Group #26103663 — subgroups

Subgroups — #26103663

Order 12032.

216 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 47 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	#20241334	C₂	1	2	yes	maximal
2	C₄₇×#1956	C₂	1	1	yes	maximal, characteristic
3	#15688849	—	2	2	no	—
4	C₂×C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
5	C₂×#103×C₄₇	—	2	2	no	—
6	C₂×C₄×C₈×C₄₇	C₄	1	1	yes	characteristic
7	C₄₇×#384	C₂²	1	1	yes	characteristic
8	C₂²⋊(C₄₇⋊C₈)	—	4	2	no	—
9	C₂²×Q₈×C₄₇	—	2	1	no	—
10	C₂²×C₈×C₄₇	—	2	1	no	—
11	C₄⋊Q₈×C₄₇	—	2	1	no	—
12	C₄⋊Q₈×C₄₇	—	4	1	no	—
13	C₄⋊Q₈×C₄₇	C₈	1	2	yes	complemented
14	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
15	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
16	C₂×C₄⋊C₄×C₄₇	D₄	1	2	yes	—
17	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
18	#103×C₄₇	—	4	4	no	—
19	C₄×C₈×C₄₇	—	4	1	no	—
20	C₂×Q₈×C₄₇	—	2	2	no	—
21	C₂×Q₈×C₄₇	—	4	2	no	—
22	C₂×Q₈×C₄₇	—	4	2	no	—
23	C₂×Q₈×C₄₇	—	4	1	no	—
24	C₂²×C₄×C₄₇	—	2	1	no	—
25	C₂²×C₄×C₄₇	—	2	2	no	—
26	C₂²×C₄×C₄₇	—	4	1	no	—
27	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₄₇	—	2	2	no	—
29	C₂×C₈×C₄₇	—	4	1	no	—
30	C₂×C₈×C₄₇	—	4	1	no	—
31	C₄⋊C₄×C₄₇	—	4	2	no	—
32	C₄⋊C₄×C₄₇	—	4	2	no	—
33	C₄⋊C₄×C₄₇	—	4	2	no	—
34	C₄⋊C₄×C₄₇	—	8	2	no	—
35	C₄²×C₄₇	—	2	1	no	—
36	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
37	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
38	C₂×C₄₇⋊C₈	—	8	2	no	—
39	C₂³×C₄₇	#99	1	1	yes	characteristic
40	Q₈×C₄₇	—	4	4	no	—
41	Q₈×C₄₇	—	8	2	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	1	no	—
45	C₂×C₄×C₄₇	—	2	1	no	—
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	—	2	1	no	—
48	C₂×C₄×C₄₇	—	4	2	no	—
49	C₂×C₄×C₄₇	—	4	2	no	—
50	C₂×C₄×C₄₇	—	8	1	no	—
51	C₂×C₄×C₄₇	—	8	1	no	—
52	C₂×C₄×C₄₇	—	8	2	no	—
53	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#100	1	1	yes	characteristic
55	C₈×C₄₇	—	4	2	no	—
56	C₄₇⋊C₈	—	16	2	no	—
57	#7355	—	47	1	no	maximal, Sylow
58	C₂²×C₄₇	—	2	1	no	—
59	C₂²×C₄₇	—	2	1	no	—
60	C₂²×C₄₇	#323	1	1	yes	characteristic
61	C₂²×C₄₇	#353	1	1	yes	characteristic
62	C₂²×C₄₇	#355	1	1	yes	characteristic
63	C₄×C₄₇	—	2	2	no	—
64	C₄×C₄₇	—	2	2	no	—
65	C₄×C₄₇	—	2	1	no	—
66	C₄×C₄₇	—	2	1	no	—
67	C₄×C₄₇	—	8	2	no	—
68	C₄×C₄₇	—	8	2	no	—
69	#1956	D₄₇	1	1	yes	characteristic
70	#1325	—	47	2	no	—
71	C₂×C₄₇	—	2	1	no	—
72	C₂×C₄₇	—	2	1	no	—
73	C₂×C₄₇	#1318	1	1	yes	characteristic
74	C₂×C₄₇	#1409	1	1	yes	characteristic
75	C₂×C₄₇	#1411	1	1	yes	characteristic
76	C₂×C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic
77	C₂×#103	—	2	2	no	—
78	C₂×C₄×C₈	Dic₄₇	1	1	yes	characteristic
79	#384	C₂×D₄₇	1	1	yes	characteristic
80	#324	—	94	2	no	—
81	C₄₇	#7355	1	1	yes	characteristic, complemented, Sylow
82	C₂²×Q₈	—	2	1	no	—
83	C₂²×C₈	—	2	1	no	—
84	C₄⋊Q₈	—	2	1	no	—
85	C₄⋊Q₈	—	4	1	no	—
86	C₄⋊Q₈	C₄₇⋊C₈	1	2	yes	complemented
87	C₂×C₄⋊C₄	—	4	1	no	—
88	C₂×C₄⋊C₄	—	4	1	no	—
89	C₂×C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
90	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
91	#103	—	4	4	no	—
92	C₂²⋊C₈	—	188	2	no	—
93	C₄×C₈	—	4	1	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	4	2	no	—
96	C₂×Q₈	—	4	2	no	—
97	C₂×Q₈	—	4	1	no	—
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	4	1	no	—
101	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	4	1	no	—
104	C₂×C₈	—	4	1	no	—
105	C₂×C₈	—	376	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	—	8	2	no	—
110	C₄²	—	2	1	no	—
111	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
112	C₄²	#10625329	1	1	yes	characteristic
113	C₂³	#13093939	1	1	yes	characteristic
114	Q₈	—	4	4	no	—
115	Q₈	—	8	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	8	1	no	—
125	C₂×C₄	—	8	1	no	—
126	C₂×C₄	—	8	2	no	—
127	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
128	C₂×C₄	#13093941	1	1	yes	characteristic
129	C₈	—	4	2	no	—
130	C₈	—	752	2	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	#15688847	1	1	yes	characteristic
134	C₂²	#15688916	1	1	yes	characteristic
135	C₂²	#15688920	1	1	yes	characteristic
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	8	2	no	—
141	C₄	—	8	2	no	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#20241320	1	1	yes	characteristic
145	C₂	#20241553	1	1	yes	characteristic
146	C₂	#20241557	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20241334

