Group #27142702 — subgroups

Subgroups — #27142702

Order 12416.

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

normal subgroups only

This group has 252 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×C₄×#11715895	C₂	1	1	yes	maximal, characteristic
2	#20260601	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	C₂²	1	2	yes	—
7	C₄×#11715895	—	2	2	no	—
8	#15690717	—	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	C₂²×C₄×C₉₇	—	2	2	no	—
20	C₂²×C₄×C₉₇	D₄	1	2	yes	—
21	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₉₇	—	2	8	no	—
23	C₄⋊C₄×C₉₇	—	2	2	no	—
24	C₄⋊C₄×C₉₇	C₂×C₄	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	—	2	4	no	—
30	C₂×C₄×C₉₇	—	4	4	no	—
31	C₂×C₄×C₉₇	C₂²⋊C₄	1	4	yes	—
32	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₉₇	#41	1	2	yes	—
36	C₂×Dic₉₇	—	4	2	no	—
37	C₂×Dic₉₇	—	4	4	no	—
38	#11715895	—	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	—	4	4	no	—
45	C₄×C₉₇	C₄×Q₈	1	2	yes	—
46	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
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	D₉₇	1	1	yes	characteristic
55	C₂×C₄×C₈	—	97	1	no	—
56	#382	—	97	1	no	—
57	#340	—	97	4	no	—
58	C₂²×C₈	—	97	2	no	—
59	#125	—	2	2	no	—
60	#125	Dic₉₇	1	4	yes	complemented
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄⋊C₄	—	97	2	no	—
63	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
64	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
65	C₄×C₈	—	194	2	no	—
66	#95	—	194	2	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	194	2	no	—
69	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
70	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
71	C₂×C₈	—	97	4	no	—
72	C₂×C₈	—	194	2	no	—
73	C₂×C₈	—	194	2	no	—
74	C₄⋊C₄	—	2	8	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	194	4	no	—
77	C₄⋊C₄	C₂×Dic₉₇	1	4	yes	—
78	C₄²	C₂×Dic₉₇	1	2	yes	—
79	C₄²	C₂²⋊D₉₇	1	2	yes	—
80	C₂³	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	388	2	no	—
85	C₂×C₄	—	388	4	no	—
86	C₂×C₄	C₂²⋊Dic₉₇	1	4	yes	—
87	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
89	C₂×C₄	Q₈×D₉₇	1	2	yes	—
90	C₂×C₄	#13095512	1	2	yes	—
91	C₈	—	194	4	no	—
92	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
93	C₂²	#15690799	1	1	yes	characteristic
94	C₂²	#15690808	1	1	yes	characteristic
95	C₂²	#15690830	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	4	4	no	—
98	C₄	—	776	2	no	—
99	C₄	Q₈×Dic₉₇	1	2	yes	—
100	C₄	#15690786	1	2	yes	—
101	C₂	#20260642	1	1	yes	characteristic
102	C₂	#20260731	1	2	yes	—
103	C₂	#20261092	1	2	yes	—
104	C₂	#20261134	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#11715895

C₂

yes

maximal, characteristic

#20260601

C₂

yes

maximal, characteristic

#20260497

C₂

yes

maximal

C₂×#125×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₄×#11715895

—

#15690717

—

#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

—

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

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

#135

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄×Q₈

yes

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

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

D₉₇

yes

characteristic

C₂×C₄×C₈

—

#382

—

#340

—

C₂²×C₈

—

#125

—

#125

Dic₉₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄×C₈

—

194

—

#95

—

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

—

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

—

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

—

C₂²

C₂×C₂²⋊Dic₉₇

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

C₄

—

776

—

C₄

Q₈×Dic₉₇

yes

—

100

C₄

#15690786

yes

—

101

C₂

#20260642

yes

characteristic

102

C₂

#20260731

yes

—

103

C₂

#20261092

yes

—

104

C₂

#20261134

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	#1991
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
53. C₉₇	#1991	semidirect
60. #125	Dic₉₇	semidirect

H ≅ G/N

type

10. #125×C₉₇

C₄

semidirect

53. C₉₇

#1991

semidirect

60. #125

Dic₉₇

semidirect

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

subgroup	index
C₂×C₄×#11715895	2
#20260601	2
#20260497	2
C₂×#125×C₉₇	2
#1991	97

subgroup

index

C₂×#125×C₉₇

Subgroups — #27142702

Conjugacy classes of subgroups

Complements

Maximal subgroups