Group #27142701 — subgroups

Subgroups — #27142701

Order 12416.

252 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 109 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×#15690802	C₂	1	1	yes	maximal, characteristic
2	#20260601	C₂	1	1	yes	maximal, characteristic
3	#20260498	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
5	#15690802	—	2	2	no	—
6	#15690802	C₄	1	4	yes	complemented
7	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
9	C₂×#13095490	—	2	2	no	—
10	#15690717	—	2	2	no	—
11	C₂²×C₈×C₉₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₉₇	—	2	2	no	—
14	C₂²×Dic₉₇	—	2	2	no	—
15	C₂²×Dic₉₇	—	2	2	no	—
16	C₄⋊Dic₉₇	—	2	2	no	—
17	C₄⋊Dic₉₇	—	2	4	no	—
18	C₄⋊Dic₉₇	C₂×C₄	1	4	yes	—
19	#13095490	—	2	8	no	—
20	C₂²×C₄×C₉₇	D₄	1	2	yes	—
21	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₉₇	—	2	2	no	—
23	C₂×C₈×C₉₇	—	2	2	no	—
24	C₂×C₈×C₉₇	Q₈	1	4	yes	—
25	C₄²×C₉₇	C₂×C₄	1	2	yes	—
26	C₄²×C₉₇	D₄	1	2	yes	—
27	C₂³×C₉₇	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₉₇	—	2	4	no	—
29	C₂×C₄×C₉₇	C₂²⋊C₄	1	4	yes	—
30	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
33	C₂×C₄×C₉₇	#41	1	2	yes	—
34	C₂×Dic₉₇	—	2	4	no	—
35	C₂×Dic₉₇	—	4	4	no	—
36	C₂×Dic₉₇	—	4	2	no	—
37	C₂×Dic₉₇	—	4	4	no	—
38	C₈×C₉₇	—	2	4	no	—
39	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₉₇	#124	1	1	yes	characteristic
41	C₂²×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₉₇	#135	1	4	yes	—
43	C₄×C₉₇	—	2	4	no	—
44	C₄×C₉₇	C₄×Q₈	1	2	yes	—
45	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
46	Dic₉₇	—	4	4	no	—
47	Dic₉₇	—	8	2	no	—
48	C₂×C₉₇	#391	1	1	yes	characteristic
49	C₂×C₉₇	#416	1	2	yes	—
50	C₂×C₉₇	#488	1	2	yes	—
51	C₂×C₉₇	#499	1	2	yes	—
52	#1991	—	97	1	no	maximal, Sylow
53	C₉₇	#1991	1	1	yes	characteristic, complemented, Sylow
54	C₂×#125	—	97	1	no	—
55	C₂×C₄×C₈	D₉₇	1	1	yes	characteristic
56	#382	—	97	1	no	—
57	#340	—	97	4	no	—
58	C₂²×C₈	C₂×D₉₇	1	2	yes	—
59	#125	—	97	4	no	—
60	#125	—	194	2	no	—
61	C₂×C₄⋊C₄	—	97	2	no	—
62	C₂×C₄⋊C₄	—	97	2	no	—
63	C₂×C₄⋊C₄	—	194	2	no	—
64	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
65	C₄×C₈	—	2	2	no	—
66	#95	—	194	2	no	—
67	C₂²×C₄	—	194	2	no	—
68	C₂²×C₄	—	194	2	no	—
69	C₂²×C₄	D₃₈₈	1	2	yes	—
70	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	Dic₁₉₄	1	4	yes	—
74	C₄⋊C₄	—	97	4	no	—
75	C₄⋊C₄	—	194	4	no	—
76	C₄⋊C₄	—	194	8	no	—
77	C₄⋊C₄	—	194	2	no	—
78	C₄²	C₄×D₉₇	1	2	yes	—
79	C₄²	C₂²⋊D₉₇	1	2	yes	—
80	C₂³	C₂×D₃₈₈	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	388	2	no	—
83	C₂×C₄	—	388	4	no	—
84	C₂×C₄	—	388	4	no	—
85	C₂×C₄	—	194	4	no	—
86	C₂×C₄	#13095488	1	4	yes	—
87	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
89	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
90	C₂×C₄	#13095511	1	2	yes	—
91	C₈	—	2	4	no	—
92	C₂²	C₂×#13095488	1	1	yes	characteristic
93	C₂²	#15690798	1	1	yes	characteristic
94	C₂²	#15690810	1	1	yes	characteristic
95	C₂²	#15690832	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	776	2	no	—
98	C₄	—	388	4	no	—
99	C₄	C₄×Dic₁₉₄	1	2	yes	—
100	C₄	#15690789	1	2	yes	—
101	C₂	#20260641	1	1	yes	characteristic
102	C₂	#20260732	1	2	yes	—
103	C₂	#20261090	1	2	yes	—
104	C₂	#20261137	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690802

