Group #27139566 — subgroups

Subgroups — #27139566

Order 12416.

152 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 77 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	#20260788	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690721	C₂	1	1	yes	maximal, characteristic
3	#431×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11715895	C₄	1	2	yes	—
5	C₄⋊#11715895	—	2	1	no	—
6	C₂²⋊#11715895	—	2	1	no	—
7	#15690721	—	2	1	no	—
8	#15690721	—	2	1	no	—
9	C₄×#11715895	C₄	1	2	yes	—
10	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₉₇	—	2	1	no	—
12	C₂²⋊C₈×C₉₇	—	2	1	no	—
13	C₄×C₈×C₉₇	C₄	1	2	yes	—
14	C₂×#11715895	—	2	2	no	—
15	C₂×#11715895	—	2	2	no	—
16	C₂×#11715895	—	2	2	no	—
17	C₂×#11715895	C₈	1	4	yes	complemented
18	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₉₇	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₉₇	—	2	2	no	—
22	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₉₇	D₄	1	1	yes	characteristic
25	C₄²×C₉₇	D₄	1	1	yes	characteristic
26	C₂³×C₉₇	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₉₇	—	4	1	no	—
28	C₂×C₄×C₉₇	—	4	1	no	—
29	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₉₇	M₁₆	1	2	yes	—
37	C₈×C₉₇	—	4	2	no	—
38	#11715895	—	2	4	no	—
39	#11715895	—	4	2	no	—
40	C₂²×C₉₇	—	2	1	no	—
41	C₂²×C₉₇	—	2	1	no	—
42	C₂²×C₉₇	#95	1	1	yes	characteristic
43	C₂²×C₉₇	C₄⋊C₈	1	2	yes	—
44	C₄×C₉₇	—	2	1	no	—
45	C₄×C₉₇	—	2	1	no	—
46	C₄×C₉₇	—	2	1	no	—
47	C₄×C₉₇	—	2	1	no	—
48	C₄×C₉₇	C₄×C₈	1	1	yes	characteristic
49	C₄×C₉₇	#97	1	1	yes	characteristic
50	C₄×C₉₇	C₂²⋊C₈	1	2	yes	—
51	C₂×C₉₇	—	2	2	no	—
52	C₂×C₉₇	#336	1	1	yes	characteristic
53	C₂×C₉₇	H(3,ℤ/4)	1	1	yes	characteristic
54	C₂×C₉₇	#338	1	1	yes	characteristic
55	#1275	—	97	1	no	maximal, Sylow
56	C₉₇	#1275	1	1	yes	characteristic, complemented, Sylow
57	#431	—	97	1	no	—
58	#431	D₉₇	1	1	yes	characteristic
59	C₂×#97	—	97	1	no	—
60	C₂²×C₈	—	97	2	no	—
61	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
62	C₄⋊C₈	—	2	1	no	—
63	C₄⋊C₈	—	194	1	no	—
64	C₂²⋊C₈	—	2	1	no	—
65	C₂²⋊C₈	—	194	1	no	—
66	#97	—	194	1	no	—
67	#97	—	194	1	no	—
68	C₄×C₈	—	97	2	no	—
69	C₄×C₈	Dic₉₇	1	2	yes	—
70	C₂²×C₄	Dic₁₉₄	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
72	C₂²×C₄	C₂²⋊D₉₇	1	1	yes	characteristic
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	97	4	no	—
75	C₂×C₈	—	194	2	no	—
76	C₂×C₈	—	194	2	no	—
77	C₂×C₈	—	194	2	no	—
78	C₄²	C₄×D₉₇	1	1	yes	characteristic
79	C₄²	D₃₈₈	1	1	yes	characteristic
80	C₄²	C₂×Dic₉₇	1	1	yes	characteristic
81	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
82	C₂³	#13095490	1	1	yes	characteristic
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	C₄×Dic₉₇	1	1	yes	characteristic
86	C₂×C₄	#13095488	1	1	yes	characteristic
87	C₂×C₄	#13095488	1	1	yes	characteristic
88	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
89	C₂×C₄	#13095490	1	1	yes	characteristic
90	C₂×C₄	C₄⋊Dic₉₇	1	1	yes	characteristic
91	C₂×C₄	C₈×D₉₇	1	2	yes	—
92	C₂×C₄	#13095494	1	2	yes	—
93	C₈	—	4	2	no	—
94	C₈	—	388	2	no	—
95	C₈	—	194	4	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	#15690717	1	1	yes	characteristic
99	C₂²	#15690739	1	2	yes	—
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	C₈×Dic₉₇	1	1	yes	characteristic
105	C₄	#15690720	1	1	yes	characteristic
106	C₄	#15690722	1	2	yes	—
107	C₂	—	2	2	no	—
108	C₂	#20260489	1	1	yes	characteristic
109	C₂	#20260491	1	1	yes	characteristic
110	C₂	#20260493	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260788

