Group #14544363 — subgroups

Subgroups — D₈×#3624 #14544363

Order 2176.

196 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 74 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	D₄×#3624	C₂	1	2	yes	maximal, complemented
2	#11763672	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄⋊#3624	C₂	1	2	yes	maximal, complemented
4	C₈×#3624	C₂	1	1	yes	maximal, characteristic, complemented
5	D₈×Dic₁₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#3624	—	2	2	no	—
7	C₂×D₈×C₁₇	C₄	1	1	yes	characteristic
8	D₄×Dic₁₇	C₂²	1	2	yes	—
9	C₂²⋊#3624	—	2	2	no	—
10	C₄⋊#3624	C₂²	1	2	yes	—
11	C₄×#3624	C₂²	1	1	yes	characteristic
12	C₄×#3624	C₂²	1	1	yes	characteristic
13	#10587503	C₄	1	2	yes	—
14	#10587489	C₄	1	1	yes	characteristic
15	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₁₇	C₂×C₄	1	2	yes	—
17	C₂²×Dic₁₇	—	2	2	no	—
18	C₂×#3624	—	2	1	no	—
19	C₂×#3624	—	2	2	no	—
20	C₂×#3624	—	4	4	no	—
21	C₂×#3624	D₄	1	1	yes	characteristic
22	C₂×#3624	D₄	1	1	yes	characteristic
23	D₈×C₁₇	C₈	1	4	yes	complemented
24	C₂×C₈×C₁₇	C₂×C₄	1	1	yes	characteristic
25	C₂²⋊Dic₁₇	—	2	2	no	—
26	C₄⋊Dic₁₇	C₂×C₄	1	2	yes	—
27	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
28	C₂×#3622	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₁₇	—	2	2	no	—
30	D₄×C₁₇	—	2	2	no	—
31	D₄×C₁₇	C₂×C₈	1	4	yes	complemented
32	C₂×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₁₇	—	2	2	no	—
34	C₂×Dic₁₇	—	4	2	no	—
35	C₂×Dic₁₇	—	4	2	no	—
36	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₁₇	#41	1	1	yes	characteristic
38	#3624	—	2	2	no	—
39	#3624	—	2	1	no	—
40	#3624	—	4	2	no	—
41	#3624	D₈	1	2	yes	direct product factor
42	C₈×C₁₇	C₂×C₈	1	2	yes	complemented
43	#3622	—	2	1	no	—
44	C₈×D₈	—	17	1	no	maximal, Sylow
45	C₂²×C₁₇	—	2	4	no	—
46	C₂²×C₁₇	—	4	2	no	—
47	C₂²×C₁₇	—	4	2	no	—
48	C₂²×C₁₇	C₄×D₄	1	1	yes	characteristic
