Group #27141641 — subgroups

Subgroups — #27141641

Order 12416.

220 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂×#15690745	C₂	1	2	yes	maximal
2	M₁₆×Dic₉₇	C₂	1	2	yes	maximal
3	C₂×#15690720	C₂	1	1	yes	maximal, characteristic
4	#20260500	C₂	1	1	yes	maximal, characteristic
5	#341×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095495	C₂²	1	2	yes	—
7	C₂²×#11715895	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
9	#15690745	C₄	1	8	yes	complemented
10	#15690720	—	2	2	no	—
11	#15690720	C₄	1	4	yes	—
12	C₈×Dic₉₇	—	2	2	no	—
13	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₉₇	Q₈	1	2	yes	—
16	#13095495	—	2	2	no	—
17	#13095495	C₂×C₄	1	4	yes	—
18	C₂×#11715895	—	2	2	no	—
19	C₂×#11715895	C₂×C₄	1	2	yes	—
20	C₂×#11715895	C₂×C₄	1	4	yes	—
21	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
22	C₄×Dic₉₇	D₄	1	2	yes	—
23	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
24	M₁₆×C₉₇	—	2	2	no	—
25	M₁₆×C₉₇	C₂×C₄	1	4	yes	—
26	C₂×C₈×C₉₇	—	2	2	no	—
27	C₂×C₈×C₉₇	—	2	2	no	—
28	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
29	C₂³×C₉₇	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
32	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₉₇	#41	1	2	yes	—
34	C₂×Dic₉₇	—	4	2	no	—
35	C₂×Dic₉₇	C₄⋊C₄	1	4	yes	—
36	C₈×C₉₇	—	2	2	no	—
37	C₈×C₉₇	—	2	4	no	—
38	#11715895	—	2	4	no	—
39	#11715895	C₄²	1	4	yes	—
40	C₂²×C₉₇	—	2	2	no	—
41	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₉₇	C₄×Q₈	1	2	yes	—
43	C₄×C₉₇	C₂×C₄²	1	1	yes	characteristic
44	C₄×C₉₇	#117	1	1	yes	characteristic
45	C₄×C₉₇	C₄×D₄	1	2	yes	—
46	Dic₉₇	—	2	4	no	—
47	C₂×C₉₇	—	2	2	no	—
48	C₂×C₉₇	C₄×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₉₇	#430	1	2	yes	—
50	#1778	—	97	1	no	maximal, Sylow
51	C₉₇	#1778	1	1	yes	characteristic, complemented, Sylow
52	C₂×#108	—	97	2	no	—
53	C₄×M₁₆	—	97	2	no	—
54	C₂×#97	—	97	1	no	—
55	#341	—	97	1	no	—
56	#341	D₉₇	1	1	yes	characteristic
57	C₂×M₁₆	—	97	2	no	—
58	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
59	C₂²×C₈	—	97	1	no	—
60	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
61	C₂×C₄²	—	97	1	no	—
62	#108	—	97	8	no	—
63	#97	—	97	4	no	—
64	#97	—	194	2	no	—
65	C₄×C₈	—	194	2	no	—
66	C₂²×C₄	—	97	2	no	—
67	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	97	4	no	—
70	M₁₆	—	194	2	no	—
71	M₁₆	C₄×D₉₇	1	4	yes	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	97	2	no	—
75	C₂×C₈	—	97	4	no	—
76	C₂×C₈	—	194	2	no	—
77	C₂×C₈	C₄×D₉₇	1	2	yes	—
78	C₄²	—	97	2	no	—
79	C₄²	—	97	2	no	—
80	C₂³	Q₈×D₉₇	1	1	yes	characteristic
81	C₂×C₄	—	97	4	no	—
82	C₂×C₄	—	388	2	no	—
83	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
84	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
85	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
86	C₂×C₄	#13095510	1	2	yes	—
87	C₈	—	2	2	no	—
88	C₈	—	2	4	no	—
89	C₈	—	97	4	no	—
90	C₈	—	194	4	no	—
91	C₂²	—	2	2	no	—
92	C₂²	C₄⋊C₄×D₉₇	1	1	yes	characteristic
93	C₂²	#15690775	1	2	yes	—
94	C₄	—	194	4	no	—
95	C₄	C₄²×D₉₇	1	1	yes	characteristic
96	C₄	#15690766	1	1	yes	characteristic
97	C₄	#15690770	1	2	yes	—
98	C₂	—	2	2	no	—
99	C₂	#20260586	1	1	yes	characteristic
100	C₂	#20260782	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690745

