Group #11763524 — subgroups

Subgroups — #11763524

Order 1088.

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

normal subgroups only

This group has 236 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×D₄×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#10587624	C₂	1	2	yes	maximal, complemented
3	#10587616	C₂	1	2	yes	maximal, complemented
4	#10587612	C₂	1	2	yes	maximal, complemented
5	#10587578	C₂	1	2	yes	maximal, complemented
6	#10587571	C₂	1	4	yes	maximal, complemented
7	C₄×D₆₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#10587555	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₄×C₁₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₁₇	C₂²	1	2	yes	—
11	C₂×C₂²⋊D₁₇	C₂²	1	4	yes	—
12	C₂×D₆₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₇	C₂²	1	2	yes	—
14	C₂×Dic₃₄	C₂²	1	2	yes	—
15	C₄⋊C₄×C₁₇	C₂²	1	1	yes	characteristic
16	C₂²⋊C₄×C₁₇	C₂²	1	2	yes	—
17	C₄²×C₁₇	C₂²	1	1	yes	characteristic
18	C₂²⋊Dic₁₇	C₂²	1	4	yes	—
19	#63227	C₂²	1	2	yes	—
20	#63227	C₂²	1	4	yes	—
21	C₄⋊Dic₁₇	C₂²	1	2	yes	—
22	C₄⋊Dic₁₇	C₂²	1	2	yes	—
23	C₄⋊Dic₁₇	C₂²	1	1	yes	characteristic
24	C₄×Dic₁₇	C₂²	1	4	yes	—
25	C₂³×C₁₇	C₂³	1	2	yes	—
26	C₂²×D₁₇	C₂³	1	2	yes	—
27	D₄×C₁₇	—	2	2	no	—
28	C₂×C₄×C₁₇	—	2	4	no	—
29	C₂×C₄×C₁₇	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₁₇	C₂³	1	2	yes	—
31	C₂×C₄×C₁₇	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₁₇	C₂³	1	1	yes	characteristic
33	C₂²⋊D₁₇	—	4	4	no	—
34	C₂×Dic₁₇	C₂³	1	2	yes	—
35	C₂×Dic₁₇	C₂³	1	4	yes	—
36	D₆₈	—	2	2	no	—
37	C₄×D₁₇	—	2	4	no	—
38	Dic₃₄	—	4	2	no	—
39	C₂²×C₁₇	—	2	2	no	—
40	C₂²×C₁₇	—	2	2	no	—
41	C₂²×C₁₇	—	2	2	no	—
42	C₂²×C₁₇	C₂⁴	1	1	yes	characteristic
43	C₂×D₁₇	—	2	2	no	—
44	C₂×D₁₇	—	2	2	no	—
45	C₂×D₁₇	—	2	2	no	—
46	C₄×C₁₇	—	2	2	no	—
47	C₄×C₁₇	—	2	1	no	—
48	C₄×C₁₇	#41	1	2	yes	—
49	C₄×C₁₇	#41	1	2	yes	—
50	Dic₁₇	—	2	2	no	—
51	Dic₁₇	—	2	4	no	—
52	#555	—	17	1	no	maximal, Sylow
53	C₂×C₁₇	—	4	2	no	—
54	C₂×C₁₇	C₂×#41	1	1	yes	characteristic
55	C₂×C₁₇	C₂×#41	1	1	yes	characteristic
56	C₂×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
57	D₁₇	—	4	2	no	—
58	C₄⋊Q₈	—	17	1	no	—
59	#124	—	17	4	no	—
60	#121	—	17	2	no	—
61	#121	—	17	2	no	—
62	C₄×D₄	—	17	1	no	—
63	C₄×D₄	D₁₇	1	1	yes	characteristic, complemented
64	#117	—	17	2	no	—
65	#117	—	17	2	no	—
66	C₁₇	#555	1	1	yes	characteristic, complemented, Sylow
67	C₂×Q₈	—	17	2	no	—
68	C₂×D₄	—	17	4	no	—
69	C₂×D₄	—	17	1	no	—
70	C₂×D₄	C₂×D₁₇	1	1	yes	characteristic
71	C₂²×C₄	—	17	2	no	—
72	C₂²×C₄	C₂×D₁₇	1	2	yes	—
73	C₄⋊C₄	—	17	1	no	—
74	C₄⋊C₄	—	17	2	no	—
75	C₄⋊C₄	—	17	2	no	—
76	C₄⋊C₄	C₂×D₁₇	1	1	yes	characteristic
77	C₂²⋊C₄	—	17	4	no	—
78	C₂²⋊C₄	—	17	4	no	—
79	C₂²⋊C₄	—	17	2	no	—
80	C₂²⋊C₄	C₂×D₁₇	1	2	yes	—
81	C₄²	—	17	4	no	—
82	C₄²	C₂×D₁₇	1	1	yes	characteristic
83	C₂³	—	17	2	no	—
84	C₂³	C₂²×D₁₇	1	2	yes	—
85	Q₈	—	68	2	no	—
86	D₄	—	2	2	no	—
87	D₄	—	34	2	no	—
88	D₄	—	68	4	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	17	2	no	—
91	C₂×C₄	—	17	4	no	—
92	C₂×C₄	—	34	4	no	—
93	C₂×C₄	C₂²×D₁₇	1	2	yes	—
94	C₂×C₄	C₂²×D₁₇	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₁₇	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₁₇	1	1	yes	characteristic
97	C₂²	—	2	2	no	—
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	34	2	no	—
101	C₂²	—	34	2	no	—
102	C₂²	—	34	2	no	—
103	C₂²	C₂³×D₁₇	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	34	2	no	—
107	C₄	—	34	4	no	—
108	C₄	#63252	1	2	yes	—
109	C₄	#63254	1	2	yes	—
110	C₂	—	4	2	no	—
111	C₂	—	68	2	no	—
112	C₂	C₂×#63252	1	1	yes	characteristic
113	C₂	C₂×#63254	1	1	yes	characteristic
114	C₂	#10587696	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₄×C₁₇

