Group #14529605 — subgroups

Subgroups — #14529605

Order 2176.

308 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 113 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×#10587506	C₂	1	4	yes	maximal
2	C₂×#10587474	C₂	1	1	yes	maximal, characteristic
3	#11762675	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₁₇	C₂²	1	2	yes	—
6	C₁₇×C₄⋊Q₈	—	2	2	no	—
7	C₁₇×C₄⋊Q₈	C₄	1	4	yes	complemented
8	C₂×C₄⋊C₄×C₁₇	—	2	2	no	—
9	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
11	C₂²×#3622	C₂²	1	2	yes	—
12	#10587506	—	2	8	no	—
13	#10587502	—	2	2	no	—
14	#10587474	—	2	2	no	—
15	C₂×Q₈×C₁₇	—	2	2	no	—
16	C₂×Q₈×C₁₇	—	2	4	no	—
17	C₂×Q₈×C₁₇	—	2	4	no	—
18	C₂×Q₈×C₁₇	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₁₇	—	2	2	no	—
20	C₂²×C₄×C₁₇	D₄	1	2	yes	—
21	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₁₇	—	2	2	no	—
23	C₄⋊C₄×C₁₇	—	2	8	no	—
24	C₄²×C₁₇	C₂×C₄	1	2	yes	—
25	C₄²×C₁₇	D₄	1	2	yes	—
26	C₄⋊Dic₁₇	—	2	4	no	—
27	C₂×#3622	—	2	2	no	—
28	C₂×#3622	—	2	2	no	—
29	C₂×#3622	D₄	1	4	yes	—
30	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₁₇	—	2	8	no	—
32	Q₈×C₁₇	—	4	4	no	—
33	C₂×C₄×C₁₇	—	2	4	no	—
34	C₂×C₄×C₁₇	—	2	4	no	—
35	C₂×C₄×C₁₇	—	4	2	no	—
36	C₂×C₄×C₁₇	—	4	2	no	—
37	C₂×C₄×C₁₇	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇	#41	1	2	yes	—
42	C₂×Dic₁₇	—	4	2	no	—
43	C₂×Dic₁₇	—	4	4	no	—
44	#3622	—	2	4	no	—
45	#1960	—	17	1	no	maximal, Sylow
46	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₇	#124	1	1	yes	characteristic
48	C₂²×C₁₇	#127	1	1	yes	characteristic
49	C₂²×C₁₇	#137	1	4	yes	—
50	C₄×C₁₇	—	2	4	no	—
51	C₄×C₁₇	—	4	4	no	—
52	C₄×C₁₇	C₄×D₄	1	2	yes	—
53	C₄×C₁₇	#121	1	2	yes	—
54	Dic₁₇	—	8	2	no	—
55	C₂×C₄⋊Q₈	D₁₇	1	1	yes	characteristic
56	C₂×#103	—	17	4	no	—
57	C₂×#97	—	17	1	no	—
58	#382	—	17	1	no	—
59	C₂×C₁₇	#390	1	1	yes	characteristic
60	C₂×C₁₇	#419	1	2	yes	—
61	C₂×C₁₇	#491	1	2	yes	—
62	C₂×C₁₇	#497	1	2	yes	—
63	C₂²×Q₈	C₂×D₁₇	1	2	yes	—
64	C₂²×C₈	—	17	2	no	—
65	C₄⋊Q₈	—	2	2	no	—
66	C₄⋊Q₈	Dic₁₇	1	4	yes	complemented
67	C₂×C₄⋊C₄	—	2	2	no	—
68	C₂×C₄⋊C₄	—	17	2	no	—
69	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
70	#103	—	34	8	no	—
71	#97	—	34	2	no	—
72	#95	—	34	2	no	—
73	C₁₇	#1960	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₁₇	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	34	2	no	—
80	C₂²×C₄	C₂²⋊D₁₇	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
82	C₂×C₈	—	17	4	no	—
83	C₂×C₈	—	34	2	no	—
84	C₂×C₈	—	34	2	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	34	4	no	—
87	C₄²	C₂×Dic₁₇	1	2	yes	—
88	C₄²	C₂²⋊D₁₇	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	68	2	no	—
97	C₂×C₄	—	68	4	no	—
98	C₂×C₄	C₂²⋊Dic₁₇	1	4	yes	—
99	C₂×C₄	D₄×D₁₇	1	2	yes	—
100	C₂×C₄	#63254	1	2	yes	—
101	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
102	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
103	C₈	—	34	4	no	—
104	C₂²	#10587622	1	1	yes	characteristic
105	C₂²	#10587626	1	1	yes	characteristic
106	C₂²	#10587628	1	4	yes	—
107	C₂²	C₂×C₂²⋊Dic₁₇	1	1	yes	characteristic
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	136	2	no	—
111	C₄	D₄×Dic₁₇	1	2	yes	—
112	C₄	#10587624	1	2	yes	—
113	C₂	#11762712	1	1	yes	characteristic
114	C₂	#11762818	1	2	yes	—
115	C₂	#11763177	1	2	yes	—
116	C₂	#11763201	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587506

