Group #14545123 — subgroups

Subgroups — #14545123

Order 2176.

168 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 74 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	#11763865	C₂	1	2	yes	maximal
2	#11763732	C₂	1	2	yes	maximal, complemented
3	#11763731	C₂	1	2	yes	maximal, complemented
4	C₄×C₈×D₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×D₁₇	C₄	1	1	yes	characteristic
6	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
7	C₄²×D₁₇	C₄	1	1	yes	characteristic
8	#10587538	C₂²	1	4	yes	—
9	#10587530	—	2	2	no	—
10	#10587529	—	2	2	no	—
11	C₄×C₈×C₁₇	C₄	1	1	yes	characteristic
12	C₈×Dic₁₇	C₄	1	1	yes	characteristic
13	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
14	C₄×#3622	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₁₇	—	2	1	no	—
16	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₄×Dic₁₇	—	2	1	no	—
21	C₄×Dic₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×#3622	C₂×C₄	1	1	yes	characteristic
23	C₂×#3622	C₂³	1	1	yes	characteristic
24	C₈×D₁₇	C₈	1	2	yes	—
25	C₈×D₁₇	C₈	1	2	yes	—
26	C₈×D₁₇	D₄	1	1	yes	characteristic
27	C₈×D₁₇	D₄	1	1	yes	characteristic
28	C₈×D₁₇	Q₈	1	1	yes	characteristic
29	C₈×D₁₇	Q₈	1	1	yes	characteristic
30	#63216	—	2	4	no	—
31	C₂²×D₁₇	M₁₆	1	1	yes	characteristic
32	C₂×C₄×C₁₇	—	2	1	no	—
33	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×Dic₁₇	—	2	1	no	—
35	C₂×Dic₁₇	M₁₆	1	1	yes	characteristic
36	C₄×D₁₇	—	2	4	no	—
37	C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
38	C₄×D₁₇	C₄⋊C₄	1	1	yes	characteristic
39	C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
40	C₄×D₁₇	C₂×C₈	1	1	yes	characteristic
41	C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
42	C₈×C₁₇	C₄⋊C₄	1	1	yes	characteristic
43	C₈×C₁₇	C₂×C₈	1	2	yes	—
44	#3622	C₂×C₈	1	2	yes	—
45	#3622	C₂×D₄	1	1	yes	characteristic
46	#3622	C₂×Q₈	1	1	yes	characteristic
47	C₂×#364	—	17	1	no	maximal, Sylow
48	C₂²×C₁₇	C₂×M₁₆	1	1	yes	characteristic
49	C₂×D₁₇	—	2	2	no	—
50	C₂×D₁₇	C₄⋊C₈	1	1	yes	characteristic
51	C₂×D₁₇	C₄⋊C₈	1	1	yes	characteristic
52	C₄×C₁₇	—	2	2	no	—
53	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
55	Dic₁₇	—	2	2	no	—
56	Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
57	Dic₁₇	C₄⋊C₈	1	1	yes	characteristic
58	C₂×M₃₂	—	17	2	no	—
59	C₂×C₄×C₈	—	17	1	no	—
60	#364	—	17	2	no	—
61	#364	—	17	2	no	—
62	C₂×C₁₇	—	2	1	no	—
63	C₂×C₁₇	C₂×C₄⋊C₈	1	1	yes	characteristic
64	D₁₇	—	2	1	no	—
65	D₁₇	#364	1	2	yes	complemented
66	C₂²×C₈	—	17	1	no	—
67	C₂²×C₈	—	17	1	no	—
68	C₂×C₄²	—	17	1	no	—
69	M₃₂	—	17	4	no	—
70	M₃₂	—	34	2	no	—
71	C₂×C₁₆	—	34	2	no	—
72	C₄×C₈	—	17	1	no	—
73	C₄×C₈	—	17	1	no	—
74	C₄×C₈	—	17	1	no	—
75	C₄×C₈	#595	1	1	yes	characteristic
76	C₁₇	C₂×#364	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₄	—	17	1	no	—
78	C₂²×C₄	—	34	1	no	—
79	C₂×C₈	—	17	1	no	—
80	C₂×C₈	—	17	1	no	—
81	C₂×C₈	—	17	2	no	—
82	C₂×C₈	—	17	2	no	—
83	C₂×C₈	—	17	1	no	—
84	C₂×C₈	—	17	1	no	—
85	C₂×C₈	—	17	1	no	—
86	C₂×C₈	—	17	1	no	—
87	C₂×C₈	C₂×#595	1	1	yes	characteristic
88	C₂×C₈	C₂×#595	1	1	yes	characteristic
89	C₄²	—	17	1	no	—
90	C₄²	—	34	1	no	—
91	C₄²	C₂×#595	1	1	yes	characteristic
92	C₁₆	—	34	4	no	—
93	C₂³	—	17	1	no	—
94	C₂×C₄	—	2	1	no	—
95	C₂×C₄	—	17	1	no	—
96	C₂×C₄	—	17	1	no	—
97	C₂×C₄	—	17	1	no	—
98	C₂×C₄	—	17	1	no	—
99	C₂×C₄	—	17	1	no	—
100	C₂×C₄	—	34	4	no	—
101	C₂×C₄	—	34	1	no	—
102	C₂×C₄	C₂²×#595	1	1	yes	characteristic
103	C₈	—	17	2	no	—
104	C₈	—	17	1	no	—
105	C₈	—	17	1	no	—
106	C₈	#63242	1	2	yes	—
107	C₈	C₄⋊#595	1	1	yes	characteristic
108	C₈	C₄⋊#595	1	1	yes	characteristic
109	C₂²	—	17	1	no	—
110	C₂²	—	17	1	no	—
111	C₂²	—	34	2	no	—
112	C₂²	D₁₇⋊M₁₆	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	17	1	no	—
115	C₄	—	17	1	no	—
116	C₄	—	34	2	no	—
117	C₄	C₂×#63242	1	1	yes	characteristic
118	C₄	C₂×C₄⋊#595	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	17	2	no	—
121	C₂	—	34	1	no	—
122	C₂	#11763810	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763865

