Group #20260836 — subgroups

Subgroups — #20260836

Order 6208.

120 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 57 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	Q₈×Dic₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₉₇	C₂	1	1	yes	maximal, characteristic
3	#15690744	C₂	1	1	yes	maximal, characteristic, complemented
4	#15690734	C₂	1	1	yes	maximal, characteristic, complemented
5	#15690731	C₂	1	1	yes	maximal, characteristic
6	#15690720	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×SD₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₉₇	—	2	1	no	—
9	C₂×#11715895	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₉₇	—	2	1	no	—
11	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₉₇	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₉₇	—	2	1	no	—
14	C₄×Dic₉₇	—	2	1	no	—
15	C₄×Dic₉₇	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₉₇	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₉₇	C₂²	1	1	yes	characteristic
18	SD₁₆×C₉₇	C₄	1	4	yes	complemented
19	C₂×C₈×C₉₇	C₂²	1	1	yes	characteristic
20	C₂³×C₉₇	—	2	1	no	—
21	Q₈×C₉₇	—	2	1	no	—
22	Q₈×C₉₇	C₂×C₄	1	2	yes	complemented
23	D₄×C₉₇	—	2	1	no	—
24	D₄×C₉₇	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₉₇	—	2	1	no	—
26	C₂×C₄×C₉₇	C₂³	1	1	yes	characteristic
27	C₂×Dic₉₇	—	2	1	no	—
28	C₂×Dic₉₇	—	2	1	no	—
29	C₂×Dic₉₇	—	4	2	no	—
30	C₂×Dic₉₇	D₄	1	1	yes	characteristic
31	C₂×Dic₉₇	D₄	1	1	yes	characteristic
32	C₈×C₉₇	C₂×C₄	1	2	yes	complemented
33	#11715895	—	2	1	no	—
34	C₂²×C₉₇	—	2	2	no	—
35	C₂²×C₉₇	—	4	1	no	—
36	C₂²×C₉₇	—	4	1	no	—
37	C₂²×C₉₇	C₂×D₄	1	1	yes	characteristic
38	C₄×C₉₇	—	2	2	no	—
39	C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
40	C₄×C₉₇	#41	1	1	yes	characteristic
41	Dic₉₇	—	2	1	no	—
42	Dic₉₇	—	2	1	no	—
43	Dic₉₇	—	4	1	no	—
44	Dic₉₇	—	4	1	no	—
45	C₂×C₉₇	—	4	2	no	—
46	C₂×C₉₇	C₄×D₄	1	1	yes	characteristic
47	C₂×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
48	C₂×C₉₇	#137	1	1	yes	characteristic
49	C₉₇	#440	1	1	yes	characteristic, complemented, Sylow
50	#440	—	97	1	no	maximal, Sylow
51	C₂×SD₁₆	D₉₇	1	1	yes	characteristic
52	C₄×Q₈	—	97	1	no	—
53	C₄×D₄	—	97	1	no	—
54	#107	—	97	1	no	—
55	#103	—	97	1	no	—
56	#102	—	97	1	no	—
57	#97	—	97	1	no	—
58	C₂×Q₈	C₂×D₉₇	1	1	yes	characteristic
59	C₂×D₄	C₂×D₉₇	1	1	yes	characteristic
60	C₂²×C₄	—	194	1	no	—
61	SD₁₆	Dic₉₇	1	4	yes	complemented
62	C₂×C₈	—	97	1	no	—
63	C₂×C₈	C₂×D₉₇	1	1	yes	characteristic
64	C₄⋊C₄	—	97	1	no	—
65	C₄⋊C₄	—	97	1	no	—
66	C₄⋊C₄	—	194	1	no	—
67	C₂²⋊C₄	—	194	1	no	—
68	C₄²	—	97	1	no	—
69	C₄²	—	194	1	no	—
70	C₂³	—	2	1	no	—
71	Q₈	—	2	1	no	—
72	Q₈	C₂×Dic₉₇	1	2	yes	complemented
73	D₄	—	2	1	no	—
74	D₄	C₂×Dic₉₇	1	2	yes	—
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	97	1	no	—
77	C₂×C₄	—	97	1	no	—
78	C₂×C₄	—	388	2	no	—
79	C₂×C₄	—	194	1	no	—
80	C₂×C₄	—	194	1	no	—
81	C₂×C₄	C₂²×D₉₇	1	1	yes	characteristic
82	C₈	—	194	1	no	—
83	C₈	C₂×Dic₉₇	1	2	yes	complemented
84	C₂²	—	2	2	no	—
85	C₂²	—	4	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	D₄×D₉₇	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	388	1	no	—
90	C₄	—	388	1	no	—
91	C₄	—	194	1	no	—
92	C₄	—	194	1	no	—
93	C₄	C₂²×Dic₉₇	1	1	yes	characteristic
94	C₄	#13095510	1	1	yes	characteristic
95	C₂	—	4	2	no	—
96	C₂	D₄×Dic₉₇	1	1	yes	characteristic
97	C₂	#15690837	1	1	yes	characteristic
98	C₂	#15690839	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₉₇

