Group #26103112 — subgroups

Subgroups — #26103112

Order 12032.

262 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 151 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	C₄×#13093964	C₂	1	2	yes	maximal
2	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic
3	C₂×#13093964	C₄	1	4	yes	—
4	#15688899	—	2	2	no	—
5	C₄×C₄₇⋊C₁₆	—	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	#13093964	—	2	4	no	—
10	#13093964	C₈	1	8	yes	complemented
11	C₂×C₄₇⋊C₁₆	—	2	4	no	—
12	C₂²×C₈×C₄₇	C₈	1	4	yes	—
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	—	2	2	no	—
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	C₂×C₈	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	M₁₆	1	2	yes	—
31	C₄₇⋊C₁₆	—	2	8	no	—
32	C₂³×C₄₇	C₄×C₈	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	#97	1	2	yes	—
37	C₂×C₄×C₄₇	#97	1	1	yes	characteristic
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	#7139	—	47	1	no	maximal, Sylow
44	C₂²×C₄₇	—	2	2	no	—
45	C₂²×C₄₇	C₈²	1	1	yes	characteristic
46	C₂²×C₄₇	#322	1	1	yes	characteristic
47	C₂²×C₄₇	#336	1	1	yes	characteristic
48	C₄×C₄₇	—	2	2	no	—
49	C₄×C₄₇	#322	1	2	yes	—
50	C₄×C₄₇	#336	1	2	yes	—
51	C₄×C₄₇	#349	1	4	yes	—
52	C₄×M₃₂	—	47	2	no	—
53	C₂×#322	D₄₇	1	1	yes	characteristic
54	C₂×C₄₇	—	2	1	no	—
55	C₂×C₄₇	—	2	1	no	—
56	C₂×C₄₇	#1273	1	1	yes	characteristic
57	C₂×C₄₇	#1377	1	2	yes	—
58	C₂×M₃₂	—	47	4	no	—
59	C₂×C₄×C₈	Dic₄₇	1	2	yes	—
60	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
61	#346	—	94	2	no	—
62	C₄×C₁₆	—	94	2	no	—
63	#322	—	2	2	no	—
64	C₄₇	#7139	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₈	C₄₇⋊C₈	1	4	yes	—
66	C₂²×C₈	C₄×D₄₇	1	2	yes	—
67	C₂×C₄²	C₂×Dic₄₇	1	1	yes	characteristic
68	M₃₂	—	47	8	no	—
69	M₃₂	—	94	4	no	—
70	C₂×C₁₆	—	94	4	no	—
71	C₄×C₈	—	2	2	no	—
72	C₄×C₈	—	2	2	no	—
73	C₄×C₈	Dic₉₄	1	1	yes	characteristic
74	C₄×C₈	D₁₈₈	1	1	yes	characteristic
75	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
76	C₂²×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
77	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
78	C₂×C₈	—	2	4	no	—
79	C₂×C₈	—	4	4	no	—
80	C₂×C₈	C₈×D₄₇	1	4	yes	—
81	C₂×C₈	#10625324	1	4	yes	—
82	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
83	C₂×C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
84	C₄²	#10625329	1	2	yes	—
85	C₄²	C₄⋊Dic₄₇	1	1	yes	characteristic
86	C₄²	C₂²⋊Dic₄₇	1	1	yes	characteristic
87	C₁₆	—	94	8	no	—
88	C₂³	C₄×C₄₇⋊C₈	1	1	yes	characteristic
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
91	C₂×C₄	C₈×Dic₄₇	1	2	yes	—
92	C₂×C₄	#13093934	1	2	yes	—
93	C₂×C₄	#13093935	1	1	yes	characteristic
94	C₂×C₄	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
95	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
96	C₈	—	4	4	no	—
97	C₈	(C₂×D₄₇)⋊C₈	1	4	yes	—
98	C₈	Dic₄₇⋊C₈	1	4	yes	—
99	C₂²	—	2	2	no	—
100	C₂²	C₈×C₄₇⋊C₈	1	1	yes	characteristic
101	C₂²	#15688844	1	1	yes	characteristic
102	C₂²	#15688878	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	#15688845	1	2	yes	—
105	C₄	#15688877	1	2	yes	—
106	C₄	#15688905	1	4	yes	—
107	C₂	—	2	1	no	—
108	C₂	—	2	1	no	—
109	C₂	C₄₇⋊#1273	1	1	yes	characteristic
110	C₂	#20241472	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13093964

