Group #20252412 — subgroups

Subgroups — #20252412

Order 6016.

500 conjugacy classes of subgroups; only the 297 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 297 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	D₄×C₄₇⋊C₈	C₂	1	8	yes	maximal, complemented
2	#15689176	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₄₇⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄₇⋊C₈	C₂²	1	8	yes	complemented
6	C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	complemented
7	C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
8	C₂²⋊(C₄₇⋊C₈)	C₂²	1	8	yes	complemented
9	C₄×C₄₇⋊C₈	C₂²	1	4	yes	complemented
10	C₂²×D₄×C₄₇	C₄	1	1	yes	characteristic
11	C₂³×C₄×C₄₇	C₂²	1	2	yes	—
12	C₄×D₄×C₄₇	C₄	1	4	yes	—
13	C₄×D₄×C₄₇	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
15	C₂×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
16	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂⁴×C₄₇	C₂×C₄	1	2	yes	—
18	C₂×D₄×C₄₇	C₈	1	8	yes	complemented
19	C₂×D₄×C₄₇	C₂×C₄	1	4	yes	—
20	C₂²×C₄×C₄₇	—	4	4	no	—
21	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
27	C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
28	C₂²⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
29	C₄²×C₄₇	C₂×C₄	1	2	yes	—
30	C₄²×C₄₇	C₂³	1	2	yes	—
31	C₂×C₄₇⋊C₈	—	4	8	no	—
32	C₂×C₄₇⋊C₈	C₂³	1	8	yes	—
33	C₂³×C₄₇	—	4	2	no	—
34	C₂³×C₄₇	—	4	2	no	—
35	C₂³×C₄₇	C₂×C₈	1	8	yes	complemented
36	C₂³×C₄₇	C₂²×C₄	1	4	yes	—
37	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	D₄×C₄₇	C₂×C₈	1	16	yes	complemented
39	C₂×C₄×C₄₇	—	2	8	no	—
40	C₂×C₄×C₄₇	—	2	2	no	—
41	C₂×C₄×C₄₇	—	4	4	no	—
42	C₂×C₄×C₄₇	—	4	2	no	—
43	C₂×C₄×C₄₇	—	4	1	no	—
44	C₂×C₄×C₄₇	C₂×C₈	1	4	yes	complemented
45	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₄₇⋊C₈	—	2	8	no	—
51	C₂²×C₄₇	—	2	1	no	—
52	C₂²×C₄₇	—	2	4	no	—
53	C₂²×C₄₇	—	2	1	no	—
54	C₂²×C₄₇	—	2	4	no	—
55	C₂²×C₄₇	—	4	8	no	—
56	C₂²×C₄₇	C₂²×C₈	1	2	yes	—
57	C₂²×C₄₇	C₂²×C₈	1	8	yes	—
58	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
59	C₄×C₄₇	—	2	1	no	—
60	C₄×C₄₇	—	2	4	no	—
61	C₄×C₄₇	—	2	2	no	—
62	C₄×C₄₇	C₂²×C₈	1	4	yes	—
63	C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
64	#2972	—	47	1	no	maximal, Sylow
65	C₂×C₄₇	—	2	2	no	—
66	C₂×C₄₇	—	2	8	no	—
67	C₂×C₄₇	#518	1	1	yes	characteristic
68	C₂×C₄₇	C₂³×C₈	1	1	yes	characteristic
69	C₂×C₄₇	#568	1	1	yes	characteristic
70	C₂×C₄×D₄	D₄₇	1	1	yes	characteristic
71	C₈×D₄	—	47	8	no	—
72	#431	—	47	2	no	—
73	C₂×C₂²⋊C₈	—	47	4	no	—
74	C₄₇	#2972	1	1	yes	characteristic, complemented, Sylow
75	C₂²×D₄	Dic₄₇	1	1	yes	characteristic
76	C₂³×C₄	C₂×D₄₇	1	2	yes	—
77	C₂²×C₈	—	47	8	no	—
78	C₄×D₄	Dic₄₇	1	4	yes	—
79	C₄×D₄	C₂×D₄₇	1	4	yes	—
80	C₂×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
81	C₂×C₂²⋊C₄	Dic₄₇	1	2	yes	—
82	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
83	C₄⋊C₈	—	47	4	no	—
84	C₂²⋊C₈	—	47	4	no	—
85	C₂²⋊C₈	—	47	8	no	—
86	C₄×C₈	—	47	4	no	—
87	C₂⁴	C₂×Dic₄₇	1	2	yes	—
88	C₂×D₄	C₄₇⋊C₈	1	8	yes	complemented
89	C₂×D₄	C₂×Dic₄₇	1	4	yes	—
90	C₂²×C₄	—	4	4	no	—
91	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
93	C₂²×C₄	C₂×Dic₄₇	1	4	yes	—
94	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
95	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
97	C₂×C₈	—	47	8	no	—
98	C₂×C₈	—	188	8	no	—
99	C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
100	C₂²⋊C₄	C₂×Dic₄₇	1	8	yes	—
101	C₄²	C₂×Dic₄₇	1	2	yes	—
102	C₄²	C₂²×D₄₇	1	2	yes	—
103	C₂³	—	4	2	no	—
104	C₂³	—	4	2	no	—
105	C₂³	C₂×C₄₇⋊C₈	1	8	yes	complemented
106	C₂³	C₂²×Dic₄₇	1	4	yes	—
107	C₂³	C₂²×Dic₄₇	1	1	yes	characteristic
108	D₄	C₂×C₄₇⋊C₈	1	16	yes	complemented
109	C₂×C₄	—	2	8	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	C₂×C₄₇⋊C₈	1	4	yes	complemented
115	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
116	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
117	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
118	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
119	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
120	C₈	—	94	8	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	4	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	4	8	no	—
126	C₂²	C₂²×C₄₇⋊C₈	1	2	yes	—
127	C₂²	C₂²×C₄₇⋊C₈	1	8	yes	—
128	C₂²	C₂³×Dic₄₇	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	4	no	—
131	C₄	—	2	2	no	—
132	C₄	C₂²×C₄₇⋊C₈	1	4	yes	—
133	C₄	#13094069	1	2	yes	—
134	C₂	—	2	2	no	—
135	C₂	—	2	8	no	—
136	C₂	#15689598	1	1	yes	characteristic
137	C₂	C₂³×C₄₇⋊C₈	1	1	yes	characteristic
138	C₂	#15689890	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₄×C₄₇⋊C₈

