Group #14528952 — subgroups

Subgroups — #14528952

Order 2176.

256 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 141 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×#10587489	C₂	1	2	yes	maximal
2	C₂×#10587488	C₂	1	2	yes	maximal
3	#11762679	C₂	1	1	yes	maximal, characteristic
4	#11762675	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₇×#97	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₁₇	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₇	—	2	2	no	—
9	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
12	C₁₇×#97	C₄	1	4	yes	complemented
13	#10587502	—	2	2	no	—
14	#10587489	—	2	4	no	—
15	#10587488	—	2	4	no	—
16	C₂²×C₄×C₁₇	D₄	1	2	yes	—
17	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₁₇	—	2	1	no	—
19	C₂²×Dic₁₇	—	2	1	no	—
20	C₂²×Dic₁₇	—	2	2	no	—
21	C₂×C₈×C₁₇	C₂×C₄	1	4	yes	—
22	C₂×C₈×C₁₇	D₄	1	4	yes	—
23	C₂×C₈×C₁₇	Q₈	1	4	yes	—
24	C₄²×C₁₇	C₂×C₄	1	2	yes	—
25	C₄²×C₁₇	Q₈	1	2	yes	—
26	C₄⋊Dic₁₇	—	2	2	no	—
27	C₄⋊Dic₁₇	—	2	2	no	—
28	C₄⋊Dic₁₇	—	2	2	no	—
29	C₄⋊Dic₁₇	—	2	2	no	—
30	C₄⋊Dic₁₇	—	4	4	no	—
31	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₇	—	2	2	no	—
33	C₂×C₄×C₁₇	—	2	2	no	—
34	C₂×C₄×C₁₇	C₄⋊C₄	1	4	yes	—
35	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×Dic₁₇	—	4	1	no	—
40	C₂×Dic₁₇	—	4	2	no	—
41	C₂×Dic₁₇	—	4	1	no	—
42	C₂×Dic₁₇	—	4	2	no	—
43	C₂×Dic₁₇	—	4	4	no	—
44	C₂×Dic₁₇	—	4	2	no	—
45	C₈×C₁₇	C₄⋊C₄	1	8	yes	complemented
46	#1847	—	17	1	no	maximal, Sylow
47	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₇	#127	1	1	yes	characteristic
49	C₂²×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
51	C₂²×C₁₇	#137	1	2	yes	—
52	C₄×C₁₇	—	2	4	no	—
53	C₄×C₁₇	C₂×C₄⋊C₄	1	2	yes	—
54	C₄×C₁₇	C₄⋊Q₈	1	2	yes	—
55	Dic₁₇	—	8	1	no	—
56	Dic₁₇	—	8	2	no	—
57	Dic₁₇	—	8	1	no	—
58	C₂×#107	—	17	2	no	—
59	C₂×#106	—	17	2	no	—
60	C₂×#97	D₁₇	1	1	yes	characteristic
61	#384	—	17	1	no	—
62	#382	—	17	1	no	—
63	C₂×C₁₇	#384	1	1	yes	characteristic
64	C₂×C₁₇	#428	1	2	yes	—
65	C₂×C₁₇	#496	1	1	yes	characteristic
66	C₂×C₁₇	#497	1	1	yes	characteristic
67	C₂×C₁₇	#501	1	2	yes	—
68	C₂²×C₈	C₂×D₁₇	1	2	yes	—
69	C₂×C₄⋊C₄	—	17	1	no	—
70	C₂×C₄⋊C₄	—	17	1	no	—
71	C₂×C₄⋊C₄	—	17	2	no	—
72	C₂×C₄⋊C₄	—	34	2	no	—
73	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
74	#107	—	34	4	no	—
75	#106	—	34	4	no	—
76	#97	Dic₁₇	1	4	yes	complemented
77	#95	—	34	2	no	—
78	C₁₇	#1847	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₄	—	34	1	no	—
80	C₂²×C₄	—	34	1	no	—
81	C₂²×C₄	—	34	2	no	—
82	C₂²×C₄	D₆₈	1	2	yes	—
83	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
84	C₂×C₈	Dic₃₄	1	4	yes	—
85	C₂×C₈	D₆₈	1	4	yes	—
86	C₂×C₈	C₂×Dic₁₇	1	4	yes	—
87	C₄⋊C₄	—	34	2	no	—
88	C₄⋊C₄	—	34	2	no	—
89	C₄⋊C₄	—	34	2	no	—
90	C₄⋊C₄	—	34	2	no	—
91	C₄⋊C₄	—	68	4	no	—
92	C₄²	Dic₃₄	1	2	yes	—
93	C₄²	C₂×Dic₁₇	1	2	yes	—
94	C₂³	C₂×D₆₈	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	68	1	no	—
98	C₂×C₄	—	68	2	no	—
99	C₂×C₄	—	68	1	no	—
100	C₂×C₄	—	68	4	no	—
101	C₂×C₄	—	68	2	no	—
102	C₂×C₄	—	68	2	no	—
103	C₂×C₄	C₄⋊Dic₁₇	1	4	yes	—
104	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
105	C₂×C₄	C₂×Dic₃₄	1	2	yes	—
106	C₂×C₄	C₂×D₆₈	1	2	yes	—
107	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
108	C₈	C₄⋊Dic₁₇	1	8	yes	complemented
109	C₂²	#10587555	1	1	yes	characteristic
110	C₂²	#10587560	1	1	yes	characteristic
111	C₂²	#10587594	1	2	yes	—
112	C₂²	#10587595	1	2	yes	—
113	C₂²	C₂×C₄⋊Dic₁₇	1	1	yes	characteristic
114	C₄	—	2	4	no	—
115	C₄	—	136	1	no	—
116	C₄	—	136	2	no	—
117	C₄	—	136	1	no	—
118	C₄	#10587555	1	2	yes	—
119	C₄	C₂×C₄⋊Dic₁₇	1	2	yes	—
120	C₂	#11762679	1	1	yes	characteristic
121	C₂	#11762851	1	2	yes	—
122	C₂	#11763198	1	1	yes	characteristic
123	C₂	#11763203	1	1	yes	characteristic
124	C₂	#11763218	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587489

