Group #27164829 — subgroups

Subgroups — #27164829

Order 12736.

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	#20264002	C₂	1	1	yes	maximal, characteristic, complemented
2	#20263985	C₂	1	3	yes	maximal, complemented
3	#20263981	C₂	1	3	yes	maximal, complemented
4	#20263975	C₂	1	3	yes	maximal, complemented
5	C₂×#15691874	C₂	1	3	yes	maximal, complemented
6	#20263962	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#15691886	C₂²	1	3	yes	complemented
9	#15691885	—	2	3	no	—
10	C₂×Dic₃₉₈	C₂²	1	3	yes	complemented
11	Q₈×D₁₉₉	—	2	1	no	—
12	#15691876	C₂²	1	3	yes	complemented
13	#15691874	C₂²	1	3	yes	complemented
14	#15691874	C₂²	1	6	yes	complemented
15	#15691873	C₂²	1	1	yes	characteristic, complemented
16	#15691871	C₂²	1	3	yes	complemented
17	#15691870	C₂²	1	3	yes	complemented
18	C₂×C₁₉₉⋊C₈	C₂²	1	3	yes	complemented
19	#41×C₁₉₉	—	2	1	no	—
20	#41×C₁₉₉	C₂²	1	3	yes	—
21	#41×C₁₉₉	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₁₉₉	—	2	3	no	—
23	C₂×D₄×C₁₉₉	C₂²	1	3	yes	—
24	Dic₃₉₈	—	2	3	no	—
25	Dic₃₉₈	C₂³	1	3	yes	—
26	D₇₉₆	C₂³	1	1	yes	characteristic
27	C₂²⋊D₁₉₉	—	2	3	no	—
28	C₂×Dic₁₉₉	—	2	3	no	—
29	C₄×D₁₉₉	—	2	3	no	—
30	C₁₉₉⋊C₈	C₂³	1	1	yes	characteristic
31	C₁₉₉⋊C₈	C₂³	1	3	yes	—
32	C₂³×C₁₉₉	—	2	3	no	—
33	Q₈×C₁₉₉	D₄	1	1	yes	characteristic
34	Q₈×C₁₉₉	C₂³	1	1	yes	characteristic
35	D₄×C₁₉₉	—	2	3	no	—
36	D₄×C₁₉₉	—	2	3	no	—
37	D₄×C₁₉₉	D₄	1	3	yes	—
38	D₄×C₁₉₉	C₂³	1	3	yes	—
39	C₂×C₄×C₁₉₉	—	2	3	no	—
40	C₂×C₄×C₁₉₉	C₂³	1	3	yes	—
41	C₂²×C₁₉₉	—	2	3	no	—
42	C₂²×C₁₉₉	—	8	3	no	—
43	C₂²×C₁₉₉	C₂×D₄	1	3	yes	—
44	C₂×D₁₉₉	—	2	1	no	—
45	C₄×C₁₉₉	—	2	1	no	—
46	C₄×C₁₉₉	C₂×D₄	1	3	yes	—
47	C₄×C₁₉₉	C₂⁴	1	1	yes	characteristic
48	Dic₁₉₉	—	2	3	no	—
49	C₂×C₁₉₉	—	2	3	no	—
50	C₂×C₁₉₉	—	4	3	no	—
51	C₂×C₁₉₉	C₂²×D₄	1	1	yes	characteristic
52	D₁₉₉	—	4	1	no	—
53	C₁₉₉	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	199	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	199	1	no	—
56	H(5,𝔽₂)	D₁₉₉	1	1	yes	characteristic, complemented
57	#137	—	199	3	no	—
58	AGL(1,ℤ/8)	—	199	3	no	—
59	#135	—	199	3	no	—
60	C₂×SD₁₆	—	199	3	no	—
61	#131	—	199	1	no	—
62	#41	—	2	1	no	—
63	#41	—	398	3	no	—
64	#41	—	199	3	no	—
65	#41	C₂×D₁₉₉	1	3	yes	—
66	#41	C₂×D₁₉₉	1	1	yes	characteristic
67	C₂×Q₈	—	398	1	no	—
68	C₂×Q₈	—	199	3	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₁₉₉	1	3	yes	—
71	Q₁₆	—	199	3	no	—
72	SD₁₆	—	199	1	no	—
73	SD₁₆	—	199	3	no	—
74	SD₁₆	—	199	6	no	—
75	D₈	—	199	3	no	—
76	M₁₆	—	199	3	no	—
77	C₂×C₈	—	199	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	398	3	no	—
80	Q₈	—	199	3	no	—
81	Q₈	C₂²⋊D₁₉₉	1	1	yes	characteristic
82	Q₈	C₂²×D₁₉₉	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	398	3	no	—
86	D₄	—	199	1	no	—
87	D₄	C₂²⋊D₁₉₉	1	3	yes	—
88	D₄	C₂²×D₁₉₉	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	398	3	no	—
91	C₂×C₄	—	398	3	no	—
92	C₂×C₄	C₂²×D₁₉₉	1	3	yes	—
93	C₈	—	199	1	no	—
94	C₈	—	199	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	398	1	no	—
98	C₂²	C₂×C₂²⋊D₁₉₉	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	398	3	no	—
101	C₄	C₂×C₂²⋊D₁₉₉	1	3	yes	—
102	C₄	C₂³×D₁₉₉	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	796	1	no	—
106	C₂	C₂²×C₂²⋊D₁₉₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20264002