C₂

yes

maximal

C₂×#10587474

C₂

yes

maximal, characteristic

#11762675

C₂

yes

maximal, characteristic

C₂×C₁₇×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₇

C₂²

yes

—

C₁₇×C₄⋊Q₈

—

C₁₇×C₄⋊Q₈

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₇

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂²×#3622

C₂²

yes

—

#10587506

—

#10587502

—

#10587474

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

C₂×C₄

yes

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

D₄

yes

—

C₄⋊Dic₁₇

—

C₂×#3622

—

C₂×#3622

—

C₂×#3622

D₄

yes

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

#3622

—

#1960

—

maximal, Sylow

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₄×D₄

yes

—

C₄×C₁₇

#121

yes

—

Dic₁₇

—

C₂×C₄⋊Q₈

D₁₇

yes

characteristic

C₂×#103

—

C₂×#97

—

#382

—

C₂×C₁₇

#390

yes

characteristic

C₂×C₁₇

#419

yes

—

C₂×C₁₇

#491

yes

—

C₂×C₁₇

#497

yes

—

C₂²×Q₈

C₂×D₁₇

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₁₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₇

yes

—

—

—

C₁₇

#1960

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×Dic₁₇

yes

—

C₄²

C₂²⋊D₁₇

yes

—

C₂³

C₂×C₂²⋊D₁₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₁₇

yes

—

C₂×C₄

D₄×D₁₇

yes

—

100

C₂×C₄

#63254

yes

—

101

C₂×C₄

C₂²×Dic₁₇

yes

characteristic

102

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

103

C₈

—

104

C₂²

#10587622

yes

characteristic

105

C₂²

#10587626

yes

characteristic

106

C₂²

#10587628

yes

—

107

C₂²

C₂×C₂²⋊Dic₁₇

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

136

—

111

C₄

D₄×Dic₁₇

yes

—

112

C₄

#10587624

yes

—

113

C₂

#11762712

yes

characteristic

114

C₂

#11762818

yes

—

115

C₂

#11763177

yes

—

116

C₂

#11763201

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×C₄⋊Q₈	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₇
hypocenter	C₁₇	#1960
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₇

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂×C₁₇×C₄⋊Q₈

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

cosocle

C₂²×C₄×C₁₇

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₇

hypocenter

C₁₇

#1960

LCS

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
7. C₁₇×C₄⋊Q₈	C₄	semidirect
66. C₄⋊Q₈	Dic₁₇	semidirect
73. C₁₇	#1960	semidirect

H ≅ G/N

type

7. C₁₇×C₄⋊Q₈

C₄

semidirect

66. C₄⋊Q₈

Dic₁₇

semidirect

73. C₁₇

#1960

semidirect

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

subgroup	index
C₂×#10587506	2
C₂×#10587474	2
#11762675	2
C₂×C₁₇×C₄⋊Q₈	2
#1960	17

subgroup

index

C₂×C₁₇×C₄⋊Q₈

Subgroups — #14529605

Conjugacy classes of subgroups

Complements

Maximal subgroups