Group #34000871 — subgroups

Subgroups — #34000871

Order 18688.

138 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 48 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	#22625036	C₂	1	1	yes	maximal, characteristic, complemented
2	#22624962	C₂	1	1	yes	maximal, characteristic, complemented
3	D₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556641	—	2	1	no	—
5	#17989450	—	2	1	no	—
6	D₈×Dic₇₃	—	2	1	no	—
7	#17988413	C₄	1	1	yes	characteristic
8	#17988388	—	2	1	no	—
9	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
10	C₂×D₁₆×C₇₃	C₄	1	1	yes	characteristic
11	C₄⋊#10596833	—	4	1	no	—
12	#14556641	—	2	2	no	—
13	D₄×Dic₇₃	—	4	1	no	—
14	C₂×#11924003	D₄	1	1	yes	characteristic
15	#14556525	—	2	1	no	—
16	#14556512	—	4	1	no	—
17	C₈×Dic₇₃	D₄	1	1	yes	characteristic
18	C₂×D₈×C₇₃	—	2	1	no	—
19	D₁₆×C₇₃	—	2	1	no	—
20	D₁₆×C₇₃	C₈	1	2	yes	complemented
21	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×#10596833	—	8	1	no	—
23	C₂²×Dic₇₃	—	8	1	no	—
24	C₂×#10596831	SD₁₆	1	1	yes	characteristic
25	C₄⋊Dic₇₃	—	4	1	no	—
26	C₂²⋊Dic₇₃	—	8	1	no	—
27	C₄×Dic₇₃	D₈	1	1	yes	characteristic
28	#11924003	—	2	1	no	—
29	C₂×D₄×C₇₃	—	4	1	no	—
30	D₈×C₇₃	—	2	2	no	—
31	D₈×C₇₃	—	4	1	no	—
32	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₁₆×C₇₃	C₂×C₈	1	1	yes	characteristic
34	C₁₆×C₇₃	M₁₆	1	1	yes	characteristic
35	C₂³×C₇₃	—	8	1	no	—
36	D₄×C₇₃	—	4	2	no	—
37	D₄×C₇₃	—	8	1	no	—
38	C₂×C₄×C₇₃	#102	1	1	yes	characteristic
39	C₂×Dic₇₃	—	8	1	no	—
40	C₂×Dic₇₃	—	16	1	no	—
41	C₂×Dic₇₃	—	16	1	no	—
42	C₂×Dic₇₃	D₁₆	1	1	yes	characteristic
43	C₂×Dic₇₃	SD₃₂	1	1	yes	characteristic
44	#10596833	—	16	1	no	—
45	C₈×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
46	C₈×C₇₃	#104	1	1	yes	characteristic
47	#10596831	—	2	1	no	—
48	C₂²×C₇₃	—	8	2	no	—
49	C₂²×C₇₃	—	16	1	no	—
50	C₂²×C₇₃	—	16	1	no	—
51	C₂²×C₇₃	#357	1	1	yes	characteristic
52	C₄×C₇₃	D₄⋊C₈	1	1	yes	characteristic
53	C₄×C₇₃	#359	1	1	yes	characteristic
54	Dic₇₃	—	2	1	no	—
55	Dic₇₃	—	16	1	no	—
