Group #22609352 — subgroups

Subgroups — #22609352

Order 9344.

166 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 55 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	#17988937	C₂	1	1	yes	maximal, characteristic, complemented
2	#17988792	C₂	1	1	yes	maximal, characteristic
3	D₄×#10596831	C₂	1	1	yes	maximal, characteristic
4	#17988337	C₂	1	1	yes	maximal, characteristic, complemented
5	#17988334	C₂	1	1	yes	maximal, characteristic, complemented
6	#17988333	C₂	1	1	yes	maximal, characteristic, complemented
7	#474×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#11924016	—	2	1	no	—
9	C₂²×#10596831	—	2	1	no	—
10	#14556591	C₂²	1	1	yes	characteristic
11	C₄⋊#10596831	C₂²	1	1	yes	characteristic
12	C₄⋊#10596831	C₂²	1	1	yes	characteristic
13	#14556515	—	2	1	no	—
14	#14556515	—	2	1	no	—
15	#14556514	—	2	1	no	—
16	C₂²⋊#10596831	—	2	1	no	—
17	C₄×#10596831	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₇₃	C₂²	1	1	yes	characteristic
20	#107×C₇₃	—	2	1	no	—
21	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
22	#102×C₇₃	—	2	1	no	—
23	C₂×Dic₁₄₆	—	2	1	no	—
24	#11924016	—	4	2	no	—
25	C₂×#10596831	—	2	1	no	—
26	C₂×#10596831	—	2	1	no	—
27	C₂×#10596831	—	4	2	no	—
28	C₂×#10596831	D₄	1	1	yes	characteristic
29	C₂×#10596831	D₄	1	1	yes	characteristic
30	C₄⋊Dic₇₃	—	2	1	no	—
31	C₄⋊Dic₇₃	—	2	1	no	—
32	C₂×Q₈×C₇₃	—	2	1	no	—
33	C₂×D₄×C₇₃	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₇₃	—	2	1	no	—
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	D₄	1	1	yes	characteristic
39	C₂²⋊C₄×C₇₃	—	2	1	no	—
40	C₄²×C₇₃	C₂³	1	1	yes	characteristic
41	C₂³×C₇₃	—	2	1	no	—
42	Q₈×C₇₃	—	4	1	no	—
43	Q₈×C₇₃	—	4	1	no	—
44	D₄×C₇₃	—	2	1	no	—
45	D₄×C₇₃	SD₁₆	1	2	yes	—
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	—	4	1	no	—
48	C₂×C₄×C₇₃	—	4	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	C₂×Dic₇₃	—	4	1	no	—
54	Dic₁₄₆	—	4	1	no	—
55	Dic₁₄₆	—	4	1	no	—
56	C₈×C₇₃	—	4	1	no	—
57	#10596831	—	2	1	no	—
58	#10596831	—	2	1	no	—
59	#10596831	—	4	1	no	—
60	#10596831	—	4	1	no	—
61	C₂²×C₇₃	—	2	2	no	—
62	C₂²×C₇₃	—	4	1	no	—
63	C₂²×C₇₃	—	4	1	no	—
64	C₂²×C₇₃	#121	1	1	yes	characteristic
65	C₄×C₇₃	—	2	1	no	—
66	C₄×C₇₃	—	4	1	no	—
67	C₄×C₇₃	—	8	1	no	—
68	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
69	C₄×C₇₃	#135	1	1	yes	characteristic
70	C₄×C₇₃	#135	1	1	yes	characteristic
71	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
72	Dic₇₃	—	8	1	no	—
73	C₂×C₇₃	—	4	2	no	—
74	C₂×C₇₃	#463	1	1	yes	characteristic
75	C₂×C₇₃	#465	1	1	yes	characteristic
76	C₂×C₇₃	#473	1	1	yes	characteristic
77	#1679	—	73	1	no	maximal, Sylow
78	C₇₃	#1679	1	1	yes	characteristic, complemented, Sylow
79	#487	—	73	1	no	—
80	#474	D₇₃	1	1	yes	characteristic
81	#461	—	73	1	no	—
82	C₈×D₄	—	73	1	no	—
83	#334	—	73	1	no	—
84	#332	—	73	1	no	—
85	#332	—	73	1	no	—
86	C₂×SD₁₆	—	146	1	no	—
87	C₂²×C₈	—	146	1	no	—
88	C₄⋊Q₈	—	73	1	no	—
89	C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
90	C₄×D₄	C₂×D₇₃	1	1	yes	characteristic
91	#107	—	2	1	no	—
92	C₄⋊C₈	—	73	1	no	—
93	C₄⋊C₈	—	73	1	no	—
94	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
95	#103	—	146	1	no	—
96	#103	—	146	1	no	—
97	#103	—	146	1	no	—
98	#102	—	2	1	no	—
99	C₂²⋊C₈	—	146	1	no	—
100	C₄×C₈	—	73	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	146	1	no	—
103	C₂×D₄	C₂²⋊D₇₃	1	1	yes	characteristic
104	C₂²×C₄	—	2	1	no	—
105	SD₁₆	—	292	2	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	292	2	no	—
108	C₂×C₈	—	73	1	no	—
109	C₂×C₈	—	73	1	no	—
110	C₂×C₈	—	146	1	no	—
111	C₂×C₈	—	146	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	146	1	no	—
115	C₄⋊C₄	—	146	1	no	—
116	C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
117	C₂²⋊C₄	—	2	1	no	—
118	C₄²	C₂²×D₇₃	1	1	yes	characteristic
119	C₂³	—	2	1	no	—
120	Q₈	—	4	1	no	—
121	Q₈	—	4	1	no	—
122	Q₈	—	292	1	no	—
123	Q₈	—	292	1	no	—
124	D₄	—	2	1	no	—
125	D₄	#11924015	1	2	yes	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	292	1	no	—
131	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
132	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
133	C₂×C₄	#11924027	1	1	yes	characteristic
134	C₈	—	4	1	no	—
135	C₈	—	292	1	no	—
136	C₈	—	292	1	no	—
137	C₈	—	146	1	no	—
138	C₈	—	146	1	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	#14556563	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	8	1	no	—
146	C₄	—	584	1	no	—
147	C₄	C₂×#11924015	1	1	yes	characteristic
148	C₄	#14556610	1	1	yes	characteristic
149	C₄	#14556611	1	1	yes	characteristic
150	C₄	#14556615	1	1	yes	characteristic
151	C₂	—	4	2	no	—
152	C₂	#17988800	1	1	yes	characteristic
153	C₂	#17988818	1	1	yes	characteristic
154	C₂	#17988860	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988937

