Group #18010988 — subgroups

Subgroups — #18010988

Order 4736.

144 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 58 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	#14558592	C₂	1	2	yes	maximal, complemented
2	#14558455	C₂	1	1	yes	maximal, characteristic, complemented
3	#14558432	C₂	1	2	yes	maximal, complemented
4	#14558422	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557649	C₂	1	1	yes	maximal, characteristic
6	#11924611	—	2	2	no	—
7	#11924610	C₂²	1	2	yes	—
8	D₃₇⋊C₁₆	—	2	2	no	—
9	C₂×#10596927	C₂²	1	1	yes	characteristic
10	C₂×#10596927	C₂²	1	1	yes	characteristic
11	#11924558	C₄	1	1	yes	characteristic
12	#11924544	C₂²	1	2	yes	—
13	#11924462	C₄	1	2	yes	—
14	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
15	#108×C₃₇	C₄	1	1	yes	characteristic
16	#10596962	C₂×C₄	1	2	yes	—
17	M₁₆×C₃₇	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₃₇	C₂×C₄	1	1	yes	characteristic
19	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂×#64436	C₂³	1	1	yes	characteristic
21	Dic₁₄₈	C₈	1	1	yes	characteristic
22	D₂₉₆	C₈	1	1	yes	characteristic
23	#10596930	C₈	1	2	yes	—
24	#10596929	—	2	2	no	—
25	C₈×D₃₇	—	2	2	no	—
26	#10596927	—	2	2	no	—
27	#10596927	—	2	1	no	—
28	#10596927	D₄	1	2	yes	—
29	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
30	C₂²⋊D₃₇	—	2	2	no	—
31	C₂×Dic₃₇	—	2	1	no	—
32	D₁₄₈	C₂×C₈	1	2	yes	—
33	C₄×D₃₇	—	2	2	no	—
34	Dic₇₄	C₂×C₈	1	2	yes	—
35	C₈×C₃₇	—	2	2	no	—
36	C₈×C₃₇	C₂×C₈	1	1	yes	characteristic
37	C₈×C₃₇	C₂×C₈	1	1	yes	characteristic
38	#64436	C₂×D₄	1	1	yes	characteristic
39	#64436	#41	1	1	yes	characteristic
40	C₂²×C₃₇	#131	1	1	yes	characteristic
41	C₂×D₃₇	—	2	2	no	—
42	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
43	C₄×C₃₇	C₂²×C₈	1	1	yes	characteristic
44	Dic₃₇	—	2	2	no	—
45	Dic₃₇	—	4	1	no	—
46	#2171	—	37	1	no	maximal, Sylow
47	C₂×C₃₇	—	2	1	no	—
48	C₂×C₃₇	C₈×D₄	1	1	yes	characteristic
49	D₃₇	—	4	2	no	—
50	#504	—	37	2	no	—
51	#443	—	37	1	no	—
52	#364	—	37	1	no	—
53	#350	—	37	2	no	—
54	#346	—	37	1	no	—
55	C₃₇	#2171	1	1	yes	characteristic, complemented, Sylow
56	#135	—	37	1	no	—
57	#131	—	37	2	no	—
58	M₃₂	—	37	2	no	—
59	M₃₂	—	74	2	no	—
60	C₂×C₁₆	—	37	1	no	—
61	C₂×C₁₆	—	37	1	no	—
62	C₂×C₁₆	—	74	2	no	—
63	#108	#3866	1	1	yes	characteristic
64	#104	—	37	2	no	—
65	C₄×C₈	—	37	1	no	—
66	#41	—	37	2	no	—
67	Q₁₆	—	37	1	no	—
68	SD₁₆	—	37	2	no	—
69	D₈	—	37	1	no	—
70	M₁₆	—	74	2	no	—
71	M₁₆	C₂×#3866	1	2	yes	—
72	C₂×C₈	—	37	1	no	—
73	C₂×C₈	—	74	2	no	—
74	C₂×C₈	C₂×#3866	1	1	yes	characteristic
75	C₄²	—	37	1	no	—
76	C₁₆	—	37	2	no	—
77	C₁₆	—	74	1	no	—
78	C₁₆	—	74	2	no	—
79	Q₈	—	37	2	no	—
80	D₄	—	37	2	no	—
81	D₄	—	74	2	no	—
82	C₂×C₄	—	74	2	no	—
83	C₂×C₄	—	74	1	no	—
84	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
85	C₈	—	2	2	no	—
86	C₈	—	37	1	no	—
87	C₈	—	37	1	no	—
88	C₈	D₃₇⋊C₈	1	1	yes	characteristic
89	C₈	D₃₇⋊C₈	1	1	yes	characteristic
90	C₂²	—	74	2	no	—
91	C₂²	#11924624	1	1	yes	characteristic
92	C₄	—	74	2	no	—
93	C₄	—	148	1	no	—
94	C₄	D₄×#3866	1	1	yes	characteristic
95	C₄	C₂×D₃₇⋊C₈	1	1	yes	characteristic
96	C₂	—	2	1	no	—
97	C₂	—	148	2	no	—
98	C₂	#14558561	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558592

