146	2	no	—
91	#102	—	146	2	no	—
92	#99	—	146	4	no	—
93	#41	—	146	2	no	—
94	#41	—	146	2	no	—
95	#41	—	146	2	no	—
96	#41	—	146	2	no	—
97	C₂×Q₈	—	73	2	no	—
98	C₂×D₄	—	73	2	no	—
99	C₂×D₄	—	146	2	no	—
100	C₂²×C₄	—	146	2	no	—
101	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
102	Q₁₆	—	146	2	no	—
103	SD₁₆	—	146	4	no	—
104	D₈	—	146	2	no	—
105	M₁₆	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	C₂²⋊D₇₃	1	2	yes	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	146	2	no	—
111	C₂²⋊C₄	—	146	2	no	—
112	C₂²⋊C₄	D₂₉₂	1	2	yes	—
113	C₄²	—	2	1	no	—
114	C₄²	—	146	2	no	—
115	C₂³	—	146	2	no	—
116	C₂³	C₂×D₂₉₂	1	1	yes	characteristic
117	Q₈	—	146	2	no	—
118	Q₈	—	146	2	no	—
119	D₄	—	292	2	no	—
120	D₄	—	292	2	no	—
121	D₄	—	146	2	no	—
122	D₄	—	146	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	292	2	no	—
125	C₂×C₄	—	292	2	no	—
126	C₂×C₄	—	292	2	no	—
127	C₂×C₄	—	146	2	no	—
128	C₂×C₄	D₄×D₇₃	1	2	yes	—
129	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
130	C₂×C₄	#11924028	1	1	yes	characteristic
131	C₂×C₄	#11924029	1	2	yes	—
132	C₈	—	2	2	no	—
133	C₈	—	4	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	—	292	2	no	—
136	C₂²	—	584	2	no	—
137	C₂²	#14556565	1	1	yes	characteristic
138	C₂²	#14556566	1	2	yes	—
139	C₄	—	4	2	no	—
140	C₄	—	292	2	no	—
141	C₄	—	584	2	no	—
142	C₄	#14556559	1	1	yes	characteristic
143	C₄	#14556576	1	2	yes	—
144	C₄	#14556580	1	1	yes	characteristic
145	C₂	—	2	1	no	—
146	C₂	—	2	2	no	—
147	C₂	—	584	2	no	—
148	C₂	#17988504	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989387

C₂

yes

maximal, characteristic

#17988584

C₂

yes

maximal, complemented

#17988560

C₂

yes

maximal, complemented

#17988344

C₂

yes

maximal, characteristic, complemented

#424×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924028

C₂²

yes

—

—

—

—

C₂²

yes

—

#14556516

—

#14556513

—

#14556505

—

C₂×M₁₆×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

#117×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

—

—

—

—

—

C₂×Dic₁₄₆

D₄

yes

—

C₂×D₂₉₂

D₄

yes

—

C₂×C₂²⋊D₇₃

—

C₂×C₄×D₇₃

—

Dic₂₉₂

—

#11924017

—

D₅₈₄

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

—

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

D₄

yes

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

C₂²⋊D₇₃

—

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#121

yes

characteristic

C₂²×C₇₃

#121

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#120

yes

characteristic

C₄×C₇₃

#123

yes

—

C₄×C₇₃

#124

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#394

yes

characteristic

D₇₃

—

#2053

—

maximal, Sylow

C₇₃

#2053

yes

characteristic, complemented, Sylow

#575

—

#424

D₇₃

yes

characteristic, complemented

#416

—

#410

—

H(3,ℤ/4)

—

C₂×#41

—

#137

—

146

—

AGL(1,ℤ/8)

—

146

—

#135

—

146

—

C₂×M₁₆

C₂×D₇₃

yes

characteristic

C₂²×C₈

C₂×D₇₃

yes

characteristic

#117

—

#117

C₂×D₇₃

yes

characteristic

C₄⋊C₈

—

#103

—

146

—

#102

—

146

—

#99

—

146

—

#41

—

146

—

#41

—

146

—

#41

—

146

—

#41

—

146

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

146

—

100

C₂²×C₄

—

146

—

101

C₂²×C₄

C₂²×D₇₃

yes

characteristic

102

Q₁₆

—

146

—

103

SD₁₆

—

146

—

104

D₈

—

146

—

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²⋊D₇₃

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

146

—

111

C₂²⋊C₄

—

146

—

112

C₂²⋊C₄

D₂₉₂

yes

—

113

C₄²

—

114

C₄²

—

146

—

115

C₂³

—

146

—

116

C₂³

C₂×D₂₉₂

yes

characteristic

117

Q₈

—

146

—

118

Q₈

—

146

—

119

D₄

—

292

—

120

D₄

—

292

—

121

D₄

—

146

—

122

D₄

—

146

—

123

C₂×C₄

—

124

C₂×C₄

—

292

—

125

C₂×C₄

—

292

—

126

C₂×C₄

—

292

—

127

C₂×C₄

—

146

—

128

C₂×C₄

D₄×D₇₃

yes

—

129

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

130

C₂×C₄

#11924028

yes

characteristic

131

C₂×C₄

#11924029

yes

—

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

292

—

136

C₂²

—

584

—

137

C₂²

#14556565

yes

characteristic

138

C₂²

#14556566

yes

—

139

C₄

—

140

C₄

—

292

—

141

C₄

—

584

—

142

C₄

#14556559

yes

characteristic

143

C₄

#14556576

yes

—

144

C₄

#14556580

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

584

—

148

C₂

#17988504

yes

characteristic

	subgroup	G/H
center	C₄	#14556580
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	#424×C₇₃	C₂
socle	C₂×C₇₃	#394
hypercenter	#424	D₇₃
hypocenter	C₇₃	#2053
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₄

#14556580

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

#424×C₇₃

C₂

C₂×C₇₃

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
2. #17988584	C₂	semidirect
3. #17988560	C₂	semidirect
4. #17988344	C₂	semidirect
5. #424×C₇₃	C₂	semidirect
75. C₇₃	#2053	semidirect
77. #424	D₇₃	semidirect

H ≅ G/N

type

2. #17988584

C₂

semidirect

3. #17988560

C₂

semidirect

4. #17988344

C₂

semidirect

5. #424×C₇₃

C₂

semidirect

75. C₇₃

#2053

semidirect

77. #424

D₇₃

semidirect

Subgroups — #22611237

Conjugacy classes of subgroups

Complements