Group #18010980 — subgroups

Subgroups — #18010980

Order 4736.

162 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 74 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×#11924610	C₂	1	2	yes	maximal, complemented
2	C₄⋊#10596927	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#10596927	C₂	1	2	yes	maximal, complemented
4	C₄×#10596927	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×#64436	C₂	1	1	yes	maximal, characteristic
6	#11924610	—	2	4	no	—
7	C₂×#10596927	C₂²	1	2	yes	—
8	C₂×#10596927	C₂²	1	1	yes	characteristic
9	C₂×#10596927	C₂²	1	1	yes	characteristic
10	C₂²×#64436	C₂²	1	2	yes	—
11	C₄⋊#64436	C₄	1	1	yes	characteristic
12	C₂²⋊#64436	C₄	1	2	yes	—
13	C₄×#64436	C₂²	1	1	yes	characteristic
14	C₄×D₄×C₃₇	C₄	1	1	yes	characteristic
15	C₂×D₄×C₃₇	C₈	1	1	yes	characteristic
16	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
17	C₄⋊C₄×C₃₇	C₈	1	1	yes	characteristic
18	C₂²⋊C₄×C₃₇	C₈	1	2	yes	—
19	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂×#64436	—	2	2	no	—
21	C₂×#64436	—	2	2	no	—
22	C₂×#64436	C₂×C₄	1	2	yes	—
23	C₂×#64436	C₂×C₄	1	1	yes	characteristic
24	C₂×#64436	C₂³	1	1	yes	characteristic
25	#10596927	—	2	2	no	—
26	#10596927	—	2	1	no	—
27	#10596927	D₄	1	2	yes	complemented
28	C₂³×C₃₇	C₂×C₈	1	2	yes	—
29	D₄×C₃₇	—	2	2	no	—
30	C₂×C₄×C₃₇	—	2	2	no	—
31	C₂×C₄×C₃₇	—	2	2	no	—
32	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
33	C₂×C₄×C₃₇	C₂×C₈	1	2	yes	—
34	C₂×C₄×C₃₇	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
36	#64436	—	2	2	no	—
37	#64436	—	2	1	no	—
38	#64436	C₂×D₄	1	1	yes	characteristic
39	#64436	#41	1	1	yes	characteristic
40	C₂²×C₃₇	—	2	2	no	—
41	C₂²×C₃₇	—	2	2	no	—
42	C₂²×C₃₇	—	2	2	no	—
43	C₂²×C₃₇	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₃₇	—	2	2	no	—
45	C₄×C₃₇	—	2	1	no	—
46	C₄×C₃₇	M₃₂	1	2	yes	complemented
47	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
48	C₄×C₃₇	#131	1	1	yes	characteristic
49	#2169	—	37	1	no	maximal, Sylow
50	C₂×C₃₇	—	4	2	no	—
51	C₂×C₃₇	C₈×D₄	1	1	yes	characteristic
52	C₂×C₃₇	C₂×M₃₂	1	1	yes	characteristic
53	C₂×C₃₇	#504	1	1	yes	characteristic
54	C₂×M₃₂	—	37	2	no	—
55	C₈×D₄	—	37	1	no	—
56	C₄⋊C₁₆	—	37	1	no	—
57	C₂²⋊C₁₆	—	37	2	no	—
58	C₄×C₁₆	—	37	1	no	—
59	C₃₇	#2169	1	1	yes	characteristic, complemented, Sylow
60	C₂²×C₈	—	37	2	no	—
61	C₄×D₄	#3866	1	1	yes	characteristic
62	M₃₂	—	74	4	no	—
63	C₂×C₁₆	—	37	1	no	—
64	C₂×C₁₆	—	37	1	no	—
65	C₂×C₁₆	—	37	2	no	—
66	C₄⋊C₈	—	37	1	no	—
67	C₂²⋊C₈	—	37	2	no	—
68	C₄×C₈	—	37	1	no	—
69	C₂×D₄	#64438	1	1	yes	characteristic
70	C₂²×C₄	C₂×#3866	1	2	yes	—
71	C₂×C₈	—	37	1	no	—
72	C₂×C₈	—	37	1	no	—
73	C₂×C₈	—	37	2	no	—
74	C₂×C₈	—	74	2	no	—
75	C₂×C₈	—	74	2	no	—
76	C₄⋊C₄	#64438	1	1	yes	characteristic
77	C₂²⋊C₄	#64438	1	2	yes	—
78	C₄²	C₂×#3866	1	1	yes	characteristic
79	C₁₆	—	37	2	no	—
80	C₁₆	—	74	1	no	—
81	C₁₆	—	74	2	no	—
82	C₂³	C₂×#64438	1	2	yes	—
83	D₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	C₂×#64438	1	1	yes	characteristic
87	C₂×C₄	C₂×#64438	1	2	yes	—
88	C₂×C₄	C₂×#64438	1	1	yes	characteristic
89	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
90	C₈	—	37	1	no	—
91	C₈	—	37	1	no	—
92	C₈	—	74	2	no	—
93	C₈	—	74	1	no	—
94	C₂²	—	2	2	no	—
95	C₂²	—	2	2	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₂²×#64438	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	2	1	no	—
100	C₄	#11924610	1	2	yes	complemented
101	C₄	D₄×#3866	1	1	yes	characteristic
102	C₄	#11924623	1	1	yes	characteristic
103	C₂	—	4	2	no	—
104	C₂	D₄×#64438	1	1	yes	characteristic
105	C₂	C₂×#11924610	1	1	yes	characteristic
106	C₂	#14558591	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924610