C₂

yes

maximal, complemented

#15689176

C₂

yes

maximal, complemented

C₂×C₂²⋊(C₄₇⋊C₈)

C₂

yes

maximal, complemented

C₂×C₄×D₄×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

complemented

C₄⋊(C₄₇⋊C₈)

C₂²

yes

complemented

C₂²⋊(C₄₇⋊C₈)

C₂²

yes

—

C₂²⋊(C₄₇⋊C₈)

C₂²

yes

complemented

C₄×C₄₇⋊C₈

C₂²

yes

complemented

C₂²×D₄×C₄₇

C₄

yes

characteristic

C₂³×C₄×C₄₇

C₂²

yes

—

C₄×D₄×C₄₇

C₄

yes

—

C₄×D₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

C₄

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

C₂×C₄

yes

—

C₂×D₄×C₄₇

C₈

yes

complemented

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂³

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×C₈

yes

complemented

C₂³×C₄₇

C₂²×C₄

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

D₄×C₄₇

C₂×C₈

yes

complemented

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₈

yes

complemented

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×C₈

yes

—

C₂²×C₄₇

C₂²×C₈

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²×C₈

yes

—

C₄×C₄₇

H(5,𝔽₂)

yes

—

#2972

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#518

yes

characteristic

C₂×C₄₇

C₂³×C₈

yes

characteristic

C₂×C₄₇

#568

yes

characteristic

C₂×C₄×D₄

D₄₇

yes

characteristic

C₈×D₄

—

#431

—

C₂×C₂²⋊C₈

—

C₄₇

#2972

yes

characteristic, complemented, Sylow

C₂²×D₄

Dic₄₇

yes

characteristic

C₂³×C₄

C₂×D₄₇

yes

—

C₂²×C₈

—

C₄×D₄

Dic₄₇

yes

—

C₄×D₄

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

Dic₄₇

yes

characteristic

C₂×C₂²⋊C₄

Dic₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂⁴

C₂×Dic₄₇

yes

—

C₂×D₄

C₄₇⋊C₈

yes

complemented

C₂×D₄

C₂×Dic₄₇

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₂²×C₄

C₂²×D₄₇

yes

—

C₂×C₈

—

C₂×C₈

—

188

—

C₄⋊C₄

C₂×Dic₄₇

yes

—

100

C₂²⋊C₄

C₂×Dic₄₇

yes

—

101

C₄²

C₂×Dic₄₇

yes

—

102

C₄²

C₂²×D₄₇

yes

—

103

C₂³

—

104

C₂³

—

105

C₂³

C₂×C₄₇⋊C₈

yes

complemented

106

C₂³

C₂²×Dic₄₇

yes

—

107

C₂³

C₂²×Dic₄₇

yes

characteristic

108

D₄

C₂×C₄₇⋊C₈

yes

complemented

