Group #14528945 — subgroups

Subgroups — #14528945

Order 2176.

248 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 117 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×#10587478	C₂	1	2	yes	maximal
2	C₂×#10587488	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
4	#11762674	C₂	1	2	yes	maximal
5	C₂×C₁₇×#106	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₇	C₂²	1	2	yes	—
9	C₂×C₄×Dic₁₇	C₂²	1	1	yes	characteristic
10	C₂²×#3622	C₂²	1	1	yes	characteristic
11	C₁₇×#106	—	2	2	no	—
12	#10587502	—	2	4	no	—
13	#10587488	—	2	2	no	—
14	C₈×Dic₁₇	C₄	1	4	yes	complemented
15	#10587478	—	2	4	no	—
16	C₂²×C₄×C₁₇	—	2	2	no	—
17	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₁₇	—	2	2	no	—
19	C₂²×Dic₁₇	D₄	1	2	yes	—
20	C₂×C₈×C₁₇	C₂×C₄	1	2	yes	—
21	C₂×C₈×C₁₇	D₄	1	2	yes	—
22	C₂×C₈×C₁₇	Q₈	1	2	yes	—
23	C₄⋊C₄×C₁₇	—	2	2	no	—
24	C₄⋊C₄×C₁₇	—	2	2	no	—
25	C₄⋊Dic₁₇	—	2	2	no	—
26	C₄⋊Dic₁₇	—	2	2	no	—
27	C₄×Dic₁₇	C₂×C₄	1	2	yes	—
28	C₄×Dic₁₇	Q₈	1	2	yes	—
29	C₂×#3622	C₂×C₄	1	2	yes	—
30	C₂×#3622	D₄	1	2	yes	—
31	C₂×#3622	Q₈	1	2	yes	—
32	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₁₇	—	4	2	no	—
34	C₂×C₄×C₁₇	—	4	4	no	—
35	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₁₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×Dic₁₇	—	2	4	no	—
42	C₂×Dic₁₇	—	4	2	no	—
43	C₂×Dic₁₇	—	4	4	no	—
44	C₂×Dic₁₇	C₄⋊C₄	1	4	yes	—
45	C₈×C₁₇	C₄⋊C₄	1	4	yes	—
46	#3622	C₄⋊C₄	1	4	yes	—
47	#1846	—	17	1	no	maximal, Sylow
48	C₂²×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₇	#127	1	1	yes	characteristic
50	C₂²×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₁₇	#135	1	4	yes	—
52	C₄×C₁₇	—	8	2	no	—
53	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
56	C₄×C₁₇	C₄⋊Q₈	1	1	yes	characteristic
57	Dic₁₇	—	2	4	no	—
58	Dic₁₇	—	8	2	no	—
59	C₂×#107	—	17	2	no	—
60	C₂×#106	—	17	1	no	—
61	C₂×#106	D₁₇	1	1	yes	characteristic
62	C₂×C₄×C₈	—	17	1	no	—
63	#382	—	17	2	no	—
64	C₂×C₁₇	#384	1	1	yes	characteristic
65	C₂×C₁₇	#427	1	2	yes	—
66	C₂×C₁₇	#495	1	2	yes	—
67	C₂×C₁₇	#499	1	2	yes	—
68	C₂²×C₈	—	17	1	no	—
69	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
70	C₂×C₄⋊C₄	—	17	2	no	—
71	C₂×C₄⋊C₄	C₂×D₁₇	1	2	yes	—
72	C₂×C₄²	—	17	1	no	—
73	#107	—	34	4	no	—
74	#106	—	2	2	no	—
75	#106	—	34	2	no	—
76	C₄×C₈	—	17	4	no	—
77	#95	—	34	4	no	—
78	C₁₇	#1846	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	17	2	no	—
81	C₂²×C₄	—	34	2	no	—
82	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
83	C₂×C₈	—	17	2	no	—
84	C₂×C₈	—	17	2	no	—
85	C₂×C₈	—	17	2	no	—
86	C₂×C₈	Dic₃₄	1	2	yes	—
87	C₂×C₈	C₄×D₁₇	1	2	yes	—
88	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	34	2	no	—
92	C₄⋊C₄	—	34	2	no	—
93	C₄²	—	17	2	no	—
94	C₄²	—	17	2	no	—
95	C₂³	D₄×D₁₇	1	1	yes	characteristic
96	C₂×C₄	—	4	4	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	17	4	no	—
99	C₂×C₄	—	34	4	no	—
100	C₂×C₄	—	68	4	no	—
101	C₂×C₄	—	68	2	no	—
102	C₂×C₄	C₂×Dic₃₄	1	1	yes	characteristic
103	C₂×C₄	C₂×C₄×D₁₇	1	1	yes	characteristic
104	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
105	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₇	1	1	yes	characteristic
107	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
108	C₈	—	17	4	no	—
109	C₈	#63225	1	4	yes	—
110	C₂²	#10587574	1	1	yes	characteristic
111	C₂²	C₄⋊C₄×D₁₇	1	1	yes	characteristic
112	C₂²	#10587602	1	4	yes	—
113	C₂²	#10587626	1	1	yes	characteristic
114	C₄	—	8	2	no	—
115	C₄	—	34	4	no	—
116	C₄	—	136	2	no	—
117	C₄	#10587574	1	1	yes	characteristic
118	C₄	C₄⋊C₄×D₁₇	1	1	yes	characteristic
119	C₄	C₂×#63225	1	1	yes	characteristic
120	C₄	#10587630	1	1	yes	characteristic
121	C₂	#11762678	1	1	yes	characteristic
122	C₂	#11762844	1	2	yes	—
123	C₂	#11763190	1	2	yes	—
124	C₂	#11763209	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10587478

