Group #33000642 — subgroups

Subgroups — #33000642

Order 18688.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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	#22612206	C₂	1	1	yes	maximal, characteristic
2	#22612184	C₂	1	1	yes	maximal, characteristic
3	#22612159	C₂	1	2	yes	maximal
4	#22609528	C₂	1	2	yes	maximal
5	C₈×C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₈₄	—	2	2	no	—
7	#17988992	C₂²	1	2	yes	—
8	#17988985	—	2	2	no	—
9	#17988967	C₂²	1	2	yes	—
10	#17988938	—	2	2	no	—
11	#17988413	—	2	2	no	—
12	#17988392	—	2	4	no	—
13	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₈²×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₉₂	—	2	2	no	—
17	C₂×Dic₂₉₂	—	2	2	no	—
18	#14556591	—	2	2	no	—
19	#14556591	—	2	2	no	—
20	Dic₅₈₄	—	4	4	no	—
21	#14556525	—	2	2	no	—
22	#14556525	—	2	2	no	—
23	#14556523	—	4	4	no	—
24	#14556516	—	4	4	no	—
25	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
26	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
27	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₄₆	—	4	2	no	—
31	C₂×Dic₁₄₆	—	4	2	no	—
32	Dic₂₉₂	—	4	4	no	—
33	Dic₂₉₂	—	4	4	no	—
34	C₄⋊Dic₇₃	—	4	2	no	—
35	C₄⋊Dic₇₃	—	4	2	no	—
36	C₄⋊Dic₇₃	—	4	2	no	—
37	C₄⋊Dic₇₃	—	4	2	no	—
38	C₂×C₈×C₇₃	D₈	1	2	yes	—
39	C₂×C₈×C₇₃	D₈	1	2	yes	—
40	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₇₃	—	2	2	no	—
44	C₁₆×C₇₃	SD₁₆	1	4	yes	—
45	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
46	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₇₃	—	8	2	no	—
49	C₂×Dic₇₃	—	8	2	no	—
50	Dic₁₄₆	—	8	2	no	—
51	Dic₁₄₆	—	8	2	no	—
52	Dic₁₄₆	—	8	4	no	—
53	C₈×C₇₃	—	2	2	no	—
54	C₈×C₇₃	Q₃₂	1	4	yes	complemented
55	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
56	C₈×C₇₃	#135	1	2	yes	—
57	C₂²×C₇₃	#493	1	1	yes	characteristic
58	C₄×C₇₃	#492	1	1	yes	characteristic
59	C₄×C₇₃	#495	1	1	yes	characteristic
60	C₄×C₇₃	C₂×Q₃₂	1	2	yes	—
61	C₄×C₇₃	#508	1	2	yes	—
62	Dic₇₃	—	16	2	no	—
63	Dic₇₃	—	16	2	no	—
64	#12504	—	73	1	no	maximal, Sylow
65	C₂×C₇₃	#1706	1	1	yes	characteristic
66	C₂×C₇₃	#2247	1	1	yes	characteristic
67	C₂×C₇₃	#2249	1	1	yes	characteristic
68	#2252	—	73	1	no	—
69	#2247	—	73	1	no	—
70	#2241	—	73	2	no	—
71	#1707	—	73	2	no	—
72	C₈×C₁₆	D₇₃	1	1	yes	characteristic
73	C₇₃	#12504	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₃₂	—	146	2	no	—
75	#500	—	73	2	no	—
76	#498	—	146	2	no	—
77	#494	—	73	2	no	—
78	#487	—	146	2	no	—
79	#366	—	146	2	no	—
80	#358	—	146	4	no	—
81	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
82	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
83	C₈²	C₂×D₇₃	1	1	yes	characteristic
84	C₂×Q₁₆	—	146	2	no	—
85	C₂×Q₁₆	—	146	2	no	—
86	C₄⋊Q₈	—	146	2	no	—
87	C₄⋊Q₈	—	146	2	no	—
88	Q₃₂	—	292	4	no	—
89	C₂×C₁₆	D₂₉₂	1	2	yes	—
90	C₂×C₁₆	D₂₉₂	1	2	yes	—
91	#107	—	146	2	no	—
92	#107	—	146	2	no	—
93	#106	—	292	4	no	—
94	#103	—	292	4	no	—
95	C₄×C₈	D₂₉₂	1	1	yes	characteristic
96	C₄×C₈	D₂₉₂	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
98	C₂×Q₈	—	292	2	no	—
99	C₂×Q₈	—	292	2	no	—
100	Q₁₆	—	292	4	no	—
101	Q₁₆	—	292	4	no	—
102	C₂×C₈	D₅₈₄	1	2	yes	—
103	C₂×C₈	D₅₈₄	1	2	yes	—
104	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
105	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
106	C₄⋊C₄	—	292	2	no	—
107	C₄⋊C₄	—	292	2	no	—
108	C₄⋊C₄	—	292	2	no	—
109	C₄⋊C₄	—	292	2	no	—
110	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	#11924017	1	4	yes	—
113	Q₈	—	584	2	no	—
114	Q₈	—	584	2	no	—
115	Q₈	—	584	4	no	—
116	C₂×C₄	—	584	2	no	—
117	C₂×C₄	—	584	2	no	—
118	C₂×C₄	#14556588	1	1	yes	characteristic
119	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
120	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	Dic₅₈₄	1	4	yes	complemented
123	C₈	C₂×#11924017	1	2	yes	—
124	C₈	#14556613	1	2	yes	—
125	C₂²	#17988964	1	1	yes	characteristic
126	C₄	—	1168	2	no	—
127	C₄	—	1168	2	no	—
128	C₄	#17988960	1	1	yes	characteristic
129	C₄	#17988971	1	1	yes	characteristic
130	C₄	C₂×Dic₅₈₄	1	2	yes	—
131	C₄	#17989019	1	2	yes	—
132	C₂	#22609523	1	1	yes	characteristic
133	C₂	#22612184	1	1	yes	characteristic
134	C₂	#22612192	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612206

