Group #27141491 — subgroups

Subgroups — #27141491

Order 12416.

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

normal subgroups only

This group has 200 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	#20260690	C₂	1	2	yes	maximal
2	#20260493	C₂	1	1	yes	maximal, characteristic
3	#20260492	C₂	1	1	yes	maximal, characteristic
4	#20260491	C₂	1	2	yes	maximal
5	#405×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095495	C₂²	1	2	yes	—
7	#15690769	C₂²	1	2	yes	—
8	#15690720	C₄	1	4	yes	complemented
9	C₈×Dic₉₇	C₄	1	4	yes	complemented
10	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
12	#117×C₉₇	C₂²	1	1	yes	characteristic
13	#97×C₉₇	C₄	1	2	yes	complemented
14	C₄×C₈×C₉₇	C₄	1	2	yes	complemented
15	#13095495	—	2	2	no	—
16	#13095495	—	2	2	no	—
17	C₂×#11715895	C₂×C₄	1	4	yes	—
18	C₄⋊Dic₉₇	—	2	2	no	—
19	C₂²⋊Dic₉₇	—	2	2	no	—
20	C₄×Dic₉₇	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
22	M₁₆×C₉₇	—	2	2	no	—
23	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
28	C₄⋊C₄×C₉₇	—	2	1	no	—
29	C₂²⋊C₄×C₉₇	—	2	1	no	—
30	C₄²×C₉₇	C₂×C₄	1	2	yes	—
31	C₂³×C₉₇	#41	1	1	yes	characteristic
32	C₂×C₄×C₉₇	—	2	2	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	1	yes	characteristic
36	C₂×C₄×C₉₇	#41	1	2	yes	—
37	C₂×Dic₉₇	—	2	4	no	—
38	C₈×C₉₇	—	2	2	no	—
39	C₈×C₉₇	C₄²	1	2	yes	—
40	C₈×C₉₇	C₄²	1	1	yes	characteristic
41	C₈×C₉₇	C₄²	1	1	yes	characteristic
42	#11715895	—	2	4	no	—
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	2	no	—
47	C₄×C₉₇	C₂×C₄²	1	1	yes	characteristic
48	C₄×C₉₇	#117	1	2	yes	—
49	C₄×C₉₇	#117	1	1	yes	characteristic
50	Dic₉₇	—	4	4	no	—
51	C₂×C₉₇	—	2	2	no	—
52	C₂×C₉₇	—	2	1	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	2	no	—
57	#405	D₉₇	1	1	yes	characteristic
58	#338	—	97	1	no	—
59	H(3,ℤ/4)	—	97	1	no	—
60	H(3,ℤ/4)	—	97	2	no	—
61	C₂×M₁₆	—	97	2	no	—
62	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
63	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
64	#117	—	97	2	no	—
65	#117	C₂×D₉₇	1	1	yes	characteristic
66	#97	—	97	4	no	—
67	#97	Dic₉₇	1	2	yes	complemented
68	C₄×C₈	—	97	4	no	—
69	C₄×C₈	Dic₉₇	1	2	yes	complemented
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	4	no	—
75	C₂×C₈	C₄×D₉₇	1	2	yes	—
76	C₂×C₈	C₄×D₉₇	1	2	yes	—
77	C₂×C₈	C₂×Dic₉₇	1	2	yes	—
78	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
79	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
80	C₄⋊C₄	—	2	1	no	—
81	C₄⋊C₄	—	194	2	no	—
82	C₂²⋊C₄	—	2	1	no	—
83	C₂²⋊C₄	—	194	2	no	—
84	C₄²	—	97	4	no	—
85	C₄²	C₂×Dic₉₇	1	2	yes	—
86	C₂³	#13095511	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	194	4	no	—
89	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
90	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
91	C₂×C₄	#13095511	1	1	yes	characteristic
92	C₂×C₄	#13095511	1	2	yes	—
93	C₈	—	2	2	no	—
94	C₈	—	194	4	no	—
95	C₈	C₄×Dic₉₇	1	1	yes	characteristic
96	C₈	C₄×Dic₉₇	1	2	yes	—
97	C₈	C₄×Dic₉₇	1	1	yes	characteristic
98	C₂²	—	4	1	no	—
99	C₂²	#15690767	1	2	yes	—
100	C₂²	#15690769	1	1	yes	characteristic
101	C₄	—	4	2	no	—
102	C₄	—	388	4	no	—
103	C₄	C₂×C₄×Dic₉₇	1	1	yes	characteristic
104	C₄	#15690767	1	2	yes	—
105	C₄	#15690769	1	1	yes	characteristic
106	C₂	—	2	1	no	—
107	C₂	—	2	2	no	—
108	C₂	#20260581	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260690

