Group #33000753 — subgroups

Subgroups — #33000753

Order 18688.

274 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22612184	C₂	1	2	yes	maximal
2	#22612159	C₂	1	2	yes	maximal
3	#22609549	C₂	1	1	yes	maximal, characteristic
4	#22609528	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1312	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₈₄	—	2	4	no	—
7	#17988992	—	2	1	no	—
8	#17988992	—	2	1	no	—
9	#17988992	C₂²	1	1	yes	characteristic
10	#17988985	—	2	1	no	—
11	#17988967	C₂²	1	1	yes	characteristic
12	#17988967	C₂²	1	2	yes	—
13	#17988938	—	2	1	no	—
14	#17988392	—	2	4	no	—
15	C₄×C₁₆×C₇₃	C₂²	1	2	yes	—
16	C₈²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₉₂	—	2	2	no	—
18	C₂×Dic₂₉₂	—	2	2	no	—
19	C₂×Dic₂₉₂	—	2	2	no	—
20	#14556591	—	2	1	no	—
21	#14556591	—	2	1	no	—
22	#14556591	—	2	2	no	—
23	Dic₅₈₄	—	4	8	no	—
24	#14556525	—	2	2	no	—
25	#14556525	—	4	2	no	—
26	#14556525	—	4	2	no	—
27	#14556523	—	4	2	no	—
28	#14556516	—	4	2	no	—
29	C₂×C₁₆×C₇₃	D₄	1	4	yes	—
30	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₁₄₆	—	4	1	no	—
34	C₂×Dic₁₄₆	—	4	2	no	—
35	C₂×Dic₁₄₆	—	4	1	no	—
36	Dic₂₉₂	—	4	4	no	—
37	Dic₂₉₂	—	4	4	no	—
38	Dic₂₉₂	—	4	4	no	—
39	C₄⋊Dic₇₃	—	4	1	no	—
40	C₄⋊Dic₇₃	—	4	1	no	—
41	C₄⋊Dic₇₃	—	4	1	no	—
42	C₄⋊Dic₇₃	—	4	1	no	—
43	C₄⋊Dic₇₃	—	4	2	no	—
44	C₄⋊Dic₇₃	—	4	2	no	—
45	C₂×C₈×C₇₃	D₈	1	2	yes	—
46	C₂×C₈×C₇₃	D₈	1	2	yes	—
47	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
48	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
49	C₁₆×C₇₃	—	2	4	no	—
50	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
51	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₇₃	—	8	1	no	—
54	C₂×Dic₇₃	—	8	2	no	—
55	C₂×Dic₇₃	—	8	1	no	—
56	Dic₁₄₆	—	8	2	no	—
57	Dic₁₄₆	—	8	2	no	—
58	Dic₁₄₆	—	8	2	no	—
59	Dic₁₄₆	—	8	2	no	—
60	C₈×C₇₃	—	2	2	no	—
61	C₈×C₇₃	Q₃₂	1	4	yes	complemented
62	C₈×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
63	C₂²×C₇₃	#493	1	1	yes	characteristic
64	C₄×C₇₃	#496	1	2	yes	—
65	C₄×C₇₃	C₂×Q₃₂	1	2	yes	—
66	C₄×C₇₃	#510	1	2	yes	—
67	Dic₇₃	—	16	1	no	—
68	Dic₇₃	—	16	2	no	—
69	Dic₇₃	—	16	1	no	—
70	#12528	—	73	1	no	maximal, Sylow
