Group #14526679 — subgroups

Subgroups — #14526679

Order 2176.

148 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 77 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	#11762862	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#3622	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#423	C₂	1	1	yes	maximal, characteristic
4	C₂×M₁₆×C₁₇	—	2	1	no	—
5	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
6	C₂²×#3622	—	2	1	no	—
7	C₁₇×C₄⋊C₈	—	2	1	no	—
8	C₁₇×C₄⋊C₈	C₄	1	2	yes	—
9	C₂²⋊#3622	C₄	1	2	yes	—
10	C₄⋊#3622	—	2	1	no	—
11	C₄⋊#3622	—	2	1	no	—
12	C₄⋊#3622	C₄	1	2	yes	—
13	C₄×#3622	—	2	1	no	—
14	C₂²×C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₁₇	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₇	Q₈	1	1	yes	characteristic
17	M₁₆×C₁₇	—	4	1	no	—
18	M₁₆×C₁₇	—	4	1	no	—
19	C₂×C₈×C₁₇	—	2	2	no	—
20	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₁₇	D₄	1	1	yes	characteristic
23	C₄²×C₁₇	D₄	1	1	yes	characteristic
24	C₂×#3622	—	2	2	no	—
25	C₂×#3622	—	2	2	no	—
26	C₂×#3622	—	4	1	no	—
27	C₂×#3622	—	4	1	no	—
28	C₂³×C₁₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₇	—	2	2	no	—
30	C₂×C₄×C₁₇	—	4	1	no	—
31	C₂×C₄×C₁₇	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₇	D₈	1	1	yes	characteristic
38	C₂×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
39	C₂×C₄×C₁₇	SD₁₆	1	1	yes	characteristic
40	C₂×C₄×C₁₇	Q₁₆	1	1	yes	characteristic
41	C₈×C₁₇	—	4	2	no	—
42	#3622	—	4	2	no	—
43	#3622	—	4	2	no	—
44	#1301	—	17	1	no	maximal, Sylow
45	C₂²×C₁₇	—	2	1	no	—
46	C₂²×C₁₇	—	2	1	no	—
47	C₂²×C₁₇	#95	1	1	yes	characteristic
48	C₂²×C₁₇	#106	1	1	yes	characteristic
49	C₂²×C₁₇	#107	1	1	yes	characteristic
50	C₄×C₁₇	—	2	1	no	—
51	C₄×C₁₇	—	2	1	no	—
52	C₄×C₁₇	—	2	1	no	—
53	C₄×C₁₇	#102	1	1	yes	characteristic
54	C₄×C₁₇	#102	1	1	yes	characteristic
55	C₄×C₁₇	#103	1	1	yes	characteristic
56	C₄×C₁₇	#103	1	1	yes	characteristic
57	C₄×C₁₇	#108	1	2	yes	—
58	#431	—	17	1	no	—
59	#423	D₁₇	1	1	yes	characteristic
60	C₂×C₄⋊C₈	—	17	1	no	—
61	C₂×C₁₇	—	2	1	no	—
62	C₂×C₁₇	—	2	1	no	—
63	C₂×C₁₇	#340	1	1	yes	characteristic
64	C₂×C₁₇	#341	1	1	yes	characteristic
65	C₂×C₁₇	#344	1	1	yes	characteristic
66	C₂×M₁₆	—	2	1	no	—
67	C₂²×C₈	—	34	1	no	—
68	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
69	C₄⋊C₈	—	2	1	no	—
70	C₄⋊C₈	—	17	2	no	—
71	C₄⋊C₈	—	34	1	no	—
72	C₄⋊C₈	—	34	1	no	—
73	C₄⋊C₈	Dic₁₇	1	2	yes	—
74	C₂²⋊C₈	—	17	2	no	—
75	C₄×C₈	—	34	1	no	—
76	C₁₇	#1301	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₄	Dic₃₄	1	1	yes	characteristic
78	C₂²×C₄	C₄×D₁₇	1	1	yes	characteristic
79	C₂²×C₄	C₂²⋊D₁₇	1	1	yes	characteristic
80	M₁₆	—	4	1	no	—
81	M₁₆	—	4	1	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	34	2	no	—
84	C₂×C₈	—	34	2	no	—
85	C₂×C₈	—	68	1	no	—
86	C₂×C₈	—	68	1	no	—
87	C₄²	C₄×D₁₇	1	1	yes	characteristic
88	C₄²	D₆₈	1	1	yes	characteristic
89	C₄²	C₂×Dic₁₇	1	1	yes	characteristic
90	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
91	C₂³	#63225	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	C₄×Dic₁₇	1	1	yes	characteristic
95	C₂×C₄	#63225	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₁₇	1	1	yes	characteristic
97	C₂×C₄	#63227	1	1	yes	characteristic
98	C₂×C₄	#63227	1	1	yes	characteristic
99	C₂×C₄	#63228	1	1	yes	characteristic
100	C₂×C₄	#63229	1	1	yes	characteristic
101	C₂×C₄	#63230	1	1	yes	characteristic
102	C₂×C₄	#63231	1	1	yes	characteristic
103	C₂×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
104	C₈	—	4	2	no	—
105	C₈	—	68	2	no	—
106	C₈	—	68	2	no	—
107	C₂²	—	2	1	no	—
108	C₂²	—	2	1	no	—
109	C₂²	#10587478	1	1	yes	characteristic
110	C₂²	#10587479	1	1	yes	characteristic
111	C₂²	#10587502	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	#10587480	1	1	yes	characteristic
116	C₄	#10587481	1	1	yes	characteristic
117	C₄	#10587493	1	2	yes	—
118	C₄	#10587503	1	1	yes	characteristic
119	C₄	#10587506	1	1	yes	characteristic
120	C₂	—	2	1	no	—
121	C₂	—	2	1	no	—
122	C₂	#11762571	1	1	yes	characteristic
123	C₂	#11762573	1	1	yes	characteristic
124	C₂	#11762580	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11762862

