Group #14529628 — subgroups

Subgroups — #14529628

Order 2176.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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×#10587555	C₂	1	1	yes	maximal, characteristic
2	#11762833	C₂	1	1	yes	maximal, characteristic
3	C₂×#10587495	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₄	C₂²	1	2	yes	—
6	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
7	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₁₇	C₄	1	4	yes	—
9	C₂×#63223	C₂²	1	2	yes	—
10	#10587555	—	2	2	no	—
11	#10587555	—	2	1	no	—
12	#10587555	—	2	1	no	—
13	C₁₇×#97	—	2	1	no	—
14	C₄×C₈×C₁₇	—	2	1	no	—
15	#10587495	—	2	8	no	—
16	C₄⋊#3622	—	2	2	no	—
17	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
19	C₂²×Dic₁₇	C₂×C₄	1	4	yes	—
20	C₂×Dic₃₄	—	2	4	no	—
21	C₂×Dic₃₄	—	2	2	no	—
22	C₂×Dic₃₄	—	2	2	no	—
23	C₂×Dic₃₄	—	2	4	no	—
24	M₁₆×C₁₇	—	2	2	no	—
25	M₁₆×C₁₇	D₄	1	4	yes	—
26	C₂×C₈×C₁₇	—	2	2	no	—
27	C₄²×C₁₇	D₄	1	2	yes	—
28	C₄²×C₁₇	D₄	1	1	yes	characteristic
29	C₄²×C₁₇	D₄	1	1	yes	characteristic
30	C₄⋊Dic₁₇	—	2	4	no	—
31	C₄⋊Dic₁₇	—	2	4	no	—
32	#63223	—	2	4	no	—
33	C₂×#3622	—	2	2	no	—
34	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
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	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇	#41	1	2	yes	—
42	C₂×Dic₁₇	—	2	4	no	—
43	C₂×Dic₁₇	—	2	8	no	—
44	Dic₃₄	—	4	4	no	—
45	Dic₃₄	—	4	2	no	—
46	Dic₃₄	—	4	2	no	—
47	C₈×C₁₇	—	2	4	no	—
48	#3622	—	4	2	no	—
49	#1966	—	17	1	no	maximal, Sylow
50	C₂²×C₁₇	—	2	2	no	—
51	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₇	C₄×D₄	1	2	yes	—
53	C₄×C₁₇	—	2	2	no	—
54	C₄×C₁₇	#101	1	4	yes	complemented
55	C₄×C₁₇	#121	1	2	yes	—
56	C₄×C₁₇	#124	1	1	yes	characteristic
57	C₄×C₁₇	#127	1	1	yes	characteristic
58	Dic₁₇	—	4	4	no	—
59	C₂×C₄⋊Q₈	—	17	1	no	—
60	#423	—	17	1	no	—
61	C₂×#101	—	17	4	no	—
62	C₄×M₁₆	D₁₇	1	1	yes	characteristic
63	C₂×C₁₇	—	2	2	no	—
64	C₂×C₁₇	#390	1	1	yes	characteristic
65	C₂×C₁₇	C₂×#101	1	2	yes	—
66	C₂²×Q₈	—	17	2	no	—
67	C₂×M₁₆	—	17	2	no	—
68	C₂×M₁₆	C₂×D₁₇	1	2	yes	—
69	C₄⋊Q₈	—	34	2	no	—
70	C₄⋊Q₈	—	34	1	no	—
71	C₄⋊Q₈	—	34	1	no	—
72	C₂×C₄⋊C₄	—	17	4	no	—
73	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
74	C₄⋊C₈	—	34	2	no	—
75	#101	—	34	8	no	—
76	#97	—	2	1	no	—
77	C₄×C₈	—	2	1	no	—
78	C₁₇	#1966	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₈	—	34	4	no	—
80	C₂×Q₈	—	34	2	no	—
81	C₂×Q₈	—	34	4	no	—
82	C₂×Q₈	—	34	2	no	—
83	C₂²×C₄	—	17	4	no	—
84	C₂²×C₄	C₄×D₁₇	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	M₁₆	—	34	4	no	—
88	M₁₆	D₆₈	1	4	yes	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	34	2	no	—
91	C₄⋊C₄	—	34	4	no	—
92	C₄⋊C₄	—	34	4	no	—
93	C₄²	D₆₈	1	2	yes	—
94	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
95	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
96	C₂³	C₂×C₄×D₁₇	1	1	yes	characteristic
97	Q₈	—	68	2	no	—
98	Q₈	—	68	2	no	—
99	Q₈	—	68	4	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	34	8	no	—
102	C₂×C₄	—	34	4	no	—
103	C₂×C₄	#63227	1	2	yes	—
104	C₂×C₄	#63227	1	2	yes	—
105	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
106	C₂×C₄	C₂×D₆₈	1	2	yes	—
107	C₂×C₄	#63252	1	2	yes	—
108	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
109	C₈	—	2	4	no	—
110	C₈	—	68	2	no	—
111	C₂²	—	2	2	no	—
112	C₂²	C₄×D₆₈	1	2	yes	—
113	C₂²	C₂×#63227	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	68	4	no	—
116	C₄	#10587495	1	4	yes	complemented
117	C₄	#10587560	1	1	yes	characteristic
118	C₄	#10587561	1	1	yes	characteristic
119	C₄	#10587616	1	2	yes	—
120	C₂	—	2	2	no	—
121	C₂	#11762711	1	1	yes	characteristic
122	C₂	C₂×#10587495	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587555