C₂

yes

maximal, complemented

#14558455

C₂

yes

maximal, characteristic, complemented

#14558432

C₂

yes

maximal, complemented

#14558422

C₂

yes

maximal, characteristic, complemented

#14557649

C₂

yes

maximal, characteristic

#11924611

—

#11924610

C₂²

yes

—

D₃₇⋊C₁₆

—

C₂×#10596927

C₂²

yes

characteristic

C₂×#10596927

C₂²

yes

characteristic

#11924558

C₄

yes

characteristic

#11924544

C₂²

yes

—

#11924462

C₄

yes

—

C₈×Dic₃₇

C₂²

yes

characteristic

#108×C₃₇

C₄

yes

characteristic

#10596962

C₂×C₄

yes

—

M₁₆×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₂×#64436

C₂³

yes

characteristic

Dic₁₄₈

C₈

yes

characteristic

D₂₉₆

C₈

yes

characteristic

#10596930

C₈

yes

—

#10596929

—

C₈×D₃₇

—

#10596927

—

#10596927

—

#10596927

D₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

D₁₄₈

C₂×C₈

yes

—

C₄×D₃₇

—

Dic₇₄

C₂×C₈

yes

—

C₈×C₃₇

—

C₈×C₃₇

C₂×C₈

yes

characteristic

C₈×C₃₇

C₂×C₈

yes

characteristic

#64436

C₂×D₄

yes

yes

C₂²×C₃₇

yes

C₂×D₃₇

—

C₄×C₃₇

C₄×D₄

yes

characteristic

C₄×C₃₇

C₂²×C₈

yes

characteristic

Dic₃₇

—

Dic₃₇

—

#2171

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₈×D₄

yes

characteristic

D₃₇

—

—

—

—

—

—

C₃₇

#2171

yes

characteristic, complemented, Sylow

#135

—

#131

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

yes

—

C₄×C₈

—

#41

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

C₂×#3866

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#3866

yes

characteristic

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×#3866

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

D₃₇⋊C₈

yes

characteristic

C₈

D₃₇⋊C₈

yes

characteristic

C₂²

—

C₂²

#11924624

yes

characteristic

C₄

—

C₄

—

148

—

C₄

D₄×#3866

yes

characteristic

C₄

C₂×D₃₇⋊C₈

yes

characteristic

C₂

—

C₂

—

148

—

C₂

#14558561

yes

characteristic

	subgroup	G/H
center	C₄	D₄×#3866
commutator	C₄×C₃₇	C₂²×C₈
Frattini	C₂×C₄	C₂²×#3866
Fitting	#108×C₃₇	C₄
socle	C₂×C₃₇	C₈×D₄
prime radical	#11924558	C₄
cosocle	C₂×#64436	C₂³
hypercenter	#108	#3866
hypocenter	C₃₇	#2171
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

subgroup

G/H

center

C₄

D₄×#3866

commutator

C₄×C₃₇

C₂²×C₈

Frattini

C₂×C₄

C₂²×#3866

Fitting

#108×C₃₇

C₄

socle

C₂×C₃₇

C₈×D₄

C₄

C₂³

C₃₇

G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

N	H ≅ G/N	type
1. #14558592	C₂	semidirect
2. #14558455	C₂	semidirect
3. #14558432	C₂	semidirect
4. #14558422	C₂	semidirect
55. C₃₇	#2171	semidirect

H ≅ G/N

type

1. #14558592

C₂

semidirect

2. #14558455

C₂

semidirect

3. #14558432

C₂

semidirect

4. #14558422

C₂

semidirect

55. C₃₇

#2171

semidirect

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

subgroup	index
#14558592	2
#14558455	2
#14558432	2
#14558422	2
#14557649	2
#2171	37

subgroup

index

Subgroups — #18010988

Conjugacy classes of subgroups

Complements

Maximal subgroups