Group #20251734 — subgroups

Subgroups — #20251734

Order 6016.

924 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 924 conjugacy classes of subgroups — too many to draw. Only its 209 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×#13094021	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13093999	C₂	1	6	yes	maximal, complemented
3	C₂×C₄²×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689024	C₂	1	6	yes	maximal, complemented
5	C₄₇×#384	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₈₈	C₂²	1	6	yes	—
7	C₂²×D₁₈₈	C₂²	1	4	yes	—
8	C₂²×C₄×D₄₇	C₂²	1	3	yes	—
9	#13094021	—	2	4	no	—
10	#13093999	—	2	24	no	—
11	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	12	yes	—
12	C₂×C₄×Dic₄₇	C₂²	1	3	yes	—
13	C₄²×D₄₇	—	2	4	no	—
14	C₂×C₄⋊C₄×C₄₇	C₂²	1	6	yes	—
15	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂³×D₄₇	C₂³	1	4	yes	—
17	C₂³×D₄₇	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
19	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
20	C₂²×Dic₄₇	C₂³	1	3	yes	—
21	C₂×D₁₈₈	—	2	24	no	—
22	C₂×D₁₈₈	—	2	12	no	—
23	C₂×D₁₈₈	—	2	12	no	—
24	C₂×D₁₈₈	—	2	12	no	—
25	C₂×C₄×D₄₇	—	2	6	no	—
26	C₂×C₄×D₄₇	—	2	6	no	—
27	C₂×C₄×D₄₇	D₄	1	12	yes	complemented
28	C₄⋊C₄×C₄₇	—	2	12	no	—
29	C₄²×C₄₇	D₄	1	4	yes	complemented
30	(C₂×D₄₇)⋊C₄	—	2	24	no	—
31	C₄×Dic₄₇	—	2	6	no	—
32	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
33	C₂²×D₄₇	—	2	1	no	—
34	C₂²×D₄₇	—	2	3	no	—
35	C₂²×D₄₇	—	2	12	no	—
36	C₂²×D₄₇	—	2	4	no	—
37	C₂²×D₄₇	—	2	12	no	—
38	C₂²×D₄₇	C₂×D₄	1	6	yes	—
39	C₂×C₄×C₄₇	—	2	12	no	—
40	C₂×C₄×C₄₇	C₂×D₄	1	6	yes	—
41	C₂×C₄×C₄₇	C₂×D₄	1	6	yes	—
42	C₂×C₄×C₄₇	#41	1	6	yes	—
43	C₂×Dic₄₇	—	2	3	no	—
44	C₂×Dic₄₇	—	2	3	no	—
45	C₂×Dic₄₇	C₂×D₄	1	6	yes	—
46	D₁₈₈	—	4	16	no	—
47	D₁₈₈	—	4	24	no	—
48	C₄×D₄₇	—	2	24	no	—
49	C₂²×C₄₇	C₂²×D₄	1	3	yes	—
50	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
51	C₂²×C₄₇	C₂×#41	1	3	yes	—
52	C₂×D₄₇	—	2	6	no	—
53	C₂×D₄₇	—	2	6	no	—
54	C₂×D₄₇	—	4	12	no	—
55	C₂×D₄₇	—	4	4	no	—
56	C₂×D₄₇	—	4	12	no	—
57	C₂×D₄₇	#127	1	4	yes	complemented
58	C₄×C₄₇	—	4	4	no	—
59	C₄×C₄₇	#121	1	12	yes	complemented
60	Dic₄₇	—	2	6	no	—
61	#2856	—	47	1	no	maximal, Sylow
62	C₂×C₄₇	C₂×#121	1	3	yes	—
63	C₂×C₄₇	C₂×#127	1	1	yes	characteristic
64	C₂×C₄₇	#532	1	3	yes	—
65	D₄₇	—	2	4	no	—
66	D₄₇	—	8	4	no	—
67	C₂×#127	—	47	1	no	—
68	C₂×#121	—	47	6	no	—
69	C₂²×C₄²	—	47	1	no	—
70	#390	—	47	6	no	—
71	#384	D₄₇	1	1	yes	characteristic, complemented
72	C₄₇	#2856	1	1	yes	characteristic, complemented, Sylow
73	C₂²×D₄	—	47	4	no	—
74	C₂²×D₄	—	47	6	no	—
75	C₂³×C₄	—	47	3	no	—
76	#127	—	94	4	no	—
77	#121	—	94	24	no	—
78	C₂×C₄⋊C₄	C₂×D₄₇	1	6	yes	—
79	C₂×C₂²⋊C₄	—	47	12	no	—
80	C₂×C₄²	—	47	3	no	—
81	C₂×C₄²	—	94	4	no	—
82	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
83	C₂⁴	—	47	1	no	—
84	C₂⁴	—	47	4	no	—
85	C₂×D₄	—	94	24	no	—
86	C₂×D₄	—	94	12	no	—
87	C₂×D₄	—	94	12	no	—
88	C₂×D₄	—	94	12	no	—
89	C₂²×C₄	—	47	3	no	—
90	C₂²×C₄	—	47	12	no	—
91	C₂²×C₄	—	94	6	no	—
92	C₂²×C₄	—	94	6	no	—
93	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
94	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
95	C₄⋊C₄	—	2	12	no	—
96	C₂²⋊C₄	—	94	24	no	—
97	C₄²	—	94	6	no	—
98	C₄²	D₁₈₈	1	4	yes	complemented
99	C₂³	—	47	6	no	—
100	C₂³	—	94	1	no	—
101	C₂³	—	94	3	no	—
102	C₂³	—	94	12	no	—
103	C₂³	—	94	4	no	—
104	C₂³	—	94	12	no	—
105	C₂³	C₂³×D₄₇	1	1	yes	characteristic
106	D₄	—	188	16	no	—
107	D₄	—	188	24	no	—
108	C₂×C₄	—	2	12	no	—
109	C₂×C₄	—	47	6	no	—
110	C₂×C₄	—	94	3	no	—
111	C₂×C₄	—	94	3	no	—
112	C₂×C₄	—	94	24	no	—
113	C₂×C₄	C₂×D₁₈₈	1	6	yes	—
114	C₂×C₄	D₄×D₄₇	1	6	yes	—
115	C₂×C₄	#10625354	1	6	yes	—
116	C₂²	—	47	4	no	—
117	C₂²	—	94	6	no	—
118	C₂²	—	94	6	no	—
119	C₂²	—	188	12	no	—
120	C₂²	—	188	4	no	—
121	C₂²	—	188	12	no	—
122	C₂²	C₂×D₄×D₄₇	1	3	yes	—
123	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
124	C₂²	C₂×#10625354	1	3	yes	—
125	C₄	—	4	4	no	—
126	C₄	—	94	6	no	—
127	C₄	#13093999	1	12	yes	complemented
128	C₂	—	94	4	no	—
129	C₂	—	376	4	no	—
130	C₂	C₂×#13093999	1	3	yes	—
131	C₂	#127×D₄₇	1	1	yes	characteristic
132	C₂	#15689674	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094021

