Group #20251660 — subgroups

Subgroups — #20251660

Order 6016.

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

normal subgroups only

This group has 356 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	#15689065	C₂	1	4	yes	maximal
2	#15689062	C₂	1	2	yes	maximal
3	#15689044	C₂	1	4	yes	maximal
4	#15689043	C₂	1	2	yes	maximal
5	#15688989	C₂	1	2	yes	maximal
6	C₄₇×#382	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₉₄	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
12	C₄₇⋊#95	C₂²	1	4	yes	—
13	C₄₇⋊#95	C₂²	1	4	yes	—
14	C₄₇⋊#95	C₂²	1	2	yes	—
15	C₄₇⋊#95	C₂²	1	4	yes	—
16	C₄₇⋊#95	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	#95×C₄₇	C₂²	1	4	yes	—
21	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
22	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
23	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
25	C₂²×Dic₄₇	C₂³	1	4	yes	—
26	C₂²×Dic₄₇	C₂³	1	4	yes	—
27	C₂×Dic₉₄	—	4	4	no	—
28	C₂×Dic₉₄	—	4	2	no	—
29	C₄⋊C₄×C₄₇	—	4	1	no	—
30	C₄⋊C₄×C₄₇	—	4	1	no	—
31	C₄²×C₄₇	Q₈	1	4	yes	complemented
32	C₄⋊Dic₄₇	—	4	2	no	—
33	C₄⋊Dic₄₇	—	4	2	no	—
34	C₄⋊Dic₄₇	—	4	2	no	—
35	Dic₄₇⋊C₄	—	4	4	no	—
36	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
37	C₂×C₄×C₄₇	—	2	2	no	—
38	C₂×C₄×C₄₇	—	2	1	no	—
39	C₂×C₄×C₄₇	—	2	2	no	—
40	C₂×C₄×C₄₇	—	2	1	no	—
41	C₂×C₄×C₄₇	—	2	4	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	4	no	—
45	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
47	C₂×Dic₄₇	—	2	4	no	—
48	C₂×Dic₄₇	—	2	4	no	—
49	C₂×Dic₄₇	—	2	4	no	—
50	C₂×Dic₄₇	—	2	4	no	—
51	C₂×Dic₄₇	—	2	4	no	—
52	C₂×Dic₄₇	—	2	4	no	—
53	Dic₉₄	—	4	8	no	—
54	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
55	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
56	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
57	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
58	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
59	C₄×C₄₇	—	2	4	no	—
60	C₄×C₄₇	—	2	2	no	—
61	C₄×C₄₇	—	4	2	no	—
62	C₄×C₄₇	—	4	2	no	—
63	Dic₄₇	—	4	4	no	—
64	Dic₄₇	—	4	4	no	—
65	#2844	—	47	1	no	maximal, Sylow
66	C₂×C₄₇	#544	1	2	yes	—
67	C₂×C₄₇	#544	1	1	yes	characteristic
68	C₂×C₄₇	H(3,𝔽₄)	1	1	yes	characteristic
69	C₂×C₄₇	#563	1	1	yes	characteristic
70	C₂×C₄₇	#563	1	2	yes	—
71	#400	—	47	2	no	—
72	#400	—	47	4	no	—
73	#395	—	47	2	no	—
74	#395	—	47	4	no	—
75	#382	—	47	2	no	—
76	#382	D₄₇	1	1	yes	characteristic
77	C₄₇	#2844	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₈	—	47	2	no	—
79	C₂×C₄⋊C₄	—	47	4	no	—
80	C₂×C₄⋊C₄	—	47	2	no	—
81	C₂×C₄⋊C₄	—	47	2	no	—
82	C₂×C₄⋊C₄	—	47	2	no	—
83	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
84	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
85	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
86	#95	—	47	4	no	—
87	#95	—	47	4	no	—
88	#95	—	47	2	no	—
89	#95	—	47	4	no	—
90	#95	—	47	2	no	—
91	#95	C₂×D₄₇	1	4	yes	—
92	C₂×Q₈	—	188	2	no	—
93	C₂×Q₈	—	188	4	no	—
94	C₂²×C₄	—	47	4	no	—
95	C₂²×C₄	—	47	4	no	—
96	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
97	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
98	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
100	C₄⋊C₄	—	4	1	no	—
101	C₄⋊C₄	—	4	1	no	—
102	C₄⋊C₄	—	188	4	no	—
103	C₄⋊C₄	—	188	2	no	—
104	C₄⋊C₄	—	188	2	no	—
105	C₄⋊C₄	—	188	2	no	—
106	C₄²	Dic₉₄	1	4	yes	complemented
107	C₂³	C₂³×D₄₇	1	1	yes	characteristic
108	Q₈	—	188	8	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	94	4	no	—
118	C₂×C₄	—	94	4	no	—
119	C₂×C₄	—	94	4	no	—
120	C₂×C₄	—	94	4	no	—
121	C₂×C₄	—	94	4	no	—
122	C₂×C₄	—	94	4	no	—
123	C₂×C₄	C₂×Dic₉₄	1	4	yes	—
124	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
125	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
126	C₂²	#13094069	1	1	yes	characteristic
127	C₂²	#13094070	1	2	yes	—
128	C₂²	#13094071	1	2	yes	—
129	C₂²	#13094072	1	1	yes	characteristic
130	C₄	—	2	4	no	—
131	C₄	—	2	2	no	—
132	C₄	—	4	2	no	—
133	C₄	—	4	2	no	—
134	C₄	—	188	4	no	—
135	C₄	—	188	4	no	—
136	C₂	#15689751	1	2	yes	—
137	C₂	#15689754	1	1	yes	characteristic
138	C₂	#15689865	1	1	yes	characteristic
139	C₂	#15689873	1	1	yes	characteristic
140	C₂	#15689877	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689065

