Group #26109622 — subgroups

Subgroups — #26109622

Order 12032.

486 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 131 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	#20243456	C₂	1	2	yes	maximal
2	#20241231	C₂	1	4	yes	maximal
3	C₂×C₄₇×#383	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
5	#15688878	—	2	4	no	—
6	C₂²×C₄²×C₄₇	C₂²	1	1	yes	characteristic
7	C₄₇×#383	—	2	4	no	—
8	C₂×#95×C₄₇	C₄	1	2	yes	—
9	C₂×#95×C₄₇	C₄	1	2	yes	—
10	C₂×#95×C₄₇	C₂²	1	2	yes	—
11	C₂²×C₄₇⋊C₈	—	2	4	no	—
12	C₂²⋊(C₄₇⋊C₈)	—	2	8	no	—
13	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₄×C₄₇	C₂×C₄	1	4	yes	—
15	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₄×C₄₇	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₄₇	—	2	4	no	—
18	C₂×C₄²×C₄₇	D₄	1	4	yes	—
19	#95×C₄₇	—	2	4	no	—
20	#95×C₄₇	—	2	4	no	—
21	#95×C₄₇	—	2	4	no	—
22	C₂⁴×C₄₇	C₂²×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	—	2	2	no	—
24	C₂²×C₄×C₄₇	—	2	2	no	—
25	C₂²×C₄×C₄₇	—	2	4	no	—
26	C₂²×C₄×C₄₇	—	2	4	no	—
27	C₂²×C₄×C₄₇	—	2	8	no	—
28	C₂²×C₄×C₄₇	—	4	2	no	—
29	C₂²×C₄×C₄₇	—	4	2	no	—
30	C₂²×C₄×C₄₇	—	4	4	no	—
31	C₂²×C₄×C₄₇	—	4	4	no	—
32	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
35	C₂²×C₄×C₄₇	#41	1	4	yes	—
36	C₄²×C₄₇	—	2	8	no	—
37	C₂×C₄₇⋊C₈	—	4	8	no	—
38	C₂×C₄₇⋊C₈	—	4	4	no	—
39	C₂³×C₄₇	—	2	2	no	—
40	C₂³×C₄₇	—	2	2	no	—
41	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
42	C₂³×C₄₇	#117	1	2	yes	—
43	C₂³×C₄₇	#131	1	4	yes	—
44	C₂×C₄×C₄₇	—	2	4	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	—	2	4	no	—
49	C₂×C₄×C₄₇	—	4	4	no	—
50	C₂×C₄×C₄₇	—	4	4	no	—
51	C₂×C₄×C₄₇	—	4	4	no	—
52	C₂×C₄×C₄₇	—	4	4	no	—
53	C₂×C₄×C₄₇	—	4	4	no	—
54	C₂×C₄×C₄₇	—	4	4	no	—
55	C₂×C₄×C₄₇	—	8	8	no	—
56	C₂×C₄×C₄₇	#123	1	4	yes	—
57	C₄₇⋊C₈	—	8	4	no	—
58	#8705	—	47	1	no	maximal, Sylow
59	C₂²×C₄₇	—	2	2	no	—
60	C₂²×C₄₇	—	2	2	no	—
61	C₂²×C₄₇	—	2	8	no	—
62	C₂²×C₄₇	#381	1	1	yes	characteristic
63	C₂²×C₄₇	#408	1	2	yes	—
64	C₂²×C₄₇	#410	1	2	yes	—
65	C₂²×C₄₇	#413	1	2	yes	—
66	C₂²×C₄₇	#433	1	4	yes	—
67	C₄×C₄₇	—	2	4	no	—
68	C₄×C₄₇	—	4	2	no	—
69	C₄×C₄₇	—	4	2	no	—
70	C₄×C₄₇	—	8	4	no	—
71	C₂×#383	D₄₇	1	1	yes	characteristic
72	#1817	—	47	2	no	—
73	#1280	—	47	4	no	—
74	C₂×C₄₇	—	2	4	no	—
75	C₂×C₄₇	#1473	1	4	yes	—
76	C₂×C₄₇	#1818	1	1	yes	characteristic
77	C₂×C₄₇	#1820	1	1	yes	characteristic
78	C₂×C₄₇	#1822	1	1	yes	characteristic
79	C₂²×C₄²	C₂×D₄₇	1	1	yes	characteristic
80	C₂×C₂²⋊C₈	—	47	4	no	—
81	#383	—	2	4	no	—
82	C₂×#95	Dic₄₇	1	2	yes	—
83	C₂×#95	Dic₄₇	1	2	yes	—
84	C₂×#95	C₂×D₄₇	1	2	yes	—
85	#336	—	94	4	no	—
86	C₄₇	#8705	1	1	yes	characteristic, complemented, Sylow
87	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
88	C₂³×C₄	C₂×Dic₄₇	1	4	yes	—
89	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
90	C₂³×C₄	C₂²×D₄₇	1	1	yes	characteristic
91	C₂²×C₈	—	94	4	no	—
92	C₂×C₄²	—	2	4	no	—
93	C₂×C₄²	C₂²⋊D₄₇	1	4	yes	—
94	C₂²⋊C₈	—	94	8	no	—
95	#95	—	2	4	no	—
96	#95	—	2	4	no	—
97	#95	—	2	4	no	—
98	C₂⁴	C₂²×Dic₄₇	1	1	yes	characteristic
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	2	4	no	—
103	C₂²×C₄	—	2	8	no	—
104	C₂²×C₄	—	4	2	no	—
105	C₂²×C₄	—	4	2	no	—
106	C₂²×C₄	—	4	4	no	—
107	C₂²×C₄	—	4	4	no	—
108	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
109	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
110	C₂²×C₄	#10625350	1	4	yes	—
111	C₂²×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
112	C₂×C₈	—	188	8	no	—
113	C₂×C₈	—	188	4	no	—
114	C₄²	—	2	8	no	—
115	C₂³	—	2	2	no	—
116	C₂³	—	2	2	no	—
117	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
118	C₂³	#13093983	1	2	yes	—
119	C₂³	#13094031	1	4	yes	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	4	no	—
126	C₂×C₄	—	4	4	no	—
127	C₂×C₄	—	4	4	no	—
128	C₂×C₄	—	4	4	no	—
129	C₂×C₄	—	4	4	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	—	8	8	no	—
132	C₂×C₄	#13094007	1	4	yes	—
133	C₈	—	376	4	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	2	8	no	—
137	C₂²	#15688986	1	1	yes	characteristic
138	C₂²	#15689090	1	2	yes	—
139	C₂²	#15689097	1	2	yes	—
140	C₂²	#15689107	1	2	yes	—
141	C₂²	#15689184	1	4	yes	—
142	C₄	—	2	4	no	—
143	C₄	—	4	2	no	—
144	C₄	—	4	2	no	—
145	C₄	—	8	4	no	—
146	C₂	—	2	4	no	—
147	C₂	#20241757	1	4	yes	—
148	C₂	#20243459	1	1	yes	characteristic
149	C₂	#20243465	1	1	yes	characteristic
150	C₂	#20243471	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243456