49	C₄×C₁₇	C₂²×C₈	1	1	yes	characteristic
50	C₄×C₁₇	#131	1	1	yes	characteristic
51	Dic₁₇	—	2	1	no	—
52	Dic₁₇	—	4	2	no	—
53	Dic₁₇	C₂×D₈	1	1	yes	characteristic
54	Dic₁₇	#135	1	1	yes	characteristic
55	C₄×D₈	—	17	1	no	—
56	C₈×D₄	—	17	2	no	—
57	#335	—	17	1	no	—
58	D₄⋊C₈	—	17	2	no	—
59	C₈²	—	17	1	no	—
60	C₂×C₁₇	—	4	4	no	—
61	C₂×C₁₇	C₈×D₄	1	1	yes	characteristic
62	C₂×C₁₇	C₄×D₈	1	1	yes	characteristic
63	C₂×C₁₇	#443	1	1	yes	characteristic
64	C₂×D₈	#595	1	1	yes	characteristic
65	C₂²×C₈	—	34	2	no	—
66	C₄×D₄	—	17	2	no	—
67	#107	—	17	1	no	—
68	C₄⋊C₈	—	17	2	no	—
69	#102	—	17	2	no	—
70	C₂²⋊C₈	—	34	2	no	—
71	C₄×C₈	—	17	1	no	—
72	C₄×C₈	—	17	1	no	—
73	C₄×C₈	—	17	1	no	—
74	C₁₇	C₈×D₈	1	1	yes	characteristic, complemented, Sylow
75	C₂×D₄	C₂×#595	1	2	yes	—
76	C₂²×C₄	—	34	2	no	—
77	D₈	#3624	1	4	yes	direct product factor
78	C₂×C₈	—	17	1	no	—
79	C₂×C₈	—	17	1	no	—
80	C₂×C₈	—	17	1	no	—
81	C₂×C₈	—	34	1	no	—
82	C₂×C₈	—	34	2	no	—
83	C₂×C₈	—	68	4	no	—
84	C₂×C₈	C₂×#595	1	1	yes	characteristic
85	C₄⋊C₄	—	17	2	no	—
86	C₂²⋊C₄	—	34	2	no	—
87	C₄²	—	17	1	no	—
88	C₂³	—	2	2	no	—
89	D₄	—	2	2	no	—
90	D₄	C₂×#3624	1	4	yes	complemented
91	C₂×C₄	—	17	1	no	—
92	C₂×C₄	—	17	1	no	—
93	C₂×C₄	—	34	2	no	—
94	C₂×C₄	—	68	2	no	—
95	C₂×C₄	—	68	2	no	—
96	C₂×C₄	C₂²×#595	1	1	yes	characteristic
97	C₈	—	17	2	no	—
98	C₈	—	34	1	no	—
99	C₈	—	34	2	no	—
100	C₈	—	34	1	no	—
101	C₈	—	68	2	no	—
102	C₈	C₂×#3624	1	2	yes	complemented
103	C₂²	—	2	4	no	—
104	C₂²	—	4	2	no	—
105	C₂²	—	4	2	no	—
106	C₂²	D₄×#595	1	1	yes	characteristic
107	C₄	—	17	1	no	—
108	C₄	—	17	1	no	—
109	C₄	—	34	1	no	—
110	C₄	—	68	2	no	—
111	C₄	#10587678	1	1	yes	characteristic
112	C₄	C₂²×#3624	1	1	yes	characteristic
113	C₂	—	4	4	no	—
114	C₂	D₄×#3624	1	1	yes	characteristic
115	C₂	D₈×#595	1	1	yes	characteristic
116	C₂	#11763851	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×#3624

