Group #26103108 — subgroups

Subgroups — #26103108

Order 12032.

266 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 159 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	C₂×#15688899	C₂	1	2	yes	maximal
2	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄₇⋊C₁₆	C₄	1	4	yes	—
4	#15688899	—	2	2	no	—
5	#15688899	—	2	2	no	—
6	C₂×C₄×C₈×C₄₇	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
8	C₄₇×#322	—	2	2	no	—
9	C₂×C₄₇⋊C₁₆	—	2	4	no	—
10	C₂×C₄₇⋊C₁₆	—	2	4	no	—
11	C₂×C₄₇⋊C₁₆	C₈	1	8	yes	complemented
12	C₂²×C₈×C₄₇	—	2	2	no	—
13	C₂²×C₈×C₄₇	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₄₇	—	2	2	no	—
16	C₄×C₈×C₄₇	C₈	1	4	yes	—
17	C₄×C₈×C₄₇	D₄	1	2	yes	—
18	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
19	C₄×C₈×C₄₇	Q₈	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	M₁₆	1	2	yes	—
22	C₂×C₈×C₄₇	—	2	4	no	—
23	C₂×C₈×C₄₇	—	4	4	no	—
24	C₂×C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
25	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₄₇	C₂×C₈	1	4	yes	—
27	C₂×C₈×C₄₇	M₁₆	1	4	yes	—
28	C₄²×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₄²×C₄₇	C₄⋊C₄	1	1	yes	characteristic
30	C₄²×C₄₇	C₂×C₈	1	2	yes	—
31	C₄₇⋊C₁₆	—	2	8	no	—
32	C₂³×C₄₇	#97	1	1	yes	characteristic
33	C₂×C₄×C₄₇	—	2	4	no	—
34	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
35	C₂×C₄×C₄₇	C₄×C₈	1	2	yes	—
36	C₂×C₄×C₄₇	C₄×C₈	1	1	yes	characteristic
37	C₂×C₄×C₄₇	#97	1	2	yes	—
38	C₂×C₄×C₄₇	C₂²⋊C₈	1	2	yes	—
39	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
40	C₈×C₄₇	—	4	4	no	—
41	C₈×C₄₇	C₂²⋊C₈	1	4	yes	—
42	C₈×C₄₇	C₄⋊C₈	1	4	yes	—
43	#7137	—	47	1	no	maximal, Sylow
44	C₂²×C₄₇	#322	1	2	yes	—
45	C₂²×C₄₇	#336	1	1	yes	characteristic
46	C₂²×C₄₇	#347	1	4	yes	—
47	C₄×C₄₇	—	2	2	no	—
48	C₄×C₄₇	—	2	2	no	—
49	C₄×C₄₇	C₈²	1	1	yes	characteristic
50	C₄×C₄₇	#322	1	1	yes	characteristic
51	C₄×C₄₇	#336	1	2	yes	—
52	C₂×#346	—	47	2	no	—
53	C₂×#322	D₄₇	1	1	yes	characteristic
54	C₂×C₄₇	#1273	1	1	yes	characteristic
55	C₂×C₄₇	#1313	1	4	yes	—
56	C₂×C₄₇	#1375	1	2	yes	—
57	C₂²×C₁₆	—	47	4	no	—
58	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
59	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
60	#346	—	94	2	no	—
61	#346	—	94	2	no	—
62	#322	—	2	2	no	—
63	C₄₇	#7137	1	1	yes	characteristic, complemented, Sylow
64	C₂²×C₈	—	2	2	no	—
65	C₂²×C₈	C₄×D₄₇	1	2	yes	—
66	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
67	C₂×C₁₆	—	47	8	no	—
68	C₂×C₁₆	—	94	4	no	—
69	C₂×C₁₆	—	94	4	no	—
70	C₄×C₈	—	2	2	no	—
71	C₄×C₈	C₄₇⋊C₈	1	4	yes	—
72	C₄×C₈	Dic₉₄	1	1	yes	characteristic
73	C₄×C₈	D₁₈₈	1	1	yes	characteristic
74	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
75	C₂²×C₄	#10625329	1	2	yes	—
76	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
77	C₂×C₈	—	2	4	no	—
78	C₂×C₈	—	4	4	no	—
79	C₂×C₈	C₈×D₄₇	1	4	yes	—
80	C₂×C₈	#10625324	1	4	yes	—
81	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
82	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
83	C₄²	C₂×C₄₇⋊C₈	1	2	yes	—
84	C₄²	C₄⋊Dic₄₇	1	1	yes	characteristic
85	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
86	C₁₆	—	94	8	no	—
87	C₂³	#13093935	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
90	C₂×C₄	C₈×Dic₄₇	1	2	yes	—
91	C₂×C₄	C₄×C₄₇⋊C₈	1	1	yes	characteristic
92	C₂×C₄	#13093934	1	2	yes	—
93	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
94	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
95	C₈	—	4	4	no	—
96	C₈	(C₂×D₄₇)⋊C₈	1	4	yes	—
97	C₈	Dic₄₇⋊C₈	1	4	yes	—
98	C₂²	#15688845	1	2	yes	—
99	C₂²	#15688878	1	1	yes	characteristic
100	C₂²	#15688901	1	4	yes	—
101	C₄	—	2	2	no	—
102	C₄	—	2	2	no	—
103	C₄	C₈×C₄₇⋊C₈	1	1	yes	characteristic
104	C₄	#15688844	1	1	yes	characteristic
105	C₄	#15688877	1	2	yes	—
106	C₂	C₄₇⋊#1273	1	1	yes	characteristic
107	C₂	#20241309	1	4	yes	—
108	C₂	#20241468	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688899