C₂

yes

maximal

#20241231

C₂

yes

maximal

C₂×C₄₇×#383

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₄₇⋊C₈)

C₂²

yes

—

#15688878

—

C₂²×C₄²×C₄₇

C₂²

yes

characteristic

C₄₇×#383

—

C₂×#95×C₄₇

C₄

yes

—

C₂×#95×C₄₇

C₄

yes

—

C₂×#95×C₄₇

C₂²

yes

—

C₂²×C₄₇⋊C₈

—

C₂²⋊(C₄₇⋊C₈)

—

C₂³×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₇

C₂×C₄

yes

—

C₂³×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

—

#95×C₄₇

—

#95×C₄₇

—

#95×C₄₇

—

C₂⁴×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

#41

yes

—

C₄²×C₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

#117

yes

—

C₂³×C₄₇

#131

yes

—

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×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#123

yes

—

C₄₇⋊C₈

—

#8705

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#381

yes

characteristic

C₂²×C₄₇

#408

yes

—

C₂²×C₄₇

#410

yes

—

C₂²×C₄₇

#413

yes

—

C₂²×C₄₇

#433

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₂×#383

D₄₇

yes

characteristic

#1817

—

#1280

—

C₂×C₄₇

—

C₂×C₄₇

#1473

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂²×C₄²

C₂×D₄₇

yes

characteristic

C₂×C₂²⋊C₈

—

#383

—

C₂×#95

Dic₄₇

yes

—

C₂×#95

Dic₄₇

yes

—

C₂×#95

C₂×D₄₇

yes

—

#336

—

C₄₇

#8705

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×Dic₄₇

yes

characteristic

C₂³×C₄

C₂×Dic₄₇

yes

—

C₂³×C₄

C₂×Dic₄₇

yes

characteristic

C₂³×C₄

C₂²×D₄₇

yes

characteristic

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

C₂²⋊D₄₇

yes

—

C₂²⋊C₈

—

#95

—

#95

—

#95

—

C₂⁴

C₂²×Dic₄₇

yes

characteristic

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

109

C₂²×C₄

C₂²⋊Dic₄₇

yes

—

110

C₂²×C₄

#10625350

yes

—

111

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

—

112

C₂×C₈

—

188

—

113

C₂×C₈

—

188

—

114

C₄²

—

115

C₂³

—

116

C₂³

—

117

C₂³

C₂×C₂²⋊Dic₄₇

yes

characteristic

118

C₂³

#13093983

yes

—

119

C₂³

#13094031

yes

—

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

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#13094007

yes

—

133

C₈

—

376

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

#15688986

yes

characteristic

138

C₂²

#15689090

yes

—

139

C₂²

#15689097

yes

—

140

C₂²

#15689107

yes

—

141

C₂²

#15689184

yes

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₂

—

147

C₂

#20241757

yes

—

148

C₂

#20243459

yes

characteristic

149

C₂

#20243465

yes

characteristic

150

C₂

#20243471

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₄₇
commutator, prime radical	C₂⁴×C₄₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₇
Fitting	C₂×C₄₇×#383	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
cosocle	C₂³×C₄×C₄₇	C₂³
hypercenter	C₂×#383	D₄₇
hypocenter	C₄₇	#8705
LCS	G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#383

subgroup

G/H

center

C₂³

C₂×C₂²⋊Dic₄₇

commutator, prime radical

C₂⁴×C₄₇

C₂²×C₄

Frattini

C₂³×C₄

C₂²×D₄₇

Fitting

C₂×C₄₇×#383

C₂

socle

C₂³×C₄₇

C₂×C₂²⋊C₄

cosocle

C₂³×C₄×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂⁴×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×#383

N	H ≅ G/N	type
86. C₄₇	#8705	semidirect

H ≅ G/N

type

86. C₄₇

#8705

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20243456	2
#20241231	2
C₂×C₄₇×#383	2
#8705	47

subgroup

index

#20243456

#20241231

C₂×C₄₇×#383

#8705

Subgroups — #26109622

Conjugacy classes of subgroups

Complements

Maximal subgroups