Group #26119044 — subgroups

Subgroups — #26119044

Order 12032.

478 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 89 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	#20254782	C₂	1	3	yes	maximal
2	#20244364	C₂	1	3	yes	maximal
3	C₄₇×#1309	C₂	1	1	yes	maximal, characteristic
4	#15689680	C₂²	1	3	yes	—
5	#15689409	—	2	6	no	—
6	#15689381	—	2	6	no	—
7	#15689334	—	2	6	no	—
8	C₄²×Dic₄₇	C₂²	1	1	yes	characteristic
9	#15688884	—	2	6	no	—
10	C₄₇×#423	C₂²	1	3	yes	—
11	C₂×Dic₁₈₈	—	2	6	no	—
12	#13094024	D₄	1	6	yes	—
13	#13094016	—	2	3	no	—
14	#13094003	—	2	6	no	—
15	C₄×Dic₉₄	—	2	6	no	—
16	#13093983	—	2	3	no	—
17	C₂×C₄×Dic₄₇	—	2	3	no	—
18	#13093956	—	2	6	no	—
19	#13093949	—	2	12	no	—
20	C₂×M₁₆×C₄₇	D₄	1	6	yes	—
21	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
22	C₄⋊C₈×C₄₇	—	2	6	no	—
23	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
24	C₂²×Dic₄₇	—	4	1	no	—
25	C₂×Dic₉₄	—	2	6	no	—
26	M₁₆×C₄₇	—	4	6	no	—
27	C₂×C₈×C₄₇	—	2	6	no	—
28	C₄²×C₄₇	C₂×D₄	1	3	yes	—
29	C₄²×C₄₇	#41	1	1	yes	characteristic
30	C₂²⋊Dic₄₇	—	4	3	no	—
31	C₄⋊Dic₄₇	—	2	6	no	—
32	C₄⋊Dic₄₇	—	2	6	no	—
33	Dic₄₇⋊C₄	—	4	6	no	—
34	C₄×Dic₄₇	—	2	6	no	—
35	C₄×Dic₄₇	—	4	3	no	—
36	C₄×Dic₄₇	—	4	3	no	—
37	Dic₁₈₈	—	8	6	no	—
38	C₂³×C₄₇	#127	1	1	yes	characteristic
39	C₂×C₄×C₄₇	—	2	3	no	—
40	C₂×C₄×C₄₇	—	2	3	no	—
41	C₂×C₄×C₄₇	#120	1	3	yes	—
42	C₂×C₄×C₄₇	#121	1	3	yes	—
43	C₂×Dic₄₇	—	4	6	no	—
44	C₂×Dic₄₇	—	4	6	no	—
45	Dic₉₄	—	4	6	no	—
46	Dic₉₄	—	4	6	no	—
47	C₈×C₄₇	—	4	6	no	—
48	#10303	—	47	1	no	maximal, Sylow
49	C₂²×C₄₇	—	2	3	no	—
50	C₂²×C₄₇	#392	1	1	yes	characteristic
51	C₄×C₄₇	—	2	3	no	—
52	C₄×C₄₇	#455	1	3	yes	—
53	C₄×C₄₇	#456	1	3	yes	—
54	Dic₄₇	—	4	4	no	—
55	Dic₄₇	—	8	6	no	—
56	#3255	—	47	3	no	—
57	#1994	—	47	3	no	—
58	#1309	D₄₇	1	1	yes	characteristic
59	C₂×C₄₇	—	4	1	no	—
60	C₂×C₄₇	#2007	1	3	yes	—
61	#533	—	47	3	no	—
62	#475	—	94	6	no	—
63	#470	—	94	6	no	—
64	#462	—	94	6	no	—
65	#423	C₂×D₄₇	1	3	yes	—
66	C₄³	—	47	1	no	—
67	#339	—	94	6	no	—
68	C₄₇	#10303	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₁₆	—	94	6	no	—
70	C₂×M₁₆	D₁₈₈	1	6	yes	—
71	C₄⋊Q₈	—	47	6	no	—
72	#125	—	94	3	no	—
73	C₂²⋊Q₈	—	94	6	no	—
74	C₄×Q₈	—	94	6	no	—
75	#117	—	94	3	no	—
76	C₂×C₄²	—	94	3	no	—
77	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
78	#106	—	94	6	no	—
79	C₄⋊C₈	—	2	6	no	—
80	#103	—	94	12	no	—
81	C₂×Q₈	—	94	6	no	—
82	C₂²×C₄	—	188	1	no	—
83	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
84	Q₁₆	—	376	6	no	—
85	M₁₆	—	4	6	no	—
86	C₂×C₈	—	2	6	no	—
87	C₄⋊C₄	—	94	6	no	—
88	C₄⋊C₄	—	94	6	no	—
89	C₄⋊C₄	—	188	6	no	—
90	C₂²⋊C₄	—	188	3	no	—
91	C₄²	—	94	6	no	—
92	C₄²	—	188	3	no	—
93	C₄²	—	188	3	no	—
94	C₄²	D₄×D₄₇	1	3	yes	—
95	C₄²	#10625354	1	1	yes	characteristic
96	C₂³	#13094021	1	1	yes	characteristic
97	Q₈	—	188	6	no	—
98	Q₈	—	188	6	no	—
99	C₂×C₄	—	2	3	no	—
100	C₂×C₄	—	2	3	no	—
101	C₂×C₄	—	188	6	no	—
102	C₂×C₄	—	188	6	no	—
103	C₂×C₄	#13093994	1	3	yes	—
104	C₂×C₄	#13093999	1	3	yes	—
105	C₈	—	4	6	no	—
106	C₂²	—	2	3	no	—
107	C₂²	#15689033	1	1	yes	characteristic
108	C₄	—	2	3	no	—
109	C₄	—	376	6	no	—
110	C₄	—	188	4	no	—
111	C₄	#15689303	1	3	yes	—
112	C₄	#15689308	1	3	yes	—
113	C₂	—	4	1	no	—
114	C₂	#20244437	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20254782

