Group #11786180 — subgroups

Subgroups — #11786180

Order 1152.

558 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 158 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×C₄⋊#622	C₂	1	6	yes	maximal
2	C₄²×C₃⋊Dic₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×#622	C₂²	1	4	yes	—
4	C₂×C₃²×C₄²	C₄	1	1	yes	characteristic
5	C₂×C₄×C₃⋊Dic₃	C₄	1	3	yes	—
6	C₂×C₄×C₃⋊Dic₃	C₂²	1	3	yes	—
7	C₄⋊#622	—	2	12	no	—
8	C₄²×Dic₃	—	6	2	no	—
9	C₂²×C₃²×C₄	C₂×C₄	1	3	yes	—
10	C₂²×C₃⋊Dic₃	C₂×C₄	1	3	yes	—
11	C₂²×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
12	C₂×#622	—	2	12	no	—
13	C₃²×C₄²	C₈	1	4	yes	complemented
14	C₄×C₃⋊Dic₃	—	2	6	no	—
15	C₄×C₃⋊Dic₃	D₄	1	6	yes	—
16	C₄×C₃⋊Dic₃	Q₈	1	6	yes	—
17	#1842	—	9	1	no	maximal, Sylow
18	C₂×C₃×C₄²	—	2	2	no	—
19	C₂×C₄×Dic₃	—	6	6	no	—
20	C₂×C₄×Dic₃	—	6	6	no	—
21	C₂³×C₃²	C₂²×C₄	1	1	yes	characteristic
22	C₂×C₃²×C₄	C₄⋊C₄	1	6	yes	—
23	C₂×C₃²×C₄	C₂×C₈	1	6	yes	—
24	C₂×C₃²×C₄	M₁₆	1	6	yes	—
25	C₂×C₃⋊Dic₃	—	2	6	no	—
26	C₂×C₃⋊Dic₃	C₄⋊C₄	1	6	yes	—
27	C₂×C₃⋊Dic₃	C₂×D₄	1	3	yes	—
28	C₂×C₃⋊Dic₃	C₂×Q₈	1	3	yes	—
29	#622	—	4	4	no	—
30	C₂×C₄⋊C₈	—	9	6	no	—
31	C₄³	—	9	1	no	—
32	C₂²×C₃×C₄	—	2	6	no	—
33	C₂²×Dic₃	—	6	6	no	—
34	C₂²×Dic₃	—	6	2	no	—
35	C₃×C₄²	—	2	8	no	—
36	C₄×Dic₃	—	6	12	no	—
37	C₄×Dic₃	—	6	12	no	—
38	C₄×Dic₃	—	6	24	no	—
39	C₂²×C₃²	C₂×C₄⋊C₄	1	3	yes	—
40	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
41	C₂²×C₃²	C₂×M₁₆	1	3	yes	—
42	C₃²×C₄	C₄⋊C₈	1	12	yes	complemented
43	C₃⋊Dic₃	—	2	6	no	—
44	C₃⋊Dic₃	#127	1	1	yes	characteristic
45	C₃⋊Dic₃	C₄⋊Q₈	1	3	yes	—
46	C₂²×C₈	—	9	4	no	—
47	C₂×C₄²	—	9	3	no	—
48	C₂×C₄²	—	9	3	no	—
49	C₂×C₄²	#157	1	1	yes	characteristic
50	C₄⋊C₈	—	18	12	no	—
51	C₂³×C₃	—	2	2	no	—
52	C₂×C₃×C₄	—	2	12	no	—
53	C₂×C₃×C₄	—	2	12	no	—
54	C₂×C₃×C₄	—	2	12	no	—
55	C₂×Dic₃	—	6	12	no	—
56	C₂×Dic₃	—	6	6	no	—
57	C₂×Dic₃	—	6	6	no	—
58	C₂×Dic₃	—	6	24	no	—
59	C₂×C₃²	#384	1	1	yes	characteristic
60	C₂×C₃²	C₂×C₄⋊C₈	1	3	yes	—
61	C₂×C₃²	#423	1	3	yes	—
62	C₂²×C₄	—	9	1	no	—
63	C₂²×C₄	—	9	3	no	—
64	C₂²×C₄	C₂×#157	1	3	yes	—
65	C₂×C₈	—	18	12	no	—
66	C₄²	—	9	6	no	—
67	C₄²	—	9	6	no	—
68	C₄²	—	18	6	no	—
69	C₄²	#622	1	4	yes	complemented
70	C₂²×C₃	—	2	6	no	—
71	C₂²×C₃	—	2	6	no	—
72	C₂²×C₃	—	2	2	no	—
73	C₃×C₄	—	2	24	no	—
74	Dic₃	—	6	2	no	—
75	Dic₃	—	6	6	no	—
76	Dic₃	—	6	24	no	—
77	C₃²	#1842	1	1	yes	characteristic, complemented, Sylow
78	C₂³	C₂²×#157	1	1	yes	characteristic
79	C₂×C₄	—	9	3	no	—
80	C₂×C₄	—	9	3	no	—
81	C₂×C₄	—	9	6	no	—
82	C₂×C₄	—	18	6	no	—
83	C₂×C₄	#3788	1	6	yes	—
84	C₂×C₄	C₄⋊#157	1	6	yes	—
85	C₂×C₄	C₂×#622	1	6	yes	—
86	C₈	—	36	4	no	—
87	C₂×C₃	—	2	6	no	—
88	C₂×C₃	—	2	6	no	—
89	C₂×C₃	—	2	2	no	—
90	C₂²	C₂×#3788	1	3	yes	—
91	C₂²	C₂×C₄⋊#157	1	3	yes	—
92	C₂²	C₂²×#622	1	1	yes	characteristic
93	C₄	—	9	1	no	—
94	C₄	—	9	3	no	—
95	C₄	—	18	6	no	—
96	C₄	C₄⋊#622	1	12	yes	complemented
97	C₃	—	2	2	no	—
98	C₂	#10593728	1	3	yes	—
99	C₂	#10593735	1	1	yes	characteristic
100	C₂	C₂×C₄⋊#622	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#622

