Group #14537873 — subgroups

Subgroups — #14537873

Order 2176.

364 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 221 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	#11762940	C₂	1	6	yes	maximal, complemented
2	Q₈×#3622	C₂	1	2	yes	maximal, complemented
3	#11762866	C₂	1	3	yes	maximal
4	#11762862	C₂	1	3	yes	maximal
5	C₂×C₄×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₇	C₄	1	1	yes	characteristic
7	C₄×Q₈×C₁₇	C₄	1	3	yes	—
8	C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
9	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₁₇	C₂²	1	3	yes	—
11	C₂×C₄⋊C₄×C₁₇	C₄	1	3	yes	—
12	C₂×C₄²×C₁₇	C₂²	1	3	yes	—
13	C₂²⋊#3622	C₂²	1	1	yes	characteristic
14	C₂²⋊#3622	C₂²	1	3	yes	—
15	C₄⋊#3622	C₂²	1	6	yes	—
16	C₄⋊#3622	C₂²	1	6	yes	—
17	#10587474	C₂²	1	6	yes	—
18	C₄×#3622	C₂²	1	6	yes	—
19	C₂×Q₈×C₁₇	—	2	4	no	—
20	C₂×Q₈×C₁₇	C₂×C₄	1	3	yes	—
21	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₇	C₂×C₄	1	3	yes	—
23	C₂²×C₄×C₁₇	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₇	C₂×C₄	1	6	yes	—
26	C₄⋊C₄×C₁₇	C₂×C₄	1	3	yes	—
27	C₄⋊C₄×C₁₇	C₂×C₄	1	3	yes	—
28	C₄²×C₁₇	C₂×C₄	1	3	yes	—
29	C₄²×C₁₇	C₂×C₄	1	3	yes	—
30	C₄²×C₁₇	C₂³	1	3	yes	—
31	C₄²×C₁₇	C₂³	1	3	yes	—
32	C₂×#3622	C₂³	1	2	yes	—
33	C₂×#3622	C₂³	1	6	yes	—
34	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₁₇	—	2	6	no	—
36	Q₈×C₁₇	M₁₆	1	4	yes	—
37	C₂×C₄×C₁₇	—	2	6	no	—
38	C₂×C₄×C₁₇	—	2	6	no	—
39	C₂×C₄×C₁₇	—	2	2	no	—
40	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₇	C₂²×C₄	1	3	yes	—
42	C₂×C₄×C₁₇	C₂²×C₄	1	3	yes	—
43	C₂×C₄×C₁₇	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₁₇	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
46	#3622	—	2	2	no	—
47	#3622	—	2	6	no	—
48	#2984	—	17	1	no	maximal, Sylow
49	C₂²×C₁₇	—	2	1	no	—
50	C₂²×C₁₇	—	2	1	no	—
51	C₂²×C₁₇	#131	1	2	yes	—
52	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
53	C₄×C₁₇	—	2	1	no	—
54	C₄×C₁₇	—	2	3	no	—
55	C₄×C₁₇	—	2	3	no	—
56	C₄×C₁₇	—	2	3	no	—
57	C₄×C₁₇	C₂×M₁₆	1	6	yes	—
58	C₄×C₁₇	2₋¹⁺⁴	1	2	yes	—