C₂

yes

maximal, characteristic, complemented

#20263985

C₂

yes

maximal, complemented

#20263981

C₂

yes

maximal, complemented

#20263975

C₂

yes

maximal, complemented

C₂×#15691874

C₂

yes

maximal, complemented

#20263962

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₁₉₉

C₂

yes

maximal, characteristic, complemented

#15691886

C₂²

yes

complemented

#15691885

—

C₂×Dic₃₉₈

C₂²

yes

complemented

Q₈×D₁₉₉

—

#15691876

C₂²

yes

complemented

#15691874

C₂²

yes

complemented

#15691874

C₂²

yes

complemented

#15691873

C₂²

yes

characteristic, complemented

#15691871

C₂²

yes

complemented

#15691870

C₂²

yes

complemented

C₂×C₁₉₉⋊C₈

C₂²

yes

complemented

#41×C₁₉₉

—

#41×C₁₉₉

C₂²

yes

—

#41×C₁₉₉

C₂²

yes

characteristic

C₂×D₄×C₁₉₉

—

C₂×D₄×C₁₉₉

C₂²

yes

—

Dic₃₉₈

—

Dic₃₉₈

C₂³

yes

—

D₇₉₆

C₂³

yes

characteristic

C₂²⋊D₁₉₉

—

C₂×Dic₁₉₉

—

C₄×D₁₉₉

—

C₁₉₉⋊C₈

C₂³

yes

characteristic

C₁₉₉⋊C₈

C₂³

yes

—

C₂³×C₁₉₉

—

Q₈×C₁₉₉

D₄

yes

characteristic

Q₈×C₁₉₉

C₂³

yes

characteristic

D₄×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

D₄

yes

—

D₄×C₁₉₉

C₂³

yes

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂³

yes

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₂×D₄

yes

—

C₂×D₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂×D₄

yes

—

C₄×C₁₉₉

C₂⁴

yes

characteristic

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

C₂²×D₄

yes

characteristic

D₁₉₉

—

C₁₉₉

#577

yes

characteristic, complemented, Sylow

#577

—

199

maximal, Sylow

2₋¹⁺⁴

—

199

—

H(5,𝔽₂)

D₁₉₉

yes

characteristic, complemented

#137

—

199

—

AGL(1,ℤ/8)

