Group #14544318 — subgroups

Subgroups — #14544318

Order 2176.

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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	#11763832	C₂	1	2	yes	maximal
2	#11763669	C₂	1	4	yes	maximal, complemented
3	C₄⋊Q₈×D₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈×D₁₇	C₄	1	2	yes	—
5	C₁₇×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	#10587630	C₂²	1	2	yes	—
7	C₄⋊C₄×D₁₇	—	2	2	no	—
8	#10587574	—	2	2	no	—
9	C₄²×D₁₇	C₂²	1	1	yes	characteristic
10	#10587555	C₄	1	1	yes	characteristic
11	#10587544	C₂²	1	4	yes	—
12	#10587543	—	2	2	no	—
13	#10587542	—	2	2	no	—
14	C₂×Q₈×C₁₇	C₂×C₄	1	2	yes	—
15	Q₈×D₁₇	—	4	4	no	—
16	C₂×C₄×D₁₇	—	2	2	no	—
17	C₂×C₄×D₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×D₁₇	D₄	1	2	yes	—
19	C₂×Dic₃₄	—	2	2	no	—
20	C₂×Dic₃₄	C₂×C₄	1	2	yes	—
21	C₂×#3624	—	2	4	no	—
22	C₄⋊C₄×C₁₇	—	2	2	no	—
23	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
24	C₄⋊Dic₁₇	—	2	2	no	—
25	#63225	—	2	4	no	—
26	C₄×Dic₁₇	D₄	1	2	yes	—
27	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
28	C₂²×D₁₇	C₂²⋊C₄	1	1	yes	characteristic
29	Q₈×C₁₇	—	4	2	no	—
30	C₂×C₄×C₁₇	—	2	2	no	—
31	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₁₇	—	2	2	no	—
34	C₂×Dic₁₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₁₇	C₂×D₄	1	2	yes	—
36	C₄×D₁₇	—	2	2	no	—
37	C₄×D₁₇	—	2	4	no	—
38	C₄×D₁₇	—	4	4	no	—
39	Dic₃₄	—	4	4	no	—
40	Dic₃₄	—	4	2	no	—
41	#3624	—	4	4	no	—
42	#1558	—	17	1	no	maximal, Sylow
43	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
44	C₂×D₁₇	—	2	2	no	—
45	C₂×D₁₇	#100	1	2	yes	—
46	C₄×C₁₇	—	2	1	no	—
47	C₄×C₁₇	—	2	2	no	—
48	C₄×C₁₇	—	4	2	no	—
49	Dic₁₇	—	2	1	no	—
50	Dic₁₇	—	4	2	no	—
51	Dic₁₇	#137	1	4	yes	—
52	C₂×C₄⋊Q₈	—	17	1	no	—
53	#432	—	17	2	no	—
54	#333	—	17	4	no	—
55	C₂×C₁₇	C₂×#100	1	1	yes	characteristic
56	C₂×C₁₇	#419	1	2	yes	—
57	D₁₇	—	2	2	no	—
58	C₂²×Q₈	—	17	2	no	—
59	C₄⋊Q₈	—	17	2	no	—
60	C₄⋊Q₈	—	17	1	no	—
61	C₄⋊Q₈	—	34	2	no	—
62	C₄⋊Q₈	#595	1	1	yes	characteristic
63	C₂×C₄⋊C₄	—	34	2	no	—
64	C₂×C₄²	—	17	1	no	—
65	C₄⋊C₈	—	17	4	no	—
66	C₂²⋊C₈	—	34	2	no	—
67	#97	—	34	2	no	—
68	C₁₇	#1558	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₈	—	17	2	no	—
70	C₂×Q₈	—	34	2	no	—
71	C₂×Q₈	—	68	4	no	—
72	C₂×Q₈	C₂×#595	1	2	yes	—
73	C₂²×C₄	—	17	2	no	—
74	C₂²×C₄	—	17	1	no	—
75	C₂²×C₄	—	34	2	no	—
76	C₂×C₈	—	34	4	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	34	4	no	—
79	C₄⋊C₄	—	34	2	no	—
80	C₄²	—	17	2	no	—
81	C₄²	—	17	1	no	—
82	C₄²	C₂×#595	1	1	yes	characteristic
83	C₂³	—	17	1	no	—
84	Q₈	—	4	2	no	—
85	Q₈	—	68	2	no	—
86	Q₈	—	68	4	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	17	2	no	—
89	C₂×C₄	—	17	1	no	—
90	C₂×C₄	—	34	4	no	—
91	C₂×C₄	—	34	2	no	—
92	C₂×C₄	—	34	2	no	—
93	C₂×C₄	—	68	4	no	—
94	C₂×C₄	C₂²⋊#595	1	2	yes	—
95	C₂×C₄	C₂²×#595	1	1	yes	characteristic
96	C₈	—	68	4	no	—
97	C₂²	—	17	2	no	—
98	C₂²	—	34	2	no	—
99	C₂²	C₂×C₂²⋊#595	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	2	no	—
102	C₄	—	4	2	no	—
103	C₄	—	17	4	no	—
104	C₄	—	34	1	no	—
105	C₄	—	68	2	no	—
106	C₂	—	34	2	no	—
107	C₂	D₁₇⋊#100	1	1	yes	characteristic
108	C₂	#11763802	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#11763832

