Group #11785988 — subgroups

Subgroups — #11785988

Order 1152.

270 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 71 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	#10593733	C₂	1	1	yes	maximal, characteristic
2	#10593726	C₂	1	2	yes	maximal
3	#10590090	C₂	1	4	yes	maximal
4	C₂×C₄×#157	C₂²	1	2	yes	—
5	C₂×#3787	C₂²	1	4	yes	—
6	C₄²×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	#63781	C₄	1	4	yes	—
8	#63779	—	2	2	no	—
9	C₄⋊#622	—	2	2	no	—
10	C₂²×#157	C₂×C₄	1	4	yes	—
11	C₂×C₄×C₃⋊D₃	C₂×C₄	1	2	yes	—
12	C₂×C₄×C₃⋊D₃	C₂³	1	1	yes	characteristic
13	C₂×#622	—	2	4	no	—
14	C₄×#157	—	2	4	no	—
15	#3787	—	2	8	no	—
16	C₃²×C₄²	Q₈	1	1	yes	characteristic
17	C₄×C₃⋊Dic₃	D₄	1	2	yes	—
18	C₄×C₃⋊Dic₃	Q₈	1	1	yes	characteristic
19	#1832	—	9	1	no	maximal, Sylow
20	C₄²×D₃	—	12	1	no	—
21	C₂²×C₃⋊D₃	C₂²×C₄	1	1	yes	characteristic
22	C₂×#157	—	2	4	no	—
23	C₂×#157	—	2	8	no	—
24	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
25	C₂×C₃²×C₄	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₃⋊Dic₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₃⋊Dic₃	C₂×D₄	1	1	yes	characteristic
28	C₄×C₃⋊D₃	—	2	4	no	—
29	C₄×C₃⋊D₃	#41	1	4	yes	—
30	#622	—	4	4	no	—
31	C₂×C₄⋊C₈	—	9	2	no	—
32	#376	—	9	1	no	—
33	#336	—	9	4	no	—
34	C₂×C₄×D₃	—	12	2	no	—
35	C₂×C₄×D₃	—	12	1	no	—
36	C₃×C₄²	—	4	1	no	—
37	C₄×Dic₃	—	12	2	no	—
38	C₄×Dic₃	—	12	1	no	—
39	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₃⋊D₃	#117	1	2	yes	—
41	C₂×C₃⋊D₃	#131	1	4	yes	—
42	#157	—	4	4	no	—
43	C₃²×C₄	—	2	2	no	—
44	C₃²×C₄	#125	1	2	yes	—
45	C₃⋊Dic₃	—	2	2	no	—
46	C₃⋊Dic₃	#124	1	2	yes	—
47	C₂²×C₈	—	9	4	no	—
48	C₂×C₄²	—	9	1	no	—
49	C₂×C₄²	—	9	2	no	—
50	C₄⋊C₈	—	18	2	no	—
51	C₄⋊C₈	—	18	2	no	—
52	#95	—	9	4	no	—
53	C₂²×D₃	—	12	1	no	—
54	C₂×C₃×C₄	—	4	1	no	—
55	C₂×C₃×C₄	—	4	2	no	—
56	C₂×Dic₃	—	12	2	no	—
57	C₂×Dic₃	—	12	1	no	—
58	C₄×D₃	—	12	4	no	—
59	C₄×D₃	—	12	8	no	—
60	C₂×C₃²	#382	1	1	yes	characteristic
61	C₂×C₃²	#424	1	2	yes	—
62	C₃⋊D₃	#433	1	4	yes	—
63	C₂²×C₄	—	9	1	no	—
64	C₂²×C₄	—	9	2	no	—
65	C₂²×C₄	—	9	4	no	—
66	C₂×C₈	—	18	4	no	—
67	C₂×C₈	—	18	8	no	—
68	C₄²	—	9	1	no	—
69	C₄²	—	9	2	no	—
70	C₄²	—	18	4	no	—
71	C₄²	PSU(3,𝔽₂)	1	1	yes	characteristic
72	C₂²×C₃	—	4	1	no	—
73	C₂×D₃	—	12	2	no	—
74	C₂×D₃	—	12	4	no	—
75	C₃×C₄	—	4	4	no	—
76	C₃×C₄	—	4	2	no	—
77	Dic₃	—	12	2	no	—
78	Dic₃	—	12	4	no	—
79	C₃²	#1832	1	1	yes	characteristic, complemented, Sylow
80	C₂³	—	9	1	no	—
81	C₂×C₄	—	9	1	no	—
82	C₂×C₄	—	9	4	no	—
83	C₂×C₄	—	9	2	no	—
84	C₂×C₄	—	18	4	no	—
85	C₂×C₄	—	18	4	no	—
86	C₂×C₄	—	18	8	no	—
87	C₂×C₄	#3777	1	2	yes	—
88	C₂×C₄	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic
89	C₈	—	36	4	no	—
90	C₂×C₃	—	4	2	no	—
91	C₂×C₃	—	4	1	no	—
92	D₃	—	12	4	no	—
93	C₂²	—	9	4	no	—
94	C₂²	—	9	2	no	—
95	C₂²	C₂×#3777	1	1	yes	characteristic
96	C₄	—	2	2	no	—
97	C₄	—	9	2	no	—
98	C₄	—	18	2	no	—
99	C₄	—	36	4	no	—
100	C₄	#64243	1	2	yes	—
101	C₃	—	4	1	no	—
102	C₂	—	9	4	no	—
103	C₂	#10593540	1	2	yes	—
104	C₂	#10593545	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10593733