C₂

yes

maximal, complemented

#11763672

C₂

yes

maximal, characteristic, complemented

D₄⋊#3624

C₂

yes

maximal, complemented

C₈×#3624

C₂

yes

maximal, characteristic, complemented

D₈×Dic₁₇

C₂

yes

maximal, characteristic

C₂²×#3624

—

C₂×D₈×C₁₇

C₄

yes

characteristic

D₄×Dic₁₇

C₂²

yes

—

C₂²⋊#3624

—

C₄⋊#3624

C₂²

yes

—

C₄×#3624

C₂²

yes

characteristic

C₄×#3624

C₂²

yes

characteristic

#10587503

C₄

yes

—

#10587489

C₄

yes

characteristic

C₈×Dic₁₇

C₂²

yes

characteristic

C₂×D₄×C₁₇

C₂×C₄

yes

—

C₂²×Dic₁₇

—

—

—

—

D₄

yes

characteristic

C₂×#3624

D₄

yes

characteristic

D₈×C₁₇

C₈

yes

complemented

C₂×C₈×C₁₇

C₂×C₄

yes

characteristic

C₂²⋊Dic₁₇

—

C₄⋊Dic₁₇

C₂×C₄

yes

—

C₄×Dic₁₇

C₂³

yes

characteristic

C₂×#3622

C₂×C₄

yes

characteristic

C₂³×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

C₂×C₈

yes

complemented

C₂×C₄×C₁₇

C₂²×C₄

yes

characteristic

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

—

C₂×Dic₁₇

C₂×D₄

yes

C₂×Dic₁₇

yes

—

—

—

D₈

yes

direct product factor

C₈×C₁₇

C₂×C₈

yes

complemented

#3622

—

C₈×D₈

—

maximal, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₄×D₄

yes

characteristic

C₄×C₁₇

C₂²×C₈

yes

characteristic

C₄×C₁₇

#131

yes

characteristic

Dic₁₇

—

Dic₁₇

—

Dic₁₇

C₂×D₈

yes

characteristic

Dic₁₇

#135

yes

characteristic

C₄×D₈

—

C₈×D₄

—

#335

—

D₄⋊C₈

—

C₈²

—

C₂×C₁₇

—

C₂×C₁₇

C₈×D₄

yes

characteristic

C₂×C₁₇

C₄×D₈

yes

C₂×C₁₇

yes

C₂×D₈

yes

C₂²×C₈

—

C₄×D₄

—

#107

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₁₇

C₈×D₈

yes

characteristic, complemented, Sylow

C₂×D₄

C₂×#595

yes

—

C₂²×C₄

—

D₈

#3624

yes

direct product factor

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#595

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₂³

—

D₄

—

D₄

C₂×#3624

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×#595

yes

characteristic

C₈

—

C₈

—

C₈

—

100

C₈

—

101

C₈

—

102

C₈

C₂×#3624

yes

complemented

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

D₄×#595

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#10587678

yes

characteristic

112

C₄

C₂²×#3624

yes

characteristic

113

C₂

—

114

C₂

D₄×#3624

yes

characteristic

115

C₂

D₈×#595

yes

characteristic

116

C₂

#11763851

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×#595
commutator	C₄×C₁₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#595
Fitting, prime radical	C₂×D₈×C₁₇	C₄
socle	C₂²×C₁₇	C₄×D₄
cosocle	C₄×Dic₁₇	C₂³
hypercenter	C₂×D₈	#595
hypocenter	C₁₇	C₈×D₈
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

subgroup

G/H

center

C₂²

D₄×#595

commutator

C₄×C₁₇

C₂²×C₈

Frattini

C₂×C₄

C₂²×#595

Fitting, prime radical

C₂×D₈×C₁₇

C₄

socle

C₂²×C₁₇

C₄×D₄

cosocle

C₄×Dic₁₇

C₂³

hypercenter

C₂×D₈

#595

hypocenter

C₁₇

C₈×D₈

LCS

G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

N	H ≅ G/N	type
1. D₄×#3624	C₂	semidirect
2. #11763672	C₂	semidirect
3. D₄⋊#3624	C₂	semidirect
4. C₈×#3624	C₂	semidirect
23. D₈×C₁₇	C₈	semidirect
31. D₄×C₁₇	C₂×C₈	semidirect
41. #3624	D₈	direct (N × H)
42. C₈×C₁₇	C₂×C₈	semidirect
74. C₁₇	C₈×D₈	semidirect
77. D₈	#3624	direct (N × H)
90. D₄	C₂×#3624	semidirect
102. C₈	C₂×#3624	semidirect

H ≅ G/N

type

1. D₄×#3624

C₂

semidirect

2. #11763672

C₂

semidirect

3. D₄⋊#3624

C₂

semidirect

4. C₈×#3624

C₂

semidirect

23. D₈×C₁₇

C₈

semidirect

31. D₄×C₁₇

C₂×C₈

semidirect

41. #3624

D₈

direct (N × H)

42. C₈×C₁₇

C₂×C₈

semidirect

74. C₁₇

C₈×D₈

semidirect

77. D₈

#3624

direct (N × H)

90. D₄

C₂×#3624

semidirect

102. C₈

C₂×#3624

semidirect

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

subgroup	index
D₄×#3624	2
#11763672	2
D₄⋊#3624	2
C₈×#3624	2
D₈×Dic₁₇	2
C₈×D₈	17

subgroup

index

D₈×Dic₁₇

C₈×D₈

Subgroups — D₈×#3624 #14544363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D8×#3624 #14544363

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₈×#3624 #14544363