Group #26103104 — subgroups

Subgroups — #26103104

Order 12032.

202 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20241470	C₂	1	2	yes	maximal
2	C₄₇×#1991	C₂	1	1	yes	maximal, characteristic
3	C₂×#13093964	—	2	2	no	—
4	C₂×#125×C₄₇	C₄	1	1	yes	characteristic
5	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
6	C₄₇×#382	C₄	1	1	yes	characteristic
7	C₄₇×#340	—	2	2	no	—
8	#13093964	—	2	4	no	—
9	#13093964	—	4	2	no	—
10	C₂×C₄₇⋊C₁₆	—	4	2	no	—
11	C₂²×C₈×C₄₇	D₄	1	2	yes	—
12	#125×C₄₇	—	2	1	no	—
13	#125×C₄₇	—	4	1	no	—
14	#125×C₄₇	C₈	1	2	yes	—
15	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
16	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
17	C₂×C₄⋊C₄×C₄₇	—	4	1	no	—
18	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₄×C₈×C₄₇	—	4	1	no	—
20	#95×C₄₇	—	4	1	no	—
21	C₂²×C₄×C₄₇	—	2	1	no	—
22	C₂²×C₄×C₄₇	—	4	1	no	—
23	C₂²×C₄×C₄₇	—	4	1	no	—
24	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₈×C₄₇	—	2	2	no	—
26	C₂×C₈×C₄₇	—	2	2	no	—
27	C₂×C₈×C₄₇	SD₁₆	1	2	yes	—
28	C₂×C₈×C₄₇	Q₁₆	1	2	yes	—
29	C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₄⋊C₄×C₄₇	—	4	2	no	—
31	C₄⋊C₄×C₄₇	—	4	2	no	—
32	C₄⋊C₄×C₄₇	—	4	1	no	—
33	C₄⋊C₄×C₄₇	—	4	2	no	—
34	C₄²×C₄₇	—	2	1	no	—
35	C₄²×C₄₇	C₂×C₈	1	1	yes	characteristic
36	C₄²×C₄₇	M₁₆	1	1	yes	characteristic
37	C₄₇⋊C₁₆	—	4	4	no	—
38	C₂³×C₄₇	#100	1	1	yes	characteristic
39	C₂×C₄×C₄₇	—	2	1	no	—
40	C₂×C₄×C₄₇	—	2	1	no	—
41	C₂×C₄×C₄₇	—	4	2	no	—
42	C₂×C₄×C₄₇	—	4	2	no	—
43	C₂×C₄×C₄₇	—	8	2	no	—
44	C₂×C₄×C₄₇	—	8	2	no	—
45	C₂×C₄×C₄₇	—	8	1	no	—
46	C₂×C₄×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
47	C₂×C₄×C₄₇	#99	1	1	yes	characteristic
48	C₂×C₄×C₄₇	#103	1	2	yes	—
49	C₂×C₄×C₄₇	#104	1	2	yes	—
50	C₈×C₄₇	—	2	2	no	—
51	C₈×C₄₇	—	2	2	no	—
52	#7135	—	47	1	no	maximal, Sylow
53	C₂²×C₄₇	—	2	2	no	—
54	C₂²×C₄₇	#323	1	1	yes	characteristic
55	C₂²×C₄₇	#329	1	1	yes	characteristic
56	C₂²×C₄₇	#333	1	1	yes	characteristic
57	C₄×C₄₇	—	4	1	no	—
58	C₄×C₄₇	—	4	1	no	—
59	C₄×C₄₇	—	8	2	no	—
60	C₄×C₄₇	—	16	1	no	—
61	C₄×C₄₇	Q₈⋊C₈	1	2	yes	—
62	C₄×C₄₇	#328	1	2	yes	—
63	#1991	D₄₇	1	1	yes	characteristic
64	#1376	—	47	2	no	—
65	C₂×C₄₇	—	2	1	no	—
66	C₂×C₄₇	—	2	1	no	—
67	C₂×C₄₇	#1272	1	1	yes	characteristic
68	C₂×C₄₇	#1366	1	2	yes	—
69	C₂×#125	Dic₄₇	1	1	yes	characteristic
70	C₂×M₃₂	—	94	2	no	—
71	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
72	#382	Dic₄₇	1	1	yes	characteristic
73	#340	—	2	2	no	—
74	C₄₇	#7135	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
76	#125	—	2	1	no	—
77	#125	—	4	1	no	—
78	#125	C₄₇⋊C₈	1	2	yes	—
79	C₂×C₄⋊C₄	—	2	1	no	—
80	C₂×C₄⋊C₄	—	2	1	no	—
81	C₂×C₄⋊C₄	—	4	1	no	—
82	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
83	M₃₂	—	94	4	no	—
84	M₃₂	—	188	2	no	—
85	C₂×C₁₆	—	188	2	no	—
86	C₄×C₈	—	4	1	no	—
87	#95	—	4	1	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	4	1	no	—
90	C₂²×C₄	—	4	1	no	—
91	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	#10625336	1	2	yes	—
95	C₂×C₈	#10625337	1	2	yes	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄²	—	2	1	no	—
102	C₄²	C₂×C₄₇⋊C₈	1	1	yes	characteristic
103	C₄²	#10625329	1	1	yes	characteristic
104	C₁₆	—	188	4	no	—
105	C₂³	#13093941	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	8	2	no	—
112	C₂×C₄	—	8	1	no	—
113	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
114	C₂×C₄	#13093939	1	1	yes	characteristic
115	C₂×C₄	#13093948	1	2	yes	—
116	C₂×C₄	#13093951	1	2	yes	—
117	C₈	—	2	2	no	—
118	C₈	—	2	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	#15688847	1	1	yes	characteristic
121	C₂²	#15688863	1	1	yes	characteristic
122	C₂²	#15688872	1	1	yes	characteristic
123	C₄	—	4	1	no	—
124	C₄	—	4	1	no	—
125	C₄	—	8	2	no	—
126	C₄	—	16	1	no	—
127	C₄	Q₈⋊(C₄₇⋊C₈)	1	2	yes	—
128	C₄	#15688860	1	2	yes	—
129	C₂	—	2	1	no	—
130	C₂	—	2	1	no	—
131	C₂	#20241214	1	1	yes	characteristic
132	C₂	#20241449	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20241470