71	C₂×C₇₃	#1721	1	1	yes	characteristic
72	C₂×C₇₃	#2247	1	1	yes	characteristic
73	C₂×C₇₃	#2251	1	1	yes	characteristic
74	#2247	—	73	2	no	—
75	#2241	—	73	2	no	—
76	#1713	—	73	1	no	—
77	#1707	—	73	1	no	—
78	#1312	D₇₃	1	1	yes	characteristic
79	C₇₃	#12528	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₃₂	—	146	4	no	—
81	#500	—	73	1	no	—
82	#500	—	146	1	no	—
83	#500	—	146	1	no	—
84	#498	—	146	1	no	—
85	#494	—	73	1	no	—
86	#494	—	73	2	no	—
87	#487	—	146	1	no	—
88	#358	—	146	4	no	—
89	C₄×C₁₆	C₂×D₇₃	1	2	yes	—
90	C₈²	C₂×D₇₃	1	1	yes	characteristic
91	C₂×Q₁₆	—	146	2	no	—
92	C₂×Q₁₆	—	146	2	no	—
93	C₂×Q₁₆	—	146	2	no	—
94	C₄⋊Q₈	—	146	1	no	—
95	C₄⋊Q₈	—	146	1	no	—
96	C₄⋊Q₈	—	146	2	no	—
97	Q₃₂	—	292	8	no	—
98	C₂×C₁₆	D₂₉₂	1	4	yes	—
99	#107	—	292	2	no	—
100	#107	—	292	2	no	—
101	#107	—	146	2	no	—
102	#106	—	292	2	no	—
103	#103	—	292	2	no	—
104	C₄×C₈	D₂₉₂	1	1	yes	characteristic
105	C₄×C₈	D₂₉₂	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
107	C₂×Q₈	—	292	1	no	—
108	C₂×Q₈	—	292	2	no	—
109	C₂×Q₈	—	292	1	no	—
110	Q₁₆	—	292	4	no	—
111	Q₁₆	—	292	4	no	—
112	Q₁₆	—	292	4	no	—
113	C₂×C₈	D₅₈₄	1	2	yes	—
114	C₂×C₈	D₅₈₄	1	2	yes	—
115	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
116	C₄⋊C₄	—	292	1	no	—
117	C₄⋊C₄	—	292	1	no	—
118	C₄⋊C₄	—	292	1	no	—
119	C₄⋊C₄	—	292	1	no	—
120	C₄⋊C₄	—	292	2	no	—
121	C₄⋊C₄	—	292	2	no	—
122	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
123	C₁₆	—	2	4	no	—
124	Q₈	—	584	2	no	—
125	Q₈	—	584	2	no	—
126	Q₈	—	584	2	no	—
127	Q₈	—	584	2	no	—
128	C₂×C₄	—	584	1	no	—
129	C₂×C₄	—	584	2	no	—
130	C₂×C₄	—	584	1	no	—
131	C₂×C₄	#14556588	1	1	yes	characteristic
132	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
133	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	Dic₅₈₄	1	4	yes	complemented
136	C₈	#14556619	1	4	yes	—
137	C₂²	#17988964	1	1	yes	characteristic
138	C₄	—	1168	1	no	—
139	C₄	—	1168	2	no	—
140	C₄	—	1168	1	no	—
141	C₄	#17988976	1	2	yes	—
142	C₄	C₂×Dic₅₈₄	1	2	yes	—
143	C₄	#17989030	1	2	yes	—
144	C₂	#22609597	1	1	yes	characteristic
145	C₂	#22612184	1	1	yes	characteristic
146	C₂	#22612202	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612184

