Group #15689830 — subgroups

Subgroups — #15689830

Order 3008.

228 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 115 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	#13094024	C₂	1	1	yes	maximal, characteristic, complemented
2	#13094015	C₂	1	2	yes	maximal, complemented
3	#13094003	C₂	1	2	yes	maximal, complemented
4	#13094001	C₂	1	4	yes	maximal, complemented
5	C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×Dic₄₇	C₂	1	2	yes	maximal
7	C₂×C₄⋊Dic₄₇	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
9	C₂×D₄×C₄₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₄₇	C₂²	1	2	yes	—
11	C₂²×Dic₄₇	C₂²	1	4	yes	—
12	C₂×Dic₉₄	C₂²	1	1	yes	characteristic
13	C₂×Dic₉₄	C₂²	1	2	yes	—
14	C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
16	C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂²⋊Dic₄₇	C₂²	1	4	yes	—
18	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
19	C₄⋊Dic₄₇	C₂²	1	4	yes	—
20	C₄⋊Dic₄₇	C₂²	1	2	yes	—
21	C₄⋊Dic₄₇	C₂²	1	2	yes	—
22	Dic₄₇⋊C₄	C₂²	1	2	yes	—
23	Dic₄₇⋊C₄	C₂²	1	4	yes	—
24	C₄×Dic₄₇	C₂²	1	2	yes	—
25	C₂³×C₄₇	C₂³	1	2	yes	—
26	D₄×C₄₇	Q₈	1	4	yes	complemented
27	C₂×C₄×C₄₇	—	2	4	no	—
28	C₂×C₄×C₄₇	C₂³	1	2	yes	—
29	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×Dic₄₇	—	4	4	no	—
33	C₂×Dic₄₇	C₂³	1	2	yes	—
34	C₂×Dic₄₇	C₂³	1	2	yes	—
35	C₂×Dic₄₇	C₂³	1	4	yes	—
36	Dic₉₄	—	2	2	no	—
37	Dic₉₄	—	4	2	no	—
38	C₂²×C₄₇	—	2	2	no	—
39	C₂²×C₄₇	—	2	2	no	—
40	C₂²×C₄₇	C₂×Q₈	1	4	yes	—
41	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
42	C₄×C₄₇	—	2	2	no	—
43	C₄×C₄₇	—	2	1	no	—
44	C₄×C₄₇	C₂×Q₈	1	2	yes	—
45	C₄×C₄₇	#41	1	2	yes	—
46	Dic₄₇	—	2	2	no	—
47	Dic₄₇	—	2	4	no	—
48	Dic₄₇	—	2	2	no	—
49	C₂×C₄₇	—	2	4	no	—
50	C₂×C₄₇	C₂²×Q₈	1	1	yes	characteristic
51	C₂×C₄₇	C₂×#41	1	1	yes	characteristic
52	C₂×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
53	#554	—	47	1	no	maximal, Sylow
54	C₄₇	#554	1	1	yes	characteristic, complemented, Sylow
55	C₄⋊Q₈	—	47	1	no	—
56	#125	—	47	2	no	—
57	C₂²⋊Q₈	—	47	2	no	—
58	C₂²⋊Q₈	—	47	4	no	—
59	C₄×Q₈	—	47	1	no	—
60	C₄×D₄	—	47	2	no	—
61	C₄×D₄	D₄₇	1	1	yes	characteristic
62	C₂×C₄⋊C₄	—	47	2	no	—
63	C₂×Q₈	—	47	1	no	—
64	C₂×Q₈	—	47	2	no	—
65	C₂×D₄	C₂×D₄₇	1	1	yes	characteristic
66	C₂²×C₄	—	47	4	no	—
67	C₂²×C₄	C₂×D₄₇	1	2	yes	—
68	C₄⋊C₄	—	47	2	no	—
69	C₄⋊C₄	—	47	4	no	—
70	C₄⋊C₄	—	47	1	no	—
71	C₄⋊C₄	—	47	4	no	—
72	C₄⋊C₄	—	47	2	no	—
73	C₄⋊C₄	—	47	2	no	—
74	C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
75	C₂²⋊C₄	—	47	4	no	—
76	C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
77	C₄²	—	47	2	no	—
78	C₄²	C₂×D₄₇	1	1	yes	characteristic
79	C₂³	C₂²×D₄₇	1	2	yes	—
80	Q₈	—	94	2	no	—
81	Q₈	—	188	2	no	—
82	D₄	Dic₉₄	1	4	yes	complemented
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	47	2	no	—
85	C₂×C₄	—	47	2	no	—
86	C₂×C₄	—	47	4	no	—
87	C₂×C₄	—	188	4	no	—
88	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₄₇	1	2	yes	—
90	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
91	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
92	C₂²	—	2	2	no	—
93	C₂²	—	2	2	no	—
94	C₂²	C₂×Dic₉₄	1	4	yes	—
95	C₂²	C₂³×D₄₇	1	1	yes	characteristic
96	C₄	—	2	2	no	—
97	C₄	—	2	1	no	—
98	C₄	—	94	2	no	—
99	C₄	—	94	2	no	—
100	C₄	—	94	4	no	—
101	C₄	C₂×Dic₉₄	1	2	yes	—
102	C₄	#10625352	1	2	yes	—
103	C₂	—	2	4	no	—
104	C₂	C₂²×Dic₉₄	1	1	yes	characteristic
105	C₂	C₂×#10625352	1	1	yes	characteristic
106	C₂	#13094070	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13094024

