Group #33007000 — subgroups

Subgroups — #33007000

Order 18688.

232 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 75 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×Dic₁₁₆₈	C₂	1	2	yes	maximal, complemented
2	C₂×#17988426	C₂	1	2	yes	maximal
3	#22612192	C₂	1	1	yes	maximal, characteristic
4	#22612184	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₅₈₄	—	2	1	no	—
7	C₂×Dic₅₈₄	C₂²	1	2	yes	—
8	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic
9	C₂×#14556536	—	2	1	no	—
10	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic
11	#17988971	—	2	1	no	—
12	#17988967	—	2	1	no	—
13	Dic₁₁₆₈	—	2	4	no	—
14	#17988426	—	2	4	no	—
15	C₂×C₃₂×C₇₃	C₂²	1	2	yes	—
16	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₉₂	—	2	2	no	—
18	C₂×Dic₂₉₂	—	2	1	no	—
19	C₂×Dic₂₉₂	—	4	1	no	—
20	C₂×#11924017	—	4	1	no	—
21	C₂×D₅₈₄	—	2	1	no	—
22	#14556591	—	4	1	no	—
23	#14556579	—	4	1	no	—
24	Dic₅₈₄	—	2	2	no	—
25	Dic₅₈₄	—	2	2	no	—
26	Dic₅₈₄	—	2	2	no	—
27	Dic₅₈₄	—	4	2	no	—
28	#14556536	—	4	2	no	—
29	D₁₁₆₈	—	2	2	no	—
30	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₃₂×C₇₃	D₄	1	4	yes	—
34	C₂×Dic₁₄₆	—	4	1	no	—
35	C₂×Dic₁₄₆	—	4	2	no	—
36	C₂×D₂₉₂	—	4	1	no	—
37	Dic₂₉₂	—	4	2	no	—
38	Dic₂₉₂	—	4	1	no	—
39	Dic₂₉₂	—	4	1	no	—
40	Dic₂₉₂	—	4	2	no	—
41	Dic₂₉₂	—	8	2	no	—
42	#11924017	—	8	2	no	—
43	D₅₈₄	—	4	1	no	—
44	D₅₈₄	—	4	1	no	—
45	C₄⋊Dic₇₃	—	8	2	no	—
46	#11924005	—	8	2	no	—
47	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
48	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₇₃	D₈	1	1	yes	characteristic
50	C₁₆×C₇₃	D₈	1	2	yes	—
51	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
52	C₂²×D₇₃	—	8	1	no	—
53	C₂×C₄×C₇₃	D₁₆	1	2	yes	—
54	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
55	C₂×Dic₇₃	—	8	1	no	—
56	C₂×Dic₇₃	—	8	2	no	—
57	D₂₉₂	—	8	1	no	—
58	D₂₉₂	—	8	1	no	—
59	Dic₁₄₆	—	8	1	no	—
60	Dic₁₄₆	—	8	2	no	—
61	Dic₁₄₆	—	8	1	no	—
62	Dic₁₄₆	—	8	2	no	—
63	C₈×C₇₃	D₁₆	1	2	yes	—
64	C₈×C₇₃	#127	1	1	yes	characteristic
65	C₈×C₇₃	C₂×D₈	1	1	yes	characteristic
66	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
67	C₂×D₇₃	—	16	1	no	—
68	C₂×D₇₃	—	16	2	no	—
69	C₄×C₇₃	—	2	2	no	—
70	C₄×C₇₃	#493	1	1	yes	characteristic
71	C₄×C₇₃	C₂×D₁₆	1	1	yes	characteristic
72	Dic₇₃	—	16	1	no	—
73	Dic₇₃	—	16	2	no	—
