Group #41003589 — subgroups

Subgroups — #41003589

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	#27212364	C₂	1	1	yes	maximal, characteristic, complemented
2	#27212361	C₂	1	2	yes	maximal, complemented
3	#27212360	C₂	1	2	yes	maximal, complemented
4	#27212356	C₂	1	2	yes	maximal, complemented
5	#27212355	C₂	1	1	yes	maximal, characteristic
6	#27212351	C₂	1	2	yes	maximal
7	#27212344	C₂	1	2	yes	maximal, complemented
8	#27212342	C₂	1	2	yes	maximal, complemented
9	C₄⋊Q₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₈₄₂	C₂²	1	1	yes	characteristic
11	C₂×D₁₆₈₄	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄₂₁	C₂²	1	2	yes	—
13	C₄⋊Dic₄₂₁	C₂²	1	2	yes	—
14	#20269750	C₂²	1	2	yes	—
15	#20269750	C₂²	1	2	yes	—
16	#20269750	C₂²	1	2	yes	—
17	#20269750	C₂²	1	2	yes	—
18	#20269750	C₂²	1	2	yes	—
19	#20269748	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	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
25	C₄×Dic₄₂₁	C₂²	1	1	yes	characteristic
26	C₂×Q₈×C₄₂₁	C₂²	1	1	yes	characteristic
27	C₂×Q₈×C₄₂₁	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₄₂₁	C₂²	1	2	yes	—
29	C₄⋊C₄×C₄₂₁	C₂²	1	2	yes	—
30	C₄²×C₄₂₁	C₂²	1	1	yes	characteristic
31	C₂²×D₄₂₁	C₂³	1	2	yes	—
32	Dic₈₄₂	—	4	1	no	—
33	D₁₆₈₄	—	4	1	no	—
34	C₂×Dic₄₂₁	C₂³	1	2	yes	—
35	C₂×Dic₄₂₁	C₂³	1	2	yes	—
36	C₂×Dic₄₂₁	C₂³	1	2	yes	—
37	C₄×D₄₂₁	—	4	2	no	—
38	Q₈×C₄₂₁	—	4	1	no	—
39	Q₈×C₄₂₁	—	4	1	no	—
40	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
41	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₄₂₁	C₂³	1	2	yes	—
45	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
46	C₂×D₄₂₁	—	2	2	no	—
47	C₂×D₄₂₁	—	2	4	no	—
48	C₄×C₄₂₁	—	2	1	no	—
49	C₄×C₄₂₁	—	2	2	no	—
50	C₄×C₄₂₁	—	2	1	no	—
51	C₄×C₄₂₁	—	2	1	no	—
52	C₄×C₄₂₁	—	2	2	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	2	yes	—
58	D₄₂₁	—	4	2	no	—
59	C₄₂₁	#563	1	1	yes	characteristic, complemented, Sylow
60	#563	—	421	1	no	maximal, Sylow
61	C₄⋊Q₈	—	421	1	no	—
62	C₄⋊Q₈	D₄₂₁	1	1	yes	characteristic, complemented
63	#126	—	421	2	no	—
64	#126	—	421	2	no	—
65	#125	—	421	2	no	—
66	#124	—	421	1	no	—
67	#123	—	421	2	no	—
68	C₂²⋊Q₈	—	421	2	no	—
69	C₂²⋊Q₈	—	421	2	no	—
70	C₂×Q₈	—	421	1	no	—
71	C₂×Q₈	C₂×D₄₂₁	1	1	yes	characteristic
72	C₂×Q₈	C₂×D₄₂₁	1	1	yes	characteristic
73	C₂×D₄	—	421	1	no	—
74	C₂²×C₄	—	421	2	no	—
75	C₄⋊C₄	—	421	2	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	2	no	—
81	C₄⋊C₄	C₂×D₄₂₁	1	2	yes	—
82	C₄⋊C₄	C₂×D₄₂₁	1	2	yes	—
83	C₂²⋊C₄	—	421	2	no	—
84	C₂²⋊C₄	—	421	2	no	—
85	C₂²⋊C₄	—	421	2	no	—
86	C₂²⋊C₄	—	421	2	no	—
87	C₂²⋊C₄	—	421	2	no	—
88	C₄²	—	421	1	no	—
89	C₄²	—	421	1	no	—
90	C₄²	C₂×D₄₂₁	1	1	yes	characteristic
91	C₂³	—	421	2	no	—
92	Q₈	—	4	1	no	—
93	Q₈	—	4	1	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	1	yes	characteristic
101	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₄₂₁	1	2	yes	—
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	4	no	—
107	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	2	no	—
113	C₄	—	842	2	no	—
114	C₄	—	842	2	no	—
115	C₄	—	842	2	no	—
116	C₂	—	1684	2	no	—
117	C₂	#27212411	1	1	yes	characteristic
118	C₂	#27212414	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27212364