C₂

yes

maximal

#15689062

C₂

yes

maximal

#15689044

C₂

yes

maximal

#15689043

C₂

yes

maximal

#15688989

C₂

yes

maximal

C₄₇×#382

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

Q₈

yes

complemented

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

C₂²×C₄₇

C₂²×Q₈

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

#2844

—

maximal, Sylow

C₂×C₄₇

#544

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

—

—

—

—

D₄₇

yes

characteristic

C₄₇

#2844

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂×C₄²

C₂×D₄₇

yes

—

—

—

—

—

C₂×D₄₇

yes

—

C₂×Q₈

—

188

—

C₂×Q₈

—

188

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₂²×C₄

C₂²×D₄₇

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

188

—

103

C₄⋊C₄

—

188

—

104

C₄⋊C₄

—

188

—

105

C₄⋊C₄

—

188

—

106

C₄²

Dic₉₄

yes

complemented

107

C₂³

C₂³×D₄₇

yes

characteristic

108

Q₈

—

188

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂×Dic₉₄

yes

—

124

C₂×C₄

C₂×Dic₉₄

yes

—

125

C₂²

C₂²×Dic₉₄

yes

characteristic

126

C₂²

#13094069

yes

characteristic

127

C₂²

#13094070

yes

—

128

C₂²

#13094071

yes

—

129

C₂²

#13094072

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

188

—

135

C₄

—

188

—

136

C₂

#15689751

yes

—

137

C₂

#15689754

yes

characteristic

138

C₂

#15689865

yes

characteristic

139

C₂

#15689873

yes

characteristic

140

C₂

#15689877

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄₇×#382	C₂
hypercenter	#382	D₄₇
hypocenter	C₄₇	#2844
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₄₇×#382

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
31. C₄²×C₄₇	Q₈	semidirect
77. C₄₇	#2844	semidirect
106. C₄²	Dic₉₄	semidirect

H ≅ G/N

type

31. C₄²×C₄₇

Q₈

semidirect

77. C₄₇

#2844

semidirect

106. C₄²

Dic₉₄

semidirect

Subgroups — #20251660

Conjugacy classes of subgroups

Complements