C₂

yes

maximal

C₄²×C₃⋊Dic₃

C₂

yes

maximal, characteristic

C₂²×#622

C₂²

yes

—

C₂×C₃²×C₄²

C₄

yes

characteristic

C₂×C₄×C₃⋊Dic₃

C₄

yes

—

C₂×C₄×C₃⋊Dic₃

C₂²

yes

—

C₄⋊#622

—

C₄²×Dic₃

—

C₂²×C₃²×C₄

C₂×C₄

yes

—

C₂²×C₃⋊Dic₃

C₂×C₄

yes

—

C₂²×C₃⋊Dic₃

C₂³

yes

characteristic

C₂×#622

—

C₃²×C₄²

C₈

yes

complemented

C₄×C₃⋊Dic₃

—

C₄×C₃⋊Dic₃

D₄

yes

—

C₄×C₃⋊Dic₃

Q₈

yes

—

#1842

—

maximal, Sylow

C₂×C₃×C₄²

—

C₂×C₄×Dic₃

—

C₂×C₄×Dic₃

—

C₂³×C₃²

C₂²×C₄

yes

characteristic

C₂×C₃²×C₄

C₄⋊C₄

yes

—

C₂×C₃²×C₄

C₂×C₈

yes

—

C₂×C₃²×C₄

M₁₆

yes

—

C₂×C₃⋊Dic₃

—

C₂×C₃⋊Dic₃

C₄⋊C₄

yes

—

C₂×C₃⋊Dic₃

C₂×D₄

yes

—

C₂×C₃⋊Dic₃

C₂×Q₈

yes

—

#622

—

C₂×C₄⋊C₈

—

C₄³

—

C₂²×C₃×C₄

—

C₂²×Dic₃

—

C₂²×Dic₃

—

C₃×C₄²

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂²×C₃²

C₂×C₄⋊C₄

yes

—

C₂²×C₃²

C₂²×C₈

yes

characteristic

C₂²×C₃²

C₂×M₁₆

yes

—

C₃²×C₄

C₄⋊C₈

yes

complemented

C₃⋊Dic₃

—

C₃⋊Dic₃

#127

yes

characteristic

C₃⋊Dic₃

C₄⋊Q₈

yes

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

#157

yes

characteristic

C₄⋊C₈

—

C₂³×C₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×C₃²

#384

yes

characteristic

C₂×C₃²

C₂×C₄⋊C₈

yes

—

C₂×C₃²

#423

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#157

yes

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#622

yes

complemented

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₃×C₄

—

Dic₃

—

Dic₃

—

Dic₃

—

C₃²

#1842

yes

characteristic, complemented, Sylow

C₂³

C₂²×#157

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#3788

yes

—

C₂×C₄

C₄⋊#157

yes

—

C₂×C₄

C₂×#622

yes

—

C₈

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂²

C₂×#3788

yes

—

C₂²

C₂×C₄⋊#157

yes

—

C₂²

C₂²×#622

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₄⋊#622

yes

complemented

C₃

—

C₂

#10593728

yes

—

C₂

#10593735

yes

characteristic

100

C₂

C₂×C₄⋊#622

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#157
commutator	C₂²×C₃²	C₂²×C₈
Fitting	C₂×C₃²×C₄²	C₄
socle, prime radical	C₂³×C₃²	C₂²×C₄
cosocle	C₂²×C₃⋊Dic₃	C₂³
hypercenter	C₂×C₄²	#157
hypocenter	C₃²	#1842
LCS	G ▷ C₂²×C₃² ▷ C₃²
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

subgroup

G/H

center, Frattini

C₂³

C₂²×#157

commutator

C₂²×C₃²

C₂²×C₈

Fitting

C₂×C₃²×C₄²

C₄

socle, prime radical

C₂³×C₃²

C₂²×C₄

cosocle

C₂²×C₃⋊Dic₃

C₂³

C₂×C₄²

C₃²

G ▷ C₂²×C₃² ▷ C₃²

C₁ ◁ C₂³ ◁ C₂×C₄²

N	H ≅ G/N	type
13. C₃²×C₄²	C₈	semidirect
42. C₃²×C₄	C₄⋊C₈	semidirect
69. C₄²	#622	semidirect
77. C₃²	#1842	semidirect
96. C₄	C₄⋊#622	semidirect

H ≅ G/N

type

13. C₃²×C₄²

C₈

semidirect

42. C₃²×C₄

C₄⋊C₈

semidirect

69. C₄²

#622

semidirect

77. C₃²

#1842

semidirect

96. C₄

C₄⋊#622

semidirect

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

subgroup	index
C₂×C₄⋊#622	2
C₄²×C₃⋊Dic₃	2
#1842	9

subgroup

index

C₂×C₄⋊#622

C₄²×C₃⋊Dic₃

#1842

Subgroups — #11786180

Conjugacy classes of subgroups

Complements

Maximal subgroups