C₂

yes

maximal

C₂×C₄₇×#322

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

C₄

yes

—

#15688899

—

#15688899

—

C₂×C₄×C₈×C₄₇

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#322

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

C₈

yes

complemented

C₂²×C₈×C₄₇

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₈

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

Q₈

yes

characteristic

C₂²×C₄×C₄₇

C₄²

yes

characteristic

C₂²×C₄×C₄₇

M₁₆

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

M₁₆

yes

—

C₄²×C₄₇

C₂²⋊C₄

yes

characteristic

C₄²×C₄₇

C₄⋊C₄

yes

characteristic

C₄²×C₄₇

C₂×C₈

yes

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

#97

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#95

yes

characteristic

C₂×C₄×C₄₇

C₄×C₈

yes

—

C₂×C₄×C₄₇

C₄×C₈

yes

characteristic

C₂×C₄×C₄₇

#97

yes

—

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₂²⋊C₈

yes

—

C₈×C₄₇

C₄⋊C₈

yes

—

#7137

—

maximal, Sylow

C₂²×C₄₇

#322

yes

—

C₂²×C₄₇

#336

yes

characteristic

C₂²×C₄₇

#347

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₈²

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×#346

—

C₂×#322

D₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#1375

yes

—

C₂²×C₁₆

—

C₂×C₄×C₈

Dic₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

—

—

C₄₇

#7137

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₄₇

yes

—

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

C₄₇⋊C₈

yes

—

C₄×C₈

Dic₉₄

yes

characteristic

C₄×C₈

D₁₈₈

yes

characteristic

C₄×C₈

C₂²⋊D₄₇

yes

—

C₂²×C₄

#10625329

yes

—

C₂²×C₄

C₄×Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×D₄₇

yes

—

C₂×C₈

#10625324

yes

—

C₂×C₈

Dic₄₇⋊C₄

yes

—

C₂×C₈

(C₂×D₄₇)⋊C₄

yes

—

C₄²

C₂×C₄₇⋊C₈

yes

—

C₄²

C₄⋊Dic₄₇

yes

characteristic

C₄²

C₂²⋊Dic₄₇

yes

characteristic

C₁₆

—

C₂³

#13093935

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄₇⋊#95

yes

characteristic

C₂×C₄

C₈×Dic₄₇

yes

—

C₂×C₄

C₄×C₄₇⋊C₈

yes

characteristic

C₂×C₄

#13093934

yes

—

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

—

C₂×C₄

C₄⋊(C₄₇⋊C₈)

yes

—

C₈

—

C₈

(C₂×D₄₇)⋊C₈

yes

—

C₈

Dic₄₇⋊C₈

yes

—

C₂²

#15688845

yes

—

C₂²

#15688878

yes

characteristic

100

C₂²

#15688901

yes

—

101

C₄

—

102

C₄

—

103

C₄

C₈×C₄₇⋊C₈

yes

characteristic

104

C₄

#15688844

yes

characteristic

105

C₄

#15688877

yes

—

106

C₂

C₄₇⋊#1273

yes

characteristic

107

C₂

#20241309

yes

—

108

C₂

#20241468

yes

—

	subgroup	G/H
center	C₂²×C₄	C₄×Dic₄₇
commutator	C₄×C₄₇	C₈²
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle, prime radical	C₂³×C₄₇	#97
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	#7137
LCS	G ▷ C₄×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#322

subgroup

G/H

center

C₂²×C₄

C₄×Dic₄₇

commutator

C₄×C₄₇

C₈²

Frattini

C₂×C₄×C₈

C₂×D₄₇

Fitting

C₂×C₄₇×#322

C₂

socle, prime radical

C₂³×C₄₇

#97

cosocle

C₂×C₄×C₈×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₄ ◁ C₂×#322

N	H ≅ G/N	type
11. C₂×C₄₇⋊C₁₆	C₈	semidirect
63. C₄₇	#7137	semidirect

H ≅ G/N

type

11. C₂×C₄₇⋊C₁₆

C₈

semidirect

63. C₄₇

#7137

semidirect

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

subgroup	index
C₂×#15688899	2
C₂×C₄₇×#322	2
#7137	47

subgroup

index

C₂×#15688899

C₂×C₄₇×#322

#7137

Subgroups — #26103108

Conjugacy classes of subgroups

Complements

Maximal subgroups