Group #26120963 — subgroups

Subgroups — #26120963

Order 12032.

328 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 79 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	#20242179	C₂	1	3	yes	maximal, complemented
2	#20242057	C₂	1	3	yes	maximal
3	C₄₇×#1309	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13094021	C₂²	1	1	yes	characteristic
5	#15689176	C₂²	1	3	yes	—
6	#15688875	—	2	3	no	—
7	#15688873	—	2	3	no	—
8	#15688866	—	2	3	no	—
9	#15688866	—	2	3	no	—
10	C₄₇×#423	C₂²	1	3	yes	—
11	C₂²×D₁₈₈	—	2	3	no	—
12	#13094021	—	2	3	no	—
13	#13094021	Q₈	1	2	yes	—
14	C₄⋊(C₄₇⋊C₈)	D₄	1	3	yes	—
15	C₄⋊(C₄₇⋊C₈)	D₄	1	3	yes	—
16	C₂²⋊(C₄₇⋊C₈)	—	2	3	no	—
17	C₄×C₄₇⋊C₈	—	2	3	no	—
18	C₂×M₁₆×C₄₇	—	2	3	no	—
19	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
20	C₄⋊C₈×C₄₇	—	2	3	no	—
21	C₄⋊C₈×C₄₇	—	2	3	no	—
22	C₂³×D₄₇	—	4	1	no	—
23	C₂²×C₄×C₄₇	#41	1	3	yes	—
24	C₂×D₁₈₈	—	2	6	no	—
25	C₂×D₁₈₈	—	4	3	no	—
26	C₂×D₁₈₈	—	8	3	no	—
27	C₂×D₁₈₈	—	8	3	no	—
28	M₁₆×C₄₇	—	8	3	no	—
29	C₂×C₈×C₄₇	—	4	3	no	—
30	C₄²×C₄₇	C₂×D₄	1	3	yes	—
31	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄₇⋊C₈	—	2	3	no	—
33	C₂×C₄₇⋊C₈	—	2	3	no	—
34	C₂³×C₄₇	#126	1	1	yes	characteristic
35	C₂²×D₄₇	—	4	2	no	—
36	C₂²×D₄₇	—	8	6	no	—
37	C₂×C₄×C₄₇	—	2	3	no	—
38	C₂×C₄×C₄₇	—	2	3	no	—
39	C₂×C₄×C₄₇	#121	1	3	yes	—
40	C₂×C₄×C₄₇	C₂²⋊Q₈	1	3	yes	—
41	D₁₈₈	—	8	6	no	—
42	D₁₈₈	—	8	3	no	—
43	D₁₈₈	—	8	3	no	—
44	C₈×C₄₇	—	8	3	no	—
45	C₄₇⋊C₈	—	4	3	no	—
46	C₄₇⋊C₈	—	4	3	no	—
47	#10612	—	47	1	no	maximal, Sylow
48	C₂²×C₄₇	—	2	3	no	—
49	C₂²×C₄₇	#396	1	1	yes	characteristic
50	C₂×D₄₇	—	8	6	no	—
51	C₂×D₄₇	—	16	4	no	—
52	C₄×C₄₇	—	2	3	no	—
53	C₄×C₄₇	#471	1	3	yes	—
54	C₄×C₄₇	#472	1	3	yes	—
55	#1574	—	47	3	no	—
56	#1545	—	47	3	no	—
57	#1309	D₄₇	1	1	yes	characteristic, complemented
58	C₂×C₄₇	—	4	1	no	—
59	C₂×C₄₇	#2063	1	3	yes	—
60	D₄₇	—	16	2	no	—
61	C₂×#127	—	47	1	no	—
62	#431	—	47	3	no	—
63	#423	C₂×D₄₇	1	3	yes	—
64	#335	—	94	3	no	—
65	#334	—	94	3	no	—
66	#331	—	94	3	no	—
67	#331	—	94	3	no	—
68	C₄₇	#10612	1	1	yes	characteristic, complemented, Sylow
69	C₂²×D₄	—	94	3	no	—
70	C₂×M₁₆	—	2	3	no	—
71	#127	—	47	2	no	—
72	#127	—	94	3	no	—
73	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
74	C₄⋊C₈	—	2	3	no	—
75	C₄⋊C₈	—	2	3	no	—
76	C₄⋊C₈	—	47	3	no	—
77	C₄⋊C₈	—	47	3	no	—
78	C₂²⋊C₈	—	94	3	no	—
79	C₄×C₈	—	94	3	no	—
80	C₂⁴	—	188	1	no	—
81	C₂×D₄	—	94	6	no	—
82	C₂×D₄	—	376	3	no	—
83	C₂×D₄	—	376	3	no	—
84	C₂×D₄	—	188	3	no	—
85	C₂²×C₄	#10625350	1	3	yes	—
86	M₁₆	—	8	3	no	—
87	C₂×C₈	—	4	3	no	—
88	C₂×C₈	—	94	3	no	—
89	C₂×C₈	—	94	3	no	—
90	C₄²	D₄×D₄₇	1	3	yes	—
91	C₄²	Q₈×D₄₇	1	1	yes	characteristic
92	C₂³	—	376	6	no	—
93	C₂³	—	188	2	no	—
94	C₂³	#13094018	1	1	yes	characteristic
95	D₄	—	376	6	no	—
96	D₄	—	376	3	no	—
97	D₄	—	376	3	no	—
98	C₂×C₄	—	2	3	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	#13093997	1	3	yes	—
101	C₂×C₄	#13094002	1	3	yes	—
102	C₈	—	8	3	no	—
103	C₈	—	188	3	no	—
104	C₈	—	188	3	no	—
105	C₂²	—	2	3	no	—
106	C₂²	—	376	6	no	—
107	C₂²	—	752	4	no	—
108	C₂²	#15689047	1	1	yes	characteristic
109	C₄	—	2	3	no	—
110	C₄	#15689389	1	3	yes	—
111	C₄	#15689392	1	3	yes	—
112	C₂	—	4	1	no	—
113	C₂	—	752	2	no	—
114	C₂	#20244749	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20242179

