Group #27142706 — subgroups

Subgroups — #27142706

Order 12416.

260 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 113 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×#15690721	C₂	1	1	yes	maximal, characteristic
2	#20260605	C₂	1	1	yes	maximal, characteristic
3	#20260497	C₂	1	4	yes	maximal
4	C₂×#125×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11715895	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₉₇	—	2	2	no	—
7	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
8	#15690721	—	2	2	no	—
9	#125×C₉₇	—	2	2	no	—
10	#125×C₉₇	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₉₇	—	2	2	no	—
12	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₉₇	—	2	2	no	—
15	C₂×#11715895	—	2	2	no	—
16	C₂×#11715895	—	2	2	no	—
17	C₂×#11715895	Q₈	1	4	yes	—
18	C₄⋊Dic₉₇	—	2	4	no	—
19	C₄⋊Dic₉₇	—	4	4	no	—
20	C₂²×C₄×C₉₇	—	2	2	no	—
21	C₂²×C₄×C₉₇	D₄	1	2	yes	—
22	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₉₇	—	2	8	no	—
24	C₄⋊C₄×C₉₇	—	2	2	no	—
25	C₄⋊C₄×C₉₇	C₂×C₄	1	4	yes	—
26	C₄²×C₉₇	C₂×C₄	1	2	yes	—
27	C₄²×C₉₇	D₄	1	2	yes	—
28	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₉₇	—	2	2	no	—
30	C₂×C₄×C₉₇	—	2	2	no	—
31	C₂×C₄×C₉₇	—	2	4	no	—
32	C₂×C₄×C₉₇	—	4	2	no	—
33	C₂×C₄×C₉₇	—	4	2	no	—
34	C₂×C₄×C₉₇	C₂²⋊C₄	1	4	yes	—
35	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₉₇	#41	1	2	yes	—
39	C₂×Dic₉₇	—	4	2	no	—
40	C₂×Dic₉₇	—	4	4	no	—
41	#11715895	—	2	4	no	—
42	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂²×C₉₇	#124	1	1	yes	characteristic
44	C₂²×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
45	C₂²×C₉₇	AGL(1,ℤ/8)	1	2	yes	—
46	C₂²×C₉₇	#137	1	2	yes	—
47	C₄×C₉₇	—	2	4	no	—
48	C₄×C₉₇	—	4	4	no	—
49	C₄×C₉₇	C₄×Q₈	1	2	yes	—
50	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
51	Dic₉₇	—	8	2	no	—
52	C₂×C₉₇	#391	1	1	yes	characteristic
53	C₂×C₉₇	#417	1	2	yes	—
54	C₂×C₉₇	#490	1	1	yes	characteristic
55	C₂×C₉₇	#491	1	1	yes	characteristic
56	C₂×C₉₇	#501	1	2	yes	—
57	#1992	—	97	1	no	maximal, Sylow
58	C₉₇	#1992	1	1	yes	characteristic, complemented, Sylow
59	C₂×#125	D₉₇	1	1	yes	characteristic
60	C₂×#97	—	97	1	no	—
61	#384	—	97	1	no	—
62	#340	—	97	4	no	—
63	C₂²×C₈	—	97	2	no	—
64	#125	—	2	2	no	—
65	#125	Dic₉₇	1	4	yes	complemented
66	C₂×C₄⋊C₄	—	2	2	no	—
67	C₂×C₄⋊C₄	—	97	2	no	—
68	C₂×C₄⋊C₄	—	194	2	no	—
69	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
70	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
71	#97	—	194	2	no	—
72	C₂²×C₄	—	2	2	no	—
73	C₂²×C₄	—	194	2	no	—
74	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
75	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
76	C₂×C₈	—	97	4	no	—
77	C₂×C₈	—	194	2	no	—
78	C₂×C₈	—	194	2	no	—
79	C₄⋊C₄	—	2	8	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	388	4	no	—
82	C₄⋊C₄	—	194	4	no	—
83	C₄⋊C₄	C₂×Dic₉₇	1	4	yes	—
84	C₄²	C₂×Dic₉₇	1	2	yes	—
85	C₄²	C₂²⋊D₉₇	1	2	yes	—
86	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	388	2	no	—
93	C₂×C₄	—	388	4	no	—
94	C₂×C₄	C₂²⋊Dic₉₇	1	4	yes	—
95	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
96	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
97	C₂×C₄	Q₈×D₉₇	1	2	yes	—
98	C₂×C₄	#13095512	1	2	yes	—
99	C₈	—	194	4	no	—
100	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
101	C₂²	#15690799	1	1	yes	characteristic
102	C₂²	#15690808	1	1	yes	characteristic
103	C₂²	#15690835	1	2	yes	—
104	C₂²	#15690840	1	2	yes	—
105	C₄	—	2	4	no	—
106	C₄	—	4	4	no	—
107	C₄	—	776	2	no	—
108	C₄	Q₈×Dic₉₇	1	2	yes	—
109	C₄	#15690786	1	2	yes	—
110	C₂	#20260642	1	1	yes	characteristic
111	C₂	#20260736	1	2	yes	—
112	C₂	#20261101	1	1	yes	characteristic
113	C₂	#20261105	1	1	yes	characteristic
114	C₂	#20261145	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690721

