Group #22609573 — subgroups

Subgroups — #22609573

Order 9344.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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	#17988964	C₂	1	1	yes	maximal, characteristic
2	#17988961	C₂	1	1	yes	maximal, characteristic, complemented
3	#17988935	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988935	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988935	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988309	C₂	1	1	yes	maximal, characteristic, complemented
7	#487×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924015	—	2	2	no	—
9	C₂×D₅₈₄	—	2	2	no	—
10	#14556588	C₂²	1	1	yes	characteristic
11	#14556588	C₂²	1	1	yes	characteristic
12	#14556511	—	2	2	no	—
13	#14556511	—	2	2	no	—
14	#14556511	—	2	2	no	—
15	C₄×#10596831	C₂²	1	1	yes	characteristic
16	C₄×#10596831	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
19	#103×C₇₃	—	2	2	no	—
20	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×D₂₉₂	—	2	1	no	—
22	C₂×D₂₉₂	—	2	1	no	—
23	C₂×D₂₉₂	—	2	1	no	—
24	C₂×D₂₉₂	—	2	1	no	—
25	C₂×D₂₉₂	—	2	1	no	—
26	C₂×D₂₉₂	—	2	1	no	—
27	#11924015	—	4	4	no	—
28	D₅₈₄	—	4	4	no	—
29	C₂×#10596831	D₄	1	2	yes	—
30	C₂×#10596831	D₄	1	2	yes	—
31	C₂×Q₈×C₇₃	—	2	1	no	—
32	C₂×Q₈×C₇₃	—	2	1	no	—
33	C₂×C₈×C₇₃	D₄	1	2	yes	—
34	C₄⋊C₄×C₇₃	—	2	1	no	—
35	C₄⋊C₄×C₇₃	—	2	1	no	—
36	C₄⋊C₄×C₇₃	—	2	1	no	—
37	C₄⋊C₄×C₇₃	—	2	1	no	—
38	C₄²×C₇₃	C₂³	1	1	yes	characteristic
39	C₂²×D₇₃	—	4	1	no	—
40	C₂²×D₇₃	—	4	1	no	—
41	Q₈×C₇₃	—	4	2	no	—
42	Q₈×C₇₃	—	4	2	no	—
43	C₂×C₄×C₇₃	—	4	1	no	—
44	C₂×C₄×C₇₃	—	4	1	no	—
45	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
48	D₂₉₂	—	4	2	no	—
49	D₂₉₂	—	4	2	no	—
50	D₂₉₂	—	4	2	no	—
51	D₂₉₂	—	4	2	no	—
52	D₂₉₂	—	4	2	no	—
53	D₂₉₂	—	4	2	no	—
54	C₈×C₇₃	—	2	2	no	—
55	#10596831	—	2	2	no	—
56	#10596831	—	2	2	no	—
57	C₂²×C₇₃	#127	1	1	yes	characteristic
58	C₂×D₇₃	—	8	1	no	—
59	C₂×D₇₃	—	8	1	no	—
60	C₂×D₇₃	—	8	1	no	—
61	C₂×D₇₃	—	8	1	no	—
62	C₂×D₇₃	—	8	1	no	—
63	C₂×D₇₃	—	8	1	no	—
64	C₄×C₇₃	—	8	1	no	—
65	C₄×C₇₃	—	8	1	no	—
66	C₄×C₇₃	#135	1	2	yes	—
67	C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
68	C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
69	C₂×C₇₃	#495	1	1	yes	characteristic
70	C₂×C₇₃	#496	1	1	yes	characteristic
71	C₂×C₇₃	#496	1	1	yes	characteristic
72	D₇₃	—	16	1	no	—
73	D₇₃	—	16	1	no	—
74	#1717	—	73	1	no	maximal, Sylow
75	C₇₃	#1717	1	1	yes	characteristic, complemented, Sylow
76	#493	—	73	1	no	—
77	#492	—	73	1	no	—
78	#487	D₇₃	1	1	yes	characteristic, complemented
79	#486	—	73	1	no	—
80	#486	—	73	1	no	—
81	#486	—	73	1	no	—
82	#322	—	73	1	no	—
83	C₂×SD₁₆	—	146	2	no	—
84	C₂×D₈	—	146	2	no	—
85	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
86	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
87	#127	—	73	1	no	—
88	#127	—	73	1	no	—
89	#103	—	2	2	no	—
90	#102	—	146	2	no	—
91	#102	—	146	2	no	—
92	#102	—	146	2	no	—
93	C₄×C₈	—	73	1	no	—
94	C₄×C₈	—	73	1	no	—
95	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	2	1	no	—
98	C₂×D₄	—	146	1	no	—
99	C₂×D₄	—	146	1	no	—
100	C₂×D₄	—	146	1	no	—
101	C₂×D₄	—	146	1	no	—
102	C₂×D₄	—	146	1	no	—
103	C₂×D₄	—	146	1	no	—
104	SD₁₆	—	292	4	no	—
105	D₈	—	292	4	no	—
106	C₂×C₈	—	73	2	no	—
107	C₂×C₈	—	73	2	no	—
108	C₂×C₈	C₂²⋊D₇₃	1	2	yes	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄²	C₂²×D₇₃	1	1	yes	characteristic
114	C₂³	—	292	1	no	—
115	C₂³	—	292	1	no	—
116	Q₈	—	4	2	no	—
117	Q₈	—	4	2	no	—
118	D₄	—	292	2	no	—
119	D₄	—	292	2	no	—
120	D₄	—	292	2	no	—
121	D₄	—	292	2	no	—
122	D₄	—	292	2	no	—
123	D₄	—	292	2	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
127	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	146	2	no	—
131	C₈	—	146	2	no	—
132	C₂²	—	584	1	no	—
133	C₂²	—	584	1	no	—
134	C₂²	—	584	1	no	—
135	C₂²	—	584	1	no	—
136	C₂²	—	584	1	no	—
137	C₂²	—	584	1	no	—
138	C₂²	#14556587	1	1	yes	characteristic
139	C₄	—	8	1	no	—
140	C₄	—	8	1	no	—
141	C₄	#14556614	1	2	yes	—
142	C₄	#14556616	1	2	yes	—
143	C₄	#14556618	1	2	yes	—
144	C₂	—	1168	1	no	—
145	C₂	—	1168	1	no	—
146	C₂	#17988972	1	1	yes	characteristic
147	C₂	#17988973	1	1	yes	characteristic
148	C₂	#17988977	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988964