56	Dic₇₃	D₃₂	1	1	yes	characteristic
57	Dic₇₃	SD₆₄	1	1	yes	characteristic
58	#7383	—	73	1	no	maximal, Sylow
59	C₂×C₇₃	—	16	2	no	—
60	C₂×C₇₃	#1331	1	1	yes	characteristic
61	C₂×C₇₃	#1415	1	1	yes	characteristic
62	C₂×C₇₃	#1417	1	1	yes	characteristic
63	C₄×D₁₆	—	73	1	no	—
64	C₄×C₃₂	—	73	1	no	—
65	#1371	—	73	1	no	—
66	C₇₃	#7383	1	1	yes	characteristic, complemented, Sylow
67	C₂×D₁₆	#64408	1	1	yes	characteristic
68	C₄×D₈	—	146	1	no	—
69	C₂×C₃₂	—	146	1	no	—
70	#366	—	73	1	no	—
71	#357	—	146	1	no	—
72	C₄×C₁₆	—	73	1	no	—
73	#335	—	146	1	no	—
74	C₂×D₈	—	2	1	no	—
75	C₄×D₄	—	292	1	no	—
76	D₁₆	—	2	1	no	—
77	D₁₆	#10596833	1	2	yes	complemented
78	C₂×C₁₆	—	73	1	no	—
79	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
80	#107	—	146	1	no	—
81	C₄⋊C₈	—	292	1	no	—
82	#102	—	292	1	no	—
83	C₄×C₈	—	73	1	no	—
84	C₃₂	—	146	2	no	—
85	C₂×D₄	—	4	1	no	—
86	C₂²×C₄	—	584	1	no	—
87	D₈	—	2	2	no	—
88	D₈	—	4	1	no	—
89	C₂×C₈	—	584	1	no	—
90	C₂×C₈	—	73	1	no	—
91	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
92	C₄⋊C₄	—	292	1	no	—
93	C₂²⋊C₄	—	584	1	no	—
94	C₄²	—	73	1	no	—
95	C₁₆	—	146	1	no	—
96	C₁₆	C₂×#10596833	1	1	yes	characteristic
97	C₁₆	#11924037	1	1	yes	characteristic
98	C₂³	—	8	1	no	—
99	D₄	—	4	2	no	—
100	D₄	—	8	1	no	—
101	C₂×C₄	—	584	1	no	—
102	C₂×C₄	—	73	1	no	—
103	C₂×C₄	—	73	1	no	—
104	C₂×C₄	—	1168	1	no	—
105	C₂×C₄	—	1168	1	no	—
106	C₂×C₄	#14556653	1	1	yes	characteristic
107	C₈	—	1168	1	no	—
108	C₈	—	146	1	no	—
109	C₈	C₂²⋊#10596833	1	1	yes	characteristic
110	C₈	#14556655	1	1	yes	characteristic
111	C₂²	—	8	2	no	—
112	C₂²	—	16	1	no	—
113	C₂²	—	16	1	no	—
114	C₂²	#17989499	1	1	yes	characteristic
115	C₄	—	73	1	no	—
116	C₄	—	73	1	no	—
117	C₄	—	1168	1	no	—
118	C₄	—	146	1	no	—
119	C₄	D₄⋊#10596833	1	1	yes	characteristic
120	C₄	#17989501	1	1	yes	characteristic
121	C₂	—	16	2	no	—
122	C₂	#22624885	1	1	yes	characteristic
123	C₂	#22625080	1	1	yes	characteristic
124	C₂	#22625082	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22625036

