Group #27140644 — subgroups

Subgroups — #27140644

Order 12416.

148 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 63 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	#20260862	C₂	1	2	yes	maximal
2	C₄×Dic₃₈₈	C₂	1	1	yes	maximal, characteristic
3	#20260465	C₂	1	2	yes	maximal
4	#20260456	C₂	1	1	yes	maximal, characteristic
5	#334×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	#15690739	—	2	2	no	—
10	#15690735	C₄	1	2	yes	—
11	#15690720	—	2	2	no	—
12	C₄×#11715895	C₂²	1	1	yes	characteristic
13	C₄×#11715895	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
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	1	no	—
19	C₂×#11715895	D₄	1	2	yes	—
20	C₄⋊Dic₉₇	C₂×C₄	1	2	yes	—
21	#13095490	—	2	2	no	—
22	C₄×Dic₉₇	—	2	2	no	—
23	C₂×C₈×C₉₇	—	2	2	no	—
24	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
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	—	4	2	no	—
36	C₈×C₉₇	M₁₆	1	2	yes	—
37	#11715895	—	2	2	no	—
38	#11715895	—	4	1	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	D₉₇	1	1	yes	characteristic
54	Q₈⋊C₈	—	97	2	no	—
55	#322	—	97	1	no	—
56	C₂×Q₁₆	—	97	1	no	—
57	C₄×Q₈	—	97	2	no	—
58	#107	—	97	1	no	—
59	C₄⋊C₈	—	194	2	no	—
60	C₄⋊C₈	C₂×D₉₇	1	2	yes	—
61	#103	—	97	2	no	—
62	#97	—	194	2	no	—
63	C₄×C₈	—	97	1	no	—
64	C₄×C₈	—	97	1	no	—
65	C₄×C₈	C₂×D₉₇	1	1	yes	characteristic
66	C₂×Q₈	—	97	2	no	—
67	Q₁₆	—	194	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	97	2	no	—
70	C₂×C₈	—	194	1	no	—
71	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
72	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
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	D₄×D₉₇	1	1	yes	characteristic
83	C₂×C₄	#13095512	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	4	2	no	—
86	C₈	—	388	1	no	—
87	C₈	—	194	2	no	—
88	C₈	#13095494	1	2	yes	—
89	C₂²	#15690772	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	#15690818	1	1	yes	characteristic
95	C₄	#15690839	1	2	yes	—
96	C₂	#20260812	1	1	yes	characteristic
97	C₂	#20260841	1	1	yes	characteristic
98	C₂	#20260856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260862

C₂

yes

maximal

C₄×Dic₃₈₈

C₂

yes

maximal, characteristic

#20260465

C₂

yes

maximal

#20260456

C₂

yes

maximal, characteristic

#334×C₉₇

C₂

yes

maximal, characteristic

C₂×Dic₃₈₈

C₄

yes

characteristic

C₄×Dic₁₉₄

C₂²

yes

—

#15690744

C₄

yes

characteristic

#15690739

—

#15690735

C₄

yes

—

#15690720

—

C₄×#11715895

C₂²

yes

characteristic

C₄×#11715895

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

C₂²

yes

—

C₄×C₈×C₉₇

C₂²

yes

characteristic

C₂×Dic₁₉₄

C₂×C₄

yes

—

Dic₃₈₈

—

C₂×#11715895

—

C₂×#11715895

D₄

yes

—

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

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

M₁₆

yes

—

#11715895

—

#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

D₉₇

yes

characteristic

Q₈⋊C₈

—

#322

—

C₂×Q₁₆

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

194

—

C₄⋊C₈

C₂×D₉₇

yes

—

#103

—

#97

—

194

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₉₇

yes

characteristic

C₂×Q₈

—

Q₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

C₄×D₉₇

yes

characteristic

C₂×C₈

C₄×D₉₇

yes

characteristic

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

D₄×D₉₇

yes

characteristic

C₂×C₄

#13095512

yes

characteristic

C₈

—

C₈

—

C₈

—

388

—

C₈

—

194

—

C₈

#13095494

yes

—

C₂²

#15690772

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	D₄×D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₄²	C₂²×D₉₇
Fitting	#334×C₉₇	C₂
socle, prime radical	C₂²×C₉₇	C₄×D₄
cosocle	C₄²×C₉₇	C₂³
hypercenter	#334	D₉₇
hypocenter	C₉₇	#1593
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₄²

C₂²×D₉₇

Fitting

#334×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 ◁ C₄² ◁ #334

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
#20260862	2
C₄×Dic₃₈₈	2
#20260465	2
#20260456	2
#334×C₉₇	2
#1593	97

subgroup

index

C₄×Dic₃₈₈

#334×C₉₇

Subgroups — #27140644

Conjugacy classes of subgroups

Complements

Maximal subgroups