Group #20251640 — subgroups

Subgroups — #20251640

Order 6016.

580 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 125 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	#15689036	C₂	1	6	yes	maximal, complemented
2	#15689035	C₂	1	3	yes	maximal, complemented
3	#15689034	C₂	1	2	yes	maximal, complemented
4	#15688978	C₂	1	3	yes	maximal
5	C₄₇×#393	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×D₄₇	C₂²	1	1	yes	characteristic
7	C₂²×Dic₉₄	C₂²	1	3	yes	complemented
8	C₂×C₂²⋊Dic₄₇	C₂²	1	3	yes	—
9	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	6	yes	—
10	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	6	yes	—
11	C₄₇⋊#95	C₂²	1	6	yes	complemented
12	C₄₇⋊#95	C₂²	1	3	yes	complemented
13	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₂²⋊C₄×C₄₇	C₂²	1	3	yes	—
15	#95×C₄₇	C₂²	1	3	yes	complemented
16	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
17	C₂³×D₄₇	—	4	3	no	—
18	C₂³×D₄₇	—	4	3	no	—
19	C₂³×D₄₇	Q₈	1	4	yes	complemented
20	C₂³×D₄₇	C₂³	1	2	yes	—
21	C₂×Q₈×C₄₇	—	4	3	no	—
22	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
23	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
24	C₂²×Dic₄₇	C₂³	1	6	yes	—
25	C₂×Dic₉₄	—	4	3	no	—
26	C₂×Dic₉₄	—	4	6	no	—
27	C₂²⋊C₄×C₄₇	—	4	3	no	—
28	C₂²⋊Dic₄₇	—	4	3	no	—
29	(C₂×D₄₇)⋊C₄	—	4	6	no	—
30	(C₂×D₄₇)⋊C₄	—	4	6	no	—
31	C₂³×C₄₇	—	2	3	no	—
32	C₂³×C₄₇	—	2	3	no	—
33	C₂³×C₄₇	C₂×Q₈	1	2	yes	—
34	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
35	C₂²×D₄₇	—	2	6	no	—
36	C₂²×D₄₇	—	2	6	no	—
37	C₂²×D₄₇	—	4	12	no	—
38	C₂²×D₄₇	—	4	12	no	—
39	C₂²×D₄₇	—	4	4	no	—
40	C₂²×D₄₇	C₂×Q₈	1	4	yes	—
41	Q₈×C₄₇	—	4	4	no	—
42	C₂×C₄×C₄₇	—	2	6	no	—
43	C₂×C₄×C₄₇	—	2	6	no	—
44	C₂×C₄×C₄₇	—	2	3	no	—
45	C₂×C₄×C₄₇	—	2	3	no	—
46	C₂×Dic₄₇	—	2	6	no	—
47	C₂×Dic₄₇	—	2	6	no	—
48	C₂×Dic₄₇	—	2	6	no	—
49	Dic₉₄	—	4	12	no	—
50	C₂²×C₄₇	—	2	6	no	—
51	C₂²×C₄₇	—	4	3	no	—
52	C₂²×C₄₇	—	4	1	no	—
53	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
54	C₂²×C₄₇	H(5,𝔽₂)	1	3	yes	—
55	C₂²×C₄₇	H(5,𝔽₂)	1	3	yes	—
56	C₂×D₄₇	—	2	12	no	—
57	C₂×D₄₇	—	4	16	no	—
58	C₂×D₄₇	—	4	6	no	—
59	C₂×D₄₇	—	4	2	no	—
60	C₄×C₄₇	—	4	3	no	—
61	C₄×C₄₇	—	4	3	no	—
62	Dic₄₇	—	4	6	no	—
63	#2840	—	47	1	no	maximal, Sylow
64	C₂×C₄₇	—	4	2	no	—
65	C₂×C₄₇	#543	1	3	yes	—
66	C₂×C₄₇	H(3,𝔽₄)	1	3	yes	—
67	C₂×C₄₇	H(3,𝔽₄)	1	1	yes	characteristic
68	D₄₇	—	4	4	no	—
69	#393	—	47	2	no	—
70	#393	—	47	3	no	—
71	#393	—	47	6	no	—
72	#393	D₄₇	1	1	yes	characteristic, complemented
73	#379	—	47	3	no	—
74	C₄₇	#2840	1	1	yes	characteristic, complemented, Sylow
75	C₂⁵	—	47	1	no	—
76	C₂²×Q₈	—	47	3	no	—
77	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic, complemented
78	C₂×C₂²⋊C₄	—	47	3	no	—
79	C₂×C₂²⋊C₄	—	47	6	no	—
80	C₂×C₂²⋊C₄	—	47	6	no	—
81	C₂×C₂²⋊C₄	C₂×D₄₇	1	3	yes	—
82	#95	—	47	6	no	—
83	#95	—	47	3	no	—
84	#95	C₂×D₄₇	1	3	yes	complemented
85	C₂⁴	—	47	2	no	—
86	C₂⁴	—	47	4	no	—
87	C₂⁴	—	188	3	no	—
88	C₂⁴	—	188	3	no	—
89	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
90	C₂×Q₈	—	4	3	no	—
91	C₂×Q₈	—	188	3	no	—
92	C₂×Q₈	—	188	6	no	—
93	C₂²×C₄	—	47	6	no	—
94	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
95	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
96	C₂²⋊C₄	—	4	3	no	—
97	C₂²⋊C₄	—	188	6	no	—
98	C₂²⋊C₄	—	188	6	no	—
99	C₂²⋊C₄	—	188	3	no	—
100	C₂³	—	2	3	no	—
101	C₂³	—	2	3	no	—
102	C₂³	—	47	4	no	—
103	C₂³	—	94	6	no	—
104	C₂³	—	94	6	no	—
105	C₂³	—	188	12	no	—
106	C₂³	—	188	12	no	—
107	C₂³	—	188	4	no	—
108	C₂³	Q₈×D₄₇	1	2	yes	—
109	C₂³	C₂³×D₄₇	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	188	12	no	—
112	C₂×C₄	—	2	3	no	—
113	C₂×C₄	—	2	3	no	—
114	C₂×C₄	—	2	6	no	—
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	94	6	no	—
117	C₂×C₄	—	94	6	no	—
118	C₂×C₄	—	94	6	no	—
119	C₂²	—	2	6	no	—
120	C₂²	—	4	3	no	—
121	C₂²	—	4	1	no	—
122	C₂²	—	94	12	no	—
123	C₂²	—	188	16	no	—
124	C₂²	—	188	6	no	—
125	C₂²	—	188	2	no	—
126	C₂²	C₂×Q₈×D₄₇	1	1	yes	characteristic
127	C₂²	#13094069	1	3	yes	—
128	C₂²	#13094070	1	3	yes	—
129	C₄	—	4	3	no	—
130	C₄	—	4	3	no	—
131	C₄	—	188	6	no	—
132	C₂	—	4	2	no	—
133	C₂	—	188	4	no	—
134	C₂	#15689748	1	3	yes	—
135	C₂	#15689865	1	3	yes	—
136	C₂	#15689866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689036

