Group #15689872 — subgroups

Subgroups — #15689872

Order 3008.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	#13094022	C₂	1	1	yes	maximal, characteristic, complemented
2	#13094019	C₂	1	2	yes	maximal, complemented
3	#13094018	C₂	1	2	yes	maximal, complemented
4	#13094014	C₂	1	2	yes	maximal, complemented
5	#13094013	C₂	1	1	yes	maximal, characteristic
6	#13094009	C₂	1	2	yes	maximal
7	#13094002	C₂	1	2	yes	maximal, complemented
8	#13094000	C₂	1	2	yes	maximal, complemented
9	C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×D₁₈₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄₇	C₂²	1	2	yes	—
14	C₂×Dic₉₄	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
16	C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
17	C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
19	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
20	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
23	C₄⋊Dic₄₇	C₂²	1	2	yes	—
24	Dic₄₇⋊C₄	C₂²	1	2	yes	—
25	Dic₄₇⋊C₄	C₂²	1	2	yes	—
26	Dic₄₇⋊C₄	C₂²	1	2	yes	—
27	Dic₄₇⋊C₄	C₂²	1	2	yes	—
28	Dic₄₇⋊C₄	C₂²	1	2	yes	—
29	C₄×Dic₄₇	C₂²	1	1	yes	characteristic
30	C₄×Dic₄₇	C₂²	1	1	yes	characteristic
31	C₂²×D₄₇	C₂³	1	2	yes	—
32	Q₈×C₄₇	—	4	1	no	—
33	Q₈×C₄₇	—	4	1	no	—
34	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₄₇	C₂³	1	2	yes	—
37	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₄₇	C₂³	1	2	yes	—
39	C₂×Dic₄₇	C₂³	1	2	yes	—
40	C₂×Dic₄₇	C₂³	1	2	yes	—
41	C₂×Dic₄₇	C₂³	1	2	yes	—
42	D₁₈₈	—	4	1	no	—
43	C₄×D₄₇	—	4	2	no	—
44	Dic₉₄	—	4	1	no	—
45	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
46	C₂×D₄₇	—	2	2	no	—
47	C₂×D₄₇	—	2	4	no	—
48	C₄×C₄₇	—	2	1	no	—
49	C₄×C₄₇	—	2	2	no	—
50	C₄×C₄₇	—	2	1	no	—
51	C₄×C₄₇	—	2	1	no	—
52	C₄×C₄₇	—	2	2	no	—
53	Dic₄₇	—	2	2	no	—
54	Dic₄₇	—	2	2	no	—
55	Dic₄₇	—	2	2	no	—
56	C₂×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₄₇	2₋¹⁺⁴	1	2	yes	—
58	D₄₇	—	4	2	no	—
59	#563	—	47	1	no	maximal, Sylow
60	C₄₇	#563	1	1	yes	characteristic, complemented, Sylow
61	C₄⋊Q₈	—	47	1	no	—
62	C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
63	#126	—	47	2	no	—
64	#126	—	47	2	no	—
65	#125	—	47	2	no	—
66	#124	—	47	1	no	—
67	#123	—	47	2	no	—
68	C₂²⋊Q₈	—	47	2	no	—
69	C₂²⋊Q₈	—	47	2	no	—
70	C₂×Q₈	—	47	1	no	—
71	C₂×Q₈	C₂×D₄₇	1	1	yes	characteristic
72	C₂×Q₈	C₂×D₄₇	1	1	yes	characteristic
73	C₂×D₄	—	47	1	no	—
74	C₂²×C₄	—	47	2	no	—
75	C₄⋊C₄	—	47	2	no	—
76	C₄⋊C₄	—	47	2	no	—
77	C₄⋊C₄	—	47	2	no	—
78	C₄⋊C₄	—	47	2	no	—
79	C₄⋊C₄	—	47	2	no	—
80	C₄⋊C₄	—	47	2	no	—
81	C₄⋊C₄	C₂×D₄₇	1	2	yes	—
82	C₄⋊C₄	C₂×D₄₇	1	2	yes	—
83	C₂²⋊C₄	—	47	2	no	—
84	C₂²⋊C₄	—	47	2	no	—
85	C₂²⋊C₄	—	47	2	no	—
86	C₂²⋊C₄	—	47	2	no	—
87	C₂²⋊C₄	—	47	2	no	—
88	C₄²	—	47	1	no	—
89	C₄²	—	47	1	no	—
90	C₄²	C₂×D₄₇	1	1	yes	characteristic
91	C₂³	—	47	2	no	—
92	Q₈	—	4	1	no	—
93	Q₈	—	4	1	no	—
94	Q₈	—	188	1	no	—
95	D₄	—	188	1	no	—
96	C₂×C₄	—	47	2	no	—
97	C₂×C₄	—	47	2	no	—
98	C₂×C₄	—	47	2	no	—
99	C₂×C₄	—	188	2	no	—
100	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₄₇	1	2	yes	—
103	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₄₇	1	2	yes	—
105	C₂²	—	94	2	no	—
106	C₂²	—	94	4	no	—
107	C₂²	C₂³×D₄₇	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	2	no	—
113	C₄	—	94	2	no	—
114	C₄	—	94	2	no	—
115	C₄	—	94	2	no	—
116	C₂	—	188	2	no	—
117	C₂	#13094069	1	1	yes	characteristic
118	C₂	#13094072	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#13094022

