Group #20256363 — subgroups

Subgroups — #20256363

Order 6016.

460 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 141 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	#15689795	C₂	1	2	yes	maximal, complemented
2	#15689500	C₂	1	3	yes	maximal, complemented
3	#15689319	C₂	1	6	yes	maximal, complemented
4	C₄×D₃₇₆	C₂	1	3	yes	maximal, complemented
5	C₈×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625354	—	2	2	no	—
7	C₂×D₃₇₆	C₂²	1	3	yes	—
8	C₂×D₃₇₆	C₂²	1	6	yes	—
9	#13094021	C₂²	1	6	yes	—
10	#13093999	—	2	6	no	—
11	C₄×D₁₈₈	C₂²	1	6	yes	—
12	#13093958	C₂²	1	1	yes	characteristic
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	#10625354	—	4	8	no	—
19	C₂×D₁₈₈	—	2	6	no	—
20	C₂×D₁₈₈	—	2	6	no	—
21	C₂×D₁₈₈	C₂³	1	6	yes	—
22	C₂×C₄×D₄₇	—	2	6	no	—
23	C₂×C₈×C₄₇	C₂³	1	3	yes	—
24	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₄₇	D₄	1	3	yes	—
26	C₄²×C₄₇	C₂³	1	3	yes	—
27	(C₂×D₄₇)⋊C₄	—	2	6	no	—
28	C₄⋊Dic₄₇	C₂³	1	2	yes	—
29	D₃₇₆	—	2	6	no	—
30	D₃₇₆	—	4	6	no	—
31	C₂²×D₄₇	—	2	6	no	—
32	Q₈×C₄₇	D₈	1	4	yes	complemented
33	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
34	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
35	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
36	C₂×Dic₄₇	—	2	2	no	—
37	D₁₈₈	—	2	6	no	—
38	D₁₈₈	—	2	6	no	—
39	D₁₈₈	—	4	12	no	—
40	D₁₈₈	—	8	6	no	—
41	C₄×D₄₇	—	4	12	no	—
42	C₈×C₄₇	—	2	3	no	—
43	C₈×C₄₇	#41	1	2	yes	complemented
44	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
45	C₂×D₄₇	—	4	6	no	—
46	C₂×D₄₇	—	4	6	no	—
47	C₂×D₄₇	—	4	6	no	—
48	C₄×C₄₇	—	2	3	no	—
49	C₄×C₄₇	C₂×D₈	1	6	yes	—
50	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
51	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
52	Dic₄₇	—	4	2	no	—
53	#3391	—	47	1	no	maximal, Sylow
54	C₂×C₄₇	#550	1	1	yes	characteristic
55	C₂×C₄₇	C₂²×D₈	1	1	yes	characteristic
56	C₂×C₄₇	#576	1	1	yes	characteristic
57	D₄₇	—	8	6	no	—
58	#550	—	47	2	no	—
59	#493	—	47	3	no	—
60	#459	—	47	6	no	—
61	C₈×Q₈	D₄₇	1	1	yes	characteristic, complemented
62	C₄×D₈	—	47	3	no	—
63	C₄₇	#3391	1	1	yes	characteristic, complemented, Sylow
64	C₂×#41	—	94	2	no	—
65	C₂×D₈	—	47	3	no	—
66	C₂×D₈	—	47	6	no	—
67	#127	—	47	6	no	—
68	#121	—	94	6	no	—
69	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic
70	C₄×D₄	—	47	6	no	—
71	#107	—	47	1	no	—
72	C₄⋊C₈	C₂×D₄₇	1	3	yes	—
73	#102	—	47	6	no	—
74	C₄×C₈	C₂×D₄₇	1	3	yes	—
75	#41	—	188	8	no	—
76	C₂×Q₈	D₁₈₈	1	1	yes	characteristic
77	C₂×D₄	—	47	6	no	—
78	C₂×D₄	—	94	6	no	—
79	C₂×D₄	—	94	6	no	—
80	C₂²×C₄	—	94	6	no	—
81	D₈	—	94	6	no	—
82	D₈	—	188	6	no	—
83	C₂×C₈	C₂²×D₄₇	1	3	yes	—
84	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
85	C₄⋊C₄	—	47	2	no	—
86	C₄⋊C₄	D₁₈₈	1	3	yes	—
87	C₂²⋊C₄	—	94	6	no	—
88	C₄²	C₂²×D₄₇	1	3	yes	—
89	C₂³	—	94	6	no	—
90	Q₈	D₃₇₆	1	4	yes	complemented
91	D₄	—	94	6	no	—
92	D₄	—	94	6	no	—
93	D₄	—	376	6	no	—
94	D₄	—	188	12	no	—
95	C₂×C₄	—	94	2	no	—
96	C₂×C₄	—	188	12	no	—
97	C₂×C₄	C₂×D₁₈₈	1	3	yes	—
98	C₂×C₄	C₂×D₁₈₈	1	3	yes	—
99	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
100	C₈	—	2	3	no	—
101	C₈	#10625354	1	2	yes	complemented
102	C₂²	—	188	6	no	—
103	C₂²	—	188	6	no	—
104	C₂²	—	188	6	no	—
105	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
106	C₄	—	2	3	no	—
107	C₄	—	188	2	no	—
108	C₄	C₂×D₃₇₆	1	6	yes	—
109	C₄	C₂×#10625354	1	1	yes	characteristic
110	C₄	#13094070	1	1	yes	characteristic
111	C₂	—	376	6	no	—
112	C₂	#15689795	1	1	yes	characteristic
113	C₂	C₂²×D₃₇₆	1	1	yes	characteristic
114	C₂	#15689928	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689795

