Group #27143312 — subgroups

Subgroups — #27143312

Order 12416.

158 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 59 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	#20260784	C₂	1	1	yes	maximal, characteristic
2	#20260758	C₂	1	1	yes	maximal, characteristic
3	#20260505	C₂	1	2	yes	maximal
4	#20260494	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095495	C₂²	1	2	yes	—
7	C₂×#13095495	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
9	#15690746	—	2	2	no	—
10	#15690739	—	2	2	no	—
11	C₂×M₁₆×C₉₇	C₂²	1	1	yes	characteristic
12	#117×C₉₇	C₂²	1	2	yes	—
13	C₂²×Dic₉₇	—	2	1	no	—
14	#13095495	—	2	2	no	—
15	#13095495	—	2	2	no	—
16	#13095495	—	2	2	no	—
17	C₂×#11715895	—	2	1	no	—
18	C₂×#11715895	—	2	2	no	—
19	C₄×Dic₉₇	—	2	1	no	—
20	C₄×Dic₉₇	Q₈	1	1	yes	characteristic
21	C₄×Dic₉₇	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
23	M₁₆×C₉₇	—	2	2	no	—
24	C₂×C₈×C₉₇	D₄	1	2	yes	—
25	C₄⋊C₄×C₉₇	—	2	2	no	—
26	C₂²⋊C₄×C₉₇	—	2	2	no	—
27	C₄²×C₉₇	—	2	2	no	—
28	C₂³×C₉₇	#41	1	1	yes	characteristic
29	C₂×C₄×C₉₇	—	4	2	no	—
30	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₉₇	#41	1	2	yes	—
33	C₂×C₄×C₉₇	#41	1	2	yes	—
34	C₂×Dic₉₇	—	2	1	no	—
35	C₂×Dic₉₇	—	2	1	no	—
36	C₂×Dic₉₇	—	4	2	no	—
37	C₈×C₉₇	—	2	2	no	—
38	#11715895	—	4	1	no	—
39	#11715895	—	4	2	no	—
40	C₂²×C₉₇	—	4	1	no	—
41	C₂²×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
42	C₂²×C₉₇	#126	1	2	yes	—
43	C₄×C₉₇	—	8	2	no	—
44	C₄×C₉₇	#123	1	1	yes	characteristic
45	C₄×C₉₇	#124	1	1	yes	characteristic
46	C₄×C₉₇	#125	1	2	yes	—
47	Dic₉₇	—	2	2	no	—
48	Dic₉₇	—	4	1	no	—
49	C₂×C₉₇	—	2	1	no	—
50	C₂×C₉₇	—	2	2	no	—
51	C₂×C₉₇	#400	1	1	yes	characteristic
52	#2102	—	97	1	no	maximal, Sylow
53	C₉₇	#2102	1	1	yes	characteristic, complemented, Sylow
54	#430	—	97	1	no	—
55	#423	—	97	1	no	—
56	#344	—	97	2	no	—
57	#339	—	97	2	no	—
58	H(3,ℤ/4)	D₉₇	1	1	yes	characteristic
59	C₂×M₁₆	—	97	2	no	—
60	C₂×M₁₆	—	97	1	no	—
61	C₂×M₁₆	C₂×D₉₇	1	1	yes	characteristic
62	#117	C₂×D₉₇	1	2	yes	—
63	C₂×C₄²	—	97	1	no	—
64	#108	—	194	2	no	—
65	C₄⋊C₈	—	194	2	no	—
66	C₂²×C₄	—	194	1	no	—
67	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	194	2	no	—
70	M₁₆	—	194	2	no	—
71	M₁₆	—	194	2	no	—
72	C₂×C₈	—	194	1	no	—
73	C₂×C₈	—	194	2	no	—
74	C₂×C₈	D₃₈₈	1	2	yes	—
75	C₄⋊C₄	—	2	2	no	—
76	C₂²⋊C₄	—	2	2	no	—
77	C₄²	—	2	2	no	—
78	C₄²	—	97	1	no	—
79	C₄²	—	97	1	no	—
80	C₄²	—	194	1	no	—
81	C₂³	#13095510	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	388	2	no	—
84	C₂×C₄	—	194	1	no	—
85	C₂×C₄	—	194	1	no	—
86	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
87	C₂×C₄	Q₈×D₉₇	1	1	yes	characteristic
88	C₂×C₄	#13095510	1	2	yes	—
89	C₂×C₄	#13095511	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	388	1	no	—
92	C₈	—	388	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#15690790	1	1	yes	characteristic
95	C₂²	#15690803	1	2	yes	—
96	C₄	—	8	2	no	—
97	C₄	—	388	1	no	—
98	C₄	—	194	2	no	—
99	C₄	#15690794	1	1	yes	characteristic
100	C₄	#15690798	1	1	yes	characteristic
101	C₄	#15690800	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	—	2	1	no	—
104	C₂	#20260675	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260784