C₂

yes

maximal, characteristic

C₂×C₄⋊#3622

C₂

yes

maximal, characteristic

C₁₇×#423

C₂

yes

maximal, characteristic

C₂×M₁₆×C₁₇

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂²×#3622

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

C₄

yes

—

C₂²⋊#3622

C₄

yes

—

C₄⋊#3622

—

C₄⋊#3622

—

C₄⋊#3622

C₄

yes

—

C₄×#3622

—

C₂²×C₄×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

Q₈

yes

characteristic

M₁₆×C₁₇

—

M₁₆×C₁₇

—

C₂×C₈×C₁₇

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄²×C₁₇

D₄

yes

characteristic

C₄²×C₁₇

D₄

yes

—

—

—

—

C₂³×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₄²

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₇

D₈

yes

characteristic

C₂×C₄×C₁₇

SD₁₆

yes

characteristic

C₂×C₄×C₁₇

SD₁₆

yes

characteristic

C₂×C₄×C₁₇

Q₁₆

yes

characteristic

C₈×C₁₇

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₂²×C₁₇

yes

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

#431

—

#423

D₁₇

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

Dic₁₇

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₁₇

#1301

yes

characteristic, complemented, Sylow

C₂²×C₄

Dic₃₄

yes

characteristic

C₂²×C₄

C₄×D₁₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₄×D₁₇

yes

characteristic

C₄²

D₆₈

yes

characteristic

C₄²

C₂×Dic₁₇

yes

characteristic

C₄²

C₂²⋊D₁₇

yes

characteristic

C₂³

#63225

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₇

yes

characteristic

C₂×C₄

#63225

yes

characteristic

C₂×C₄

C₄⋊Dic₁₇

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

100

C₂×C₄

#63229

yes

characteristic

101

C₂×C₄

#63230

yes

characteristic

102

C₂×C₄

#63231

yes

characteristic

103

C₂×C₄

C₂²⋊Dic₁₇

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

—

107

C₂²

—

108

C₂²

—

109

C₂²

#10587478

yes

characteristic

110

C₂²

#10587479

yes

characteristic

111

C₂²

#10587502

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#10587480

yes

characteristic

116

C₄

#10587481

yes

characteristic

117

C₄

#10587493

yes

—

118

C₄

#10587503

yes

characteristic

119

C₄

#10587506

yes

characteristic

120

C₂

—

121

C₂

—

122

C₂

#11762571

yes

characteristic

123

C₂

#11762573

yes

characteristic

124

C₂

#11762580

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₁₇
commutator	C₂×C₄×C₁₇	C₄²
Frattini	C₂×C₄²	C₂×D₁₇
Fitting	C₁₇×#423	C₂
socle	C₂²×C₁₇	#95
prime radical	C₂³×C₁₇	C₄⋊C₄
cosocle	C₂×C₄²×C₁₇	C₂²
hypercenter	#423	D₁₇
hypocenter	C₁₇	#1301
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₁₇

commutator

C₂×C₄×C₁₇

C₄²

Frattini

C₂×C₄²

C₂×D₁₇

Fitting

C₁₇×#423

C₂

socle

C₂²×C₁₇

#95

prime radical

C₂³×C₁₇

C₄⋊C₄

cosocle

C₂×C₄²×C₁₇

C₂²

D₁₇

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
76. C₁₇	#1301	semidirect

H ≅ G/N

type

76. C₁₇

#1301

semidirect

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

subgroup	index
#11762862	2
C₂×C₄⋊#3622	2
C₁₇×#423	2
#1301	17

subgroup

index

#11762862

C₂×C₄⋊#3622

C₁₇×#423

#1301

Subgroups — #14526679

Conjugacy classes of subgroups

Complements

Maximal subgroups