C₂

yes

maximal, characteristic

#22612184

C₂

yes

maximal, characteristic

#22612159

C₂

yes

maximal

#22609528

C₂

yes

maximal

C₈×C₁₆×C₇₃

C₂

yes

maximal, characteristic

C₂×Dic₅₈₄

—

#17988992

C₂²

yes

—

#17988985

—

#17988967

C₂²

yes

—

#17988938

—

#17988413

—

#17988392

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₈²×C₇₃

C₂²

yes

characteristic

C₂×Dic₂₉₂

—

C₂×Dic₂₉₂

—

#14556591

—

#14556591

—

Dic₅₈₄

—

—

—

—

—

C₂×C₁₆×C₇₃

D₄

yes

—

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

—

Dic₂₉₂

—

Dic₂₉₂

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

D₈

yes

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

SD₁₆

yes

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

Q₃₂

yes

complemented

C₈×C₇₃

C₂×SD₁₆

yes

—

C₈×C₇₃

#135

yes

—

C₂²×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂×Q₃₂

yes

—

C₄×C₇₃

#508

yes

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

—

C₈×C₁₆

D₇₃

yes

characteristic

C₇₃

#12504

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

146

—

#500

—

#498

—

146

—

#494

—

#487

—

146

—

#366

—

146

—

#358

—

146

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₈²

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

146

—

C₂×Q₁₆

—

146

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

—

146

—

Q₃₂

—

292

—

C₂×C₁₆

D₂₉₂

yes

—

C₂×C₁₆

D₂₉₂

yes

—

#107

—

146

—

#107

—

146

—

#106

—

292

—

#103

—

292

—

C₄×C₈

D₂₉₂

yes

characteristic

C₄×C₈

D₂₉₂

yes

characteristic

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

292

—

C₂×Q₈

—

292

—

100

Q₁₆

—

292

—

101

Q₁₆

—

292

—

102

C₂×C₈

D₅₈₄

yes

—

103

C₂×C₈

D₅₈₄

yes

—

104

C₂×C₈

C₂×D₂₉₂

yes

characteristic

105

C₂×C₈

C₂×D₂₉₂

yes

characteristic

106

C₄⋊C₄

—

292

—

107

C₄⋊C₄

—

292

—

108

C₄⋊C₄

—

292

—

109

C₄⋊C₄

—

292

—

110

C₄²

C₂×D₂₉₂

yes

characteristic

111

C₁₆

—

112

C₁₆

#11924017

yes

—

113

Q₈

—

584

—

114

Q₈

—

584

—

115

Q₈

—

584

—

116

C₂×C₄

—

584

—

117

C₂×C₄

—

584

—

118

C₂×C₄

#14556588

yes

characteristic

119

C₂×C₄

C₂×D₅₈₄

yes

characteristic

120

C₂×C₄

C₂×D₅₈₄

yes

characteristic

121

C₈

—

122

C₈

Dic₅₈₄

yes

complemented

123

C₈

C₂×#11924017

yes

—

124

C₈

#14556613

yes

—

125

C₂²

#17988964

yes

characteristic

126

C₄

—

1168

—

127

C₄

—

1168

—

128

C₄

#17988960

yes

characteristic

129

C₄

#17988971

yes

characteristic

130

C₄

C₂×Dic₅₈₄

yes

—

131

C₄

#17989019

yes

—

132

C₂

#22609523

yes

characteristic

133

C₂

#22612184

yes

characteristic

134

C₂

#22612192

yes

characteristic

	subgroup	G/H
center	C₂²	#17988964
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₈×C₁₆×C₇₃	C₂
socle, prime radical	C₂²×C₇₃	#493
hypercenter	C₈×C₁₆	D₇₃
hypocenter	C₇₃	#12504
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#17988964

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₈×C₁₆×C₇₃

C₂

socle, prime radical

C₂²×C₇₃

#493

hypercenter

C₈×C₁₆

D₇₃

hypocenter

C₇₃

#12504

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
54. C₈×C₇₃	Q₃₂	semidirect
73. C₇₃	#12504	semidirect
122. C₈	Dic₅₈₄	semidirect

H ≅ G/N

type

54. C₈×C₇₃

Q₃₂

semidirect

73. C₇₃

#12504

semidirect

122. C₈

Dic₅₈₄

semidirect

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

subgroup	index
#22612206	2
#22612184	2
#22612159	2
#22609528	2
C₈×C₁₆×C₇₃	2
#12504	73

subgroup

index

C₈×C₁₆×C₇₃

Subgroups — #33000642

Conjugacy classes of subgroups

Complements

Maximal subgroups