Group #27140646 — subgroups

Subgroups — #27140646

Order 12416.

148 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 65 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	#20260865	C₂	1	2	yes	maximal
2	C₄×Dic₃₈₈	C₂	1	1	yes	maximal, characteristic
3	#20260486	C₂	1	1	yes	maximal, characteristic
4	#20260467	C₂	1	2	yes	maximal
5	#322×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₈₈	C₄	1	1	yes	characteristic
7	C₄×Dic₁₉₄	C₂²	1	2	yes	—
8	#15690744	C₄	1	1	yes	characteristic
9	C₄⋊#11715895	C₂²	1	2	yes	—
10	#15690739	—	2	2	no	—
11	#15690735	C₄	1	2	yes	—
12	#15690720	—	2	2	no	—
13	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₉₄	C₂×C₄	1	2	yes	—
17	Dic₃₈₈	—	2	2	no	—
18	C₂×#11715895	—	2	2	no	—
19	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
20	#13095490	—	2	2	no	—
21	C₄×Dic₉₇	—	2	2	no	—
22	C₂×C₈×C₉₇	—	2	1	no	—
23	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₉₇	D₄	1	2	yes	—
26	C₄²×C₉₇	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
30	C₂×Dic₉₇	—	2	2	no	—
31	C₂×Dic₉₇	—	2	2	no	—
32	Dic₁₉₄	—	2	2	no	—
33	Dic₁₉₄	—	2	2	no	—
34	C₈×C₉₇	—	2	1	no	—
35	C₈×C₉₇	—	2	2	no	—
36	C₈×C₉₇	—	4	1	no	—
37	C₈×C₉₇	M₁₆	1	2	yes	—
38	#11715895	—	4	2	no	—
39	C₂²×C₉₇	C₄×D₄	1	1	yes	characteristic
40	C₄×C₉₇	—	2	1	no	—
41	C₄×C₉₇	C₂×M₁₆	1	1	yes	characteristic
42	C₄×C₉₇	#131	1	1	yes	characteristic
43	C₄×C₉₇	#137	1	2	yes	—
44	Dic₉₇	—	4	2	no	—
45	Dic₉₇	—	4	2	no	—
46	C₂×C₉₇	#436	1	1	yes	characteristic
47	C₂×C₉₇	#441	1	1	yes	characteristic
48	C₂×C₉₇	#444	1	1	yes	characteristic
49	#1593	—	97	1	no	maximal, Sylow
50	C₉₇	#1593	1	1	yes	characteristic, complemented, Sylow
51	#446	—	97	2	no	—
52	C₄×Q₁₆	—	97	1	no	—
53	#334	—	97	1	no	—
54	Q₈⋊C₈	—	97	2	no	—
55	#322	D₉₇	1	1	yes	characteristic
56	C₂×Q₁₆	—	97	1	no	—
57	C₄×Q₈	—	97	2	no	—
58	#107	—	97	1	no	—
59	C₄⋊C₈	—	97	2	no	—
60	C₄⋊C₈	—	194	2	no	—
61	#103	—	97	2	no	—
62	#97	—	194	2	no	—
63	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
64	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
65	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
66	C₂×Q₈	—	97	2	no	—
67	Q₁₆	—	194	2	no	—
68	C₂×C₈	—	2	1	no	—
69	C₂×C₈	—	194	2	no	—
70	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
71	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
72	C₂×C₈	D₃₈₈	1	2	yes	—
73	C₄⋊C₄	—	97	2	no	—
74	C₄⋊C₄	—	194	2	no	—
75	C₄²	—	194	2	no	—
76	C₄²	C₂²×D₉₇	1	1	yes	characteristic
77	Q₈	—	194	2	no	—
78	Q₈	—	194	2	no	—
79	C₂×C₄	—	194	2	no	—
80	C₂×C₄	—	194	2	no	—
81	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
82	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
83	C₂×C₄	#13095511	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	2	2	no	—
86	C₈	—	4	1	no	—
87	C₈	—	388	2	no	—
88	C₈	#13095494	1	2	yes	—
89	C₂²	C₄×D₃₈₈	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	388	2	no	—
92	C₄	—	388	2	no	—
93	C₄	C₂×#13095494	1	1	yes	characteristic
94	C₄	#15690816	1	1	yes	characteristic
95	C₄	#15690843	1	2	yes	—
96	C₂	#20260814	1	1	yes	characteristic
97	C₂	#20260843	1	1	yes	characteristic
98	C₂	#20260858	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260865