C₂

yes

maximal, characteristic

#20260601

C₂

yes

maximal, characteristic

#20260498

C₂

yes

maximal

C₂×C₄×C₈×C₉₇

C₂

yes

maximal, characteristic

#15690802

—

#15690802

C₄

yes

complemented

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×#13095490

—

#15690717

—

C₂²×C₈×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄×C₈×C₉₇

—

C₂²×Dic₉₇

—

C₂²×Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

C₂×C₄

yes

—

#13095490

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

Q₈

yes

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂×D₄

yes

characteristic

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

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₈×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

#124

yes

characteristic

C₂²×C₉₇

C₄⋊Q₈

yes

characteristic

C₂²×C₉₇

#135

yes

—

C₄×C₉₇

—

C₄×C₉₇

C₄×Q₈

yes

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

#391

yes

characteristic

C₂×C₉₇

#416

yes

—

C₂×C₉₇

#488

yes

—

C₂×C₉₇

#499

yes

—

#1991

—

maximal, Sylow

C₉₇

#1991

yes

characteristic, complemented, Sylow

C₂×#125

—

C₂×C₄×C₈

D₉₇

yes

characteristic

#382

—

#340

—

C₂²×C₈

C₂×D₉₇

yes

—

#125

—

#125

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

194

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄×C₈

—

#95

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

D₃₈₈

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

Dic₁₉₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄²

C₄×D₉₇

yes

—

C₄²

C₂²⋊D₉₇

yes

—

C₂³

C₂×D₃₈₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

388

—

C₂×C₄

—

388

—

C₂×C₄

—

388

—

C₂×C₄

—

194

—

C₂×C₄

#13095488

yes

—

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

C₂×C₄

C₂×Dic₁₉₄

yes

—

C₂×C₄

#13095511

yes

—

C₈

—

C₂²

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

776

—

C₄

—

388

—

C₄

C₄×Dic₁₉₄

yes

—

100

C₄

#15690789

yes

—

101

C₂

#20260641

yes

characteristic

102

C₂

#20260732

yes

—

103

C₂

#20261090

yes

—

104

C₂

#20261137

yes

—

	subgroup	G/H
center	C₂³	C₂×D₃₈₈
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×C₄×C₈×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₂×D₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×C₄×C₈	D₉₇
hypocenter	C₉₇	#1991
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₃₈₈

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂×C₄×C₈×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₂×D₄

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₂×C₄×C₈

D₉₇

hypocenter

C₉₇

#1991

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
6. #15690802	C₄	semidirect
53. C₉₇	#1991	semidirect

H ≅ G/N

type

6. #15690802

C₄

semidirect

53. C₉₇

#1991

semidirect

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

subgroup	index
C₂×#15690802	2
#20260601	2
#20260498	2
C₂×C₄×C₈×C₉₇	2
#1991	97

subgroup

index

C₂×C₄×C₈×C₉₇

Subgroups — #27142701

Conjugacy classes of subgroups

Complements

Maximal subgroups