Group #15689686 — subgroups

Subgroups — #15689686

Order 3008.

334 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 103 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×#10625352	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#13094020	C₂	1	1	yes	maximal, characteristic, complemented
4	#13094012	C₂	1	1	yes	maximal, characteristic, complemented
5	#13094011	C₂	1	1	yes	maximal, characteristic, complemented
6	#13093996	C₂	1	4	yes	maximal, complemented
7	#13093992	C₂	1	4	yes	maximal, complemented
8	#13093981	C₂	1	1	yes	maximal, characteristic, complemented
9	#127×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄₇	C₂²	1	2	yes	—
11	C₂×D₄×C₄₇	C₂²	1	1	yes	characteristic
12	C₂×D₄×C₄₇	C₂²	1	1	yes	characteristic, complemented
13	C₂×D₄×C₄₇	C₂²	1	4	yes	complemented
14	C₂×C₂²⋊D₄₇	C₂²	1	2	yes	—
15	C₂×C₂²⋊D₄₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊D₄₇	C₂²	1	4	yes	complemented
17	C₂²×Dic₄₇	C₂²	1	2	yes	complemented
18	#10625352	—	2	4	no	—
19	D₄×D₄₇	—	2	4	no	—
20	C₂×D₁₈₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄₇	C₂²	1	1	yes	characteristic
22	C₂×Dic₉₄	C₂²	1	1	yes	characteristic, complemented
23	C₄²×C₄₇	C₂²	1	1	yes	characteristic, complemented
24	C₂²⋊Dic₄₇	C₂²	1	4	yes	complemented
25	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
26	(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	complemented
27	Dic₄₇⋊C₄	C₂²	1	2	yes	complemented
28	C₄×Dic₄₇	C₂²	1	1	yes	characteristic, complemented
29	C₂³×C₄₇	C₂³	1	2	yes	—
30	C₂³×C₄₇	C₂³	1	2	yes	—
31	C₂²×D₄₇	—	2	4	no	—
32	C₂²×D₄₇	—	4	4	no	—
33	C₂²×D₄₇	C₂³	1	2	yes	—
34	C₂²×D₄₇	C₂³	1	1	yes	characteristic
35	D₄×C₄₇	—	2	2	no	—
36	D₄×C₄₇	—	2	2	no	—
37	D₄×C₄₇	—	4	4	no	—
38	C₂×C₄×C₄₇	C₂³	1	2	yes	—
39	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
40	C₂²⋊D₄₇	—	2	4	no	—
41	C₂²⋊D₄₇	—	2	4	no	—
42	C₂²⋊D₄₇	—	4	4	no	—
43	C₂×Dic₄₇	—	2	4	no	—
44	C₂×Dic₄₇	C₂³	1	2	yes	—
45	C₂×Dic₄₇	C₂³	1	2	yes	—
46	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
47	D₁₈₈	—	2	2	no	—
48	C₄×D₄₇	D₄	1	4	yes	complemented
49	Dic₉₄	—	2	2	no	—
50	C₂²×C₄₇	—	2	2	no	—
51	C₂²×C₄₇	—	2	2	no	—
52	C₂²×C₄₇	—	2	4	no	—
53	C₂²×C₄₇	—	2	4	no	—
54	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
55	C₂×D₄₇	—	2	2	no	—
56	C₂×D₄₇	—	2	2	no	—
57	C₂×D₄₇	—	2	4	no	—
58	C₂×D₄₇	—	4	8	no	—
59	C₂×D₄₇	C₂×D₄	1	2	yes	—
60	C₄×C₄₇	—	2	2	no	—
61	C₄×C₄₇	C₂×D₄	1	2	yes	—
62	Dic₄₇	—	2	2	no	—
63	Dic₄₇	—	2	2	no	—
64	Dic₄₇	C₂×D₄	1	2	yes	—
65	C₂×C₄₇	—	4	2	no	—
66	C₂×C₄₇	—	4	2	no	—
67	C₂×C₄₇	C₂²×D₄	1	1	yes	characteristic
68	C₂×C₄₇	H(5,𝔽₂)	1	2	yes	—
69	D₄₇	—	2	2	no	—
70	D₄₇	—	4	2	no	—
71	#535	—	47	1	no	maximal, Sylow
72	C₄₇	#535	1	1	yes	characteristic, complemented, Sylow
73	C₂×#41	—	47	1	no	—
74	C₂²×D₄	—	47	1	no	—
75	#127	—	47	1	no	—
76	#127	D₄₇	1	1	yes	characteristic, complemented
77	#124	—	47	1	no	—
78	#124	—	47	1	no	—
79	#121	—	47	4	no	—
80	#120	—	47	4	no	—
81	#117	—	47	1	no	—
82	C₂⁴	—	47	2	no	—
83	#41	—	94	4	no	—
84	C₂×Q₈	—	47	1	no	—
85	C₂×D₄	—	47	2	no	—
86	C₂×D₄	—	47	2	no	—
87	C₂×D₄	—	47	4	no	—
88	C₂×D₄	—	47	1	no	—
89	C₂×D₄	—	94	4	no	—
90	C₂×D₄	C₂×D₄₇	1	1	yes	characteristic
91	C₂×D₄	C₂×D₄₇	1	1	yes	characteristic, complemented
92	C₂×D₄	C₂×D₄₇	1	4	yes	complemented
93	C₂²×C₄	—	47	1	no	—
94	C₂²×C₄	—	47	2	no	—
95	C₄⋊C₄	—	47	2	no	—
96	C₂²⋊C₄	—	47	4	no	—
97	C₂²⋊C₄	—	47	2	no	—
98	C₂²⋊C₄	—	47	4	no	—
99	C₄²	—	47	1	no	—
100	C₄²	C₂×D₄₇	1	1	yes	characteristic, complemented
101	C₂³	—	47	1	no	—
102	C₂³	—	47	2	no	—
103	C₂³	—	94	4	no	—
104	C₂³	—	188	4	no	—
105	C₂³	C₂²×D₄₇	1	2	yes	—
106	C₂³	C₂²×D₄₇	1	2	yes	—
107	Q₈	—	94	2	no	—
108	D₄	—	2	2	no	—
109	D₄	—	2	2	no	—
110	D₄	—	4	4	no	—
111	D₄	—	94	4	no	—
112	D₄	—	94	4	no	—
113	D₄	—	94	2	no	—
114	D₄	—	188	4	no	—
115	C₂×C₄	—	47	1	no	—
116	C₂×C₄	—	47	4	no	—
117	C₂×C₄	—	47	2	no	—
118	C₂×C₄	—	47	2	no	—
119	C₂×C₄	—	94	4	no	—
120	C₂×C₄	C₂²×D₄₇	1	2	yes	—
121	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	2	4	no	—
126	C₂²	—	47	2	no	—
127	C₂²	—	94	2	no	—
128	C₂²	—	94	2	no	—
129	C₂²	—	94	4	no	—
130	C₂²	—	188	8	no	—
131	C₂²	C₂³×D₄₇	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	47	2	no	—
134	C₄	—	94	2	no	—
135	C₄	—	94	2	no	—
136	C₄	D₄×D₄₇	1	2	yes	—
137	C₂	—	4	2	no	—
138	C₂	—	4	2	no	—
139	C₂	—	94	2	no	—
140	C₂	—	188	2	no	—
141	C₂	C₂×D₄×D₄₇	1	1	yes	characteristic
142	C₂	#13094069	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625352