C₂

yes

maximal, characteristic

C₂×#15690721

C₂

yes

maximal, characteristic

#431×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₄

yes

—

—

—

—

—

C₄

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

—

C₂²⋊C₈×C₉₇

—

C₄×C₈×C₉₇

C₄

yes

—

—

—

—

C₈

yes

complemented

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

Q₈

yes

characteristic

C₂×C₈×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₄²×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

D₄

yes

characteristic

C₂³×C₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄²

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

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

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×C₈

yes

—

C₂×C₄×C₉₇

M₁₆

yes

—

C₈×C₉₇

—

#11715895

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#95

yes

characteristic

C₂²×C₉₇

C₄⋊C₈

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄×C₈

yes

characteristic

C₄×C₉₇

#97

yes

characteristic

C₄×C₉₇

C₂²⋊C₈

yes

—

C₂×C₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

#431

—

#431

D₉₇

yes

characteristic

C₂×#97

—

C₂²×C₈

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

194

—

C₂²⋊C₈

—

C₂²⋊C₈

—

194

—

#97

—

194

—

#97

—

194

—

C₄×C₈

—

C₄×C₈

Dic₉₇

yes

—

C₂²×C₄

Dic₁₉₄

yes

characteristic

C₂²×C₄

C₄×D₉₇

yes

characteristic

C₂²×C₄

C₂²⋊D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄²

C₄×D₉₇

yes

characteristic

C₄²

D₃₈₈

yes

characteristic

C₄²

C₂×Dic₉₇

yes

characteristic

C₄²

C₂²⋊D₉₇

yes

characteristic

C₂³

#13095490

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₉₇

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₉₇

yes

characteristic

C₂×C₄

#13095490

yes

characteristic

C₂×C₄

C₄⋊Dic₉₇

yes

characteristic

C₂×C₄

C₈×D₉₇

yes

—

C₂×C₄

#13095494

yes

—

C₈

—

C₈

—

388

—

C₈

—

194

—

C₂²

—

C₂²

—

C₂²

#15690717

yes

characteristic

C₂²

#15690739

yes

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

C₈×Dic₉₇

yes

characteristic

105

C₄

#15690720

yes

characteristic

106

C₄

#15690722

yes

—

107

C₂

—

108

C₂

#20260489

yes

characteristic

109

C₂

#20260491

yes

characteristic

110

C₂

#20260493

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₉₇
commutator	C₄×C₉₇	C₄×C₈
Frattini	C₂×C₄²	C₂×D₉₇
Fitting	#431×C₉₇	C₂
socle	C₂²×C₉₇	#95
prime radical	C₂³×C₉₇	C₄⋊C₄
cosocle	C₂×C₄²×C₉₇	C₂²
hypercenter	#431	D₉₇
hypocenter	C₉₇	#1275
LCS	G ▷ C₄×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center

C₂×C₄

C₄×Dic₉₇

commutator

C₄×C₉₇

C₄×C₈

Frattini

C₂×C₄²

C₂×D₉₇

Fitting

#431×C₉₇

C₂

socle

C₂²×C₉₇

#95

prime radical

C₂³×C₉₇

C₄⋊C₄

cosocle

C₂×C₄²×C₉₇

C₂²

D₉₇

C₉₇

G ▷ C₄×C₉₇ ▷ C₉₇

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
17. C₂×#11715895	C₈	semidirect
56. C₉₇	#1275	semidirect

H ≅ G/N

type

17. C₂×#11715895

C₈

semidirect

56. C₉₇

#1275

semidirect

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

subgroup	index
#20260788	2
C₂×#15690721	2
#431×C₉₇	2
#1275	97

subgroup

index

#20260788

C₂×#15690721

#431×C₉₇

#1275

Subgroups — #27139566

Conjugacy classes of subgroups

Complements

Maximal subgroups