Group #27141490 — subgroups

Subgroups — #27141490

Order 12416.

200 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 93 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	#20260693	C₂	1	1	yes	maximal, characteristic
2	#20260691	C₂	1	2	yes	maximal
3	#20260493	C₂	1	1	yes	maximal, characteristic
4	#20260491	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095495	C₂²	1	2	yes	—
7	C₂²×#11715895	C₂²	1	1	yes	characteristic
8	#15690769	C₂²	1	1	yes	characteristic
9	#15690721	C₄	1	4	yes	complemented
10	#15690720	C₄	1	2	yes	complemented
11	C₄×#11715895	C₄	1	4	yes	complemented
12	C₈×Dic₉₇	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
14	#117×C₉₇	C₂²	1	2	yes	—
15	#13095495	—	2	2	no	—
16	#13095495	—	2	2	no	—
17	C₂×#11715895	C₂×C₄	1	4	yes	—
18	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
19	C₂×#11715895	C₂×C₄	1	2	yes	—
20	C₂×#11715895	C₂×C₄	1	1	yes	characteristic
21	C₂×#11715895	C₂×C₄	1	2	yes	—
22	C₄⋊Dic₉₇	—	2	1	no	—
23	C₂²⋊Dic₉₇	—	2	1	no	—
24	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₉₇	—	2	2	no	—
27	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
28	C₄⋊C₄×C₉₇	—	2	2	no	—
29	C₂²⋊C₄×C₉₇	—	2	2	no	—
30	C₄²×C₉₇	C₂×C₄	1	4	yes	—
31	C₂³×C₉₇	#41	1	1	yes	characteristic
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	#41	1	2	yes	—
36	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
37	C₂×Dic₉₇	—	2	2	no	—
38	C₈×C₉₇	—	2	2	no	—
39	#11715895	—	2	4	no	—
40	#11715895	C₄²	1	1	yes	characteristic
41	#11715895	C₄²	1	1	yes	characteristic
42	#11715895	C₄²	1	2	yes	—
43	C₂²×C₉₇	—	4	1	no	—
44	C₂²×C₉₇	#117	1	2	yes	—
45	C₂²×C₉₇	#117	1	1	yes	characteristic
46	C₄×C₉₇	—	4	4	no	—
47	C₄×C₉₇	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₉₇	#117	1	1	yes	characteristic
49	C₄×C₉₇	#117	1	2	yes	—
50	Dic₉₇	—	4	2	no	—
51	C₂×C₉₇	—	2	1	no	—
52	C₂×C₉₇	—	2	2	no	—
53	C₂×C₉₇	#376	1	1	yes	characteristic
54	#1747	—	97	1	no	maximal, Sylow
55	C₉₇	#1747	1	1	yes	characteristic, complemented, Sylow
56	#405	—	97	1	no	—
57	#405	—	97	2	no	—
58	#338	—	97	1	no	—
59	H(3,ℤ/4)	—	97	2	no	—
60	H(3,ℤ/4)	D₉₇	1	1	yes	characteristic
61	C₂×M₁₆	—	97	2	no	—
62	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
63	C₂²×C₈	—	97	1	no	—
64	#117	—	97	1	no	—
65	#117	C₂×D₉₇	1	2	yes	—
66	#97	—	97	2	no	—
67	#97	—	97	4	no	—
68	C₄×C₈	—	97	2	no	—
69	C₄×C₈	—	97	4	no	—
70	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
71	M₁₆	—	2	2	no	—
72	M₁₆	—	194	2	no	—
73	M₁₆	—	194	2	no	—
74	C₂×C₈	—	97	1	no	—
75	C₂×C₈	—	97	2	no	—
76	C₂×C₈	—	97	1	no	—
77	C₂×C₈	—	97	2	no	—
78	C₂×C₈	—	97	4	no	—
79	C₂×C₈	C₄×D₉₇	1	2	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	194	1	no	—
82	C₂²⋊C₄	—	2	2	no	—
83	C₂²⋊C₄	—	194	1	no	—
84	C₄²	—	97	2	no	—
85	C₄²	C₄×D₉₇	1	4	yes	—
86	C₂³	#13095510	1	1	yes	characteristic
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	194	2	no	—
89	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
91	C₂×C₄	#13095510	1	2	yes	—
92	C₂×C₄	#13095512	1	1	yes	characteristic
93	C₈	—	2	2	no	—
94	C₈	—	97	1	no	—
95	C₈	—	97	1	no	—
96	C₈	—	97	2	no	—
97	C₈	—	194	4	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#15690766	1	2	yes	—
100	C₂²	#15690768	1	1	yes	characteristic
101	C₄	—	4	4	no	—
102	C₄	—	388	2	no	—
103	C₄	C₄²×D₉₇	1	1	yes	characteristic
104	C₄	#15690766	1	1	yes	characteristic
105	C₄	#15690768	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	—	2	1	no	—
108	C₂	#20260580	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260693

