Group #17957978 — subgroups

Subgroups — C₈×#10587529 #17957978

Order 4352.

398 conjugacy classes of subgroups; only the 236 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 236 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	C₄×#10587529	C₂	1	2	yes	maximal
2	C₈×#63216	C₂	1	4	yes	maximal, complemented
3	C₈²×D₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#10587529	C₄	1	4	yes	—
5	#11763695	C₄	1	4	yes	—
6	C₄×#63216	C₂²	1	4	yes	—
7	C₄×C₈×D₁₇	C₄	1	2	yes	—
8	C₄×C₈×D₁₇	C₂²	1	1	yes	characteristic
9	C₈×#3622	C₄	1	1	yes	characteristic
10	C₈×#3622	C₂²	1	2	yes	—
11	C₈²×C₁₇	C₄	1	1	yes	characteristic
12	C₂×C₈×D₁₇	C₈	1	4	yes	—
13	C₂×C₈×D₁₇	C₂×C₄	1	2	yes	—
14	C₄²×D₁₇	C₂×C₄	1	1	yes	characteristic
15	C₂×#63216	C₂×C₄	1	8	yes	—
16	#10587529	C₈	1	16	yes	direct product factor
17	C₄×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
18	C₄×C₈×C₁₇	C₂×C₄	1	2	yes	—
19	C₈×Dic₁₇	C₈	1	4	yes	—
20	C₈×Dic₁₇	C₂×C₄	1	2	yes	—
21	C₄×#3622	C₂×C₄	1	2	yes	—
22	C₄×#3622	C₂³	1	1	yes	characteristic
23	C₂×C₄×D₁₇	C₄²	1	1	yes	characteristic
24	C₂×C₄×D₁₇	C₂×C₈	1	2	yes	—
25	C₂×C₈×C₁₇	C₄²	1	2	yes	—
26	C₂×C₈×C₁₇	C₂×C₈	1	4	yes	—
27	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
28	C₄×Dic₁₇	C₄²	1	1	yes	characteristic
29	C₄×Dic₁₇	C₂×C₈	1	2	yes	—
30	C₂×#3622	C₂×C₈	1	4	yes	—
31	C₂×#3622	C₂²×C₄	1	2	yes	—
32	C₈×D₁₇	C₁₆	1	16	yes	complemented
33	C₈×D₁₇	C₂×C₈	1	8	yes	—
34	#63216	C₂×C₈	1	16	yes	complemented
35	C₂×C₈×C₁₆	—	17	1	no	maximal, Sylow
36	C₂²×D₁₇	C₄×C₈	1	1	yes	characteristic
37	C₂×C₄×C₁₇	C₂×C₄²	1	1	yes	characteristic
38	C₂×C₄×C₁₇	C₂²×C₈	1	2	yes	—
39	C₂×Dic₁₇	C₄×C₈	1	1	yes	characteristic
40	C₂×Dic₁₇	C₄×C₈	1	2	yes	—
41	C₄×D₁₇	C₄×C₈	1	4	yes	—
42	C₄×D₁₇	C₂×C₁₆	1	8	yes	—
43	C₈×C₁₇	C₄×C₈	1	4	yes	—
44	C₈×C₁₇	C₂×C₁₆	1	8	yes	complemented
45	#3622	C₂×C₁₆	1	8	yes	complemented