—

199

—

#135

—

199

—

C₂×SD₁₆

—

199

—

#131

—

199

—

#41

—

#41

—

398

—

#41

—

199

—

#41

C₂×D₁₉₉

yes

—

#41

C₂×D₁₉₉

yes

characteristic

C₂×Q₈

—

398

—

C₂×Q₈

—

199

—

C₂×D₄

—

C₂×D₄

C₂×D₁₉₉

yes

—

Q₁₆

—

199

—

SD₁₆

—

199

—

SD₁₆

—

199

—

SD₁₆

—

199

—

D₈

—

199

—

M₁₆

—

199

—

C₂×C₈

—

199

—

C₂³

—

Q₈

—

398

—

Q₈

—

199

—

Q₈

C₂²⋊D₁₉₉

yes

characteristic

Q₈

C₂²×D₁₉₉

yes

characteristic

D₄

—

D₄

—

D₄

—

398

—

D₄

—

199

—

D₄

C₂²⋊D₁₉₉

yes

—

D₄

C₂²×D₁₉₉

yes

—

C₂×C₄

—

C₂×C₄

—

398

—

C₂×C₄

—

398

—

C₂×C₄

C₂²×D₁₉₉

yes

—

C₈

—

199

—

C₈

—

199

—

C₂²

—

C₂²

—

C₂²

—

398

—

C₂²

C₂×C₂²⋊D₁₉₉

yes

—

C₄

—

100

C₄

—

398

—

101

C₄

C₂×C₂²⋊D₁₉₉

yes

—

102

C₄

C₂³×D₁₉₉

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

796

—

106

C₂

C₂²×C₂²⋊D₁₉₉

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₁₉₉
commutator, cosocle	C₄×C₁₉₉	C₂⁴
Frattini	C₄	C₂³×D₁₉₉
Fitting	H(5,𝔽₂)×C₁₉₉	C₂
socle	C₂×C₁₉₉	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₁₉₉
hypocenter	C₁₉₉	#577
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₁₉₉

commutator, cosocle

C₄×C₁₉₉

C₂⁴

Frattini

C₄

C₂³×D₁₉₉

Fitting

H(5,𝔽₂)×C₁₉₉

C₂

socle

C₂×C₁₉₉

C₂²×D₄

D₁₉₉

C₁₉₉

G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #20264002	C₂	semidirect
2. #20263985	C₂	semidirect
3. #20263981	C₂	semidirect
4. #20263975	C₂	semidirect
5. C₂×#15691874	C₂	semidirect
6. #20263962	C₂	semidirect
7. H(5,𝔽₂)×C₁₉₉	C₂	semidirect
8. #15691886	C₂²	semidirect
10. C₂×Dic₃₉₈	C₂²	semidirect
12. #15691876	C₂²	semidirect
13. #15691874	C₂²	semidirect
14. #15691874	C₂²	semidirect
15. #15691873	C₂²	semidirect
16. #15691871	C₂²	semidirect
17. #15691870	C₂²	semidirect
18. C₂×C₁₉₉⋊C₈	C₂²	semidirect
53. C₁₉₉	#577	semidirect
56. H(5,𝔽₂)	D₁₉₉	semidirect

H ≅ G/N

type

1. #20264002

C₂

semidirect

2. #20263985

C₂

semidirect

3. #20263981

C₂

semidirect

4. #20263975

C₂

semidirect

5. C₂×#15691874

C₂

semidirect

6. #20263962

C₂

semidirect

7. H(5,𝔽₂)×C₁₉₉

C₂

semidirect

8. #15691886

C₂²

semidirect

10. C₂×Dic₃₉₈

C₂²

semidirect

12. #15691876

C₂²

semidirect

13. #15691874

C₂²

semidirect

14. #15691874

C₂²

semidirect

15. #15691873

C₂²

semidirect

16. #15691871

C₂²

semidirect

17. #15691870

C₂²

semidirect

18. C₂×C₁₉₉⋊C₈

C₂²

semidirect

53. C₁₉₉

#577

semidirect

56. H(5,𝔽₂)

D₁₉₉

semidirect

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

subgroup	index
#20264002	2
#20263985	2
#20263981	2
#20263975	2
C₂×#15691874	2
#20263962	2
H(5,𝔽₂)×C₁₉₉	2
#577	199

subgroup

index

H(5,𝔽₂)×C₁₉₉

199

Subgroups — #27164829

Conjugacy classes of subgroups

Complements

Maximal subgroups