C₂

yes

maximal, characteristic

#20260758

C₂

yes

maximal, characteristic

#20260505

C₂

yes

maximal

#20260494

C₂

yes

maximal

H(3,ℤ/4)×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

C₂×#13095495

C₂²

yes

characteristic

C₂×C₄×Dic₉₇

C₂²

yes

characteristic

#15690746

—

#15690739

—

C₂×M₁₆×C₉₇

C₂²

yes

characteristic

#117×C₉₇

C₂²

yes

—

C₂²×Dic₉₇

—

—

—

—

—

—

C₄×Dic₉₇

—

C₄×Dic₉₇

Q₈

yes

characteristic

C₄×Dic₉₇

Q₈

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

C₂×C₈×C₉₇

D₄

yes

—

C₄⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₂³×C₉₇

#41

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₈×C₉₇

—

#11715895

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₉₇

#126

yes

—

C₄×C₉₇

—

C₄×C₉₇

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

—

Dic₉₇

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

—

—

—

—

D₉₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

characteristic

#117

C₂×D₉₇

yes

—

C₂×C₄²

—

#108

—

194

—

C₄⋊C₈

—

194

—

C₂²×C₄

—

194

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

194

—

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

D₃₈₈

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

194

—

C₂³

#13095510

yes

characteristic

C₂×C₄

—

C₂×C₄

—

388

—

C₂×C₄

—

194

—

C₂×C₄

—

194

—

C₂×C₄

C₂×D₃₈₈

yes

characteristic

C₂×C₄

Q₈×D₉₇

yes

characteristic

C₂×C₄

#13095510

yes

—

C₂×C₄

#13095511

yes

—

C₈

—

C₈

—

388

—

C₈

—

388

—

C₂²

—

C₂²

#15690790

yes

characteristic

C₂²

#15690803

yes

—

C₄

—

C₄

—

388

—

C₄

—

194

—

C₄

#15690794

yes

characteristic

100

C₄

#15690798

yes

characteristic

101

C₄

#15690800

yes

—

102

C₂

—

103

C₂

—

104

C₂

#20260675

yes

characteristic

	subgroup	G/H
center	C₄	#15690798
commutator, cosocle	C₂²×C₄×C₉₇	C₂³
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	H(3,ℤ/4)×C₉₇	C₂
socle	C₂×C₉₇	#400
prime radical	C₂³×C₉₇	#41
hypercenter	H(3,ℤ/4)	D₉₇
hypocenter	C₉₇	#2102
LCS	G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#15690798

commutator, cosocle

C₂²×C₄×C₉₇

C₂³

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

H(3,ℤ/4)×C₉₇

C₂

C₂×C₉₇

C₂³×C₉₇

D₉₇

C₉₇

G ▷ C₂²×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
53. C₉₇	#2102	semidirect

H ≅ G/N

type

53. C₉₇

#2102

semidirect

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

subgroup	index
#20260784	2
#20260758	2
#20260505	2
#20260494	2
H(3,ℤ/4)×C₉₇	2
#2102	97

subgroup

index

H(3,ℤ/4)×C₉₇

Subgroups — #27143312

Conjugacy classes of subgroups

Complements

Maximal subgroups