Group #33006780 — subgroups

Subgroups — #33006780

Order 18688.

200 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 65 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
2	#22612184	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₅₈₄	C₂	1	1	yes	maximal, characteristic
4	#22608113	C₂	1	2	yes	maximal
5	C₇₃×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₈₄	—	2	1	no	—
7	C₂×Dic₅₈₄	C₂²	1	2	yes	—
8	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic
9	#17988967	—	2	1	no	—
10	C₄×Dic₂₉₂	—	2	1	no	—
11	Dic₁₁₆₈	—	2	4	no	—
12	#17988413	C₂²	1	1	yes	characteristic
13	#17988392	—	2	1	no	—
14	C₂×C₃₂×C₇₃	C₂²	1	2	yes	—
15	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₉₂	—	2	1	no	—
17	C₂×Dic₂₉₂	—	2	2	no	—
18	C₂×Dic₂₉₂	—	4	1	no	—
19	#14556591	—	4	1	no	—
20	C₄×Dic₁₄₆	—	4	1	no	—
21	Dic₅₈₄	—	2	2	no	—
22	Dic₅₈₄	—	2	1	no	—
23	Dic₅₈₄	—	2	2	no	—
24	Dic₅₈₄	—	4	2	no	—
25	Dic₅₈₄	D₄	1	2	yes	—
26	#14556525	—	2	1	no	—
27	#14556516	—	4	1	no	—
28	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
29	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
30	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₃₂×C₇₃	—	2	2	no	—
32	C₂×Dic₁₄₆	—	4	1	no	—
33	C₂×Dic₁₄₆	—	4	2	no	—
34	Dic₂₉₂	—	2	2	no	—
35	Dic₂₉₂	—	4	2	no	—
36	Dic₂₉₂	—	4	1	no	—
37	Dic₂₉₂	—	4	2	no	—
38	Dic₂₉₂	—	8	2	no	—
39	C₄⋊Dic₇₃	—	4	1	no	—
40	C₄⋊Dic₇₃	—	8	1	no	—
41	C₄⋊Dic₇₃	—	8	1	no	—
42	#11924007	—	8	1	no	—
43	C₄×Dic₇₃	—	8	1	no	—
44	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
45	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₄²×C₇₃	D₈	1	1	yes	characteristic
47	C₁₆×C₇₃	—	2	1	no	—
48	C₁₆×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₇₃	#41	1	1	yes	characteristic
50	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
51	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₇₃	—	8	1	no	—
54	C₂×Dic₇₃	—	8	1	no	—
55	C₂×Dic₇₃	—	8	2	no	—
56	Dic₁₄₆	—	4	2	no	—
57	Dic₁₄₆	—	8	1	no	—
58	Dic₁₄₆	—	8	2	no	—
59	Dic₁₄₆	—	8	2	no	—
60	C₈×C₇₃	—	2	1	no	—
61	C₈×C₇₃	#121	1	1	yes	characteristic
62	C₈×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
63	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
64	C₄×C₇₃	—	2	1	no	—
65	C₄×C₇₃	Q₆₄	1	2	yes	complemented
66	C₄×C₇₃	#459	1	1	yes	characteristic
67	C₄×C₇₃	#509	1	1	yes	characteristic
68	Dic₇₃	—	8	2	no	—
69	Dic₇₃	—	16	2	no	—