C₂

yes

maximal

#20260493

C₂

yes

maximal, characteristic

#20260492

C₂

yes

maximal, characteristic

#20260491

C₂

yes

maximal

#405×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

#15690769

C₂²

yes

—

#15690720

C₄

yes

complemented

C₈×Dic₉₇

C₄

yes

complemented

C₂×M₁₆×C₉₇

C₂²

yes

characteristic

C₂²×C₈×C₉₇

C₂²

yes

characteristic

#117×C₉₇

C₂²

yes

characteristic

#97×C₉₇

C₄

yes

complemented

C₄×C₈×C₉₇

C₄

yes

—

—

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

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

C₂×C₄

yes

characteristic

C₂×C₈×C₉₇

C₂×C₄

yes

characteristic

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

C₂×C₄×C₉₇

yes

C₂×C₄×C₉₇

yes

—

C₂×Dic₉₇

—

C₈×C₉₇

—

C₈×C₉₇

C₄²

yes

—

C₈×C₉₇

C₄²

yes

characteristic

C₈×C₉₇

C₄²

yes

characteristic

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

#117

yes

—

C₂²×C₉₇

#117

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

C₂×C₄²

yes

characteristic

C₄×C₉₇

#117

yes

—

C₄×C₉₇

#117

yes

characteristic

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

#405

—

#405

D₉₇

yes

—

—

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

characteristic

C₂²×C₈

C₂×D₉₇

yes

characteristic

#117

—

#117

C₂×D₉₇

yes

characteristic

#97

—

#97

Dic₉₇

yes

complemented

C₄×C₈

—

C₄×C₈

Dic₉₇

yes

complemented

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

C₄×D₉₇

yes

—

C₂×C₈

C₄×D₉₇

yes

—

C₂×C₈

C₂×Dic₉₇

yes

—

C₂×C₈

C₂×Dic₉₇

yes

characteristic

C₂×C₈

C₂×Dic₉₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₂²⋊C₄

—

C₂²⋊C₄

—

194

—

C₄²

—

C₄²

C₂×Dic₉₇

yes

—

C₂³

#13095511

yes

characteristic

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

C₂×C₄×D₉₇

yes

—

C₂×C₄

C₂²×Dic₉₇

yes

C₂×C₄

yes

C₂×C₄

yes

—

C₈

—

C₈

—

194

—

C₈

C₄×Dic₉₇

yes

characteristic

C₈

C₄×Dic₉₇

yes

—

C₈

C₄×Dic₉₇

yes

characteristic

C₂²

—

C₂²

#15690767

yes

—

100

C₂²

#15690769

yes

characteristic

101

C₄

—

102

C₄

—

388

—

103

C₄

C₂×C₄×Dic₉₇

yes

characteristic

104

C₄

#15690767

yes

—

105

C₄

#15690769

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

#20260581

yes

characteristic

	subgroup	G/H
center	C₈	C₄×Dic₉₇
commutator	C₄×C₉₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#405×C₉₇	C₂
socle	C₂×C₉₇	#376
prime radical	C₂³×C₉₇	#41
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#405	D₉₇
hypocenter	C₉₇	#1747
LCS	G ▷ C₄×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₈ ◁ #405

subgroup

G/H

center

C₈

C₄×Dic₉₇

commutator

C₄×C₉₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#405×C₉₇

C₂

C₂×C₉₇

C₂³×C₉₇

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₄×C₉₇ ▷ C₉₇

C₁ ◁ C₈ ◁ #405

N	H ≅ G/N	type
8. #15690720	C₄	semidirect
9. C₈×Dic₉₇	C₄	semidirect
13. #97×C₉₇	C₄	semidirect
14. C₄×C₈×C₉₇	C₄	semidirect
55. C₉₇	#1747	semidirect
67. #97	Dic₉₇	semidirect
69. C₄×C₈	Dic₉₇	semidirect

H ≅ G/N

type

8. #15690720

C₄

semidirect

9. C₈×Dic₉₇

C₄

semidirect

13. #97×C₉₇

C₄

semidirect

14. C₄×C₈×C₉₇

C₄

semidirect

55. C₉₇

#1747

semidirect

67. #97

Dic₉₇

semidirect

69. C₄×C₈

Dic₉₇

semidirect

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

subgroup	index
#20260690	2
#20260493	2
#20260492	2
#20260491	2
#405×C₉₇	2
#1747	97

subgroup

index

#405×C₉₇

Subgroups — #27141491

Conjugacy classes of subgroups

Complements

Maximal subgroups