Group #14545914 — subgroups

Subgroups — #14545914

Order 2176.

160 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 57 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	Q₈×#3633	C₂	1	2	yes	maximal
2	#11763898	C₂	1	1	yes	maximal, characteristic
3	Q₈⋊#3633	C₂	1	2	yes	maximal
4	#11763893	C₂	1	1	yes	maximal, characteristic
5	Q₁₆×#595	C₂	1	1	yes	maximal, characteristic
6	Q₈×#595	C₂²	1	2	yes	—
7	C₄⋊#3633	—	2	2	no	—
8	C₄⋊#3633	C₂²	1	2	yes	—
9	C₄×#3633	—	2	2	no	—
10	C₄×#3633	C₂²	1	1	yes	characteristic
11	C₄×#3633	C₂²	1	1	yes	characteristic
12	Q₁₆×D₁₇	C₄	1	1	yes	characteristic
13	#10587549	C₄	1	2	yes	—
14	#10587534	C₄	1	1	yes	characteristic
15	C₈×#595	C₂²	1	1	yes	characteristic
16	C₂×#3633	—	2	2	no	—
17	C₂×#3633	—	2	1	no	—
18	C₂×#3633	D₄	1	2	yes	—
19	Q₈×D₁₇	C₂×C₄	1	2	yes	—
20	C₄⋊#595	—	2	2	no	—
21	C₄⋊#595	C₂×C₄	1	2	yes	—
22	C₄×#595	—	2	2	no	—
23	C₄×#595	C₂³	1	1	yes	characteristic
24	#63242	C₂×C₄	1	1	yes	characteristic
25	Q₁₆×C₁₇	C₈	1	1	yes	characteristic, complemented
26	#63231	C₈	1	2	yes	complemented
27	Dic₆₈	C₈	1	1	yes	characteristic, complemented
28	C₈×D₁₇	C₂×C₄	1	1	yes	characteristic
29	C₂×#595	—	2	2	no	—
30	C₂×#595	C₂×D₄	1	1	yes	characteristic
31	C₂×#595	#41	1	1	yes	characteristic
32	#3633	—	2	2	no	—
33	#3633	—	4	1	no	—
34	#3633	—	4	2	no	—
35	Q₈×C₁₇	C₂×C₈	1	2	yes	—
36	C₄×D₁₇	—	2	2	no	—
37	C₄×D₁₇	C₂²×C₄	1	1	yes	characteristic
38	Dic₃₄	—	2	2	no	—
39	Dic₃₄	C₂×C₈	1	2	yes	—
40	#3624	—	2	1	no	—
41	C₈×C₁₇	C₂×C₈	1	1	yes	characteristic
42	#3622	C₂×C₈	1	1	yes	characteristic
43	#1579	—	17	1	no	maximal, Sylow
44	C₂×D₁₇	C₄×D₄	1	1	yes	characteristic
45	#595	—	2	1	no	—
46	#595	—	4	2	no	—
47	#595	#137	1	2	yes	—
48	C₄×C₁₇	—	2	2	no	—
49	C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
50	Dic₁₇	—	2	2	no	—
51	Dic₁₇	#131	1	1	yes	characteristic
52	C₈×Q₈	—	17	2	no	—
53	C₄×Q₁₆	—	17	1	no	—
54	#334	—	17	1	no	—
55	Q₈⋊C₈	—	17	2	no	—
56	#322	—	17	1	no	—
57	C₂×C₁₇	C₈×D₄	1	1	yes	characteristic
58	D₁₇	#441	1	1	yes	characteristic
59	D₁₇	#444	1	1	yes	characteristic
60	C₂×Q₁₆	—	17	1	no	—
61	C₄×Q₈	—	17	2	no	—
62	#107	—	17	1	no	—
63	C₄⋊C₈	—	17	2	no	—
64	C₄⋊C₈	—	34	2	no	—
65	#103	—	17	2	no	—
66	C₄×C₈	—	17	1	no	—
67	C₄×C₈	—	17	1	no	—
68	C₄×C₈	—	17	1	no	—
69	C₄×C₈	—	34	2	no	—
70	C₁₇	#1579	1	1	yes	characteristic, complemented, Sylow
71	C₂×Q₈	—	17	2	no	—
72	Q₁₆	—	17	2	no	—
73	Q₁₆	—	17	1	no	—
74	Q₁₆	#3633	1	1	yes	characteristic, complemented
75	C₂×C₈	—	17	1	no	—
76	C₂×C₈	—	17	1	no	—
77	C₂×C₈	—	17	2	no	—
78	C₂×C₈	—	34	1	no	—
79	C₂×C₈	—	34	2	no	—
80	C₄⋊C₄	—	17	2	no	—
81	C₄⋊C₄	—	34	2	no	—
82	C₄²	—	17	1	no	—
83	C₄²	—	34	2	no	—
84	Q₈	—	17	2	no	—
85	Q₈	—	34	2	no	—
86	Q₈	C₂×#3633	1	2	yes	—
87	C₂×C₄	—	17	1	no	—
88	C₂×C₄	—	17	1	no	—
89	C₂×C₄	—	17	1	no	—
90	C₂×C₄	—	34	2	no	—
91	C₂×C₄	—	34	2	no	—
92	C₈	—	17	1	no	—
93	C₈	—	34	1	no	—
94	C₈	—	34	2	no	—
95	C₈	—	68	2	no	—
96	C₈	—	68	1	no	—
97	C₈	C₂×#3633	1	1	yes	characteristic
98	C₂²	—	17	1	no	—
99	C₄	—	2	2	no	—
100	C₄	—	17	2	no	—
101	C₄	—	17	1	no	—
102	C₄	—	34	1	no	—
103	C₄	—	34	2	no	—
104	C₄	—	68	2	no	—
105	C₄	C₂²×#3633	1	1	yes	characteristic
106	C₂	—	17	1	no	—
107	C₂	—	17	1	no	—
108	C₂	D₄×#3633	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#3633