C₂

yes

maximal, complemented

#15689035

C₂

yes

maximal, complemented

#15689034

C₂

yes

maximal, complemented

#15688978

C₂

yes

maximal

C₄₇×#393

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₄₇

C₂²

yes

characteristic

C₂²×Dic₉₄

C₂²

yes

complemented

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

complemented

C₄₇⋊#95

C₂²

yes

complemented

C₂²×Q₈×C₄₇

C₂²

yes

characteristic, complemented

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

#95×C₄₇

C₂²

yes

complemented

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂³×D₄₇

—

C₂³×D₄₇

—

C₂³×D₄₇

Q₈

yes

complemented

C₂³×D₄₇

C₂³

yes

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂²⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×Q₈

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂×Q₈

yes

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

#2840

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#543

yes

—

C₂×C₄₇

H(3,𝔽₄)

yes

—

C₂×C₄₇

H(3,𝔽₄)

yes

characteristic

D₄₇

—

—

—

—

D₄₇

yes

characteristic, complemented

#379

—

C₄₇

#2840

yes

characteristic, complemented, Sylow

C₂⁵

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₄₇

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

#95

—

#95

—

#95

C₂×D₄₇

yes

complemented

C₂⁴

—

C₂⁴

—

C₂⁴

—

188

—

C₂⁴

—

188

—

C₂⁴

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

188

—

C₂×Q₈

—

188

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

188

—

C₂²⋊C₄

—

188

—

C₂²⋊C₄

—

188

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

188

—

106

C₂³

—

188

—

107

C₂³

—

188

—

108

C₂³

Q₈×D₄₇

yes

—

109

C₂³

C₂³×D₄₇

yes

characteristic

110

Q₈

—

111

Q₈

—

188

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

188

—

124

C₂²

—

188

—

125

C₂²

—

188

—

126

C₂²

C₂×Q₈×D₄₇

yes

characteristic

127

C₂²

#13094069

yes

—

128

C₂²

#13094070

yes

—

129

C₄

—

130

C₄

—

131

C₄

—

188

—

132

C₂

—

133

C₂

—

188

—

134

C₂

#15689748

yes

—

135

C₂

#15689865

yes

—

136

C₂

#15689866

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄₇×#393	C₂
hypercenter	#393	D₄₇
hypocenter	C₄₇	#2840
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #393

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₄₇×#393

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #393

N	H ≅ G/N	type
1. #15689036	C₂	semidirect
2. #15689035	C₂	semidirect
3. #15689034	C₂	semidirect
5. C₄₇×#393	C₂	semidirect
7. C₂²×Dic₉₄	C₂²	semidirect
11. C₄₇⋊#95	C₂²	semidirect
12. C₄₇⋊#95	C₂²	semidirect
13. C₂²×Q₈×C₄₇	C₂²	semidirect
15. #95×C₄₇	C₂²	semidirect
19. C₂³×D₄₇	Q₈	semidirect
72. #393	D₄₇	semidirect
74. C₄₇	#2840	semidirect
77. C₂²×Q₈	C₂×D₄₇	semidirect
84. #95	C₂×D₄₇	semidirect

H ≅ G/N

type

1. #15689036

C₂

semidirect

2. #15689035

C₂

semidirect

3. #15689034

C₂

semidirect

5. C₄₇×#393

C₂

semidirect

7. C₂²×Dic₉₄

C₂²

semidirect

11. C₄₇⋊#95

C₂²

semidirect

12. C₄₇⋊#95

C₂²

semidirect

13. C₂²×Q₈×C₄₇

C₂²

semidirect

15. #95×C₄₇

C₂²

semidirect

19. C₂³×D₄₇

Q₈

semidirect

72. #393

D₄₇

semidirect

74. C₄₇

#2840

semidirect

77. C₂²×Q₈

C₂×D₄₇

semidirect

84. #95

C₂×D₄₇

semidirect

Subgroups — #20251640

Conjugacy classes of subgroups

Complements