Group #17034712 — subgroups

Subgroups — #17034712

Order 4352.

1118 conjugacy classes of subgroups; only the 571 normal subgroups are shown.

normal subgroups only

This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 571 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×#11762833	C₂	1	6	yes	maximal
2	#14528938	C₂	1	8	yes	maximal, complemented
3	C₂×C₁₇×#376	C₂	1	1	yes	maximal, characteristic
4	C₂²×#63223	C₂²	1	4	yes	—
5	#11762833	—	2	24	no	—
6	C₂×C₄⋊#3622	C₂²	1	24	yes	—
7	C₂²×C₄²×C₁₇	C₂²	1	3	yes	—
8	C₁₇×#376	C₄	1	4	yes	—
9	C₁₇×#376	C₂²	1	4	yes	—
10	C₂×C₁₇×#95	C₄	1	4	yes	—
11	C₂³×C₄×C₁₇	C₂×C₄	1	6	yes	—
12	C₂³×C₄×C₁₇	C₂³	1	1	yes	characteristic
13	C₂×C₄²×C₁₇	C₂×C₄	1	6	yes	—
14	C₂×C₄²×C₁₇	D₄	1	12	yes	—
15	C₂×C₄²×C₁₇	Q₈	1	12	yes	—
16	C₂×C₄²×C₁₇	C₂³	1	6	yes	—
17	C₂×#63223	—	2	24	no	—
18	C₂²×#3622	C₂³	1	8	yes	—
19	C₁₇×#95	C₂×C₄	1	16	yes	—
20	C₄⋊#3622	—	2	48	no	—
21	C₂⁴×C₁₇	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₇	—	4	12	no	—
23	C₂²×C₄×C₁₇	C₄⋊C₄	1	24	yes	—
24	C₂²×C₄×C₁₇	C₂²×C₄	1	12	yes	—
25	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₇	C₂×D₄	1	6	yes	—
27	C₂²×C₄×C₁₇	C₂×Q₈	1	6	yes	—
28	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
29	C₄²×C₁₇	C₄⋊C₄	1	24	yes	—
30	C₄²×C₁₇	C₂×D₄	1	12	yes	—
31	C₄²×C₁₇	C₂×Q₈	1	12	yes	—
32	#63223	—	4	16	no	—
33	C₂×#3622	—	2	24	no	—
34	#20792	—	17	1	no	maximal, Sylow
35	C₂³×C₁₇	—	2	4	no	—
36	C₂³×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
37	C₂³×C₁₇	C₂³×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₇	—	2	24	no	—
39	C₂×C₄×C₁₇	—	4	24	no	—
40	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	24	yes	—
41	C₂×C₄×C₁₇	C₂×C₄⋊C₄	1	6	yes	—
42	C₂×C₄×C₁₇	#127	1	4	yes	—
43	C₂×C₄×C₁₇	C₄⋊Q₈	1	12	yes	—
44	C₂×C₄×C₁₇	C₂²×D₄	1	3	yes	—
45	C₂×C₄×C₁₇	C₂²×Q₈	1	3	yes	—
46	#3622	—	4	8	no	—
47	C₂×#423	—	17	6	no	—
48	C₂×#376	D₁₇	1	1	yes	characteristic
49	#1843	—	17	8	no	—
50	C₂²×C₁₇	—	2	12	no	—
