Group #11763532 — subgroups

Subgroups — #11763532

Order 1088.

212 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 99 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×Q₈×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#10587632	C₂	1	2	yes	maximal, complemented
3	#10587584	C₂	1	4	yes	maximal, complemented
4	#10587578	C₂	1	2	yes	maximal, complemented
5	#10587561	C₂	1	2	yes	maximal
6	C₄×D₆₈	C₂	1	1	yes	maximal, characteristic
7	#10587555	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Dic₃₄	C₂	1	2	yes	maximal, complemented
9	C₂×Q₈×C₁₇	C₂²	1	1	yes	characteristic
10	C₂×D₆₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₇	C₂²	1	2	yes	—
12	C₂×Dic₃₄	C₂²	1	2	yes	—
13	C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₁₇	C₂²	1	2	yes	—
15	C₄²×C₁₇	C₂²	1	2	yes	—
16	C₄²×C₁₇	C₂²	1	1	yes	characteristic
17	#63227	C₂²	1	2	yes	—
18	#63227	C₂²	1	4	yes	—
19	#63227	C₂²	1	4	yes	—
20	C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
21	C₄⋊Dic₁₇	C₂²	1	2	yes	—
22	C₄⋊Dic₁₇	C₂²	1	2	yes	—
23	#63225	C₂²	1	4	yes	—
24	C₄×Dic₁₇	C₂²	1	4	yes	—
25	C₂²×D₁₇	C₂³	1	2	yes	—
26	Q₈×C₁₇	—	2	2	no	—
27	C₂×C₄×C₁₇	C₂³	1	2	yes	—
28	C₂×C₄×C₁₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₁₇	C₂³	1	2	yes	—
30	C₂×C₄×C₁₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₁₇	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₇	C₂³	1	4	yes	—
33	C₂×Dic₁₇	C₂³	1	2	yes	—
34	D₆₈	—	2	2	no	—
35	C₄×D₁₇	—	2	4	no	—
36	Dic₃₄	—	2	4	no	—
37	C₂²×C₁₇	C₂⁴	1	1	yes	characteristic
38	C₂×D₁₇	—	2	2	no	—
39	C₂×D₁₇	—	2	2	no	—
40	C₂×D₁₇	—	2	2	no	—
41	C₄×C₁₇	—	2	2	no	—
42	C₄×C₁₇	—	2	2	no	—
43	C₄×C₁₇	—	2	1	no	—
44	C₄×C₁₇	#41	1	2	yes	—
45	C₄×C₁₇	#41	1	2	yes	—
46	Dic₁₇	—	2	4	no	—
47	Dic₁₇	—	2	2	no	—
48	#556	—	17	1	no	maximal, Sylow
49	C₂×C₁₇	C₂×#41	1	1	yes	characteristic
50	C₂×C₁₇	C₂×#41	1	1	yes	characteristic
51	C₂×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
52	D₁₇	—	4	2	no	—
53	C₄⋊Q₈	—	17	1	no	—
54	#126	—	17	4	no	—
55	#124	—	17	2	no	—
56	C₂²⋊Q₈	—	17	2	no	—
57	C₄×Q₈	—	17	2	no	—
58	C₄×Q₈	D₁₇	1	1	yes	characteristic, complemented
59	C₄×D₄	—	17	1	no	—
60	#117	—	17	2	no	—
61	C₁₇	#556	1	1	yes	characteristic, complemented, Sylow
62	C₂×Q₈	—	17	2	no	—
63	C₂×Q₈	C₂×D₁₇	1	1	yes	characteristic
64	C₂×D₄	—	17	1	no	—
65	C₂²×C₄	—	17	2	no	—
66	C₄⋊C₄	—	17	2	no	—
67	C₄⋊C₄	—	17	1	no	—
68	C₄⋊C₄	—	17	4	no	—
69	C₄⋊C₄	—	17	2	no	—
70	C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
71	C₄⋊C₄	C₂×D₁₇	1	2	yes	—
72	C₂²⋊C₄	—	17	4	no	—
73	C₂²⋊C₄	—	17	2	no	—
74	C₂²⋊C₄	—	17	4	no	—
75	C₄²	—	17	4	no	—
76	C₄²	C₂×D₁₇	1	1	yes	characteristic
77	C₄²	C₂×D₁₇	1	2	yes	—
78	C₂³	—	17	2	no	—
79	Q₈	—	2	2	no	—
80	Q₈	—	34	4	no	—
81	D₄	—	34	2	no	—
82	C₂×C₄	—	17	2	no	—
83	C₂×C₄	—	17	4	no	—
84	C₂×C₄	—	34	4	no	—
85	C₂×C₄	C₂²×D₁₇	1	2	yes	—
86	C₂×C₄	C₂²×D₁₇	1	1	yes	characteristic
87	C₂×C₄	C₂²×D₁₇	1	2	yes	—
88	C₂×C₄	C₂²×D₁₇	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₁₇	1	1	yes	characteristic
90	C₂²	—	34	2	no	—
91	C₂²	—	34	2	no	—
92	C₂²	—	34	2	no	—
93	C₂²	C₂³×D₁₇	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	2	2	no	—
96	C₄	—	2	1	no	—
97	C₄	—	34	2	no	—
98	C₄	—	34	4	no	—
99	C₄	#63252	1	2	yes	—
100	C₄	#63256	1	2	yes	—
101	C₂	—	68	2	no	—
102	C₂	C₂×#63252	1	1	yes	characteristic
103	C₂	C₂×#63256	1	1	yes	characteristic
104	C₂	#10587697	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Q₈×C₁₇