C₂

yes

maximal, complemented

#15689500

C₂

yes

maximal, complemented

#15689319

C₂

yes

maximal, complemented

C₄×D₃₇₆

C₂

yes

maximal, complemented

C₈×Q₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#10625354

—

C₂×D₃₇₆

C₂²

yes

—

C₂×D₃₇₆

C₂²

yes

—

#13094021

C₂²

yes

—

#13093999

—

C₄×D₁₈₈

C₂²

yes

—

#13093958

C₂²

yes

characteristic

#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

#10625354

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂³

yes

—

C₂×C₄×D₄₇

—

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

C₂³

yes

—

D₃₇₆

—

D₃₇₆

—

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

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

#41

yes

complemented

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

—

C₄×C₄₇

C₂×#41

yes

characteristic

C₄×C₄₇

H(5,𝔽₂)

yes

characteristic

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₂²×D₈

yes

characteristic

C₂×C₄₇

#576

yes

characteristic

D₄₇

—

#550

—

#493

—

#459

—

C₈×Q₈

D₄₇

yes

characteristic, complemented

C₄×D₈

—

C₄₇

#3391

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×D₈

—

C₂×D₈

—

#127

—

#121

—

C₄×Q₈

C₂×D₄₇

yes

characteristic

C₄×D₄

—

#107

—

C₄⋊C₈

C₂×D₄₇

yes

—

#102

—

C₄×C₈

C₂×D₄₇

yes

—

#41

—

188

—

C₂×Q₈

D₁₈₈

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

D₈

—

D₈

—

188

—

C₂×C₈

C₂²×D₄₇

yes

—

C₂×C₈

C₂²×D₄₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

D₁₈₈

yes

—

C₂²⋊C₄

—

C₄²

C₂²×D₄₇

yes

—

C₂³

—

Q₈

D₃₇₆

yes

complemented

D₄

—

D₄

—

D₄

—

376

—

D₄

—

188

—

C₂×C₄

—

C₂×C₄

—

188

—

C₂×C₄

C₂×D₁₈₈

yes

—

C₂×C₄

C₂×D₁₈₈

yes

—

C₂×C₄

C₂³×D₄₇

yes

characteristic

100

C₈

—

101

C₈

#10625354

yes

complemented

102

C₂²

—

188

—

103

C₂²

—

188

—

104

C₂²

—

188

—

105

C₂²

C₂²×D₁₈₈

yes

characteristic

106

C₄

—

107

C₄

—

188

—

108

C₄

C₂×D₃₇₆

yes

—

109

C₄

C₂×#10625354

yes

characteristic

110

C₄

#13094070

yes

characteristic

111

C₂

—

376

—

112

C₂

#15689795

yes

characteristic

113

C₂

C₂²×D₃₇₆

yes

characteristic

114

C₂

#15689928

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	#3391
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

#3391

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. #15689795	C₂	semidirect
2. #15689500	C₂	semidirect
3. #15689319	C₂	semidirect
4. C₄×D₃₇₆	C₂	semidirect
5. C₈×Q₈×C₄₇	C₂	semidirect
32. Q₈×C₄₇	D₈	semidirect
43. C₈×C₄₇	#41	semidirect
61. C₈×Q₈	D₄₇	semidirect
63. C₄₇	#3391	semidirect
90. Q₈	D₃₇₆	semidirect
101. C₈	#10625354	semidirect

H ≅ G/N

type

1. #15689795

C₂

semidirect

2. #15689500

C₂

semidirect

3. #15689319

C₂

semidirect

4. C₄×D₃₇₆

C₂

semidirect

5. C₈×Q₈×C₄₇

C₂

semidirect

32. Q₈×C₄₇

D₈

semidirect

43. C₈×C₄₇

#41

semidirect

61. C₈×Q₈

D₄₇

semidirect

63. C₄₇

#3391

semidirect

90. Q₈

D₃₇₆

semidirect

101. C₈

#10625354

semidirect

Subgroups — #20256363

Conjugacy classes of subgroups

Complements