51	C₂²×C₁₇	#384	1	4	yes	—
52	C₂²×C₁₇	C₂²×C₄⋊C₄	1	3	yes	—
53	C₂²×C₁₇	#568	1	4	yes	—
54	C₄×C₁₇	—	2	24	no	—
55	C₄×C₁₇	#384	1	4	yes	—
56	C₄×C₁₇	C₂×#127	1	1	yes	characteristic
57	C₄×C₁₇	C₂×C₄⋊Q₈	1	3	yes	—
58	C₂²×M₁₆	—	17	4	no	—
59	C₂²×C₄²	C₂×D₁₇	1	3	yes	—
60	#423	—	34	24	no	—
61	C₂×C₄⋊C₈	—	17	24	no	—
62	#376	Dic₁₇	1	4	yes	—
63	#376	C₂×D₁₇	1	4	yes	—
64	C₂×#95	Dic₁₇	1	4	yes	—
65	C₂×C₁₇	—	2	4	no	—
66	C₂×C₁₇	C₂×#384	1	1	yes	characteristic
67	C₂×C₁₇	#2905	1	6	yes	—
68	C₂³×C₄	C₂×Dic₁₇	1	6	yes	—
69	C₂³×C₄	C₂²×D₁₇	1	1	yes	characteristic
70	C₂×M₁₆	—	34	24	no	—
71	C₂²×C₈	—	17	8	no	—
72	C₂×C₄²	Dic₃₄	1	12	yes	—
73	C₂×C₄²	D₆₈	1	12	yes	—
74	C₂×C₄²	C₂×Dic₁₇	1	6	yes	—
75	C₂×C₄²	C₂²×D₁₇	1	6	yes	—
76	C₄⋊C₈	—	34	48	no	—
77	#95	C₂×Dic₁₇	1	16	yes	—
78	C₁₇	#20792	1	1	yes	characteristic, complemented, Sylow
79	C₂⁴	C₂²×Dic₁₇	1	1	yes	characteristic
80	C₂²×C₄	—	4	12	no	—
81	C₂²×C₄	C₄⋊Dic₁₇	1	24	yes	—
82	C₂²×C₄	C₂×Dic₃₄	1	6	yes	—
83	C₂²×C₄	C₂×D₆₈	1	6	yes	—
84	C₂²×C₄	C₂²×Dic₁₇	1	12	yes	—
85	C₂²×C₄	C₂²×Dic₁₇	1	1	yes	characteristic
86	C₂²×C₄	C₂³×D₁₇	1	1	yes	characteristic
87	M₁₆	—	68	16	no	—
88	C₂×C₈	—	34	24	no	—
89	C₄²	C₄⋊Dic₁₇	1	24	yes	—
90	C₄²	C₂×Dic₃₄	1	12	yes	—
91	C₄²	C₂×D₆₈	1	12	yes	—
92	C₂³	—	2	4	no	—
93	C₂³	C₂×C₄⋊Dic₁₇	1	6	yes	—
94	C₂³	C₂³×Dic₁₇	1	1	yes	characteristic
95	C₂×C₄	—	2	24	no	—
96	C₂×C₄	—	4	24	no	—
97	C₂×C₄	#10587555	1	12	yes	—
98	C₂×C₄	#10587560	1	4	yes	—
99	C₂×C₄	C₂×C₄⋊Dic₁₇	1	24	yes	—
100	C₂×C₄	C₂×C₄⋊Dic₁₇	1	6	yes	—
101	C₂×C₄	C₂²×Dic₃₄	1	3	yes	—
102	C₂×C₄	C₂²×D₆₈	1	3	yes	—
103	C₈	—	68	8	no	—
104	C₂²	—	2	12	no	—
105	C₂²	#11762679	1	4	yes	—
106	C₂²	C₂²×C₄⋊Dic₁₇	1	3	yes	—
107	C₂²	#11763576	1	4	yes	—
108	C₄	—	2	24	no	—
109	C₄	#11762679	1	4	yes	—
110	C₄	C₂×#10587560	1	1	yes	characteristic
111	C₄	C₂×#10587555	1	3	yes	—
112	C₂	—	2	4	no	—
113	C₂	C₂×#11762679	1	1	yes	characteristic
114	C₂	#14537348	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11762833

