Group #14528154 — subgroups

Subgroups — #14528154

Order 2176.

156 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 61 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	#11763189	C₂	1	1	yes	maximal, characteristic
2	#11763094	C₂	1	2	yes	maximal
3	#11762562	C₂	1	1	yes	maximal, characteristic
4	#11762556	C₂	1	2	yes	maximal
5	C₈×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×Dic₆₈	—	2	1	no	—
8	C₂×Dic₆₈	—	2	1	no	—
9	#10587555	C₂²	1	2	yes	—
10	C₁₇×C₄⋊C₈	—	2	1	no	—
11	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₁₇	—	2	1	no	—
13	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
14	#10587506	—	2	2	no	—
15	#10587479	—	2	2	no	—
16	C₄⋊#3622	C₂²	1	2	yes	—
17	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
18	C₂×Dic₃₄	—	2	2	no	—
19	C₂×C₈×C₁₇	—	2	1	no	—
20	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₇	D₄	1	1	yes	characteristic
22	C₄⋊C₄×C₁₇	—	2	1	no	—
23	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
24	C₄²×C₁₇	—	2	1	no	—
25	C₄²×C₁₇	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₁₇	—	2	2	no	—
27	C₄⋊Dic₁₇	—	2	2	no	—
28	C₂×#3622	—	2	2	no	—
29	Dic₆₈	—	4	2	no	—
30	Dic₆₈	—	4	2	no	—
31	Q₈×C₁₇	—	2	1	no	—
32	Q₈×C₁₇	Q₁₆	1	2	yes	complemented
33	C₂×C₄×C₁₇	—	2	1	no	—
34	C₂×C₄×C₁₇	—	2	1	no	—
35	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₇	#41	1	1	yes	characteristic
38	C₂×Dic₁₇	—	4	2	no	—
39	Dic₃₄	—	4	2	no	—
40	Dic₃₄	—	4	2	no	—
41	C₈×C₁₇	—	2	1	no	—
42	C₈×C₁₇	—	4	1	no	—
43	C₈×C₁₇	SD₁₆	1	2	yes	—
44	#3622	—	4	2	no	—
45	#1670	—	17	1	no	maximal, Sylow
46	C₂²×C₁₇	#121	1	1	yes	characteristic
47	C₄×C₁₇	—	2	2	no	—
48	C₄×C₁₇	—	2	1	no	—
49	C₄×C₁₇	—	4	1	no	—
50	C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
51	C₄×C₁₇	C₂×Q₁₆	1	1	yes	characteristic
52	C₄×C₁₇	#135	1	1	yes	characteristic
53	C₄×C₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
54	Dic₁₇	—	8	2	no	—
55	#494	—	17	1	no	—
56	#475	—	17	2	no	—
57	C₈×Q₈	D₁₇	1	1	yes	characteristic
58	#335	—	17	1	no	—
59	#332	—	17	2	no	—
60	C₂×C₁₇	#460	1	1	yes	characteristic
61	C₂×C₁₇	#467	1	1	yes	characteristic
62	C₂×C₁₇	#473	1	1	yes	characteristic
63	C₂×Q₁₆	—	34	1	no	—
64	C₂×Q₁₆	—	34	1	no	—
65	C₄⋊Q₈	—	17	2	no	—
66	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
67	#106	—	34	2	no	—
68	C₄⋊C₈	—	2	1	no	—
69	C₄⋊C₈	—	17	2	no	—
70	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
71	#103	—	34	2	no	—
72	C₄×C₈	—	2	1	no	—
73	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
74	C₁₇	#1670	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₈	—	34	2	no	—
76	C₂×Q₈	D₆₈	1	1	yes	characteristic
77	Q₁₆	—	68	2	no	—
78	Q₁₆	—	68	2	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	34	2	no	—
81	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
82	C₂×C₈	C₂²⋊D₁₇	1	1	yes	characteristic
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	34	2	no	—
85	C₄⋊C₄	—	34	2	no	—
86	C₄⋊C₄	D₆₈	1	1	yes	characteristic
87	C₄²	—	2	1	no	—
88	C₄²	C₂²×D₁₇	1	1	yes	characteristic
89	Q₈	—	2	1	no	—
90	Q₈	—	68	2	no	—
91	Q₈	—	68	2	no	—
92	Q₈	Dic₆₈	1	2	yes	complemented
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	68	2	no	—
96	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
97	C₂×C₄	#63252	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	—	4	1	no	—
101	C₈	—	68	2	no	—
102	C₈	#63230	1	2	yes	—
103	C₂²	#10587616	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	4	1	no	—
107	C₄	—	136	2	no	—
108	C₄	#10587590	1	1	yes	characteristic
109	C₄	C₂×Dic₆₈	1	1	yes	characteristic
110	C₄	C₂×#63230	1	1	yes	characteristic
111	C₄	#10587635	1	1	yes	characteristic
112	C₂	#11763009	1	1	yes	characteristic
113	C₂	#11763050	1	1	yes	characteristic
114	C₂	#11763084	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763189

