Group #28384391 — subgroups

Subgroups — #28384391

Order 14272.

258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 107 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	#20297351	C₂	1	1	yes	maximal, characteristic, complemented
2	#20297334	C₂	1	3	yes	maximal, complemented
3	#20297330	C₂	1	3	yes	maximal, complemented
4	#20297324	C₂	1	3	yes	maximal, complemented
5	C₂×#16813158	C₂	1	3	yes	maximal, complemented
6	#20297311	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#16813170	C₂²	1	3	yes	complemented
9	#16813169	—	2	3	no	—
10	C₂×Dic₄₄₆	C₂²	1	3	yes	complemented
11	Q₈×D₂₂₃	—	2	1	no	—
12	#16813160	C₂²	1	3	yes	complemented
13	#16813158	C₂²	1	3	yes	complemented
14	#16813158	C₂²	1	6	yes	complemented
15	#16813157	C₂²	1	1	yes	characteristic, complemented
16	#16813155	C₂²	1	3	yes	complemented
17	#16813154	C₂²	1	3	yes	complemented
18	C₂×C₂₂₃⋊C₈	C₂²	1	3	yes	complemented
19	#41×C₂₂₃	—	2	1	no	—
20	#41×C₂₂₃	C₂²	1	3	yes	—
21	#41×C₂₂₃	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₂₂₃	—	2	3	no	—
23	C₂×D₄×C₂₂₃	C₂²	1	3	yes	—
24	Dic₄₄₆	—	2	3	no	—
25	Dic₄₄₆	C₂³	1	3	yes	—
26	D₈₉₂	C₂³	1	1	yes	characteristic
27	C₂²⋊D₂₂₃	—	2	3	no	—
28	C₂×Dic₂₂₃	—	2	3	no	—
29	C₄×D₂₂₃	—	2	3	no	—
30	C₂₂₃⋊C₈	C₂³	1	1	yes	characteristic
31	C₂₂₃⋊C₈	C₂³	1	3	yes	—
32	C₂³×C₂₂₃	—	2	3	no	—
33	Q₈×C₂₂₃	D₄	1	1	yes	characteristic
34	Q₈×C₂₂₃	C₂³	1	1	yes	characteristic
35	D₄×C₂₂₃	—	2	3	no	—
36	D₄×C₂₂₃	—	2	3	no	—
37	D₄×C₂₂₃	D₄	1	3	yes	—
38	D₄×C₂₂₃	C₂³	1	3	yes	—
39	C₂×C₄×C₂₂₃	—	2	3	no	—
40	C₂×C₄×C₂₂₃	C₂³	1	3	yes	—
41	C₂²×C₂₂₃	—	2	3	no	—
42	C₂²×C₂₂₃	—	8	3	no	—
43	C₂²×C₂₂₃	C₂×D₄	1	3	yes	—
44	C₂×D₂₂₃	—	2	1	no	—
45	C₄×C₂₂₃	—	2	1	no	—
46	C₄×C₂₂₃	C₂×D₄	1	3	yes	—
47	C₄×C₂₂₃	C₂⁴	1	1	yes	characteristic
48	Dic₂₂₃	—	2	3	no	—
49	C₂×C₂₂₃	—	2	3	no	—
50	C₂×C₂₂₃	—	4	3	no	—
51	C₂×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
52	D₂₂₃	—	4	1	no	—
53	C₂₂₃	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	223	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	223	1	no	—
56	H(5,𝔽₂)	D₂₂₃	1	1	yes	characteristic, complemented
57	#137	—	223	3	no	—
58	AGL(1,ℤ/8)	—	223	3	no	—
59	#135	—	223	3	no	—
60	C₂×SD₁₆	—	223	3	no	—
61	#131	—	223	1	no	—
62	#41	—	2	1	no	—
63	#41	—	446	3	no	—
64	#41	—	223	3	no	—
65	#41	C₂×D₂₂₃	1	3	yes	—
66	#41	C₂×D₂₂₃	1	1	yes	characteristic
67	C₂×Q₈	—	446	1	no	—
68	C₂×Q₈	—	223	3	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₂₂₃	1	3	yes	—
71	Q₁₆	—	223	3	no	—
72	SD₁₆	—	223	1	no	—
73	SD₁₆	—	223	3	no	—
74	SD₁₆	—	223	6	no	—
75	D₈	—	223	3	no	—
76	M₁₆	—	223	3	no	—
77	C₂×C₈	—	223	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	446	3	no	—
80	Q₈	—	223	3	no	—
81	Q₈	C₂²⋊D₂₂₃	1	1	yes	characteristic
82	Q₈	C₂²×D₂₂₃	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	446	3	no	—
86	D₄	—	223	1	no	—
87	D₄	C₂²⋊D₂₂₃	1	3	yes	—
88	D₄	C₂²×D₂₂₃	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	446	3	no	—
91	C₂×C₄	—	446	3	no	—
92	C₂×C₄	C₂²×D₂₂₃	1	3	yes	—
93	C₈	—	223	1	no	—
94	C₈	—	223	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	446	1	no	—
98	C₂²	C₂×C₂²⋊D₂₂₃	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	446	3	no	—
101	C₄	C₂×C₂²⋊D₂₂₃	1	3	yes	—
102	C₄	C₂³×D₂₂₃	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	892	1	no	—
106	C₂	C₂²×C₂²⋊D₂₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20297351