C₂

yes

maximal, characteristic, complemented

#27212361

C₂

yes

maximal, complemented

#27212360

C₂

yes

maximal, complemented

#27212356

C₂

yes

maximal, complemented

#27212355

C₂

yes

maximal, characteristic

#27212351

C₂

yes

maximal

#27212344

C₂

yes

maximal, complemented

#27212342

C₂

yes

maximal, complemented

C₄⋊Q₈×C₄₂₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₈₄₂

C₂²

yes

characteristic

C₂×D₁₆₈₄

C₂²

yes

characteristic

C₂×C₄×D₄₂₁

C₂²

yes

—

C₄⋊Dic₄₂₁

C₂²

yes

—

#20269750

C₂²

yes

—

#20269750

C₂²

yes

—

#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

characteristic

C₄×Dic₄₂₁

C₂²

yes

characteristic

C₂×Q₈×C₄₂₁

C₂²

yes

characteristic

C₂×Q₈×C₄₂₁

C₂²

yes

characteristic

C₄⋊C₄×C₄₂₁

C₂²

yes

—

C₄⋊C₄×C₄₂₁

C₂²

yes

—

C₄²×C₄₂₁

C₂²

yes

characteristic

C₂²×D₄₂₁

C₂³

yes

—

Dic₈₄₂

—

D₁₆₈₄

—

C₂×Dic₄₂₁

C₂³

yes

—

C₂×Dic₄₂₁

C₂³

yes

—

C₂×Dic₄₂₁

C₂³

yes

—

C₄×D₄₂₁

—

Q₈×C₄₂₁

—

Q₈×C₄₂₁

—

C₂×C₄×C₄₂₁

C₂³

yes

—

C₂×C₄×C₄₂₁

C₂³

yes

characteristic

C₂×C₄×C₄₂₁

C₂³

yes

characteristic

C₂×C₄×C₄₂₁

C₂³

yes

characteristic

C₂×C₄×C₄₂₁

C₂³

yes

—

C₂²×C₄₂₁

C₂⁴

yes

characteristic

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

—

D₄₂₁

—

C₄₂₁

#563

yes

characteristic, complemented, Sylow

#563

—

421

maximal, Sylow

C₄⋊Q₈

—

421

—

C₄⋊Q₈

D₄₂₁

yes

characteristic, complemented

#126

—

421

—

#126

—

421

—

#125

—

421

—

#124

—

421

—

#123

—

421

—

C₂²⋊Q₈

—

421

—

C₂²⋊Q₈

—

421

—

C₂×Q₈

—

421

—

C₂×Q₈

C₂×D₄₂₁

yes

characteristic

C₂×Q₈

C₂×D₄₂₁

yes

characteristic

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

—

C₄⋊C₄

C₂×D₄₂₁

yes

—

C₂²⋊C₄

—

421

—

C₂²⋊C₄

—

421

—

C₂²⋊C₄

—

421

—

C₂²⋊C₄

—

421

—

C₂²⋊C₄

—

421

—

C₄²

—

421

—

C₄²

—

421

—

C₄²

C₂×D₄₂₁

yes

characteristic

C₂³

—

421

—

Q₈

—

Q₈

—

Q₈

—

1684

—

D₄

—

1684

—

C₂×C₄

—

1684

—

C₂×C₄

—

421

—

C₂×C₄

—

421

—

C₂×C₄

—

421

—

100

C₂×C₄

C₂²×D₄₂₁

yes

characteristic

101

C₂×C₄

C₂²×D₄₂₁

yes

characteristic

102

C₂×C₄

C₂²×D₄₂₁

yes

—

103

C₂×C₄

C₂²×D₄₂₁

yes

characteristic

104

C₂×C₄

C₂²×D₄₂₁

yes

—

105

C₂²

—

842

—

106

C₂²

—

842

—

107

C₂²

C₂³×D₄₂₁

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

842

—

114

C₄

—

842

—

115

C₄

—

842

—

116

C₂

—

1684

—

117

C₂

#27212411

yes

characteristic

118

C₂

#27212414

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₂₁
commutator, socle, cosocle	C₂²×C₄₂₁	C₂⁴
Fitting	C₄⋊Q₈×C₄₂₁	C₂
prime radical	C₂×D₁₆₈₄	C₂²
hypercenter	C₄⋊Q₈	D₄₂₁
hypocenter	C₄₂₁	#563
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₂₁

commutator, socle, cosocle

C₂²×C₄₂₁

C₂⁴

Fitting

C₄⋊Q₈×C₄₂₁

C₂

prime radical

C₂×D₁₆₈₄

C₂²

hypercenter

C₄⋊Q₈

D₄₂₁

C₄₂₁

G ▷ C₂²×C₄₂₁ ▷ C₄₂₁

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #27212364	C₂	semidirect
2. #27212361	C₂	semidirect
3. #27212360	C₂	semidirect
4. #27212356	C₂	semidirect
7. #27212344	C₂	semidirect
8. #27212342	C₂	semidirect
9. C₄⋊Q₈×C₄₂₁	C₂	semidirect
59. C₄₂₁	#563	semidirect
62. C₄⋊Q₈	D₄₂₁	semidirect

H ≅ G/N

type

1. #27212364

C₂

semidirect

2. #27212361

C₂

semidirect

3. #27212360

C₂

semidirect

4. #27212356

C₂

semidirect

7. #27212344

C₂

semidirect

8. #27212342

C₂

semidirect

9. C₄⋊Q₈×C₄₂₁

C₂

semidirect

59. C₄₂₁

#563

semidirect

62. C₄⋊Q₈

D₄₂₁

semidirect

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

subgroup	index
#27212364	2
#27212361	2
#27212360	2
#27212356	2
#27212355	2
#27212351	2
#27212344	2
#27212342	2
C₄⋊Q₈×C₄₂₁	2
#563	421

subgroup

index

C₄⋊Q₈×C₄₂₁

421

Subgroups — #41003589

Conjugacy classes of subgroups

Complements

Maximal subgroups