Group #20257015 — subgroups

Subgroups — #20257015

Order 6016.

1202 conjugacy classes of subgroups; only the 491 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 491 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	#15689839	C₂	1	9	yes	maximal, complemented
2	#15689795	C₂	1	6	yes	maximal, complemented
3	#15689674	C₂	1	9	yes	maximal, complemented
4	C₄×#10625354	C₂	1	6	yes	maximal, complemented
5	Q₈²×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625354	C₂²	1	6	yes	—
7	C₂×#10625354	C₂²	1	6	yes	—
8	#13094021	C₂²	1	9	yes	—
9	#13094013	C₂²	1	18	yes	—
10	#13093999	C₂²	1	36	yes	—
11	Q₈×Dic₄₇	C₂²	1	2	yes	—
12	C₄×D₁₈₈	C₂²	1	18	yes	—
13	#13093986	C₂²	1	18	yes	—
14	#13093982	C₂²	1	18	yes	—
15	C₄²×D₄₇	C₂²	1	9	yes	—
16	C₄⋊Q₈×C₄₇	C₂²	1	9	yes	—
17	C₄×Q₈×C₄₇	C₂²	1	6	yes	—
18	C₂×Q₈×C₄₇	C₂³	1	6	yes	—
19	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
20	#10625354	—	2	24	no	—
21	#10625354	—	4	12	no	—
22	C₂×D₁₈₈	C₂³	1	18	yes	—
23	C₂×D₁₈₈	C₂³	1	18	yes	—
24	C₂×C₄×D₄₇	C₂³	1	18	yes	—
25	C₂×C₄×D₄₇	C₂³	1	9	yes	—
26	C₄⋊C₄×C₄₇	C₂³	1	18	yes	—
27	C₄²×C₄₇	C₂³	1	9	yes	—
28	(C₂×D₄₇)⋊C₄	C₂³	1	36	yes	—
29	C₄⋊Dic₄₇	C₂³	1	6	yes	—
30	C₄×Dic₄₇	C₂³	1	6	yes	—
31	C₄×Dic₄₇	C₂³	1	9	yes	—
32	C₂²×D₄₇	C₂⁴	1	9	yes	—
33	Q₈×C₄₇	—	4	6	no	—
34	Q₈×C₄₇	#41	1	8	yes	complemented
35	C₂×C₄×C₄₇	C₂⁴	1	9	yes	—
36	C₂×C₄×C₄₇	C₂⁴	1	6	yes	—
37	C₂×Dic₄₇	C₂⁴	1	6	yes	—
38	C₂×Dic₄₇	C₂⁴	1	1	yes	characteristic
39	D₁₈₈	—	2	36	no	—
40	D₁₈₈	—	4	18	no	—
41	C₄×D₄₇	—	2	36	no	—
42	C₄×D₄₇	—	2	18	no	—
43	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
44	C₂×D₄₇	—	2	18	no	—
45	C₂×D₄₇	—	2	9	no	—
46	C₄×C₄₇	—	2	9	no	—
47	C₄×C₄₇	C₂×#41	1	12	yes	—
48	Dic₄₇	—	2	6	no	—
49	Dic₄₇	H(5,𝔽₂)	1	2	yes	—
50	#3483	—	47	1	no	maximal, Sylow
51	C₂×C₄₇	C₂²×#41	1	2	yes	—
52	C₂×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
53	D₄₇	—	4	9	no	—
54	Q₈²	D₄₇	1	1	yes	characteristic, complemented
55	#555	—	47	9	no	—
56	#550	—	47	6	no	—
57	#532	—	47	9	no	—
58	C₄×#41	—	47	6	no	—
59	C₄₇	#3483	1	1	yes	characteristic, complemented, Sylow
60	C₂×#41	—	47	6	no	—
61	C₂×#41	—	47	6	no	—
62	C₄⋊Q₈	C₂×D₄₇	1	9	yes	—
63	#127	—	47	9	no	—
64	#124	—	47	18	no	—
65	#121	—	47	36	no	—
66	C₄×Q₈	—	47	2	no	—
67	C₄×Q₈	C₂×D₄₇	1	6	yes	—
68	C₄×D₄	—	47	18	no	—
69	C₄×D₄	—	47	18	no	—
70	#117	—	47	18	no	—
71	C₂×C₄²	—	47	9	no	—
72	#41	—	94	24	no	—
73	#41	—	188	12	no	—
74	C₂×Q₈	C₂²×D₄₇	1	6	yes	—
75	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
76	C₂×D₄	—	47	18	no	—
77	C₂×D₄	—	47	18	no	—
78	C₂²×C₄	—	47	18	no	—
79	C₂²×C₄	—	47	9	no	—
80	C₄⋊C₄	—	47	6	no	—
81	C₄⋊C₄	C₂²×D₄₇	1	18	yes	—
82	C₂²⋊C₄	—	47	36	no	—
83	C₄²	—	47	6	no	—
84	C₄²	—	47	9	no	—
85	C₄²	C₂²×D₄₇	1	9	yes	—
86	C₂³	—	47	9	no	—
87	Q₈	—	4	6	no	—
88	Q₈	#10625354	1	8	yes	complemented
89	D₄	—	94	36	no	—
90	D₄	—	188	18	no	—
91	C₂×C₄	—	47	1	no	—
92	C₂×C₄	—	47	6	no	—
93	C₂×C₄	—	94	18	no	—
94	C₂×C₄	—	94	36	no	—
95	C₂×C₄	C₂³×D₄₇	1	9	yes	—
96	C₂×C₄	C₂³×D₄₇	1	6	yes	—
97	C₂²	—	94	18	no	—
98	C₂²	—	94	9	no	—
99	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
100	C₄	—	2	9	no	—
101	C₄	—	47	2	no	—
102	C₄	—	94	6	no	—
103	C₄	C₂×#10625354	1	12	yes	—
104	C₂	—	188	9	no	—
105	C₂	C₂²×#10625354	1	2	yes	—
106	C₂	H(5,𝔽₂)×D₄₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689839

