Group #20251385 — subgroups

Subgroups — #20251385

Order 6016.

712 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13093993	C₂	1	2	yes	maximal, complemented
2	#15689048	C₂	1	3	yes	maximal, complemented
3	#15689039	C₂	1	6	yes	maximal, complemented
4	#15689032	C₂	1	3	yes	maximal, complemented
5	C₄₇×#379	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic, complemented
7	C₂²×C₂²⋊D₄₇	C₂²	1	6	yes	—
8	#13093993	—	2	8	no	—
9	C₂×C₄⋊Dic₄₇	C₂²	1	3	yes	complemented
10	C₂×C₂²⋊Dic₄₇	C₂²	1	6	yes	—
11	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	6	yes	complemented
12	C₄₇⋊#95	C₂²	1	3	yes	complemented
13	C₄₇⋊#95	C₂²	1	3	yes	complemented
14	C₂⁵×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₄₇	C₂²	1	3	yes	—
16	C₂×C₂²⋊C₄×C₄₇	C₂²	1	3	yes	—
17	C₂⁴×C₄₇	—	4	3	no	—
18	C₂⁴×C₄₇	—	4	3	no	—
19	C₂⁴×C₄₇	D₄	1	4	yes	complemented
20	C₂⁴×C₄₇	C₂³	1	3	yes	—
21	C₂³×D₄₇	C₂³	1	2	yes	—
22	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
24	C₂×C₂²⋊D₄₇	—	2	12	no	—
25	C₂×C₂²⋊D₄₇	—	4	6	no	—
26	C₂×C₂²⋊D₄₇	—	4	6	no	—
27	C₂²×Dic₄₇	C₂³	1	6	yes	—
28	C₂×D₁₈₈	—	4	3	no	—
29	C₂²⋊C₄×C₄₇	—	4	3	no	—
30	C₂²⋊C₄×C₄₇	—	4	3	no	—
31	C₂²⋊Dic₄₇	—	2	12	no	—
32	(C₂×D₄₇)⋊C₄	—	4	6	no	—
33	C₄⋊Dic₄₇	—	4	3	no	—
34	C₂³×C₄₇	—	2	3	no	—
35	C₂³×C₄₇	—	2	3	no	—
36	C₂³×C₄₇	—	2	6	no	—
37	C₂³×C₄₇	—	2	6	no	—
38	C₂³×C₄₇	—	4	12	no	—
39	C₂³×C₄₇	—	4	12	no	—
40	C₂³×C₄₇	—	4	4	no	—
41	C₂³×C₄₇	C₂×D₄	1	6	yes	—
42	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
43	C₂²×D₄₇	—	2	2	no	—
44	C₂²×D₄₇	—	2	6	no	—
45	C₂²×D₄₇	—	2	6	no	—
46	C₂×C₄×C₄₇	—	2	3	no	—
47	C₂×C₄×C₄₇	—	2	6	no	—
48	C₂×C₄×C₄₇	—	2	3	no	—
49	C₂²⋊D₄₇	—	4	24	no	—
50	C₂×Dic₄₇	—	2	6	no	—
51	C₂×Dic₄₇	—	2	6	no	—
52	C₂×Dic₄₇	—	2	6	no	—
53	D₁₈₈	—	4	4	no	—
54	C₂²×C₄₇	—	2	12	no	—
55	C₂²×C₄₇	—	2	6	no	—
56	C₂²×C₄₇	—	4	16	no	—
57	C₂²×C₄₇	—	4	6	no	—
58	C₂²×C₄₇	—	4	3	no	—
59	C₂²×C₄₇	—	4	3	no	—
60	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
61	C₂²×C₄₇	H(5,𝔽₂)	1	3	yes	—
62	C₂²×C₄₇	H(5,𝔽₂)	1	3	yes	—
63	C₂×D₄₇	—	4	6	no	—
64	C₂×D₄₇	—	4	6	no	—
65	C₂×D₄₇	—	4	2	no	—
66	C₄×C₄₇	—	4	3	no	—
67	C₄×C₄₇	—	4	1	no	—
68	Dic₄₇	—	4	6	no	—