70	Dic₇₃	—	16	1	no	—
71	#13688	—	73	1	no	maximal, Sylow
72	C₂×C₇₃	#2206	1	1	yes	characteristic
73	C₂×C₇₃	C₂×Q₆₄	1	1	yes	characteristic
74	C₂×C₇₃	#2264	1	1	yes	characteristic
75	C₂×Q₆₄	—	73	2	no	—
76	#2247	—	73	1	no	—
77	C₄×Q₃₂	—	73	1	no	—
78	C₄⋊C₃₂	D₇₃	1	1	yes	characteristic
79	#1416	—	73	2	no	—
80	C₇₃	#13688	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₃₂	—	73	1	no	—
82	C₂×Q₃₂	—	73	2	no	—
83	C₂×Q₃₂	—	146	1	no	—
84	#494	—	146	1	no	—
85	C₄×Q₁₆	—	146	1	no	—
86	Q₆₄	—	146	4	no	—
87	C₂×C₃₂	C₂×D₇₃	1	2	yes	—
88	#366	—	73	1	no	—
89	#358	—	146	1	no	—
90	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
91	C₂×Q₁₆	—	292	1	no	—
92	C₂×Q₁₆	—	146	2	no	—
93	C₂×Q₁₆	—	146	1	no	—
94	C₄⋊Q₈	—	292	1	no	—
95	C₄×Q₈	—	292	1	no	—
96	Q₃₂	—	292	2	no	—
97	Q₃₂	—	73	2	no	—
98	Q₃₂	—	146	2	no	—
99	Q₃₂	—	146	2	no	—
100	Q₃₂	—	146	1	no	—
101	C₂×C₁₆	D₂₉₂	1	1	yes	characteristic
102	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
103	#107	—	146	1	no	—
104	#103	—	292	1	no	—
105	C₄×C₈	D₂₉₂	1	1	yes	characteristic
106	C₃₂	—	2	2	no	—
107	C₂×Q₈	—	292	1	no	—
108	C₂×Q₈	—	292	2	no	—
109	Q₁₆	—	292	2	no	—
110	Q₁₆	—	292	1	no	—
111	Q₁₆	—	292	2	no	—
112	Q₁₆	—	584	2	no	—
113	Q₁₆	—	146	2	no	—
114	C₂×C₈	D₅₈₄	1	1	yes	characteristic
115	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
116	C₄⋊C₄	—	292	1	no	—
117	C₄⋊C₄	—	584	1	no	—
118	C₄⋊C₄	—	584	1	no	—
119	C₄⋊C₄	—	584	1	no	—
120	C₄²	—	584	1	no	—
121	C₄²	D₅₈₄	1	1	yes	characteristic
122	C₁₆	—	2	1	no	—
123	C₁₆	D₄×D₇₃	1	1	yes	characteristic
124	C₁₆	#11924029	1	1	yes	characteristic
125	Q₈	—	292	2	no	—
126	Q₈	—	584	1	no	—
127	Q₈	—	584	2	no	—
128	Q₈	—	584	2	no	—
129	C₂×C₄	—	584	1	no	—
130	C₂×C₄	—	584	1	no	—
131	C₂×C₄	—	584	2	no	—
132	C₂×C₄	D₁₁₆₈	1	1	yes	characteristic
133	C₂×C₄	D₁₁₆₈	1	1	yes	characteristic
134	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
135	C₈	—	2	1	no	—
136	C₈	#14556566	1	1	yes	characteristic
137	C₈	#14556619	1	1	yes	characteristic
138	C₂²	C₂×D₁₁₆₈	1	1	yes	characteristic
139	C₄	—	2	1	no	—
140	C₄	—	584	2	no	—
141	C₄	—	1168	1	no	—
142	C₄	—	1168	2	no	—
143	C₄	Dic₁₁₆₈	1	2	yes	complemented
144	C₄	#17988783	1	1	yes	characteristic
145	C₄	#17989025	1	1	yes	characteristic
146	C₂	#22611967	1	1	yes	characteristic
147	C₂	C₂×Dic₁₁₆₈	1	1	yes	characteristic
148	C₂	#22612251	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₁₆₈

