Group #27142484 — subgroups

Subgroups — #27142484

Order 12416.

208 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 93 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×#15690746	C₂	1	1	yes	maximal, characteristic
2	#20260773	C₂	1	1	yes	maximal, characteristic
3	#20260761	C₂	1	2	yes	maximal
4	#20260495	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095495	C₂²	1	2	yes	—
7	#15690769	C₂²	1	2	yes	—
8	#15690746	—	2	2	no	—
9	#15690744	C₄	1	2	yes	complemented
10	#15690742	C₄	1	2	yes	complemented
11	#15690739	—	2	4	no	—
12	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₉₇	—	2	2	no	—
16	#13095495	—	2	2	no	—
17	#13095495	—	2	2	no	—
18	C₂×#11715895	—	2	2	no	—
19	C₄⋊Dic₉₇	C₂×C₄	1	4	yes	—
20	C₂²⋊Dic₉₇	—	2	2	no	—
21	C₄×Dic₉₇	—	2	2	no	—
22	C₂²×C₄×C₉₇	—	2	1	no	—
23	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₉₇	—	2	1	no	—
25	C₂×C₈×C₉₇	—	2	1	no	—
26	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₉₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₉₇	D₄	1	2	yes	—
29	C₂×C₈×C₉₇	D₄	1	2	yes	—
30	C₂×C₈×C₉₇	Q₈	1	2	yes	—
31	C₄²×C₉₇	—	2	1	no	—
32	C₄²×C₉₇	Q₈	1	2	yes	—
33	C₂³×C₉₇	#41	1	1	yes	characteristic
34	C₂×C₄×C₉₇	—	2	4	no	—
35	C₂×C₄×C₉₇	—	2	2	no	—
36	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
42	C₂×Dic₉₇	—	2	4	no	—
43	C₈×C₉₇	—	2	2	no	—
44	C₈×C₉₇	C₄⋊C₄	1	2	yes	—
45	C₈×C₉₇	C₄⋊C₄	1	2	yes	—
46	#11715895	—	4	2	no	—
47	C₂²×C₉₇	—	2	2	no	—
48	C₂²×C₉₇	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₉₇	#125	1	1	yes	characteristic
51	C₄×C₉₇	—	2	4	no	—
52	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₉₇	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₉₇	#121	1	1	yes	characteristic
55	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₉₇	—	4	4	no	—
57	C₂×C₉₇	—	2	1	no	—
58	C₂×C₉₇	—	2	1	no	—
59	C₂×C₉₇	#389	1	1	yes	characteristic
60	C₂×C₉₇	#443	1	2	yes	—
61	#1945	—	97	1	no	maximal, Sylow
62	C₉₇	#1945	1	1	yes	characteristic, complemented, Sylow
63	C₂×#108	—	97	1	no	—
64	#427	—	97	1	no	—
65	#424	—	97	2	no	—
66	C₂×C₄×C₈	D₉₇	1	1	yes	characteristic
67	#339	—	97	2	no	—
68	C₂×M₁₆	—	97	2	no	—
69	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
71	#117	—	97	2	no	—
72	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
73	#108	—	194	2	no	—
74	#107	—	97	2	no	—
75	#106	—	97	2	no	—
76	C₄⋊C₈	—	194	4	no	—
77	C₄×C₈	—	2	2	no	—
78	C₂²×C₄	—	2	1	no	—
79	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
80	M₁₆	—	194	2	no	—
81	M₁₆	—	194	2	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	194	2	no	—
85	C₂×C₈	Dic₁₉₄	1	2	yes	—
86	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
87	C₂×C₈	C₄×D₉₇	1	1	yes	characteristic
88	C₂×C₈	D₃₈₈	1	2	yes	—
89	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
90	C₄⋊C₄	—	97	4	no	—
91	C₂²⋊C₄	—	194	2	no	—
92	C₄²	—	2	1	no	—
93	C₄²	—	194	2	no	—
94	C₄²	Dic₁₉₄	1	2	yes	—
95	C₂³	#13095511	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	194	4	no	—
99	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
101	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
102	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
104	C₂×C₄	#13095511	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	388	2	no	—
107	C₈	#13095490	1	2	yes	—
108	C₈	#13095490	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	C₄×D₃₈₈	1	1	yes	characteristic
111	C₂²	#15690789	1	1	yes	characteristic
112	C₂²	#15690802	1	1	yes	characteristic
113	C₄	—	2	4	no	—
114	C₄	—	388	4	no	—
115	C₄	C₂×#13095490	1	1	yes	characteristic
116	C₄	C₄×Dic₁₉₄	1	1	yes	characteristic
117	C₄	#15690784	1	1	yes	characteristic
118	C₄	#15690810	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#20260633	1	1	yes	characteristic
122	C₂	#20260853	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690746

