Group #42119626 — subgroups

Subgroups — #42119626

Order 27584.

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

normal subgroups only

This group has 334 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, complemented
2	#28352185	C₂	1	2	yes	maximal, complemented
3	#28352176	C₂	1	2	yes	maximal, complemented
4	#28352160	C₂	1	4	yes	maximal, complemented
5	#28352158	C₂	1	4	yes	maximal, complemented
6	#28352148	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₄×C₄₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	#20293800	—	2	4	no	—
9	C₂×Dic₈₆₂	C₂²	1	1	yes	characteristic, complemented
10	C₂×D₁₇₂₄	C₂²	1	1	yes	characteristic, complemented
11	C₂×C₂²⋊D₄₃₁	C₂²	1	2	yes	—
12	C₂×C₂²⋊D₄₃₁	C₂²	1	8	yes	complemented
13	C₂×C₄×D₄₃₁	C₂²	1	2	yes	complemented
14	C₄⋊Dic₄₃₁	C₂²	1	2	yes	complemented
15	C₂²⋊Dic₄₃₁	C₂²	1	2	yes	—
16	C₂²⋊Dic₄₃₁	C₂²	1	8	yes	complemented
17	C₄×Dic₄₃₁	C₂²	1	2	yes	complemented
18	C₂⁴×C₄₃₁	C₂²	1	2	yes	—
19	C₂×D₄×C₄₃₁	—	2	4	no	—
20	C₂×D₄×C₄₃₁	C₂²	1	2	yes	—
21	C₂×D₄×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	C₂²⋊D₄₃₁	—	4	8	no	—
28	C₂×Dic₄₃₁	C₂³	1	4	yes	—
29	C₂×Dic₄₃₁	C₂³	1	2	yes	—
30	C₄×D₄₃₁	—	2	4	no	—
31	C₂³×C₄₃₁	—	2	4	no	—
32	C₂³×C₄₃₁	—	4	4	no	—
33	C₂³×C₄₃₁	C₂³	1	4	yes	—
34	C₂³×C₄₃₁	C₂³	1	1	yes	characteristic
35	D₄×C₄₃₁	—	2	4	no	—
36	D₄×C₄₃₁	—	2	4	no	—
37	C₂×C₄×C₄₃₁	D₄	1	4	yes	complemented
38	C₂×C₄×C₄₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₄₃₁	C₂³	1	1	yes	characteristic
40	C₂²×C₄₃₁	—	2	2	no	—
41	C₂²×C₄₃₁	—	2	4	no	—
42	C₂²×C₄₃₁	—	2	8	no	—
43	C₂²×C₄₃₁	—	4	8	no	—
44	C₂²×C₄₃₁	C₂×D₄	1	2	yes	—
45	C₂²×C₄₃₁	C₂⁴	1	1	yes	characteristic
46	C₂×D₄₃₁	—	2	2	no	—
47	C₂×D₄₃₁	—	2	4	no	—
48	C₄×C₄₃₁	C₂×D₄	1	2	yes	—
49	C₄×C₄₃₁	C₂×D₄	1	2	yes	—
50	Dic₄₃₁	—	2	2	no	—
51	Dic₄₃₁	—	2	4	no	—
52	C₂×C₄₃₁	—	2	2	no	—
53	C₂×C₄₃₁	—	4	4	no	—
54	C₂×C₄₃₁	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₄₃₁	H(5,𝔽₂)	1	2	yes	—
56	D₄₃₁	—	4	2	no	—
57	C₄₃₁	#535	1	1	yes	characteristic, complemented, Sylow
58	#535	—	431	1	no	maximal, Sylow
59	C₂×#41	—	431	1	no	—
60	C₂²×D₄	D₄₃₁	1	1	yes	characteristic, complemented
61	#127	—	431	2	no	—
62	#124	—	431	2	no	—
63	#121	—	431	4	no	—
64	#120	—	431	4	no	—
65	#117	—	431	1	no	—
66	C₂⁴	C₂×D₄₃₁	1	2	yes	—
67	#41	—	862	4	no	—
68	C₂×Q₈	—	431	1	no	—
69	C₂×D₄	—	2	4	no	—
70	C₂×D₄	—	431	2	no	—
71	C₂×D₄	—	431	8	no	—
72	C₂×D₄	—	431	1	no	—
73	C₂×D₄	C₂×D₄₃₁	1	2	yes	—
74	C₂×D₄	C₂×D₄₃₁	1	2	yes	—
75	C₂²×C₄	—	431	2	no	—
76	C₂²×C₄	C₂×D₄₃₁	1	1	yes	characteristic
77	C₄⋊C₄	—	431	2	no	—
78	C₂²⋊C₄	—	431	2	no	—
79	C₂²⋊C₄	—	431	8	no	—
80	C₄²	—	431	2	no	—
81	C₂³	—	2	4	no	—
82	C₂³	—	4	4	no	—
83	C₂³	—	431	2	no	—
84	C₂³	C₂²×D₄₃₁	1	4	yes	—
85	C₂³	C₂²×D₄₃₁	1	1	yes	characteristic
86	Q₈	—	862	2	no	—
87	D₄	—	2	4	no	—
88	D₄	—	2	4	no	—
89	D₄	—	862	4	no	—
90	D₄	—	862	2	no	—
91	D₄	—	1724	8	no	—
92	C₂×C₄	—	862	4	no	—
93	C₂×C₄	—	431	4	no	—
94	C₂×C₄	—	431	2	no	—
95	C₂×C₄	C₂²⋊D₄₃₁	1	4	yes	complemented
96	C₂×C₄	C₂²×D₄₃₁	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₄₃₁	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	—	2	4	no	—
100	C₂²	—	2	8	no	—
101	C₂²	—	4	8	no	—
102	C₂²	—	862	2	no	—
103	C₂²	—	862	4	no	—
104	C₂²	C₂×C₂²⋊D₄₃₁	1	2	yes	—
105	C₂²	C₂³×D₄₃₁	1	1	yes	characteristic
106	C₄	—	862	2	no	—
107	C₄	—	862	4	no	—
108	C₄	C₂×C₂²⋊D₄₃₁	1	2	yes	—
109	C₄	C₂×C₂²⋊D₄₃₁	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	4	4	no	—
112	C₂	—	1724	2	no	—
113	C₂	C₂²×C₂²⋊D₄₃₁	1	1	yes	characteristic
114	C₂	#28352234	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20293800