69	#2812	—	47	1	no	maximal, Sylow
70	C₂×C₄₇	—	4	6	no	—
71	C₂×C₄₇	#534	1	3	yes	—
72	C₂×C₄₇	#560	1	3	yes	—
73	C₂×C₄₇	H(3,𝔽₄)	1	1	yes	characteristic
74	D₄₇	—	8	2	no	—
75	C₂×#120	—	47	2	no	—
76	#397	—	47	3	no	—
77	#394	—	47	6	no	—
78	#392	—	47	3	no	—
79	#379	D₄₇	1	1	yes	characteristic, complemented
80	C₄₇	#2812	1	1	yes	characteristic, complemented, Sylow
81	C₂⁵	C₂×D₄₇	1	1	yes	characteristic
82	C₂²×D₄	—	47	6	no	—
83	C₂²×D₄	—	47	1	no	—
84	#120	—	94	8	no	—
85	C₂×C₄⋊C₄	—	47	3	no	—
86	C₂×C₂²⋊C₄	—	47	6	no	—
87	C₂×C₂²⋊C₄	—	47	6	no	—
88	C₂×C₂²⋊C₄	C₂×D₄₇	1	3	yes	—
89	C₂×C₂²⋊C₄	C₂×D₄₇	1	3	yes	—
90	#95	—	47	3	no	—
91	#95	—	47	3	no	—
92	C₂⁴	—	4	3	no	—
93	C₂⁴	—	4	3	no	—
94	C₂⁴	—	47	2	no	—
95	C₂⁴	D₁₈₈	1	4	yes	complemented
96	C₂⁴	C₂²×D₄₇	1	3	yes	—
97	C₂×D₄	—	94	12	no	—
98	C₂×D₄	—	188	3	no	—
99	C₂×D₄	—	188	6	no	—
100	C₂×D₄	—	188	6	no	—
101	C₂²×C₄	—	47	6	no	—
102	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
103	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
104	C₄⋊C₄	—	188	3	no	—
105	C₂²⋊C₄	—	4	3	no	—
106	C₂²⋊C₄	—	4	3	no	—
107	C₂²⋊C₄	—	94	12	no	—
108	C₂²⋊C₄	—	188	6	no	—
109	C₂³	—	2	3	no	—
110	C₂³	—	2	3	no	—
111	C₂³	—	2	6	no	—
112	C₂³	—	2	6	no	—
113	C₂³	—	4	12	no	—
114	C₂³	—	4	12	no	—
115	C₂³	—	4	4	no	—
116	C₂³	—	94	2	no	—
117	C₂³	—	94	6	no	—
118	C₂³	—	94	6	no	—
119	C₂³	C₂×D₁₈₈	1	6	yes	—
120	C₂³	C₂³×D₄₇	1	1	yes	characteristic
121	D₄	—	188	4	no	—
122	D₄	—	188	24	no	—
123	C₂×C₄	—	2	6	no	—
124	C₂×C₄	—	2	3	no	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	94	6	no	—
127	C₂×C₄	—	94	6	no	—
128	C₂×C₄	—	94	6	no	—
129	C₂²	—	2	12	no	—
130	C₂²	—	2	6	no	—
131	C₂²	—	4	16	no	—
132	C₂²	—	4	6	no	—
133	C₂²	—	4	3	no	—
134	C₂²	—	4	3	no	—
135	C₂²	—	188	6	no	—
136	C₂²	—	188	2	no	—
137	C₂²	—	188	6	no	—
138	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
139	C₂²	#13094069	1	3	yes	—
140	C₂²	#13094069	1	3	yes	—
141	C₄	—	4	3	no	—
142	C₄	—	4	1	no	—
143	C₄	—	188	6	no	—
144	C₂	—	4	6	no	—
145	C₂	—	376	2	no	—
146	C₂	#15689685	1	3	yes	—
147	C₂	#15689862	1	3	yes	—
148	C₂	#15689866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093993

