Group #14545678 — subgroups

Subgroups — #14545678

Order 2176.

396 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 204 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	#11763858	C₂	1	6	yes	maximal, complemented
2	Q₈×#3624	C₂	1	2	yes	maximal, complemented
3	#11763828	C₂	1	3	yes	maximal
4	#11763827	C₂	1	3	yes	maximal
5	C₄×Q₈×D₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×D₁₇	C₄	1	1	yes	characteristic
7	C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
8	Q₈×Dic₁₇	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×D₁₇	C₄	1	3	yes	—
10	#10587573	C₂²	1	3	yes	—
11	C₄²×D₁₇	C₂²	1	3	yes	—
12	C₄×Dic₃₄	C₄	1	3	yes	—
13	#10587544	C₂²	1	6	yes	—
14	#10587543	C₂²	1	1	yes	characteristic
15	#10587543	C₂²	1	3	yes	—
16	C₄⋊#3624	C₂²	1	6	yes	—
17	C₄×#3624	C₂²	1	6	yes	—
18	C₄×#3624	C₂²	1	6	yes	—
19	C₂×Q₈×C₁₇	C₂×C₄	1	1	yes	characteristic
20	Q₈×D₁₇	C₈	1	8	yes	complemented
21	C₂×C₄×D₁₇	C₂×C₄	1	3	yes	—
22	C₂×C₄×D₁₇	C₂×C₄	1	3	yes	—
23	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
24	C₂×Dic₃₄	C₂×C₄	1	3	yes	—
25	C₂×#3624	C₂³	1	6	yes	—
26	C₂×#3624	C₂³	1	2	yes	—
27	C₄⋊C₄×C₁₇	C₂×C₄	1	3	yes	—
28	C₄²×C₁₇	C₂×C₄	1	3	yes	—
29	C₄⋊Dic₁₇	C₂×C₄	1	3	yes	—
30	#63225	C₂×C₄	1	6	yes	—
31	C₄×Dic₁₇	C₂×C₄	1	3	yes	—
32	C₄×Dic₁₇	C₂³	1	3	yes	—
33	C₄×Dic₁₇	C₂³	1	3	yes	—
34	C₂²×D₁₇	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₁₇	C₂×C₈	1	4	yes	—
36	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₇	C₂²×C₄	1	3	yes	—
38	C₂×C₄×C₁₇	C₂²×C₄	1	3	yes	—
39	C₂×Dic₁₇	C₂²×C₄	1	3	yes	—
40	C₂×Dic₁₇	C₂²×C₄	1	3	yes	—
41	C₂×Dic₁₇	C₂⁴	1	1	yes	characteristic
42	C₄×D₁₇	—	2	2	no	—
43	C₄×D₁₇	—	4	3	no	—
44	C₄×D₁₇	C₂×C₈	1	12	yes	—
45	Dic₃₄	C₂×C₈	1	12	yes	—
46	#3624	—	2	6	no	—
47	#3624	—	2	2	no	—
48	#2982	—	17	1	no	maximal, Sylow
49	C₂²×C₁₇	C₂³×C₄	1	1	yes	characteristic
50	C₂×D₁₇	—	2	1	no	—
51	C₂×D₁₇	—	2	1	no	—
52	C₂×D₁₇	C₂²×C₈	1	2	yes	—
53	C₄×C₁₇	—	2	1	no	—
54	C₄×C₁₇	—	2	3	no	—
55	C₄×C₁₇	C₂²×C₈	1	6	yes	—
56	Dic₁₇	—	2	3	no	—
57	Dic₁₇	C₂²×C₈	1	6	yes	—
58	Dic₁₇	2₋¹⁺⁴	1	2	yes	—
59	C₂×C₄×Q₈	—	17	1	no	—
60	C₈×Q₈	—	17	2	no	—
61	C₈×Q₈	—	17	6	no	—
62	#431	—	17	3	no	—
63	#431	—	17	3	no	—
64	C₂×C₁₇	#519	1	1	yes	characteristic
65	C₂×C₁₇	C₂³×C₈	1	1	yes	characteristic
66	C₂×C₁₇	#568	1	1	yes	characteristic
67	D₁₇	—	2	2	no	—
68	C₂²×Q₈	—	17	1	no	—
69	C₄×Q₈	—	17	1	no	—
70	C₄×Q₈	—	17	3	no	—
71	C₄×Q₈	—	17	3	no	—
72	C₄×Q₈	#595	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	17	3	no	—
74	C₂×C₄²	—	17	3	no	—
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	C₄×C₈	—	17	6	no	—
80	C₄×C₈	—	17	6	no	—
81	C₁₇	#2982	1	1	yes	characteristic, complemented, Sylow
82	C₂×Q₈	—	17	8	no	—
83	C₂×Q₈	—	17	3	no	—
84	C₂×Q₈	C₂×#595	1	1	yes	characteristic
85	C₂²×C₄	—	17	3	no	—
86	C₂²×C₄	—	17	1	no	—
87	C₂²×C₄	—	17	3	no	—
88	C₂×C₈	—	17	6	no	—
89	C₂×C₈	—	17	2	no	—
90	C₄⋊C₄	—	17	6	no	—
91	C₄⋊C₄	—	17	3	no	—
92	C₄⋊C₄	C₂×#595	1	3	yes	—
93	C₄²	—	17	3	no	—
94	C₄²	—	17	3	no	—
95	C₄²	—	17	3	no	—
96	C₄²	C₂×#595	1	3	yes	—
97	C₂³	—	17	1	no	—
98	Q₈	—	17	12	no	—
99	Q₈	#63242	1	4	yes	—
100	C₂×C₄	—	17	1	no	—
101	C₂×C₄	—	17	3	no	—
102	C₂×C₄	—	17	12	no	—
103	C₂×C₄	—	17	3	no	—
104	C₂×C₄	—	34	2	no	—
105	C₂×C₄	—	68	3	no	—
106	C₂×C₄	C₂²×#595	1	1	yes	characteristic
107	C₂×C₄	C₂²×#595	1	3	yes	—
108	C₂×C₄	C₂²×#595	1	3	yes	—
109	C₈	—	34	6	no	—
110	C₈	—	34	2	no	—
111	C₂²	—	17	2	no	—
112	C₂²	—	34	1	no	—
113	C₂²	—	34	1	no	—
114	C₂²	C₂³×#595	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	3	no	—
117	C₄	—	17	2	no	—
118	C₄	—	17	6	no	—
119	C₄	—	34	3	no	—
120	C₄	C₂×#63242	1	6	yes	—
121	C₂	—	34	2	no	—
122	C₂	#11763877	1	1	yes	characteristic
123	C₂	C₂²×#63242	1	1	yes	characteristic
124	C₂	#11763888	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763858

