Group #27139613 — subgroups

Subgroups — #27139613

Order 12416.

184 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 61 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	#20260699	C₂	1	2	yes	maximal
2	C₂×C₂²⋊C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#13095495	—	2	2	no	—
4	C₂²⋊#11715895	—	2	2	no	—
5	C₂²⋊#11715895	C₄	1	4	yes	—
6	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₉₇	C₄	1	2	yes	—
8	C₂²⋊C₈×C₉₇	—	2	2	no	—
9	#13095495	—	4	4	no	—
10	C₂×#11715895	—	2	4	no	—
11	C₂⁴×C₉₇	Q₈	1	1	yes	characteristic
12	C₂²×C₄×C₉₇	—	4	2	no	—
13	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₉₇	D₄	1	2	yes	—
17	C₂×C₈×C₉₇	—	2	2	no	—
18	C₂×C₈×C₉₇	—	4	2	no	—
19	C₂³×C₉₇	—	2	2	no	—
20	C₂³×C₉₇	—	4	1	no	—
21	C₂³×C₉₇	—	4	1	no	—
22	C₂³×C₉₇	C₄⋊C₄	1	2	yes	—
23	C₂³×C₉₇	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₉₇	—	2	2	no	—
25	C₂×C₄×C₉₇	—	2	4	no	—
26	C₂×C₄×C₉₇	—	4	2	no	—
27	C₂×C₄×C₉₇	—	4	1	no	—
28	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
29	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
30	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
31	C₈×C₉₇	—	4	2	no	—
32	#11715895	—	4	4	no	—
33	C₂²×C₉₇	—	2	1	no	—
34	C₂²×C₉₇	—	2	1	no	—
35	C₂²×C₉₇	—	2	2	no	—
36	C₂²×C₉₇	—	2	2	no	—
37	C₂²×C₉₇	—	2	2	no	—
38	C₂²×C₉₇	—	4	4	no	—
39	C₂²×C₉₇	#95	1	1	yes	characteristic
40	C₂²×C₉₇	#108	1	2	yes	—
41	C₄×C₉₇	—	2	2	no	—
42	C₄×C₉₇	—	2	1	no	—
43	C₄×C₉₇	#99	1	2	yes	—
44	C₂×C₉₇	—	2	2	no	—
45	C₂×C₉₇	—	4	2	no	—
46	C₂×C₉₇	#338	1	1	yes	characteristic
47	C₂×C₉₇	#341	1	1	yes	characteristic
48	C₂×C₉₇	#342	1	1	yes	characteristic
49	#1293	—	97	1	no	maximal, Sylow
50	C₉₇	#1293	1	1	yes	characteristic, complemented, Sylow
51	#407	—	97	2	no	—
52	C₂×C₂²⋊C₈	D₉₇	1	1	yes	characteristic
53	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
54	C₂×M₁₆	—	194	2	no	—
55	C₂²×C₈	Dic₉₇	1	2	yes	—
56	C₂²⋊C₈	—	2	2	no	—
57	C₂²⋊C₈	—	97	4	no	—
58	C₂²⋊C₈	—	194	2	no	—
59	C₂⁴	Dic₁₉₄	1	1	yes	characteristic
60	C₂²×C₄	—	4	2	no	—
61	C₂²×C₄	C₄×D₉₇	1	2	yes	—
62	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
63	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
64	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
65	M₁₆	—	388	4	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	4	2	no	—
68	C₂×C₈	—	194	4	no	—
69	C₂³	—	2	2	no	—
70	C₂³	—	4	1	no	—
71	C₂³	—	4	1	no	—
72	C₂³	#13095490	1	2	yes	—
73	C₂³	C₄⋊Dic₉₇	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	C₄×Dic₉₇	1	1	yes	characteristic
79	C₂×C₄	#13095488	1	2	yes	—
80	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
81	C₈	—	4	2	no	—
82	C₈	—	388	4	no	—
83	C₂²	—	2	1	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	2	2	no	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	4	4	no	—
89	C₂²	#15690717	1	1	yes	characteristic
90	C₂²	#15690746	1	2	yes	—
91	C₄	—	2	2	no	—
92	C₄	—	2	1	no	—
93	C₄	#15690725	1	2	yes	—
94	C₂	—	2	2	no	—
95	C₂	—	4	2	no	—
96	C₂	#20260493	1	1	yes	characteristic
97	C₂	#20260500	1	1	yes	characteristic
98	C₂	#20260502	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260699

C₂

yes

maximal

C₂×C₂²⋊C₈×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

—

C₂²⋊#11715895

—

C₂²⋊#11715895

C₄

yes

—

C₂³×C₄×C₉₇

C₂²

yes

characteristic

C₂²×C₈×C₉₇

C₄

yes

—

C₂²⋊C₈×C₉₇

—

#13095495

—

C₂×#11715895

—

C₂⁴×C₉₇

Q₈

yes

characteristic

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

C₄⋊C₄

yes

—

C₂³×C₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄²

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#95

yes

characteristic

C₂²×C₉₇

#108

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#99

yes

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

#407

—

C₂×C₂²⋊C₈

D₉₇

yes

characteristic

C₂³×C₄

C₂×D₉₇

yes

characteristic

C₂×M₁₆

—

194

—

C₂²×C₈

Dic₉₇

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

194

—

C₂⁴

Dic₁₉₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄×D₉₇

yes

—

C₂²×C₄

D₃₈₈

yes

characteristic

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂²⋊D₉₇

yes

—

M₁₆

—

388

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂³

—

C₂³

—

C₂³

—

C₂³

#13095490

yes

—

C₂³

C₄⋊Dic₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₉₇

yes

characteristic

C₂×C₄

#13095488

yes

—

C₂×C₄

C₂²⋊Dic₉₇

yes

characteristic

C₈

—

C₈

—

388

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#15690717

yes

characteristic

C₂²

#15690746

yes

—

C₄

—

C₄

—

C₄

#15690725

yes

—

C₂

—

C₂

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	#15690717
commutator	C₂×C₄×C₉₇	C₄²
Frattini	C₂³×C₄	C₂×D₉₇
Fitting	C₂×C₂²⋊C₈×C₉₇	C₂
socle	C₂²×C₉₇	#95
prime radical	C₂⁴×C₉₇	Q₈
cosocle	C₂³×C₄×C₉₇	C₂²
hypercenter	C₂×C₂²⋊C₈	D₉₇
hypocenter	C₉₇	#1293
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂²

#15690717

commutator

C₂×C₄×C₉₇

C₄²

Frattini

C₂³×C₄

C₂×D₉₇

Fitting

C₂×C₂²⋊C₈×C₉₇

C₂

socle

C₂²×C₉₇

#95

prime radical

C₂⁴×C₉₇

Q₈

cosocle

C₂³×C₄×C₉₇

C₂²

hypercenter

C₂×C₂²⋊C₈

D₉₇

hypocenter

C₉₇

#1293

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
50. C₉₇	#1293	semidirect

H ≅ G/N

type

50. C₉₇

#1293

semidirect

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

subgroup	index
#20260699	2
C₂×C₂²⋊C₈×C₉₇	2
#1293	97

subgroup

index

#20260699

C₂×C₂²⋊C₈×C₉₇

#1293

Subgroups — #27139613

Conjugacy classes of subgroups

Complements

Maximal subgroups