Group #27142728 — subgroups

Subgroups — #27142728

Order 12416.

228 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 89 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×#15690802	C₂	1	1	yes	maximal, characteristic
2	C₄×#13095495	C₂	1	1	yes	maximal, characteristic
3	#20260503	C₂	1	4	yes	maximal
4	#423×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095495	C₂²	1	2	yes	—
6	#15690802	—	2	1	no	—
7	#15690802	—	2	1	no	—
8	#15690802	—	2	2	no	—
9	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
10	C₂×#13095490	C₄	1	4	yes	—
11	#15690721	—	2	1	no	—
12	C₄×#11715895	—	2	1	no	—
13	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₉₇	—	2	2	no	—
16	C₂²×Dic₉₇	C₂×C₄	1	4	yes	—
17	#13095495	—	2	2	no	—
18	#13095495	Q₈	1	4	yes	—
19	C₂×#11715895	—	2	2	no	—
20	C₄⋊Dic₉₇	—	2	2	no	—
21	C₄⋊Dic₉₇	—	2	2	no	—
22	#13095490	—	2	4	no	—
23	#13095490	—	2	4	no	—
24	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₉₇	—	2	4	no	—
27	C₂×C₈×C₉₇	—	2	2	no	—
28	C₄²×C₉₇	D₄	1	1	yes	characteristic
29	C₄²×C₉₇	D₄	1	1	yes	characteristic
30	C₄²×C₉₇	D₄	1	2	yes	—
31	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₉₇	—	4	2	no	—
33	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₉₇	#41	1	2	yes	—
39	C₂×Dic₉₇	—	2	4	no	—
40	C₂×Dic₉₇	—	4	4	no	—
41	C₈×C₉₇	—	4	2	no	—
42	#11715895	—	2	4	no	—
43	C₂²×C₉₇	—	2	2	no	—
44	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₉₇	C₄×Q₈	1	2	yes	—
46	C₄×C₉₇	—	2	2	no	—
47	C₄×C₉₇	—	2	2	no	—
48	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
49	C₄×C₉₇	#124	1	1	yes	characteristic
50	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
51	Dic₉₇	—	4	4	no	—
52	C₂×C₉₇	—	2	2	no	—
53	C₂×C₉₇	#391	1	1	yes	characteristic
54	C₂×C₉₇	#413	1	2	yes	—
55	#1998	—	97	1	no	maximal, Sylow
56	C₉₇	#1998	1	1	yes	characteristic, complemented, Sylow
57	C₂×#125	—	97	1	no	—
58	#423	D₉₇	1	1	yes	characteristic
59	C₄×M₁₆	—	97	1	no	—
60	#343	—	97	4	no	—
61	C₂×M₁₆	—	97	2	no	—
62	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
63	#125	—	194	1	no	—
64	#125	—	194	1	no	—
65	#125	—	194	2	no	—
66	C₂×C₄⋊C₄	—	97	4	no	—
67	C₂×C₄⋊C₄	—	97	2	no	—
68	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
69	C₄⋊C₈	—	2	2	no	—
70	#97	—	194	1	no	—
71	C₄×C₈	—	194	1	no	—
72	C₂²×C₄	—	97	4	no	—
73	C₂²×C₄	C₄×D₉₇	1	2	yes	—
74	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
75	M₁₆	—	2	4	no	—
76	M₁₆	—	97	4	no	—
77	M₁₆	—	194	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	194	2	no	—
80	C₄⋊C₄	—	194	4	no	—
81	C₄⋊C₄	—	194	4	no	—
82	C₄⋊C₄	—	194	2	no	—
83	C₄⋊C₄	—	194	2	no	—
84	C₄²	D₃₈₈	1	1	yes	characteristic
85	C₄²	D₃₈₈	1	1	yes	characteristic
86	C₄²	C₂²⋊D₉₇	1	2	yes	—
87	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	388	4	no	—
90	C₂×C₄	—	194	4	no	—
91	C₂×C₄	#13095488	1	2	yes	—
92	C₂×C₄	#13095488	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
94	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
95	C₂×C₄	Q₈×D₉₇	1	2	yes	—
96	C₂×C₄	#13095510	1	2	yes	—
97	C₈	—	4	2	no	—
98	C₈	—	194	4	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×#13095488	1	1	yes	characteristic
101	C₂²	#15690775	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	388	4	no	—
105	C₄	#15690790	1	2	yes	—
106	C₄	#15690797	1	1	yes	characteristic
107	C₄	#15690808	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#20260640	1	1	yes	characteristic
110	C₂	#20260717	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690802

C₂

yes

maximal, characteristic

C₄×#13095495

C₂

yes

maximal, characteristic

#20260503

C₂

yes

maximal

#423×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

#15690802

—

#15690802

—

#15690802

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×#13095490

C₄

yes

—

#15690721

—

C₄×#11715895

—

C₂×M₁₆×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

—

C₂²×Dic₉₇

C₂×C₄

yes

—

#13095495

—

#13095495

Q₈

yes

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

#13095490

—

#13095490

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₄²×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

C₄×Q₈

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

C₄×C₉₇

#124

yes

characteristic

C₄×C₉₇

C₄⋊Q₈

yes

characteristic

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#391

yes

characteristic

C₂×C₉₇

#413

yes

—

#1998

—

maximal, Sylow

C₉₇

#1998

yes

characteristic, complemented, Sylow

C₂×#125

—

#423

D₉₇

yes

characteristic

C₄×M₁₆

—

#343

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

—

#125

—

194

—

#125

—

194

—

#125

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

#97

—

194

—

C₄×C₈

—

194

—

C₂²×C₄

—

C₂²×C₄

C₄×D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

194

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄²

D₃₈₈

yes

characteristic

C₄²

D₃₈₈

yes

characteristic

C₄²

C₂²⋊D₉₇

yes

—

C₂³

C₂×C₄×D₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

388

—

C₂×C₄

—

194

—

C₂×C₄

#13095488

yes

—

C₂×C₄

#13095488

yes

—

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

C₂×C₄

C₂×D₃₈₈

yes

characteristic

C₂×C₄

Q₈×D₉₇

yes

—

C₂×C₄

#13095510

yes

—

C₈

—

C₈

—

194

—

C₂²

—

100

C₂²

C₂×#13095488

yes

characteristic

101

C₂²

#15690775

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

388

—

105

C₄

#15690790

yes

—

106

C₄

#15690797

yes

characteristic

107

C₄

#15690808

yes

characteristic

108

C₂

—

109

C₂

#20260640

yes

characteristic

110

C₂

#20260717

yes

—

	subgroup	G/H
center	C₂²	C₂×#13095488
commutator, prime radical	C₂³×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#423×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#423	D₉₇
hypocenter	C₉₇	#1998
LCS	G ▷ C₂³×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×#13095488

commutator, prime radical

C₂³×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#423×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂³×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
56. C₉₇	#1998	semidirect

H ≅ G/N

type

56. C₉₇

#1998

semidirect

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

subgroup	index
C₂×#15690802	2
C₄×#13095495	2
#20260503	2
#423×C₉₇	2
#1998	97

subgroup

index

#423×C₉₇

Subgroups — #27142728

Conjugacy classes of subgroups

Complements

Maximal subgroups