C₂

yes

maximal, characteristic, complemented

#28352185

C₂

yes

maximal, complemented

#28352176

C₂

yes

maximal, complemented

#28352160

C₂

yes

maximal, complemented

#28352158

C₂

yes

maximal, complemented

#28352148

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₄₃₁

C₂

yes

maximal, characteristic, complemented

#20293800

—

C₂×Dic₈₆₂

C₂²

yes

characteristic, complemented

C₂×D₁₇₂₄

C₂²

yes

characteristic, complemented

C₂×C₂²⋊D₄₃₁

C₂²

yes

—

C₂×C₂²⋊D₄₃₁

C₂²

yes

complemented

C₂×C₄×D₄₃₁

C₂²

yes

complemented

C₄⋊Dic₄₃₁

C₂²

yes

complemented

C₂²⋊Dic₄₃₁

C₂²

yes

—

C₂²⋊Dic₄₃₁

C₂²

yes

complemented

C₄×Dic₄₃₁

C₂²

yes

complemented

C₂⁴×C₄₃₁

C₂²

yes

—

C₂×D₄×C₄₃₁

—

C₂×D₄×C₄₃₁

C₂²

yes

—

C₂×D₄×C₄₃₁

C₂²

yes

—

C₂²×C₄×C₄₃₁

C₂²

yes

characteristic

C₂²×D₄₃₁

C₂³

yes

—

Dic₈₆₂

—

D₁₇₂₄

—

C₂²⋊D₄₃₁

—

C₂²⋊D₄₃₁

—

C₂×Dic₄₃₁

C₂³

yes

—

C₂×Dic₄₃₁

C₂³

yes

—

C₄×D₄₃₁

—

C₂³×C₄₃₁

—

C₂³×C₄₃₁

—

C₂³×C₄₃₁

C₂³

yes

—

C₂³×C₄₃₁

C₂³

yes

characteristic

D₄×C₄₃₁

—

D₄×C₄₃₁

—

C₂×C₄×C₄₃₁

D₄

yes

complemented

C₂×C₄×C₄₃₁

C₂³

yes

characteristic

C₂×C₄×C₄₃₁

C₂³

yes

characteristic

C₂²×C₄₃₁

—

C₂²×C₄₃₁

—

C₂²×C₄₃₁

—

C₂²×C₄₃₁

—

C₂²×C₄₃₁

C₂×D₄

yes

—

C₂²×C₄₃₁

C₂⁴

yes

characteristic

C₂×D₄₃₁

—

C₂×D₄₃₁

—

C₄×C₄₃₁

C₂×D₄

yes

—

C₄×C₄₃₁

C₂×D₄

yes

—

Dic₄₃₁

—

Dic₄₃₁

—

C₂×C₄₃₁

—

C₂×C₄₃₁

—

C₂×C₄₃₁

C₂²×D₄

yes

characteristic

C₂×C₄₃₁

H(5,𝔽₂)

