Group #22609281 — subgroups

Subgroups — #22609281

Order 9344.

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

normal subgroups only

This group has 172 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	#17988961	C₂	1	1	yes	maximal, characteristic
2	#17988787	C₂	1	1	yes	maximal, characteristic
3	Q₈×#10596831	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988337	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988333	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988330	C₂	1	1	yes	maximal, characteristic
7	#475×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11924015	—	2	1	no	—
9	C₂×#11924015	—	2	1	no	—
10	C₂×#11924015	—	2	1	no	—
11	#14556588	C₂²	1	1	yes	characteristic
12	C₄⋊#10596831	—	2	1	no	—
13	C₄⋊#10596831	C₂²	1	1	yes	characteristic
14	C₄⋊#10596831	C₂²	1	1	yes	characteristic
15	#14556511	—	2	1	no	—
16	C₄×#10596831	—	2	1	no	—
17	C₄×#10596831	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
20	#106×C₇₃	—	2	1	no	—
21	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
22	#103×C₇₃	—	2	1	no	—
23	C₂×D₂₉₂	—	2	1	no	—
24	C₂×D₂₉₂	—	2	1	no	—
25	C₂×D₂₉₂	—	2	1	no	—
26	#11924015	—	4	2	no	—
27	#11924015	—	4	2	no	—
28	#11924015	—	4	2	no	—
29	C₂×#10596831	—	2	1	no	—
30	C₂×#10596831	—	2	1	no	—
31	C₂×#10596831	D₄	1	1	yes	characteristic
32	C₂×#10596831	D₄	1	1	yes	characteristic
33	C₂×Q₈×C₇₃	—	2	1	no	—
34	C₂×Q₈×C₇₃	D₄	1	1	yes	characteristic
35	C₂×C₈×C₇₃	—	2	1	no	—
36	C₄⋊C₄×C₇₃	—	2	1	no	—
37	C₄⋊C₄×C₇₃	—	2	1	no	—
38	C₄⋊C₄×C₇₃	—	2	1	no	—
39	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
40	C₄²×C₇₃	—	2	1	no	—
41	C₄²×C₇₃	C₂³	1	1	yes	characteristic
42	C₂²×D₇₃	—	4	1	no	—
43	Q₈×C₇₃	—	2	1	no	—
44	Q₈×C₇₃	—	4	1	no	—
45	Q₈×C₇₃	—	4	1	no	—
46	Q₈×C₇₃	SD₁₆	1	2	yes	complemented
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	2	1	no	—
49	C₂×C₄×C₇₃	—	4	1	no	—
50	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
53	D₂₉₂	—	4	1	no	—
54	D₂₉₂	—	4	1	no	—
55	D₂₉₂	—	4	1	no	—
56	D₂₉₂	—	4	1	no	—
57	D₂₉₂	—	8	1	no	—
58	C₈×C₇₃	—	4	1	no	—
59	#10596831	—	2	1	no	—
60	#10596831	—	4	1	no	—
61	#10596831	—	4	1	no	—
62	#10596831	SD₁₆	1	2	yes	complemented
63	C₂²×C₇₃	#121	1	1	yes	characteristic
64	C₂×D₇₃	—	8	1	no	—
65	C₂×D₇₃	—	8	1	no	—
66	C₂×D₇₃	—	8	1	no	—
67	C₄×C₇₃	—	2	2	no	—
68	C₄×C₇₃	—	2	1	no	—
69	C₄×C₇₃	—	4	1	no	—
70	C₄×C₇₃	—	8	1	no	—
71	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
72	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
73	C₄×C₇₃	#135	1	1	yes	characteristic
74	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
75	C₂×C₇₃	#460	1	1	yes	characteristic
76	C₂×C₇₃	#465	1	1	yes	characteristic
77	C₂×C₇₃	#471	1	1	yes	characteristic
78	D₇₃	—	16	1	no	—
79	#1666	—	73	1	no	maximal, Sylow
80	C₇₃	#1666	1	1	yes	characteristic, complemented, Sylow
81	#492	—	73	1	no	—
82	#475	D₇₃	1	1	yes	characteristic, complemented
83	#460	—	73	1	no	—
84	C₈×Q₈	—	73	1	no	—
85	#334	—	73	1	no	—
86	#332	—	73	1	no	—
87	#331	—	73	1	no	—
88	C₂×SD₁₆	—	146	1	no	—
89	C₂×SD₁₆	—	146	1	no	—
90	C₂×SD₁₆	—	146	1	no	—
91	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
92	#127	—	73	1	no	—
93	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
94	#106	—	2	1	no	—
95	C₄⋊C₈	—	73	1	no	—
96	C₄⋊C₈	—	73	1	no	—
97	C₄⋊C₈	—	146	1	no	—
98	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
99	#103	—	2	1	no	—
100	#102	—	146	1	no	—
101	C₄×C₈	—	73	1	no	—
102	C₄×C₈	—	146	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	C₂²⋊D₇₃	1	1	yes	characteristic
105	C₂×D₄	—	146	1	no	—
106	C₂×D₄	—	146	1	no	—
107	C₂×D₄	—	146	1	no	—
108	SD₁₆	—	292	2	no	—
109	SD₁₆	—	292	2	no	—
110	SD₁₆	—	292	2	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	73	1	no	—
113	C₂×C₈	—	73	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	—	2	1	no	—
119	C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
120	C₄²	—	2	1	no	—
121	C₄²	C₂²×D₇₃	1	1	yes	characteristic
122	C₂³	—	292	1	no	—
123	Q₈	—	2	1	no	—
124	Q₈	—	4	1	no	—
125	Q₈	—	4	1	no	—
126	Q₈	#11924015	1	2	yes	complemented
127	D₄	—	292	1	no	—
128	D₄	—	292	1	no	—
129	D₄	—	292	1	no	—
130	D₄	—	292	1	no	—
131	D₄	—	584	1	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
136	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
137	C₂×C₄	#11924027	1	1	yes	characteristic
138	C₈	—	4	1	no	—
139	C₈	—	292	1	no	—
140	C₈	—	292	1	no	—
141	C₈	—	73	2	no	—
142	C₈	—	146	1	no	—
143	C₂²	—	584	1	no	—
144	C₂²	—	584	1	no	—
145	C₂²	—	584	1	no	—
146	C₂²	#14556563	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	1	no	—
149	C₄	—	4	1	no	—
150	C₄	—	8	1	no	—
151	C₄	SD₁₆×D₇₃	1	1	yes	characteristic
152	C₄	C₂×#11924015	1	1	yes	characteristic
153	C₄	#14556611	1	1	yes	characteristic
154	C₄	#14556618	1	1	yes	characteristic
155	C₂	—	1168	1	no	—
156	C₂	#17988785	1	1	yes	characteristic
157	C₂	#17988818	1	1	yes	characteristic
158	C₂	#17988852	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988961

