Group #27141869 — subgroups

Subgroups — #27141869

Order 12416.

224 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 101 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	#382×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	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
11	#95×C₉₇	—	2	2	no	—
12	C₂²×Dic₉₇	—	2	2	no	—
13	C₂×#11715895	—	2	2	no	—
14	C₂×#11715895	—	2	2	no	—
15	C₂×#11715895	C₂×C₄	1	4	yes	—
16	C₄⋊Dic₉₇	—	2	4	no	—
17	C₄⋊Dic₉₇	—	4	4	no	—
18	C₂²×C₄×C₉₇	—	2	2	no	—
19	C₂²×C₄×C₉₇	D₄	1	2	yes	—
20	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₉₇	—	2	4	no	—
22	C₄²×C₉₇	C₂×C₄	1	2	yes	—
23	C₄²×C₉₇	Q₈	1	2	yes	—
24	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
25	C₂×C₄×C₉₇	—	2	2	no	—
26	C₂×C₄×C₉₇	—	2	2	no	—
27	C₂×C₄×C₉₇	—	4	2	no	—
28	C₂×C₄×C₉₇	—	4	2	no	—
29	C₂×C₄×C₉₇	—	4	2	no	—
30	C₂×C₄×C₉₇	C₄⋊C₄	1	4	yes	—
31	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₉₇	#41	1	2	yes	—
34	C₂×C₄×C₉₇	#41	1	2	yes	—
35	C₂×Dic₉₇	—	4	2	no	—
36	C₂×Dic₉₇	—	4	4	no	—
37	#11715895	—	2	4	no	—
38	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₉₇	#124	1	1	yes	characteristic
40	C₂²×C₉₇	#124	1	1	yes	characteristic
41	C₂²×C₉₇	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₉₇	#137	1	2	yes	—
43	C₄×C₉₇	—	2	4	no	—
44	C₄×C₉₇	—	8	2	no	—
45	C₄×C₉₇	#117	1	2	yes	—
46	C₄×C₉₇	#125	1	2	yes	—
47	Dic₉₇	—	8	2	no	—
48	C₂×C₉₇	#382	1	1	yes	characteristic
49	C₂×C₉₇	#428	1	2	yes	—
50	C₂×C₉₇	#489	1	2	yes	—
51	C₂×C₉₇	#490	1	1	yes	characteristic
52	C₂×C₉₇	#491	1	1	yes	characteristic
53	#1836	—	97	1	no	maximal, Sylow
54	C₉₇	#1836	1	1	yes	characteristic, complemented, Sylow
55	C₂×#97	—	97	1	no	—
56	#384	—	97	1	no	—
57	#382	D₉₇	1	1	yes	characteristic
58	#340	—	97	4	no	—
59	C₂²×C₈	—	97	2	no	—
60	C₂×C₄⋊C₄	—	97	2	no	—
61	C₂×C₄⋊C₄	—	194	2	no	—
62	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
63	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
64	#97	—	97	4	no	—
65	#95	—	2	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	194	2	no	—
68	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
69	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
70	C₂×C₈	—	97	4	no	—
71	C₂×C₈	—	194	2	no	—
72	C₂×C₈	—	194	2	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	388	4	no	—
75	C₄⋊C₄	—	194	4	no	—
76	C₄²	Dic₁₉₄	1	2	yes	—
77	C₄²	C₄×D₉₇	1	2	yes	—
78	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	388	2	no	—
85	C₂×C₄	—	388	4	no	—
86	C₂×C₄	#13095490	1	4	yes	—
87	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
89	C₂×C₄	#13095510	1	2	yes	—
90	C₂×C₄	#13095512	1	2	yes	—
91	C₈	—	194	4	no	—
92	C₂²	C₂×#13095490	1	1	yes	characteristic
93	C₂²	#15690797	1	1	yes	characteristic
94	C₂²	#15690799	1	1	yes	characteristic
95	C₂²	#15690835	1	2	yes	—
96	C₂²	#15690840	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	2	no	—
99	C₄	—	776	2	no	—
100	C₄	#15690768	1	2	yes	—
101	C₄	#15690801	1	2	yes	—
102	C₂	#20260600	1	1	yes	characteristic
103	C₂	#20260778	1	2	yes	—
104	C₂	#20261094	1	2	yes	—
105	C₂	#20261101	1	1	yes	characteristic
106	C₂	#20261105	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690721

C₂

yes

maximal, characteristic

#20260605

C₂

yes

maximal, characteristic

#20260497

C₂

yes

maximal

#382×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₂×C₄⋊Dic₉₇

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

#15690721

C₄

yes

complemented

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#95×C₉₇

—

C₂²×Dic₉₇

—

C₂×#11715895

—

C₂×#11715895

—

C₂×#11715895

C₂×C₄

yes

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

Q₈

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

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

#11715895

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

—

C₂²×C₉₇

#137

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#117

yes

—

C₄×C₉₇

#125

yes

—

Dic₉₇

—

C₂×C₉₇

#382

yes

characteristic

C₂×C₉₇

#428

yes

—

C₂×C₉₇

#489

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×#97

—

#384

—

#382

D₉₇

yes

characteristic

#340

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#97

—

#95

—

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

—

388

—

C₄⋊C₄

—

194

—

C₄²

Dic₁₉₄

yes

—

C₄²

C₄×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

#13095490

yes

—

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

C₂×C₄

#13095510

yes

—

C₂×C₄

#13095512

yes

—

C₈

—

194

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#15690840

yes

—

C₄

—

C₄

—

C₄

—

776

—

100

C₄

#15690768

yes

—

101

C₄

#15690801

yes

—

102

C₂

#20260600

yes

characteristic

103

C₂

#20260778

yes

—

104

C₂

#20261094

yes

—

105

C₂

#20261101

yes

characteristic

106

C₂

#20261105

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#382×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#382	D₉₇
hypocenter	C₉₇	#1836
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#382×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 ◁ #382

N	H ≅ G/N	type
8. #15690721	C₄	semidirect
54. C₉₇	#1836	semidirect

H ≅ G/N

type

8. #15690721

C₄

semidirect

54. C₉₇

#1836

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
#382×C₉₇	2
#1836	97

subgroup

index

#382×C₉₇

Subgroups — #27141869

Conjugacy classes of subgroups

Complements

Maximal subgroups