C₂

yes

maximal, characteristic, complemented

#13094019

C₂

yes

maximal, complemented

#13094018

C₂

yes

maximal, complemented

#13094014

C₂

yes

maximal, complemented

#13094013

C₂

yes

maximal, characteristic

#13094009

C₂

yes

maximal

#13094002

C₂

yes

maximal, complemented

#13094000

C₂

yes

maximal, complemented

C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

C₂²

yes

characteristic

C₂×D₁₈₈

C₂²

yes

characteristic

C₂×C₄×D₄₇

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

—

(C₂×D₄₇)⋊C₄

C₂²

yes

—

(C₂×D₄₇)⋊C₄

C₂²

yes

—

(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₄⋊Dic₄₇

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

characteristic

C₄×Dic₄₇

C₂²

yes

characteristic

C₂²×D₄₇

C₂³

yes

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

C₂²×C₄₇

C₂⁴

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

C₂×C₄₇

H(5,𝔽₂)

yes

characteristic

C₂×C₄₇

2₋¹⁺⁴

yes

—

D₄₇

—

#563

—

maximal, Sylow

C₄₇

#563

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

D₄₇

yes

characteristic, complemented

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄₇

yes

characteristic

C₂×Q₈

C₂×D₄₇

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄₇

yes

—

C₄⋊C₄

C₂×D₄₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₄₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

188

—

D₄

—

188

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

188

—

100

C₂×C₄

C₂²×D₄₇

yes

characteristic

101

C₂×C₄

C₂²×D₄₇

yes

characteristic

102

C₂×C₄

C₂²×D₄₇

yes

—

103

C₂×C₄

C₂²×D₄₇

yes

characteristic

104

C₂×C₄

C₂²×D₄₇

yes

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂³×D₄₇

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₂

—

188

—

117

C₂

#13094069

yes

characteristic

118

C₂

#13094072

yes

—

	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	#563
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

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #13094022	C₂	semidirect
2. #13094019	C₂	semidirect
3. #13094018	C₂	semidirect
4. #13094014	C₂	semidirect
7. #13094002	C₂	semidirect
8. #13094000	C₂	semidirect
9. C₄⋊Q₈×C₄₇	C₂	semidirect
60. C₄₇	#563	semidirect
62. C₄⋊Q₈	D₄₇	semidirect

H ≅ G/N

type

1. #13094022

C₂

semidirect

2. #13094019

C₂

semidirect

3. #13094018

C₂

semidirect

4. #13094014

C₂

semidirect

7. #13094002

C₂

semidirect

8. #13094000

C₂

semidirect

9. C₄⋊Q₈×C₄₇

C₂

semidirect

60. C₄₇

#563

semidirect

62. C₄⋊Q₈

D₄₇

semidirect

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

subgroup	index
#13094022	2
#13094019	2
#13094018	2
#13094014	2
#13094013	2
#13094009	2
#13094002	2
#13094000	2
C₄⋊Q₈×C₄₇	2
#563	47

subgroup

index

C₄⋊Q₈×C₄₇

Subgroups — #15689872

Conjugacy classes of subgroups

Complements

Maximal subgroups