C₂

yes

maximal, complemented

C₄⋊#10596927

C₂

yes

maximal, characteristic, complemented

C₂²⋊#10596927

C₂

yes

maximal, complemented

C₄×#10596927

C₂

yes

maximal, characteristic, complemented

D₄×#64436

C₂

yes

maximal, characteristic

#11924610

—

C₂×#10596927

C₂²

yes

—

C₂×#10596927

C₂²

yes

characteristic

C₂×#10596927

C₂²

yes

characteristic

C₂²×#64436

C₂²

yes

—

C₄⋊#64436

C₄

yes

characteristic

C₂²⋊#64436

C₄

yes

—

C₄×#64436

C₂²

yes

characteristic

C₄×D₄×C₃₇

C₄

yes

characteristic

C₂×D₄×C₃₇

C₈

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₄⋊C₄×C₃₇

C₈

yes

characteristic

C₂²⋊C₄×C₃₇

C₈

yes

—

C₄²×C₃₇

C₂×C₄

yes

—

—

C₂×C₄

yes

—

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

C₂³

yes

—

—

D₄

yes

complemented

C₂³×C₃₇

C₂×C₈

yes

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₃₇

C₂×C₈

yes

—

C₂×C₄×C₃₇

C₂×C₈

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

—

—

C₂×D₄

yes

yes

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂²×C₈

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

M₃₂

yes

complemented

C₄×C₃₇

C₄×D₄

yes

C₄×C₃₇

yes

—

C₂×C₃₇

—

C₂×C₃₇

C₈×D₄

yes

characteristic

C₂×C₃₇

C₂×M₃₂

yes

characteristic

C₂×C₃₇

#504

yes

characteristic

C₂×M₃₂

—

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₃₇

#2169

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄×D₄

#3866

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×D₄

#64438

yes

characteristic

C₂²×C₄

C₂×#3866

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

#64438

yes

characteristic

C₂²⋊C₄

#64438

yes

—

C₄²

C₂×#3866

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

C₂×#64438

yes

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×#64438

yes

characteristic

C₂×C₄

C₂×#64438

yes

—

C₂×C₄

C₂×#64438

yes

characteristic

C₂×C₄

C₂²×#3866

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×#64438

yes

characteristic

C₄

—

C₄

—

100

C₄

#11924610

yes

complemented

101

C₄

D₄×#3866

yes

characteristic

102

C₄

#11924623

yes

characteristic

103

C₂

—

104

C₂

D₄×#64438

yes

characteristic

105

C₂

C₂×#11924610

yes

characteristic

106

C₂

#14558591

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#3866
commutator, socle	C₂²×C₃₇	C₂²×C₈
Fitting	C₄×D₄×C₃₇	C₄
prime radical	C₂×D₄×C₃₇	C₈
cosocle	C₂×#64436	C₂³
hypercenter	C₄×D₄	#3866
hypocenter	C₃₇	#2169
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×D₄

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#3866

commutator, socle

C₂²×C₃₇

C₂²×C₈

Fitting

C₄×D₄×C₃₇

C₄

prime radical

C₂×D₄×C₃₇

C₈

C₂³

C₄×D₄

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂×C₄ ◁ C₄×D₄

N	H ≅ G/N	type
1. C₂×#11924610	C₂	semidirect
2. C₄⋊#10596927	C₂	semidirect
3. C₂²⋊#10596927	C₂	semidirect
4. C₄×#10596927	C₂	semidirect
27. #10596927	D₄	semidirect
46. C₄×C₃₇	M₃₂	semidirect
59. C₃₇	#2169	semidirect
100. C₄	#11924610	semidirect

H ≅ G/N

type

1. C₂×#11924610

C₂

semidirect

2. C₄⋊#10596927

C₂

semidirect

3. C₂²⋊#10596927

C₂

semidirect

4. C₄×#10596927

C₂

semidirect

27. #10596927

D₄

semidirect

46. C₄×C₃₇

M₃₂

semidirect

59. C₃₇

#2169

semidirect

100. C₄

#11924610

semidirect

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

subgroup	index
C₂×#11924610	2
C₄⋊#10596927	2
C₂²⋊#10596927	2
C₄×#10596927	2
D₄×#64436	2
#2169	37

subgroup

index

Subgroups — #18010980

Conjugacy classes of subgroups

Complements

Maximal subgroups