Group #15689920 — subgroups

Subgroups — #15689920

Order 3008.

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

normal subgroups only

This group has 262 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	#41×D₄₇	C₂	1	2	yes	maximal, complemented
2	#13094056	C₂	1	2	yes	maximal, complemented
3	#13094053	C₂	1	2	yes	maximal, complemented
4	#13094051	C₂	1	2	yes	maximal, complemented
5	#13094048	C₂	1	2	yes	maximal, complemented
6	#13094046	C₂	1	1	yes	maximal, characteristic, complemented
7	#13094044	C₂	1	2	yes	maximal, complemented
8	C₂×#10625324	C₂	1	1	yes	maximal, characteristic, complemented
9	#135×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#41×C₄₇	C₂²	1	2	yes	complemented
11	#10625354	C₂²	1	2	yes	—
12	Q₈×D₄₇	C₂²	1	2	yes	complemented
13	#10625352	—	2	2	no	—
14	#10625352	C₂²	1	2	yes	—
15	D₄×D₄₇	—	2	2	no	—
16	D₄×D₄₇	C₂²	1	2	yes	complemented
17	#10625350	—	2	2	no	—
18	#10625350	C₂²	1	2	yes	complemented
19	C₂×C₄×D₄₇	—	2	2	no	—
20	C₂×C₄×D₄₇	C₂²	1	1	yes	characteristic
21	Q₁₆×C₄₇	C₂²	1	1	yes	characteristic, complemented
22	SD₁₆×C₄₇	C₂²	1	2	yes	complemented
23	D₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
24	C₂×C₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
25	#10625337	C₂²	1	2	yes	complemented
26	#10625336	C₂²	1	2	yes	complemented
27	#10625335	C₂²	1	2	yes	complemented
28	#10625334	C₂²	1	2	yes	complemented
29	C₂×C₄₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
30	Dic₁₈₈	C₂²	1	1	yes	characteristic, complemented
31	D₃₇₆	C₂²	1	1	yes	characteristic, complemented
32	#10625325	C₂²	1	2	yes	complemented
33	#10625324	C₂²	1	2	yes	complemented
34	#10625324	C₂²	1	2	yes	complemented
35	C₂²×D₄₇	—	2	2	no	—
36	C₂²×D₄₇	D₄	1	1	yes	characteristic
37	Q₈×C₄₇	C₂³	1	2	yes	—
38	D₄×C₄₇	—	2	2	no	—
39	D₄×C₄₇	C₂³	1	2	yes	—
40	C₂×C₄×C₄₇	—	2	2	no	—
41	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂²⋊D₄₇	—	2	2	no	—
43	C₂²⋊D₄₇	—	2	2	no	—
44	C₂²⋊D₄₇	—	2	2	no	—
45	C₂×Dic₄₇	—	2	2	no	—
46	C₂×Dic₄₇	D₄	1	1	yes	characteristic
47	D₁₈₈	—	2	2	no	—
48	D₁₈₈	C₂³	1	2	yes	—
49	C₄×D₄₇	—	2	2	no	—
50	C₄×D₄₇	—	2	2	no	—
51	C₄×D₄₇	—	2	2	no	—
52	C₄×D₄₇	D₄	1	2	yes	—
53	C₄×D₄₇	C₂³	1	2	yes	—
54	Dic₉₄	—	2	2	no	—
55	Dic₉₄	C₂³	1	2	yes	—
56	C₈×C₄₇	C₂³	1	1	yes	characteristic
57	C₈×C₄₇	C₂³	1	1	yes	characteristic
58	C₄₇⋊C₈	C₂³	1	2	yes	—
59	C₂²×C₄₇	—	2	2	no	—
60	C₂²×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂×D₄₇	—	2	2	no	—