C₂

yes

maximal, characteristic, complemented

C₂×#13093999

C₂

yes

maximal, complemented

C₂×C₄²×D₄₇

C₂

yes

maximal, characteristic, complemented

#15689024

C₂

yes

maximal, complemented

C₄₇×#384

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

—

C₂²×D₁₈₈

C₂²

yes

—

C₂²×C₄×D₄₇

C₂²

yes

—

#13094021

—

#13093999

—

C₂×(C₂×D₄₇)⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄²×D₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

C₂³

yes

—

C₂³×D₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

D₄

yes

complemented

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

complemented

(C₂×D₄₇)⋊C₄

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₂²×C₄₇

C₂²×D₄

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

#127

yes

complemented

C₄×C₄₇

—

C₄×C₄₇

#121

yes

complemented

Dic₄₇

—

#2856

—

maximal, Sylow

C₂×C₄₇

C₂×#121

yes

—

C₂×C₄₇

C₂×#127

yes

characteristic

C₂×C₄₇

#532

yes

—

D₄₇

—

D₄₇

—

C₂×#127

—

C₂×#121

—

C₂²×C₄²

—

#390

—

#384

D₄₇

yes

characteristic, complemented

C₄₇

#2856

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂³×C₄

—

#127

—

#121

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₂⁴

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

D₁₈₈

yes

complemented

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

C₂³×D₄₇

yes

characteristic

106

D₄

—

188

—

107

D₄

—

188

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×D₁₈₈

yes

—

114

C₂×C₄

D₄×D₄₇

yes

—

115

C₂×C₄

#10625354

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

188

—

120

C₂²

—

188

—

121

C₂²

—

188

—

122

C₂²

C₂×D₄×D₄₇

yes

—

123

C₂²

C₂²×D₁₈₈

yes

characteristic

124

C₂²

C₂×#10625354

yes

—

125

C₄

—

126

C₄

—

127

C₄

#13093999

yes

complemented

128

C₂

—

129

C₂

—

376

—

130

C₂

C₂×#13093999

yes

—

131

C₂

#127×D₄₇

yes

characteristic

132

C₂

#15689674

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄₇×#384	C₂
hypercenter	#384	D₄₇
hypocenter	C₄₇	#2856
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₄₇×#384

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
1. C₂×#13094021	C₂	semidirect
2. C₂×#13093999	C₂	semidirect
3. C₂×C₄²×D₄₇	C₂	semidirect
4. #15689024	C₂	semidirect
5. C₄₇×#384	C₂	semidirect
27. C₂×C₄×D₄₇	D₄	semidirect
29. C₄²×C₄₇	D₄	semidirect
57. C₂×D₄₇	#127	semidirect
59. C₄×C₄₇	#121	semidirect
71. #384	D₄₇	semidirect
72. C₄₇	#2856	semidirect
98. C₄²	D₁₈₈	semidirect
127. C₄	#13093999	semidirect

H ≅ G/N

type

1. C₂×#13094021

C₂

semidirect

2. C₂×#13093999

C₂

semidirect

3. C₂×C₄²×D₄₇

C₂

semidirect

4. #15689024

C₂

semidirect

5. C₄₇×#384

C₂

semidirect

27. C₂×C₄×D₄₇

D₄

semidirect

29. C₄²×C₄₇

D₄

semidirect

57. C₂×D₄₇

#127

semidirect

59. C₄×C₄₇

#121

semidirect

71. #384

D₄₇

semidirect

72. C₄₇

#2856

semidirect

98. C₄²

D₁₈₈

semidirect

127. C₄

#13093999

semidirect

Subgroups — #20251734

Conjugacy classes of subgroups

Complements