Group #27139829 — subgroups

Subgroups — #27139829

Order 12416.

128 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 73 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	C₂²×#13095486	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690746	C₂	1	1	yes	maximal, characteristic
3	C₂×#108×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095495	—	2	1	no	—
5	C₂×#13095486	—	2	1	no	—
6	C₂×#13095486	—	2	1	no	—
7	C₂×#13095486	C₄	1	2	yes	—
8	#15690746	—	2	1	no	—
9	#15690746	C₄	1	2	yes	—
10	C₂×M₁₆×C₉₇	—	2	1	no	—
11	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
12	#108×C₉₇	—	2	1	no	—
13	#108×C₉₇	C₄	1	2	yes	—
14	#13095495	—	2	2	no	—
15	#13095495	—	4	1	no	—
16	C₂×#11715895	—	4	1	no	—
17	#13095486	—	2	2	no	—
18	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
19	M₁₆×C₉₇	—	2	2	no	—
20	M₁₆×C₉₇	—	4	1	no	—
21	C₂×C₈×C₉₇	—	4	1	no	—
22	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
27	C₂×C₈×C₉₇	Q₈	1	1	yes	characteristic
28	C₂³×C₉₇	Q₁₆	1	1	yes	characteristic
29	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
33	C₂×C₄×C₉₇	SD₁₆	1	1	yes	characteristic
34	C₂×C₄×C₉₇	SD₁₆	1	1	yes	characteristic
35	C₈×C₉₇	—	4	2	no	—
36	C₈×C₉₇	C₄²	1	1	yes	characteristic
37	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
38	C₈×C₉₇	C₄⋊C₄	1	1	yes	characteristic
39	C₈×C₉₇	C₄⋊C₄	1	1	yes	characteristic
40	#11715895	—	4	2	no	—
41	C₂²×C₉₇	—	2	2	no	—
42	C₂²×C₉₇	#103	1	1	yes	characteristic
43	C₂²×C₉₇	#103	1	1	yes	characteristic
44	C₂²×C₉₇	#107	1	1	yes	characteristic
45	C₄×C₉₇	#95	1	1	yes	characteristic
46	C₄×C₉₇	#102	1	1	yes	characteristic
47	C₄×C₉₇	#102	1	1	yes	characteristic
48	C₄×C₉₇	#106	1	1	yes	characteristic
49	C₂×C₉₇	—	2	1	no	—
50	C₂×C₉₇	—	2	1	no	—
51	C₂×C₉₇	#340	1	1	yes	characteristic
52	C₂×C₉₇	#368	1	2	yes	—
53	#1381	—	97	1	no	maximal, Sylow
54	C₉₇	#1381	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₁₆	—	97	1	no	—
56	C₂×#108	—	97	1	no	—
57	C₂×#108	D₉₇	1	1	yes	characteristic
58	C₂×M₁₆	—	2	1	no	—
59	C₂×M₁₆	—	194	1	no	—
60	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
61	C₂×C₁₆	—	97	2	no	—
62	C₂×C₁₆	—	194	1	no	—
63	C₂×C₁₆	—	194	1	no	—
64	#108	—	2	1	no	—
65	#108	—	97	2	no	—
66	#108	—	194	1	no	—
67	#108	Dic₉₇	1	2	yes	—
68	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	4	1	no	—
71	M₁₆	—	388	1	no	—
72	M₁₆	—	194	2	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	—	388	1	no	—
75	C₂×C₈	Dic₁₉₄	1	1	yes	characteristic
76	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
77	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
78	C₂×C₈	D₃₈₈	1	1	yes	characteristic
79	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
80	C₂×C₈	C₂²⋊D₉₇	1	1	yes	characteristic
81	C₁₆	—	194	2	no	—
82	C₂³	#13095501	1	1	yes	characteristic
83	C₂×C₄	#13095488	1	1	yes	characteristic
84	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
85	C₂×C₄	#13095490	1	1	yes	characteristic
86	C₂×C₄	#13095496	1	1	yes	characteristic
87	C₂×C₄	#13095498	1	1	yes	characteristic
88	C₂×C₄	#13095499	1	1	yes	characteristic
89	C₈	—	4	2	no	—
90	C₈	—	388	2	no	—
91	C₈	C₄×Dic₉₇	1	1	yes	characteristic
92	C₈	#13095488	1	1	yes	characteristic
93	C₈	#13095490	1	1	yes	characteristic
94	C₈	C₄⋊Dic₉₇	1	1	yes	characteristic
95	C₂²	—	2	2	no	—
96	C₂²	#15690733	1	1	yes	characteristic
97	C₂²	#15690734	1	1	yes	characteristic
98	C₂²	#15690743	1	1	yes	characteristic
99	C₄	#15690717	1	1	yes	characteristic
100	C₄	#15690730	1	1	yes	characteristic
101	C₄	#15690731	1	1	yes	characteristic
102	C₄	#15690741	1	1	yes	characteristic
103	C₂	—	2	1	no	—
104	C₂	—	2	1	no	—
105	C₂	#20260497	1	1	yes	characteristic
106	C₂	#20260564	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13095486

