Group #27157662 — subgroups

Subgroups — #27157662

Order 12416.

180 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 66 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	#20261740	C₂	1	1	yes	maximal, characteristic
2	#20261739	C₂	1	1	yes	maximal, characteristic
3	#20261611	C₂	1	4	yes	maximal
4	#97×D₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095521	C₂²	1	2	yes	—
6	C₂×#13095521	C₂²	1	2	yes	—
7	#15690877	—	2	2	no	—
8	C₄⋊#11715897	—	2	2	no	—
9	C₂×C₈×D₉₇	C₂²	1	2	yes	—
10	C₄²×D₉₇	C₂²	1	1	yes	characteristic
11	#15690721	C₄	1	1	yes	characteristic
12	#15690720	C₄	1	2	yes	—
13	#97×C₉₇	C₄	1	1	yes	characteristic
14	#13095521	—	2	4	no	—
15	#13095521	—	2	4	no	—
16	C₂×#11715897	—	2	2	no	—
17	C₂×#11715897	—	2	2	no	—
18	C₂×C₄×D₉₇	Q₈	1	2	yes	—
19	C₂×C₄×D₉₇	C₂³	1	1	yes	characteristic
20	C₂×#11715895	C₂×C₄	1	2	yes	—
21	C₈×D₉₇	—	2	2	no	—
22	C₈×D₉₇	—	2	2	no	—
23	C₄×Dic₉₇	C₂×C₄	1	1	yes	characteristic
24	C₄×Dic₉₇	D₄	1	2	yes	—
25	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
26	C₄²×C₉₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×D₉₇	C₂×Q₈	1	1	yes	characteristic
28	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
29	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×Dic₉₇	C₄⋊C₄	1	2	yes	—
31	C₂×Dic₉₇	C₂×D₄	1	1	yes	characteristic
32	C₄×D₉₇	—	4	2	no	—
33	C₄×D₉₇	#41	1	2	yes	—
34	C₄×D₉₇	#41	1	2	yes	—
35	#11715897	—	4	2	no	—
36	#11715897	—	4	2	no	—
37	C₈×C₉₇	—	2	2	no	—
38	#11715895	—	2	2	no	—
39	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂×D₉₇	—	2	2	no	—
41	C₂×D₉₇	#125	1	2	yes	—
42	C₄×C₉₇	—	2	2	no	—
43	C₄×C₉₇	#117	1	2	yes	—
44	Dic₉₇	—	2	2	no	—
45	Dic₉₇	#124	1	1	yes	characteristic
46	Dic₉₇	#124	1	1	yes	characteristic
47	C₂×C₉₇	#382	1	1	yes	characteristic
48	C₂×C₉₇	#430	1	2	yes	—
49	D₉₇	—	2	2	no	—
50	#1838	—	97	1	no	maximal, Sylow
51	C₉₇	#1838	1	1	yes	characteristic, complemented, Sylow
52	#423	—	97	1	no	—
53	#423	—	97	1	no	—
54	C₂×#97	—	97	1	no	—
55	#341	—	97	4	no	—
56	C₂×M₁₆	—	97	2	no	—
57	C₂×M₁₆	—	97	2	no	—
58	C₂²×C₈	—	97	2	no	—
59	C₂×C₄²	—	97	1	no	—
60	C₄⋊C₈	—	194	2	no	—
61	C₄⋊C₈	—	194	2	no	—
62	#97	—	97	2	no	—
63	#97	—	97	1	no	—
64	#97	#87604	1	1	yes	characteristic
65	C₂²×C₄	—	97	1	no	—
66	C₂²×C₄	—	97	2	no	—
67	M₁₆	—	194	4	no	—
68	M₁₆	—	194	4	no	—
69	C₂×C₈	—	97	2	no	—
70	C₂×C₈	—	194	2	no	—
71	C₂×C₈	—	194	2	no	—
72	C₂×C₈	—	194	2	no	—
73	C₂×C₈	—	194	2	no	—
74	C₂×C₈	C₂×#87604	1	2	yes	—
75	C₄²	—	97	2	no	—
76	C₄²	—	97	1	no	—
77	C₄²	C₂×#87604	1	1	yes	characteristic
78	C₂³	—	97	1	no	—
79	C₂×C₄	—	97	1	no	—
80	C₂×C₄	—	97	2	no	—
81	C₂×C₄	—	97	2	no	—
82	C₂×C₄	—	97	2	no	—
83	C₂×C₄	—	388	2	no	—
84	C₂×C₄	C₄⋊#87604	1	2	yes	—
85	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₈	—	388	2	no	—
88	C₈	—	388	2	no	—
89	C₈	—	194	2	no	—
90	C₂²	—	97	2	no	—
91	C₂²	—	194	2	no	—
92	C₂²	C₂×C₄⋊#87604	1	1	yes	characteristic
93	C₄	—	2	2	no	—
94	C₄	—	97	1	no	—
95	C₄	—	97	1	no	—
96	C₄	—	194	2	no	—
97	C₄	#15690889	1	2	yes	—
98	C₂	—	194	2	no	—
99	C₂	#20261679	1	1	yes	characteristic
100	C₂	#20261750	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20261740