C₂

yes

maximal

C₂×C₄₇×#322

C₂

yes

maximal, characteristic

C₂×#13093964

C₄

yes

—

#15688899

—

C₄×C₄₇⋊C₁₆

—

C₂×C₄×C₈×C₄₇

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#322

—

#13093964

—

#13093964

C₈

yes

complemented

C₂×C₄₇⋊C₁₆

—

C₂²×C₈×C₄₇

C₈

yes

—

C₂²×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

—

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

C₂×C₈

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

M₁₆

yes

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₄×C₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#95

yes

characteristic

C₂×C₄×C₄₇

C₄×C₈

yes

—

C₂×C₄×C₄₇

#97

yes

—

C₂×C₄×C₄₇

#97

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₈

yes

—

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₂²⋊C₈

yes

—

C₈×C₄₇

C₄⋊C₈

yes

—

#7139

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₈²

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

#322

yes

—

C₄×C₄₇

#336

yes

—

C₄×C₄₇

#349

yes

—

C₄×M₃₂

—

C₂×#322

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1273

yes

characteristic

C₂×C₄₇

#1377

yes

—

C₂×M₃₂

—

C₂×C₄×C₈

Dic₄₇

yes

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#346

—

C₄×C₁₆

—

#322

—

C₄₇

#7139

yes

characteristic, complemented, Sylow

C₂²×C₈

C₄₇⋊C₈

yes

—

C₂²×C₈

C₄×D₄₇

yes

—

C₂×C₄²

C₂×Dic₄₇

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₉₄

yes

characteristic

C₄×C₈

D₁₈₈

yes

characteristic

C₄×C₈

C₂²⋊D₄₇

yes

—

C₂²×C₄

C₂×C₄₇⋊C₈

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

#10625329

yes

—

C₄²

C₄⋊Dic₄₇

yes

characteristic

C₄²

C₂²⋊Dic₄₇

yes

characteristic

C₁₆

—

C₂³

C₄×C₄₇⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄₇⋊#95

yes

characteristic

C₂×C₄

C₈×Dic₄₇

yes

—

C₂×C₄

#13093934

yes

—

C₂×C₄

#13093935

yes

characteristic

C₂×C₄

C₂²⋊(C₄₇⋊C₈)

yes

—

C₂×C₄

C₄⋊(C₄₇⋊C₈)

yes

—

C₈

—

C₈

(C₂×D₄₇)⋊C₈

yes

—

C₈

Dic₄₇⋊C₈

yes

—

C₂²

—

100

C₂²

C₈×C₄₇⋊C₈

yes

characteristic

101

C₂²

#15688844

yes

characteristic

102

C₂²

#15688878

yes

characteristic

103

C₄

—

104

C₄

#15688845

yes

—

105

C₄

#15688877

yes

—

106

C₄

#15688905

yes

—

107

C₂

—

108

C₂

—

109

C₂

C₄₇⋊#1273

yes

characteristic

110

C₂

#20241472

yes

—

	subgroup	G/H
center	C₄²	C₂²⋊Dic₄₇
commutator	C₂²×C₄₇	C₈²
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle	C₂²×C₄₇	#336
prime radical	C₂³×C₄₇	C₄×C₈
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	#7139
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×#322

subgroup

G/H

center

C₄²

C₂²⋊Dic₄₇

commutator

C₂²×C₄₇

C₈²

Frattini

C₂×C₄×C₈

C₂×D₄₇

Fitting

C₂×C₄₇×#322

C₂

socle

C₂²×C₄₇

#336

prime radical

C₂³×C₄₇

C₄×C₈

cosocle

C₂×C₄×C₈×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×#322

N	H ≅ G/N	type
10. #13093964	C₈	semidirect
64. C₄₇	#7139	semidirect

H ≅ G/N

type

10. #13093964

C₈

semidirect

64. C₄₇

#7139

semidirect

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

subgroup	index
C₄×#13093964	2
C₂×C₄₇×#322	2
#7139	47

subgroup

index

C₄×#13093964

C₂×C₄₇×#322

#7139

Subgroups — #26103112

Conjugacy classes of subgroups

Complements

Maximal subgroups