Group #26110092 — subgroups

Subgroups — #26110092

Order 12032.

338 conjugacy classes of subgroups; only the 187 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 187 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	#20243560	C₂	1	2	yes	maximal
2	#20242147	C₂	1	4	yes	maximal
3	C₂×C₄₇×#322	C₂	1	1	yes	maximal, characteristic
4	C₂×#13093960	—	2	4	no	—
5	#15689147	C₂²	1	4	yes	—
6	#15688876	—	2	8	no	—
7	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₈×C₄₇	C₂²	1	2	yes	—
9	C₄₇×#322	C₄	1	4	yes	—
10	C₂×#10625329	—	2	4	no	—
11	#13093960	—	4	8	no	—
12	C₄⋊(C₄₇⋊C₈)	—	2	8	no	—
13	C₂²×C₈×C₄₇	D₄	1	2	yes	—
14	C₂²×C₈×C₄₇	Q₈	1	4	yes	—
15	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
17	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₄₇	D₄	1	4	yes	—
19	C₄×C₈×C₄₇	Q₈	1	2	yes	—
20	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	C₂×Q₈	1	2	yes	—
22	C₂×C₈×C₄₇	—	4	4	no	—
23	C₂×C₈×C₄₇	C₄⋊C₄	1	8	yes	—
24	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
25	C₂×C₈×C₄₇	D₈	1	4	yes	—
26	C₂×C₈×C₄₇	Q₁₆	1	4	yes	—
27	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
28	C₄²×C₄₇	C₂×D₄	1	2	yes	—
29	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
30	#10625329	—	4	8	no	—
31	C₂×C₄₇⋊C₈	—	4	4	no	—
32	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
33	C₂×C₄×C₄₇	—	2	4	no	—
34	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
37	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
38	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
39	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
40	C₈×C₄₇	—	2	8	no	—
41	C₈×C₄₇	#107	1	8	yes	—
42	C₄₇⋊C₈	—	8	4	no	—
43	#8826	—	47	1	no	maximal, Sylow
44	C₂²×C₄₇	—	2	2	no	—
45	C₂²×C₄₇	#384	1	1	yes	characteristic
46	C₂²×C₄₇	#500	1	2	yes	—
47	C₄×C₄₇	—	2	2	no	—
48	C₄×C₄₇	C₂×#107	1	2	yes	—
49	C₄×C₄₇	#430	1	4	yes	—
50	C₄×C₄₇	#493	1	1	yes	characteristic
51	C₄×C₄₇	#494	1	1	yes	characteristic
52	#1848	—	47	2	no	—
53	#1567	—	47	4	no	—
54	C₂×#322	D₄₇	1	1	yes	characteristic
55	C₂×C₄₇	—	2	2	no	—
56	C₂×C₄₇	#1845	1	1	yes	characteristic
57	C₂×C₄₇	#1849	1	2	yes	—
58	C₂×#108	—	94	4	no	—
59	#423	—	47	4	no	—
60	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
61	C₂×C₄×C₈	C₂×D₄₇	1	2	yes	—
62	#335	—	94	8	no	—
63	#322	Dic₄₇	1	4	yes	—
64	C₄₇	#8826	1	1	yes	characteristic, complemented, Sylow
65	C₂×M₁₆	—	94	4	no	—
66	C₂²×C₈	Dic₉₄	1	4	yes	—
67	C₂²×C₈	D₁₈₈	1	2	yes	—
68	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
69	#108	—	188	8	no	—
70	C₄⋊C₈	—	94	8	no	—
71	C₄×C₈	Dic₉₄	1	2	yes	—
72	C₄×C₈	D₁₈₈	1	4	yes	—
73	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
74	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
75	C₂²×C₄	C₂×Dic₉₄	1	2	yes	—
76	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
77	M₁₆	—	188	8	no	—
78	C₂×C₈	—	4	4	no	—
79	C₂×C₈	—	188	4	no	—
80	C₂×C₈	D₃₇₆	1	4	yes	—
81	C₂×C₈	Dic₁₈₈	1	4	yes	—
82	C₂×C₈	C₄⋊Dic₄₇	1	8	yes	—
83	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
84	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
85	C₄²	C₂×D₁₈₈	1	2	yes	—
86	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
87	C₂³	#13094024	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄⋊Dic₄₇	1	2	yes	—
91	C₂×C₄	#13094021	1	1	yes	characteristic
92	C₂×C₄	#13094024	1	2	yes	—
93	C₂×C₄	C₂×D₃₇₆	1	2	yes	—
94	C₂×C₄	C₂×Dic₁₈₈	1	2	yes	—
95	C₈	—	2	8	no	—
96	C₈	—	376	4	no	—
97	C₈	#13093958	1	8	yes	—
98	C₂²	—	2	2	no	—
99	C₂²	#15688993	1	1	yes	characteristic
100	C₂²	#15689528	1	2	yes	—
101	C₄	—	2	2	no	—
102	C₄	C₂×#13093958	1	2	yes	—
103	C₄	#15689172	1	4	yes	—
104	C₄	#15689500	1	1	yes	characteristic
105	C₄	#15689503	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	#20243548	1	1	yes	characteristic
108	C₂	#20243563	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243560

