Group #15689788 — subgroups

Subgroups — #15689788

Order 3008.

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

normal subgroups only

This group has 280 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	C₂×Q₈×D₄₇	C₂	1	2	yes	maximal, complemented
2	#13094023	C₂	1	2	yes	maximal, complemented
3	#13094002	C₂	1	4	yes	maximal, complemented
4	#13094000	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₁₈₈	C₂	1	1	yes	maximal, characteristic
7	#13093986	C₂	1	2	yes	maximal
8	C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Q₈×C₄₇	C₂²	1	2	yes	—
10	Q₈×D₄₇	—	2	8	no	—
11	C₂×D₁₈₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄₇	C₂²	1	4	yes	—
13	C₂×C₄×D₄₇	C₂²	1	2	yes	—
14	C₂×Dic₉₄	C₂²	1	4	yes	—
15	C₂×Dic₉₄	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
17	C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
18	C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
20	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
21	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
22	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
23	Dic₄₇⋊C₄	C₂²	1	2	yes	—
24	Dic₄₇⋊C₄	C₂²	1	4	yes	—
25	C₄×Dic₄₇	C₂²	1	2	yes	—
26	C₂²×D₄₇	C₂³	1	2	yes	—
27	Q₈×C₄₇	—	2	4	no	—
28	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₄₇	C₂³	1	4	yes	—
30	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×Dic₄₇	C₂³	1	2	yes	—
33	C₂×Dic₄₇	C₂³	1	2	yes	—
34	C₂×Dic₄₇	C₂³	1	2	yes	—
35	D₁₈₈	Q₈	1	4	yes	complemented
36	C₄×D₄₇	—	2	4	no	—
37	C₄×D₄₇	—	2	8	no	—
38	Dic₉₄	—	2	8	no	—
39	Dic₉₄	D₄	1	4	yes	complemented
40	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
41	C₂×D₄₇	—	2	2	no	—
42	C₂×D₄₇	—	2	2	no	—
43	C₂×D₄₇	C₂×Q₈	1	4	yes	complemented
44	C₄×C₄₇	—	2	4	no	—
45	C₄×C₄₇	—	2	1	no	—
46	C₄×C₄₇	C₂×D₄	1	2	yes	—
47	C₄×C₄₇	C₂×Q₈	1	2	yes	complemented
48	Dic₄₇	—	2	2	no	—
49	Dic₄₇	—	2	2	no	—
50	Dic₄₇	C₂×D₄	1	4	yes	—
51	C₂×C₄₇	C₂²×D₄	1	1	yes	characteristic
52	C₂×C₄₇	C₂²×Q₈	1	1	yes	characteristic
53	C₂×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
54	D₄₇	—	2	4	no	—
55	D₄×Q₈	—	47	1	no	maximal, Sylow
56	C₄₇	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
57	C₂²×Q₈	—	47	2	no	—
58	C₄⋊Q₈	—	47	2	no	—
59	C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
60	C₂²⋊Q₈	—	47	2	no	—
61	C₂²⋊Q₈	—	47	4	no	—
62	C₄×Q₈	—	47	1	no	—
63	C₄×D₄	—	47	2	no	—
64	C₄×D₄	—	47	1	no	—
65	C₂×Q₈	—	47	1	no	—
66	C₂×Q₈	—	47	4	no	—
67	C₂×Q₈	—	94	8	no	—
68	C₂×Q₈	C₂×D₄₇	1	2	yes	—
69	C₂×D₄	—	47	1	no	—
70	C₂²×C₄	—	47	4	no	—
71	C₂²×C₄	—	47	2	no	—
72	C₄⋊C₄	—	47	2	no	—
73	C₄⋊C₄	—	47	4	no	—
74	C₄⋊C₄	—	47	1	no	—
75	C₄⋊C₄	—	47	1	no	—
76	C₄⋊C₄	C₂×D₄₇	1	2	yes	—
77	C₄⋊C₄	C₂×D₄₇	1	2	yes	—
78	C₂²⋊C₄	—	47	4	no	—
79	C₂²⋊C₄	—	47	2	no	—
80	C₄²	—	47	2	no	—
81	C₄²	C₂×D₄₇	1	1	yes	characteristic
82	C₂³	—	47	2	no	—
83	Q₈	—	2	4	no	—
84	Q₈	—	47	4	no	—
85	Q₈	—	94	8	no	—
86	D₄	—	47	4	no	—
87	C₂×C₄	—	47	2	no	—
88	C₂×C₄	—	47	2	no	—
89	C₂×C₄	—	47	2	no	—
90	C₂×C₄	—	94	8	no	—
91	C₂×C₄	—	94	4	no	—
92	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
93	C₂×C₄	C₂²×D₄₇	1	4	yes	—
94	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
96	C₂²	—	47	4	no	—
97	C₂²	—	94	2	no	—
98	C₂²	—	94	2	no	—
99	C₂²	C₂³×D₄₇	1	1	yes	characteristic
100	C₄	—	2	4	no	—
101	C₄	—	2	1	no	—
102	C₄	—	47	4	no	—
103	C₄	—	94	2	no	—
104	C₄	—	94	2	no	—
105	C₄	D₄×D₄₇	1	2	yes	—
106	C₄	Q₈×D₄₇	1	2	yes	complemented
107	C₂	—	94	4	no	—
108	C₂	C₂×D₄×D₄₇	1	1	yes	characteristic
109	C₂	C₂×Q₈×D₄₇	1	1	yes	characteristic
110	C₂	#13094072	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×D₄₇

