Group #14529762 — subgroups

Subgroups — #14529762

Order 2176.

284 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 105 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	#11762833	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₁₇	C₂	1	1	yes	maximal, characteristic
3	#11762568	C₂	1	4	yes	maximal
4	C₁₇×#533	C₂	1	1	yes	maximal, characteristic
5	C₁₇×C₄⋊Q₈	C₄	1	2	yes	complemented
6	C₁₇×#125	C₄	1	2	yes	complemented
7	C₁₇×C₂²⋊Q₈	—	2	2	no	—
8	C₄×Q₈×C₁₇	—	2	2	no	—
9	C₁₇×#117	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₇	—	2	2	no	—
12	C₂×C₄×Dic₁₇	C₂²	1	2	yes	—
13	C₂×#63223	C₂²	1	2	yes	—
14	C₄⋊#3622	—	2	2	no	—
15	C₂×Q₈×C₁₇	—	2	2	no	—
16	C₂²×C₄×C₁₇	D₄	1	2	yes	—
17	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₁₇	—	2	2	no	—
19	C₄⋊C₄×C₁₇	—	2	4	no	—
20	C₄⋊C₄×C₁₇	—	2	2	no	—
21	C₄⋊C₄×C₁₇	C₂×C₄	1	4	yes	—
22	C₂²⋊C₄×C₁₇	—	2	2	no	—
23	C₄²×C₁₇	—	2	2	no	—
24	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
26	C₄²×C₁₇	D₄	1	2	yes	—
27	C₄×Dic₁₇	—	2	8	no	—
28	C₄×Dic₁₇	—	2	2	no	—
29	C₄×Dic₁₇	Q₈	1	4	yes	complemented
30	#63223	—	2	4	no	—
31	C₂×#3622	—	2	2	no	—
32	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
33	Q₈×C₁₇	—	4	4	no	—
34	C₂×C₄×C₁₇	—	2	4	no	—
35	C₂×C₄×C₁₇	—	2	4	no	—
36	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₇	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₁₇	#41	1	2	yes	—
42	C₂×Dic₁₇	—	2	8	no	—
43	C₂×Dic₁₇	—	2	4	no	—
44	#3622	—	4	2	no	—
45	#1994	—	17	1	no	maximal, Sylow
46	C₂²×C₁₇	—	2	2	no	—
47	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₇	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₁₇	—	2	2	no	—
50	C₄×C₁₇	—	4	4	no	—
51	C₄×C₁₇	#104	1	4	yes	—
52	C₄×C₁₇	C₄×Q₈	1	2	yes	—
53	C₄×C₁₇	#124	1	1	yes	characteristic
54	C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
55	Dic₁₇	—	2	8	no	—
56	#533	D₁₇	1	1	yes	characteristic
57	#423	—	17	1	no	—
58	C₄³	—	17	1	no	—
59	#339	—	17	4	no	—
60	C₂×C₁₇	—	2	2	no	—
61	C₂×C₁₇	#391	1	1	yes	characteristic
62	C₂×C₁₇	C₂×#104	1	2	yes	—
63	C₂×M₁₆	—	17	2	no	—
64	C₄⋊Q₈	Dic₁₇	1	2	yes	complemented
65	#125	Dic₁₇	1	2	yes	complemented
66	C₂²⋊Q₈	—	2	2	no	—
67	C₄×Q₈	—	2	2	no	—
68	#117	C₂×D₁₇	1	2	yes	—
69	C₂×C₄²	—	17	2	no	—
70	C₂×C₄²	—	34	2	no	—
71	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
72	C₄⋊C₈	—	34	2	no	—
73	C₁₇	#1994	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₈	—	2	2	no	—
75	C₂²×C₄	—	34	2	no	—
76	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
77	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
78	M₁₆	—	34	4	no	—
79	C₂×C₈	—	34	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	C₂×Dic₁₇	1	4	yes	—
83	C₂²⋊C₄	—	2	2	no	—
84	C₄²	—	2	2	no	—
85	C₄²	—	17	4	no	—
86	C₄²	—	34	8	no	—
87	C₄²	—	34	2	no	—
88	C₄²	C₂×Dic₁₇	1	1	yes	characteristic
89	C₄²	C₂×Dic₁₇	1	1	yes	characteristic
90	C₄²	C₂²⋊D₁₇	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
92	Q₈	—	4	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	34	8	no	—
96	C₂×C₄	—	34	4	no	—
97	C₂×C₄	C₂²⋊Dic₁₇	1	2	yes	—
98	C₂×C₄	C₂²⋊Dic₁₇	1	2	yes	—
99	C₂×C₄	Q₈×D₁₇	1	2	yes	—
100	C₂×C₄	#63256	1	2	yes	—
101	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
103	C₈	—	68	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	#10587632	1	2	yes	—
106	C₂²	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
107	C₄	—	2	2	no	—
108	C₄	—	4	4	no	—
109	C₄	—	34	8	no	—
110	C₄	#10587508	1	4	yes	—
111	C₄	#10587630	1	1	yes	characteristic
112	C₄	Q₈×Dic₁₇	1	2	yes	—
113	C₄	#10587633	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#11762716	1	1	yes	characteristic
116	C₂	C₂×#10587508	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11762833