C₂

yes

maximal, characteristic, complemented

#20297334

C₂

yes

maximal, complemented

#20297330

C₂

yes

maximal, complemented

#20297324

C₂

yes

maximal, complemented

C₂×#16813158

C₂

yes

maximal, complemented

#20297311

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₂₂₃

C₂

yes

maximal, characteristic, complemented

#16813170

C₂²

yes

complemented

#16813169

—

C₂×Dic₄₄₆

C₂²

yes

complemented

Q₈×D₂₂₃

—

#16813160

C₂²

yes

complemented

#16813158

C₂²

yes

complemented

#16813158

C₂²

yes

complemented

#16813157

C₂²

yes

characteristic, complemented

#16813155

C₂²

yes

complemented

#16813154

C₂²

yes

complemented

C₂×C₂₂₃⋊C₈

C₂²

yes

complemented

#41×C₂₂₃

—

#41×C₂₂₃

C₂²

yes

—

#41×C₂₂₃

C₂²

yes

characteristic

C₂×D₄×C₂₂₃

—

C₂×D₄×C₂₂₃

C₂²

yes

—

Dic₄₄₆

—

Dic₄₄₆

C₂³

yes

—

D₈₉₂

C₂³

yes

characteristic

C₂²⋊D₂₂₃

—

C₂×Dic₂₂₃

—

C₄×D₂₂₃

—

C₂₂₃⋊C₈

C₂³

yes

characteristic

C₂₂₃⋊C₈

C₂³

yes

—

C₂³×C₂₂₃

—

Q₈×C₂₂₃

D₄

yes

characteristic

Q₈×C₂₂₃

C₂³

yes

characteristic

D₄×C₂₂₃

—

D₄×C₂₂₃

—

D₄×C₂₂₃

D₄

yes

—

D₄×C₂₂₃

C₂³

yes

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂³

yes

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

C₂×D₄

yes

—

C₂×D₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

C₂×D₄

yes

—

C₄×C₂₂₃

C₂⁴

yes

characteristic

Dic₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

C₂²×D₄

yes

characteristic

D₂₂₃

—

C₂₂₃

#577

yes

characteristic, complemented, Sylow

#577

—

223

maximal, Sylow

2₋¹⁺⁴

—

223

—

H(5,𝔽₂)

D₂₂₃

yes

characteristic, complemented

#137

—

223

—

AGL(1,ℤ/8)