74	#13730	—	73	1	no	maximal, Sylow
75	C₂×C₇₃	#2246	1	1	yes	characteristic
76	C₂×C₇₃	#2264	1	2	yes	—
77	D₇₃	—	32	1	no	—
78	C₂×Q₆₄	—	73	2	no	—
79	C₂×SD₆₄	—	73	2	no	—
80	#2249	—	73	1	no	—
81	#2247	—	73	1	no	—
82	#1397	D₇₃	1	1	yes	characteristic, complemented
83	C₇₃	#13730	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₃₂	—	73	2	no	—
85	C₂×Q₃₂	—	73	1	no	—
86	C₂×Q₃₂	—	146	1	no	—
87	C₂×SD₃₂	—	146	1	no	—
88	C₂×D₁₆	—	73	1	no	—
89	#495	—	146	1	no	—
90	#494	—	146	1	no	—
91	Q₆₄	—	146	4	no	—
92	SD₆₄	—	146	4	no	—
93	C₂×C₃₂	C₂×D₇₃	1	2	yes	—
94	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
95	C₂×Q₁₆	—	292	1	no	—
96	C₂×Q₁₆	—	146	2	no	—
97	C₂×Q₁₆	—	146	1	no	—
98	C₂×SD₁₆	—	292	1	no	—
99	C₂×D₈	—	146	1	no	—
100	C₄⋊Q₈	—	292	1	no	—
101	#124	—	292	1	no	—
102	Q₃₂	—	292	2	no	—
103	Q₃₂	—	146	2	no	—
104	Q₃₂	—	146	2	no	—
105	Q₃₂	—	146	2	no	—
106	SD₃₂	—	292	2	no	—
107	D₁₆	—	146	2	no	—
108	C₂×C₁₆	D₂₉₂	1	1	yes	characteristic
109	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
110	C₄×C₈	D₂₉₂	1	1	yes	characteristic
111	C₃₂	D₂₉₂	1	4	yes	—
112	C₂×Q₈	—	292	1	no	—
113	C₂×Q₈	—	292	2	no	—
114	C₂×D₄	—	292	1	no	—
115	Q₁₆	—	292	2	no	—
116	Q₁₆	—	292	1	no	—
117	Q₁₆	—	292	1	no	—
118	Q₁₆	—	292	2	no	—
119	Q₁₆	—	584	2	no	—
120	SD₁₆	—	584	2	no	—
121	D₈	—	292	1	no	—
122	D₈	—	292	1	no	—
123	C₂×C₈	D₅₈₄	1	1	yes	characteristic
124	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
125	C₄⋊C₄	—	584	2	no	—
126	C₂²⋊C₄	—	584	2	no	—
127	C₄²	D₅₈₄	1	1	yes	characteristic
128	C₁₆	D₅₈₄	1	2	yes	—
129	C₁₆	C₂×D₂₉₂	1	2	yes	—
130	C₂³	—	584	1	no	—
131	Q₈	—	584	1	no	—
132	Q₈	—	584	2	no	—
133	Q₈	—	584	1	no	—
134	Q₈	—	584	2	no	—
135	D₄	—	584	1	no	—
136	D₄	—	584	1	no	—
137	C₂×C₄	—	584	1	no	—
138	C₂×C₄	—	584	2	no	—
139	C₂×C₄	D₁₁₆₈	1	2	yes	—
140	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
141	C₈	D₁₁₆₈	1	2	yes	—
142	C₈	#14556588	1	1	yes	characteristic
143	C₈	C₂×D₅₈₄	1	1	yes	characteristic
144	C₂²	—	1168	2	no	—
145	C₂²	—	1168	1	no	—
146	C₂²	C₂×D₁₁₆₈	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	1168	1	no	—
149	C₄	—	1168	2	no	—
150	C₄	#17988964	1	1	yes	characteristic
151	C₄	C₂×D₁₁₆₈	1	1	yes	characteristic
152	C₂	—	2336	1	no	—
153	C₂	#22612181	1	1	yes	characteristic
154	C₂	#22612251	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₁₆₈

