Group #42119633 — subgroups

Subgroups — #42119633

Order 27584.

270 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 111 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×#20293800	C₂	1	1	yes	maximal, characteristic
2	#28352187	C₂	1	2	yes	maximal, complemented
3	#28352178	C₂	1	2	yes	maximal, complemented
4	#28352172	C₂	1	4	yes	maximal, complemented
5	#28352165	C₂	1	4	yes	maximal, complemented
6	#28352148	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×C₄₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#20293800	—	2	4	no	—
9	C₂×Dic₈₆₂	C₂²	1	1	yes	characteristic
10	C₂×D₁₇₂₄	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊D₄₃₁	C₂²	1	2	yes	—
12	C₂×C₄×D₄₃₁	C₂²	1	2	yes	—
13	C₄⋊Dic₄₃₁	C₂²	1	2	yes	—
14	Dic₄₃₁⋊C₄	C₂²	1	8	yes	—
15	C₂²⋊Dic₄₃₁	C₂²	1	2	yes	—
16	(C₂×D₄₃₁)⋊C₄	C₂²	1	8	yes	—
17	C₄×Dic₄₃₁	C₂²	1	2	yes	—
18	C₂×Q₈×C₄₃₁	—	2	4	no	—
19	C₂×Q₈×C₄₃₁	C₂²	1	2	yes	—
20	C₂×Q₈×C₄₃₁	C₂²	1	2	yes	—
21	C₂²×C₄×C₄₃₁	C₂²	1	2	yes	—
22	C₂²×C₄×C₄₃₁	C₂²	1	1	yes	characteristic
23	C₂²×D₄₃₁	C₂³	1	2	yes	—
24	Dic₈₆₂	—	2	2	no	—
25	D₁₇₂₄	—	2	2	no	—
26	C₂²⋊D₄₃₁	—	2	4	no	—
27	C₂×Dic₄₃₁	C₂³	1	4	yes	—
28	C₂×Dic₄₃₁	C₂³	1	2	yes	—
29	C₄×D₄₃₁	—	2	4	no	—
30	C₂³×C₄₃₁	C₂³	1	1	yes	characteristic
31	Q₈×C₄₃₁	—	2	4	no	—
32	Q₈×C₄₃₁	—	2	4	no	—
33	C₂×C₄×C₄₃₁	—	2	4	no	—
34	C₂×C₄×C₄₃₁	D₄	1	4	yes	—
35	C₂×C₄×C₄₃₁	C₂³	1	4	yes	—
36	C₂×C₄×C₄₃₁	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₄₃₁	C₂³	1	1	yes	characteristic
38	C₂²×C₄₃₁	—	2	2	no	—
39	C₂²×C₄₃₁	C₂×D₄	1	2	yes	—
40	C₂²×C₄₃₁	C₂⁴	1	1	yes	characteristic
41	C₂×D₄₃₁	—	2	2	no	—
42	C₂×D₄₃₁	—	2	4	no	—
43	C₄×C₄₃₁	—	2	4	no	—
44	C₄×C₄₃₁	C₂×D₄	1	2	yes	—
45	C₄×C₄₃₁	C₂×D₄	1	2	yes	—
46	Dic₄₃₁	—	2	2	no	—
47	Dic₄₃₁	—	2	4	no	—
48	C₂×C₄₃₁	—	2	2	no	—
49	C₂×C₄₃₁	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₄₃₁	2₋¹⁺⁴	1	2	yes	—
51	D₄₃₁	—	4	2	no	—
52	C₄₃₁	#536	1	1	yes	characteristic, complemented, Sylow
53	#536	—	431	1	no	maximal, Sylow
54	C₂×#41	—	431	1	no	—
55	C₂²×Q₈	D₄₃₁	1	1	yes	characteristic, complemented
56	C₄⋊Q₈	—	431	2	no	—
57	#124	—	431	2	no	—
58	#123	—	431	4	no	—
59	C₂²⋊Q₈	—	431	4	no	—
60	#117	—	431	1	no	—
61	#41	—	862	4	no	—
62	C₂×Q₈	—	2	4	no	—
63	C₂×Q₈	—	431	1	no	—
64	C₂×Q₈	C₂×D₄₃₁	1	2	yes	—
65	C₂×Q₈	C₂×D₄₃₁	1	2	yes	—
66	C₂×D₄	—	431	2	no	—
67	C₂×D₄	—	431	1	no	—
68	C₂²×C₄	—	431	2	no	—
69	C₂²×C₄	C₂×D₄₃₁	1	2	yes	—
70	C₂²×C₄	C₂×D₄₃₁	1	1	yes	characteristic
71	C₄⋊C₄	—	431	2	no	—
72	C₄⋊C₄	—	431	8	no	—
73	C₂²⋊C₄	—	431	2	no	—
74	C₂²⋊C₄	—	431	8	no	—
75	C₄²	—	431	2	no	—
76	C₂³	—	431	2	no	—
77	C₂³	C₂²×D₄₃₁	1	1	yes	characteristic
78	Q₈	—	2	4	no	—
79	Q₈	—	2	4	no	—
80	Q₈	—	862	2	no	—
81	D₄	—	862	4	no	—
82	D₄	—	862	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	862	4	no	—
85	C₂×C₄	—	431	4	no	—
86	C₂×C₄	—	431	2	no	—
87	C₂×C₄	C₂²⋊D₄₃₁	1	4	yes	—
88	C₂×C₄	C₂²×D₄₃₁	1	4	yes	—
89	C₂×C₄	C₂²×D₄₃₁	1	1	yes	characteristic
90	C₂×C₄	C₂²×D₄₃₁	1	1	yes	characteristic
91	C₂²	—	2	2	no	—
92	C₂²	—	862	2	no	—
93	C₂²	—	862	4	no	—
94	C₂²	C₂×C₂²⋊D₄₃₁	1	2	yes	—
95	C₂²	C₂³×D₄₃₁	1	1	yes	characteristic
96	C₄	—	2	4	no	—
97	C₄	—	862	2	no	—
98	C₄	—	862	4	no	—
99	C₄	C₂×C₂²⋊D₄₃₁	1	2	yes	—
100	C₄	C₂×C₂²⋊D₄₃₁	1	2	yes	—
101	C₂	—	2	2	no	—
102	C₂	—	1724	2	no	—
103	C₂	C₂²×C₂²⋊D₄₃₁	1	1	yes	characteristic
104	C₂	#28352237	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20293800