C₂

yes

maximal

#14528938

C₂

yes

maximal, complemented

C₂×C₁₇×#376

C₂

yes

maximal, characteristic

C₂²×#63223

C₂²

yes

—

#11762833

—

C₂×C₄⋊#3622

C₂²

yes

—

C₂²×C₄²×C₁₇

C₂²

yes

—

C₁₇×#376

C₄

yes

—

C₁₇×#376

C₂²

yes

—

C₂×C₁₇×#95

C₄

yes

—

C₂³×C₄×C₁₇

C₂×C₄

yes

—

C₂³×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₄²×C₁₇

C₂×C₄

yes

—

C₂×C₄²×C₁₇

D₄

yes

—

C₂×C₄²×C₁₇

Q₈

yes

—

C₂×C₄²×C₁₇

C₂³

yes

—

C₂×#63223

—

C₂²×#3622

C₂³

yes

—

C₁₇×#95

C₂×C₄

yes

—

C₄⋊#3622

—

C₂⁴×C₁₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₇

—

C₂²×C₄×C₁₇

C₄⋊C₄

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

—

C₂²×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×D₄

yes

—

C₂²×C₄×C₁₇

C₂×Q₈

yes

—

C₂²×C₄×C₁₇

C₂⁴

yes

characteristic

C₄²×C₁₇

C₄⋊C₄

yes

—

C₄²×C₁₇

C₂×D₄

yes

—

C₄²×C₁₇

C₂×Q₈

yes

—

—

—

—

C₂³×C₁₇

—

C₂³×C₁₇

C₂×C₄⋊C₄

yes

—

C₂³×C₁₇

C₂³×C₄

yes

characteristic

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₇

#127

yes

—

C₂×C₄×C₁₇

C₄⋊Q₈

yes

—

C₂×C₄×C₁₇

C₂²×D₄

yes

—

C₂×C₄×C₁₇

C₂²×Q₈

yes

—

#3622

—

C₂×#423

—

C₂×#376

D₁₇

yes

characteristic

#1843

—

C₂²×C₁₇

—

C₂²×C₁₇

#384

yes

—

C₂²×C₁₇

C₂²×C₄⋊C₄

yes

—

C₂²×C₁₇

#568

yes

—

C₄×C₁₇

—

C₄×C₁₇

#384

yes

—

C₄×C₁₇

C₂×#127

yes

characteristic

C₄×C₁₇

C₂×C₄⋊Q₈

yes

—

C₂²×M₁₆

—

C₂²×C₄²

C₂×D₁₇

yes

—

#423

—

C₂×C₄⋊C₈

—

#376

Dic₁₇

yes

—

#376

C₂×D₁₇

yes

—

C₂×#95

Dic₁₇

yes

—

C₂×C₁₇

—

C₂×C₁₇

C₂×#384

yes

characteristic

C₂×C₁₇

#2905

yes

—

C₂³×C₄

C₂×Dic₁₇

yes

—

C₂³×C₄

C₂²×D₁₇

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

Dic₃₄

yes

—

C₂×C₄²

D₆₈

yes

—

C₂×C₄²

C₂×Dic₁₇

yes

—

C₂×C₄²

C₂²×D₁₇

yes

—

C₄⋊C₈

—

#95

C₂×Dic₁₇

yes

—

C₁₇

#20792

yes

characteristic, complemented, Sylow

C₂⁴

C₂²×Dic₁₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄⋊Dic₁₇

yes

—

C₂²×C₄

C₂×Dic₃₄

yes

—

C₂²×C₄

C₂×D₆₈

yes

—

C₂²×C₄

C₂²×Dic₁₇

yes

—

C₂²×C₄

C₂²×Dic₁₇

yes

characteristic

C₂²×C₄

C₂³×D₁₇

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄²

C₄⋊Dic₁₇

yes

—

C₄²

C₂×Dic₃₄

yes

—

C₄²

C₂×D₆₈

yes

—

C₂³

—

C₂³

C₂×C₄⋊Dic₁₇

yes

—

C₂³

C₂³×Dic₁₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#10587555

yes

—

C₂×C₄

#10587560

yes

—

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

—

100

C₂×C₄

C₂×C₄⋊Dic₁₇

yes

—

101

C₂×C₄

C₂²×Dic₃₄

yes

—

102

C₂×C₄

C₂²×D₆₈

yes

—

103

C₈

—

104

C₂²

—

105

C₂²

#11762679

yes

—

106

C₂²

C₂²×C₄⋊Dic₁₇

yes

—

107

C₂²

#11763576

yes

—

108

C₄

—

109

C₄

#11762679

yes

—

110

C₄

C₂×#10587560

yes

characteristic

111

C₄

C₂×#10587555

yes

—

112

C₂

—

113

C₂

C₂×#11762679

yes

characteristic

114

C₂

#14537348

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₁₇
commutator, socle	C₂³×C₁₇	C₂³×C₄
Fitting	C₂×C₁₇×#376	C₂
prime radical	C₂⁴×C₁₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₇	C₂⁴
hypercenter	C₂×#376	D₁₇
hypocenter	C₁₇	#20792
LCS	G ▷ C₂³×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₁₇

commutator, socle

C₂³×C₁₇

C₂³×C₄

Fitting

C₂×C₁₇×#376

C₂

prime radical

C₂⁴×C₁₇

C₂²×C₄

cosocle

C₂²×C₄×C₁₇

C₂⁴

D₁₇

C₁₇

G ▷ C₂³×C₁₇ ▷ C₁₇

C₁ ◁ C₂²×C₄ ◁ C₂×#376

N	H ≅ G/N	type
2. #14528938	C₂	semidirect
78. C₁₇	#20792	semidirect

H ≅ G/N

type

2. #14528938

C₂

semidirect

78. C₁₇

#20792

semidirect

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

subgroup	index
C₂×#11762833	2
#14528938	2
C₂×C₁₇×#376	2
#20792	17

subgroup

index

C₂×#11762833

#14528938

C₂×C₁₇×#376

#20792

Subgroups — #17034712

Conjugacy classes of subgroups

Complements

Maximal subgroups