C₂

yes

maximal, characteristic, complemented

#10587632

C₂

yes

maximal, complemented

#10587584

C₂

yes

maximal, complemented

#10587578

C₂

yes

maximal, complemented

#10587561

C₂

yes

maximal

C₄×D₆₈

C₂

yes

maximal, characteristic

#10587555

C₂

yes

maximal, characteristic, complemented

C₄×Dic₃₄

C₂

yes

maximal, complemented

C₂×Q₈×C₁₇

C₂²

yes

characteristic

C₂×D₆₈

C₂²

yes

characteristic

C₂×C₄×D₁₇

C₂²

yes

—

C₂×Dic₃₄

C₂²

yes

—

C₄⋊C₄×C₁₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₇

C₂²

yes

—

C₄²×C₁₇

C₂²

yes

—

C₄²×C₁₇

C₂²

yes

characteristic

#63227

C₂²

yes

—

#63227

C₂²

yes

—

#63227

C₂²

yes

—

C₄⋊Dic₁₇

C₂²

yes

characteristic

C₄⋊Dic₁₇

C₂²

yes

—

C₄⋊Dic₁₇

C₂²

yes

—

#63225

C₂²

yes

—

C₄×Dic₁₇

C₂²

yes

—

C₂²×D₁₇

C₂³

yes

—

Q₈×C₁₇

—

C₂×C₄×C₁₇

C₂³

yes

—

C₂×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₄×C₁₇

C₂³

yes

—

C₂×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₄×C₁₇

C₂³

yes

characteristic

C₂×Dic₁₇

C₂³

yes

—

C₂×Dic₁₇

C₂³

yes

—

D₆₈

—

C₄×D₁₇

—

Dic₃₄

—

C₂²×C₁₇

C₂⁴

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#41

yes

—

C₄×C₁₇

#41

yes

—

Dic₁₇

—

Dic₁₇

—

#556

—

maximal, Sylow

C₂×C₁₇

C₂×#41

yes

characteristic

C₂×C₁₇

C₂×#41

yes

characteristic

C₂×C₁₇

2₋¹⁺⁴

yes

characteristic

D₁₇

—

C₄⋊Q₈

—

#126

—

#124

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

D₁₇

yes

characteristic, complemented

C₄×D₄

—

#117

—

C₁₇

#556

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

C₂×D₁₇

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₁₇

yes

characteristic

C₄⋊C₄

C₂×D₁₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₁₇

yes

characteristic

C₄²

C₂×D₁₇

yes

—

C₂³

—

Q₈

—

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×D₁₇

yes

—

C₂×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₄

C₂²×D₁₇

yes

—

C₂×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₄

C₂²×D₁₇

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂³×D₁₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

#63252

yes

—

100

C₄

#63256

yes

—

101

C₂

—

102

C₂

C₂×#63252

yes

characteristic

103

C₂

C₂×#63256

yes

characteristic

104

C₂

#10587697

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₇
commutator, socle, cosocle	C₂²×C₁₇	C₂⁴
Fitting	C₄×Q₈×C₁₇	C₂
prime radical	C₂×D₆₈	C₂²
hypercenter	C₄×Q₈	D₁₇
hypocenter	C₁₇	#556
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₇

commutator, socle, cosocle

C₂²×C₁₇

C₂⁴

Fitting

C₄×Q₈×C₁₇

C₂

prime radical

C₂×D₆₈

C₂²

hypercenter

C₄×Q₈

D₁₇

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. C₄×Q₈×C₁₇	C₂	semidirect
2. #10587632	C₂	semidirect
3. #10587584	C₂	semidirect
4. #10587578	C₂	semidirect
7. #10587555	C₂	semidirect
8. C₄×Dic₃₄	C₂	semidirect
58. C₄×Q₈	D₁₇	semidirect
61. C₁₇	#556	semidirect

H ≅ G/N

type

1. C₄×Q₈×C₁₇

C₂

semidirect

2. #10587632

C₂

semidirect

3. #10587584

C₂

semidirect

4. #10587578

C₂

semidirect

7. #10587555

C₂

semidirect

8. C₄×Dic₃₄

C₂

semidirect

58. C₄×Q₈

D₁₇

semidirect

61. C₁₇

#556

semidirect

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

subgroup	index
C₄×Q₈×C₁₇	2
#10587632	2
#10587584	2
#10587578	2
#10587561	2
C₄×D₆₈	2
#10587555	2
C₄×Dic₃₄	2
#556	17

subgroup

index

C₄×Q₈×C₁₇

C₄×D₆₈

C₄×Dic₃₄

Subgroups — #11763532

Conjugacy classes of subgroups

Complements

Maximal subgroups