C₂

yes

maximal, characteristic

#20260605

C₂

yes

maximal, characteristic

#20260497

C₂

yes

maximal

C₂×#125×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

#15690721

—

#125×C₉₇

—

#125×C₉₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂²×Dic₉₇

—

C₂×#11715895

—

C₂×#11715895

—

C₂×#11715895

Q₈

yes

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

#11715895

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

#124

yes

characteristic

C₂²×C₉₇

C₄⋊Q₈

yes

characteristic

C₂²×C₉₇

AGL(1,ℤ/8)

yes

—

C₂²×C₉₇

#137

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄×Q₈

yes

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

Dic₉₇

—

C₂×C₉₇

#391

yes

characteristic

C₂×C₉₇

#417

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

#1992

—

maximal, Sylow

C₉₇

#1992

yes

characteristic, complemented, Sylow

C₂×#125

D₉₇

yes

characteristic

C₂×#97

—

#384

—

#340

—

C₂²×C₈

—

#125

—

#125

Dic₉₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#97

—

194

—

C₂²×C₄

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

388

—

C₄⋊C₄

—

194

—

C₄⋊C₄

C₂×Dic₉₇

yes

—

C₄²

C₂×Dic₉₇

yes

—

C₄²

C₂²⋊D₉₇

yes

—

C₂³

C₂×C₂²⋊D₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

388

—

C₂×C₄

—

388

—

C₂×C₄

C₂²⋊Dic₉₇

yes

—

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

C₂×C₄

Q₈×D₉₇

yes

—

C₂×C₄

#13095512

yes

—

C₈

—

194

—

100

C₂²

C₂×C₂²⋊Dic₉₇

yes

characteristic

101

C₂²

#15690799

yes

characteristic

102

C₂²

#15690808

yes

characteristic

103

C₂²

#15690835

yes

—

104

C₂²

#15690840

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

776

—

108

C₄

Q₈×Dic₉₇

yes

—

109

C₄

#15690786

yes

—

110

C₂

#20260642

yes

characteristic

111

C₂

#20260736

yes

—

112

C₂

#20261101

yes

characteristic

113

C₂

#20261105

yes

characteristic

114

C₂

#20261145

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×#125×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×#125	D₉₇
hypocenter	C₉₇	#1992
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂×#125×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
10. #125×C₉₇	C₄	semidirect
58. C₉₇	#1992	semidirect
65. #125	Dic₉₇	semidirect

H ≅ G/N

type

10. #125×C₉₇

C₄

semidirect

58. C₉₇

#1992

semidirect

65. #125

Dic₉₇

semidirect

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

subgroup	index
C₂×#15690721	2
#20260605	2
#20260497	2
C₂×#125×C₉₇	2
#1992	97

subgroup

index

C₂×#125×C₉₇

Subgroups — #27142706

Conjugacy classes of subgroups

Complements

Maximal subgroups