Group #11772198 — subgroups

Subgroups — #11772198

Order 1152.

147 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 147 conjugacy classes of subgroups — too many to draw. Only its 66 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	#10589327	C₂	1	1	yes	maximal, characteristic
2	C₃×#4463	C₂	1	2	yes	maximal
3	#67567	—	3	2	no	maximal
4	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
5	C₂×C₃⋊(C₃⋊C₁₆)	C₄	1	1	yes	characteristic
6	C₂×C₃⋊(C₃⋊C₁₆)	C₄	1	1	yes	characteristic
7	C₃×#812	—	2	2	no	—
8	C₃×#364	—	3	2	no	—
9	#4463	D₃	1	2	yes	—
10	#4461	—	3	2	no	—
11	#4461	—	6	1	no	—
12	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
13	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
14	C₃²×C₄²	Q₈	1	1	yes	characteristic
15	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
16	C₃⋊(C₃⋊C₁₆)	C₈	1	2	yes	—
17	C₃×C₃⋊C₁₆	—	4	2	no	—
18	#1369	—	9	1	no	Sylow
19	C₃×M₃₂	—	6	2	no	—
20	C₃×C₄×C₈	—	2	1	no	—
21	C₃×C₄×C₈	C₂×D₃	1	2	yes	—
22	#812	—	2	2	no	—
23	C₂×C₃⋊C₁₆	—	3	2	no	—
24	C₂×C₃⋊C₁₆	—	3	2	no	—
25	C₂×C₃⋊C₁₆	—	6	1	no	—
26	C₂×C₃⋊C₁₆	—	6	1	no	—
27	C₂×C₃²×C₄	—	2	1	no	—
28	C₂×C₃²×C₄	C₄⋊C₄	1	1	yes	characteristic
29	C₃²×C₈	C₂×C₈	1	1	yes	characteristic
30	C₃²×C₈	M₁₆	1	1	yes	characteristic
31	C₃²×C₈	SD₁₆	1	2	yes	—
32	#364	—	3	2	no	—
33	#346	—	9	1	no	—
34	C₂×C₃×C₈	—	2	1	no	—
35	C₂×C₃×C₈	—	2	1	no	—
36	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
37	C₂×C₃×C₈	#67	1	2	yes	—
38	C₃×C₄²	—	2	1	no	—
39	C₃×C₄²	Dic₆	1	2	yes	—
40	C₃×C₁₆	—	12	2	no	—
41	C₃⋊C₁₆	—	3	4	no	—
42	C₃⋊C₁₆	—	4	2	no	—
43	C₃⋊C₁₆	—	6	2	no	—
44	C₃⋊C₁₆	—	6	2	no	—
45	C₃⋊C₁₆	—	6	2	no	—
46	C₂²×C₃²	#108	1	1	yes	characteristic
47	C₃²×C₄	—	4	1	no	—
48	C₃²×C₄	C₄⋊C₈	1	1	yes	characteristic
49	C₃²×C₄	#106	1	1	yes	characteristic
50	M₃₂	—	6	2	no	—
51	C₂×C₁₆	—	9	1	no	—
52	C₂×C₁₆	—	9	1	no	—
53	C₄×C₈	D₃²	1	1	yes	characteristic
54	C₂×C₃×C₄	—	2	2	no	—
55	C₂×C₃×C₄	—	2	2	no	—
56	C₂×C₃×C₄	—	2	1	no	—
57	C₂×C₃×C₄	Dic₃⋊C₄	1	2	yes	—
58	C₃×C₈	—	2	1	no	—
59	C₃×C₈	—	2	1	no	—
60	C₃×C₈	—	2	2	no	—
61	C₃×C₈	D₃×C₈	1	2	yes	—
62	C₃×C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
63	C₃×C₈	#214	1	2	yes	—
64	C₃×C₈	#215	1	2	yes	—
65	C₂×C₃²	—	2	1	no	—
66	C₂×C₃²	#334	1	1	yes	characteristic
67	C₂×C₈	#624	1	1	yes	characteristic
68	C₂×C₈	#625	1	1	yes	characteristic
69	C₄²	#627	1	1	yes	characteristic
70	C₁₆	—	9	2	no	—
71	C₁₆	—	12	2	no	—
72	C₁₆	—	18	1	no	—
73	C₂²×C₃	—	2	1	no	—
74	C₂²×C₃	#822	1	2	yes	—
75	C₃×C₄	—	2	1	no	—
76	C₃×C₄	—	2	1	no	—
77	C₃×C₄	—	4	2	no	—
78	C₃×C₄	—	4	2	no	—
79	C₃×C₄	#808	1	2	yes	—
80	C₃×C₄	Dic₃⋊C₈	1	2	yes	—
81	C₃²	#1369	1	1	yes	characteristic, complemented, Sylow
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	#3724	1	1	yes	characteristic
84	C₈	#3710	1	1	yes	characteristic
85	C₈	#3712	1	1	yes	characteristic
86	C₈	#3715	1	2	yes	—
87	C₂×C₃	—	2	2	no	—
88	C₂×C₃	—	2	2	no	—
89	C₂×C₃	—	2	1	no	—
90	C₂×C₃	#4480	1	2	yes	—
91	C₂²	#63578	1	1	yes	characteristic
92	C₄	—	4	1	no	—
93	C₄	#63572	1	1	yes	characteristic
94	C₄	#63574	1	1	yes	characteristic
95	C₃	—	2	1	no	—
96	C₃	#67567	1	2	yes	complemented
97	C₂	—	2	1	no	—
98	C₂	#10589098	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10589327