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×C₄₇⋊C₈

yes

complemented

115

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

116

C₂×C₄

C₂²×Dic₄₇

yes

—

117

C₂×C₄

C₂²×Dic₄₇

yes

—

118

C₂×C₄

C₂²×Dic₄₇

yes

—

119

C₂×C₄

C₂³×D₄₇

yes

characteristic

120

C₈

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

C₂²×C₄₇⋊C₈

yes

—

127

C₂²

C₂²×C₄₇⋊C₈

yes

—

128

C₂²

C₂³×Dic₄₇

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₂²×C₄₇⋊C₈

yes

—

133

C₄

#13094069

yes

—

134

C₂

—

135

C₂

—

136

C₂

#15689598

yes

characteristic

137

C₂

C₂³×C₄₇⋊C₈

yes

characteristic

138

C₂

#15689890

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator	C₂×C₄₇	C₂³×C₈
Fitting	C₂×C₄×D₄×C₄₇	C₂
socle	C₂²×C₄₇	C₂³×C₄
hypercenter	C₂×C₄×D₄	D₄₇
hypocenter	C₄₇	#2972
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₄₇

commutator

C₂×C₄₇

C₂³×C₈

Fitting

C₂×C₄×D₄×C₄₇

C₂

socle

C₂²×C₄₇

C₂³×C₄

hypercenter

C₂×C₄×D₄

D₄₇

C₄₇

G ▷ C₂×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

N	H ≅ G/N	type
1. D₄×C₄₇⋊C₈	C₂	semidirect
2. #15689176	C₂	semidirect
3. C₂×C₂²⋊(C₄₇⋊C₈)	C₂	semidirect
5. C₂²×C₄₇⋊C₈	C₂²	semidirect
6. C₄⋊(C₄₇⋊C₈)	C₂²	semidirect
8. C₂²⋊(C₄₇⋊C₈)	C₂²	semidirect
9. C₄×C₄₇⋊C₈	C₂²	semidirect
18. C₂×D₄×C₄₇	C₈	semidirect
35. C₂³×C₄₇	C₂×C₈	semidirect
38. D₄×C₄₇	C₂×C₈	semidirect
44. C₂×C₄×C₄₇	C₂×C₈	semidirect
74. C₄₇	#2972	semidirect
88. C₂×D₄	C₄₇⋊C₈	semidirect
105. C₂³	C₂×C₄₇⋊C₈	semidirect
108. D₄	C₂×C₄₇⋊C₈	semidirect
114. C₂×C₄	C₂×C₄₇⋊C₈	semidirect

H ≅ G/N

type

1. D₄×C₄₇⋊C₈

C₂

semidirect

2. #15689176

C₂

semidirect

3. C₂×C₂²⋊(C₄₇⋊C₈)

C₂

semidirect

5. C₂²×C₄₇⋊C₈

C₂²

semidirect

6. C₄⋊(C₄₇⋊C₈)

C₂²

semidirect

8. C₂²⋊(C₄₇⋊C₈)

C₂²

semidirect

9. C₄×C₄₇⋊C₈

C₂²

semidirect

18. C₂×D₄×C₄₇

C₈

semidirect

35. C₂³×C₄₇

C₂×C₈

semidirect

38. D₄×C₄₇

C₂×C₈

semidirect

44. C₂×C₄×C₄₇

C₂×C₈

semidirect

74. C₄₇

#2972

semidirect

88. C₂×D₄

C₄₇⋊C₈

semidirect

105. C₂³

C₂×C₄₇⋊C₈

semidirect

108. D₄

C₂×C₄₇⋊C₈

semidirect

114. C₂×C₄

C₂×C₄₇⋊C₈

semidirect

Subgroups — #20252412

Conjugacy classes of subgroups

Complements