62	C₂×D₄₇	—	4	1	no	—
63	C₂×D₄₇	—	4	4	no	—
64	C₂×D₄₇	C₂×D₄	1	2	yes	—
65	C₄×C₄₇	—	2	2	no	—
66	C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
67	C₄×C₄₇	C₂⁴	1	1	yes	characteristic
68	Dic₄₇	—	2	2	no	—
69	Dic₄₇	C₂×D₄	1	2	yes	—
70	C₂×C₄₇	—	2	1	no	—
71	C₂×C₄₇	—	4	2	no	—
72	C₂×C₄₇	C₂²×D₄	1	1	yes	characteristic
73	D₄₇	—	2	2	no	—
74	D₄₇	—	4	2	no	—
75	#575	—	47	1	no	maximal, Sylow
76	C₄₇	#575	1	1	yes	characteristic, complemented, Sylow
77	C₂×#41	—	47	2	no	—
78	#137	—	47	2	no	—
79	#137	—	47	2	no	—
80	AGL(1,ℤ/8)	—	47	2	no	—
81	AGL(1,ℤ/8)	—	47	2	no	—
82	#135	—	47	2	no	—
83	#135	—	47	1	no	—
84	#135	D₄₇	1	1	yes	characteristic, complemented
85	C₂×M₁₆	—	47	1	no	—
86	#41	—	47	2	no	—
87	#41	—	47	2	no	—
88	#41	—	47	2	no	—
89	#41	—	94	2	no	—
90	#41	—	94	2	no	—
91	#41	C₂×D₄₇	1	2	yes	complemented
92	C₂×Q₈	—	47	2	no	—
93	C₂×D₄	—	47	2	no	—
94	C₂×D₄	—	94	2	no	—
95	C₂²×C₄	—	47	1	no	—
96	C₂²×C₄	—	94	2	no	—
97	Q₁₆	—	47	2	no	—
98	Q₁₆	—	47	1	no	—
99	Q₁₆	C₂×D₄₇	1	1	yes	characteristic, complemented
100	SD₁₆	—	47	2	no	—
101	SD₁₆	—	47	2	no	—
102	SD₁₆	—	47	2	no	—
103	SD₁₆	C₂×D₄₇	1	2	yes	complemented
104	D₈	—	47	2	no	—
105	D₈	—	47	1	no	—
106	D₈	C₂×D₄₇	1	1	yes	characteristic, complemented
107	M₁₆	—	47	2	no	—
108	M₁₆	—	47	2	no	—
109	C₂×C₈	—	47	1	no	—
110	C₂×C₈	C₂×D₄₇	1	1	yes	characteristic, complemented
111	C₂³	—	47	1	no	—
112	C₂³	—	94	2	no	—
113	Q₈	—	47	2	no	—
114	Q₈	—	94	2	no	—
115	Q₈	C₂²×D₄₇	1	2	yes	—
116	D₄	—	2	2	no	—
117	D₄	—	47	2	no	—
118	D₄	—	94	2	no	—
119	D₄	—	94	2	no	—
120	D₄	—	94	2	no	—
121	D₄	—	94	2	no	—
122	D₄	C₂²×D₄₇	1	2	yes	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	47	1	no	—
125	C₂×C₄	—	47	2	no	—
126	C₂×C₄	—	47	2	no	—
127	C₂×C₄	—	94	2	no	—
128	C₂×C₄	—	94	2	no	—
129	C₂×C₄	—	94	2	no	—
130	C₂×C₄	—	94	2	no	—
131	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
132	C₈	—	47	2	no	—
133	C₈	C₂²×D₄₇	1	1	yes	characteristic
134	C₈	C₂²×D₄₇	1	1	yes	characteristic
135	C₂²	—	2	2	no	—
136	C₂²	—	47	2	no	—
137	C₂²	—	94	2	no	—
138	C₂²	—	188	1	no	—
139	C₂²	—	188	4	no	—
140	C₂²	D₄×D₄₇	1	1	yes	characteristic
141	C₄	—	2	2	no	—
142	C₄	—	47	2	no	—
143	C₄	—	94	2	no	—
144	C₄	D₄×D₄₇	1	1	yes	characteristic
145	C₄	C₂³×D₄₇	1	1	yes	characteristic
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	—	94	2	no	—
149	C₂	—	188	2	no	—
150	C₂	C₂×D₄×D₄₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#41×D₄₇

