Group #41003594 — subgroups

Subgroups — #41003594

Order 26944.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	#27212365	C₂	1	2	yes	maximal, complemented
2	#27212361	C₂	1	2	yes	maximal, complemented
3	#27212361	C₂	1	2	yes	maximal, complemented
4	#27212358	C₂	1	1	yes	maximal, characteristic, complemented
5	#27212354	C₂	1	1	yes	maximal, characteristic
6	#27212351	C₂	1	2	yes	maximal
7	#27212346	C₂	1	2	yes	maximal, complemented
8	#27212344	C₂	1	2	yes	maximal, complemented
9	#125×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₈₄₂	C₂²	1	2	yes	—
11	C₂×Dic₈₄₂	C₂²	1	1	yes	characteristic
12	C₂×D₁₆₈₄	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄₂₁	C₂²	1	2	yes	—
14	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₄₂₁	C₂²	1	2	yes	—
16	C₄⋊Dic₄₂₁	C₂²	1	1	yes	characteristic
17	#20269750	C₂²	1	2	yes	—
18	#20269750	C₂²	1	2	yes	—
19	#20269750	C₂²	1	2	yes	—
20	#20269748	C₂²	1	2	yes	—
21	#20269748	C₂²	1	2	yes	—
22	#20269748	C₂²	1	2	yes	—
23	#20269748	C₂²	1	2	yes	—
24	#20269748	C₂²	1	2	yes	—
25	C₄×Dic₄₂₁	C₂²	1	2	yes	—
26	C₄⋊C₄×C₄₂₁	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×C₄₂₁	C₂²	1	2	yes	—
28	C₄⋊C₄×C₄₂₁	C₂²	1	2	yes	—
29	C₄⋊C₄×C₄₂₁	C₂²	1	1	yes	characteristic
30	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
31	C₂²×D₄₂₁	C₂³	1	2	yes	—
32	Dic₈₄₂	—	4	2	no	—
33	Dic₈₄₂	—	4	1	no	—
34	D₁₆₈₄	—	4	1	no	—
35	C₂×Dic₄₂₁	C₂³	1	2	yes	—
36	C₂×Dic₄₂₁	C₂³	1	2	yes	—
37	C₂×Dic₄₂₁	C₂³	1	2	yes	—
38	C₄×D₄₂₁	—	4	2	no	—
39	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
43	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
44	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
45	C₂×D₄₂₁	—	2	2	no	—
46	C₂×D₄₂₁	—	2	2	no	—
47	C₂×D₄₂₁	—	2	2	no	—
48	C₄×C₄₂₁	—	2	2	no	—
49	C₄×C₄₂₁	—	2	1	no	—
50	C₄×C₄₂₁	—	2	1	no	—
51	C₄×C₄₂₁	—	2	2	no	—
52	C₄×C₄₂₁	—	2	1	no	—
53	Dic₄₂₁	—	2	2	no	—
54	Dic₄₂₁	—	2	2	no	—
55	Dic₄₂₁	—	2	2	no	—
56	C₂×C₄₂₁	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₄₂₁	2₋¹⁺⁴	1	1	yes	characteristic
58	C₂×C₄₂₁	2₋¹⁺⁴	1	1	yes	characteristic
59	D₄₂₁	—	4	2	no	—
60	C₄₂₁	#563	1	1	yes	characteristic, complemented, Sylow
61	#563	—	421	1	no	maximal, Sylow
62	C₄⋊Q₈	—	421	2	no	—
63	#126	—	421	2	no	—
64	#126	—	421	2	no	—
65	#125	—	421	1	no	—
66	#125	D₄₂₁	1	1	yes	characteristic, complemented
67	#124	—	421	1	no	—
68	#123	—	421	2	no	—
69	C₂²⋊Q₈	—	421	2	no	—
70	C₂²⋊Q₈	—	421	2	no	—
71	C₂×Q₈	—	421	2	no	—
72	C₂×Q₈	—	421	1	no	—
73	C₂×D₄	—	421	1	no	—
74	C₂²×C₄	—	421	2	no	—
75	C₄⋊C₄	—	421	1	no	—
76	C₄⋊C₄	—	421	2	no	—
77	C₄⋊C₄	—	421	2	no	—
78	C₄⋊C₄	—	421	2	no	—
79	C₄⋊C₄	—	421	2	no	—
80	C₄⋊C₄	—	421	1	no	—
81	C₄⋊C₄	C₂×D₄₂₁	1	1	yes	characteristic
82	C₄⋊C₄	C₂×D₄₂₁	1	2	yes	—
83	C₄⋊C₄	C₂×D₄₂₁	1	2	yes	—
84	C₄⋊C₄	C₂×D₄₂₁	1	1	yes	characteristic
85	C₂²⋊C₄	—	421	2	no	—
86	C₂²⋊C₄	—	421	2	no	—
87	C₂²⋊C₄	—	421	2	no	—
88	C₂²⋊C₄	—	421	2	no	—
89	C₂²⋊C₄	—	421	2	no	—
90	C₄²	—	421	2	no	—
91	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
92	C₂³	—	421	2	no	—
93	Q₈	—	1684	2	no	—
94	Q₈	—	1684	1	no	—
95	D₄	—	1684	1	no	—
96	C₂×C₄	—	1684	2	no	—
97	C₂×C₄	—	421	2	no	—
98	C₂×C₄	—	421	2	no	—
99	C₂×C₄	—	421	2	no	—
100	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
101	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
103	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
105	C₂²	—	842	2	no	—
106	C₂²	—	842	2	no	—
107	C₂²	—	842	2	no	—
108	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	842	2	no	—
115	C₄	—	842	2	no	—
116	C₄	—	842	2	no	—
117	C₂	—	1684	2	no	—
118	C₂	#27212412	1	1	yes	characteristic
119	C₂	#27212413	1	1	yes	characteristic
120	C₂	#27212414	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27212365