C₂

yes

maximal, characteristic

#28352187

C₂

yes

maximal, complemented

#28352178

C₂

yes

maximal, complemented

#28352172

C₂

yes

maximal, complemented

#28352165

C₂

yes

maximal, complemented

#28352148

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₄₃₁

C₂

yes

maximal, characteristic, complemented

#20293800

—

C₂×Dic₈₆₂

C₂²

yes

characteristic

C₂×D₁₇₂₄

C₂²

yes

characteristic

C₂×C₂²⋊D₄₃₁

C₂²

yes

—

C₂×C₄×D₄₃₁

C₂²

yes

—

C₄⋊Dic₄₃₁

C₂²

yes

—

Dic₄₃₁⋊C₄

C₂²

yes

—

C₂²⋊Dic₄₃₁

C₂²

yes

—

(C₂×D₄₃₁)⋊C₄

C₂²

yes

—

C₄×Dic₄₃₁

C₂²

yes

—

C₂×Q₈×C₄₃₁

—

C₂×Q₈×C₄₃₁

C₂²

yes

—

C₂×Q₈×C₄₃₁

C₂²

yes

—

C₂²×C₄×C₄₃₁

C₂²

yes

—

C₂²×C₄×C₄₃₁

C₂²

yes

characteristic

C₂²×D₄₃₁

C₂³

yes

—

Dic₈₆₂

—

D₁₇₂₄

—

C₂²⋊D₄₃₁

—

C₂×Dic₄₃₁

C₂³

yes

—

C₂×Dic₄₃₁

C₂³

yes

—

C₄×D₄₃₁

—

C₂³×C₄₃₁

C₂³

yes

characteristic

Q₈×C₄₃₁

—

Q₈×C₄₃₁

—

C₂×C₄×C₄₃₁

—

C₂×C₄×C₄₃₁

D₄

yes

—

C₂×C₄×C₄₃₁

C₂³

yes

—

C₂×C₄×C₄₃₁

C₂³

yes

characteristic

C₂×C₄×C₄₃₁

C₂³

yes

characteristic

C₂²×C₄₃₁

—

C₂²×C₄₃₁

C₂×D₄

yes

—

C₂²×C₄₃₁

C₂⁴

yes

characteristic

C₂×D₄₃₁

—

C₂×D₄₃₁

—

C₄×C₄₃₁

—

C₄×C₄₃₁

C₂×D₄

yes

—

C₄×C₄₃₁

C₂×D₄

yes

—

Dic₄₃₁

—

Dic₄₃₁

—

C₂×C₄₃₁

—

C₂×C₄₃₁

C₂²×D₄

yes

characteristic

C₂×C₄₃₁

2₋¹⁺⁴

yes

—

D₄₃₁

—

C₄₃₁

#536

yes

characteristic, complemented, Sylow

#536

—

431

maximal, Sylow

C₂×#41

—

431

—

C₂²×Q₈

D₄₃₁

yes

characteristic, complemented

C₄⋊Q₈

—

431

—

#124

—

431

—

#123

—

431

—

C₂²⋊Q₈

—

431

—

#117

—

431

—

#41

—

862

—

C₂×Q₈

—

C₂×Q₈

—

431

—

C₂×Q₈

C₂×D₄₃₁

yes

—

C₂×Q₈

C₂×D₄₃₁

yes

—

C₂×D₄

—

431

—

C₂×D₄

—

431

—

C₂²×C₄

—

431

—

C₂²×C₄

C₂×D₄₃₁

yes

—

C₂²×C₄

C₂×D₄₃₁

yes

characteristic

C₄⋊C₄

—

431

—

C₄⋊C₄

—

431

—

C₂²⋊C₄

—

431

—

C₂²⋊C₄

—

431

—

C₄²

—

431

—

C₂³

—

