Group #14527539 — subgroups

Subgroups — #14527539

Order 2176.

184 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂×C₄×#3622	C₂	1	1	yes	maximal, characteristic
3	#11762559	C₂	1	4	yes	maximal, complemented
4	C₁₇×#423	C₂	1	1	yes	maximal, characteristic
5	C₂×M₁₆×C₁₇	—	2	1	no	—
6	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×#63223	—	2	1	no	—
8	C₂²×#3622	C₄	1	2	yes	—
9	C₁₇×C₄⋊C₈	—	2	1	no	—
10	C₁₇×C₄⋊C₈	C₂²	1	2	yes	—
11	C₄⋊#3622	—	2	1	no	—
12	C₄⋊#3622	C₂²	1	2	yes	—
13	C₄×#3622	C₄	1	1	yes	characteristic
14	C₄×#3622	C₄	1	1	yes	characteristic
15	C₄×#3622	C₂²	1	2	yes	—
16	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₁₇	Q₈	1	1	yes	characteristic
19	M₁₆×C₁₇	—	4	2	no	—
20	C₂×C₈×C₁₇	—	2	2	no	—
21	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₁₇	D₄	1	1	yes	characteristic
23	C₄²×C₁₇	Q₈	1	1	yes	characteristic
24	C₄²×C₁₇	C₂³	1	1	yes	characteristic
25	#63223	—	4	2	no	—
26	C₂×#3622	—	2	4	no	—
27	C₂×#3622	—	2	2	no	—
28	C₂×#3622	C₂×C₄	1	2	yes	—
29	C₂×#3622	C₂×C₄	1	2	yes	—
30	C₂³×C₁₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₇	—	2	2	no	—
32	C₂×C₄×C₁₇	—	2	2	no	—
33	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇	SD₁₆	1	2	yes	—
37	C₂×C₄×C₁₇	SD₁₆	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	1	yes	characteristic
41	C₈×C₁₇	—	4	2	no	—
42	#3622	—	2	2	no	—
43	#3622	—	4	2	no	—
44	#3622	M₁₆	1	4	yes	complemented
45	#1566	—	17	1	no	maximal, Sylow
46	C₂²×C₁₇	—	2	1	no	—
47	C₂²×C₁₇	—	2	1	no	—
48	C₂²×C₁₇	#106	1	2	yes	—
49	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₁₇	—	2	1	no	—
51	C₄×C₁₇	—	2	1	no	—
52	C₄×C₁₇	#106	1	2	yes	—
53	C₄×C₁₇	#108	1	2	yes	—
54	C₄×C₁₇	C₂×M₁₆	1	2	yes	—
55	C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
56	C₄×C₁₇	C₂×SD₁₆	1	1	yes	characteristic
57	#423	—	17	1	no	—
58	#423	D₁₇	1	1	yes	characteristic
59	C₂×C₄×C₈	—	17	1	no	—
60	#334	—	17	4	no	—
61	C₂×C₁₇	—	2	2	no	—
62	C₂×C₁₇	#423	1	1	yes	characteristic
63	C₂×C₁₇	C₂×#106	1	1	yes	characteristic
64	C₂×C₁₇	C₂×#108	1	1	yes	characteristic
65	C₂×M₁₆	—	2	1	no	—
66	C₂×M₁₆	—	34	1	no	—
67	C₂²×C₈	—	17	2	no	—
68	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	—	17	2	no	—
71	C₄⋊C₈	—	34	1	no	—
72	C₄⋊C₈	C₂×D₁₇	1	2	yes	—
73	C₄×C₈	—	17	1	no	—
74	C₄×C₈	—	17	1	no	—
75	C₄×C₈	—	17	2	no	—
76	C₁₇	#1566	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₄	Dic₃₄	1	1	yes	characteristic
78	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
80	M₁₆	—	4	2	no	—
81	M₁₆	—	68	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	17	2	no	—
84	C₂×C₈	—	17	2	no	—
85	C₂×C₈	—	34	4	no	—
86	C₂×C₈	—	34	2	no	—
87	C₄²	Dic₃₄	1	1	yes	characteristic
88	C₄²	C₄×D₁₇	1	1	yes	characteristic
89	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
90	C₄²	C₂²×D₁₇	1	1	yes	characteristic
91	C₂³	#63225	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	#63225	1	1	yes	characteristic
95	C₂×C₄	#63225	1	1	yes	characteristic
96	C₂×C₄	#63225	1	1	yes	characteristic
97	C₂×C₄	#63229	1	2	yes	—
98	C₂×C₄	#63230	1	2	yes	—
99	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
102	C₈	—	4	2	no	—
103	C₈	—	17	4	no	—
104	C₈	—	34	2	no	—
105	C₈	—	68	2	no	—
106	C₂²	—	2	1	no	—
107	C₂²	—	2	1	no	—
108	C₂²	#10587479	1	2	yes	—
109	C₂²	C₂×#63225	1	1	yes	characteristic
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	#10587479	1	2	yes	—
113	C₄	#10587493	1	2	yes	—
114	C₄	M₁₆×D₁₇	1	2	yes	—
115	C₄	C₂×#63229	1	1	yes	characteristic
116	C₄	C₂×#63230	1	1	yes	characteristic
117	C₂	—	2	2	no	—
118	C₂	#11762832	1	1	yes	characteristic
119	C₂	C₂×#10587479	1	1	yes	characteristic
120	C₂	C₂×#10587493	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11762833

