Group #14528458 — subgroups

Subgroups — #14528458

Order 2176.

176 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 65 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	#11763214	C₂	1	1	yes	maximal, characteristic
2	#11763212	C₂	1	1	yes	maximal, characteristic
3	#11763188	C₂	1	1	yes	maximal, characteristic
4	#11763161	C₂	1	1	yes	maximal, characteristic
5	#11763159	C₂	1	1	yes	maximal, characteristic
6	#11762530	C₂	1	1	yes	maximal, characteristic
7	C₁₇×#498	C₂	1	1	yes	maximal, characteristic
8	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₁₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×#63231	—	2	2	no	—
11	#10587555	C₂²	1	1	yes	characteristic
12	#10587555	C₂²	1	1	yes	characteristic
13	C₁₇×#106	—	2	2	no	—
14	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
15	#10587506	—	2	2	no	—
16	#10587489	—	2	2	no	—
17	#10587481	—	2	2	no	—
18	#10587478	—	2	2	no	—
19	C₄×#3622	C₂²	1	1	yes	characteristic
20	C₄×#3622	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₁₇	—	2	1	no	—
22	C₂×Q₈×C₁₇	—	2	1	no	—
23	C₂×Dic₃₄	—	2	1	no	—
24	C₂×Dic₃₄	—	2	1	no	—
25	C₂×C₈×C₁₇	D₄	1	2	yes	—
26	C₄⋊C₄×C₁₇	—	2	1	no	—
27	C₄⋊C₄×C₁₇	—	2	1	no	—
28	C₄⋊C₄×C₁₇	—	2	1	no	—
29	C₄⋊C₄×C₁₇	—	2	1	no	—
30	C₄²×C₁₇	C₂³	1	1	yes	characteristic
31	#63231	—	4	4	no	—
32	C₄⋊Dic₁₇	—	2	1	no	—
33	C₄⋊Dic₁₇	—	2	1	no	—
34	C₄⋊Dic₁₇	—	2	1	no	—
35	C₄⋊Dic₁₇	—	2	1	no	—
36	C₂×#3622	D₄	1	2	yes	—
37	C₂×#3622	D₄	1	2	yes	—
38	Q₈×C₁₇	—	4	2	no	—
39	Q₈×C₁₇	—	4	2	no	—
40	C₂×C₄×C₁₇	—	4	1	no	—
41	C₂×C₄×C₁₇	—	4	1	no	—
42	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₁₇	—	4	1	no	—
46	C₂×Dic₁₇	—	4	1	no	—
47	Dic₃₄	—	4	2	no	—
48	Dic₃₄	—	4	2	no	—
49	C₈×C₁₇	Q₁₆	1	4	yes	complemented
50	#3622	—	2	2	no	—
51	#3622	—	2	2	no	—
52	#1723	—	17	1	no	maximal, Sylow
53	C₂²×C₁₇	#127	1	1	yes	characteristic
54	C₄×C₁₇	—	8	1	no	—
55	C₄×C₁₇	—	8	1	no	—
56	C₄×C₁₇	C₂×Q₁₆	1	2	yes	—
57	C₄×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
58	C₄×C₁₇	#137	1	2	yes	—
59	Dic₁₇	—	8	1	no	—
60	Dic₁₇	—	8	1	no	—
61	#500	—	17	1	no	—
62	#500	—	17	1	no	—
63	#498	D₁₇	1	1	yes	characteristic
64	#494	—	17	1	no	—
65	#487	—	17	1	no	—
66	#487	—	17	1	no	—
67	#322	—	17	1	no	—
68	C₂×C₁₇	#494	1	1	yes	characteristic
69	C₂×C₁₇	#496	1	1	yes	characteristic
70	C₂×C₁₇	#497	1	1	yes	characteristic
71	C₂×Q₁₆	—	34	2	no	—
72	C₄⋊Q₈	—	17	1	no	—
73	C₄⋊Q₈	—	17	1	no	—
74	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
75	C₄⋊Q₈	C₂×D₁₇	1	1	yes	characteristic
76	#107	—	34	2	no	—
77	#107	—	34	2	no	—
78	#106	—	2	2	no	—
79	#103	—	34	2	no	—
80	#103	—	34	2	no	—
81	C₄×C₈	—	17	1	no	—
82	C₄×C₈	—	17	1	no	—
83	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
84	C₁₇	#1723	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₈	—	2	1	no	—
86	C₂×Q₈	—	2	1	no	—
87	C₂×Q₈	—	34	1	no	—
88	C₂×Q₈	—	34	1	no	—
89	Q₁₆	—	68	4	no	—
90	C₂×C₈	—	17	2	no	—
91	C₂×C₈	—	17	2	no	—
92	C₂×C₈	C₂²⋊D₁₇	1	2	yes	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	1	no	—
96	C₄⋊C₄	—	2	1	no	—
97	C₄⋊C₄	—	34	1	no	—
98	C₄⋊C₄	—	34	1	no	—
99	C₄⋊C₄	—	34	1	no	—
100	C₄⋊C₄	—	34	1	no	—
101	C₄²	C₂²×D₁₇	1	1	yes	characteristic
102	Q₈	—	4	2	no	—
103	Q₈	—	4	2	no	—
104	Q₈	—	68	2	no	—
105	Q₈	—	68	2	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	68	1	no	—
109	C₂×C₄	—	68	1	no	—
110	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
111	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₇	1	1	yes	characteristic
113	C₈	—	34	2	no	—
114	C₈	—	34	2	no	—
115	C₈	#63231	1	4	yes	complemented
116	C₂²	#10587626	1	1	yes	characteristic
117	C₄	—	8	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	136	1	no	—
120	C₄	—	136	1	no	—
121	C₄	#10587600	1	2	yes	—
122	C₄	#10587601	1	2	yes	—
123	C₄	C₂×#63231	1	2	yes	—
124	C₂	#11763188	1	1	yes	characteristic
125	C₂	#11763195	1	1	yes	characteristic
126	C₂	#11763200	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11763214