59	C₂×C₄×Q₈	D₁₇	1	1	yes	characteristic
60	#446	—	17	6	no	—
61	C₈×Q₈	—	17	2	no	—
62	#432	—	17	3	no	—
63	#431	—	17	3	no	—
64	C₂×C₁₇	—	2	2	no	—
65	C₂×C₁₇	#519	1	1	yes	characteristic
66	C₂×C₁₇	C₂²×M₁₆	1	1	yes	characteristic
67	C₂×C₁₇	C₂×#131	1	1	yes	characteristic
68	C₂²×Q₈	Dic₁₇	1	1	yes	characteristic
69	C₄×Q₈	Dic₁₇	1	3	yes	—
70	C₄×Q₈	Dic₁₇	1	1	yes	characteristic
71	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
72	C₄×Q₈	C₂×D₁₇	1	3	yes	—
73	C₂×C₄⋊C₄	Dic₁₇	1	3	yes	—
74	C₂×C₄²	C₂×D₁₇	1	3	yes	—
75	C₄⋊C₈	—	17	6	no	—
76	C₄⋊C₈	—	17	6	no	—
77	C₂²⋊C₈	—	17	1	no	—
78	C₂²⋊C₈	—	17	3	no	—
79	#97	—	17	6	no	—
80	C₄×C₈	—	17	6	no	—
81	C₁₇	#2984	1	1	yes	characteristic, complemented, Sylow
82	C₂×Q₈	—	2	4	no	—
83	C₂×Q₈	C₂×Dic₁₇	1	3	yes	—
84	C₂×Q₈	C₂×Dic₁₇	1	1	yes	characteristic
85	C₂²×C₄	C₂×Dic₁₇	1	3	yes	—
86	C₂²×C₄	C₂×Dic₁₇	1	3	yes	—
87	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
88	C₂×C₈	—	17	2	no	—
89	C₂×C₈	—	17	6	no	—
90	C₄⋊C₄	C₂×Dic₁₇	1	6	yes	—
91	C₄⋊C₄	C₂×Dic₁₇	1	3	yes	—
92	C₄⋊C₄	C₂×Dic₁₇	1	3	yes	—
93	C₄²	C₂×Dic₁₇	1	3	yes	—
94	C₄²	C₂×Dic₁₇	1	3	yes	—
95	C₄²	C₂²×D₁₇	1	3	yes	—
96	C₄²	C₂²×D₁₇	1	3	yes	—
97	C₂³	C₂²×Dic₁₇	1	1	yes	characteristic
98	Q₈	—	2	6	no	—
99	Q₈	#63223	1	4	yes	—
100	C₂×C₄	—	2	6	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	6	no	—
103	C₂×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
104	C₂×C₄	C₂²×Dic₁₇	1	3	yes	—
105	C₂×C₄	C₂²×Dic₁₇	1	3	yes	—
106	C₂×C₄	C₂²×Dic₁₇	1	3	yes	—
107	C₂×C₄	C₂²×Dic₁₇	1	3	yes	—
108	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
109	C₈	—	34	2	no	—
110	C₈	—	34	6	no	—
111	C₂²	—	2	1	no	—
112	C₂²	—	2	1	no	—
113	C₂²	#10587634	1	2	yes	—
114	C₂²	C₂³×Dic₁₇	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	3	no	—
117	C₄	—	2	3	no	—
118	C₄	—	2	3	no	—
119	C₄	C₂×#63223	1	6	yes	—
120	C₄	#10587694	1	2	yes	—
121	C₂	—	2	2	no	—
122	C₂	#11763287	1	1	yes	characteristic
123	C₂	C₂²×#63223	1	1	yes	characteristic
124	C₂	C₂×#10587634	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11762940