C₂

yes

maximal

C₄₇×#1956

C₂

yes

maximal, characteristic

#15688849

—

C₂×C₄⋊Q₈×C₄₇

C₄

yes

characteristic

C₂×#103×C₄₇

—

C₂×C₄×C₈×C₄₇

C₄

yes

characteristic

C₄₇×#384

C₂²

yes

characteristic

C₂²⋊(C₄₇⋊C₈)

—

C₂²×Q₈×C₄₇

—

C₂²×C₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

C₈

yes

complemented

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

#103×C₄₇

—

C₄×C₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₈

yes

characteristic

C₄²×C₄₇

M₁₆

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#99

yes

characteristic

Q₈×C₄₇

—

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×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₄₇

#100

yes

characteristic

C₈×C₄₇

—

C₄₇⋊C₈

—

#7355

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

#1956

D₄₇

yes

characteristic

#1325

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄⋊Q₈

Dic₄₇

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

Dic₄₇

yes

characteristic

#384

C₂×D₄₇

yes

characteristic

#324

—

C₄₇

#7355

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₄₇⋊C₈

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₄₇

yes

—

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

#103

—

C₂²⋊C₈

—

188

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

376

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄²

—

111

C₄²

C₂×C₄₇⋊C₈

yes

characteristic

112

C₄²

#10625329

yes

characteristic

113

C₂³

#13093939

yes

characteristic

114

Q₈

—

115

Q₈

—

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

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

128

C₂×C₄

#13093941

yes

characteristic

129

C₈

—

130

C₈

—

752

—

131

C₂²

—

132

C₂²

—

133

C₂²

#15688847

yes

characteristic

134

C₂²

#15688916

yes

characteristic

135

C₂²

#15688920

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₂

—

143

C₂

—

144

C₂

#20241320

yes

characteristic

145

C₂

#20241553

yes

characteristic

146

C₂

#20241557

yes

characteristic

	subgroup	G/H
center	C₂²	#15688916
commutator	C₄²×C₄₇	C₂×C₈
Frattini	#384	C₂×D₄₇
Fitting	C₄₇×#1956	C₂
socle	C₂²×C₄₇	#353
prime radical	C₂³×C₄₇	#99
cosocle	C₄₇×#384	C₂²
hypercenter	#1956	D₄₇
hypocenter	C₄₇	#7355
LCS	G ▷ C₄²×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1956

subgroup

G/H

center

C₂²

#15688916

commutator

C₄²×C₄₇

C₂×C₈

Frattini

#384

C₂×D₄₇

Fitting

C₄₇×#1956

C₂

C₂²×C₄₇

C₂³×C₄₇

C₄₇×#384

C₂²

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #384 ◁ #1956

N	H ≅ G/N	type
13. C₄⋊Q₈×C₄₇	C₈	semidirect
81. C₄₇	#7355	semidirect
86. C₄⋊Q₈	C₄₇⋊C₈	semidirect

H ≅ G/N

type

13. C₄⋊Q₈×C₄₇

C₈

semidirect

81. C₄₇

#7355

semidirect

86. C₄⋊Q₈

C₄₇⋊C₈

semidirect

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

subgroup	index
#20241334	2
C₄₇×#1956	2
#7355	47

subgroup

index

#20241334

C₄₇×#1956

#7355

Subgroups — #26103663

Conjugacy classes of subgroups

Complements

Maximal subgroups