C₂

yes

maximal

#11763732

C₂

yes

maximal, complemented

#11763731

C₂

yes

maximal, complemented

C₄×C₈×D₁₇

C₂

yes

maximal, characteristic

C₂×C₈×D₁₇

C₄

yes

characteristic

C₂×C₈×D₁₇

C₂²

yes

characteristic

C₄²×D₁₇

C₄

yes

characteristic

#10587538

C₂²

yes

—

#10587530

—

#10587529

—

C₄×C₈×C₁₇

C₄

yes

characteristic

C₈×Dic₁₇

C₄

yes

characteristic

C₈×Dic₁₇

C₂²

yes

characteristic

C₄×#3622

C₂²

yes

characteristic

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄×Dic₁₇

—

C₄×Dic₁₇

C₂×C₄

yes

characteristic

C₂×#3622

C₂×C₄

yes

characteristic

C₂×#3622

C₂³

yes

characteristic

C₈×D₁₇

C₈

yes

—

C₈×D₁₇

C₈

yes

—

C₈×D₁₇

D₄

yes

characteristic

C₈×D₁₇

D₄

yes

characteristic

C₈×D₁₇

Q₈

yes

characteristic

C₈×D₁₇

Q₈

yes

characteristic

#63216

—

C₂²×D₁₇

M₁₆

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

M₁₆

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

C₄⋊C₄

yes

characteristic

C₄×D₁₇

C₄⋊C₄

yes

characteristic

C₄×D₁₇

C₂×C₈

yes

characteristic

C₄×D₁₇

C₂×C₈

yes

characteristic

C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₈×C₁₇

C₄⋊C₄

yes

characteristic

C₈×C₁₇

C₂×C₈

yes

—

#3622

C₂×C₈

yes

—

#3622

C₂×D₄

yes

characteristic

#3622

C₂×Q₈

yes

characteristic

C₂×#364

—

maximal, Sylow

C₂²×C₁₇

C₂×M₁₆

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

C₄⋊C₈

yes

characteristic

C₂×D₁₇

C₄⋊C₈

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₇

C₂²×C₈

yes

characteristic

Dic₁₇

—

Dic₁₇

C₄⋊C₈

yes

characteristic

Dic₁₇

C₄⋊C₈

yes

characteristic

C₂×M₃₂

—

C₂×C₄×C₈

—

#364

—

#364

—

C₂×C₁₇

—

C₂×C₁₇

C₂×C₄⋊C₈

yes

characteristic

D₁₇

—

D₁₇

#364

yes

complemented

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#595

yes

characteristic

C₁₇

C₂×#364

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

—

C₄²

C₂×#595

yes

characteristic

C₁₆

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₂²×#595

yes

characteristic

103

C₈

—

104

C₈

—

105

C₈

—

106

C₈

#63242

yes

—

107

C₈

C₄⋊#595

yes

characteristic

108

C₈

C₄⋊#595

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

—

112

C₂²

D₁₇⋊M₁₆

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

C₂×#63242

yes

characteristic

118

C₄

C₂×C₄⋊#595

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

—

122

C₂

#11763810

yes

characteristic

	subgroup	G/H
center	C₈	C₄⋊#595
commutator	C₄×C₁₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#595
Fitting	C₄×C₈×C₁₇	C₄
socle	C₂×C₁₇	C₂×C₄⋊C₈
prime radical	C₂²×D₁₇	M₁₆
cosocle	C₂×#3622	C₂³
hypercenter	C₄×C₈	#595
hypocenter	C₁₇	C₂×#364
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₈

C₄⋊#595

commutator

C₄×C₁₇

C₂²×C₈

Frattini

C₂×C₄

C₂²×#595

Fitting

C₄×C₈×C₁₇

C₄

socle

C₂×C₁₇

C₂×C₄⋊C₈

prime radical

C₂²×D₁₇

M₁₆

C₂³

C₄×C₈

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
2. #11763732	C₂	semidirect
3. #11763731	C₂	semidirect
65. D₁₇	#364	semidirect
76. C₁₇	C₂×#364	semidirect

H ≅ G/N

type

2. #11763732

C₂

semidirect

3. #11763731

C₂

semidirect

65. D₁₇

#364

semidirect

76. C₁₇

C₂×#364

semidirect

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

subgroup	index
#11763865	2
#11763732	2
#11763731	2
C₄×C₈×D₁₇	2
C₂×#364	17

subgroup

index

C₄×C₈×D₁₇

Subgroups — #14545123

Conjugacy classes of subgroups

Complements

Maximal subgroups