C₂

yes

maximal

C₂×#10587488

C₂

yes

maximal

#11762679

C₂

yes

maximal, characteristic

#11762675

C₂

yes

maximal, characteristic

C₂×C₁₇×#97

C₂

yes

maximal, characteristic

C₂²×C₈×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

—

C₂×C₄⋊Dic₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×C₄⋊Dic₁₇

C₂²

yes

characteristic

C₁₇×#97

C₄

yes

—

—

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂²×Dic₁₇

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₂×C₈×C₁₇

D₄

yes

—

C₂×C₈×C₁₇

Q₈

yes

—

C₄²×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

Q₈

yes

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×Q₈

yes

—

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₈×C₁₇

C₄⋊C₄

yes

complemented

#1847

—

maximal, Sylow

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇

#127

yes

characteristic

C₂²×C₁₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₇

AGL(1,ℤ/8)

yes

—

C₂²×C₁₇

#137

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₂×C₄⋊C₄

yes

—

C₄×C₁₇

C₄⋊Q₈

yes

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

—

C₂×#107

—

C₂×#106

—

C₂×#97

D₁₇

yes

characteristic

#384

—

#382

—

C₂×C₁₇

#384

yes

characteristic

C₂×C₁₇

#428

yes

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

—

C₂²×C₈

C₂×D₁₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₇

yes

—

—

Dic₁₇

yes

complemented

#95

—

C₁₇

#1847

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₆₈

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

Dic₃₄

yes

—

C₂×C₈

D₆₈

yes

—

C₂×C₈

C₂×Dic₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₃₄

yes

—

C₄²

C₂×Dic₁₇

yes

—

C₂³

C₂×D₆₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₄⋊Dic₁₇

yes

—

104

C₂×C₄

C₂×Dic₃₄

yes

characteristic

105

C₂×C₄

C₂×Dic₃₄

yes

—

106

C₂×C₄

C₂×D₆₈

yes

—

107

C₂×C₄

C₂²×Dic₁₇

yes

characteristic

108

C₈

C₄⋊Dic₁₇

yes

complemented

109

C₂²

#10587555

yes

characteristic

110

C₂²

#10587560

yes

characteristic

111

C₂²

#10587594

yes

—

112

C₂²

#10587595

yes

—

113

C₂²

C₂×C₄⋊Dic₁₇

yes

characteristic

114

C₄

—

115

C₄

—

136

—

116

C₄

—

136

—

117

C₄

—

136

—

118

C₄

#10587555

yes

—

119

C₄

C₂×C₄⋊Dic₁₇

yes

—

120

C₂

#11762679

yes

characteristic

121

C₂

#11762851

yes

—

122

C₂

#11763198

yes

characteristic

123

C₂

#11763203

yes

characteristic

124

C₂

#11763218

yes

—

	subgroup	G/H
center	C₂³	C₂×D₆₈
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×#97	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂×#97	D₁₇
hypocenter	C₁₇	#1847
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₆₈

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂×C₁₇×#97

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

cosocle

C₂²×C₄×C₁₇

C₂³

hypercenter

C₂×#97

D₁₇

hypocenter

C₁₇

#1847

LCS

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
12. C₁₇×#97	C₄	semidirect
45. C₈×C₁₇	C₄⋊C₄	semidirect
76. #97	Dic₁₇	semidirect
78. C₁₇	#1847	semidirect
108. C₈	C₄⋊Dic₁₇	semidirect

H ≅ G/N

type

12. C₁₇×#97

C₄

semidirect

45. C₈×C₁₇

C₄⋊C₄

semidirect

76. #97

Dic₁₇

semidirect

78. C₁₇

#1847

semidirect

108. C₈

C₄⋊Dic₁₇

semidirect

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

subgroup	index
C₂×#10587489	2
C₂×#10587488	2
#11762679	2
#11762675	2
C₂×C₁₇×#97	2
#1847	17

subgroup

index

C₂×C₁₇×#97

Subgroups — #14528952

Conjugacy classes of subgroups

Complements

Maximal subgroups