C₂

yes

maximal, characteristic, complemented

D₄×Dic₉₇

C₂

yes

maximal, characteristic

#15690744

C₂

yes

maximal, characteristic, complemented

#15690734

C₂

yes

maximal, characteristic, complemented

#15690731

C₂

yes

maximal, characteristic

#15690720

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₇

—

C₂×#11715895

C₂²

yes

characteristic

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

C₂²

yes

characteristic

C₄⋊Dic₉₇

C₂²

yes

characteristic

C₂²⋊Dic₉₇

—

C₄×Dic₉₇

—

C₄×Dic₉₇

C₂²

yes

characteristic

C₂×Q₈×C₉₇

C₂²

yes

characteristic

C₂×D₄×C₉₇

C₂²

yes

characteristic

SD₁₆×C₉₇

C₄

yes

complemented

C₂×C₈×C₉₇

C₂²

yes

characteristic

C₂³×C₉₇

—

Q₈×C₉₇

—

Q₈×C₉₇

C₂×C₄

yes

complemented

D₄×C₉₇

—

D₄×C₉₇

C₂×C₄

yes

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂³

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

D₄

yes

characteristic

C₂×Dic₉₇

D₄

yes

characteristic

C₈×C₉₇

C₂×C₄

yes

complemented

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×D₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

C₂²×C₄

yes

characteristic

C₄×C₉₇

#41

yes

characteristic

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₄×D₄

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₉₇

yes

characteristic, complemented, Sylow

#440

—

maximal, Sylow

C₂×SD₁₆

D₉₇

yes

characteristic

C₄×Q₈

—

C₄×D₄

—

—

—

—

—

C₂×Q₈

C₂×D₉₇

yes

characteristic

C₂×D₄

C₂×D₉₇

yes

characteristic

C₂²×C₄

—

194

—

SD₁₆

Dic₉₇

yes

complemented

C₂×C₈

—

C₂×C₈

C₂×D₉₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₂²⋊C₄

—

194

—

C₄²

—

C₄²

—

194

—

C₂³

—

Q₈

—

Q₈

C₂×Dic₉₇

yes

complemented

D₄

—

D₄

C₂×Dic₉₇

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

388

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

C₂²×D₉₇

yes

characteristic

C₈

—

194

—

C₈

C₂×Dic₉₇

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

D₄×D₉₇

yes

characteristic

C₄

—

C₄

—

388

—

C₄

—

388

—

C₄

—

194

—

C₄

—

194

—

C₄

C₂²×Dic₉₇

yes

characteristic

C₄

#13095510

yes

characteristic

C₂

—

C₂

D₄×Dic₉₇

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	D₄×D₉₇
commutator	C₄×C₉₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₉₇
Fitting	C₂×SD₁₆×C₉₇	C₂
socle	C₂²×C₉₇	C₂×D₄
hypercenter	C₂×SD₁₆	D₉₇
hypocenter	C₉₇	#440
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

subgroup

G/H

center

C₂²

D₄×D₉₇

commutator

C₄×C₉₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×D₉₇

Fitting

C₂×SD₁₆×C₉₇

C₂

socle

C₂²×C₉₇

C₂×D₄

hypercenter

C₂×SD₁₆

D₉₇

hypocenter

C₉₇

#440

LCS

G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

N	H ≅ G/N	type
1. Q₈×Dic₉₇	C₂	semidirect
3. #15690744	C₂	semidirect
4. #15690734	C₂	semidirect
6. #15690720	C₂	semidirect
18. SD₁₆×C₉₇	C₄	semidirect
22. Q₈×C₉₇	C₂×C₄	semidirect
32. C₈×C₉₇	C₂×C₄	semidirect
49. C₉₇	#440	semidirect
61. SD₁₆	Dic₉₇	semidirect
72. Q₈	C₂×Dic₉₇	semidirect
83. C₈	C₂×Dic₉₇	semidirect

H ≅ G/N

type

1. Q₈×Dic₉₇

C₂

semidirect

3. #15690744

C₂

semidirect

4. #15690734

C₂

semidirect

6. #15690720

C₂

semidirect

18. SD₁₆×C₉₇

C₄

semidirect

22. Q₈×C₉₇

C₂×C₄

semidirect

32. C₈×C₉₇

C₂×C₄

semidirect

49. C₉₇

#440

semidirect

61. SD₁₆

Dic₉₇

semidirect

72. Q₈

C₂×Dic₉₇

semidirect

83. C₈

C₂×Dic₉₇

semidirect

Subgroups — #20260836

Conjugacy classes of subgroups

Complements