C₂

yes

maximal, characteristic

C₂×C₄×#3622

C₂

yes

maximal, characteristic

#11762559

C₂

yes

maximal, complemented

C₁₇×#423

C₂

yes

maximal, characteristic

C₂×M₁₆×C₁₇

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×#63223

—

C₂²×#3622

C₄

yes

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

C₂²

yes

—

C₄⋊#3622

—

C₄⋊#3622

C₂²

yes

—

C₄×#3622

C₄

yes

characteristic

C₄×#3622

C₄

yes

characteristic

C₄×#3622

C₂²

yes

—

C₂²×C₄×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

Q₈

yes

characteristic

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

Q₈

yes

characteristic

C₄²×C₁₇

C₂³

yes

—

—

—

C₂×C₄

yes

—

C₂×#3622

C₂×C₄

yes

—

C₂³×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

SD₁₆

yes

—

C₂×C₄×C₁₇

SD₁₆

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₈×C₁₇

—

#3622

—

#3622

—

#3622

M₁₆

yes

complemented

#1566

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#106

yes

—

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#106

yes

—

C₄×C₁₇

#108

yes

—

C₄×C₁₇

C₂×M₁₆

yes

—

C₄×C₁₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₇

C₂×SD₁₆

yes

characteristic

#423

—

#423

D₁₇

yes

characteristic

C₂×C₄×C₈

—

#334

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₁₇

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#1566

yes

characteristic, complemented, Sylow

C₂²×C₄

Dic₃₄

yes

characteristic

C₂²×C₄

C₄×D₁₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

Dic₃₄

yes

characteristic

C₄²

C₄×D₁₇

yes

characteristic

C₄²

C₂²⋊D₁₇

yes

characteristic

C₄²

C₂²×D₁₇

yes

characteristic

C₂³

#63225

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₂×C₄

#63230

yes

—

C₂×C₄

C₂×Dic₃₄

yes

characteristic

100

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

102

C₈

—

103

C₈

—

104

C₈

—

105

C₈

—

106

C₂²

—

107

C₂²

—

108

C₂²

#10587479

yes

—

109

C₂²

C₂×#63225

yes

characteristic

110

C₄

—

111

C₄

—

112

C₄

#10587479

yes

—

113

C₄

#10587493

yes

—

114

C₄

M₁₆×D₁₇

yes

—

115

C₄

C₂×#63229

yes

characteristic

116

C₄

C₂×#63230

yes

characteristic

117

C₂

—

118

C₂

#11762832

yes

characteristic

119

C₂

C₂×#10587479

yes

characteristic

120

C₂

C₂×#10587493

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#423	C₂
socle	C₂²×C₁₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₇	C₄⋊C₄
cosocle	C₄²×C₁₇	C₂³
hypercenter	#423	D₁₇
hypocenter	C₁₇	#1566
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₇

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#423

C₂

socle

C₂²×C₁₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₇

C₄⋊C₄

cosocle

C₄²×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
3. #11762559	C₂	semidirect
44. #3622	M₁₆	semidirect
76. C₁₇	#1566	semidirect

H ≅ G/N

type

3. #11762559

C₂

semidirect

44. #3622

M₁₆

semidirect

76. C₁₇

#1566

semidirect

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

subgroup	index
#11762833	2
C₂×C₄×#3622	2
#11762559	2
C₁₇×#423	2
#1566	17

subgroup

index

#11762833

C₂×C₄×#3622

#11762559

C₁₇×#423

#1566

Subgroups — #14527539

Conjugacy classes of subgroups

Complements

Maximal subgroups