46	#3622	C₂²×C₈	1	4	yes	—
47	C₂×C₄×C₁₆	—	17	2	no	—
48	C₂×C₈²	—	17	1	no	—
49	C₈×C₁₆	—	17	4	no	—
50	C₂²×C₁₇	C₂×C₄×C₈	1	1	yes	characteristic
51	C₂×D₁₇	C₈²	1	2	yes	—
52	C₂×D₁₇	C₄×C₁₆	1	4	yes	—
53	C₄×C₁₇	C₂×C₄×C₈	1	2	yes	—
54	C₄×C₁₇	C₂²×C₁₆	1	4	yes	—
55	Dic₁₇	C₈²	1	2	yes	—
56	Dic₁₇	C₄×C₁₆	1	4	yes	—
57	C₂²×C₁₆	—	17	4	no	—
58	C₂×C₄×C₈	—	17	1	no	—
59	C₂×C₄×C₈	—	17	2	no	—
60	C₄×C₁₆	—	17	4	no	—
61	C₄×C₁₆	—	17	4	no	—
62	C₈²	—	17	2	no	—
63	C₈²	—	17	1	no	—
64	C₈²	#595	1	1	yes	characteristic
65	C₂×C₁₇	C₂×C₈²	1	1	yes	characteristic
66	C₂×C₁₇	C₂×C₄×C₁₆	1	2	yes	—
67	D₁₇	C₈×C₁₆	1	4	yes	complemented
68	C₂²×C₈	—	17	4	no	—
69	C₂²×C₈	—	17	2	no	—
70	C₂×C₄²	—	17	1	no	—
71	C₂×C₁₆	—	17	8	no	—
72	C₂×C₁₆	—	17	16	no	—
73	C₄×C₈	—	17	4	no	—
74	C₄×C₈	—	17	2	no	—
75	C₄×C₈	—	17	2	no	—
76	C₄×C₈	—	17	1	no	—
77	C₄×C₈	C₂×#595	1	2	yes	—
78	C₄×C₈	C₂×#595	1	1	yes	characteristic
79	C₁₇	C₂×C₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₄	—	17	2	no	—
81	C₂²×C₄	—	17	1	no	—
82	C₂×C₈	—	17	4	no	—
83	C₂×C₈	—	17	2	no	—
84	C₂×C₈	—	17	16	no	—
85	C₂×C₈	—	17	8	no	—
86	C₂×C₈	#63242	1	4	yes	—
87	C₂×C₈	C₄×#595	1	2	yes	—
88	C₄²	—	17	2	no	—
89	C₄²	—	17	1	no	—
90	C₄²	C₂²×#595	1	1	yes	characteristic
91	C₁₆	—	17	16	no	—
92	C₂³	—	17	1	no	—
93	C₂×C₄	—	17	2	no	—
94	C₂×C₄	—	17	1	no	—
95	C₂×C₄	—	17	4	no	—
96	C₂×C₄	—	17	8	no	—
97	C₂×C₄	C₂×#63242	1	2	yes	—
98	C₂×C₄	C₂×C₄×#595	1	1	yes	characteristic
99	C₈	—	17	4	no	—
100	C₈	—	17	8	no	—
101	C₈	#10587529	1	8	yes	direct product factor
102	C₈	C₈×#595	1	4	yes	—
103	C₂²	—	17	4	no	—
104	C₂²	—	17	2	no	—
105	C₂²	C₄×#63242	1	1	yes	characteristic
106	C₄	—	17	4	no	—
107	C₄	—	17	2	no	—
108	C₄	C₂×C₈×#595	1	2	yes	—
109	C₄	C₂×#10587529	1	4	yes	—
110	C₂	—	17	4	no	—
111	C₂	C₈×#63242	1	1	yes	characteristic
112	C₂	C₄×#10587529	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10587529