C₂

yes

maximal

C₄₇×#1991

C₂

yes

maximal, characteristic

C₂×#13093964

—

C₂×#125×C₄₇

C₄

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#382

C₄

yes

characteristic

C₄₇×#340

—

#13093964

—

#13093964

—

C₂×C₄₇⋊C₁₆

—

C₂²×C₈×C₄₇

D₄

yes

—

#125×C₄₇

—

#125×C₄₇

—

#125×C₄₇

C₈

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

—

#95×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

SD₁₆

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₄⋊C₄×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

C₄₇⋊C₁₆

—

C₂³×C₄₇

#100

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

C₂²⋊C₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#104

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

#7135

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

Q₈⋊C₈

yes

—

C₄×C₄₇

#328

yes

—

#1991

D₄₇

yes

characteristic

#1376

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1272

yes

characteristic

C₂×C₄₇

#1366

yes

—

C₂×#125

Dic₄₇

yes

characteristic

C₂×M₃₂

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#382

Dic₄₇

yes

characteristic

#340

—

C₄₇

#7135

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂²⋊D₄₇

yes

—

#125

—

#125

—

#125

C₄₇⋊C₈

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

M₃₂

—

M₃₂

—

188

—

C₂×C₁₆

—

188

—

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#10625336

yes

—

C₂×C₈

#10625337

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

C₂×C₄₇⋊C₈

yes

characteristic

103

C₄²

#10625329

yes

characteristic

104

C₁₆

—

188

—

105

C₂³

#13093941

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

114

C₂×C₄

#13093939

yes

characteristic

115

C₂×C₄

#13093948

yes

—

116

C₂×C₄

#13093951

yes

—

117

C₈

—

118

C₈

—

119

C₂²

—

120

C₂²

#15688847

yes

characteristic

121

C₂²

#15688863

yes

characteristic

122

C₂²

#15688872

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

Q₈⋊(C₄₇⋊C₈)

yes

—

128

C₄

#15688860

yes

—

129

C₂

—

130

C₂

—

131

C₂

#20241214

yes

characteristic

132

C₂

#20241449

yes

—

	subgroup	G/H
center	C₂²	#15688872
commutator	C₄²×C₄₇	C₂×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₄₇×#1991	C₂
socle	C₂²×C₄₇	#333
prime radical	C₂³×C₄₇	#100
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	#1991	D₄₇
hypocenter	C₄₇	#7135
LCS	G ▷ C₄²×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1991

subgroup

G/H

center

C₂²

#15688872

commutator

C₄²×C₄₇

C₂×C₈

Frattini

C₂×C₄×C₈

C₂×D₄₇

Fitting

C₄₇×#1991

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄×C₈×C₄₇

C₂²

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1991

N	H ≅ G/N	type
74. C₄₇	#7135	semidirect

H ≅ G/N

type

74. C₄₇

#7135

semidirect

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

subgroup	index
#20241470	2
C₄₇×#1991	2
#7135	47

subgroup

index

#20241470

C₄₇×#1991

#7135

Subgroups — #26103104

Conjugacy classes of subgroups

Complements

Maximal subgroups