yes

—

D₄₃₁

—

C₄₃₁

#535

yes

characteristic, complemented, Sylow

#535

—

431

maximal, Sylow

C₂×#41

—

431

—

C₂²×D₄

D₄₃₁

yes

characteristic, complemented

#127

—

431

—

#124

—

431

—

#121

—

431

—

#120

—

431

—

#117

—

431

—

C₂⁴

C₂×D₄₃₁

yes

—

#41

—

862

—

C₂×Q₈

—

431

—

C₂×D₄

—

C₂×D₄

—

431

—

C₂×D₄

—

431

—

C₂×D₄

—

431

—

C₂×D₄

C₂×D₄₃₁

yes

—

C₂×D₄

C₂×D₄₃₁

yes

—

C₂²×C₄

—

431

—

C₂²×C₄

C₂×D₄₃₁

yes

characteristic

C₄⋊C₄

—

431

—

C₂²⋊C₄

—

431

—

C₂²⋊C₄

—

431

—

C₄²

—

431

—

C₂³

—

C₂³

—

C₂³

—

431

—

C₂³

C₂²×D₄₃₁

yes

—

C₂³

C₂²×D₄₃₁

yes

characteristic

Q₈

—

862

—

D₄

—

D₄

—

D₄

—

862

—

D₄

—

862

—

D₄

—

1724

—

C₂×C₄

—

862

—

C₂×C₄

—

431

—

C₂×C₄

—

431

—

C₂×C₄

C₂²⋊D₄₃₁

yes

complemented

C₂×C₄

C₂²×D₄₃₁

yes

characteristic

C₂×C₄

C₂²×D₄₃₁

yes

characteristic

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

—

862

—

103

C₂²

—

862

—

104

C₂²

C₂×C₂²⋊D₄₃₁

yes

—

105

C₂²

C₂³×D₄₃₁

yes

characteristic

106

C₄

—

862

—

107

C₄

—

862

—

108

C₄

C₂×C₂²⋊D₄₃₁

yes

—

109

C₄

C₂×C₂²⋊D₄₃₁

yes

—

110

C₂

—

111

C₂

—

112

C₂

—

1724

—

113

C₂

C₂²×C₂²⋊D₄₃₁

yes

characteristic

114

C₂

#28352234

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₃₁
commutator, socle, cosocle	C₂²×C₄₃₁	C₂⁴
Fitting	C₂²×D₄×C₄₃₁	C₂
hypercenter	C₂²×D₄	D₄₃₁
hypocenter	C₄₃₁	#535
LCS	G ▷ C₂²×C₄₃₁ ▷ C₄₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₃₁

commutator, socle, cosocle

C₂²×C₄₃₁

C₂⁴

Fitting

C₂²×D₄×C₄₃₁

C₂

hypercenter

C₂²×D₄

D₄₃₁

C₄₃₁

G ▷ C₂²×C₄₃₁ ▷ C₄₃₁

C₁ ◁ C₂² ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂×#20293800	C₂	semidirect
2. #28352185	C₂	semidirect
3. #28352176	C₂	semidirect
4. #28352160	C₂	semidirect
5. #28352158	C₂	semidirect
6. #28352148	C₂	semidirect
7. C₂²×D₄×C₄₃₁	C₂	semidirect
9. C₂×Dic₈₆₂	C₂²	semidirect
10. C₂×D₁₇₂₄	C₂²	semidirect
12. C₂×C₂²⋊D₄₃₁	C₂²	semidirect
13. C₂×C₄×D₄₃₁	C₂²	semidirect
14. C₄⋊Dic₄₃₁	C₂²	semidirect
16. C₂²⋊Dic₄₃₁	C₂²	semidirect
17. C₄×Dic₄₃₁	C₂²	semidirect
37. C₂×C₄×C₄₃₁	D₄	semidirect
57. C₄₃₁	#535	semidirect
60. C₂²×D₄	D₄₃₁	semidirect
95. C₂×C₄	C₂²⋊D₄₃₁	semidirect

H ≅ G/N

type

1. C₂×#20293800

C₂

semidirect

2. #28352185

C₂

semidirect

3. #28352176

C₂

semidirect

4. #28352160

C₂

semidirect

5. #28352158

C₂

semidirect

6. #28352148

C₂

semidirect

7. C₂²×D₄×C₄₃₁

C₂

semidirect

9. C₂×Dic₈₆₂

C₂²

semidirect

10. C₂×D₁₇₂₄

C₂²

semidirect

12. C₂×C₂²⋊D₄₃₁

C₂²

semidirect

13. C₂×C₄×D₄₃₁

C₂²

semidirect

14. C₄⋊Dic₄₃₁

C₂²

semidirect

16. C₂²⋊Dic₄₃₁

C₂²

semidirect

17. C₄×Dic₄₃₁

C₂²

semidirect

37. C₂×C₄×C₄₃₁

D₄

semidirect

57. C₄₃₁

#535

semidirect

60. C₂²×D₄

D₄₃₁

semidirect

95. C₂×C₄

C₂²⋊D₄₃₁

semidirect

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

subgroup	index
C₂×#20293800	2
#28352185	2
#28352176	2
#28352160	2
#28352158	2
#28352148	2
C₂²×D₄×C₄₃₁	2
#535	431

subgroup

index

C₂²×D₄×C₄₃₁

431

Subgroups — #42119626

Conjugacy classes of subgroups

Complements

Maximal subgroups