Group #20251376 — subgroups

Subgroups — #20251376

Order 6016.

368 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 141 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	#15689062	C₂	1	2	yes	maximal
2	#15689055	C₂	1	4	yes	maximal
3	#15689030	C₂	1	2	yes	maximal
4	#15689021	C₂	1	2	yes	maximal
5	#15689010	C₂	1	4	yes	maximal
6	C₄₇×#391	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊Dic₄₇	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	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
13	C₄₇⋊#95	C₂²	1	2	yes	—
14	C₄₇⋊#95	C₂²	1	4	yes	—
15	C₄₇⋊#95	C₂²	1	4	yes	—
16	C₄₇⋊#95	C₂²	1	4	yes	—
17	C₂²×Q₈×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	C₂×Q₈×C₄₇	—	4	1	no	—
22	C₂×Q₈×C₄₇	—	4	1	no	—
23	C₂×Q₈×C₄₇	Q₈	1	4	yes	—
24	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
25	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
27	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
28	C₂²×Dic₄₇	C₂³	1	4	yes	—
29	C₂²×Dic₄₇	C₂³	1	4	yes	—
30	C₄⋊C₄×C₄₇	—	4	1	no	—
31	C₄²×C₄₇	—	4	1	no	—
32	C₄⋊Dic₄₇	—	2	4	no	—
33	C₄⋊Dic₄₇	—	4	2	no	—
34	C₄⋊Dic₄₇	—	4	2	no	—
35	C₄⋊Dic₄₇	—	4	2	no	—
36	Dic₄₇⋊C₄	—	2	8	no	—
37	C₄×Dic₄₇	—	4	2	no	—
38	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
39	Q₈×C₄₇	—	4	4	no	—
40	C₂×C₄×C₄₇	—	2	2	no	—
41	C₂×C₄×C₄₇	—	2	2	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	2	1	no	—
44	C₂×C₄×C₄₇	—	2	1	no	—
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	4	no	—
48	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
50	C₂×C₄×C₄₇	#41	1	4	yes	—
51	C₂×Dic₄₇	—	2	4	no	—
52	C₂×Dic₄₇	—	2	4	no	—
53	C₂×Dic₄₇	—	2	4	no	—
54	C₂×Dic₄₇	—	2	4	no	—
55	C₂×Dic₄₇	—	2	4	no	—
56	C₂×Dic₄₇	—	2	4	no	—
57	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
58	C₂²×C₄₇	C₂×#41	1	2	yes	—
59	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
60	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
61	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
62	C₄×C₄₇	—	2	4	no	—
63	C₄×C₄₇	—	2	4	no	—
64	C₄×C₄₇	—	2	2	no	—
65	C₄×C₄₇	—	4	2	no	—
66	Dic₄₇	—	4	4	no	—
67	Dic₄₇	—	4	4	no	—
68	#2809	—	47	1	no	maximal, Sylow
69	C₂×C₄₇	#539	1	2	yes	—
70	C₂×C₄₇	#543	1	1	yes	characteristic
71	C₂×C₄₇	#555	1	1	yes	characteristic
72	C₂×C₄₇	#557	1	2	yes	—
73	C₂×C₄₇	#563	1	1	yes	characteristic
74	#400	—	47	2	no	—
75	#398	—	47	4	no	—
76	#391	—	47	2	no	—
77	#391	D₄₇	1	1	yes	characteristic
78	#389	—	47	2	no	—
79	#387	—	47	4	no	—
80	C₄₇	#2809	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
82	C₂×C₄⋊C₄	—	47	4	no	—
83	C₂×C₄⋊C₄	—	47	2	no	—
84	C₂×C₄⋊C₄	—	47	2	no	—
85	C₂×C₄⋊C₄	—	47	2	no	—
86	C₂×C₄⋊C₄	—	47	2	no	—
87	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
88	C₂×C₄²	—	47	2	no	—
89	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
90	#95	—	47	2	no	—
91	#95	—	47	4	no	—
92	#95	—	47	4	no	—
93	#95	—	47	4	no	—
94	#95	C₂×D₄₇	1	4	yes	—
95	C₂×Q₈	—	4	1	no	—
96	C₂×Q₈	—	4	1	no	—
97	C₂×Q₈	Dic₉₄	1	4	yes	—
98	C₂²×C₄	—	47	4	no	—
99	C₂²×C₄	—	47	4	no	—
100	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
101	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
102	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
103	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
104	C₄⋊C₄	—	4	1	no	—
105	C₄⋊C₄	—	94	8	no	—
106	C₄⋊C₄	—	94	4	no	—
107	C₄⋊C₄	—	188	2	no	—
108	C₄⋊C₄	—	188	2	no	—
109	C₄⋊C₄	—	188	2	no	—
110	C₄²	—	4	1	no	—
111	C₄²	—	188	2	no	—
112	C₂³	C₂³×D₄₇	1	1	yes	characteristic
113	Q₈	—	4	4	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	94	4	no	—
123	C₂×C₄	—	94	4	no	—
124	C₂×C₄	—	94	4	no	—
125	C₂×C₄	—	94	4	no	—
126	C₂×C₄	—	94	4	no	—
127	C₂×C₄	—	94	4	no	—
128	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
129	C₂×C₄	C₂×Dic₉₄	1	4	yes	—
130	C₂×C₄	#10625354	1	4	yes	—
131	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
132	C₂²	C₂×#10625354	1	2	yes	—
133	C₂²	#13094069	1	1	yes	characteristic
134	C₂²	#13094069	1	1	yes	characteristic
135	C₂²	#13094071	1	2	yes	—
136	C₄	—	2	2	no	—
137	C₄	—	2	4	no	—
138	C₄	—	2	4	no	—
139	C₄	—	4	2	no	—
140	C₄	—	188	4	no	—
141	C₄	—	188	4	no	—
142	C₂	#15689722	1	2	yes	—
143	C₂	#15689749	1	1	yes	characteristic
144	C₂	#15689839	1	1	yes	characteristic
145	C₂	#15689852	1	2	yes	—
146	C₂	#15689873	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689062

