Group #15689867 — subgroups

Subgroups — #15689867

Order 3008.

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

normal subgroups only

This group has 220 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	#13094016	C₂	1	1	yes	maximal, characteristic, complemented
2	#13094012	C₂	1	1	yes	maximal, characteristic, complemented
3	#13094008	C₂	1	2	yes	maximal, complemented
4	#13094006	C₂	1	2	yes	maximal, complemented
5	#13094001	C₂	1	2	yes	maximal, complemented
6	#13094000	C₂	1	2	yes	maximal, complemented
7	#13093997	C₂	1	2	yes	maximal, complemented
8	#13093996	C₂	1	2	yes	maximal, complemented
9	#124×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×D₄×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₄₇	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₄₇	C₂²	1	2	yes	—
14	C₂²×Dic₄₇	C₂²	1	2	yes	—
15	C₂×D₁₈₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄₇	C₂²	1	2	yes	—
17	C₂×Dic₉₄	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
19	C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
20	C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²⋊Dic₄₇	C₂²	1	2	yes	—
22	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
25	C₄⋊Dic₄₇	C₂²	1	2	yes	—
26	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
27	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
28	Dic₄₇⋊C₄	C₂²	1	2	yes	—
29	Dic₄₇⋊C₄	C₂²	1	2	yes	—
30	Dic₄₇⋊C₄	C₂²	1	2	yes	—
31	C₂³×C₄₇	C₂³	1	2	yes	—
32	C₂²×D₄₇	C₂³	1	2	yes	—
33	Q₈×C₄₇	—	4	1	no	—
34	D₄×C₄₇	—	4	1	no	—
35	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₄₇	C₂³	1	2	yes	—
38	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂²⋊D₄₇	—	4	2	no	—
40	C₂²⋊D₄₇	—	4	2	no	—
41	C₂×Dic₄₇	—	4	2	no	—
42	C₂×Dic₄₇	C₂³	1	2	yes	—
43	C₂×Dic₄₇	C₂³	1	2	yes	—
44	C₂×Dic₄₇	C₂³	1	2	yes	—
45	D₁₈₈	—	4	1	no	—
46	C₄×D₄₇	—	4	2	no	—
47	Dic₉₄	—	4	1	no	—
48	C₂²×C₄₇	—	2	2	no	—
49	C₂²×C₄₇	—	2	2	no	—
50	C₂²×C₄₇	—	2	2	no	—
51	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
52	C₂×D₄₇	—	2	2	no	—
53	C₂×D₄₇	—	2	2	no	—
54	C₂×D₄₇	—	2	2	no	—
55	C₄×C₄₇	—	2	1	no	—
56	C₄×C₄₇	—	2	2	no	—
57	C₄×C₄₇	—	2	1	no	—
58	C₄×C₄₇	—	2	1	no	—
59	Dic₄₇	—	2	2	no	—
60	Dic₄₇	—	2	2	no	—
61	Dic₄₇	—	2	2	no	—
62	C₂×C₄₇	—	4	2	no	—
63	C₂×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
66	D₄₇	—	4	2	no	—
67	#562	—	47	1	no	maximal, Sylow
68	C₄₇	#562	1	1	yes	characteristic, complemented, Sylow
69	#125	—	47	1	no	—
70	#124	—	47	1	no	—
71	#124	D₄₇	1	1	yes	characteristic, complemented
72	#123	—	47	2	no	—
73	#123	—	47	2	no	—
74	C₂²⋊Q₈	—	47	2	no	—
75	C₂²⋊Q₈	—	47	2	no	—
76	#121	—	47	2	no	—
77	#121	—	47	2	no	—
78	C₂×Q₈	—	47	1	no	—
79	C₂×Q₈	C₂×D₄₇	1	1	yes	characteristic
80	C₂×D₄	—	47	2	no	—
81	C₂×D₄	—	47	2	no	—
82	C₂×D₄	—	47	1	no	—
83	C₂×D₄	C₂×D₄₇	1	1	yes	characteristic
84	C₂²×C₄	—	47	2	no	—
85	C₂²×C₄	—	47	2	no	—
86	C₄⋊C₄	—	47	2	no	—
87	C₄⋊C₄	—	47	2	no	—
88	C₄⋊C₄	—	47	2	no	—
89	C₄⋊C₄	—	47	2	no	—
90	C₄⋊C₄	—	47	1	no	—
91	C₄⋊C₄	—	47	1	no	—
92	C₂²⋊C₄	—	47	2	no	—
93	C₂²⋊C₄	—	47	2	no	—
94	C₂²⋊C₄	—	47	2	no	—
95	C₂²⋊C₄	—	47	2	no	—
96	C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
97	C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
98	C₄²	C₂×D₄₇	1	1	yes	characteristic
99	C₂³	—	47	2	no	—
100	C₂³	C₂²×D₄₇	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	188	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	188	1	no	—
105	D₄	—	188	2	no	—
106	D₄	—	188	2	no	—
107	C₂×C₄	—	47	2	no	—
108	C₂×C₄	—	47	2	no	—
109	C₂×C₄	—	47	2	no	—
110	C₂×C₄	—	188	2	no	—
111	C₂×C₄	—	188	2	no	—
112	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₄₇	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	94	2	no	—
120	C₂²	—	94	2	no	—
121	C₂²	—	94	2	no	—
122	C₂²	C₂³×D₄₇	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	94	2	no	—
128	C₄	—	94	2	no	—
129	C₄	—	94	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	188	2	no	—
132	C₂	#13094069	1	1	yes	characteristic
133	C₂	#13094070	1	1	yes	characteristic
134	C₂	#13094071	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13094016