C₂

yes

maximal, characteristic, complemented

#13094015

C₂

yes

maximal, complemented

#13094003

C₂

yes

maximal, complemented

#13094001

C₂

yes

maximal, complemented

C₄×Dic₉₄

C₂

yes

maximal, characteristic, complemented

D₄×Dic₄₇

C₂

yes

maximal

C₂×C₄⋊Dic₄₇

C₂

yes

maximal, complemented

C₄×D₄×C₄₇

C₂

yes

maximal, characteristic

C₂×D₄×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

C₂²

yes

—

C₂²×Dic₄₇

C₂²

yes

—

C₂×Dic₉₄

C₂²

yes

characteristic

C₂×Dic₉₄

C₂²

yes

—

C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂²⋊C₄×C₄₇

C₂²

yes

—

C₄²×C₄₇

C₂²

yes

characteristic

C₂²⋊Dic₄₇

C₂²

yes

—

C₄⋊Dic₄₇

C₂²

yes

characteristic

C₄⋊Dic₄₇

C₂²

yes

—

C₄⋊Dic₄₇

C₂²

yes

—

C₄⋊Dic₄₇

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

—

C₂³×C₄₇

C₂³

yes

—

D₄×C₄₇

Q₈

yes

complemented

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂³

yes

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

—

Dic₉₄

—

Dic₉₄

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×Q₈

yes

—

C₂²×C₄₇

C₂⁴

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×Q₈

yes

—

C₄×C₄₇

#41

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂²×Q₈

yes

characteristic

C₂×C₄₇

C₂×#41

yes

C₂×C₄₇

yes

—

C₄₇

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

D₄₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

C₂×D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₄₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₄₇

yes

—

C₄²

—

C₄²

C₂×D₄₇

yes

characteristic

C₂³

C₂²×D₄₇

yes

—

Q₈

—

Q₈

—

188

—

D₄

Dic₉₄

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

188

—

C₂×C₄

C₂²×D₄₇

yes

characteristic

C₂×C₄

C₂²×D₄₇

yes

—

C₂×C₄

C₂²×D₄₇

yes

characteristic

C₂×C₄

C₂²×D₄₇

yes

characteristic

C₂²

—

C₂²

—

C₂²

C₂×Dic₉₄

yes

—

C₂²

C₂³×D₄₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

100

C₄

—

101

C₄

C₂×Dic₉₄

yes

—

102

C₄

#10625352

yes

—

103

C₂

—

104

C₂

C₂²×Dic₉₄

yes

characteristic

105

C₂

C₂×#10625352

yes

characteristic

106

C₂

#13094070

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	C₄×D₄×C₄₇	C₂
prime radical	C₂×D₄×C₄₇	C₂²
hypercenter	C₄×D₄	D₄₇
hypocenter	C₄₇	#554
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₇

commutator, socle, cosocle

C₂²×C₄₇

C₂⁴

Fitting

C₄×D₄×C₄₇

C₂

prime radical

C₂×D₄×C₄₇

C₂²

hypercenter

C₄×D₄

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #13094024	C₂	semidirect
2. #13094015	C₂	semidirect
3. #13094003	C₂	semidirect
4. #13094001	C₂	semidirect
5. C₄×Dic₉₄	C₂	semidirect
7. C₂×C₄⋊Dic₄₇	C₂	semidirect
26. D₄×C₄₇	Q₈	semidirect
54. C₄₇	#554	semidirect
82. D₄	Dic₉₄	semidirect

H ≅ G/N

type

1. #13094024

C₂

semidirect

2. #13094015

C₂

semidirect

3. #13094003

C₂

semidirect

4. #13094001

C₂

semidirect

5. C₄×Dic₉₄

C₂

semidirect

7. C₂×C₄⋊Dic₄₇

C₂

semidirect

26. D₄×C₄₇

Q₈

semidirect

54. C₄₇

#554

semidirect

82. D₄

Dic₉₄

semidirect

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

subgroup	index
#13094024	2
#13094015	2
#13094003	2
#13094001	2
C₄×Dic₉₄	2
D₄×Dic₄₇	2
C₂×C₄⋊Dic₄₇	2
C₄×D₄×C₄₇	2
#554	47

subgroup

index

C₄×Dic₉₄

D₄×Dic₄₇

C₂×C₄⋊Dic₄₇

C₄×D₄×C₄₇

#554

Subgroups — #15689830

Conjugacy classes of subgroups

Complements

Maximal subgroups