C₂

yes

maximal, characteristic

#11762833

C₂

yes

maximal, characteristic

C₂×#10587495

C₂

yes

maximal

C₄×M₁₆×C₁₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₄

C₂²

yes

—

C₂×M₁₆×C₁₇

C₂²

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₇

C₄

yes

—

C₂×#63223

C₂²

yes

—

#10587555

—

#10587555

—

#10587555

—

C₁₇×#97

—

C₄×C₈×C₁₇

—

#10587495

—

C₄⋊#3622

—

C₂²×C₄×C₁₇

C₂×C₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

C₂×C₄

yes

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

M₁₆×C₁₇

—

M₁₆×C₁₇

D₄

yes

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

D₄

yes

—

C₄²×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

D₄

yes

characteristic

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

#63223

—

C₂×#3622

—

C₂³×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

—

#3622

—

#1966

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇

C₄×D₄

yes

—

C₄×C₁₇

—

C₄×C₁₇

#101

yes

complemented

C₄×C₁₇

#121

yes

—

C₄×C₁₇

yes

C₄×C₁₇

yes

Dic₁₇

—

C₂×C₄⋊Q₈

—

#423

—

C₂×#101

—

C₄×M₁₆

D₁₇

yes

characteristic

C₂×C₁₇

—

C₂×C₁₇

#390

yes

characteristic

C₂×C₁₇

C₂×#101

yes

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₁₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

C₄⋊C₈

—

#101

—

#97

—

C₄×C₈

—

C₁₇

#1966

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₄×D₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

D₆₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₆₈

yes

—

C₄²

C₂²⋊D₁₇

yes

characteristic

C₄²

C₂²⋊D₁₇

yes

characteristic

C₂³

C₂×C₄×D₁₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#63227

yes

—

104

C₂×C₄

#63227

yes

—

105

C₂×C₄

C₂×D₆₈

yes

characteristic

106

C₂×C₄

C₂×D₆₈

yes

—

107

C₂×C₄

#63252

yes

—

108

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

109

C₈

—

110

C₈

—

111

C₂²

—

112

C₂²

C₄×D₆₈

yes

—

113

C₂²

C₂×#63227

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

#10587495

yes

complemented

117

C₄

#10587560

yes

characteristic

118

C₄

#10587561

yes

characteristic

119

C₄

#10587616

yes

—

120

C₂

—

121

C₂

#11762711

yes

characteristic

122

C₂

C₂×#10587495

yes

—

	subgroup	G/H
center	C₂²	C₂×#63227
commutator, prime radical	C₂³×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₄×M₁₆×C₁₇	C₂
socle	C₂²×C₁₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₄×M₁₆	D₁₇
hypocenter	C₁₇	#1966
LCS	G ▷ C₂³×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×#63227

commutator, prime radical

C₂³×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₄×M₁₆×C₁₇

C₂

socle

C₂²×C₁₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₇

C₂³

hypercenter

C₄×M₁₆

D₁₇

hypocenter

C₁₇

#1966

LCS

G ▷ C₂³×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
54. C₄×C₁₇	#101	semidirect
78. C₁₇	#1966	semidirect
116. C₄	#10587495	semidirect

H ≅ G/N

type

54. C₄×C₁₇

#101

semidirect

78. C₁₇

#1966

semidirect

116. C₄

#10587495

semidirect

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

subgroup	index
C₂×#10587555	2
#11762833	2
C₂×#10587495	2
C₄×M₁₆×C₁₇	2
#1966	17

subgroup

index

C₄×M₁₆×C₁₇

Subgroups — #14529628

Conjugacy classes of subgroups

Complements

Maximal subgroups