431

—

C₂³

C₂²×D₄₃₁

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

862

—

D₄

—

862

—

D₄

—

862

—

C₂×C₄

—

C₂×C₄

—

862

—

C₂×C₄

—

431

—

C₂×C₄

—

431

—

C₂×C₄

C₂²⋊D₄₃₁

yes

—

C₂×C₄

C₂²×D₄₃₁

yes

—

C₂×C₄

C₂²×D₄₃₁

yes

characteristic

C₂×C₄

C₂²×D₄₃₁

yes

characteristic

C₂²

—

C₂²

—

862

—

C₂²

—

862

—

C₂²

C₂×C₂²⋊D₄₃₁

yes

—

C₂²

C₂³×D₄₃₁

yes

characteristic

C₄

—

C₄

—

862

—

C₄

—

862

—

C₄

C₂×C₂²⋊D₄₃₁

yes

—

100

C₄

C₂×C₂²⋊D₄₃₁

yes

—

101

C₂

—

102

C₂

—

1724

—

103

C₂

C₂²×C₂²⋊D₄₃₁

yes

characteristic

104

C₂

#28352237

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₃₁
commutator, socle, cosocle	C₂²×C₄₃₁	C₂⁴
Fitting	C₂²×Q₈×C₄₃₁	C₂
prime radical	C₂×#20293800	C₂
hypercenter	C₂²×Q₈	D₄₃₁
hypocenter	C₄₃₁	#536
LCS	G ▷ C₂²×C₄₃₁ ▷ C₄₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₃₁

commutator, socle, cosocle

C₂²×C₄₃₁

C₂⁴

Fitting

C₂²×Q₈×C₄₃₁

C₂

prime radical

C₂×#20293800

C₂

hypercenter

C₂²×Q₈

D₄₃₁

C₄₃₁

G ▷ C₂²×C₄₃₁ ▷ C₄₃₁

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
2. #28352187	C₂	semidirect
3. #28352178	C₂	semidirect
4. #28352172	C₂	semidirect
5. #28352165	C₂	semidirect
6. #28352148	C₂	semidirect
7. C₂²×Q₈×C₄₃₁	C₂	semidirect
52. C₄₃₁	#536	semidirect
55. C₂²×Q₈	D₄₃₁	semidirect

H ≅ G/N

type

2. #28352187

C₂

semidirect

3. #28352178

C₂

semidirect

4. #28352172

C₂

semidirect

5. #28352165

C₂

semidirect

6. #28352148

C₂

semidirect

7. C₂²×Q₈×C₄₃₁

C₂

semidirect

52. C₄₃₁

#536

semidirect

55. C₂²×Q₈

D₄₃₁

semidirect

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

subgroup	index
C₂×#20293800	2
#28352187	2
#28352178	2
#28352172	2
#28352165	2
#28352148	2
C₂²×Q₈×C₄₃₁	2
#536	431

subgroup

index

C₂²×Q₈×C₄₃₁

431

Subgroups — #42119633

Conjugacy classes of subgroups

Complements

Maximal subgroups