C₂

yes

maximal, complemented

#15689048

C₂

yes

maximal, complemented

#15689039

C₂

yes

maximal, complemented

#15689032

C₂

yes

maximal, complemented

C₄₇×#379

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

characteristic, complemented

C₂²×C₂²⋊D₄₇

C₂²

yes

—

#13093993

—

C₂×C₄⋊Dic₄₇

C₂²

yes

complemented

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

complemented

C₄₇⋊#95

C₂²

yes

complemented

C₄₇⋊#95

C₂²

yes

complemented

C₂⁵×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

D₄

yes

complemented

C₂⁴×C₄₇

C₂³

yes

—

C₂³×D₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂×C₂²⋊D₄₇

—

C₂×C₂²⋊D₄₇

—

C₂×C₂²⋊D₄₇

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×D₁₈₈

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

#2812

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#534

yes

—

C₂×C₄₇

#560

yes

—

C₂×C₄₇

H(3,𝔽₄)

yes

characteristic

D₄₇

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₄₇

#2812

yes

characteristic, complemented, Sylow

C₂⁵

C₂×D₄₇

yes

characteristic

C₂²×D₄

—

C₂²×D₄

—

#120

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

#95

—

#95

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

C₂⁴

D₁₈₈

yes

complemented

C₂⁴

C₂²×D₄₇

yes

—

C₂×D₄

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

100

C₂×D₄

—

188

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₄₇

yes

characteristic

103

C₂²×C₄

C₂²×D₄₇

yes

—

104

C₄⋊C₄

—

188

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

188

—

109

C₂³

—

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

119

C₂³

C₂×D₁₈₈

yes

—

120

C₂³

C₂³×D₄₇

yes

characteristic

121

D₄

—

188

—

122

D₄

—

188

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

188

—

136

C₂²

—

188

—

137

C₂²

—

188

—

138

C₂²

C₂²×D₁₈₈

yes

characteristic

139

C₂²

#13094069

yes

—

140

C₂²

#13094069

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

188

—

144

C₂

—

145

C₂

—

376

—

146

C₂

#15689685

yes

—

147

C₂

#15689862

yes

—

148

C₂

#15689866

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄₇×#379	C₂
hypercenter	#379	D₄₇
hypocenter	C₄₇	#2812
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #379

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₄₇×#379

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #379

N	H ≅ G/N	type
1. C₂×#13093993	C₂	semidirect
2. #15689048	C₂	semidirect
3. #15689039	C₂	semidirect
4. #15689032	C₂	semidirect
5. C₄₇×#379	C₂	semidirect
6. C₂²×D₁₈₈	C₂²	semidirect
9. C₂×C₄⋊Dic₄₇	C₂²	semidirect
11. C₂×(C₂×D₄₇)⋊C₄	C₂²	semidirect
12. C₄₇⋊#95	C₂²	semidirect
13. C₄₇⋊#95	C₂²	semidirect
19. C₂⁴×C₄₇	D₄	semidirect
79. #379	D₄₇	semidirect
80. C₄₇	#2812	semidirect
95. C₂⁴	D₁₈₈	semidirect

H ≅ G/N

type

1. C₂×#13093993

C₂

semidirect

2. #15689048

C₂

semidirect

3. #15689039

C₂

semidirect

4. #15689032

C₂

semidirect

5. C₄₇×#379

C₂

semidirect

6. C₂²×D₁₈₈

C₂²

semidirect

9. C₂×C₄⋊Dic₄₇

C₂²

semidirect

11. C₂×(C₂×D₄₇)⋊C₄

C₂²

semidirect

12. C₄₇⋊#95

C₂²

semidirect

13. C₄₇⋊#95

C₂²

semidirect

19. C₂⁴×C₄₇

D₄

semidirect

79. #379

D₄₇

semidirect

80. C₄₇

#2812

semidirect

95. C₂⁴

D₁₈₈

semidirect

Subgroups — #20251385

Conjugacy classes of subgroups

Complements