Group #26121420 — subgroups

Subgroups — #26121420

Order 12032.

372 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 59 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	C₄₇⋊#1304	C₂	1	3	yes	maximal
2	C₄₇⋊#1304	C₂	1	3	yes	maximal
3	C₄₇×#1304	C₂	1	1	yes	maximal, characteristic
4	#15688979	C₂²	1	1	yes	characteristic
5	#15688979	C₂²	1	3	yes	—
6	C₄₇×#379	C₂²	1	3	yes	—
7	C₂×C₂²⋊Dic₄₇	—	2	3	no	—
8	C₂×C₂²⋊Dic₄₇	—	2	3	no	—
9	C₂×C₂²⋊Dic₄₇	—	2	3	no	—
10	C₂×C₂²⋊Dic₄₇	—	2	3	no	—
11	C₂⁵×C₄₇	C₂³	1	1	yes	characteristic
12	C₂×C₂²⋊C₄×C₄₇	—	2	3	no	—
13	C₂×C₂²⋊C₄×C₄₇	—	2	3	no	—
14	C₂×C₂²⋊C₄×C₄₇	—	2	3	no	—
15	C₂⁴×C₄₇	—	8	3	no	—
16	C₂⁴×C₄₇	#41	1	3	yes	—
17	C₂⁴×C₄₇	#41	1	3	yes	—
18	C₂⁴×C₄₇	#41	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	—	4	3	no	—
20	C₂²×Dic₄₇	—	4	1	no	—
21	C₂²×Dic₄₇	—	4	3	no	—
22	C₂²⋊C₄×C₄₇	—	8	3	no	—
23	C₂²⋊C₄×C₄₇	—	8	3	no	—
24	C₂²⋊C₄×C₄₇	—	8	3	no	—
25	C₂²⋊Dic₄₇	—	8	3	no	—
26	C₂²⋊Dic₄₇	—	8	3	no	—
27	C₂²⋊Dic₄₇	—	8	3	no	—
28	C₂²⋊Dic₄₇	—	8	3	no	—
29	C₂³×C₄₇	—	4	3	no	—
30	C₂³×C₄₇	—	4	3	no	—
31	C₂³×C₄₇	—	4	3	no	—
32	C₂³×C₄₇	—	4	3	no	—
33	C₂³×C₄₇	—	4	3	no	—
34	C₂³×C₄₇	—	4	3	no	—
35	C₂³×C₄₇	—	4	3	no	—
36	C₂³×C₄₇	—	8	8	no	—
37	C₂³×C₄₇	#126	1	3	yes	—
38	C₂³×C₄₇	#126	1	1	yes	characteristic
39	C₂³×C₄₇	#126	1	3	yes	—
40	C₂×C₄×C₄₇	—	8	3	no	—
41	C₂×C₄×C₄₇	—	8	3	no	—
42	C₂×C₄×C₄₇	—	8	3	no	—
43	C₂×Dic₄₇	—	8	3	no	—
44	C₂×Dic₄₇	—	8	3	no	—
45	C₂×Dic₄₇	—	8	3	no	—
46	C₂×Dic₄₇	—	8	3	no	—
47	#10690	—	47	1	no	maximal, Sylow
48	C₂²×C₄₇	—	2	3	no	—
49	C₂²×C₄₇	—	2	3	no	—
50	C₂²×C₄₇	—	2	3	no	—
51	C₂²×C₄₇	—	2	3	no	—
52	C₂²×C₄₇	—	2	3	no	—
53	C₂²×C₄₇	—	2	3	no	—
54	C₂²×C₄₇	—	2	3	no	—
55	C₂²×C₄₇	—	8	2	no	—
56	C₂²×C₄₇	—	8	6	no	—
57	C₂²×C₄₇	—	8	6	no	—
58	C₂²×C₄₇	#401	1	1	yes	characteristic
59	C₄×C₄₇	—	16	3	no	—
60	Dic₄₇	—	16	1	no	—
61	Dic₄₇	—	16	3	no	—
62	#1304	—	47	3	no	—
63	#1304	—	47	3	no	—
64	#1304	D₄₇	1	1	yes	characteristic
65	C₂×C₄₇	—	4	1	no	—
66	C₂×C₄₇	—	4	3	no	—
67	C₂×C₄₇	—	4	3	no	—
68	C₂×C₄₇	#2104	1	3	yes	—
69	#379	—	47	1	no	—
70	#379	—	47	3	no	—
71	#379	C₂×D₄₇	1	3	yes	—
72	C₄₇	#10690	1	1	yes	characteristic, complemented, Sylow
73	C₂⁵	C₂²×D₄₇	1	1	yes	characteristic
74	C₂×C₂²⋊C₄	—	2	3	no	—
75	C₂×C₂²⋊C₄	—	2	3	no	—
76	C₂×C₂²⋊C₄	—	2	3	no	—
77	C₂×C₂²⋊C₄	—	94	3	no	—
78	C₂×C₂²⋊C₄	—	94	3	no	—
79	C₂×C₂²⋊C₄	—	94	3	no	—
80	C₂×C₂²⋊C₄	—	94	3	no	—
81	C₂⁴	—	8	3	no	—
82	C₂⁴	#10625350	1	3	yes	—
83	C₂⁴	#10625352	1	3	yes	—
84	C₂⁴	#10625354	1	1	yes	characteristic
85	C₂²×C₄	—	4	3	no	—
86	C₂²×C₄	—	188	1	no	—
87	C₂²×C₄	—	188	3	no	—
88	C₂²⋊C₄	—	8	3	no	—
89	C₂²⋊C₄	—	8	3	no	—
90	C₂²⋊C₄	—	8	3	no	—
91	C₂²⋊C₄	—	376	3	no	—
92	C₂²⋊C₄	—	376	3	no	—
93	C₂²⋊C₄	—	376	3	no	—
94	C₂²⋊C₄	—	376	3	no	—
95	C₂³	—	4	3	no	—
96	C₂³	—	4	3	no	—
97	C₂³	—	4	3	no	—
98	C₂³	—	4	3	no	—
99	C₂³	—	4	3	no	—
100	C₂³	—	4	3	no	—
101	C₂³	—	4	3	no	—
102	C₂³	—	8	8	no	—
103	C₂³	#13094017	1	3	yes	—
104	C₂³	#13094018	1	1	yes	characteristic
105	C₂³	#13094019	1	3	yes	—
106	C₂×C₄	—	8	3	no	—
107	C₂×C₄	—	8	3	no	—
108	C₂×C₄	—	8	3	no	—
109	C₂×C₄	—	376	3	no	—
110	C₂×C₄	—	376	3	no	—
111	C₂×C₄	—	376	3	no	—
112	C₂×C₄	—	376	3	no	—
113	C₂²	—	2	3	no	—
114	C₂²	—	2	3	no	—
115	C₂²	—	2	3	no	—
116	C₂²	—	2	3	no	—
117	C₂²	—	2	3	no	—
118	C₂²	—	2	3	no	—
119	C₂²	—	2	3	no	—
120	C₂²	—	8	6	no	—
121	C₂²	—	8	6	no	—
122	C₂²	—	8	2	no	—
123	C₂²	C₄₇⋊#401	1	1	yes	characteristic
124	C₄	—	16	3	no	—
125	C₄	—	752	1	no	—
126	C₄	—	752	3	no	—
127	C₂	—	4	3	no	—
128	C₂	—	4	1	no	—
129	C₂	—	4	3	no	—
130	C₂	C₄₇⋊#2104	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄₇⋊#1304