C₂

yes

maximal, characteristic, complemented

#13094012

C₂

yes

maximal, characteristic, complemented

#13094008

C₂

yes

maximal, complemented

#13094006

C₂

yes

maximal, complemented

#13094001

C₂

yes

maximal, complemented

#13094000

C₂

yes

maximal, complemented

#13093997

C₂

yes

maximal, complemented

#13093996

C₂

yes

maximal, complemented

#124×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊D₄₇

C₂²

yes

—

C₂×C₂²⋊D₄₇

C₂²

yes

—

C₂²×Dic₄₇

C₂²

yes

—

C₂×D₁₈₈

C₂²

yes

characteristic

C₂×C₄×D₄₇

C₂²

yes

—

C₂×Dic₉₄

C₂²

yes

characteristic

C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂²⋊C₄×C₄₇

C₂²

yes

—

C₄²×C₄₇

C₂²

yes

characteristic

C₂²⋊Dic₄₇

C₂²

yes

—

(C₂×D₄₇)⋊C₄

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

characteristic

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

C₂³×C₄₇

C₂³

yes

—

C₂²×D₄₇

C₂³

yes

—

Q₈×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂³

yes

—

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

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

2₋¹⁺⁴

yes

characteristic

D₄₇

—

#562

—

maximal, Sylow

C₄₇

#562

yes

characteristic, complemented, Sylow

#125

—

#124

—

#124

D₄₇

yes

characteristic, complemented

#123

—

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

#121

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₄₇

yes

—

C₂²⋊C₄

C₂×D₄₇

yes

—

C₄²

C₂×D₄₇

yes

characteristic

C₂³

—

100

C₂³

C₂²×D₄₇

yes

—

101

Q₈

—

102

Q₈

—

188

—

103

D₄

—

104

D₄

—

188

—

105

D₄

—

188

—

106

D₄

—

188

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

188

—

111

C₂×C₄

—

188

—

112

C₂×C₄

C₂²×D₄₇

yes

characteristic

113

C₂×C₄

C₂²×D₄₇

yes

characteristic

114

C₂×C₄

C₂²×D₄₇

yes

characteristic

115

C₂×C₄

C₂²×D₄₇

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂³×D₄₇

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₂

—

131

C₂

—

188

—

132

C₂

#13094069

yes

characteristic

133

C₂

#13094070

yes

characteristic

134

C₂

#13094071

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	#124×C₄₇	C₂
hypercenter	#124	D₄₇
hypocenter	C₄₇	#562
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₇

commutator, socle, cosocle

C₂²×C₄₇

C₂⁴

Fitting

#124×C₄₇

C₂

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #13094016	C₂	semidirect
2. #13094012	C₂	semidirect
3. #13094008	C₂	semidirect
4. #13094006	C₂	semidirect
5. #13094001	C₂	semidirect
6. #13094000	C₂	semidirect
7. #13093997	C₂	semidirect
8. #13093996	C₂	semidirect
9. #124×C₄₇	C₂	semidirect
68. C₄₇	#562	semidirect
71. #124	D₄₇	semidirect

H ≅ G/N

type

1. #13094016

C₂

semidirect

2. #13094012

C₂

semidirect

3. #13094008

C₂

semidirect

4. #13094006

C₂

semidirect

5. #13094001

C₂

semidirect

6. #13094000

C₂

semidirect

7. #13093997

C₂

semidirect

8. #13093996

C₂

semidirect

9. #124×C₄₇

C₂

semidirect

68. C₄₇

#562

semidirect

71. #124

D₄₇

semidirect

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

subgroup	index
#13094016	2
#13094012	2
#13094008	2
#13094006	2
#13094001	2
#13094000	2
#13093997	2
#13093996	2
#124×C₄₇	2
#562	47

subgroup

index

#124×C₄₇

Subgroups — #15689867

Conjugacy classes of subgroups

Complements

Maximal subgroups