C₂

yes

maximal, characteristic

#10593726

C₂

yes

maximal

#10590090

C₂

yes

maximal

C₂×C₄×#157

C₂²

yes

—

C₂×#3787

C₂²

yes

—

C₄²×C₃⋊D₃

C₂²

yes

characteristic

#63781

C₄

yes

—

#63779

—

C₄⋊#622

—

C₂²×#157

C₂×C₄

yes

—

C₂×C₄×C₃⋊D₃

C₂×C₄

yes

—

C₂×C₄×C₃⋊D₃

C₂³

yes

—

—

—

C₃²×C₄²

Q₈

yes

characteristic

C₄×C₃⋊Dic₃

D₄

yes

—

C₄×C₃⋊Dic₃

Q₈

yes

characteristic

#1832

—

maximal, Sylow

C₄²×D₃

—

C₂²×C₃⋊D₃

C₂²×C₄

yes

characteristic

C₂×#157

—

C₂×#157

—

C₂×C₃²×C₄

C₄⋊C₄

yes

—

C₂×C₃²×C₄

C₂×Q₈

yes

characteristic

C₂×C₃⋊Dic₃

C₄⋊C₄

yes

—

C₂×C₃⋊Dic₃

C₂×D₄

yes

characteristic

C₄×C₃⋊D₃

—

C₄×C₃⋊D₃

#41

yes

—

#622

—

C₂×C₄⋊C₈

—

#376

—

#336

—

C₂×C₄×D₃

—

C₂×C₄×D₃

—

C₃×C₄²

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂²×C₃²

C₂×C₄⋊C₄

yes

characteristic

C₂×C₃⋊D₃

#117

yes

—

C₂×C₃⋊D₃

#131

yes

—

#157

—

C₃²×C₄

—

C₃²×C₄

#125

yes

—

C₃⋊Dic₃

—

C₃⋊Dic₃

#124

yes

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

#95

—

C₂²×D₃

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₂×C₃²

#382

yes

characteristic

C₂×C₃²

#424

yes

—

C₃⋊D₃

#433

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

PSU(3,𝔽₂)

yes

characteristic

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

C₃²

#1832

yes

characteristic, complemented, Sylow

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#3777

yes

—

C₂×C₄

C₂×PSU(3,𝔽₂)

yes

characteristic

C₈

—

C₂×C₃

—

C₂×C₃

—

D₃

—

C₂²

—

C₂²

—

C₂²

C₂×#3777

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

100

C₄

#64243

yes

—

101

C₃

—

102

C₂

—

103

C₂

#10593540

yes

—

104

C₂

#10593545

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×PSU(3,𝔽₂)
commutator, prime radical	C₂²×C₃⋊D₃	C₂²×C₄
Fitting	C₃²×C₄²	Q₈
socle	C₂²×C₃²	C₂×C₄⋊C₄
cosocle	C₂×C₄×C₃⋊D₃	C₂³
hypercenter	C₄²	PSU(3,𝔽₂)
hypocenter	C₃²	#1832
LCS	G ▷ C₂²×C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

subgroup

G/H

center, Frattini

C₂×C₄

C₂×PSU(3,𝔽₂)

commutator, prime radical

C₂²×C₃⋊D₃

C₂²×C₄

Fitting

C₃²×C₄²

Q₈

socle

C₂²×C₃²

C₂×C₄⋊C₄

cosocle

C₂×C₄×C₃⋊D₃

C₂³

C₄²

C₃²

G ▷ C₂²×C₃⋊D₃ ▷ C₃²

C₁ ◁ C₂×C₄ ◁ C₄²

N	H ≅ G/N	type
79. C₃²	#1832	semidirect

H ≅ G/N

type

79. C₃²

#1832

semidirect

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

subgroup	index
#10593733	2
#10593726	2
#10590090	2
#1832	9

subgroup

index

Subgroups — #11785988

Conjugacy classes of subgroups

Complements

Maximal subgroups