C₂

yes

maximal

#22612159

C₂

yes

maximal

#22609549

C₂

yes

maximal, characteristic

#22609528

C₂

yes

maximal, characteristic

C₇₃×#1312

C₂

yes

maximal, characteristic

C₂×Dic₅₈₄

—

#17988992

—

#17988992

—

#17988992

C₂²

yes

characteristic

#17988985

—

#17988967

C₂²

yes

characteristic

#17988967

C₂²

yes

—

#17988938

—

#17988392

—

C₄×C₁₆×C₇₃

C₂²

yes

—

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

#14556591

—

#14556591

—

#14556591

—

Dic₅₈₄

—

—

—

—

—

—

C₂×C₁₆×C₇₃

D₄

yes

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

Dic₂₉₂

—

Dic₂₉₂

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

Q₃₂

yes

complemented

C₈×C₇₃

AGL(1,ℤ/8)

yes

—

C₂²×C₇₃

#493

yes

characteristic

C₄×C₇₃

#496

yes

—

C₄×C₇₃

C₂×Q₃₂

yes

—

C₄×C₇₃

#510

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

—

D₇₃

yes

characteristic

C₇₃

#12528

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

146

—

#500

—

#500

—

146

—

#500

—

146

—

#498

—

146

—

#494

—

#494

—

#487

—

146

—

#358

—

146

—

C₄×C₁₆

C₂×D₇₃

yes

—

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

146

—

C₂×Q₁₆

—

146

—

C₂×Q₁₆

—

146

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

—

146

—

Q₃₂

—

292

—

C₂×C₁₆

D₂₉₂

yes

—

#107

—

292

—

100

#107

—

292

—

101

#107

—

146

—

102

#106

—

292

—

103

#103

—

292

—

104

C₄×C₈

D₂₉₂

yes

characteristic

105

C₄×C₈

D₂₉₂

yes

characteristic

106

C₄×C₈

C₂²×D₇₃

yes

characteristic

107

C₂×Q₈

—

292

—

108

C₂×Q₈

—

292

—

109

C₂×Q₈

—

292

—

110

Q₁₆

—

292

—

111

Q₁₆

—

292

—

112

Q₁₆

—

292

—

113

C₂×C₈

D₅₈₄

yes

—

114

C₂×C₈

D₅₈₄

yes

—

115

C₂×C₈

C₂×D₂₉₂

yes

—

116

C₄⋊C₄

—

292

—

117

C₄⋊C₄

—

292

—

118

C₄⋊C₄

—

292

—

119

C₄⋊C₄

—

292

—

120

C₄⋊C₄

—

292

—

121

C₄⋊C₄

—

292

—

122

C₄²

C₂×D₂₉₂

yes

characteristic

123

C₁₆

—

124

Q₈

—

584

—

125

Q₈

—

584

—

126

Q₈

—

584

—

127

Q₈

—

584

—

128

C₂×C₄

—

584

—

129

C₂×C₄

—

584

—

130

C₂×C₄

—

584

—

131

C₂×C₄

#14556588

yes

characteristic

132

C₂×C₄

C₂×D₅₈₄

yes

characteristic

133

C₂×C₄

C₂×D₅₈₄

yes

characteristic

134

C₈

—

135

C₈

Dic₅₈₄

yes

complemented

136

C₈

#14556619

yes

—

137

C₂²

#17988964

yes

characteristic

138

C₄

—

1168

—

139

C₄

—

1168

—

140

C₄

—

1168

—

141

C₄

#17988976

yes

—

142

C₄

C₂×Dic₅₈₄

yes

—

143

C₄

#17989030

yes

—

144

C₂

#22609597

yes

characteristic

145

C₂

#22612184

yes

characteristic

146

C₂

#22612202

yes

characteristic

	subgroup	G/H
center	C₂²	#17988964
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#1312	C₂
socle, prime radical	C₂²×C₇₃	#493
hypercenter	#1312	D₇₃
hypocenter	C₇₃	#12528
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#17988964

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#1312

C₂

C₂²×C₇₃

D₇₃

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
61. C₈×C₇₃	Q₃₂	semidirect
79. C₇₃	#12528	semidirect
135. C₈	Dic₅₈₄	semidirect

H ≅ G/N

type

61. C₈×C₇₃

Q₃₂

semidirect

79. C₇₃

#12528

semidirect

135. C₈

Dic₅₈₄

semidirect

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

subgroup	index
#22612184	2
#22612159	2
#22609549	2
#22609528	2
C₇₃×#1312	2
#12528	73

subgroup

index

C₇₃×#1312

Subgroups — #33000753

Conjugacy classes of subgroups

Complements

Maximal subgroups