Group #20256242 — subgroups

Subgroups — Q₈×D₃₇₆ #20256242

Order 6016.

404 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 149 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×D₁₈₈	C₂	1	2	yes	maximal, complemented
2	#15689528	C₂	1	3	yes	maximal, complemented
3	#15689404	C₂	1	6	yes	maximal, complemented
4	C₄×D₃₇₆	C₂	1	3	yes	maximal
5	C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×D₄₇	—	2	2	no	—
7	C₂×D₃₇₆	C₂²	1	1	yes	characteristic
8	#13094024	C₂²	1	6	yes	—
9	#13094004	—	2	6	no	—
10	C₄×D₁₈₈	C₂²	1	6	yes	—
11	#13093958	C₂²	1	3	yes	—
12	#13093958	C₂²	1	6	yes	—
13	#13093946	C₂²	1	6	yes	—
14	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₄₇	C₂²	1	3	yes	—
16	C₄×C₈×C₄₇	C₂²	1	3	yes	—
17	C₂×Q₈×C₄₇	D₄	1	1	yes	characteristic
18	Q₈×D₄₇	—	4	8	no	—
19	C₂×D₁₈₈	C₂³	1	2	yes	—
20	C₂×C₄×D₄₇	—	2	6	no	—
21	C₂×Dic₉₄	—	2	6	no	—
22	C₂×C₈×C₄₇	C₂³	1	3	yes	—
23	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₄₇	D₄	1	3	yes	—
25	C₄²×C₄₇	C₂³	1	3	yes	—
26	(C₂×D₄₇)⋊C₄	—	2	6	no	—
27	C₄⋊Dic₄₇	—	2	6	no	—
28	C₄⋊Dic₄₇	C₂³	1	6	yes	—
29	D₃₇₆	Q₈	1	4	yes	direct product factor
30	C₂²×D₄₇	—	2	2	no	—
31	Q₈×C₄₇	D₈	1	4	yes	complemented
32	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
33	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
34	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₄₇	—	2	6	no	—
36	D₁₈₈	—	2	2	no	—
37	D₁₈₈	C₂×Q₈	1	4	yes	complemented
38	C₄×D₄₇	—	4	12	no	—
39	Dic₉₄	—	4	12	no	—
40	C₈×C₄₇	—	2	3	no	—
41	C₈×C₄₇	C₂×Q₈	1	2	yes	complemented
42	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
43	C₂×D₄₇	—	2	4	no	—
44	C₂×D₄₇	—	4	2	no	—
45	C₂×D₄₇	—	4	2	no	—
46	C₄×C₄₇	—	2	3	no	—
47	C₄×C₄₇	C₂×D₈	1	6	yes	—
48	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
49	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
50	Dic₄₇	—	4	6	no	—
51	Q₈×D₈	—	47	1	no	maximal, Sylow
52	C₂×C₄₇	D₄×Q₈	1	1	yes	characteristic
53	C₂×C₄₇	C₂²×D₈	1	1	yes	characteristic
54	C₂×C₄₇	#578	1	1	yes	characteristic
55	D₄₇	—	4	4	no	—
56	D₄×Q₈	—	47	2	no	—
57	#500	—	47	3	no	—
58	#474	—	47	6	no	—
59	C₈×Q₈	D₄₇	1	1	yes	characteristic, complemented
60	C₄×D₈	—	47	3	no	—
61	C₄₇	Q₈×D₈	1	1	yes	characteristic, complemented, Sylow
62	C₂²×Q₈	—	94	2	no	—
63	C₂×D₈	—	47	1	no	—
64	C₄⋊Q₈	—	47	6	no	—
65	C₂²⋊Q₈	—	94	6	no	—
66	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
67	C₄×D₄	—	47	6	no	—
68	#107	—	47	3	no	—
69	#107	—	47	6	no	—
70	C₄⋊C₈	C₂×D₄₇	1	3	yes	—
71	#102	—	47	6	no	—
72	C₄×C₈	C₂×D₄₇	1	3	yes	—
73	C₂×Q₈	—	94	6	no	—
74	C₂×Q₈	—	188	8	no	—
75	C₂×Q₈	D₁₈₈	1	1	yes	characteristic
76	C₂×D₄	—	47	2	no	—
77	C₂²×C₄	—	94	6	no	—
78	D₈	—	47	4	no	—
79	C₂×C₈	C₂²×D₄₇	1	3	yes	—
80	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
81	C₄⋊C₄	—	47	6	no	—
82	C₄⋊C₄	—	94	6	no	—
83	C₄⋊C₄	D₁₈₈	1	3	yes	—
84	C₂²⋊C₄	—	94	6	no	—
85	C₄²	C₂²×D₄₇	1	3	yes	—
86	C₂³	—	94	2	no	—
87	Q₈	—	188	12	no	—
88	Q₈	D₃₇₆	1	4	yes	direct product factor
89	D₄	—	47	4	no	—
90	D₄	—	94	2	no	—
91	C₂×C₄	—	94	6	no	—
92	C₂×C₄	—	188	12	no	—
93	C₂×C₄	C₂×D₁₈₈	1	3	yes	—
94	C₂×C₄	C₂×D₁₈₈	1	3	yes	—
95	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
96	C₈	—	2	3	no	—
97	C₈	Q₈×D₄₇	1	2	yes	complemented
98	C₂²	—	94	4	no	—
99	C₂²	—	188	2	no	—
100	C₂²	—	188	2	no	—
101	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
102	C₄	—	2	3	no	—
103	C₄	—	188	6	no	—
104	C₄	C₂×D₃₇₆	1	6	yes	—
105	C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
106	C₄	#13094071	1	1	yes	characteristic
107	C₂	—	188	4	no	—
108	C₂	Q₈×D₁₈₈	1	1	yes	characteristic
109	C₂	C₂²×D₃₇₆	1	1	yes	characteristic
110	C₂	#15689940	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₁₈₈