C₂

yes

maximal, characteristic

#20260691

C₂

yes

maximal

#20260493

C₂

yes

maximal, characteristic

#20260491

C₂

yes

maximal

H(3,ℤ/4)×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

C₂²×#11715895

C₂²

yes

characteristic

#15690769

C₂²

yes

characteristic

#15690721

C₄

yes

complemented

#15690720

C₄

yes

complemented

C₄×#11715895

C₄

yes

complemented

C₈×Dic₉₇

C₄

yes

complemented

C₂×M₁₆×C₉₇

C₂²

yes

characteristic

#117×C₉₇

C₂²

yes

—

#13095495

—

#13095495

—

C₂×#11715895

C₂×C₄

yes

—

C₂×#11715895

C₂×C₄

yes

characteristic

C₂×#11715895

C₂×C₄

yes

—

C₂×#11715895

C₂×C₄

yes

characteristic

C₂×#11715895

C₂×C₄

yes

—

C₄⋊Dic₉₇

—

C₂²⋊Dic₉₇

—

C₄×Dic₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₄⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

—

C₂³×C₉₇

#41

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

#41

yes

characteristic

C₂×Dic₉₇

—

C₈×C₉₇

—

#11715895

—

#11715895

C₄²

yes

characteristic

#11715895

C₄²

yes

characteristic

#11715895

C₄²

yes

—

C₂²×C₉₇

—

C₂²×C₉₇

#117

yes

—

C₂²×C₉₇

#117

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

C₂×C₄²

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

—

—

—

—

D₉₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

characteristic

C₂²×C₈

—

#117

—

#117

C₂×D₉₇

yes

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₉₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₂²⋊C₄

—

C₂²⋊C₄

—

194

—

C₄²

—

C₄²

C₄×D₉₇

yes

—

C₂³

#13095510

yes

characteristic

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

C₂×C₄

C₂×C₄×D₉₇

yes

—

C₂×C₄

#13095510

yes

—

C₂×C₄

#13095512

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

—

194

—

C₂²

—

C₂²

#15690766

yes

—

100

C₂²

#15690768

yes

characteristic

101

C₄

—

102

C₄

—

388

—

103

C₄

C₄²×D₉₇

yes

characteristic

104

C₄

#15690766

yes

characteristic

105

C₄

#15690768

yes

—

106

C₂

—

107

C₂

—

108

C₂

#20260580

yes

characteristic

	subgroup	G/H
center	C₄	C₄²×D₉₇
commutator	C₄×C₉₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	H(3,ℤ/4)×C₉₇	C₂
socle	C₂×C₉₇	#376
prime radical	C₂³×C₉₇	#41
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	H(3,ℤ/4)	D₉₇
hypocenter	C₉₇	#1747
LCS	G ▷ C₄×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

C₄²×D₉₇

commutator

C₄×C₉₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

H(3,ℤ/4)×C₉₇

C₂

C₂×C₉₇

C₂³×C₉₇

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₄×C₉₇ ▷ C₉₇

C₁ ◁ C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
9. #15690721	C₄	semidirect
10. #15690720	C₄	semidirect
11. C₄×#11715895	C₄	semidirect
12. C₈×Dic₉₇	C₄	semidirect
55. C₉₇	#1747	semidirect

H ≅ G/N

type

9. #15690721

C₄

semidirect

10. #15690720

C₄

semidirect

11. C₄×#11715895

C₄

semidirect

12. C₈×Dic₉₇

C₄

semidirect

55. C₉₇

#1747

semidirect

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

subgroup	index
#20260693	2
#20260691	2
#20260493	2
#20260491	2
H(3,ℤ/4)×C₉₇	2
#1747	97

subgroup

index

H(3,ℤ/4)×C₉₇

Subgroups — #27141490

Conjugacy classes of subgroups

Complements

Maximal subgroups