C₂

yes

maximal, characteristic

#11763094

C₂

yes

maximal

#11762562

C₂

yes

maximal, characteristic

#11762556

C₂

yes

maximal

C₈×Q₈×C₁₇

C₂

yes

maximal, characteristic

C₄×Q₈×C₁₇

C₂²

yes

characteristic

C₂×Dic₆₈

—

C₂×Dic₆₈

—

#10587555

C₂²

yes

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

C₂²

yes

characteristic

C₄×C₈×C₁₇

—

C₄×C₈×C₁₇

C₂²

yes

—

—

C₂²

yes

—

C₂×Q₈×C₁₇

D₄

yes

characteristic

C₂×Dic₃₄

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

D₄

yes

characteristic

C₂×C₈×C₁₇

D₄

yes

characteristic

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

—

C₄²×C₁₇

C₂³

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

—

Dic₆₈

—

Dic₆₈

—

Q₈×C₁₇

—

Q₈×C₁₇

Q₁₆

yes

complemented

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

characteristic

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

C₈×C₁₇

—

C₈×C₁₇

SD₁₆

yes

—

—

—

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₇

C₂×Q₁₆

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

#494

—

#475

—

C₈×Q₈

D₁₇

yes

characteristic

#335

—

#332

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

C₂×D₁₇

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₇

yes

characteristic

#103

—

C₄×C₈

—

C₄×C₈

C₂×D₁₇

yes

characteristic

C₁₇

#1670

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

D₆₈

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₁₇

yes

characteristic

C₂×C₈

C₂²⋊D₁₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₆₈

yes

characteristic

C₄²

—

C₄²

C₂²×D₁₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

Dic₆₈

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₆₈

yes

characteristic

C₂×C₄

#63252

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

C₈

—

100

C₈

—

101

C₈

—

102

C₈

#63230

yes

—

103

C₂²

#10587616

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

136

—

108

C₄

#10587590

yes

characteristic

109

C₄

C₂×Dic₆₈

yes

characteristic

110

C₄

C₂×#63230

yes

characteristic

111

C₄

#10587635

yes

characteristic

112

C₂

#11763009

yes

characteristic

113

C₂

#11763050

yes

characteristic

114

C₂

#11763084

yes

characteristic

	subgroup	G/H
center	C₂²	#10587616
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₈×Q₈×C₁₇	C₂
socle, prime radical	C₂²×C₁₇	#121
hypercenter	C₈×Q₈	D₁₇
hypocenter	C₁₇	#1670
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#10587616

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₈×Q₈×C₁₇

C₂

socle, prime radical

C₂²×C₁₇

#121

hypercenter

C₈×Q₈

D₁₇

hypocenter

C₁₇

#1670

LCS

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
32. Q₈×C₁₇	Q₁₆	semidirect
74. C₁₇	#1670	semidirect
92. Q₈	Dic₆₈	semidirect

H ≅ G/N

type

32. Q₈×C₁₇

Q₁₆

semidirect

74. C₁₇

#1670

semidirect

92. Q₈

Dic₆₈

semidirect

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

subgroup	index
#11763189	2
#11763094	2
#11762562	2
#11762556	2
C₈×Q₈×C₁₇	2
#1670	17

subgroup

index

C₈×Q₈×C₁₇

Subgroups — #14528154

Conjugacy classes of subgroups

Complements

Maximal subgroups