Group #15689838 — subgroups

Subgroups — #15689838

Order 3008.

236 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 99 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	#13094010	C₂	1	4	yes	maximal, complemented
3	#13093998	C₂	1	2	yes	maximal, complemented
4	#13093995	C₂	1	2	yes	maximal, complemented
5	C₄×D₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#13093983	C₂	1	2	yes	maximal, complemented
7	#13093982	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×D₄×C₄₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄×C₄₇	C₂²	1	2	yes	—
11	C₂×C₂²⋊D₄₇	C₂²	1	4	yes	—
12	C₂×D₁₈₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄₇	C₂²	1	2	yes	—
14	C₂×Dic₉₄	C₂²	1	2	yes	—
15	C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
17	C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²⋊Dic₄₇	C₂²	1	4	yes	—
19	(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
20	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
21	C₄⋊Dic₄₇	C₂²	1	2	yes	—
22	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
23	C₄⋊Dic₄₇	C₂²	1	2	yes	—
24	C₄×Dic₄₇	C₂²	1	4	yes	—
25	C₂³×C₄₇	C₂³	1	2	yes	—
26	C₂²×D₄₇	C₂³	1	2	yes	—
27	D₄×C₄₇	—	2	2	no	—
28	C₂×C₄×C₄₇	—	2	4	no	—
29	C₂×C₄×C₄₇	C₂³	1	2	yes	—
30	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂²⋊D₄₇	—	4	4	no	—
34	C₂×Dic₄₇	C₂³	1	2	yes	—
35	C₂×Dic₄₇	C₂³	1	4	yes	—
36	D₁₈₈	—	2	2	no	—
37	C₄×D₄₇	—	2	4	no	—
38	Dic₉₄	—	4	2	no	—
39	C₂²×C₄₇	—	2	2	no	—
40	C₂²×C₄₇	—	2	2	no	—
41	C₂²×C₄₇	—	2	2	no	—
42	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
43	C₂×D₄₇	—	2	2	no	—
44	C₂×D₄₇	—	2	2	no	—
45	C₂×D₄₇	—	2	2	no	—
46	C₄×C₄₇	—	2	2	no	—
47	C₄×C₄₇	—	2	1	no	—
48	C₄×C₄₇	#41	1	2	yes	—
49	C₄×C₄₇	#41	1	2	yes	—
50	Dic₄₇	—	2	2	no	—
51	Dic₄₇	—	2	4	no	—
52	C₂×C₄₇	—	4	2	no	—
53	C₂×C₄₇	C₂×#41	1	1	yes	characteristic
54	C₂×C₄₇	C₂×#41	1	1	yes	characteristic
55	C₂×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
56	D₄₇	—	4	2	no	—
57	#555	—	47	1	no	maximal, Sylow
58	C₄₇	#555	1	1	yes	characteristic, complemented, Sylow
59	C₄⋊Q₈	—	47	1	no	—
60	#124	—	47	4	no	—
61	#121	—	47	2	no	—
62	#121	—	47	2	no	—
63	C₄×D₄	—	47	1	no	—
64	C₄×D₄	D₄₇	1	1	yes	characteristic, complemented
65	#117	—	47	2	no	—
66	#117	—	47	2	no	—
67	C₂×Q₈	—	47	2	no	—
68	C₂×D₄	—	47	4	no	—
69	C₂×D₄	—	47	1	no	—
70	C₂×D₄	C₂×D₄₇	1	1	yes	characteristic
71	C₂²×C₄	—	47	2	no	—
72	C₂²×C₄	C₂×D₄₇	1	2	yes	—
73	C₄⋊C₄	—	47	1	no	—
74	C₄⋊C₄	—	47	2	no	—
75	C₄⋊C₄	—	47	2	no	—
76	C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
77	C₂²⋊C₄	—	47	4	no	—
78	C₂²⋊C₄	—	47	4	no	—
79	C₂²⋊C₄	—	47	2	no	—
80	C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
81	C₄²	—	47	4	no	—
82	C₄²	C₂×D₄₇	1	1	yes	characteristic
83	C₂³	—	47	2	no	—
84	C₂³	C₂²×D₄₇	1	2	yes	—
85	Q₈	—	188	2	no	—
86	D₄	—	2	2	no	—
87	D₄	—	94	2	no	—
88	D₄	—	188	4	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	47	2	no	—
91	C₂×C₄	—	47	4	no	—
92	C₂×C₄	—	94	4	no	—
93	C₂×C₄	C₂²×D₄₇	1	2	yes	—
94	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
95	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
96	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
97	C₂²	—	2	2	no	—
98	C₂²	—	2	2	no	—
99	C₂²	—	2	2	no	—
100	C₂²	—	94	2	no	—
101	C₂²	—	94	2	no	—
102	C₂²	—	94	2	no	—
103	C₂²	C₂³×D₄₇	1	1	yes	characteristic
104	C₄	—	2	2	no	—
105	C₄	—	2	1	no	—
106	C₄	—	94	2	no	—
107	C₄	—	94	4	no	—
108	C₄	#10625350	1	2	yes	—
109	C₄	#10625352	1	2	yes	—
110	C₂	—	4	2	no	—
111	C₂	—	188	2	no	—
112	C₂	C₂×#10625352	1	1	yes	characteristic
113	C₂	C₂×#10625350	1	1	yes	characteristic
114	C₂	#13094070	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13094024

