Group #22611074 — subgroups

Subgroups — #22611074

Order 9344.

294 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 87 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×#14556511	C₂	1	2	yes	maximal
2	#17988489	C₂	1	1	yes	maximal, characteristic
3	#17988349	C₂	1	2	yes	maximal, complemented
4	#17988342	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#125×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₂₉₂	C₂²	1	1	yes	characteristic
7	C₂²×#10596831	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
9	C₂×#11924005	—	2	2	no	—
10	#14556511	—	2	4	no	—
11	#125×C₇₃	—	2	4	no	—
12	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
13	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
15	C₂³×D₇₃	—	2	1	no	—
16	C₂×D₂₉₂	—	2	2	no	—
17	C₂×D₂₉₂	—	2	2	no	—
18	C₂×D₂₉₂	—	2	1	no	—
19	C₂×D₂₉₂	Q₈	1	2	yes	—
20	C₂²×Dic₇₃	—	2	1	no	—
21	C₂×#10596831	—	2	2	no	—
22	C₂×#10596831	—	2	2	no	—
23	C₂×#10596831	—	2	2	no	—
24	C₄⋊Dic₇₃	—	2	2	no	—
25	#11924005	—	4	4	no	—
26	C₂²×C₄×C₇₃	—	2	2	no	—
27	C₂²×C₄×C₇₃	D₄	1	2	yes	—
28	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₇₃	—	2	8	no	—
30	C₄⋊C₄×C₇₃	—	2	2	no	—
31	C₄⋊C₄×C₇₃	D₄	1	4	yes	—
32	C₄²×C₇₃	—	2	2	no	—
33	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
34	C₂²×D₇₃	—	2	2	no	—
35	C₂²×D₇₃	—	4	1	no	—
36	C₂²×D₇₃	—	4	1	no	—
37	C₂²×D₇₃	—	4	2	no	—
38	C₂²×D₇₃	—	4	2	no	—
39	C₂×C₄×C₇₃	—	2	2	no	—
40	C₂×C₄×C₇₃	—	2	4	no	—
41	C₂×C₄×C₇₃	—	2	2	no	—
42	C₂×C₄×C₇₃	—	2	2	no	—
43	C₂×C₄×C₇₃	—	4	2	no	—
44	C₂×C₄×C₇₃	—	4	2	no	—
45	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₇₃	#41	1	2	yes	—
48	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
49	C₂×Dic₇₃	—	4	1	no	—
50	C₂×Dic₇₃	—	4	2	no	—
51	D₂₉₂	—	2	4	no	—
52	D₂₉₂	—	4	2	no	—
53	#10596831	—	4	2	no	—
54	C₂²×C₇₃	#120	1	1	yes	characteristic
55	C₂²×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂²×C₇₃	#124	1	1	yes	characteristic
57	C₂²×C₇₃	#135	1	2	yes	—
58	C₂²×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
59	C₂×D₇₃	—	4	2	no	—
60	C₂×D₇₃	—	4	4	no	—
61	C₂×D₇₃	—	8	2	no	—
62	C₂×D₇₃	—	8	1	no	—
63	C₂×D₇₃	—	8	1	no	—
64	C₄×C₇₃	—	4	2	no	—
65	C₄×C₇₃	—	4	4	no	—
66	C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₇₃	#124	1	2	yes	—
68	Dic₇₃	—	8	1	no	—
69	C₂×C₇₃	#393	1	1	yes	characteristic
70	C₂×C₇₃	#449	1	2	yes	—
71	C₂×C₇₃	#478	1	2	yes	—
72	C₂×C₇₃	#488	1	1	yes	characteristic
73	C₂×C₇₃	#490	1	1	yes	characteristic
74	D₇₃	—	8	2	no	—
75	#2025	—	73	1	no	maximal, Sylow
76	C₇₃	#2025	1	1	yes	characteristic, complemented, Sylow
77	C₂×#125	D₇₃	1	1	yes	characteristic, complemented
78	C₂×#102	—	73	2	no	—
79	#390	—	73	1	no	—
80	#340	—	73	2	no	—
81	#336	—	73	1	no	—
82	C₂²×D₄	—	73	1	no	—
83	C₂²×C₈	—	73	2	no	—
84	#125	—	2	4	no	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄⋊C₄	—	73	1	no	—
87	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
88	C₂×C₂²⋊C₄	—	146	2	no	—
89	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
90	#102	—	146	4	no	—
91	C₂⁴	—	146	1	no	—
92	C₂×D₄	—	73	2	no	—
93	C₂×D₄	—	146	2	no	—
94	C₂×D₄	—	146	1	no	—
95	C₂×D₄	—	146	2	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	146	1	no	—
98	C₂²×C₄	C₂²⋊D₇₃	1	2	yes	—
99	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
100	C₂×C₈	—	146	2	no	—
101	C₂×C₈	—	146	2	no	—
102	C₂×C₈	—	146	2	no	—
103	C₄⋊C₄	—	2	8	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	146	2	no	—
106	C₄⋊C₄	C₂²⋊D₇₃	1	4	yes	—
107	C₂²⋊C₄	—	292	4	no	—
108	C₄²	—	2	2	no	—
109	C₂³	—	292	1	no	—
110	C₂³	—	292	1	no	—
111	C₂³	—	292	2	no	—
112	C₂³	—	292	2	no	—
113	C₂³	—	146	2	no	—
114	C₂³	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
115	D₄	—	292	2	no	—
116	D₄	—	146	4	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	292	1	no	—
124	C₂×C₄	—	292	2	no	—
125	C₂×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
126	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
127	C₂×C₄	#11924029	1	2	yes	—
128	C₂×C₄	#11924029	1	1	yes	characteristic
129	C₈	—	292	2	no	—
130	C₂²	—	292	4	no	—
131	C₂²	—	292	2	no	—
132	C₂²	—	584	1	no	—
133	C₂²	—	584	1	no	—
134	C₂²	—	584	2	no	—
135	C₂²	#14556560	1	1	yes	characteristic
136	C₂²	#14556567	1	1	yes	characteristic
137	C₂²	#14556580	1	1	yes	characteristic
138	C₂²	#14556611	1	2	yes	—
139	C₂²	#14556616	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	4	4	no	—
142	C₄	—	584	1	no	—
143	C₄	#14556567	1	2	yes	—
144	C₄	#14556580	1	2	yes	—
145	C₂	—	584	2	no	—
146	C₂	#17988498	1	1	yes	characteristic
147	C₂	#17988732	1	2	yes	—
148	C₂	#17988884	1	2	yes	—
149	C₂	#17988944	1	1	yes	characteristic
150	C₂	#17988953	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556511