C₂

yes

maximal, characteristic, complemented

#17988792

C₂

yes

maximal, characteristic

D₄×#10596831

C₂

yes

maximal, characteristic

#17988337

C₂

yes

maximal, characteristic, complemented

#17988334

C₂

yes

maximal, characteristic, complemented

#17988333

C₂

yes

maximal, characteristic, complemented

#474×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924016

—

C₂²×#10596831

—

#14556591

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

characteristic

C₄⋊#10596831

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₂²

yes

characteristic

C₄×D₄×C₇₃

C₂²

yes

characteristic

#107×C₇₃

—

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

#102×C₇₃

—

C₂×Dic₁₄₆

—

—

—

—

—

D₄

yes

characteristic

C₂×#10596831

D₄

yes

characteristic

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

—

C₂×D₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

—

C₂×C₈×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

SD₁₆

yes

—

C₂×C₄×C₇₃

—

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

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

—

—

—

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#121

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×SD₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#487

—

#474

D₇₃

yes

characteristic

#461

—

C₈×D₄

—

#334

—

#332

—

#332

—

C₂×SD₁₆

—

146

—

C₂²×C₈

—

146

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₇₃

yes

characteristic

C₄×D₄

C₂×D₇₃

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₃

yes

characteristic

#103

—

146

—

#103

—

146

—

#103

—

146

—

#102

—

C₂²⋊C₈

—

146

—

100

C₄×C₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

146

—

103

C₂×D₄

C₂²⋊D₇₃

yes

characteristic

104

C₂²×C₄

—

105

SD₁₆

—

292

—

106

C₂×C₈

—

107

C₂×C₈

—

292

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

146

—

111

C₂×C₈

—

146

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

146

—

115

C₄⋊C₄

—

146

—

116

C₄⋊C₄

C₂²⋊D₇₃

yes

characteristic

117

C₂²⋊C₄

—

118

C₄²

C₂²×D₇₃

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

292

—

123

Q₈

—

292

—

124

D₄

—

125

D₄

#11924015

yes

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

292

—

131

C₂×C₄

D₄×D₇₃

yes

characteristic

132

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

133

C₂×C₄

#11924027

yes

characteristic

134

C₈

—

135

C₈

—

292

—

136

C₈

—

292

—

137

C₈

—

146

—

138

C₈

—

146

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

#14556563

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

584

—

147

C₄

C₂×#11924015

yes

characteristic

148

C₄

#14556610

yes

characteristic

149

C₄

#14556611

yes

characteristic

150

C₄

#14556615

yes

characteristic

151

C₂

—

152

C₂

#17988800

yes

characteristic

153

C₂

#17988818

yes

characteristic

154

C₂

#17988860

yes

characteristic

	subgroup	G/H
center	C₂²	#14556563
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#474×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#474	D₇₃
hypocenter	C₇₃	#1679
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #474

subgroup

G/H

center

C₂²

#14556563

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#474×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #474

N	H ≅ G/N	type
1. #17988937	C₂	semidirect
4. #17988337	C₂	semidirect
5. #17988334	C₂	semidirect
6. #17988333	C₂	semidirect
78. C₇₃	#1679	semidirect

H ≅ G/N

type

1. #17988937

C₂

semidirect

4. #17988337

C₂

semidirect

5. #17988334

C₂

semidirect

6. #17988333

C₂

semidirect

78. C₇₃

#1679

semidirect

Subgroups — #22609352

Conjugacy classes of subgroups

Complements