—

223

—

#135

—

223

—

C₂×SD₁₆

—

223

—

#131

—

223

—

#41

—

#41

—

446

—

#41

—

223

—

#41

C₂×D₂₂₃

yes

—

#41

C₂×D₂₂₃

yes

characteristic

C₂×Q₈

—

446

—

C₂×Q₈

—

223

—

C₂×D₄

—

C₂×D₄

C₂×D₂₂₃

yes

—

Q₁₆

—

223

—

SD₁₆

—

223

—

SD₁₆

—

223

—

SD₁₆

—

223

—

D₈

—

223

—

M₁₆

—

223

—

C₂×C₈

—

223

—

C₂³

—

Q₈

—

446

—

Q₈

—

223

—

Q₈

C₂²⋊D₂₂₃

yes

characteristic

Q₈

C₂²×D₂₂₃

yes

characteristic

D₄

—

D₄

—

D₄

—

446

—

D₄

—

223

—

D₄

C₂²⋊D₂₂₃

yes

—

D₄

C₂²×D₂₂₃

yes

—

C₂×C₄

—

C₂×C₄

—

446

—

C₂×C₄

—

446

—

C₂×C₄

C₂²×D₂₂₃

yes

—

C₈

—

223

—

C₈

—

223

—

C₂²

—

C₂²

—

C₂²

—

446

—

C₂²

C₂×C₂²⋊D₂₂₃

yes

—

C₄

—

100

C₄

—

446

—

101

C₄

C₂×C₂²⋊D₂₂₃

yes

—

102

C₄

C₂³×D₂₂₃

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

892

—

106

C₂

C₂²×C₂²⋊D₂₂₃

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₂₂₃
commutator, cosocle	C₄×C₂₂₃	C₂⁴
Frattini	C₄	C₂³×D₂₂₃
Fitting	H(5,𝔽₂)×C₂₂₃	C₂
socle	C₂×C₂₂₃	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₂₂₃
hypocenter	C₂₂₃	#577
LCS	G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₂₂₃

commutator, cosocle

C₄×C₂₂₃

C₂⁴

Frattini

C₄

C₂³×D₂₂₃

Fitting

H(5,𝔽₂)×C₂₂₃

C₂

socle

C₂×C₂₂₃

C₂²×D₄

D₂₂₃

C₂₂₃

G ▷ C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #20297351	C₂	semidirect
2. #20297334	C₂	semidirect
3. #20297330	C₂	semidirect
4. #20297324	C₂	semidirect
5. C₂×#16813158	C₂	semidirect
6. #20297311	C₂	semidirect
7. H(5,𝔽₂)×C₂₂₃	C₂	semidirect
8. #16813170	C₂²	semidirect
10. C₂×Dic₄₄₆	C₂²	semidirect
12. #16813160	C₂²	semidirect
13. #16813158	C₂²	semidirect
14. #16813158	C₂²	semidirect
15. #16813157	C₂²	semidirect
16. #16813155	C₂²	semidirect
17. #16813154	C₂²	semidirect
18. C₂×C₂₂₃⋊C₈	C₂²	semidirect
53. C₂₂₃	#577	semidirect
56. H(5,𝔽₂)	D₂₂₃	semidirect

H ≅ G/N

type

1. #20297351

C₂

semidirect

2. #20297334

C₂

semidirect

3. #20297330

C₂

semidirect

4. #20297324

C₂

semidirect

5. C₂×#16813158

C₂

semidirect

6. #20297311

C₂

semidirect

7. H(5,𝔽₂)×C₂₂₃

C₂

semidirect

8. #16813170

C₂²

semidirect

10. C₂×Dic₄₄₆

C₂²

semidirect

12. #16813160

C₂²

semidirect

13. #16813158

C₂²

semidirect

14. #16813158

C₂²

semidirect

15. #16813157

C₂²

semidirect

16. #16813155

C₂²

semidirect

17. #16813154

C₂²

semidirect

18. C₂×C₂₂₃⋊C₈

C₂²

semidirect

53. C₂₂₃

#577

semidirect

56. H(5,𝔽₂)

D₂₂₃

semidirect

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

subgroup	index
#20297351	2
#20297334	2
#20297330	2
#20297324	2
C₂×#16813158	2
#20297311	2
H(5,𝔽₂)×C₂₂₃	2
#577	223

subgroup

index

H(5,𝔽₂)×C₂₂₃

223

Subgroups — #28384391

Conjugacy classes of subgroups

Complements

Maximal subgroups