C₂

yes

maximal

#15689055

C₂

yes

maximal

#15689030

C₂

yes

maximal

#15689021

C₂

yes

maximal

#15689010

C₂

yes

maximal

C₄₇×#391

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

C₂²

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂⁴

yes

characteristic

Q₈×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×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

#2809

—

maximal, Sylow

C₂×C₄₇

#539

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

yes

—

—

—

D₄₇

yes

characteristic

#389

—

#387

—

C₄₇

#2809

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

—

—

—

—

C₂×D₄₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₉₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

C₂²×D₄₇

yes

—

101

C₂²×C₄

C₂²×D₄₇

yes

characteristic

102

C₂²×C₄

C₂²×D₄₇

yes

—

103

C₂²×C₄

C₂²×D₄₇

yes

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

188

—

108

C₄⋊C₄

—

188

—

109

C₄⋊C₄

—

188

—

110

C₄²

—

111

C₄²

—

188

—

112

C₂³

C₂³×D₄₇

yes

characteristic

113

Q₈

—

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

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×Dic₉₄

yes

—

129

C₂×C₄

C₂×Dic₉₄

yes

—

130

C₂×C₄

#10625354

yes

—

131

C₂²

C₂²×Dic₉₄

yes

characteristic

132

C₂²

C₂×#10625354

yes

—

133

C₂²

#13094069

yes

characteristic

134

C₂²

#13094069

yes

characteristic

135

C₂²

#13094071

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

188

—

141

C₄

—

188

—

142

C₂

#15689722

yes

—

143

C₂

#15689749

yes

characteristic

144

C₂

#15689839

yes

characteristic

145

C₂

#15689852

yes

—

146

C₂

#15689873

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄₇×#391	C₂
hypercenter	#391	D₄₇
hypocenter	C₄₇	#2809
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #391

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₄₇×#391

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #391

N	H ≅ G/N	type
80. C₄₇	#2809	semidirect

H ≅ G/N

type

80. C₄₇

#2809

semidirect

Subgroups — #20251376

Conjugacy classes of subgroups

Complements