C₂

yes

maximal, characteristic, complemented

#10587624

C₂

yes

maximal, complemented

#10587616

C₂

yes

maximal, complemented

#10587612

C₂

yes

maximal, complemented

#10587578

C₂

yes

maximal, complemented

#10587571

C₂

yes

maximal, complemented

C₄×D₆₈

C₂

yes

maximal, characteristic, complemented

#10587555

C₂

yes

maximal, characteristic, complemented

C₂×D₄×C₁₇

C₂²

yes

characteristic

C₂²×C₄×C₁₇

C₂²

yes

—

C₂×C₂²⋊D₁₇

C₂²

yes

—

C₂×D₆₈

C₂²

yes

characteristic

C₂×C₄×D₁₇

C₂²

yes

—

C₂×Dic₃₄

C₂²

yes

—

C₄⋊C₄×C₁₇

C₂²

yes

characteristic

C₂²⋊C₄×C₁₇

C₂²

yes

—

C₄²×C₁₇

C₂²

yes

characteristic

C₂²⋊Dic₁₇

C₂²

yes

—

#63227

C₂²

yes

—

#63227

C₂²

yes

—

C₄⋊Dic₁₇

C₂²

yes

—

C₄⋊Dic₁₇

C₂²

yes

—

C₄⋊Dic₁₇

C₂²

yes

characteristic

C₄×Dic₁₇

C₂²

yes

—

C₂³×C₁₇

C₂³

yes

—

C₂²×D₁₇

C₂³

yes

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₄×C₁₇

C₂³

yes

—

C₂×C₄×C₁₇

C₂³

yes

characteristic

C₂×C₄×C₁₇

C₂³

yes

characteristic

C₂²⋊D₁₇

—

C₂×Dic₁₇

C₂³

yes

—

C₂×Dic₁₇

C₂³

yes

—

D₆₈

—

C₄×D₁₇

—

Dic₃₄

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂⁴

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

—

C₂×D₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#41

yes

—

C₄×C₁₇

#41

yes

—

Dic₁₇

—

Dic₁₇

—

#555

—

maximal, Sylow

C₂×C₁₇

—

C₂×C₁₇

C₂×#41

yes

characteristic

C₂×C₁₇

C₂×#41

yes

characteristic

C₂×C₁₇

H(5,𝔽₂)

yes

characteristic

D₁₇

—

C₄⋊Q₈

—

#124

—

#121

—

#121

—

C₄×D₄

—

C₄×D₄

D₁₇

yes

characteristic, complemented

#117

—

#117

—

C₁₇

#555

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₁₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₁₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₁₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₁₇

yes

—

C₄²

—

C₄²

C₂×D₁₇

yes

characteristic

C₂³

—

C₂³

C₂²×D₁₇

yes

—

Q₈

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×D₁₇

yes

—

C₂×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₄

C₂²×D₁₇

yes

characteristic

C₂×C₄

C₂²×D₁₇

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₂³×D₁₇

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

#63252

yes

—

109

C₄

#63254

yes

—

110

C₂

—

111

C₂

—

112

C₂

C₂×#63252

yes

characteristic

113

C₂

C₂×#63254

yes

characteristic

114

C₂

#10587696

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₇
commutator, socle, cosocle	C₂²×C₁₇	C₂⁴
Fitting	C₄×D₄×C₁₇	C₂
hypercenter	C₄×D₄	D₁₇
hypocenter	C₁₇	#555
LCS	G ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₇

commutator, socle, cosocle

C₂²×C₁₇

C₂⁴

Fitting

C₄×D₄×C₁₇

C₂

hypercenter

C₄×D₄

D₁₇

C₁₇

G ▷ C₂²×C₁₇ ▷ C₁₇

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. C₄×D₄×C₁₇	C₂	semidirect
2. #10587624	C₂	semidirect
3. #10587616	C₂	semidirect
4. #10587612	C₂	semidirect
5. #10587578	C₂	semidirect
6. #10587571	C₂	semidirect
7. C₄×D₆₈	C₂	semidirect
8. #10587555	C₂	semidirect
63. C₄×D₄	D₁₇	semidirect
66. C₁₇	#555	semidirect

H ≅ G/N

type

1. C₄×D₄×C₁₇

C₂

semidirect

2. #10587624

C₂

semidirect

3. #10587616

C₂

semidirect

4. #10587612

C₂

semidirect

5. #10587578

C₂

semidirect

6. #10587571

C₂

semidirect

7. C₄×D₆₈

C₂

semidirect

8. #10587555

C₂

semidirect

63. C₄×D₄

D₁₇

semidirect

66. C₁₇

#555

semidirect

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

subgroup	index
C₄×D₄×C₁₇	2
#10587624	2
#10587616	2
#10587612	2
#10587578	2
#10587571	2
C₄×D₆₈	2
#10587555	2
#555	17

subgroup

index

C₄×D₄×C₁₇

C₄×D₆₈

Subgroups — #11763524

Conjugacy classes of subgroups

Complements

Maximal subgroups