Group #15689794 — subgroups

Subgroups — #15689794

Order 3008.

312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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×#10625350	C₂	1	2	yes	maximal, complemented
2	#13094021	C₂	1	1	yes	maximal, characteristic, complemented
3	#13094020	C₂	1	2	yes	maximal, complemented
4	#13093998	C₂	1	2	yes	maximal, complemented
5	#13093997	C₂	1	4	yes	maximal, complemented
6	C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#13093986	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×C₂²⋊D₄₇	C₂²	1	4	yes	—
13	#10625350	—	2	8	no	—
14	C₂×D₁₈₈	C₂²	1	2	yes	—
15	C₂×D₁₈₈	C₂²	1	2	yes	—
16	C₂×D₁₈₈	C₂²	1	2	yes	—
17	C₂×C₄×D₄₇	C₂²	1	4	yes	—
18	C₂×Dic₉₄	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
21	C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
23	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
24	Dic₄₇⋊C₄	C₂²	1	2	yes	—
25	C₄×Dic₄₇	C₂²	1	2	yes	—
26	C₂³×C₄₇	C₂³	1	2	yes	—
27	C₂²×D₄₇	C₂³	1	4	yes	—
28	D₄×C₄₇	—	2	2	no	—
29	C₂×C₄×C₄₇	—	2	4	no	—
30	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₄₇	C₂³	1	2	yes	—
32	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
34	C₂²⋊D₄₇	—	2	8	no	—
35	C₂²⋊D₄₇	—	4	4	no	—
36	C₂×Dic₄₇	C₂³	1	2	yes	—
37	C₂×Dic₄₇	C₂³	1	2	yes	—
38	D₁₈₈	—	2	4	no	—
39	D₁₈₈	—	2	4	no	—
40	D₁₈₈	—	4	2	no	—
41	C₄×D₄₇	—	2	8	no	—
42	Dic₉₄	D₄	1	4	yes	complemented
43	C₂²×C₄₇	—	2	2	no	—
44	C₂²×C₄₇	—	2	2	no	—
45	C₂²×C₄₇	—	2	2	no	—
46	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
47	C₂×D₄₇	—	2	4	no	—
48	C₂×D₄₇	—	2	4	no	—
49	C₂×D₄₇	—	2	4	no	—
50	C₄×C₄₇	—	2	2	no	—
51	C₄×C₄₇	—	2	1	no	—
52	C₄×C₄₇	C₂×D₄	1	2	yes	—
53	C₄×C₄₇	#41	1	2	yes	complemented
54	Dic₄₇	—	2	2	no	—
55	Dic₄₇	C₂×D₄	1	4	yes	—
56	C₂×C₄₇	—	4	2	no	—
57	C₂×C₄₇	C₂²×D₄	1	1	yes	characteristic
58	C₂×C₄₇	C₂×#41	1	1	yes	characteristic
59	C₂×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
60	D₄₇	—	4	4	no	—
61	#550	—	47	1	no	maximal, Sylow
62	C₄₇	#550	1	1	yes	characteristic, complemented, Sylow
63	C₂×#41	—	47	2	no	—
64	#127	—	47	2	no	—
65	#127	—	47	1	no	—
66	#121	—	47	2	no	—
67	#121	—	47	4	no	—
68	C₄×Q₈	—	47	1	no	—
69	C₄×D₄	—	47	2	no	—
70	C₄×D₄	D₄₇	1	1	yes	characteristic, complemented
71	#41	—	94	8	no	—
72	C₂×Q₈	—	47	1	no	—
73	C₂×D₄	—	47	4	no	—
74	C₂×D₄	—	47	4	no	—
75	C₂×D₄	—	47	2	no	—
76	C₂×D₄	—	47	2	no	—
77	C₂×D₄	—	47	2	no	—
78	C₂×D₄	C₂×D₄₇	1	1	yes	characteristic
79	C₂²×C₄	—	47	4	no	—
80	C₂²×C₄	C₂×D₄₇	1	2	yes	—
81	C₄⋊C₄	—	47	2	no	—
82	C₄⋊C₄	—	47	1	no	—
83	C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
84	C₂²⋊C₄	—	47	4	no	—
85	C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
86	C₄²	—	47	2	no	—
87	C₄²	C₂×D₄₇	1	1	yes	characteristic
88	C₂³	—	47	4	no	—
89	C₂³	C₂²×D₄₇	1	2	yes	—
90	Q₈	—	47	4	no	—
91	D₄	—	2	2	no	—
92	D₄	—	94	8	no	—
93	D₄	—	94	4	no	—
94	D₄	—	94	4	no	—
95	D₄	—	188	2	no	—
96	D₄	—	188	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	47	2	no	—
99	C₂×C₄	—	47	2	no	—
100	C₂×C₄	—	94	8	no	—
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	1	yes	characteristic
105	C₂²	—	2	2	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	94	4	no	—
109	C₂²	—	94	4	no	—
110	C₂²	—	94	4	no	—
111	C₂²	C₂³×D₄₇	1	1	yes	characteristic
112	C₄	—	2	2	no	—
113	C₄	—	2	1	no	—
114	C₄	—	47	4	no	—
115	C₄	—	94	2	no	—
116	C₄	#10625350	1	2	yes	complemented
117	C₄	D₄×D₄₇	1	2	yes	—
118	C₂	—	4	2	no	—
119	C₂	—	188	4	no	—
120	C₂	C₂×D₄×D₄₇	1	1	yes	characteristic
121	C₂	C₂×#10625350	1	1	yes	characteristic
122	C₂	#13094070	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625350