C₂

yes

maximal, complemented

#27212361

C₂

yes

maximal, complemented

#27212361

C₂

yes

maximal, complemented

#27212358

C₂

yes

maximal, characteristic, complemented

#27212354

C₂

yes

maximal, characteristic

#27212351

C₂

yes

maximal

#27212346

C₂

yes

maximal, complemented

#27212344

C₂

yes

maximal, complemented

#125×C₄₂₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₈₄₂

C₂²

yes

—

C₂×Dic₈₄₂

C₂²

yes

characteristic

C₂×D₁₆₈₄

C₂²

yes

characteristic

C₂×C₄×D₄₂₁

C₂²

yes

—

C₄⋊Dic₄₂₁

C₂²

yes

characteristic

C₄⋊Dic₄₂₁

C₂²

yes

—

C₄⋊Dic₄₂₁

C₂²

yes

characteristic

#20269750

C₂²

yes

—

#20269750

C₂²

yes

—

#20269750

C₂²

yes

—

#20269748

C₂²

yes

—

#20269748

C₂²

yes

—

#20269748

C₂²

yes

—

#20269748

C₂²

yes

—

#20269748

C₂²

yes

—

C₄×Dic₄₂₁

C₂²

yes

—

C₄⋊C₄×C₄₂₁

C₂²

yes

characteristic

C₄⋊C₄×C₄₂₁

C₂²

yes

—

C₄⋊C₄×C₄₂₁

C₂²

yes

—

C₄⋊C₄×C₄₂₁

C₂²

yes

characteristic

C₄²×C₄₂₁

C₂²

yes

characteristic

C₂²×D₄₂₁

C₂³

yes

—

Dic₈₄₂

—

Dic₈₄₂

—

D₁₆₈₄

—

C₂×Dic₄₂₁

C₂³

yes

—

C₂×Dic₄₂₁

C₂³

yes

—

C₂×Dic₄₂₁

C₂³

yes

—

C₄×D₄₂₁

—

C₂×C₄×C₄₂₁

C₂³

yes

characteristic

C₂×C₄×C₄₂₁

C₂³

yes

characteristic

C₂×C₄×C₄₂₁

C₂³

yes

characteristic

C₂×C₄×C₄₂₁

C₂³

yes

—

C₂×C₄×C₄₂₁

C₂³

yes

—

C₂²×C₄₂₁

C₂⁴

yes

characteristic

C₂×D₄₂₁

—

C₂×D₄₂₁

—

C₂×D₄₂₁

—

C₄×C₄₂₁

—

C₄×C₄₂₁

—

C₄×C₄₂₁

—

C₄×C₄₂₁

—

C₄×C₄₂₁

—

Dic₄₂₁

—

Dic₄₂₁

—

Dic₄₂₁

—

C₂×C₄₂₁

H(5,𝔽₂)