C₂

yes

maximal, complemented

#13094023

C₂

yes

maximal, complemented

#13094002

C₂

yes

maximal, complemented

#13094000

C₂

yes

maximal, complemented

C₄×Dic₉₄

C₂

yes

maximal, characteristic, complemented

C₄×D₁₈₈

C₂

yes

maximal, characteristic

#13093986

C₂

yes

maximal

C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₄₇

C₂²

yes

—

Q₈×D₄₇

—

C₂×D₁₈₈

C₂²

yes

characteristic

C₂×C₄×D₄₇

C₂²

yes

—

C₂×C₄×D₄₇

C₂²

yes

—

C₂×Dic₉₄

C₂²

yes

—

C₂×Dic₉₄

C₂²

yes

characteristic

C₄⋊C₄×C₄₇

C₂²

yes

—

C₄⋊C₄×C₄₇

C₂²

yes

—

C₄²×C₄₇

C₂²

yes

characteristic

(C₂×D₄₇)⋊C₄

C₂²

yes

—

(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₄⋊Dic₄₇

C₂²

yes

characteristic

C₄⋊Dic₄₇

C₂²

yes

characteristic

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

—

C₂²×D₄₇

C₂³

yes

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

—

D₁₈₈

Q₈

yes

complemented

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

D₄

yes

complemented

C₂²×C₄₇

C₂⁴

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

C₂×Q₈

yes

complemented

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₄

yes

—

C₄×C₄₇

C₂×Q₈

yes

complemented

Dic₄₇

—

Dic₄₇

—

Dic₄₇

C₂×D₄

yes

—

C₂×C₄₇

C₂²×D₄

yes

characteristic

C₂×C₄₇

C₂²×Q₈

yes

characteristic

C₂×C₄₇

2₋¹⁺⁴

yes

characteristic

D₄₇

—

D₄×Q₈

—

maximal, Sylow

C₄₇

D₄×Q₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₄₇

yes

characteristic, complemented

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄₇

yes

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄₇

yes

—

C₄⋊C₄

C₂×D₄₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₄₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

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

—

C₂²

C₂³×D₄₇

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

D₄×D₄₇

yes

—

106

C₄

Q₈×D₄₇

yes

complemented

107

C₂

—

108

C₂

C₂×D₄×D₄₇

yes

characteristic

109

C₂

C₂×Q₈×D₄₇

yes

characteristic

110

C₂

#13094072

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	C₄⋊Q₈×C₄₇	C₂
prime radical	C₂×D₁₈₈	C₂²
hypercenter	C₄⋊Q₈	D₄₇
hypocenter	C₄₇	D₄×Q₈
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₇

commutator, socle, cosocle

C₂²×C₄₇

C₂⁴

Fitting

C₄⋊Q₈×C₄₇

C₂

prime radical

C₂×D₁₈₈

C₂²

hypercenter

C₄⋊Q₈

D₄₇

hypocenter

C₄₇

D₄×Q₈

LCS

G ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₂×Q₈×D₄₇	C₂	semidirect
2. #13094023	C₂	semidirect
3. #13094002	C₂	semidirect
4. #13094000	C₂	semidirect
5. C₄×Dic₉₄	C₂	semidirect
8. C₄⋊Q₈×C₄₇	C₂	semidirect
35. D₁₈₈	Q₈	semidirect
39. Dic₉₄	D₄	semidirect
43. C₂×D₄₇	C₂×Q₈	semidirect
47. C₄×C₄₇	C₂×Q₈	semidirect
56. C₄₇	D₄×Q₈	semidirect
59. C₄⋊Q₈	D₄₇	semidirect
106. C₄	Q₈×D₄₇	semidirect

H ≅ G/N

type

1. C₂×Q₈×D₄₇

C₂

semidirect

2. #13094023

C₂

semidirect

3. #13094002

C₂

semidirect

4. #13094000

C₂

semidirect

5. C₄×Dic₉₄

C₂

semidirect

8. C₄⋊Q₈×C₄₇

C₂

semidirect

35. D₁₈₈

Q₈

semidirect

39. Dic₉₄

D₄

semidirect

43. C₂×D₄₇

C₂×Q₈

semidirect

47. C₄×C₄₇

C₂×Q₈

semidirect

56. C₄₇

D₄×Q₈

semidirect

59. C₄⋊Q₈

D₄₇

semidirect

106. C₄

Q₈×D₄₇

semidirect

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

subgroup	index
C₂×Q₈×D₄₇	2
#13094023	2
#13094002	2
#13094000	2
C₄×Dic₉₄	2
C₄×D₁₈₈	2
#13093986	2
C₄⋊Q₈×C₄₇	2
D₄×Q₈	47

subgroup

index

C₂×Q₈×D₄₇

#13094023

#13094002

#13094000

C₄×Dic₉₄

C₄×D₁₈₈

#13093986

C₄⋊Q₈×C₄₇

D₄×Q₈

Subgroups — #15689788

Conjugacy classes of subgroups

Complements

Maximal subgroups