Group #14526673 — subgroups

Subgroups — #14526673

Order 2176.

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

normal subgroups only

This group has 150 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	#11762862	C₂	1	1	yes	maximal, characteristic
2	C₄×#63223	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#423	C₂	1	1	yes	maximal, characteristic
4	C₂×M₁₆×C₁₇	C₄	1	2	yes	—
5	C₂×C₄²×C₁₇	C₂²	1	1	yes	characteristic
6	C₂×#63223	—	2	1	no	—
7	C₁₇×C₄⋊C₈	—	2	1	no	—
8	C₁₇×C₄⋊C₈	—	2	1	no	—
9	C₂²⋊#3622	—	2	1	no	—
10	C₄⋊#3622	C₄	1	2	yes	—
11	#10587474	C₄	1	2	yes	—
12	C₄×#3622	—	2	1	no	—
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	2	no	—
18	C₂×C₈×C₁₇	—	2	2	no	—
19	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₁₇	D₄	1	1	yes	characteristic
22	C₄²×C₁₇	D₄	1	1	yes	characteristic
23	#63223	—	2	4	no	—
24	C₂×#3622	—	2	2	no	—
25	C₂×#3622	—	2	2	no	—
26	C₂³×C₁₇	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₇	—	2	1	no	—
28	C₂×C₄×C₁₇	—	2	1	no	—
29	C₂×C₄×C₁₇	—	2	2	no	—
30	C₂×C₄×C₁₇	—	2	2	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	C₈×C₁₇	—	4	2	no	—
38	#3622	—	2	4	no	—
39	#3622	—	4	2	no	—
40	#1299	—	17	1	no	maximal, Sylow
41	C₂²×C₁₇	—	2	1	no	—
42	C₂²×C₁₇	—	2	1	no	—
43	C₂²×C₁₇	—	2	1	no	—
44	C₂²×C₁₇	#95	1	1	yes	characteristic
45	C₄×C₁₇	—	2	1	no	—
46	C₄×C₁₇	—	2	1	no	—
47	C₄×C₁₇	—	2	1	no	—
48	C₄×C₁₇	#100	1	1	yes	characteristic
49	C₄×C₁₇	#101	1	1	yes	characteristic
50	C₄×C₁₇	#104	1	2	yes	—
51	C₄×C₁₇	#108	1	2	yes	—
52	#431	—	17	1	no	—
53	#423	D₁₇	1	1	yes	characteristic
54	C₄×M₁₆	—	17	1	no	—
55	C₂×C₁₇	—	4	1	no	—
56	C₂×C₁₇	#339	1	1	yes	characteristic
57	C₂×C₁₇	#341	1	1	yes	characteristic
58	C₂×C₁₇	#343	1	1	yes	characteristic
59	C₂×M₁₆	—	34	1	no	—
60	C₂×M₁₆	Dic₁₇	1	2	yes	—
61	C₂×C₄²	C₂×D₁₇	1	1	yes	characteristic
62	C₄⋊C₈	—	2	1	no	—
63	C₄⋊C₈	—	2	1	no	—
64	C₄⋊C₈	—	17	2	no	—
65	C₂²⋊C₈	—	34	1	no	—
66	#97	—	17	2	no	—
67	C₄×C₈	—	34	1	no	—
68	C₄×C₈	—	34	1	no	—
69	C₁₇	#1299	1	1	yes	characteristic, complemented, Sylow
70	C₂²×C₄	Dic₃₄	1	1	yes	characteristic
71	C₂²×C₄	D₆₈	1	1	yes	characteristic
72	C₂²×C₄	C₂×Dic₁₇	1	1	yes	characteristic
73	M₁₆	—	4	2	no	—
74	M₁₆	—	34	4	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	34	2	no	—
77	C₂×C₈	—	34	2	no	—
78	C₄²	C₄×D₁₇	1	1	yes	characteristic
79	C₄²	C₄×D₁₇	1	1	yes	characteristic
80	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
81	C₄²	C₂²⋊D₁₇	1	1	yes	characteristic
82	C₂³	C₄⋊Dic₁₇	1	1	yes	characteristic
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	C₄×Dic₁₇	1	1	yes	characteristic
88	C₂×C₄	#63225	1	1	yes	characteristic
89	C₂×C₄	#63225	1	1	yes	characteristic
90	C₂×C₄	#63227	1	1	yes	characteristic
91	C₂×C₄	#63227	1	1	yes	characteristic
92	C₂×C₄	C₂²⋊Dic₁₇	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	34	4	no	—
95	C₈	—	68	2	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	—	2	1	no	—
99	C₂²	#10587502	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	#10587493	1	2	yes	—
104	C₄	#10587496	1	2	yes	—
105	C₄	#10587504	1	1	yes	characteristic
106	C₄	#10587507	1	1	yes	characteristic
107	C₂	—	4	1	no	—
108	C₂	#11762570	1	1	yes	characteristic
109	C₂	#11762573	1	1	yes	characteristic
110	C₂	#11762578	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11762862

C₂

yes

maximal, characteristic

C₄×#63223

C₂

yes

maximal, characteristic

C₁₇×#423

C₂

yes

maximal, characteristic

C₂×M₁₆×C₁₇

C₄

yes

—

C₂×C₄²×C₁₇

C₂²

yes

characteristic

C₂×#63223

—

C₁₇×C₄⋊C₈

—

C₁₇×C₄⋊C₈

—

C₂²⋊#3622

—

C₄⋊#3622

C₄

yes

—

#10587474

C₄

yes

—

C₄×#3622

—

C₄×#3622

—

C₂²×C₄×C₁₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

Q₈

yes

characteristic

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

—

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

C₈×C₁₇

—

—

—

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

#95

yes

characteristic

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

yes

C₄×C₁₇

yes

C₄×C₁₇

yes

—

C₄×C₁₇

#108

yes

—

#431

—

#423

D₁₇

yes

characteristic

C₄×M₁₆

—

C₂×C₁₇

—

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×C₁₇

yes

C₂×M₁₆

—

C₂×M₁₆

Dic₁₇

yes

—

C₂×C₄²

C₂×D₁₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#1299

yes

characteristic, complemented, Sylow

C₂²×C₄

Dic₃₄

yes

characteristic

C₂²×C₄

D₆₈

yes

characteristic

C₂²×C₄

C₂×Dic₁₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

C₄×D₁₇

yes

characteristic

C₄²

C₄×D₁₇

yes

characteristic

C₄²

C₂²⋊D₁₇

yes

characteristic

C₄²

C₂²⋊D₁₇

yes

characteristic

C₂³

C₄⋊Dic₁₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₇

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₁₇

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#10587502

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

#10587493

yes

—

104

C₄

#10587496

yes

—

105

C₄

#10587504

yes

characteristic

106

C₄

#10587507

yes

characteristic

107

C₂

—

108

C₂

#11762570

yes

characteristic

109

C₂

#11762573

yes

characteristic

110

C₂

#11762578

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	#1299
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
69. C₁₇	#1299	semidirect

H ≅ G/N

type

69. C₁₇

#1299

semidirect

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

subgroup	index
#11762862	2
C₄×#63223	2
C₁₇×#423	2
#1299	17

subgroup

index

#11762862

C₄×#63223

C₁₇×#423

#1299

Subgroups — #14526673

Conjugacy classes of subgroups

Complements

Maximal subgroups