Group #15689842 — subgroups

Subgroups — #15689842

Order 3008.

212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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	#13094022	C₂	1	1	yes	maximal, characteristic, complemented
2	#13094017	C₂	1	2	yes	maximal, complemented
3	#13094017	C₂	1	2	yes	maximal, complemented
4	#13094011	C₂	1	1	yes	maximal, characteristic
5	#13094001	C₂	1	2	yes	maximal, complemented
6	Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic
10	#13093980	C₂	1	2	yes	maximal, complemented
11	#124×C₄₇	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×D₄×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₄₇	C₂²	1	2	yes	—
15	C₂×Dic₉₄	C₂²	1	1	yes	characteristic
16	C₂×Dic₉₄	C₂²	1	1	yes	characteristic
17	C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
18	C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
19	C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²⋊Dic₄₇	C₂²	1	2	yes	—
21	C₂²⋊Dic₄₇	C₂²	1	2	yes	—
22	C₂²⋊Dic₄₇	C₂²	1	2	yes	—
23	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
24	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₄₇	C₂²	1	2	yes	—
26	Dic₄₇⋊C₄	C₂²	1	1	yes	characteristic
27	Dic₄₇⋊C₄	C₂²	1	2	yes	—
28	Dic₄₇⋊C₄	C₂²	1	2	yes	—
29	Dic₄₇⋊C₄	C₂²	1	1	yes	characteristic
30	Dic₄₇⋊C₄	C₂²	1	2	yes	—
31	C₄×Dic₄₇	C₂²	1	2	yes	—
32	C₄×Dic₄₇	C₂²	1	2	yes	—
33	C₄×Dic₄₇	C₂²	1	1	yes	characteristic
34	C₄×Dic₄₇	C₂²	1	1	yes	characteristic
35	C₂³×C₄₇	C₂³	1	2	yes	—
36	Q₈×C₄₇	—	2	2	no	—
37	D₄×C₄₇	—	2	2	no	—
38	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₄₇	C₂³	1	2	yes	—
40	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
42	C₂×Dic₄₇	—	2	4	no	—
43	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
44	C₂×Dic₄₇	C₂³	1	2	yes	—
45	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
46	C₂×Dic₄₇	C₂³	1	2	yes	—
47	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
48	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
49	Dic₉₄	—	2	2	no	—
50	Dic₉₄	—	2	2	no	—
51	C₂²×C₄₇	—	2	2	no	—
52	C₂²×C₄₇	—	2	2	no	—
53	C₂²×C₄₇	—	2	2	no	—
54	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
55	C₄×C₄₇	—	2	1	no	—
56	C₄×C₄₇	—	2	2	no	—
57	C₄×C₄₇	—	2	1	no	—
58	C₄×C₄₇	#41	1	2	yes	—
59	Dic₄₇	—	2	1	no	—
60	Dic₄₇	—	2	2	no	—
61	Dic₄₇	—	2	2	no	—
62	Dic₄₇	—	2	1	no	—
63	Dic₄₇	—	2	1	no	—
64	Dic₄₇	#41	1	2	yes	—
65	C₂×C₄₇	—	4	2	no	—
66	C₂×C₄₇	C₂×#41	1	1	yes	characteristic
67	C₂×C₄₇	C₂×#41	1	1	yes	characteristic
68	C₂×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
69	#556	—	47	1	no	maximal, Sylow
70	C₄₇	#556	1	1	yes	characteristic, complemented, Sylow
71	C₄⋊Q₈	—	47	1	no	—
72	#126	—	47	2	no	—
73	#126	—	47	2	no	—
74	#124	—	47	1	no	—
75	#124	D₄₇	1	1	yes	characteristic
76	C₂²⋊Q₈	—	47	2	no	—
77	C₄×Q₈	—	47	1	no	—
78	C₄×Q₈	—	47	1	no	—
79	C₄×Q₈	—	47	1	no	—
80	C₄×D₄	—	47	1	no	—
81	#117	—	47	2	no	—
82	C₂×Q₈	—	47	1	no	—
83	C₂×Q₈	—	47	1	no	—
84	C₂×Q₈	C₂×D₄₇	1	1	yes	characteristic
85	C₂×D₄	C₂×D₄₇	1	1	yes	characteristic
86	C₂²×C₄	—	47	2	no	—
87	C₄⋊C₄	—	47	1	no	—
88	C₄⋊C₄	—	47	2	no	—
89	C₄⋊C₄	—	47	2	no	—
90	C₄⋊C₄	—	47	1	no	—
91	C₄⋊C₄	—	47	2	no	—
92	C₄⋊C₄	—	47	1	no	—
93	C₄⋊C₄	—	47	2	no	—
94	C₄⋊C₄	—	47	1	no	—
95	C₂²⋊C₄	—	47	2	no	—
96	C₂²⋊C₄	—	47	2	no	—
97	C₂²⋊C₄	—	47	2	no	—
98	C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
99	C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
100	C₄²	—	47	2	no	—
101	C₄²	—	47	2	no	—
102	C₄²	—	47	1	no	—
103	C₄²	—	47	1	no	—
104	C₄²	C₂×D₄₇	1	1	yes	characteristic
105	C₂³	C₂²×D₄₇	1	2	yes	—
106	Q₈	—	2	2	no	—
107	Q₈	—	94	2	no	—
108	Q₈	—	94	2	no	—
109	D₄	—	2	2	no	—
110	C₂×C₄	—	47	1	no	—
111	C₂×C₄	—	47	1	no	—
112	C₂×C₄	—	47	2	no	—
113	C₂×C₄	—	47	1	no	—
114	C₂×C₄	—	47	1	no	—
115	C₂×C₄	—	47	2	no	—
116	C₂×C₄	—	94	4	no	—
117	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
118	C₂×C₄	C₂²×D₄₇	1	2	yes	—
119	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	C₂³×D₄₇	1	1	yes	characteristic
125	C₄	—	2	1	no	—
126	C₄	—	2	2	no	—
127	C₄	—	2	1	no	—
128	C₄	—	47	2	no	—
129	C₄	—	94	2	no	—
130	C₄	—	94	1	no	—
131	C₄	—	94	2	no	—
132	C₄	—	94	1	no	—
133	C₄	—	94	1	no	—
134	C₄	#10625352	1	2	yes	—
135	C₂	—	4	2	no	—
136	C₂	#41×D₄₇	1	1	yes	characteristic
137	C₂	C₂×#10625352	1	1	yes	characteristic
138	C₂	#13094071	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13094022