C₂

yes

maximal, characteristic, complemented

#13094010

C₂

yes

maximal, complemented

#13093998

C₂

yes

maximal, complemented

#13093995

C₂

yes

maximal, complemented

C₄×D₁₈₈

C₂

yes

maximal, characteristic, complemented

#13093983

C₂

yes

maximal, complemented

#13093982

C₂

yes

maximal, complemented

C₄×D₄×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×D₄×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

C₂²

yes

—

C₂×C₂²⋊D₄₇

C₂²

yes

—

C₂×D₁₈₈

C₂²

yes

characteristic

C₂×C₄×D₄₇

C₂²

yes

—

C₂×Dic₉₄

C₂²

yes

—

C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂²⋊C₄×C₄₇

C₂²

yes

—

C₄²×C₄₇

C₂²

yes

characteristic

C₂²⋊Dic₄₇

C₂²

yes

—

(C₂×D₄₇)⋊C₄

C₂²

yes

—

(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₄⋊Dic₄₇

C₂²

yes

—

C₄⋊Dic₄₇

C₂²

yes

characteristic

C₄⋊Dic₄₇

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

—

C₂³×C₄₇

C₂³

yes

—

C₂²×D₄₇

C₂³

yes

—

D₄×C₄₇

—

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

C₂²⋊D₄₇

—

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂⁴

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#41

yes

—

C₄×C₄₇

#41

yes

—

Dic₄₇

—

Dic₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#41

yes

characteristic

C₂×C₄₇

C₂×#41

yes

characteristic

C₂×C₄₇

H(5,𝔽₂)

yes

characteristic

D₄₇

—

#555

—

maximal, Sylow

C₄₇

#555

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

#124

—

#121

—

#121

—

C₄×D₄

—

C₄×D₄

D₄₇

yes

characteristic, complemented

#117

—

#117

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₄₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₄₇

yes

—

C₄²

—

C₄²

C₂×D₄₇

yes

characteristic

C₂³

—

C₂³

C₂²×D₄₇

yes

—

Q₈

—

188

—

D₄

—

D₄

—

D₄

—

188

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×D₄₇

yes

—

C₂×C₄

C₂²×D₄₇

yes

characteristic

C₂×C₄

C₂²×D₄₇

yes

characteristic

C₂×C₄

C₂²×D₄₇

yes

characteristic

C₂²

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₂³×D₄₇

yes

characteristic

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

#10625350

yes

—

109

C₄

#10625352

yes

—

110

C₂

—

111

C₂

—

188

—

112

C₂

C₂×#10625352

yes

characteristic

113

C₂

C₂×#10625350

yes

characteristic

114

C₂

#13094070

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	C₄×D₄×C₄₇	C₂
hypercenter	C₄×D₄	D₄₇
hypocenter	C₄₇	#555
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

hypercenter

C₄×D₄

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂² ◁ C₄×D₄

N	H ≅ G/N	type
1. #13094024	C₂	semidirect
2. #13094010	C₂	semidirect
3. #13093998	C₂	semidirect
4. #13093995	C₂	semidirect
5. C₄×D₁₈₈	C₂	semidirect
6. #13093983	C₂	semidirect
7. #13093982	C₂	semidirect
8. C₄×D₄×C₄₇	C₂	semidirect
58. C₄₇	#555	semidirect
64. C₄×D₄	D₄₇	semidirect

H ≅ G/N

type

1. #13094024

C₂

semidirect

2. #13094010

C₂

semidirect

3. #13093998

C₂

semidirect

4. #13093995

C₂

semidirect

5. C₄×D₁₈₈

C₂

semidirect

6. #13093983

C₂

semidirect

7. #13093982

C₂

semidirect

8. C₄×D₄×C₄₇

C₂

semidirect

58. C₄₇

#555

semidirect

64. C₄×D₄

D₄₇

semidirect

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

subgroup	index
#13094024	2
#13094010	2
#13093998	2
#13093995	2
C₄×D₁₈₈	2
#13093983	2
#13093982	2
C₄×D₄×C₄₇	2
#555	47

subgroup

index

C₄×D₁₈₈

C₄×D₄×C₄₇

Subgroups — #15689838

Conjugacy classes of subgroups

Complements

Maximal subgroups