C₂

yes

maximal

#22612184

C₂

yes

maximal, characteristic

C₄×Dic₅₈₄

C₂

yes

maximal, characteristic

#22608113

C₂

yes

maximal

C₇₃×C₄⋊C₃₂

C₂

yes

maximal, characteristic

C₂×Dic₅₈₄

—

C₂×Dic₅₈₄

C₂²

yes

—

C₂×Dic₅₈₄

C₂²

yes

characteristic

#17988967

—

C₄×Dic₂₉₂

—

Dic₁₁₆₈

—

#17988413

C₂²

yes

characteristic

#17988392

—

C₂×C₃₂×C₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

#14556591

—

C₄×Dic₁₄₆

—

Dic₅₈₄

—

Dic₅₈₄

—

Dic₅₈₄

—

Dic₅₈₄

—

Dic₅₈₄

D₄

yes

—

#14556525

—

#14556516

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₃₂×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

C₄×Dic₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

#41

yes

characteristic

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

Q₆₄

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

C₂×Q₆₄

yes

characteristic

C₂×C₇₃

#2264

yes

characteristic

C₂×Q₆₄

—

#2247

—

C₄×Q₃₂

—

C₄⋊C₃₂

D₇₃

yes

characteristic

#1416

—

C₇₃

#13688

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

146

—

#494

—

146

—

C₄×Q₁₆

—

146

—

Q₆₄

—

146

—

C₂×C₃₂

C₂×D₇₃

yes

—

#366

—

#358

—

146

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

292

—

C₂×Q₁₆

—

146

—

C₂×Q₁₆

—

146

—

C₄⋊Q₈

—

292

—

C₄×Q₈

—

292

—

Q₃₂

—

292

—

Q₃₂

—

Q₃₂

—

146

—

Q₃₂

—

146

—

100

Q₃₂

—

146

—

101

C₂×C₁₆

D₂₉₂

yes

characteristic

102

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

103

#107

—

146

—

104

#103

—

292

—

105

C₄×C₈

D₂₉₂

yes

characteristic

106

C₃₂

—

107

C₂×Q₈

—

292

—

108

C₂×Q₈

—

292

—

109

Q₁₆

—

292

—

110

Q₁₆

—

292

—

111

Q₁₆

—

292

—

112

Q₁₆

—

584

—

113

Q₁₆

—

146

—

114

C₂×C₈

D₅₈₄

yes

characteristic

115

C₂×C₈

C₂×D₂₉₂

yes

characteristic

116

C₄⋊C₄

—

292

—

117

C₄⋊C₄

—

584

—

118

C₄⋊C₄

—

584

—

119

C₄⋊C₄

—

584

—

120

C₄²

—

584

—

121

C₄²

D₅₈₄

yes

characteristic

122

C₁₆

—

123

C₁₆

D₄×D₇₃

yes

characteristic

124

C₁₆

#11924029

yes

characteristic

125

Q₈

—

292

—

126

Q₈

—

584

—

127

Q₈

—

584

—

128

Q₈

—

584

—

129

C₂×C₄

—

584

—

130

C₂×C₄

—

584

—

131

C₂×C₄

—

584

—

132

C₂×C₄

D₁₁₆₈

yes

characteristic

133

C₂×C₄

D₁₁₆₈

yes

characteristic

134

C₂×C₄

C₂×D₅₈₄

yes

characteristic

135

C₈

—

136

C₈

#14556566

yes

characteristic

137

C₈

#14556619

yes

characteristic

138

C₂²

C₂×D₁₁₆₈

yes

characteristic

139

C₄

—

140

C₄

—

584

—

141

C₄

—

1168

—

142

C₄

—

1168

—

143

C₄

Dic₁₁₆₈

yes

complemented

144

C₄

#17988783

yes

characteristic

145

C₄

#17989025

yes

characteristic

146

C₂

#22611967

yes

characteristic

147

C₂

C₂×Dic₁₁₆₈

yes

characteristic

148

C₂

#22612251

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₁₆₈
commutator, cosocle	C₂×C₁₆×C₇₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₇₃×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₇₃	C₂×D₁₆
hypercenter	C₄⋊C₃₂	D₇₃
hypocenter	C₇₃	#13688
LCS	G ▷ C₂×C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

subgroup

G/H

center

C₂²

C₂×D₁₁₆₈

commutator, cosocle

C₂×C₁₆×C₇₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₃

Fitting

C₇₃×C₄⋊C₃₂

C₂

socle, prime radical

C₂²×C₇₃

C₂×D₁₆

hypercenter

C₄⋊C₃₂

D₇₃

hypocenter

C₇₃

#13688

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

N	H ≅ G/N	type
65. C₄×C₇₃	Q₆₄	semidirect
80. C₇₃	#13688	semidirect
143. C₄	Dic₁₁₆₈	semidirect

H ≅ G/N

type

65. C₄×C₇₃

Q₆₄

semidirect

80. C₇₃

#13688

semidirect

143. C₄

Dic₁₁₆₈

semidirect

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

subgroup	index
C₂×Dic₁₁₆₈	2
#22612184	2
C₄×Dic₅₈₄	2
#22608113	2
C₇₃×C₄⋊C₃₂	2
#13688	73

subgroup

index

C₂×Dic₁₁₆₈

#22612184

C₄×Dic₅₈₄

#22608113

C₇₃×C₄⋊C₃₂

#13688

Subgroups — #33006780

Conjugacy classes of subgroups

Complements

Maximal subgroups