C₂

yes

maximal, characteristic, complemented

#22624962

C₂

yes

maximal, characteristic, complemented

D₁₆×Dic₇₃

C₂

yes

maximal, characteristic

C₂×#14556641

—

#17989450

—

D₈×Dic₇₃

—

#17988413

C₄

yes

characteristic

#17988388

—

C₁₆×Dic₇₃

C₂²

yes

characteristic

C₂×D₁₆×C₇₃

C₄

yes

characteristic

C₄⋊#10596833

—

#14556641

—

D₄×Dic₇₃

—

C₂×#11924003

D₄

yes

characteristic

#14556525

—

#14556512

—

C₈×Dic₇₃

D₄

yes

characteristic

C₂×D₈×C₇₃

—

D₁₆×C₇₃

—

D₁₆×C₇₃

C₈

yes

complemented

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×#10596833

—

C₂²×Dic₇₃

—

C₂×#10596831

SD₁₆

yes

characteristic

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

D₈

yes

characteristic

#11924003

—

C₂×D₄×C₇₃

—

D₈×C₇₃

—

D₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₁₆×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

M₁₆

yes

characteristic

C₂³×C₇₃

—

D₄×C₇₃

—

D₄×C₇₃

—

C₂×C₄×C₇₃

#102

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

D₁₆

yes

characteristic

C₂×Dic₇₃

SD₃₂

yes

characteristic

#10596833

—

C₈×C₇₃

C₂²⋊C₈

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#357

yes

characteristic

C₄×C₇₃

D₄⋊C₈

yes

characteristic

C₄×C₇₃

#359

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

D₃₂

yes

characteristic

Dic₇₃

SD₆₄

yes

characteristic

#7383

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₄×D₁₆

—

C₄×C₃₂

—

#1371

—

C₇₃

#7383

yes

characteristic, complemented, Sylow

C₂×D₁₆

#64408

yes

characteristic

C₄×D₈

—

146

—

C₂×C₃₂

—

146

—

#366

—

#357

—

146

—

C₄×C₁₆

—

#335

—

146

—

C₂×D₈

—

C₄×D₄

—

292

—

D₁₆

—

D₁₆

#10596833

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

C₂×#64408

yes

characteristic

#107

—

146

—

C₄⋊C₈

—

292

—

#102

—

292

—

C₄×C₈

—

C₃₂

—

146

—

C₂×D₄

—

C₂²×C₄

—

584

—

D₈

—

D₈

—

C₂×C₈

—

584

—

C₂×C₈

—

C₂×C₈

C₂²⋊#64408

yes

characteristic

C₄⋊C₄

—

292

—

C₂²⋊C₄

—

584

—

C₄²

—

C₁₆

—

146

—

C₁₆

yes

C₁₆

yes

C₂³

—

D₄

—

100

D₄

—

101

C₂×C₄

—

584

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

1168

—

105

C₂×C₄

—

1168

—

106

C₂×C₄

#14556653

yes

characteristic

107

C₈

—

1168

—

108

C₈

—

146

—

109

C₈

C₂²⋊#10596833

yes

characteristic

110

C₈

#14556655

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#17989499

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

1168

—

118

C₄

—

146

—

119

C₄

D₄⋊#10596833

yes

characteristic

120

C₄

#17989501

yes

characteristic

121

C₂

—

122

C₂

#22624885

yes

characteristic

123

C₂

#22625080

yes

characteristic

124

C₂

#22625082

yes

characteristic

	subgroup	G/H
center	C₂²	#17989499
commutator	C₁₆×C₇₃	C₂×C₈
Frattini	C₂×C₁₆	C₂×#64408
Fitting, prime radical	C₂×D₁₆×C₇₃	C₄
socle	C₂²×C₇₃	#357
cosocle	C₁₆×Dic₇₃	C₂²
hypercenter	C₂×D₁₆	#64408
hypocenter	C₇₃	#7383
LCS	G ▷ C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₁₆

subgroup

G/H

center

C₂²

#17989499

commutator

C₁₆×C₇₃

C₂×C₈

Frattini

C₂×C₁₆

C₂×#64408

Fitting, prime radical

C₂×D₁₆×C₇₃

C₄

socle

C₂²×C₇₃

#357

cosocle

C₁₆×Dic₇₃

C₂²

C₂×D₁₆

C₇₃

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₁₆

N	H ≅ G/N	type
1. #22625036	C₂	semidirect
2. #22624962	C₂	semidirect
20. D₁₆×C₇₃	C₈	semidirect
66. C₇₃	#7383	semidirect
77. D₁₆	#10596833	semidirect

H ≅ G/N

type

1. #22625036

C₂

semidirect

2. #22624962

C₂

semidirect

20. D₁₆×C₇₃

C₈

semidirect

66. C₇₃

#7383

semidirect

77. D₁₆

#10596833

semidirect

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

subgroup	index
#22625036	2
#22624962	2
D₁₆×Dic₇₃	2
#7383	73

subgroup

index

#22625036

#22624962

D₁₆×Dic₇₃

#7383

Subgroups — #34000871

Conjugacy classes of subgroups

Complements

Maximal subgroups