Group #27139586 — subgroups

Subgroups — #27139586

Order 12416.

160 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 81 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	#20260788	C₂	1	1	yes	maximal, characteristic
2	C₄×#13095495	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095495	C₄	1	2	yes	—
5	C₄⋊#11715895	—	2	1	no	—
6	C₂²⋊#11715895	C₄	1	2	yes	—
7	#15690721	—	2	1	no	—
8	C₄×#11715895	—	2	1	no	—
9	C₄×#11715895	—	2	1	no	—
10	C₂²×C₈×C₉₇	C₄	1	2	yes	—
11	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₉₇	—	2	1	no	—
13	C₄⋊C₈×C₉₇	—	2	1	no	—
14	#13095495	—	2	2	no	—
15	#13095495	C₈	1	4	yes	complemented
16	C₂×#11715895	—	2	2	no	—
17	C₂×#11715895	—	2	2	no	—
18	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₉₇	—	2	2	no	—
22	C₂×C₈×C₉₇	—	4	2	no	—
23	C₄²×C₉₇	D₄	1	1	yes	characteristic
24	C₄²×C₉₇	D₄	1	1	yes	characteristic
25	C₄²×C₉₇	D₄	1	1	yes	characteristic
26	C₄²×C₉₇	Q₈	1	1	yes	characteristic
27	C₂³×C₉₇	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₉₇	—	2	1	no	—
29	C₂×C₄×C₉₇	—	2	1	no	—
30	C₂×C₄×C₉₇	—	4	1	no	—
31	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₉₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₉₇	M₁₆	1	2	yes	—
39	C₈×C₉₇	—	4	2	no	—
40	#11715895	—	2	4	no	—
41	#11715895	—	4	2	no	—
42	C₂²×C₉₇	—	2	1	no	—
43	C₂²×C₉₇	—	2	1	no	—
44	C₂²×C₉₇	#95	1	1	yes	characteristic
45	C₂²×C₉₇	C₄×C₈	1	1	yes	characteristic
46	C₂²×C₉₇	#97	1	1	yes	characteristic
47	C₄×C₉₇	—	2	1	no	—
48	C₄×C₉₇	—	2	1	no	—
49	C₄×C₉₇	—	2	1	no	—
50	C₄×C₉₇	C₂²⋊C₈	1	2	yes	—
51	C₄×C₉₇	#100	1	1	yes	characteristic
52	C₄×C₉₇	#101	1	1	yes	characteristic
53	C₄×C₉₇	C₄⋊C₈	1	2	yes	—
54	C₂×C₉₇	—	2	1	no	—
55	C₂×C₉₇	—	2	1	no	—
56	C₂×C₉₇	#336	1	1	yes	characteristic
57	C₂×C₉₇	#343	1	1	yes	characteristic
58	C₂×C₉₇	#344	1	1	yes	characteristic
59	#1283	—	97	1	no	maximal, Sylow
60	C₉₇	#1283	1	1	yes	characteristic, complemented, Sylow
61	#431	—	97	1	no	—
62	C₂×C₄⋊C₈	D₉₇	1	1	yes	characteristic
63	C₄×M₁₆	—	97	1	no	—
64	C₂×M₁₆	—	97	2	no	—
65	C₂²×C₈	Dic₉₇	1	2	yes	—
66	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
67	C₄⋊C₈	—	2	1	no	—
68	C₄⋊C₈	—	2	1	no	—
69	C₄⋊C₈	—	194	1	no	—
70	C₂²⋊C₈	—	97	2	no	—
71	#97	—	194	1	no	—
72	C₄×C₈	—	194	1	no	—
73	C₄×C₈	—	194	1	no	—
74	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
75	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
76	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
77	M₁₆	—	97	4	no	—
78	M₁₆	—	194	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	4	2	no	—
81	C₂×C₈	—	194	2	no	—
82	C₂×C₈	—	194	2	no	—
83	C₄²	Dic₁₉₄	1	1	yes	characteristic
84	C₄²	D₃₈₈	1	1	yes	characteristic
85	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
86	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
87	C₂³	C₄×Dic₉₇	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	#13095488	1	1	yes	characteristic
92	C₂×C₄	#13095488	1	1	yes	characteristic
93	C₂×C₄	C₂²⋊Dic₉₇	1	1	yes	characteristic
94	C₂×C₄	#13095490	1	1	yes	characteristic
95	C₂×C₄	#13095490	1	1	yes	characteristic
96	C₂×C₄	C₄⋊Dic₉₇	1	1	yes	characteristic
97	C₂×C₄	C₈×D₉₇	1	2	yes	—
98	C₂×C₄	#13095494	1	2	yes	—
99	C₈	—	4	2	no	—
100	C₈	—	388	2	no	—
101	C₈	—	194	4	no	—
102	C₂²	—	2	1	no	—
103	C₂²	—	2	1	no	—
104	C₂²	#15690717	1	1	yes	characteristic
105	C₂²	C₈×Dic₉₇	1	1	yes	characteristic
106	C₂²	#15690720	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	#15690722	1	2	yes	—
111	C₄	#15690727	1	1	yes	characteristic
112	C₄	#15690729	1	1	yes	characteristic
113	C₄	#15690739	1	2	yes	—
114	C₂	—	2	1	no	—
115	C₂	—	2	1	no	—
116	C₂	#20260489	1	1	yes	characteristic
117	C₂	#20260504	1	1	yes	characteristic
118	C₂	#20260506	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260788