C₂

yes

maximal, complemented

C₂×#17988426

C₂

yes

maximal

#22612192

C₂

yes

maximal, characteristic

#22612184

C₂

yes

maximal, characteristic, complemented

C₇₃×#1397

C₂

yes

maximal, characteristic, complemented

C₂×Dic₅₈₄

—

C₂×Dic₅₈₄

C₂²

yes

—

C₂×Dic₅₈₄

C₂²

yes

characteristic

C₂×#14556536

—

C₂×D₁₁₆₈

C₂²

yes

characteristic

#17988971

—

#17988967

—

Dic₁₁₆₈

—

#17988426

—

C₂×C₃₂×C₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×#11924017

—

C₂×D₅₈₄

—

#14556591

—

#14556579

—

Dic₅₈₄

—

Dic₅₈₄

—

Dic₅₈₄

—

Dic₅₈₄

—

#14556536

—

D₁₁₆₈

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₃₂×C₇₃

D₄

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×D₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

#11924017

—

D₅₈₄

—

D₅₈₄

—

C₄⋊Dic₇₃

—

#11924005

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

D₈

yes

—

C₁₆×C₇₃

C₂×D₄

yes

—

C₂²×D₇₃

—

C₂×C₄×C₇₃

D₁₆

yes

—

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

D₁₆

yes

—

C₈×C₇₃

#127

yes

characteristic

C₈×C₇₃

C₂×D₈

yes

characteristic

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#493

yes

characteristic

C₄×C₇₃

C₂×D₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

C₂×Q₆₄

—

C₂×SD₆₄

—

#2249

—

#2247

—

#1397

D₇₃

yes

characteristic, complemented

C₇₃

#13730

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

146

—

C₂×SD₃₂

—

146

—

C₂×D₁₆

—

#495

—

146

—

#494

—

146

—

Q₆₄

—

146

—

SD₆₄

—

146

—

C₂×C₃₂

C₂×D₇₃

yes

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

292

—

C₂×Q₁₆

—

146

—

C₂×Q₁₆

—

146

—

C₂×SD₁₆

—

292

—

C₂×D₈

—

146

—

100

C₄⋊Q₈

—

292

—

101

#124

—

292

—

102

Q₃₂

—

292

—

103

Q₃₂

—

146

—

104

Q₃₂

—

146

—

105

Q₃₂

—

146

—

106

SD₃₂

—

292

—

107

D₁₆

—

146

—

108

C₂×C₁₆

D₂₉₂

yes

characteristic

109

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

110

C₄×C₈

D₂₉₂

yes

characteristic

111

C₃₂

D₂₉₂

yes

—

112

C₂×Q₈

—

292

—

113

C₂×Q₈

—

292

—

114

C₂×D₄

—

292

—

115

Q₁₆

—

292

—

116

Q₁₆

—

292

—

117

Q₁₆

—

292

—

118

Q₁₆

—

292

—

119

Q₁₆

—

584

—

120

SD₁₆

—

584

—

121

D₈

—

292

—

122

D₈

—

292

—

123

C₂×C₈

D₅₈₄

yes

characteristic

124

C₂×C₈

C₂×D₂₉₂

yes

characteristic

125

C₄⋊C₄

—

584

—

126

C₂²⋊C₄

—

584

—

127

C₄²

D₅₈₄

yes

characteristic

128

C₁₆

D₅₈₄

yes

—

129

C₁₆

C₂×D₂₉₂

yes

—

130

C₂³

—

584

—

131

Q₈

—

584

—

132

Q₈

—

584

—

133

Q₈

—

584

—

134

Q₈

—

584

—

135

D₄

—

584

—

136

D₄

—

584

—

137

C₂×C₄

—

584

—

138

C₂×C₄

—

584

—

139

C₂×C₄

D₁₁₆₈

yes

—

140

C₂×C₄

C₂×D₅₈₄

yes

characteristic

141

C₈

D₁₁₆₈

yes

—

142

C₈

#14556588

yes

characteristic

143

C₈

C₂×D₅₈₄

yes

characteristic

144

C₂²

—

1168

—

145

C₂²

—

1168

—

146

C₂²

C₂×D₁₁₆₈

yes

characteristic

147

C₄

—

148

C₄

—

1168

—

149

C₄

—

1168

—

150

C₄

#17988964

yes

characteristic

151

C₄

C₂×D₁₁₆₈

yes

characteristic

152

C₂

—

2336

—

153

C₂

#22612181

yes

characteristic

154

C₂

#22612251

yes

—

	subgroup	G/H
center	C₂²	C₂×D₁₁₆₈
commutator, cosocle	C₂×C₁₆×C₇₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₇₃×#1397	C₂
socle	C₂²×C₇₃	C₂×D₁₆
prime radical	C₂×D₁₁₆₈	C₂²
hypercenter	#1397	D₇₃
hypocenter	C₇₃	#13730
LCS	G ▷ C₂×C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

subgroup

G/H

center

C₂²

C₂×D₁₁₆₈

commutator, cosocle

C₂×C₁₆×C₇₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₃

Fitting

C₇₃×#1397

C₂

socle

C₂²×C₇₃

C₂×D₁₆

prime radical

C₂×D₁₁₆₈

C₂²

D₇₃

C₇₃

G ▷ C₂×C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

N	H ≅ G/N	type
1. C₂×Dic₁₁₆₈	C₂	semidirect
4. #22612184	C₂	semidirect
5. C₇₃×#1397	C₂	semidirect
82. #1397	D₇₃	semidirect
83. C₇₃	#13730	semidirect

H ≅ G/N

type

1. C₂×Dic₁₁₆₈

C₂

semidirect

4. #22612184

C₂

semidirect

5. C₇₃×#1397

C₂

semidirect

82. #1397

D₇₃

semidirect

83. C₇₃

#13730

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×Dic₁₁₆₈	2
C₂×#17988426	2
#22612192	2
#22612184	2
C₇₃×#1397	2
#13730	73

subgroup

index

C₂×Dic₁₁₆₈

C₇₃×#1397

Subgroups — #33007000

Conjugacy classes of subgroups

Complements

Maximal subgroups