C₂

yes

maximal, characteristic

C₃×#4463

C₂

yes

maximal

#67567

—

maximal

C₃²×C₄×C₈

C₂²

yes

characteristic

C₂×C₃⋊(C₃⋊C₁₆)

C₄

yes

characteristic

C₂×C₃⋊(C₃⋊C₁₆)

C₄

yes

—

—

D₃

yes

—

#4461

—

#4461

—

C₂×C₃²×C₈

C₂×C₄

yes

characteristic

C₂×C₃²×C₈

D₄

yes

characteristic

C₃²×C₄²

Q₈

yes

characteristic

C₃⋊(C₃⋊C₁₆)

—

C₃⋊(C₃⋊C₁₆)

C₈

yes

—

C₃×C₃⋊C₁₆

—

#1369

—

Sylow

C₃×M₃₂

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₂×D₃

yes

—

#812

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₂×C₃⋊C₁₆

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₄⋊C₄

yes

characteristic

C₃²×C₈

C₂×C₈

yes

characteristic

C₃²×C₈

M₁₆

yes

characteristic

C₃²×C₈

SD₁₆

yes

—

#364

—

#346

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₄×D₃

yes

—

C₂×C₃×C₈

#67

yes

—

C₃×C₄²

—

C₃×C₄²

Dic₆

yes

—

C₃×C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

#108

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₄⋊C₈

yes

characteristic

C₃²×C₄

#106

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄×C₈

D₃²

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

Dic₃⋊C₄

yes

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

D₃×C₈

yes

—

C₃×C₈

ΓL(1,𝔽₂₅)

yes

—

C₃×C₈

#214

yes

—

C₃×C₈

#215

yes

—

C₂×C₃²

—

C₂×C₃²

yes

C₂×C₈

yes

C₂×C₈

yes

C₄²

yes

C₁₆

—

C₁₆

—

C₁₆

—

C₂²×C₃

—

C₂²×C₃

#822

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#808

yes

—

C₃×C₄

Dic₃⋊C₈

yes

—

C₃²

#1369

yes

characteristic, complemented, Sylow

C₂×C₄

—

C₂×C₄

yes

C₈

yes

C₈

yes

C₈

yes

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

#4480

yes

—

C₂²

#63578

yes

characteristic

C₄

—

C₄

yes

C₄

yes

C₃

—

C₃

#67567

yes

complemented

C₂

—

C₂

#10589098

yes

characteristic

	subgroup	G/H
center	C₄	#63574
commutator	C₃²×C₈	C₂×C₈
Frattini, hypercenter	C₄×C₈	D₃²
Fitting, cosocle	C₃²×C₄×C₈	C₂²
socle	C₂×C₃²	#334
prime radical	C₂²×C₃²	#108
hypocenter	C₃²	#1369
LCS	G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈

subgroup

G/H

center

C₄

#63574

commutator

C₃²×C₈

C₂×C₈

Frattini, hypercenter

C₄×C₈

D₃²

Fitting, cosocle

C₃²×C₄×C₈

C₂²

C₂×C₃²

C₂²×C₃²

C₃²

G ▷ C₃²×C₈ ▷ C₃²×C₄ ▷ C₂×C₃² ▷ C₃²

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈

N	H ≅ G/N	type
81. C₃²	#1369	semidirect
96. C₃	#67567	semidirect

H ≅ G/N

type

81. C₃²

#1369

semidirect

96. C₃

#67567

semidirect

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

subgroup	index
#10589327	2
C₃×#4463	2
#67567	3

subgroup

index

#10589327

C₃×#4463

#67567

Subgroups — #11772198

Conjugacy classes of subgroups

Complements

Maximal subgroups