C₂

yes

maximal

M₁₆×Dic₉₇

C₂

yes

maximal

C₂×#15690720

C₂

yes

maximal, characteristic

#20260500

C₂

yes

maximal, characteristic

#341×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

C₂²×#11715895

C₂²

yes

characteristic

C₂×C₄×Dic₉₇

C₂²

yes

characteristic

#15690745

C₄

yes

complemented

#15690720

—

#15690720

C₄

yes

—

C₈×Dic₉₇

—

C₂×M₁₆×C₉₇

C₂²

yes

—

C₂²×C₈×C₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

Q₈

yes

—

#13095495

—

#13095495

C₂×C₄

yes

—

C₂×#11715895

—

C₂×#11715895

C₂×C₄

yes

—

C₂×#11715895

C₂×C₄

yes

—

C₄×Dic₉₇

C₂×C₄

yes

—

C₄×Dic₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

M₁₆×C₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₂³×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₄⋊C₄

yes

—

C₈×C₉₇

—

C₈×C₉₇

—

#11715895

—

#11715895

C₄²

yes

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

C₄×Q₈

yes

—

C₄×C₉₇

C₂×C₄²

yes

characteristic

C₄×C₉₇

#117

yes

characteristic

C₄×C₉₇

C₄×D₄

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₉₇

#430

yes

—

#1778

—

maximal, Sylow

C₉₇

#1778

yes

characteristic, complemented, Sylow

C₂×#108

—

C₄×M₁₆

—

C₂×#97

—

#341

—

#341

D₉₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₉₇

yes

characteristic

C₂×C₄²

—

#108

—

#97

—

#97

—

194

—

C₄×C₈

—

194

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

194

—

M₁₆

C₄×D₉₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

C₄×D₉₇

yes

—

C₄²

—

C₄²

—

C₂³

Q₈×D₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

388

—

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

C₂×C₄

C₂×C₄×D₉₇

yes

—

C₂×C₄

D₄×D₉₇

yes

characteristic

C₂×C₄

#13095510

yes

—

C₈

—

C₈

—

C₈

—

C₈

—

194

—

C₂²

—

C₂²

C₄⋊C₄×D₉₇

yes

characteristic

C₂²

#15690775

yes

—

C₄

—

194

—

C₄

C₄²×D₉₇

yes

C₄

yes

C₄

yes

—

C₂

—

C₂

#20260586

yes

characteristic

100

C₂

#20260782

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₉₇
commutator	C₄×C₉₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#341×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₄⋊C₄
prime radical	C₂³×C₉₇	C₂×Q₈
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#341	D₉₇
hypocenter	C₉₇	#1778
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₉₇

commutator

C₄×C₉₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#341×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₄⋊C₄

prime radical

C₂³×C₉₇

C₂×Q₈

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
9. #15690745	C₄	semidirect
51. C₉₇	#1778	semidirect

H ≅ G/N

type

9. #15690745

C₄

semidirect

51. C₉₇

#1778

semidirect

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

subgroup	index
C₂×#15690745	2
M₁₆×Dic₉₇	2
C₂×#15690720	2
#20260500	2
#341×C₉₇	2
#1778	97

subgroup

index

M₁₆×Dic₉₇

#341×C₉₇

Subgroups — #27141641

Conjugacy classes of subgroups

Complements

Maximal subgroups