C₂

yes

maximal

#20244364

C₂

yes

maximal

C₄₇×#1309

C₂

yes

maximal, characteristic

#15689680

C₂²

yes

—

#15689409

—

#15689381

—

#15689334

—

C₄²×Dic₄₇

C₂²

yes

characteristic

#15688884

—

C₄₇×#423

C₂²

yes

—

C₂×Dic₁₈₈

—

#13094024

D₄

yes

—

#13094016

—

#13094003

—

C₄×Dic₉₄

—

#13093983

—

C₂×C₄×Dic₄₇

—

#13093956

—

#13093949

—

C₂×M₁₆×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

#41

yes

characteristic

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

Dic₁₈₈

—

C₂³×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#120

yes

—

C₂×C₄×C₄₇

#121

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

#10303

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#392

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

#455

yes

—

C₄×C₄₇

#456

yes

—

Dic₄₇

—

Dic₄₇

—

#3255

—

#1994

—

#1309

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

#2007

yes

—

—

—

—

—

C₂×D₄₇

yes

—

C₄³

—

#339

—

C₄₇

#10303

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×M₁₆

D₁₈₈

yes

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#106

—

C₄⋊C₈

—

#103

—

C₂×Q₈

—

C₂²×C₄

—

188

—

C₂²×C₄

C₂×D₁₈₈

yes

—

Q₁₆

—

376

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

188

—

C₂²⋊C₄

—

188

—

C₄²

—

C₄²

—

188

—

C₄²

—

188

—

C₄²

D₄×D₄₇

yes

—

C₄²

yes

C₂³

yes

Q₈

—

188

—

Q₈

—

188

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

188

—

102

C₂×C₄

—

188

—

103

C₂×C₄

#13093994

yes

—

104

C₂×C₄

#13093999

yes

—

105

C₈

—

106

C₂²

—

107

C₂²

#15689033

yes

characteristic

108

C₄

—

109

C₄

—

376

—

110

C₄

—

188

—

111

C₄

#15689303

yes

—

112

C₄

#15689308

yes

—

113

C₂

—

114

C₂

#20244437

yes

—

	subgroup	G/H
center	C₂²	#15689033
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1309	C₂
socle	C₂²×C₄₇	#392
prime radical	C₂³×C₄₇	#127
hypercenter	#1309	D₄₇
hypocenter	C₄₇	#10303
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

subgroup

G/H

center

C₂²

#15689033

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1309

C₂

C₂²×C₄₇

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1309

N	H ≅ G/N	type
68. C₄₇	#10303	semidirect

H ≅ G/N

type

68. C₄₇

#10303

semidirect

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

subgroup	index
#20254782	2
#20244364	2
C₄₇×#1309	2
#10303	47

subgroup

index

#20254782

#20244364

C₄₇×#1309

#10303

Subgroups — #26119044

Conjugacy classes of subgroups

Complements

Maximal subgroups