C₂

yes

maximal, complemented

#13094021

C₂

yes

maximal, characteristic, complemented

#13094020

C₂

yes

maximal, complemented

#13093998

C₂

yes

maximal, complemented

#13093997

C₂

yes

maximal, complemented

C₄×Dic₉₄

C₂

yes

maximal, characteristic, complemented

#13093986

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×C₂²⋊D₄₇

C₂²

yes

—

#10625350

—

C₂×D₁₈₈

C₂²

yes

—

C₂×D₁₈₈

C₂²

yes

—

C₂×D₁₈₈

C₂²

yes

—

C₂×C₄×D₄₇

C₂²

yes

—

C₂×Dic₉₄

C₂²

yes

characteristic

C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂²⋊C₄×C₄₇

C₂²

yes

—

C₄²×C₄₇

C₂²

yes

characteristic

(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₄⋊Dic₄₇

C₂²

yes

characteristic

Dic₄₇⋊C₄

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

—

C₂³×C₄₇

C₂³

yes

—

C₂²×D₄₇

C₂³

yes

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

D₄

yes

complemented

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂⁴

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₄

yes

—

C₄×C₄₇

#41

yes

complemented

Dic₄₇

—

Dic₄₇

C₂×D₄

yes

—

C₂×C₄₇

—

C₂×C₄₇

C₂²×D₄

yes

characteristic

C₂×C₄₇

C₂×#41

yes

characteristic

C₂×C₄₇

H(5,𝔽₂)

yes

characteristic

D₄₇

—

#550

—

maximal, Sylow

C₄₇

#550

yes

characteristic, complemented, Sylow

C₂×#41

—

—

—

—

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

D₄₇

yes

characteristic, complemented

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×D₄₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₄₇

yes

—

C₄²

—

C₄²

C₂×D₄₇

yes

characteristic

C₂³

—

C₂³

C₂²×D₄₇

yes

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

D₄

—

188

—

D₄

—

188

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

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

characteristic

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂³×D₄₇

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

#10625350

yes

complemented

117

C₄

D₄×D₄₇

yes

—

118

C₂

—

119

C₂

—

188

—

120

C₂

C₂×D₄×D₄₇

yes

characteristic

121

C₂

C₂×#10625350

yes

characteristic

122

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	#550
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. C₂×#10625350	C₂	semidirect
2. #13094021	C₂	semidirect
3. #13094020	C₂	semidirect
4. #13093998	C₂	semidirect
5. #13093997	C₂	semidirect
6. C₄×Dic₉₄	C₂	semidirect
7. #13093986	C₂	semidirect
8. C₄×D₄×C₄₇	C₂	semidirect
42. Dic₉₄	D₄	semidirect
53. C₄×C₄₇	#41	semidirect
62. C₄₇	#550	semidirect
70. C₄×D₄	D₄₇	semidirect
116. C₄	#10625350	semidirect

H ≅ G/N

type

1. C₂×#10625350

C₂

semidirect

2. #13094021

C₂

semidirect

3. #13094020

C₂

semidirect

4. #13093998

C₂

semidirect

5. #13093997

C₂

semidirect

6. C₄×Dic₉₄

C₂

semidirect

7. #13093986

C₂

semidirect

8. C₄×D₄×C₄₇

C₂

semidirect

42. Dic₉₄

D₄

semidirect

53. C₄×C₄₇

#41

semidirect

62. C₄₇

#550

semidirect

70. C₄×D₄

D₄₇

semidirect

116. C₄

#10625350

semidirect

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

subgroup	index
C₂×#10625350	2
#13094021	2
#13094020	2
#13093998	2
#13093997	2
C₄×Dic₉₄	2
#13093986	2
C₄×D₄×C₄₇	2
#550	47

subgroup

index

C₄×Dic₉₄

C₄×D₄×C₄₇

Subgroups — #15689794

Conjugacy classes of subgroups

Complements

Maximal subgroups