C₂

yes

maximal, characteristic, complemented

#13094017

C₂

yes

maximal, complemented

#13094017

C₂

yes

maximal, complemented

#13094011

C₂

yes

maximal, characteristic

#13094001

C₂

yes

maximal, complemented

Q₈×Dic₄₇

C₂

yes

maximal, characteristic, complemented

C₄×Dic₉₄

C₂

yes

maximal, characteristic, complemented

C₄×Dic₉₄

C₂

yes

maximal, characteristic, complemented

D₄×Dic₄₇

C₂

yes

maximal, characteristic

#13093980

C₂

yes

maximal, complemented

#124×C₄₇

C₂

yes

maximal, characteristic

C₂×Q₈×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

C₂²

yes

characteristic

C₂²×Dic₄₇

C₂²

yes

—

C₂×Dic₉₄

C₂²

yes

characteristic

C₂×Dic₉₄

C₂²

yes

characteristic

C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂²⋊C₄×C₄₇

C₂²

yes

—

C₄²×C₄₇

C₂²

yes

characteristic

C₂²⋊Dic₄₇

C₂²

yes

—

C₂²⋊Dic₄₇

C₂²

yes

—

C₂²⋊Dic₄₇

C₂²

yes

—

C₄⋊Dic₄₇

C₂²

yes

characteristic

C₄⋊Dic₄₇

C₂²

yes

characteristic

C₄⋊Dic₄₇

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

characteristic

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

characteristic

Dic₄₇⋊C₄

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

characteristic

C₄×Dic₄₇

C₂²

yes

characteristic

C₂³×C₄₇

C₂³

yes

—

Q₈×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

C₂³

yes

characteristic

Dic₉₄

—

Dic₉₄

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂⁴

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#41

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

#41

yes

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#41

yes

characteristic

C₂×C₄₇

C₂×#41

yes

characteristic

C₂×C₄₇

2₋¹⁺⁴

yes

—

C₄₇

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

—

—

—

D₄₇

yes

characteristic

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄₇

yes

characteristic

C₂×D₄

C₂×D₄₇

yes

characteristic

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₄₇

yes

—

C₂²⋊C₄

C₂×D₄₇

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

104

C₄²

C₂×D₄₇

yes

characteristic

105

C₂³

C₂²×D₄₇

yes

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₂²×D₄₇

yes

characteristic

118

C₂×C₄

C₂²×D₄₇

yes

—

119

C₂×C₄

C₂²×D₄₇

yes

characteristic

120

C₂×C₄

C₂²×D₄₇

yes

characteristic

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂³×D₄₇

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#10625352

yes

—

135

C₂

—

136

C₂

#41×D₄₇

yes

characteristic

137

C₂

C₂×#10625352

yes

characteristic

138

C₂

#13094071

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	#124×C₄₇	C₂
prime radical	C₂×D₄×C₄₇	C₂²
hypercenter	#124	D₄₇
hypocenter	C₄₇	#556
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

prime radical

C₂×D₄×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #13094022	C₂	semidirect
2. #13094017	C₂	semidirect
3. #13094017	C₂	semidirect
5. #13094001	C₂	semidirect
6. Q₈×Dic₄₇	C₂	semidirect
7. C₄×Dic₉₄	C₂	semidirect
8. C₄×Dic₉₄	C₂	semidirect
10. #13093980	C₂	semidirect
70. C₄₇	#556	semidirect

H ≅ G/N

type

1. #13094022

C₂

semidirect

2. #13094017

C₂

semidirect

3. #13094017

C₂

semidirect

5. #13094001

C₂

semidirect

6. Q₈×Dic₄₇

C₂

semidirect

7. C₄×Dic₉₄

C₂

semidirect

8. C₄×Dic₉₄

C₂

semidirect

10. #13093980

C₂

semidirect

70. C₄₇

#556

semidirect

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

subgroup	index
#13094022	2
#13094017	2
#13094011	2
#13094001	2
Q₈×Dic₄₇	2
C₄×Dic₉₄	2
D₄×Dic₄₇	2
#13093980	2
#124×C₄₇	2
#556	47

subgroup

index

Q₈×Dic₄₇

C₄×Dic₉₄

D₄×Dic₄₇

#13093980

#124×C₄₇

#556

Subgroups — #15689842

Conjugacy classes of subgroups

Complements

Maximal subgroups