yes

characteristic

C₂×C₄₂₁

2₋¹⁺⁴

yes

characteristic

C₂×C₄₂₁

2₋¹⁺⁴

yes

characteristic

D₄₂₁

—

C₄₂₁

#563

yes

characteristic, complemented, Sylow

#563

—

421

maximal, Sylow

C₄⋊Q₈

—

421

—

#126

—

421

—

#126

—

421

—

#125

—

421

—

#125

D₄₂₁

yes

characteristic, complemented

#124

—

421

—

#123

—

421

—

C₂²⋊Q₈

—

421

—

C₂²⋊Q₈

—

421

—

C₂×Q₈

—

421

—

C₂×Q₈

—

421

—

C₂×D₄

—

421

—

C₂²×C₄

—

421

—

C₄⋊C₄

—

421

—

C₄⋊C₄

—

421

—

C₄⋊C₄

—

421

—

C₄⋊C₄

—

421

—

C₄⋊C₄

—

421

—

C₄⋊C₄

—

421

—

C₄⋊C₄

C₂×D₄₂₁

yes

characteristic

C₄⋊C₄

C₂×D₄₂₁

yes

—

C₄⋊C₄

C₂×D₄₂₁

yes

—

C₄⋊C₄

C₂×D₄₂₁

yes

characteristic

C₂²⋊C₄

—

421

—

C₂²⋊C₄

—

421

—

C₂²⋊C₄

—

421

—

C₂²⋊C₄

—

421

—

C₂²⋊C₄

—

421

—

C₄²

—

421

—

C₄²

C₂×D₄₂₁

yes

characteristic

C₂³

—

421

—

Q₈

—

1684

—

Q₈

—

1684

—

D₄

—

1684

—

C₂×C₄

—

1684

—

C₂×C₄

—

421

—

C₂×C₄

—

421

—

C₂×C₄

—

421

—

100

C₂×C₄

C₂²×D₄₂₁

yes

—

101

C₂×C₄

C₂²×D₄₂₁

yes

characteristic

102

C₂×C₄

C₂²×D₄₂₁

yes

characteristic

103

C₂×C₄

C₂²×D₄₂₁

yes

characteristic

104

C₂×C₄

C₂²×D₄₂₁

yes

—

105

C₂²

—

842

—

106

C₂²

—

842

—

107

C₂²

—

842

—

108

C₂²

C₂³×D₄₂₁

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

842

—

115

C₄

—

842

—

116

C₄

—

842

—

117

C₂

—

1684

—

118

C₂

#27212412

yes

characteristic

119

C₂

#27212413

yes

characteristic

120

C₂

#27212414

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₂₁
commutator, socle, cosocle	C₂²×C₄₂₁	C₂⁴
Fitting	#125×C₄₂₁	C₂
prime radical	C₂×D₁₆₈₄	C₂²
hypercenter	#125	D₄₂₁
hypocenter	C₄₂₁	#563
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₂₁

commutator, socle, cosocle

C₂²×C₄₂₁

C₂⁴

Fitting

#125×C₄₂₁

C₂

prime radical

C₂×D₁₆₈₄

C₂²

D₄₂₁

C₄₂₁

G ▷ C₂²×C₄₂₁ ▷ C₄₂₁

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #27212365	C₂	semidirect
2. #27212361	C₂	semidirect
3. #27212361	C₂	semidirect
4. #27212358	C₂	semidirect
7. #27212346	C₂	semidirect
8. #27212344	C₂	semidirect
9. #125×C₄₂₁	C₂	semidirect
60. C₄₂₁	#563	semidirect
66. #125	D₄₂₁	semidirect

H ≅ G/N

type

1. #27212365

C₂

semidirect

2. #27212361

C₂

semidirect

3. #27212361

C₂

semidirect

4. #27212358

C₂

semidirect

7. #27212346

C₂

semidirect

8. #27212344

C₂

semidirect

9. #125×C₄₂₁

C₂

semidirect

60. C₄₂₁

#563

semidirect

66. #125

D₄₂₁

semidirect

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

subgroup	index
#27212365	2
#27212361	2
#27212358	2
#27212354	2
#27212351	2
#27212346	2
#27212344	2
#125×C₄₂₁	2
#563	421

subgroup

index

#125×C₄₂₁

421

Subgroups — #41003594

Conjugacy classes of subgroups

Complements

Maximal subgroups