C₂

yes

maximal, characteristic

#11763212

C₂

yes

maximal, characteristic

#11763188

C₂

yes

maximal, characteristic

#11763161

C₂

yes

maximal, characteristic

#11763159

C₂

yes

maximal, characteristic

#11762530

C₂

yes

maximal, characteristic

C₁₇×#498

C₂

yes

maximal, characteristic

C₁₇×C₄⋊Q₈

C₂²

yes

characteristic

C₁₇×C₄⋊Q₈

C₂²

yes

characteristic

C₂×#63231

—

#10587555

C₂²

yes

characteristic

#10587555

C₂²

yes

characteristic

C₁₇×#106

—

C₄×C₈×C₁₇

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₄×#3622

C₂²

yes

characteristic

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Dic₃₄

—

C₂×Dic₃₄

—

C₂×C₈×C₁₇

D₄

yes

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂³

yes

characteristic

#63231

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₄⋊Dic₁₇

—

C₂×#3622

D₄

yes

—

C₂×#3622

D₄

yes

—

Q₈×C₁₇

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

Dic₃₄

—

Dic₃₄

—

C₈×C₁₇

Q₁₆

yes

—

—

—

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

C₂×Q₁₆

yes

—

C₄×C₁₇

AGL(1,ℤ/8)

yes

—

C₄×C₁₇

#137

yes

—

Dic₁₇

—

Dic₁₇

—

#500

—

#500

—

#498

D₁₇

yes

—

—

—

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₁₇

yes

characteristic

C₄⋊Q₈

C₂×D₁₇

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₁₇

yes

characteristic

C₁₇

#1723

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

C₂²×D₁₇

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

D₄×D₁₇

yes

characteristic

111

C₂×C₄

D₄×D₁₇

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₇

yes

characteristic

113

C₈

—

114

C₈

—

115

C₈

#63231

yes

complemented

116

C₂²

#10587626

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

136

—

120

C₄

—

136

—

121

C₄

#10587600

yes

—

122

C₄

#10587601

yes

—

123

C₄

C₂×#63231

yes

—

124

C₂

#11763188

yes

characteristic

125

C₂

#11763195

yes

characteristic

126

C₂

#11763200

yes

characteristic

	subgroup	G/H
center	C₂²	#10587626
commutator, cosocle	C₄²×C₁₇	C₂³
Frattini	C₄²	C₂²×D₁₇
Fitting	C₁₇×#498	C₂
socle, prime radical	C₂²×C₁₇	#127
hypercenter	#498	D₁₇
hypocenter	C₁₇	#1723
LCS	G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#10587626

commutator, cosocle

C₄²×C₁₇

C₂³

Frattini

C₄²

C₂²×D₁₇

Fitting

C₁₇×#498

C₂

C₂²×C₁₇

D₁₇

C₁₇

G ▷ C₄²×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
49. C₈×C₁₇	Q₁₆	semidirect
84. C₁₇	#1723	semidirect
115. C₈	#63231	semidirect

H ≅ G/N

type

49. C₈×C₁₇

Q₁₆

semidirect

84. C₁₇

#1723

semidirect

115. C₈

#63231

semidirect

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

subgroup	index
#11763214	2
#11763212	2
#11763188	2
#11763161	2
#11763159	2
#11762530	2
C₁₇×#498	2
#1723	17

subgroup

index

C₁₇×#498

Subgroups — #14528458

Conjugacy classes of subgroups

Complements

Maximal subgroups