C₂

yes

maximal, complemented

Q₈×#3622

C₂

yes

maximal, complemented

#11762866

C₂

yes

maximal

#11762862

C₂

yes

maximal

C₂×C₄×Q₈×C₁₇

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₇

C₄

yes

characteristic

C₄×Q₈×C₁₇

C₄

yes

—

C₄×Q₈×C₁₇

C₄

yes

characteristic

C₄×Q₈×C₁₇

C₂²

yes

characteristic

C₄×Q₈×C₁₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₇

C₄

yes

—

C₂×C₄²×C₁₇

C₂²

yes

—

C₂²⋊#3622

C₂²

yes

characteristic

C₂²⋊#3622

C₂²

yes

—

C₄⋊#3622

C₂²

yes

—

C₄⋊#3622

C₂²

yes

—

#10587474

C₂²

yes

—

C₄×#3622

C₂²

yes

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

C₂×C₄

yes

—

C₂×Q₈×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×C₄

yes

—

C₂²×C₄×C₁₇

C₂×C₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

C₂³

yes

—

C₄²×C₁₇

C₂³

yes

—

C₂×#3622

C₂³

yes

—

C₂×#3622

C₂³

yes

—

C₂³×C₁₇

C₂²×C₄

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

M₁₆

yes

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

—

C₂×C₄×C₁₇

C₂⁴

yes

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#131

yes

—

C₂²×C₁₇

C₂³×C₄

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×M₁₆

yes

—

C₄×C₁₇

2₋¹⁺⁴

yes

—

C₂×C₄×Q₈

D₁₇

yes

characteristic

#446

—

C₈×Q₈

—

#432

—

#431

—

C₂×C₁₇

—

C₂×C₁₇

#519

yes

characteristic

C₂×C₁₇

C₂²×M₁₆

yes

characteristic

C₂×C₁₇

C₂×#131

yes

characteristic

C₂²×Q₈

Dic₁₇

yes

characteristic

C₄×Q₈

Dic₁₇

yes

—

C₄×Q₈

Dic₁₇

yes

characteristic

C₄×Q₈

C₂×D₁₇

yes

characteristic

C₄×Q₈

C₂×D₁₇

yes

—

C₂×C₄⋊C₄

Dic₁₇

yes

—

C₂×C₄²

C₂×D₁₇

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₁₇

#2984

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

C₂×Dic₁₇

yes

—

C₂×Q₈

C₂×Dic₁₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₇

yes

—

C₂²×C₄

C₂×Dic₁₇

yes

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₁₇

yes

—

C₄⋊C₄

C₂×Dic₁₇

yes

—

C₄⋊C₄

C₂×Dic₁₇

yes

—

C₄²

C₂×Dic₁₇

yes

—

C₄²

C₂×Dic₁₇

yes

—

C₄²

C₂²×D₁₇

yes

—

C₄²

C₂²×D₁₇

yes

—

C₂³

C₂²×Dic₁₇

yes

characteristic

Q₈

—

Q₈

#63223

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²×Dic₁₇

yes

characteristic

104

C₂×C₄

C₂²×Dic₁₇

yes

—

105

C₂×C₄

C₂²×Dic₁₇

yes

—

106

C₂×C₄

C₂²×Dic₁₇

yes

—

107

C₂×C₄

C₂²×Dic₁₇

yes

—

108

C₂×C₄

C₂³×D₁₇

yes

characteristic

109

C₈

—

110

C₈

—

111

C₂²

—

112

C₂²

—

113

C₂²

#10587634

yes

—

114

C₂²

C₂³×Dic₁₇

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

C₂×#63223

yes

—

120

C₄

#10587694

yes

—

121

C₂

—

122

C₂

#11763287

yes

characteristic

123

C₂

C₂²×#63223

yes

characteristic

124

C₂

C₂×#10587634

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₇
commutator, socle	C₂²×C₁₇	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₁₇	C₂
prime radical	C₂³×C₁₇	C₂²×C₄
cosocle	C₂×C₄×C₁₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₁₇
hypocenter	C₁₇	#2984
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₇

commutator, socle

C₂²×C₁₇

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₁₇

C₂

prime radical

C₂³×C₁₇

C₂²×C₄

cosocle

C₂×C₄×C₁₇

C₂⁴

hypercenter

C₂×C₄×Q₈

D₁₇

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
1. #11762940	C₂	semidirect
2. Q₈×#3622	C₂	semidirect
81. C₁₇	#2984	semidirect

H ≅ G/N

type

1. #11762940

C₂

semidirect

2. Q₈×#3622

C₂

semidirect

81. C₁₇

#2984

semidirect

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

subgroup	index
#11762940	2
Q₈×#3622	2
#11762866	2
#11762862	2
C₂×C₄×Q₈×C₁₇	2
#2984	17

subgroup

index

C₂×C₄×Q₈×C₁₇

Subgroups — #14537873

Conjugacy classes of subgroups

Complements

Maximal subgroups