C₂

yes

maximal

C₄×Dic₃₈₈

C₂

yes

maximal, characteristic

#20260486

C₂

yes

maximal, characteristic

#20260467

C₂

yes

maximal

#322×C₉₇

C₂

yes

maximal, characteristic

C₂×Dic₃₈₈

C₄

yes

characteristic

C₄×Dic₁₉₄

C₂²

yes

—

#15690744

C₄

yes

characteristic

C₄⋊#11715895

C₂²

yes

—

#15690739

—

#15690735

C₄

yes

—

#15690720

—

C₄×C₈×C₉₇

C₂²

yes

characteristic

C₄×C₈×C₉₇

C₂²

yes

characteristic

C₄×C₈×C₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

C₂×C₄

yes

—

Dic₃₈₈

—

C₂×#11715895

—

C₄⋊Dic₉₇

C₂×C₄

yes

—

#13095490

—

C₄×Dic₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

characteristic

C₂×C₈×C₉₇

C₂×C₄

yes

characteristic

C₂×C₈×C₉₇

D₄

yes

—

C₄²×C₉₇

C₂³

yes

characteristic

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

—

Dic₁₉₄

—

Dic₁₉₄

—

C₈×C₉₇

—

C₈×C₉₇

—

C₈×C₉₇

—

C₈×C₉₇

M₁₆

yes

—

#11715895

—

C₂²×C₉₇

C₄×D₄

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

C₂×M₁₆

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

#446

—

C₄×Q₁₆

—

#334

—

Q₈⋊C₈

—

#322

D₉₇

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

194

—

#103

—

#97

—

194

—

C₄×C₈

C₂×D₉₇

yes

characteristic

C₄×C₈

C₂×D₉₇

yes

characteristic

C₄×C₈

C₂×D₉₇

yes

characteristic

C₂×Q₈

—

Q₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

C₄×D₉₇

yes

characteristic

C₂×C₈

C₄×D₉₇

yes

characteristic

C₂×C₈

D₃₈₈

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄²

—

194

—

C₄²

C₂²×D₉₇

yes

characteristic

Q₈

—

194

—

Q₈

—

194

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

C₂×C₄

C₂×D₃₈₈

yes

characteristic

C₂×C₄

#13095511

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

388

—

C₈

#13095494

yes

—

C₂²

C₄×D₃₈₈

yes

characteristic

C₄

—

C₄

—

388

—

C₄

—

388

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	C₂×D₃₈₈
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₄²	C₂²×D₉₇
Fitting	#322×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	C₄×D₄
cosocle	C₄²×C₉₇	C₂³
hypercenter	#322	D₉₇
hypocenter	C₉₇	#1593
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×D₃₈₈

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₄²

C₂²×D₉₇

Fitting

#322×C₉₇

C₂

socle, prime radical

C₂²×C₉₇

C₄×D₄

cosocle

C₄²×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
50. C₉₇	#1593	semidirect

H ≅ G/N

type

50. C₉₇

#1593

semidirect

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

subgroup	index
#20260865	2
C₄×Dic₃₈₈	2
#20260486	2
#20260467	2
#322×C₉₇	2
#1593	97

subgroup

index

C₄×Dic₃₈₈

#322×C₉₇

Subgroups — #27140646

Conjugacy classes of subgroups

Complements

Maximal subgroups