C₂

yes

maximal, complemented

#15689795

C₂

yes

maximal, complemented

#15689674

C₂

yes

maximal, complemented

C₄×#10625354

C₂

yes

maximal, complemented

Q₈²×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#10625354

C₂²

yes

—

C₂×#10625354

C₂²

yes

—

#13094021

C₂²

yes

—

#13094013

C₂²

yes

—

#13093999

C₂²

yes

—

Q₈×Dic₄₇

C₂²

yes

—

C₄×D₁₈₈

C₂²

yes

—

#13093986

C₂²

yes

—

#13093982

C₂²

yes

—

C₄²×D₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

C₂²

yes

—

C₄×Q₈×C₄₇

C₂²

yes

—

C₂×Q₈×C₄₇

C₂³

yes

—

C₂×Q₈×C₄₇

C₂³

yes

—

#10625354

—

#10625354

—

C₂×D₁₈₈

C₂³

yes

—

C₂×D₁₈₈

C₂³

yes

—

C₂×C₄×D₄₇

C₂³

yes

—

C₂×C₄×D₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄²×C₄₇

C₂³

yes

—

(C₂×D₄₇)⋊C₄

C₂³

yes

—

C₄⋊Dic₄₇

C₂³

yes

—

C₄×Dic₄₇

C₂³

yes

—

C₄×Dic₄₇

C₂³

yes

—

C₂²×D₄₇

C₂⁴

yes

—

Q₈×C₄₇

—

Q₈×C₄₇

#41

yes

complemented

C₂×C₄×C₄₇

C₂⁴

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

—

C₂×Dic₄₇

C₂⁴

yes

—

C₂×Dic₄₇

C₂⁴

yes

characteristic

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

C₂²×C₄₇

C₂⁵

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#41

yes

—

Dic₄₇

—

Dic₄₇

H(5,𝔽₂)

yes

—

#3483

—

maximal, Sylow

C₂×C₄₇

C₂²×#41

yes

—

C₂×C₄₇

C₂×H(5,𝔽₂)

yes

characteristic

D₄₇

—

Q₈²

D₄₇

yes

characteristic, complemented

#555

—

#550

—

#532

—

C₄×#41

—

C₄₇

#3483

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×#41

—

C₄⋊Q₈

C₂×D₄₇

yes

—

#127

—

#124

—

#121

—

C₄×Q₈

—

C₄×Q₈

C₂×D₄₇

yes

—

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

#41

—

#41

—

188

—

C₂×Q₈

C₂²×D₄₇

yes

—

C₂×Q₈

C₂²×D₄₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₄₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₄₇

yes

—

C₂³

—

Q₈

—

Q₈

#10625354

yes

complemented

D₄

—

D₄

—

188

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂³×D₄₇

yes

—

C₂×C₄

C₂³×D₄₇

yes

—

C₂²

—

C₂²

—

C₂²

C₂⁴×D₄₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

C₂×#10625354

yes

—

104

C₂

—

188

—

105

C₂

C₂²×#10625354

yes

—

106

C₂

H(5,𝔽₂)×D₄₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Fitting	Q₈²×C₄₇	C₂
hypercenter	Q₈²	D₄₇
hypocenter	C₄₇	#3483
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ Q₈²

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₄₇

commutator, socle

C₂²×C₄₇

C₂⁵

Fitting

Q₈²×C₄₇

C₂

hypercenter

Q₈²

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂² ◁ Q₈²

N	H ≅ G/N	type
1. #15689839	C₂	semidirect
2. #15689795	C₂	semidirect
3. #15689674	C₂	semidirect
4. C₄×#10625354	C₂	semidirect
5. Q₈²×C₄₇	C₂	semidirect
34. Q₈×C₄₇	#41	semidirect
54. Q₈²	D₄₇	semidirect
59. C₄₇	#3483	semidirect
88. Q₈	#10625354	semidirect

H ≅ G/N

type

1. #15689839

C₂

semidirect

2. #15689795

C₂

semidirect

3. #15689674

C₂

semidirect

4. C₄×#10625354

C₂

semidirect

5. Q₈²×C₄₇

C₂

semidirect

34. Q₈×C₄₇

#41

semidirect

54. Q₈²

D₄₇

semidirect

59. C₄₇

#3483

semidirect

88. Q₈

#10625354

semidirect

Subgroups — #20257015

Conjugacy classes of subgroups

Complements