C₂

yes

maximal

#11763669

C₂

yes

maximal, complemented

C₄⋊Q₈×D₁₇

C₂

yes

maximal, characteristic

C₂×Q₈×D₁₇

C₄

yes

—

C₁₇×C₄⋊Q₈

C₄

yes

characteristic

#10587630

C₂²

yes

—

C₄⋊C₄×D₁₇

—

#10587574

—

C₄²×D₁₇

C₂²

yes

characteristic

#10587555

C₄

yes

characteristic

#10587544

C₂²

yes

—

#10587543

—

#10587542

—

C₂×Q₈×C₁₇

C₂×C₄

yes

—

Q₈×D₁₇

—

C₂×C₄×D₁₇

—

C₂×C₄×D₁₇

C₂×C₄

yes

characteristic

C₂×C₄×D₁₇

D₄

yes

—

C₂×Dic₃₄

—

C₂×Dic₃₄

C₂×C₄

yes

—

C₂×#3624

—

C₄⋊C₄×C₁₇

—

C₄²×C₁₇

C₂×C₄

yes

characteristic

C₄⋊Dic₁₇

—

#63225

—

C₄×Dic₁₇

D₄

yes

—

C₄×Dic₁₇

C₂³

yes

characteristic

C₂²×D₁₇

C₂²⋊C₄

yes

characteristic

Q₈×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₇

C₂×D₄

yes

—

C₄×D₁₇

—

C₄×D₁₇

—

C₄×D₁₇

—

Dic₃₄

—

Dic₃₄

—

#3624

—

#1558

—

maximal, Sylow

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₇

—

C₂×D₁₇

#100

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

Dic₁₇

—

Dic₁₇

—

Dic₁₇

#137

yes

—

C₂×C₄⋊Q₈

—

#432

—

#333

—

C₂×C₁₇

C₂×#100

yes

characteristic

C₂×C₁₇

#419

yes

—

D₁₇

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#595

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₁₇

#1558

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×#595

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×#595

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊#595

yes

—

C₂×C₄

C₂²×#595

yes

characteristic

C₈

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊#595

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₂

—

107

C₂

D₁₇⋊#100

yes

characteristic

108

C₂

#11763802

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#595
commutator	C₂×C₄×C₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#595
Fitting	C₁₇×C₄⋊Q₈	C₄
socle	C₂²×C₁₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₇	C₂²⋊C₄
cosocle	C₄×Dic₁₇	C₂³
hypercenter	C₄⋊Q₈	#595
hypocenter	C₁₇	#1558
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#595

commutator

C₂×C₄×C₁₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#595

Fitting

C₁₇×C₄⋊Q₈

C₄

socle

C₂²×C₁₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₇

C₂²⋊C₄

cosocle

C₄×Dic₁₇

C₂³

C₄⋊Q₈

C₁₇

G ▷ C₂×C₄×C₁₇ ▷ C₂²×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. #11763669	C₂	semidirect
68. C₁₇	#1558	semidirect

H ≅ G/N

type

2. #11763669

C₂

semidirect

68. C₁₇

#1558

semidirect

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

subgroup	index
#11763832	2
#11763669	2
C₄⋊Q₈×D₁₇	2
#1558	17

subgroup

index

#11763832

#11763669

C₄⋊Q₈×D₁₇

#1558

Subgroups — #14544318

Conjugacy classes of subgroups

Complements

Maximal subgroups