C₂

yes

maximal

C₂×#10587488

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₇

C₂

yes

maximal, characteristic

#11762674

C₂

yes

maximal

C₂×C₁₇×#106

C₂

yes

maximal, characteristic

C₂²×C₈×C₁₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₇

C₂²

yes

—

C₂×C₄⋊Dic₁₇

C₂²

yes

—

C₂×C₄×Dic₁₇

C₂²

yes

characteristic

C₂²×#3622

C₂²

yes

characteristic

C₁₇×#106

—

#10587502

—

#10587488

—

C₈×Dic₁₇

C₄

yes

complemented

#10587478

—

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

C₂²×Dic₁₇

—

C₂²×Dic₁₇

D₄

yes

—

C₂×C₈×C₁₇

C₂×C₄

yes

—

C₂×C₈×C₁₇

D₄

yes

—

C₂×C₈×C₁₇

Q₈

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄×Dic₁₇

C₂×C₄

yes

—

C₄×Dic₁₇

Q₈

yes

—

C₂×#3622

C₂×C₄

yes

—

C₂×#3622

D₄

yes

—

C₂×#3622

Q₈

yes

—

C₂³×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₇

C₂×Q₈

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₄⋊C₄

yes

—

C₈×C₁₇

C₄⋊C₄

yes

—

#3622

C₄⋊C₄

yes

—

#1846

—

maximal, Sylow

C₂²×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₇

#127

yes

characteristic

C₂²×C₁₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₇

#135

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₇

C₄⋊Q₈

yes

characteristic

Dic₁₇

—

Dic₁₇

—

C₂×#107

—

C₂×#106

—

C₂×#106

D₁₇

yes

characteristic

C₂×C₄×C₈

—

#382

—

C₂×C₁₇

#384

yes

characteristic

C₂×C₁₇

#427

yes

—

C₂×C₁₇

#495

yes

—

C₂×C₁₇

#499

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₁₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₇

yes

—

C₂×C₄²

—

#107

—

#106

—

#106

—

C₄×C₈

—

#95

—

C₁₇

#1846

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₃₄

yes

—

C₂×C₈

C₄×D₁₇

yes

—

C₂×C₈

C₂²⋊D₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₂³

D₄×D₁₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₂×Dic₃₄

yes

characteristic

103

C₂×C₄

C₂×C₄×D₁₇

yes

characteristic

104

C₂×C₄

D₄×D₁₇

yes

characteristic

105

C₂×C₄

Q₈×D₁₇

yes

characteristic

106

C₂×C₄

Q₈×D₁₇

yes

characteristic

107

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

108

C₈

—

109

C₈

#63225

yes

—

110

C₂²

#10587574

yes

characteristic

111

C₂²

C₄⋊C₄×D₁₇

yes

characteristic

112

C₂²

#10587602

yes

—

113

C₂²

#10587626

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

136

—

117

C₄

#10587574

yes

characteristic

118

C₄

C₄⋊C₄×D₁₇

yes

characteristic

119

C₄

C₂×#63225

yes

characteristic

120

C₄

#10587630

yes

characteristic

121

C₂

#11762678

yes

characteristic

122

C₂

#11762844

yes

—

123

C₂

#11763190

yes

—

124

C₂

#11763209

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₇
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₇
Fitting	C₂×C₁₇×#106	C₂
socle, prime radical	C₂³×C₁₇	C₂×D₄
cosocle	C₂²×C₄×C₁₇	C₂³
hypercenter	C₂×#106	D₁₇
hypocenter	C₁₇	#1846
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₁₇

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₇

Fitting

C₂×C₁₇×#106

C₂

socle, prime radical

C₂³×C₁₇

C₂×D₄

cosocle

C₂²×C₄×C₁₇

C₂³

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
14. C₈×Dic₁₇	C₄	semidirect
78. C₁₇	#1846	semidirect

H ≅ G/N

type

14. C₈×Dic₁₇

C₄

semidirect

78. C₁₇

#1846

semidirect

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

subgroup	index
C₂×#10587478	2
C₂×#10587488	2
C₂×C₈×Dic₁₇	2
#11762674	2
C₂×C₁₇×#106	2
#1846	17

subgroup

index

C₂×C₈×Dic₁₇

C₂×C₁₇×#106

Subgroups — #14528945

Conjugacy classes of subgroups

Complements

Maximal subgroups