C₂

yes

maximal, complemented

Q₈×#3624

C₂

yes

maximal, complemented

#11763828

C₂

yes

maximal

#11763827

C₂

yes

maximal

C₄×Q₈×D₁₇

C₂

yes

maximal, characteristic

C₂×Q₈×D₁₇

C₄

yes

characteristic

C₄×Q₈×C₁₇

C₄

yes

characteristic

Q₈×Dic₁₇

C₂²

yes

characteristic

C₄⋊C₄×D₁₇

C₄

yes

—

#10587573

C₂²

yes

—

C₄²×D₁₇

C₂²

yes

—

C₄×Dic₃₄

C₄

yes

—

#10587544

C₂²

yes

—

#10587543

C₂²

yes

characteristic

#10587543

C₂²

yes

—

C₄⋊#3624

C₂²

yes

—

C₄×#3624

C₂²

yes

—

C₄×#3624

C₂²

yes

—

C₂×Q₈×C₁₇

C₂×C₄

yes

characteristic

Q₈×D₁₇

C₈

yes

complemented

C₂×C₄×D₁₇

C₂×C₄

yes

—

C₂×C₄×D₁₇

C₂×C₄

yes

—

C₂×C₄×D₁₇

C₂³

yes

characteristic

C₂×Dic₃₄

C₂×C₄

yes

—

C₂×#3624

C₂³

yes

—

C₂×#3624

C₂³

yes

—

C₄⋊C₄×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

C₂×C₄

yes

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

#63225

C₂×C₄

yes

—

C₄×Dic₁₇

C₂×C₄

yes

—

C₄×Dic₁₇

C₂³

yes

—

C₄×Dic₁₇

C₂³

yes

—

C₂²×D₁₇

C₂²×C₄

yes

characteristic

Q₈×C₁₇

C₂×C₈

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

—

C₂×Dic₁₇

C₂²×C₄

yes

—

C₂×Dic₁₇

C₂²×C₄

yes

—

C₂×Dic₁₇

C₂⁴

yes

characteristic

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

C₂×C₈

yes

—

Dic₃₄

C₂×C₈

yes

—

—

—

—

C₂²×C₁₇

C₂³×C₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

C₂²×C₈

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂²×C₈

yes

—

Dic₁₇

—

Dic₁₇

C₂²×C₈

yes

—

Dic₁₇

2₋¹⁺⁴

yes

—

C₂×C₄×Q₈

—

C₈×Q₈

—

C₈×Q₈

—

#431

—

#431

—

C₂×C₁₇

#519

yes

characteristic

C₂×C₁₇

C₂³×C₈

yes

characteristic

C₂×C₁₇

#568

yes

characteristic

D₁₇

—

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

#595

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#2982

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×#595

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#595

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×#595

yes

—

C₂³

—

Q₈

—

Q₈

#63242

yes

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₂²×#595

yes

characteristic

107

C₂×C₄

C₂²×#595

yes

—

108

C₂×C₄

C₂²×#595

yes

—

109

C₈

—

110

C₈

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂³×#595

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₂×#63242

yes

—

121

C₂

—

122

C₂

#11763877

yes

characteristic

123

C₂

C₂²×#63242

yes

characteristic

124

C₂

#11763888

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#595
commutator	C₂×C₁₇	C₂³×C₈
Fitting	C₄×Q₈×C₁₇	C₄
socle	C₂²×C₁₇	C₂³×C₄
prime radical	C₂²×D₁₇	C₂²×C₄
cosocle	C₂×Dic₁₇	C₂⁴
hypercenter	C₄×Q₈	#595
hypocenter	C₁₇	#2982
LCS	G ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#595

commutator

C₂×C₁₇

C₂³×C₈

Fitting

C₄×Q₈×C₁₇

C₄

socle

C₂²×C₁₇

C₂³×C₄

prime radical

C₂²×D₁₇

C₂²×C₄

cosocle

C₂×Dic₁₇

C₂⁴

C₄×Q₈

C₁₇

G ▷ C₂×C₁₇ ▷ C₁₇

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #11763858	C₂	semidirect
2. Q₈×#3624	C₂	semidirect
20. Q₈×D₁₇	C₈	semidirect
81. C₁₇	#2982	semidirect

H ≅ G/N

type

1. #11763858

C₂

semidirect

2. Q₈×#3624

C₂

semidirect

20. Q₈×D₁₇

C₈

semidirect

81. C₁₇

#2982

semidirect

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

subgroup	index
#11763858	2
Q₈×#3624	2
#11763828	2
#11763827	2
C₄×Q₈×D₁₇	2
#2982	17

subgroup

index

C₄×Q₈×D₁₇

Subgroups — #14545678

Conjugacy classes of subgroups

Complements

Maximal subgroups