C₂

yes

maximal, characteristic

#17988787

C₂

yes

maximal, characteristic

Q₈×#10596831

C₂

yes

maximal, characteristic, complemented

#17988337

C₂

yes

maximal, characteristic, complemented

#17988333

C₂

yes

maximal, characteristic, complemented

#17988330

C₂

yes

maximal, characteristic

#475×C₇₃

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

C₄⋊#10596831

—

C₄⋊#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄×Q₈×C₇₃

C₂²

yes

characteristic

#106×C₇₃

—

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

#103×C₇₃

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

—

—

—

—

—

D₄

yes

characteristic

C₂×#10596831

D₄

yes

characteristic

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

SD₁₆

yes

complemented

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

—

—

—

SD₁₆

yes

complemented

C₂²×C₇₃

#121

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×SD₁₆

yes

characteristic

C₄×C₇₃

C₂×SD₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1666

—

maximal, Sylow

C₇₃

#1666

yes

characteristic, complemented, Sylow

#492

—

#475

D₇₃

yes

characteristic, complemented

#460

—

C₈×Q₈

—

#334

—

#332

—

#331

—

C₂×SD₁₆

—

146

—

C₂×SD₁₆

—

146

—

C₂×SD₁₆

—

146

—

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

#127

—

C₄×Q₈

C₂×D₇₃

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

146

—

C₄⋊C₈

C₂×D₇₃

yes

characteristic

#103

—

100

#102

—

146

—

101

C₄×C₈

—

102

C₄×C₈

—

146

—

103

C₂×Q₈

—

104

C₂×Q₈

C₂²⋊D₇₃

yes

characteristic

105

C₂×D₄

—

146

—

106

C₂×D₄

—

146

—

107

C₂×D₄

—

146

—

108

SD₁₆

—

292

—

109

SD₁₆

—

292

—

110

SD₁₆

—

292

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

146

—

115

C₂×C₈

—

146

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

C₂²⋊D₇₃

yes

characteristic

120

C₄²

—

121

C₄²

C₂²×D₇₃

yes

characteristic

122

C₂³

—

292

—

123

Q₈

—

124

Q₈

—

125

Q₈

—

126

Q₈

#11924015

yes

complemented

127

D₄

—

292

—

128

D₄

—

292

—

129

D₄

—

292

—

130

D₄

—

292

—

131

D₄

—

584

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

D₄×D₇₃

yes

characteristic

136

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

137

C₂×C₄

#11924027

yes

characteristic

138

C₈

—

139

C₈

—

292

—

140

C₈

—

292

—

141

C₈

—

142

C₈

—

146

—

143

C₂²

—

584

—

144

C₂²

—

584

—

145

C₂²

—

584

—

146

C₂²

#14556563

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

SD₁₆×D₇₃

yes

characteristic

152

C₄

C₂×#11924015

yes

characteristic

153

C₄

#14556611

yes

characteristic

154

C₄

#14556618

yes

characteristic

155

C₂

—

1168

—

156

C₂

#17988785

yes

characteristic

157

C₂

#17988818

yes

characteristic

158

C₂

#17988852

yes

characteristic

	subgroup	G/H
center	C₂²	#14556563
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#475×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#475	D₇₃
hypocenter	C₇₃	#1666
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

subgroup

G/H

center

C₂²

#14556563

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#475×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #475

N	H ≅ G/N	type
3. Q₈×#10596831	C₂	semidirect
4. #17988337	C₂	semidirect
5. #17988333	C₂	semidirect
7. #475×C₇₃	C₂	semidirect
46. Q₈×C₇₃	SD₁₆	semidirect
62. #10596831	SD₁₆	semidirect
80. C₇₃	#1666	semidirect
82. #475	D₇₃	semidirect
126. Q₈	#11924015	semidirect

H ≅ G/N

type

3. Q₈×#10596831

C₂

semidirect

4. #17988337

C₂

semidirect

5. #17988333

C₂

semidirect

7. #475×C₇₃

C₂

semidirect

46. Q₈×C₇₃

SD₁₆

semidirect

62. #10596831

SD₁₆

semidirect

80. C₇₃

#1666

semidirect

82. #475

D₇₃

semidirect

126. Q₈

#11924015

semidirect

Subgroups — #22609281

Conjugacy classes of subgroups

Complements