C₂

yes

maximal

#17988489

C₂

yes

maximal, characteristic

#17988349

C₂

yes

maximal, complemented

#17988342

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₉₂

C₂²

yes

characteristic

C₂²×#10596831

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

characteristic

C₂×#11924005

—

#14556511

—

#125×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂³×D₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

Q₈

yes

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

#11924005

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

D₄

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₃

#41

yes

—

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

#10596831

—

C₂²×C₇₃

#120

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₂²×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊Q₈

yes

—

C₄×C₇₃

#124

yes

—

Dic₇₃

—

C₂×C₇₃

#393

yes

characteristic

C₂×C₇₃

#449

yes

—

C₂×C₇₃

#478

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#2025

—

maximal, Sylow

C₇₃

#2025

yes

characteristic, complemented, Sylow

C₂×#125

D₇₃

yes

characteristic, complemented

—

—

—

—

C₂²×D₄

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

C₂×C₂²⋊C₄

—

146

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

#102

—

146

—

C₂⁴

—

146

—

C₂×D₄

—

C₂×D₄

—

146

—

C₂×D₄

—

146

—

C₂×D₄

—

146

—

C₂²×C₄

—

C₂²×C₄

—

146

—

C₂²×C₄

C₂²⋊D₇₃

yes

—

C₂²×C₄

C₂²×D₇₃

yes

characteristic

100

C₂×C₈

—

146

—

101

C₂×C₈

—

146

—

102

C₂×C₈

—

146

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

146

—

106

C₄⋊C₄

C₂²⋊D₇₃

yes

—

107

C₂²⋊C₄

—

292

—

108

C₄²

—

109

C₂³

—

292

—

110

C₂³

—

292

—

111

C₂³

—

292

—

112

C₂³

—

292

—

113

C₂³

—

146

—

114

C₂³

C₂×C₂²⋊D₇₃

yes

characteristic

115

D₄

—

292

—

116

D₄

—

146

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

292

—

124

C₂×C₄

—

292

—

125

C₂×C₄

C₂×C₂²⋊D₇₃

yes

—

126

C₂×C₄

Q₈×D₇₃

yes

characteristic

127

C₂×C₄

#11924029

yes

—

128

C₂×C₄

#11924029

yes

characteristic

129

C₈

—

292

—

130

C₂²

—

292

—

131

C₂²

—

292

—

132

C₂²

—

584

—

133

C₂²

—

584

—

134

C₂²

—

584

—

135

C₂²

#14556560

yes

characteristic

136

C₂²

#14556567

yes

characteristic

137

C₂²

#14556580

yes

characteristic

138

C₂²

#14556611

yes

—

139

C₂²

#14556616

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

584

—

143

C₄

#14556567

yes

—

144

C₄

#14556580

yes

—

145

C₂

—

584

—

146

C₂

#17988498

yes

characteristic

147

C₂

#17988732

yes

—

148

C₂

#17988884

yes

—

149

C₂

#17988944

yes

characteristic

150

C₂

#17988953

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₃
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×#125×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₂×#125	D₇₃
hypocenter	C₇₃	#2025
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₃

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×#125×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

D₇₃

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
3. #17988349	C₂	semidirect
4. #17988342	C₂	semidirect
5. C₂×#125×C₇₃	C₂	semidirect
76. C₇₃	#2025	semidirect
77. C₂×#125	D₇₃	semidirect

H ≅ G/N

type

3. #17988349

C₂

semidirect

4. #17988342

C₂

semidirect

5. C₂×#125×C₇₃

C₂

semidirect

76. C₇₃

#2025

semidirect

77. C₂×#125

D₇₃

semidirect

Subgroups — #22611074

Conjugacy classes of subgroups

Complements