Group #27139571 — subgroups

Subgroups — #27139571

Order 12416.

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

normal subgroups only

This group has 180 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	2	yes	maximal
2	C₂×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11715895	—	2	2	no	—
4	C₄⋊#11715895	—	2	2	no	—
5	C₄⋊#11715895	C₄	1	4	yes	—
6	C₂²×C₈×C₉₇	C₄	1	2	yes	—
7	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
8	C₄×C₈×C₉₇	—	2	2	no	—
9	C₂×#11715895	—	2	4	no	—
10	C₂×#11715895	—	4	2	no	—
11	C₂×#11715895	—	4	2	no	—
12	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₉₇	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₉₇	Q₈	1	1	yes	characteristic
15	C₂×C₈×C₉₇	—	2	2	no	—
16	C₂×C₈×C₉₇	—	2	2	no	—
17	C₂×C₈×C₉₇	C₈	1	4	yes	complemented
18	C₄²×C₉₇	C₂×C₄	1	2	yes	—
19	C₄²×C₉₇	D₄	1	2	yes	—
20	C₂³×C₉₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₉₇	—	2	2	no	—
22	C₂×C₄×C₉₇	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
24	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₉₇	M₁₆	1	2	yes	—
28	C₂×C₄×C₉₇	D₈	1	1	yes	characteristic
29	C₂×C₄×C₉₇	SD₁₆	1	2	yes	—
30	C₂×C₄×C₉₇	Q₁₆	1	1	yes	characteristic
31	C₈×C₉₇	—	2	4	no	—
32	#11715895	—	4	4	no	—
33	C₂²×C₉₇	#95	1	1	yes	characteristic
34	C₂²×C₉₇	C₄⋊C₈	1	2	yes	—
35	C₂²×C₉₇	#106	1	1	yes	characteristic
36	C₂²×C₉₇	#107	1	1	yes	characteristic
37	C₂²×C₉₇	#108	1	2	yes	—
38	C₄×C₉₇	—	2	2	no	—
39	C₄×C₉₇	C₄×C₈	1	1	yes	characteristic
40	C₄×C₉₇	#97	1	1	yes	characteristic
41	C₄×C₉₇	C₂²⋊C₈	1	2	yes	—
42	C₄×C₉₇	#102	1	2	yes	—
43	C₄×C₉₇	#103	1	2	yes	—
44	C₂×C₉₇	#334	1	2	yes	—
45	C₂×C₉₇	#335	1	2	yes	—
46	C₂×C₉₇	#336	1	1	yes	characteristic
47	C₂×C₉₇	#340	1	1	yes	characteristic
48	C₂×C₉₇	#341	1	1	yes	characteristic
49	#1277	—	97	1	no	maximal, Sylow
50	C₉₇	#1277	1	1	yes	characteristic, complemented, Sylow
51	C₂×C₄⋊C₈	—	97	2	no	—
52	C₂×C₄×C₈	D₉₇	1	1	yes	characteristic
53	C₂²×C₈	—	194	2	no	—
54	C₂²×C₈	Dic₉₇	1	2	yes	—
55	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
56	C₄⋊C₈	—	97	4	no	—
57	C₄⋊C₈	—	194	2	no	—
58	C₄×C₈	—	2	2	no	—
59	C₂²×C₄	Dic₁₉₄	1	1	yes	characteristic
60	C₂²×C₄	D₃₈₈	1	1	yes	characteristic
61	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	388	2	no	—
65	C₂×C₈	—	388	2	no	—
66	C₂×C₈	—	194	4	no	—
67	C₂×C₈	#11715895	1	4	yes	complemented
68	C₄²	C₄×D₉₇	1	2	yes	—
69	C₄²	C₂²⋊D₉₇	1	2	yes	—
70	C₂³	C₄⋊Dic₉₇	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	C₄×Dic₉₇	1	1	yes	characteristic
73	C₂×C₄	#13095488	1	2	yes	—
74	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
75	C₂×C₄	#13095490	1	2	yes	—
76	C₂×C₄	C₂×#11715895	1	2	yes	—
77	C₂×C₄	#13095495	1	2	yes	—
78	C₂×C₄	D₇₇₆	1	1	yes	characteristic
79	C₂×C₄	#13095500	1	2	yes	—
80	C₂×C₄	Dic₃₈₈	1	1	yes	characteristic
81	C₈	—	2	4	no	—
82	C₈	—	388	4	no	—
83	C₂²	#15690717	1	1	yes	characteristic
84	C₂²	C₄⋊#11715895	1	2	yes	—
85	C₂²	#15690742	1	1	yes	characteristic
86	C₂²	#15690744	1	1	yes	characteristic
87	C₂²	#15690746	1	2	yes	—
88	C₄	—	2	2	no	—
89	C₄	C₄×#11715895	1	1	yes	characteristic
90	C₄	#15690721	1	1	yes	characteristic
91	C₄	C₂²⋊#11715895	1	2	yes	—
92	C₄	#15690732	1	2	yes	—
93	C₄	#15690735	1	2	yes	—
94	C₂	#20260486	1	2	yes	—
95	C₂	#20260488	1	2	yes	—
96	C₂	#20260490	1	1	yes	characteristic
97	C₂	#20260498	1	1	yes	characteristic
98	C₂	#20260500	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11715895