C₂

yes

maximal, complemented

#20242057

C₂

yes

maximal

C₄₇×#1309

C₂

yes

maximal, characteristic, complemented

C₂×#13094021

C₂²

yes

characteristic

#15689176

C₂²

yes

—

—

—

—

—

C₄₇×#423

C₂²

yes

—

C₂²×D₁₈₈

—

#13094021

—

#13094021

Q₈

yes

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

C₄⋊(C₄₇⋊C₈)

D₄

yes

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

—

C₂×M₁₆×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₂³×D₄₇

—

C₂²×C₄×C₄₇

#41

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#126

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#121

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#10612

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#396

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#471

yes

—

C₄×C₄₇

#472

yes

—

#1574

—

#1545

—

#1309

D₄₇

yes

characteristic, complemented

C₂×C₄₇

—

C₂×C₄₇

#2063

yes

—

D₄₇

—

C₂×#127

—

#431

—

#423

C₂×D₄₇

yes

—

—

—

—

—

C₄₇

#10612

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×M₁₆

—

#127

—

#127

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂⁴

—

188

—

C₂×D₄

—

C₂×D₄

—

376

—

C₂×D₄

—

376

—

C₂×D₄

—

188

—

C₂²×C₄

#10625350

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

D₄×D₄₇

yes

—

C₄²

Q₈×D₄₇

yes

characteristic

C₂³

—

376

—

C₂³

—

188

—

C₂³

#13094018

yes

characteristic

D₄

—

376

—

D₄

—

376

—

D₄

—

376

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

#13093997

yes

—

101

C₂×C₄

#13094002

yes

—

102

C₈

—

103

C₈

—

188

—

104

C₈

—

188

—

105

C₂²

—

106

C₂²

—

376

—

107

C₂²

—

752

—

108

C₂²

#15689047

yes

characteristic

109

C₄

—

110

C₄

#15689389

yes

—

111

C₄

#15689392

yes

—

112

C₂

—

113

C₂

—

752

—

114

C₂

#20244749

yes

—

	subgroup	G/H
center	C₂²	#15689047
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1309	C₂
socle	C₂²×C₄₇	#396
prime radical	C₂×#13094021	C₂²
hypercenter	#1309	D₄₇
hypocenter	C₄₇	#10612
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#15689047

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1309

C₂

C₂²×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
1. #20242179	C₂	semidirect
3. C₄₇×#1309	C₂	semidirect
57. #1309	D₄₇	semidirect
68. C₄₇	#10612	semidirect

H ≅ G/N

type

1. #20242179

C₂

semidirect

3. C₄₇×#1309

C₂

semidirect

57. #1309

D₄₇

semidirect

68. C₄₇

#10612

semidirect

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

subgroup	index
#20242179	2
#20242057	2
C₄₇×#1309	2
#10612	47

subgroup

index

#20242179

#20242057

C₄₇×#1309

#10612

Subgroups — #26120963

Conjugacy classes of subgroups

Complements

Maximal subgroups