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₄₇

C₂

yes

maximal, characteristic, complemented

#13094020

C₂

yes

maximal, characteristic, complemented

#13094012

C₂

yes

maximal, characteristic, complemented

#13094011

C₂

yes

maximal, characteristic, complemented

#13093996

C₂

yes

maximal, complemented

#13093992

C₂

yes

maximal, complemented

#13093981

C₂

yes

maximal, characteristic, complemented

#127×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₄₇

C₂²

yes

—

C₂×D₄×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

C₂²

yes

characteristic, complemented

C₂×D₄×C₄₇

C₂²

yes

complemented

C₂×C₂²⋊D₄₇

C₂²

yes

—

C₂×C₂²⋊D₄₇

C₂²

yes

—

C₂×C₂²⋊D₄₇

C₂²

yes

complemented

C₂²×Dic₄₇

C₂²

yes

complemented

#10625352

—

D₄×D₄₇

—

C₂×D₁₈₈

C₂²

yes

characteristic

C₂×C₄×D₄₇

C₂²

yes

characteristic

C₂×Dic₉₄

C₂²

yes

characteristic, complemented

C₄²×C₄₇

C₂²

yes

characteristic, complemented

C₂²⋊Dic₄₇

C₂²

yes

complemented

(C₂×D₄₇)⋊C₄

C₂²

yes

—

(C₂×D₄₇)⋊C₄

C₂²

yes

complemented

Dic₄₇⋊C₄

C₂²

yes

complemented

C₄×Dic₄₇

C₂²

yes

characteristic, complemented

C₂³×C₄₇

C₂³

yes

—

C₂³×C₄₇

C₂³

yes

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂³

yes

—

C₂²×D₄₇

C₂³

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

C₂³

yes

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

characteristic

D₁₈₈

—

C₄×D₄₇

D₄

yes

complemented

Dic₉₄

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂⁴

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

C₂×D₄

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₄

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

C₂×D₄

yes

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂²×D₄

yes

characteristic

C₂×C₄₇

H(5,𝔽₂)