C₂

yes

maximal

C₂×C₄×C₈×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

—

C₄⋊#11715895

—

C₄⋊#11715895

C₄

yes

—

C₂²×C₈×C₉₇

C₄

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄×C₈×C₉₇

—

C₂×#11715895

—

C₂×#11715895

—

C₂×#11715895

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

characteristic

C₂²×C₄×C₉₇

Q₈

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₈

yes

complemented

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄²

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₂×C₈

yes

—

C₂×C₄×C₉₇

M₁₆

yes

—

C₂×C₄×C₉₇

D₈

yes

characteristic

C₂×C₄×C₉₇

SD₁₆

yes

—

C₂×C₄×C₉₇

Q₁₆

yes

characteristic

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

#95

yes

characteristic

C₂²×C₉₇

C₄⋊C₈

yes

—

C₂²×C₉₇

yes

C₂²×C₉₇

yes

C₂²×C₉₇

yes

—

C₄×C₉₇

—

C₄×C₉₇

C₄×C₈

yes

characteristic

C₄×C₉₇

#97

yes

characteristic

C₄×C₉₇

C₂²⋊C₈

yes

—

C₄×C₉₇

#102

yes

—

C₄×C₉₇

#103

yes

—

C₂×C₉₇

#334

yes

—

C₂×C₉₇

#335

yes

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₉₇

yes

characteristic

C₂²×C₈

—

194

—

C₂²×C₈

Dic₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

194

—

C₄×C₈

—

C₂²×C₄

Dic₁₉₄

yes

characteristic

C₂²×C₄

D₃₈₈

yes

characteristic

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

388

—

C₂×C₈

—

388

—

C₂×C₈

—

194

—

C₂×C₈

#11715895

yes

complemented

C₄²

C₄×D₉₇

yes

—

C₄²

C₂²⋊D₉₇

yes

—

C₂³

C₄⋊Dic₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄×Dic₉₇

yes

characteristic

C₂×C₄

#13095488

yes

—

C₂×C₄

C₂²⋊Dic₉₇

yes

characteristic

C₂×C₄

#13095490

yes

—

C₂×C₄

C₂×#11715895

yes

—

C₂×C₄

#13095495

yes

—

C₂×C₄

D₇₇₆

yes

characteristic

C₂×C₄

#13095500

yes

—

C₂×C₄

Dic₃₈₈

yes

characteristic

C₈

—

C₈

—

388

—

C₂²

#15690717

yes

characteristic

C₂²

C₄⋊#11715895

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

—

C₄

#15690732

yes

—

C₄

#15690735

yes

—

C₂

#20260486

yes

—

C₂

#20260488

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²×C₄	D₃₈₈
commutator	C₄×C₉₇	C₄×C₈
Frattini	C₂×C₄²	C₂×D₉₇
Fitting	C₂×C₄×C₈×C₉₇	C₂
socle, prime radical	C₂³×C₉₇	C₄⋊C₄
cosocle	C₂×C₄²×C₉₇	C₂²
hypercenter	C₂×C₄×C₈	D₉₇
hypocenter	C₉₇	#1277
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

D₃₈₈

commutator

C₄×C₉₇

C₄×C₈

Frattini

C₂×C₄²

C₂×D₉₇

Fitting

C₂×C₄×C₈×C₉₇

C₂

socle, prime radical

C₂³×C₉₇

C₄⋊C₄

cosocle

C₂×C₄²×C₉₇

C₂²

hypercenter

C₂×C₄×C₈

D₉₇

hypocenter

C₉₇

#1277

LCS

G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
17. C₂×C₈×C₉₇	C₈	semidirect
50. C₉₇	#1277	semidirect
67. C₂×C₈	#11715895	semidirect

H ≅ G/N

type

17. C₂×C₈×C₉₇

C₈

semidirect

50. C₉₇

#1277

semidirect

67. C₂×C₈

#11715895

semidirect

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

subgroup	index
C₂×C₄⋊#11715895	2
C₂×C₄×C₈×C₉₇	2
#1277	97

subgroup

index

C₂×C₄⋊#11715895

C₂×C₄×C₈×C₉₇

#1277

Subgroups — #27139571

Conjugacy classes of subgroups

Complements

Maximal subgroups