C₂

yes

maximal, complemented

#15689528

C₂

yes

maximal, complemented

#15689404

C₂

yes

maximal, complemented

C₄×D₃₇₆

C₂

yes

maximal

C₈×Q₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₄₇

—

C₂×D₃₇₆

C₂²

yes

characteristic

#13094024

C₂²

yes

—

#13094004

—

C₄×D₁₈₈

C₂²

yes

—

#13093958

C₂²

yes

—

#13093958

C₂²

yes

—

#13093946

C₂²

yes

—

C₄×Q₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

—

C₂×Q₈×C₄₇

D₄

yes

characteristic

Q₈×D₄₇

—

C₂×D₁₈₈

C₂³

yes

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

C₂³

yes

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

D₃₇₆

Q₈

yes

direct product factor

C₂²×D₄₇

—

Q₈×C₄₇

D₈

yes

complemented

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

C₂×Q₈

yes

complemented

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₂×Q₈

yes

complemented

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

—

C₄×C₄₇

C₂²×Q₈

yes

characteristic

C₄×C₄₇

2₋¹⁺⁴

yes

characteristic

Dic₄₇

—

Q₈×D₈

—

maximal, Sylow

C₂×C₄₇

D₄×Q₈

yes

characteristic

C₂×C₄₇

C₂²×D₈

yes

characteristic

C₂×C₄₇

#578

yes

characteristic

D₄₇

—

D₄×Q₈

—

#500

—

#474

—

C₈×Q₈

D₄₇

yes

characteristic, complemented

C₄×D₈

—

C₄₇

Q₈×D₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×D₈

—

C₄⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

C₂×D₄₇

yes

characteristic

C₄×D₄

—

#107

—

#107

—

C₄⋊C₈

C₂×D₄₇

yes

—

#102

—

C₄×C₈

C₂×D₄₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

188

—

C₂×Q₈

D₁₈₈

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

D₈

—

C₂×C₈

C₂²×D₄₇

yes

—

C₂×C₈

C₂²×D₄₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₁₈₈

yes

—

C₂²⋊C₄

—

C₄²

C₂²×D₄₇

yes

—

C₂³

—

Q₈

—

188

—

Q₈

D₃₇₆

yes

direct product factor

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

188

—

C₂×C₄

C₂×D₁₈₈

yes

—

C₂×C₄

C₂×D₁₈₈

yes

—

C₂×C₄

C₂³×D₄₇

yes

characteristic

C₈

—

C₈

Q₈×D₄₇

yes

complemented

C₂²

—

C₂²

—

188

—

100

C₂²

—

188

—

101

C₂²

C₂²×D₁₈₈

yes

characteristic

102

C₄

—

103

C₄

—

188

—

104

C₄

C₂×D₃₇₆

yes

—

105

C₄

C₂×Q₈×D₄₇

yes

characteristic

106

C₄

#13094071

yes

characteristic

107

C₂

—

188

—

108

C₂

Q₈×D₁₈₈

yes

characteristic

109

C₂

C₂²×D₃₇₆

yes

characteristic

110

C₂

#15689940

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₈×Q₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	C₈×Q₈	D₄₇
hypocenter	C₄₇	Q₈×D₈
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₈×Q₈×C₄₇

C₂

socle

C₂²×C₄₇

C₂²×D₄

hypercenter

C₈×Q₈

D₄₇

hypocenter

C₄₇

Q₈×D₈

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. Q₈×D₁₈₈	C₂	semidirect
2. #15689528	C₂	semidirect
3. #15689404	C₂	semidirect
5. C₈×Q₈×C₄₇	C₂	semidirect
29. D₃₇₆	Q₈	direct (N × H)
31. Q₈×C₄₇	D₈	semidirect
37. D₁₈₈	C₂×Q₈	semidirect
41. C₈×C₄₇	C₂×Q₈	semidirect
59. C₈×Q₈	D₄₇	semidirect
61. C₄₇	Q₈×D₈	semidirect
88. Q₈	D₃₇₆	direct (N × H)
97. C₈	Q₈×D₄₇	semidirect

H ≅ G/N

type

1. Q₈×D₁₈₈

C₂

semidirect

2. #15689528

C₂

semidirect

3. #15689404

C₂

semidirect

5. C₈×Q₈×C₄₇

C₂

semidirect

29. D₃₇₆

Q₈

direct (N × H)

31. Q₈×C₄₇

D₈

semidirect

37. D₁₈₈

C₂×Q₈

semidirect

41. C₈×C₄₇

C₂×Q₈

semidirect

59. C₈×Q₈

D₄₇

semidirect

61. C₄₇

Q₈×D₈

semidirect

88. Q₈

D₃₇₆

direct (N × H)

97. C₈

Q₈×D₄₇

semidirect

Subgroups — Q₈×D₃₇₆ #20256242

Conjugacy classes of subgroups

Complements

Subgroups — Q8×D376 #20256242

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×D₃₇₆ #20256242