C₂

yes

maximal, characteristic

C₂×#15690746

C₂

yes

maximal, characteristic

C₂×#108×C₉₇

C₂

yes

maximal, characteristic

—

—

—

C₄

yes

—

#15690746

—

#15690746

C₄

yes

—

C₂×M₁₆×C₉₇

—

C₂²×C₈×C₉₇

C₂²

yes

characteristic

#108×C₉₇

—

#108×C₉₇

C₄

yes

—

—

—

—

—

C₂²×C₄×C₉₇

D₄

yes

characteristic

M₁₆×C₉₇

—

M₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

characteristic

C₂×C₈×C₉₇

C₂×C₄

yes

characteristic

C₂×C₈×C₉₇

C₂×C₄

yes

characteristic

C₂×C₈×C₉₇

D₄

yes

characteristic

C₂×C₈×C₉₇

D₄

yes

characteristic

C₂×C₈×C₉₇

Q₈

yes

characteristic

C₂³×C₉₇

Q₁₆

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉₇

D₈

yes

characteristic

C₂×C₄×C₉₇

SD₁₆

yes

characteristic

C₂×C₄×C₉₇

SD₁₆

yes

characteristic

C₈×C₉₇

—

C₈×C₉₇

C₄²

yes

characteristic

C₈×C₉₇

C₂²⋊C₄

yes

characteristic

C₈×C₉₇

C₄⋊C₄

yes

characteristic

C₈×C₉₇

C₄⋊C₄

yes

characteristic

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#340

yes

characteristic

C₂×C₉₇

#368

yes

—

#1381

—

maximal, Sylow

C₉₇

#1381

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×#108

—

C₂×#108

D₉₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

194

—

C₂²×C₈

C₂×D₉₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

194

—

C₂×C₁₆

—

194

—

#108

—

#108

—

#108

—

194

—

#108

Dic₉₇

yes

—

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

388

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

388

—

C₂×C₈

Dic₁₉₄

yes

characteristic

C₂×C₈

C₄×D₉₇

yes

characteristic

C₂×C₈

C₄×D₉₇

yes

characteristic

C₂×C₈

D₃₈₈

yes

characteristic

C₂×C₈

C₂×Dic₉₇

yes

characteristic

C₂×C₈

C₂²⋊D₉₇

yes

characteristic

C₁₆

—

194

—

C₂³

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₉₇

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

388

—

C₈

C₄×Dic₉₇

yes

C₈

yes

C₈

yes

C₈

C₄⋊Dic₉₇

yes

characteristic

C₂²

—

C₂²

yes

C₂²

yes

C₂²

yes

C₄

yes

100

C₄

#15690730

yes

characteristic

101

C₄

#15690731

yes

characteristic

102

C₄

#15690741

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

#20260497

yes

characteristic

106

C₂

#20260564

yes

—

	subgroup	G/H
center	C₂×C₄	#13095496
commutator	C₈×C₉₇	C₄²
Frattini	C₂²×C₈	C₂×D₉₇
Fitting	C₂×#108×C₉₇	C₂
socle	C₂²×C₉₇	#107
prime radical	C₂³×C₉₇	Q₁₆
cosocle	C₂²×C₈×C₉₇	C₂²
hypercenter	C₂×#108	D₉₇
hypocenter	C₉₇	#1381
LCS	G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

#13095496

commutator

C₈×C₉₇

C₄²

Frattini

C₂²×C₈

C₂×D₉₇

Fitting

C₂×#108×C₉₇

C₂

socle

C₂²×C₉₇

#107

prime radical

C₂³×C₉₇

Q₁₆

cosocle

C₂²×C₈×C₉₇

C₂²

D₉₇

C₉₇

G ▷ C₈×C₉₇ ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#108

N	H ≅ G/N	type
54. C₉₇	#1381	semidirect

H ≅ G/N

type

54. C₉₇

#1381

semidirect

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

subgroup	index
C₂²×#13095486	2
C₂×#15690746	2
C₂×#108×C₉₇	2
#1381	97

subgroup

index

C₂²×#13095486

C₂×#15690746

C₂×#108×C₉₇

#1381

Subgroups — #27139829

Conjugacy classes of subgroups

Complements

Maximal subgroups