C₂

yes

maximal, characteristic

C₄²×Dic₁₇

C₂

yes

maximal, characteristic

#11762568

C₂

yes

maximal

C₁₇×#533

C₂

yes

maximal, characteristic

C₁₇×C₄⋊Q₈

C₄

yes

complemented

C₁₇×#125

C₄

yes

complemented

C₁₇×C₂²⋊Q₈

—

C₄×Q₈×C₁₇

—

C₁₇×#117

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₇

—

C₂×C₄×Dic₁₇

C₂²

yes

—

C₂×#63223

C₂²

yes

—

C₄⋊#3622

—

C₂×Q₈×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₂²⋊C₄×C₁₇

—

C₄²×C₁₇

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

D₄

yes

—

C₄×Dic₁₇

—

C₄×Dic₁₇

—

C₄×Dic₁₇

Q₈

yes

complemented

#63223

—

C₂×#3622

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

—

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

#3622

—

#1994

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇

C₂²⋊Q₈

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#104

yes

—

C₄×C₁₇

C₄×Q₈

yes

—

C₄×C₁₇

#124

yes

characteristic

C₄×C₁₇

C₄⋊Q₈

yes

characteristic

Dic₁₇

—

#533

D₁₇

yes

characteristic

#423

—

C₄³

—

#339

—

C₂×C₁₇

—

C₂×C₁₇

#391

yes

characteristic

C₂×C₁₇

C₂×#104

yes

—

C₂×M₁₆

—

C₄⋊Q₈

Dic₁₇

yes

complemented

#125

Dic₁₇

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂×D₁₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

C₄⋊C₈

—

C₁₇

#1994

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₁₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×Dic₁₇

yes

characteristic

C₄²

C₂×Dic₁₇

yes

characteristic

C₄²

C₂²⋊D₁₇

yes

—

C₂³

C₂×C₂²⋊D₁₇

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₁₇

yes

—

C₂×C₄

C₂²⋊Dic₁₇

yes

—

C₂×C₄

Q₈×D₁₇

yes

—

100

C₂×C₄

#63256

yes

—

101

C₂×C₄

C₂²×Dic₁₇

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

103

C₈

—

104

C₂²

—

105

C₂²

#10587632

yes

—

106

C₂²

C₂×C₂²⋊Dic₁₇

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#10587508

yes

—

111

C₄

#10587630

yes

characteristic

112

C₄

Q₈×Dic₁₇

yes

—

113

C₄

#10587633

yes

characteristic

114

C₂

—

115

C₂

#11762716

yes

characteristic

116

C₂

C₂×#10587508

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₁₇×#533	C₂
socle	C₂²×C₁₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₁₇	C₂×D₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	#533	D₁₇
hypocenter	C₁₇	#1994
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₇

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₁₇×#533

C₂

socle

C₂²×C₁₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₁₇

C₂×D₄

cosocle

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
5. C₁₇×C₄⋊Q₈	C₄	semidirect
6. C₁₇×#125	C₄	semidirect
29. C₄×Dic₁₇	Q₈	semidirect
64. C₄⋊Q₈	Dic₁₇	semidirect
65. #125	Dic₁₇	semidirect
73. C₁₇	#1994	semidirect

H ≅ G/N

type

5. C₁₇×C₄⋊Q₈

C₄

semidirect

6. C₁₇×#125

C₄

semidirect

29. C₄×Dic₁₇

Q₈

semidirect

64. C₄⋊Q₈

Dic₁₇

semidirect

65. #125

Dic₁₇

semidirect

73. C₁₇

#1994

semidirect

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

subgroup	index
#11762833	2
C₄²×Dic₁₇	2
#11762568	2
C₁₇×#533	2
#1994	17

subgroup

index

#11762833

C₄²×Dic₁₇

#11762568

C₁₇×#533

#1994

Subgroups — #14529762

Conjugacy classes of subgroups

Complements

Maximal subgroups