C₂

yes

maximal, characteristic

#20260773

C₂

yes

maximal, characteristic

#20260761

C₂

yes

maximal

#20260495

C₂

yes

maximal

C₂×C₄×C₈×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

#15690769

C₂²

yes

—

#15690746

—

#15690744

C₄

yes

complemented

#15690742

C₄

yes

complemented

#15690739

—

C₂²×C₈×C₉₇

C₂²

yes

characteristic

C₂²×C₈×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₄×C₈×C₉₇

—

#13095495

—

#13095495

—

C₂×#11715895

—

C₄⋊Dic₉₇

C₂×C₄

yes

—

C₂²⋊Dic₉₇

—

C₄×Dic₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

characteristic

C₂×C₈×C₉₇

C₂×C₄

yes

characteristic

C₂×C₈×C₉₇

D₄

yes

—

C₂×C₈×C₉₇

D₄

yes

—

C₂×C₈×C₉₇

Q₈

yes

—

C₄²×C₉₇

—

C₄²×C₉₇

Q₈

yes

—

C₂³×C₉₇

#41

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

#41

yes

characteristic

C₂×Dic₉₇

—

C₈×C₉₇

—

C₈×C₉₇

C₄⋊C₄

yes

—

C₈×C₉₇

C₄⋊C₄

yes

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

C₄×D₄

yes

characteristic

C₂²×C₉₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₉₇

#125

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₉₇

C₄×Q₈

yes

characteristic

C₄×C₉₇

#121

yes

characteristic

C₄×C₉₇

C₄⋊Q₈

yes

characteristic

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#389

yes

characteristic

C₂×C₉₇

#443

yes

—

#1945

—

maximal, Sylow

C₉₇

#1945

yes

characteristic, complemented, Sylow

C₂×#108

—

#427

—

#424

—

C₂×C₄×C₈

D₉₇

yes

characteristic

#339

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₉₇

yes

characteristic

C₂²×C₈

C₂×D₉₇

yes

characteristic

#117

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

#108

—

194

—

#107

—

#106

—

C₄⋊C₈

—

194

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

Dic₁₉₄

yes

—

C₂×C₈

C₄×D₉₇

yes

characteristic

C₂×C₈

C₄×D₉₇

yes

characteristic

C₂×C₈

D₃₈₈

yes

—

C₂×C₈

C₂²⋊D₉₇

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

194

—

C₄²

—

C₄²

—

194

—

C₄²

Dic₁₉₄

yes

—

C₂³

#13095511

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

194

—

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

101

C₂×C₄

C₂×D₃₈₈

yes

characteristic

102

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

103

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

104

C₂×C₄

#13095511

yes

characteristic

105

C₈

—

106

C₈

—

388

—

107

C₈

#13095490

yes

—

108

C₈

#13095490

yes

—

109

C₂²

—

110

C₂²

C₄×D₃₈₈

yes

characteristic

111

C₂²

#15690789

yes

characteristic

112

C₂²

#15690802

yes

characteristic

113

C₄

—

114

C₄

—

388

—

115

C₄

C₂×#13095490

yes

characteristic

116

C₄

C₄×Dic₁₉₄

yes

characteristic

117

C₄

#15690784

yes

characteristic

118

C₄

#15690810

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#20260633

yes

characteristic

122

C₂

#20260853

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₈₈
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×C₄×C₈×C₉₇	C₂
socle	C₂²×C₉₇	C₄×D₄
prime radical	C₂³×C₉₇	#41
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×C₄×C₈	D₉₇
hypocenter	C₉₇	#1945
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₃₈₈

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂×C₄×C₈×C₉₇

C₂

socle

C₂²×C₉₇

C₄×D₄

prime radical

C₂³×C₉₇

#41

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₂×C₄×C₈

D₉₇

hypocenter

C₉₇

#1945

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. #15690744	C₄	semidirect
10. #15690742	C₄	semidirect
62. C₉₇	#1945	semidirect

H ≅ G/N

type

9. #15690744

C₄

semidirect

10. #15690742

C₄

semidirect

62. C₉₇

#1945

semidirect

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

subgroup	index
C₂×#15690746	2
#20260773	2
#20260761	2
#20260495	2
C₂×C₄×C₈×C₉₇	2
#1945	97

subgroup

index

C₂×C₄×C₈×C₉₇

Subgroups — #27142484

Conjugacy classes of subgroups

Complements

Maximal subgroups