C₂

yes

maximal, characteristic

#17988961

C₂

yes

maximal, characteristic, complemented

#17988935

C₂

yes

maximal, characteristic, complemented

#17988935

C₂

yes

maximal, characteristic, complemented

#17988935

C₂

yes

maximal, characteristic, complemented

#17988309

C₂

yes

maximal, characteristic, complemented

#487×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924015

—

C₂×D₅₈₄

—

#14556588

C₂²

yes

characteristic

#14556588

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄×#10596831

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

#103×C₇₃

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

#11924015

—

D₅₈₄

—

C₂×#10596831

D₄

yes

—

C₂×#10596831

D₄

yes

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

#10596831

—

#10596831

—

C₂²×C₇₃

#127

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#135

yes

—

C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₄×C₇₃

AGL(1,ℤ/8)

yes

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

D₇₃

—

#1717

—

maximal, Sylow

C₇₃

#1717

yes

characteristic, complemented, Sylow

#493

—

#492

—

#487

D₇₃

yes

characteristic, complemented

—

—

—

—

C₂×SD₁₆

—

146

—

C₂×D₈

—

146

—

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

C₄⋊Q₈

C₂×D₇₃

yes

—

—

—

—

146

—

#102

—

146

—

#102

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

146

—

C₂×D₄

—

146

—

100

C₂×D₄

—

146

—

101

C₂×D₄

—

146

—

102

C₂×D₄

—

146

—

103

C₂×D₄

—

146

—

104

SD₁₆

—

292

—

105

D₈

—

292

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²⋊D₇₃

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

C₂²×D₇₃

yes

characteristic

114

C₂³

—

292

—

115

C₂³

—

292

—

116

Q₈

—

117

Q₈

—

118

D₄

—

292

—

119

D₄

—

292

—

120

D₄

—

292

—

121

D₄

—

292

—

122

D₄

—

292

—

123

D₄

—

292

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

D₄×D₇₃

yes

characteristic

127

C₂×C₄

D₄×D₇₃

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

129

C₈

—

130

C₈

—

146

—

131

C₈

—

146

—

132

C₂²

—

584

—

133

C₂²

—

584

—

134

C₂²

—

584

—

135

C₂²

—

584

—

136

C₂²

—

584

—

137

C₂²

—

584

—

138

C₂²

#14556587

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

#14556614

yes

—

142

C₄

#14556616

yes

—

143

C₄

#14556618

yes

—

144

C₂

—

1168

—

145

C₂

—

1168

—

146

C₂

#17988972

yes

characteristic

147

C₂

#17988973

yes

characteristic

148

C₂

#17988977

yes

characteristic

	subgroup	G/H
center	C₂²	#14556587
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#487×C₇₃	C₂
socle	C₂²×C₇₃	#127
hypercenter	#487	D₇₃
hypocenter	C₇₃	#1717
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#14556587

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#487×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
2. #17988961	C₂	semidirect
3. #17988935	C₂	semidirect
4. #17988935	C₂	semidirect
5. #17988935	C₂	semidirect
6. #17988309	C₂	semidirect
7. #487×C₇₃	C₂	semidirect
75. C₇₃	#1717	semidirect
78. #487	D₇₃	semidirect

H ≅ G/N

type

2. #17988961

C₂

semidirect

3. #17988935

C₂

semidirect

4. #17988935

C₂

semidirect

5. #17988935

C₂

semidirect

6. #17988309

C₂

semidirect

7. #487×C₇₃

C₂

semidirect

75. C₇₃

#1717

semidirect

78. #487

D₇₃

semidirect

Subgroups — #22609573

Conjugacy classes of subgroups

Complements