C₂

yes

maximal

#20242147

C₂

yes

maximal

C₂×C₄₇×#322

C₂

yes

maximal, characteristic

C₂×#13093960

—

#15689147

C₂²

yes

—

#15688876

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₄₇×#322

C₄

yes

—

C₂×#10625329

—

#13093960

—

C₄⋊(C₄₇⋊C₈)

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂²×C₈×C₄₇

Q₈

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×Q₈

yes

characteristic

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

—

C₈×C₄₇

—

C₈×C₄₇

#107

yes

—

C₄₇⋊C₈

—

#8826

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#384

yes

characteristic

C₂²×C₄₇

#500

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#107

yes

—

C₄×C₄₇

#430

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

—

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

#1845

yes

characteristic

C₂×C₄₇

#1849

yes

—

C₂×#108

—

#423

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×C₄×C₈

C₂×D₄₇

yes

—

#335

—

#322

Dic₄₇

yes

—

C₄₇

#8826

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

Dic₉₄

yes

—

C₂²×C₈

D₁₈₈

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#108

—

188

—

C₄⋊C₈

—

C₄×C₈

Dic₉₄

yes

—

C₄×C₈

D₁₈₈

yes

—

C₄×C₈

C₂×Dic₄₇

yes

—

C₄×C₈

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₉₄

yes

—

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

M₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

D₃₇₆

yes

—

C₂×C₈

Dic₁₈₈

yes

—

C₂×C₈

C₄⋊Dic₄₇

yes

—

C₂×C₈

C₄⋊Dic₄₇

yes

—

C₄²

C₂×Dic₉₄

yes

characteristic

C₄²

C₂×D₁₈₈

yes

—

C₄²

C₂²×Dic₄₇

yes

characteristic

C₂³

#13094024

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

—

C₂×C₄

#13094021

yes

characteristic

C₂×C₄

#13094024

yes

—

C₂×C₄

C₂×D₃₇₆

yes

—

C₂×C₄

C₂×Dic₁₈₈

yes

—

C₈

—

C₈

—

376

—

C₈

#13093958

yes

—

C₂²

—

C₂²

#15688993

yes

characteristic

100

C₂²

#15689528

yes

—

101

C₄

—

102

C₄

C₂×#13093958

yes

—

103

C₄

#15689172

yes

—

104

C₄

#15689500

yes

characteristic

105

C₄

#15689503

yes

characteristic

106

C₂

—

107

C₂

#20243548

yes

characteristic

108

C₂

#20243563

yes

—

	subgroup	G/H
center	C₂×C₄	#13094021
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₂×C₄₇×#322	C₂
socle	C₂²×C₄₇	#384
prime radical	C₂³×C₄₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	C₂×#322	D₄₇
hypocenter	C₄₇	#8826
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#13094021

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₂×C₄₇×#322

C₂

socle

C₂²×C₄₇

#384

prime radical

C₂³×C₄₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
64. C₄₇	#8826	semidirect

H ≅ G/N

type

64. C₄₇

#8826

semidirect

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

subgroup	index
#20243560	2
#20242147	2
C₂×C₄₇×#322	2
#8826	47

subgroup

index

#20243560

#20242147

C₂×C₄₇×#322

#8826

Subgroups — #26110092

Conjugacy classes of subgroups

Complements

Maximal subgroups