C₂

yes

maximal, characteristic

C₄×#13095495

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₄

yes

—

C₄⋊#11715895

—

C₂²⋊#11715895

C₄

yes

—

#15690721

—

C₄×#11715895

—

C₄×#11715895

—

C₂²×C₈×C₉₇

C₄

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

—

C₄⋊C₈×C₉₇

—

#13095495

—

#13095495

C₈

yes

complemented

C₂×#11715895

—

C₂×#11715895

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₄²×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

Q₈

yes

characteristic

C₂³×C₉₇

C₄²

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×C₈

yes

—

C₂×C₄×C₉₇

M₁₆

yes

—

C₈×C₉₇

—

#11715895

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#95

yes

characteristic

C₂²×C₉₇

C₄×C₈

yes

characteristic

C₂²×C₉₇

#97

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂²⋊C₈

yes

—

C₄×C₉₇

yes

C₄×C₉₇

yes

C₄×C₉₇

C₄⋊C₈

yes

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

#431

—

C₂×C₄⋊C₈

D₉₇

yes

characteristic

C₄×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

Dic₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

194

—

C₂²⋊C₈

—

#97

—

194

—

C₄×C₈

—

194

—

C₄×C₈

—

194

—

C₂²×C₄

C₄×D₉₇

yes

characteristic

C₂²×C₄

C₄×D₉₇

yes

characteristic

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

M₁₆

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄²

Dic₁₉₄

yes

characteristic

C₄²

D₃₈₈

yes

characteristic

C₄²

C₂²⋊D₉₇

yes

characteristic

C₄²

C₂²⋊D₉₇

yes

characteristic

C₂³

C₄×Dic₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₉₇

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₄⋊Dic₉₇

yes

characteristic

C₂×C₄

C₈×D₉₇

yes

—

C₂×C₄

#13095494

yes

—

C₈

—

100

C₈

—

388

—

101

C₈

—

194

—

102

C₂²

—

103

C₂²

—

104

C₂²

#15690717

yes

characteristic

105

C₂²

C₈×Dic₉₇

yes

characteristic

106

C₂²

#15690720

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#15690722

yes

—

111

C₄

#15690727

yes

characteristic

112

C₄

#15690729

yes

characteristic

113

C₄

#15690739

yes

—

114

C₂

—

115

C₂

—

116

C₂

#20260489

yes

characteristic

117

C₂

#20260504

yes

characteristic

118

C₂

#20260506

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₉₇
commutator	C₂²×C₉₇	C₄×C₈
Frattini	C₂×C₄²	C₂×D₉₇
Fitting	C₂×C₄⋊C₈×C₉₇	C₂
socle	C₂²×C₉₇	#95
prime radical	C₂³×C₉₇	C₄²
cosocle	C₂×C₄²×C₉₇	C₂²
hypercenter	C₂×C₄⋊C₈	D₉₇
hypocenter	C₉₇	#1283
LCS	G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₉₇

commutator

C₂²×C₉₇

C₄×C₈

Frattini

C₂×C₄²

C₂×D₉₇

Fitting

C₂×C₄⋊C₈×C₉₇

C₂

socle

C₂²×C₉₇

#95

prime radical

C₂³×C₉₇

C₄²

cosocle

C₂×C₄²×C₉₇

C₂²

hypercenter

C₂×C₄⋊C₈

D₉₇

hypocenter

C₉₇

#1283

LCS

G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
15. #13095495	C₈	semidirect
60. C₉₇	#1283	semidirect

H ≅ G/N

type

15. #13095495

C₈

semidirect

60. C₉₇

#1283

semidirect

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

subgroup	index
#20260788	2
C₄×#13095495	2
C₂×C₄⋊C₈×C₉₇	2
#1283	97

subgroup

index

#20260788

C₄×#13095495

C₂×C₄⋊C₈×C₉₇

#1283

Subgroups — #27139586

Conjugacy classes of subgroups

Complements

Maximal subgroups