C₂

yes

maximal, characteristic

#20261739

C₂

yes

maximal, characteristic

#20261611

C₂

yes

maximal

#97×D₉₇

C₂

yes

maximal, characteristic

C₂×#13095521

C₂²

yes

—

C₂×#13095521

C₂²

yes

—

#15690877

—

C₄⋊#11715897

—

C₂×C₈×D₉₇

C₂²

yes

—

C₄²×D₉₇

C₂²

yes

characteristic

#15690721

C₄

yes

characteristic

#15690720

C₄

yes

—

#97×C₉₇

C₄

yes

—

—

—

—

C₂×C₄×D₉₇

Q₈

yes

—

C₂×C₄×D₉₇

C₂³

yes

characteristic

C₂×#11715895

C₂×C₄

yes

—

C₈×D₉₇

—

C₈×D₉₇

—

C₄×Dic₉₇

C₂×C₄

yes

characteristic

C₄×Dic₉₇

D₄

yes

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

C₂×C₄

yes

characteristic

C₂²×D₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×Dic₉₇

C₄⋊C₄

yes

—

C₂×Dic₉₇

C₂×D₄

yes

characteristic

C₄×D₉₇

—

C₄×D₉₇

#41

yes

—

C₄×D₉₇

#41

yes

—

#11715897

—

#11715897

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂×D₉₇

—

C₂×D₉₇

#125

yes

—

C₄×C₉₇

—

C₄×C₉₇

#117

yes

—

Dic₉₇

—

Dic₉₇

yes

Dic₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

D₉₇

—

#1838

—

maximal, Sylow

C₉₇

#1838

yes

characteristic, complemented, Sylow

#423

—

#423

—

C₂×#97

—

#341

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂×C₄²

—

C₄⋊C₈

—

194

—

C₄⋊C₈

—

194

—

—

—

yes

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

C₂×#87604

yes

—

C₄²

—

C₄²

—

C₄²

C₂×#87604

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

388

—

C₂×C₄

C₄⋊#87604

yes

—

C₂×C₄

C₂²×#87604

yes

characteristic

C₈

—

C₈

—

388

—

C₈

—

388

—

C₈

—

194

—

C₂²

—

C₂²

—

194

—

C₂²

C₂×C₄⋊#87604

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

194

—

C₄

#15690889

yes

—

C₂

—

194

—

C₂

#20261679

yes

characteristic

100

C₂

#20261750

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#87604
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87604
Fitting	#97×C₉₇	C₄
socle	C₂²×C₉₇	C₂×C₄⋊C₄
prime radical	C₂²×D₉₇	C₂×Q₈
cosocle	C₂×C₄×D₉₇	C₂³
hypercenter	#97	#87604
hypocenter	C₉₇	#1838
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

subgroup

G/H

center

C₂²

C₂×C₄⋊#87604

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#87604

Fitting

#97×C₉₇

C₄

socle

C₂²×C₉₇

C₂×C₄⋊C₄

prime radical

C₂²×D₉₇

C₂×Q₈

cosocle

C₂×C₄×D₉₇

C₂³

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #97

N	H ≅ G/N	type
51. C₉₇	#1838	semidirect

H ≅ G/N

type

51. C₉₇

#1838

semidirect

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

subgroup	index
#20261740	2
#20261739	2
#20261611	2
#97×D₉₇	2
#1838	97

subgroup

index

#97×D₉₇

Subgroups — #27157662

Conjugacy classes of subgroups

Complements

Maximal subgroups