C₂

yes

maximal

C₈×#63216

C₂

yes

maximal, complemented

C₈²×D₁₇

C₂

yes

maximal, characteristic

C₂×#10587529

C₄

yes

—

#11763695

C₄

yes

—

C₄×#63216

C₂²

yes

—

C₄×C₈×D₁₇

C₄

yes

—

C₄×C₈×D₁₇

C₂²

yes

characteristic

C₈×#3622

C₄

yes

characteristic

C₈×#3622

C₂²

yes

—

C₈²×C₁₇

C₄

yes

characteristic

C₂×C₈×D₁₇

C₈

yes

—

C₂×C₈×D₁₇

C₂×C₄

yes

—

C₄²×D₁₇

C₂×C₄

yes

characteristic

C₂×#63216

C₂×C₄

yes

—

#10587529

C₈

yes

direct product factor

C₄×C₈×C₁₇

C₂×C₄

yes

characteristic

C₄×C₈×C₁₇

C₂×C₄

yes

—

C₈×Dic₁₇

C₈

yes

—

C₈×Dic₁₇

C₂×C₄

yes

—

C₄×#3622

C₂×C₄

yes

—

C₄×#3622

C₂³

yes

characteristic

C₂×C₄×D₁₇

C₄²

yes

characteristic

C₂×C₄×D₁₇

C₂×C₈

yes

—

C₂×C₈×C₁₇

C₄²

yes

—

C₂×C₈×C₁₇

C₂×C₈

yes

—

C₄²×C₁₇

C₂²×C₄

yes

characteristic

C₄×Dic₁₇

C₄²

yes

characteristic

C₄×Dic₁₇

C₂×C₈

yes

—

C₂×#3622

C₂×C₈

yes

—

C₂×#3622

C₂²×C₄

yes

—

C₈×D₁₇

C₁₆

yes

complemented

C₈×D₁₇

C₂×C₈

yes

—

#63216

C₂×C₈

yes

complemented

C₂×C₈×C₁₆

—

maximal, Sylow

C₂²×D₁₇

C₄×C₈

yes

characteristic

C₂×C₄×C₁₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₁₇

C₂²×C₈

yes

—

C₂×Dic₁₇

C₄×C₈

yes

characteristic

C₂×Dic₁₇

C₄×C₈

yes

—

C₄×D₁₇

C₄×C₈

yes

—

C₄×D₁₇

C₂×C₁₆

yes

—

C₈×C₁₇

C₄×C₈

yes

—

C₈×C₁₇

C₂×C₁₆

yes

complemented

#3622

C₂×C₁₆

yes

complemented

#3622

C₂²×C₈

yes

—

C₂×C₄×C₁₆

—

C₂×C₈²

—

C₈×C₁₆

—

C₂²×C₁₇

C₂×C₄×C₈

yes

characteristic

C₂×D₁₇

C₈²

yes

—

C₂×D₁₇

C₄×C₁₆

yes

—

C₄×C₁₇

C₂×C₄×C₈

yes

—

C₄×C₁₇

C₂²×C₁₆

yes

—

Dic₁₇

C₈²

yes

—

Dic₁₇

C₄×C₁₆

yes

—

C₂²×C₁₆

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

—

C₈²

—

C₈²

#595

yes

characteristic

C₂×C₁₇

C₂×C₈²

yes

characteristic

C₂×C₁₇

C₂×C₄×C₁₆

yes

—

D₁₇

C₈×C₁₆

yes

complemented

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#595

yes

—

C₄×C₈

C₂×#595

yes

characteristic

C₁₇

C₂×C₈×C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#63242

yes

—

C₂×C₈

C₄×#595

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×#595

yes

characteristic

C₁₆

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#63242

yes

—

C₂×C₄

C₂×C₄×#595

yes

characteristic

C₈

—

100

C₈

—

101

C₈

#10587529

yes

direct product factor

102

C₈

C₈×#595

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

C₄×#63242

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

C₂×C₈×#595

yes

—

109

C₄

C₂×#10587529

yes

—

110

C₂

—

111

C₂

C₈×#63242

yes

characteristic

112

C₂

C₄×#10587529

yes

—

	subgroup	G/H
center, hypercenter	C₈²	#595
commutator, hypocenter	C₁₇	C₂×C₈×C₁₆
Frattini	C₄²	C₂²×#595
Fitting	C₈²×C₁₇	C₄
socle	C₂²×C₁₇	C₂×C₄×C₈
prime radical	C₂²×D₁₇	C₄×C₈
cosocle	C₄×#3622	C₂³
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₈²

subgroup

G/H

center, hypercenter

C₈²

#595

commutator, hypocenter

C₁₇

C₂×C₈×C₁₆

Frattini

C₄²

C₂²×#595

Fitting

C₈²×C₁₇

C₄

socle

C₂²×C₁₇

C₂×C₄×C₈

prime radical

C₂²×D₁₇

C₄×C₈

C₂³

G ▷ C₁₇

C₁ ◁ C₈²

N	H ≅ G/N	type
2. C₈×#63216	C₂	semidirect
16. #10587529	C₈	direct (N × H)
32. C₈×D₁₇	C₁₆	semidirect
34. #63216	C₂×C₈	semidirect
44. C₈×C₁₇	C₂×C₁₆	semidirect
45. #3622	C₂×C₁₆	semidirect
67. D₁₇	C₈×C₁₆	semidirect
79. C₁₇	C₂×C₈×C₁₆	semidirect
101. C₈	#10587529	direct (N × H)

H ≅ G/N

type

2. C₈×#63216

C₂

semidirect

16. #10587529

C₈

direct (N × H)

32. C₈×D₁₇

C₁₆

semidirect

34. #63216

C₂×C₈

semidirect

44. C₈×C₁₇

C₂×C₁₆

semidirect

45. #3622

C₂×C₁₆

semidirect

67. D₁₇

C₈×C₁₆

semidirect

79. C₁₇

C₂×C₈×C₁₆

semidirect

101. C₈

#10587529

direct (N × H)

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

subgroup	index
C₄×#10587529	2
C₈×#63216	2
C₈²×D₁₇	2
C₂×C₈×C₁₆	17

subgroup

index

C₄×#10587529

C₈×#63216

C₈²×D₁₇

C₂×C₈×C₁₆

Subgroups — C₈×#10587529 #17957978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C8×#10587529 #17957978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×#10587529 #17957978