C₂

yes

maximal

C₄₇⋊#1304

C₂

yes

maximal

C₄₇×#1304

C₂

yes

maximal, characteristic

#15688979

C₂²

yes

characteristic

#15688979

C₂²

yes

—

C₄₇×#379

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

—

C₂×C₂²⋊Dic₄₇

—

C₂×C₂²⋊Dic₄₇

—

C₂×C₂²⋊Dic₄₇

—

C₂⁵×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₂²⋊C₄×C₄₇

—

C₂⁴×C₄₇

—

C₂⁴×C₄₇

#41

yes

—

C₂⁴×C₄₇

#41

yes

—

C₂⁴×C₄₇

#41

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₂²⋊Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

#126

yes

—

C₂³×C₄₇

#126

yes

characteristic

C₂³×C₄₇

#126

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

#10690

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#401

yes

characteristic

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

#1304

—

#1304

—

#1304

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#2104

yes

—

#379

—

#379

—

#379

C₂×D₄₇

yes

—

C₄₇

#10690

yes

characteristic, complemented, Sylow

C₂⁵

C₂²×D₄₇

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂⁴

—

C₂⁴

#10625350

yes

—

C₂⁴

#10625352

yes

—

C₂⁴

#10625354

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

188

—

C₂²×C₄

—

188

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

376

—

C₂²⋊C₄

—

376

—

C₂²⋊C₄

—

376

—

C₂²⋊C₄

—

376

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

#13094017

yes

—

104

C₂³

#13094018

yes

characteristic

105

C₂³

#13094019

yes

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

376

—

110

C₂×C₄

—

376

—

111

C₂×C₄

—

376

—

112

C₂×C₄

—

376

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

C₄₇⋊#401

yes

characteristic

124

C₄

—

125

C₄

—

752

—

126

C₄

—

752

—

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

C₄₇⋊#2104

yes

—

	subgroup	G/H
center	C₂²	C₄₇⋊#401
commutator, prime radical, cosocle	C₂⁵×C₄₇	C₂³
Frattini	C₂⁵	C₂²×D₄₇
Fitting	C₄₇×#1304	C₂
socle	C₂²×C₄₇	#401
hypercenter	#1304	D₄₇
hypocenter	C₄₇	#10690
LCS	G ▷ C₂⁵×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ #1304

subgroup

G/H

center

C₂²

C₄₇⋊#401

commutator, prime radical, cosocle

C₂⁵×C₄₇

C₂³

Frattini

C₂⁵

C₂²×D₄₇

Fitting

C₄₇×#1304

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂⁵×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂⁵ ◁ #1304

N	H ≅ G/N	type
72. C₄₇	#10690	semidirect

H ≅ G/N

type

72. C₄₇

#10690

semidirect

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

subgroup	index
C₄₇⋊#1304	2
C₄₇×#1304	2
#10690	47

subgroup

index

C₄₇⋊#1304

C₄₇×#1304

#10690

Subgroups — #26121420

Conjugacy classes of subgroups

Complements

Maximal subgroups