Group #27140474 — subgroups

Subgroups — #27140474

Order 12416.

210 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 85 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	#20260759	C₂	1	2	yes	maximal
2	#20260481	C₂	1	4	yes	maximal, complemented
3	#533×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095495	—	2	2	no	—
5	C₄⋊#11715895	—	2	2	no	—
6	C₄⋊#11715895	C₂²	1	4	yes	—
7	C₄⋊Q₈×C₉₇	C₂²	1	2	yes	—
8	#125×C₉₇	C₄	1	2	yes	—
9	C₂²⋊Q₈×C₉₇	—	2	2	no	—
10	C₄×Q₈×C₉₇	—	2	2	no	—
11	#117×C₉₇	C₄	1	2	yes	—
12	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
13	#13095495	—	4	4	no	—
14	C₂×#11715895	—	2	4	no	—
15	C₂×Q₈×C₉₇	—	2	2	no	—
16	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₉₇	D₄	1	2	yes	—
18	C₄⋊C₄×C₉₇	—	2	4	no	—
19	C₄⋊C₄×C₉₇	—	2	2	no	—
20	C₄⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
21	C₂²⋊C₄×C₉₇	—	2	2	no	—
22	C₄²×C₉₇	—	2	2	no	—
23	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₉₇	D₄	1	2	yes	—
25	C₄²×C₉₇	C₂³	1	1	yes	characteristic
26	C₂³×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
27	Q₈×C₉₇	—	4	2	no	—
28	Q₈×C₉₇	—	4	2	no	—
29	C₂×C₄×C₉₇	—	2	2	no	—
30	C₂×C₄×C₉₇	—	2	2	no	—
31	C₂×C₄×C₉₇	—	2	2	no	—
32	C₂×C₄×C₉₇	—	2	4	no	—
33	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
37	#11715895	—	4	4	no	—
38	C₂²×C₉₇	—	2	1	no	—
39	C₂²×C₉₇	—	2	2	no	—
40	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₄×C₉₇	—	2	1	no	—
42	C₄×C₉₇	—	2	2	no	—
43	C₄×C₉₇	—	4	4	no	—
44	C₄×C₉₇	#101	1	2	yes	—
45	C₄×C₉₇	AGL(1,ℤ/8)	1	2	yes	—
46	C₄×C₉₇	#137	1	2	yes	—
47	C₂×C₉₇	—	4	1	no	—
48	C₂×C₉₇	C₂×#101	1	1	yes	characteristic
49	C₂×C₉₇	#417	1	2	yes	—
50	#1561	—	97	1	no	maximal, Sylow
51	C₉₇	#1561	1	1	yes	characteristic, complemented, Sylow
52	#533	D₉₇	1	1	yes	characteristic
53	#423	—	97	2	no	—
54	#332	—	97	4	no	—
55	C₂×M₁₆	—	194	2	no	—
56	C₄⋊Q₈	C₂×D₉₇	1	2	yes	—
57	#125	Dic₉₇	1	2	yes	—
58	C₂²⋊Q₈	—	2	2	no	—
59	C₄×Q₈	—	2	2	no	—
60	#117	Dic₉₇	1	2	yes	—
61	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
62	C₄⋊C₈	—	97	4	no	—
63	C₄⋊C₈	—	194	2	no	—
64	C₂×Q₈	—	2	2	no	—
65	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
66	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
67	M₁₆	—	388	4	no	—
68	C₂×C₈	—	194	4	no	—
69	C₄⋊C₄	—	2	4	no	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	C₂×Dic₉₇	1	4	yes	—
72	C₂²⋊C₄	—	2	2	no	—
73	C₄²	—	2	2	no	—
74	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
75	C₄²	C₂²⋊D₉₇	1	2	yes	—
76	C₄²	C₂²×D₉₇	1	1	yes	characteristic
77	C₂³	C₂²⋊Dic₉₇	1	1	yes	characteristic
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	C₂²⋊Dic₉₇	1	2	yes	—
85	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
86	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
88	C₈	—	388	4	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	2	2	no	—
91	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	2	2	no	—
94	C₄	—	4	4	no	—
95	C₄	#15690729	1	2	yes	—
96	C₄	#15690835	1	2	yes	—
97	C₄	#15690840	1	2	yes	—
98	C₂	—	4	1	no	—
99	C₂	C₂×#15690729	1	1	yes	characteristic
100	C₂	#20260736	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20260759

C₂

yes

maximal

#20260481

C₂

yes

maximal, complemented

#533×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

—

C₄⋊#11715895

—

C₄⋊#11715895

C₂²

yes

—

C₄⋊Q₈×C₉₇

C₂²

yes

—

#125×C₉₇

C₄

yes

—

C₂²⋊Q₈×C₉₇

—

C₄×Q₈×C₉₇

—

#117×C₉₇

C₄

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#13095495

—

C₂×#11715895

—

C₂×Q₈×C₉₇

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

C₂²⋊C₄

yes

characteristic

Q₈×C₉₇

—

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#101

yes

—

C₄×C₉₇

AGL(1,ℤ/8)

yes

—

C₄×C₉₇

#137

yes

—

C₂×C₉₇

—

C₂×C₉₇

C₂×#101

yes

characteristic

C₂×C₉₇

#417

yes

—

#1561

—

maximal, Sylow

C₉₇

#1561

yes

characteristic, complemented, Sylow

#533

D₉₇

yes

characteristic

#423

—

#332

—

C₂×M₁₆

—

194

—

C₄⋊Q₈

C₂×D₉₇

yes

—

#125

Dic₉₇

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

Dic₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

194

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂²⋊D₉₇

yes

—

M₁₆

—

388

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₉₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×Dic₉₇

yes

characteristic

C₄²

C₂²⋊D₉₇

yes

—

C₄²

C₂²×D₉₇

yes

characteristic

C₂³

C₂²⋊Dic₉₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₉₇

yes

—

C₂×C₄

C₂²⋊Dic₉₇

yes

characteristic

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

C₈

—

388

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊Dic₉₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#15690729

yes

—

C₄

#15690835

yes

—

C₄

#15690840

yes

—

C₂

—

C₂

C₂×#15690729

yes

characteristic

100

C₂

#20260736

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₄²	C₂²×D₉₇
Fitting	#533×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₉₇	C₂²⋊C₄
cosocle	C₄²×C₉₇	C₂³
hypercenter	#533	D₉₇
hypocenter	C₉₇	#1561
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₄²

C₂²×D₉₇

Fitting

#533×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₉₇

C₂²⋊C₄

cosocle

C₄²×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂²×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #20260481	C₂	semidirect
51. C₉₇	#1561	semidirect

H ≅ G/N

type

2. #20260481

C₂

semidirect

51. C₉₇

#1561

semidirect

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

subgroup	index
#20260759	2
#20260481	2
#533×C₉₇	2
#1561	97

subgroup

index

#20260759

#20260481

#533×C₉₇

#1561

Subgroups — #27140474

Conjugacy classes of subgroups

Complements

Maximal subgroups