yes

—

D₄₇

—

D₄₇

—

#535

—

maximal, Sylow

C₄₇

#535

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×D₄

—

#127

—

#127

D₄₇

yes

characteristic, complemented

—

—

—

—

—

C₂⁴

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₄₇

yes

characteristic

C₂×D₄

C₂×D₄₇

yes

characteristic, complemented

C₂×D₄

C₂×D₄₇

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

100

C₄²

C₂×D₄₇

yes

characteristic, complemented

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

188

—

105

C₂³

C₂²×D₄₇

yes

—

106

C₂³

C₂²×D₄₇

yes

—

107

Q₈

—

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

—

113

D₄

—

114

D₄

—

188

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂²×D₄₇

yes

—

121

C₂×C₄

C₂²×D₄₇

yes

characteristic

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

188

—

131

C₂²

C₂³×D₄₇

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

D₄×D₄₇

yes

—

137

C₂

—

138

C₂

—

139

C₂

—

140

C₂

—

188

—

141

C₂

C₂×D₄×D₄₇

yes

characteristic

142

C₂

#13094069

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	#127×C₄₇	C₂
hypercenter	#127	D₄₇
hypocenter	C₄₇	#535
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #127

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₇

commutator, socle, cosocle

C₂²×C₄₇

C₂⁴

Fitting

#127×C₄₇

C₂

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂² ◁ #127

N	H ≅ G/N	type
1. C₂×#10625352	C₂	semidirect
2. C₂×D₄×D₄₇	C₂	semidirect
3. #13094020	C₂	semidirect
4. #13094012	C₂	semidirect
5. #13094011	C₂	semidirect
6. #13093996	C₂	semidirect
7. #13093992	C₂	semidirect
8. #13093981	C₂	semidirect
9. #127×C₄₇	C₂	semidirect
12. C₂×D₄×C₄₇	C₂²	semidirect
13. C₂×D₄×C₄₇	C₂²	semidirect
16. C₂×C₂²⋊D₄₇	C₂²	semidirect
17. C₂²×Dic₄₇	C₂²	semidirect
22. C₂×Dic₉₄	C₂²	semidirect
23. C₄²×C₄₇	C₂²	semidirect
24. C₂²⋊Dic₄₇	C₂²	semidirect
26. (C₂×D₄₇)⋊C₄	C₂²	semidirect
27. Dic₄₇⋊C₄	C₂²	semidirect
28. C₄×Dic₄₇	C₂²	semidirect
48. C₄×D₄₇	D₄	semidirect
72. C₄₇	#535	semidirect
76. #127	D₄₇	semidirect
91. C₂×D₄	C₂×D₄₇	semidirect
92. C₂×D₄	C₂×D₄₇	semidirect
100. C₄²	C₂×D₄₇	semidirect

H ≅ G/N

type

1. C₂×#10625352

C₂

semidirect

2. C₂×D₄×D₄₇

C₂

semidirect

3. #13094020

C₂

semidirect

4. #13094012

C₂

semidirect

5. #13094011

C₂

semidirect

6. #13093996

C₂

semidirect

7. #13093992

C₂

semidirect

8. #13093981

C₂

semidirect

9. #127×C₄₇

C₂

semidirect

12. C₂×D₄×C₄₇

C₂²

semidirect

13. C₂×D₄×C₄₇

C₂²

semidirect

16. C₂×C₂²⋊D₄₇

C₂²

semidirect

17. C₂²×Dic₄₇

C₂²

semidirect

22. C₂×Dic₉₄

C₂²

semidirect

23. C₄²×C₄₇

C₂²

semidirect

24. C₂²⋊Dic₄₇

C₂²

semidirect

26. (C₂×D₄₇)⋊C₄

C₂²

semidirect

27. Dic₄₇⋊C₄

C₂²

semidirect

28. C₄×Dic₄₇

C₂²

semidirect

48. C₄×D₄₇

D₄

semidirect

72. C₄₇

#535

semidirect

76. #127

D₄₇

semidirect

91. C₂×D₄

C₂×D₄₇

semidirect

92. C₂×D₄

C₂×D₄₇

semidirect

100. C₄²

C₂×D₄₇

semidirect

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

subgroup	index
C₂×#10625352	2
C₂×D₄×D₄₇	2
#13094020	2
#13094012	2
#13094011	2
#13093996	2
#13093992	2
#13093981	2
#127×C₄₇	2
#535	47

subgroup

index

C₂×D₄×D₄₇

#127×C₄₇

Subgroups — #15689686

Conjugacy classes of subgroups

Complements

Maximal subgroups