C₂

yes

maximal, complemented

#13094056

C₂

yes

maximal, complemented

#13094053

C₂

yes

maximal, complemented

#13094051

C₂

yes

maximal, complemented

#13094048

C₂

yes

maximal, complemented

#13094046

C₂

yes

maximal, characteristic, complemented

#13094044

C₂

yes

maximal, complemented

C₂×#10625324

C₂

yes

maximal, characteristic, complemented

#135×C₄₇

C₂

yes

maximal, characteristic, complemented

#41×C₄₇

C₂²

yes

complemented

#10625354

C₂²

yes

—

Q₈×D₄₇

C₂²

yes

complemented

#10625352

—

#10625352

C₂²

yes

—

D₄×D₄₇

—

D₄×D₄₇

C₂²

yes

complemented

#10625350

—

#10625350

C₂²

yes

complemented

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

C₂²

yes

characteristic

Q₁₆×C₄₇

C₂²

yes

characteristic, complemented

SD₁₆×C₄₇

C₂²

yes

complemented

D₈×C₄₇

C₂²

yes

characteristic, complemented

C₂×C₈×C₄₇

C₂²

yes

characteristic, complemented

#10625337

C₂²

yes

complemented

#10625336

C₂²

yes

complemented

#10625335

C₂²

yes

complemented

#10625334

C₂²

yes

complemented

C₂×C₄₇⋊C₈

C₂²

yes

characteristic, complemented

Dic₁₈₈

C₂²

yes

characteristic, complemented

D₃₇₆

C₂²

yes

characteristic, complemented

#10625325

C₂²

yes

complemented

#10625324

C₂²

yes

complemented

#10625324

C₂²

yes

complemented

C₂²×D₄₇

—

C₂²×D₄₇

D₄

yes

characteristic

Q₈×C₄₇

C₂³

yes

—

D₄×C₄₇

—

D₄×C₄₇

C₂³

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

D₄

yes

characteristic

D₁₈₈

—

D₁₈₈

C₂³

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

D₄

yes

—

C₄×D₄₇

C₂³

yes

—

Dic₉₄

—

Dic₉₄

C₂³

yes

—

C₈×C₄₇

C₂³

yes

characteristic

C₈×C₄₇

C₂³

yes

characteristic

C₄₇⋊C₈

C₂³

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

C₂×D₄

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₄

yes

characteristic

C₄×C₄₇

C₂⁴

yes

characteristic

Dic₄₇

—

Dic₄₇

C₂×D₄

yes

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂²×D₄

yes

characteristic

D₄₇

—

D₄₇

—

#575

—

maximal, Sylow

C₄₇

#575

yes

characteristic, complemented, Sylow

C₂×#41

—

—

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₂×M₁₆

—

—

—

—

—

—

C₂×D₄₇

yes

complemented

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

Q₁₆

C₂×D₄₇

yes

characteristic, complemented

100

SD₁₆

—

101

SD₁₆

—

102

SD₁₆

—

103

SD₁₆

C₂×D₄₇

yes

complemented

104

D₈

—

105

D₈

—

106

D₈

C₂×D₄₇

yes

characteristic, complemented

107

M₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

C₂×D₄₇

yes

characteristic, complemented

111

C₂³

—

112

C₂³

—

113

Q₈

—

114

Q₈

—

115

Q₈

C₂²×D₄₇

yes

—

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

C₂²×D₄₇

yes

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₂²×D₄₇

yes

characteristic

132

C₈

—

133

C₈

C₂²×D₄₇

yes

characteristic

134

C₈

C₂²×D₄₇

yes

characteristic

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

188

—

139

C₂²

—

188

—

140

C₂²

D₄×D₄₇

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

D₄×D₄₇

yes

characteristic

145

C₄

C₂³×D₄₇

yes

characteristic

146

C₂

—

147

C₂

—

148

C₂

—

149

C₂

—

188

—

150

C₂

C₂×D₄×D₄₇

yes

characteristic

	subgroup	G/H
center	C₄	D₄×D₄₇
commutator, cosocle	C₄×C₄₇	C₂⁴
Frattini	C₄	C₂³×D₄₇
Fitting	#135×C₄₇	C₂
socle	C₂×C₄₇	C₂²×D₄
hypercenter	#135	D₄₇
hypocenter	C₄₇	#575
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