C₂

yes

maximal

#11763898

C₂

yes

maximal, characteristic

Q₈⋊#3633

C₂

yes

maximal

#11763893

C₂

yes

maximal, characteristic

Q₁₆×#595

C₂

yes

maximal, characteristic

Q₈×#595

C₂²

yes

—

C₄⋊#3633

—

C₄⋊#3633

C₂²

yes

—

C₄×#3633

—

C₄×#3633

C₂²

yes

characteristic

C₄×#3633

C₂²

yes

characteristic

Q₁₆×D₁₇

C₄

yes

characteristic

#10587549

C₄

yes

—

#10587534

C₄

yes

characteristic

C₈×#595

C₂²

yes

—

—

D₄

yes

—

Q₈×D₁₇

C₂×C₄

yes

—

C₄⋊#595

—

C₄⋊#595

C₂×C₄

yes

—

C₄×#595

—

C₄×#595

C₂³

yes

characteristic

#63242

C₂×C₄

yes

characteristic

Q₁₆×C₁₇

C₈

yes

characteristic, complemented

#63231

C₈

yes

complemented

Dic₆₈

C₈

yes

characteristic, complemented

C₈×D₁₇

C₂×C₄

yes

characteristic

C₂×#595

—

C₂×#595

C₂×D₄

yes

yes

—

—

—

Q₈×C₁₇

C₂×C₈

yes

—

C₄×D₁₇

—

C₄×D₁₇

C₂²×C₄

yes

characteristic

Dic₃₄

—

Dic₃₄

C₂×C₈

yes

—

#3624

—

C₈×C₁₇

C₂×C₈

yes

characteristic

#3622

C₂×C₈

yes

characteristic

#1579

—

maximal, Sylow

C₂×D₁₇

C₄×D₄

yes

—

—

yes

—

C₄×C₁₇

—

C₄×C₁₇

C₂²×C₈

yes

characteristic

Dic₁₇

—

Dic₁₇

#131

yes

characteristic

C₈×Q₈

—

C₄×Q₁₆

—

#334

—

Q₈⋊C₈

—

#322

—

C₂×C₁₇

C₈×D₄

yes

D₁₇

yes

D₁₇

yes

C₂×Q₁₆

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₁₇

#1579

yes

characteristic, complemented, Sylow

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

#3633

yes

characteristic, complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

Q₈

—

Q₈

—

Q₈

C₂×#3633

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

C₈

C₂×#3633

yes

characteristic

C₂²

—

C₄

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₂²×#3633

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

D₄×#3633

yes

characteristic

	subgroup	G/H
center	C₂	D₄×#3633
commutator	C₄×C₁₇	C₂²×C₈
Frattini	C₄	C₂²×#3633
Fitting	Q₁₆×C₁₇	C₈
socle	C₂×C₁₇	C₈×D₄
prime radical	C₂×D₁₇	C₄×D₄
cosocle	C₄×#595	C₂³
hypercenter	Q₁₆	#3633
hypocenter	C₁₇	#1579
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

subgroup

G/H

center

C₂

D₄×#3633

commutator

C₄×C₁₇

C₂²×C₈

Frattini

C₄

C₂²×#3633

Fitting

Q₁₆×C₁₇

C₈

socle

C₂×C₁₇

C₈×D₄

prime radical

C₂×D₁₇

C₄×D₄

C₂³

Q₁₆

C₁₇

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

N	H ≅ G/N	type
25. Q₁₆×C₁₇	C₈	semidirect
26. #63231	C₈	semidirect
27. Dic₆₈	C₈	semidirect
70. C₁₇	#1579	semidirect
74. Q₁₆	#3633	semidirect

H ≅ G/N

type

25. Q₁₆×C₁₇

C₈

semidirect

26. #63231

C₈

semidirect

27. Dic₆₈

C₈

semidirect

70. C₁₇

#1579

semidirect

74. Q₁₆

#3633

semidirect

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

subgroup	index
Q₈×#3633	2
#11763898	2
Q₈⋊#3633	2
#11763893	2
Q₁₆×#595	2
#1579	17

subgroup

index

Q₁₆×#595

Subgroups — #14545914

Conjugacy classes of subgroups

Complements

Maximal subgroups