subgroup

G/H

center

C₄

D₄×D₄₇

commutator, cosocle

C₄×C₄₇

C₂⁴

Frattini

C₄

C₂³×D₄₇

Fitting

#135×C₄₇

C₂

socle

C₂×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

N	H ≅ G/N	type
1. #41×D₄₇	C₂	semidirect
2. #13094056	C₂	semidirect
3. #13094053	C₂	semidirect
4. #13094051	C₂	semidirect
5. #13094048	C₂	semidirect
6. #13094046	C₂	semidirect
7. #13094044	C₂	semidirect
8. C₂×#10625324	C₂	semidirect
9. #135×C₄₇	C₂	semidirect
10. #41×C₄₇	C₂²	semidirect
12. Q₈×D₄₇	C₂²	semidirect
16. D₄×D₄₇	C₂²	semidirect
18. #10625350	C₂²	semidirect
21. Q₁₆×C₄₇	C₂²	semidirect
22. SD₁₆×C₄₇	C₂²	semidirect
23. D₈×C₄₇	C₂²	semidirect
24. C₂×C₈×C₄₇	C₂²	semidirect
25. #10625337	C₂²	semidirect
26. #10625336	C₂²	semidirect
27. #10625335	C₂²	semidirect
28. #10625334	C₂²	semidirect
29. C₂×C₄₇⋊C₈	C₂²	semidirect
30. Dic₁₈₈	C₂²	semidirect
31. D₃₇₆	C₂²	semidirect
32. #10625325	C₂²	semidirect
33. #10625324	C₂²	semidirect
34. #10625324	C₂²	semidirect
76. C₄₇	#575	semidirect
84. #135	D₄₇	semidirect
91. #41	C₂×D₄₇	semidirect
99. Q₁₆	C₂×D₄₇	semidirect
103. SD₁₆	C₂×D₄₇	semidirect
106. D₈	C₂×D₄₇	semidirect
110. C₂×C₈	C₂×D₄₇	semidirect

H ≅ G/N

type

1. #41×D₄₇

C₂

semidirect

2. #13094056

C₂

semidirect

3. #13094053

C₂

semidirect

4. #13094051

C₂

semidirect

5. #13094048

C₂

semidirect

6. #13094046

C₂

semidirect

7. #13094044

C₂

semidirect

8. C₂×#10625324

C₂

semidirect

9. #135×C₄₇

C₂

semidirect

10. #41×C₄₇

C₂²

semidirect

12. Q₈×D₄₇

C₂²

semidirect

16. D₄×D₄₇

C₂²

semidirect

18. #10625350

C₂²

semidirect

21. Q₁₆×C₄₇

C₂²

semidirect

22. SD₁₆×C₄₇

C₂²

semidirect

23. D₈×C₄₇

C₂²

semidirect

24. C₂×C₈×C₄₇

C₂²

semidirect

25. #10625337

C₂²

semidirect

26. #10625336

C₂²

semidirect

27. #10625335

C₂²

semidirect

28. #10625334

C₂²

semidirect

29. C₂×C₄₇⋊C₈

C₂²

semidirect

30. Dic₁₈₈

C₂²

semidirect

31. D₃₇₆

C₂²

semidirect

32. #10625325

C₂²

semidirect

33. #10625324

C₂²

semidirect

34. #10625324

C₂²

semidirect

76. C₄₇

#575

semidirect

84. #135

D₄₇

semidirect

91. #41

C₂×D₄₇

semidirect

99. Q₁₆

C₂×D₄₇

semidirect

103. SD₁₆

C₂×D₄₇

semidirect

106. D₈

C₂×D₄₇

semidirect

110. C₂×C₈

C₂×D₄₇

semidirect

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

subgroup	index
#41×D₄₇	2
#13094056	2
#13094053	2
#13094051	2
#13094048	2
#13094046	2
#13094044	2
C₂×#10625324	2
#135×C₄₇	2
#575	47

subgroup

index

#41×D₄₇

#135×C₄₇

Subgroups — #15689920

Conjugacy classes of subgroups

Complements

Maximal subgroups