Group #27143357 — subgroups

Subgroups — #27143357

Order 12416.

220 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 105 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×#15690750	C₂	1	2	yes	maximal, complemented
2	#20260517	C₂	1	4	yes	maximal, complemented
3	C₂×#131×C₉₇	C₂	1	1	yes	maximal, characteristic
4	#15690750	—	2	2	no	—
5	#15690750	C₂²	1	4	yes	—
6	C₂×#13095486	—	2	2	no	—
7	C₂×#41×C₉₇	C₄	1	1	yes	characteristic
8	#131×C₉₇	—	2	2	no	—
9	#131×C₉₇	—	2	2	no	—
10	C₂×M₁₆×C₉₇	C₄	1	1	yes	characteristic
11	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
12	C₂²×C₈×C₉₇	C₄	1	2	yes	—
13	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
14	#13095486	—	2	4	no	—
15	#41×C₉₇	—	2	2	no	—
16	#41×C₉₇	—	2	2	no	—
17	C₂×Q₈×C₉₇	C₈	1	1	yes	characteristic
18	C₂×D₄×C₉₇	C₈	1	2	yes	—
19	C₂×D₄×C₉₇	C₈	1	1	yes	characteristic
20	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
22	M₁₆×C₉₇	—	2	2	no	—
23	M₁₆×C₉₇	—	2	4	no	—
24	C₂×C₈×C₉₇	—	2	2	no	—
25	C₂×C₈×C₉₇	—	2	2	no	—
26	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
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	C₂³	1	1	yes	characteristic
31	C₂³×C₉₇	C₂×C₈	1	2	yes	—
32	C₂³×C₉₇	C₂×C₈	1	1	yes	characteristic
33	Q₈×C₉₇	—	2	2	no	—
34	D₄×C₉₇	—	2	4	no	—
35	D₄×C₉₇	—	2	2	no	—
36	C₂×C₄×C₉₇	—	2	2	no	—
37	C₂×C₄×C₉₇	—	2	2	no	—
38	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₉₇	C₂×C₈	1	2	yes	—
40	C₂×C₄×C₉₇	C₂×C₈	1	1	yes	characteristic
41	C₂×C₄×C₉₇	M₁₆	1	2	yes	—
42	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
43	C₈×C₉₇	—	2	2	no	—
44	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₈×C₉₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₈×C₉₇	C₂×D₄	1	2	yes	—
47	C₂²×C₉₇	—	2	2	no	—
48	C₂²×C₉₇	—	4	1	no	—
49	C₂²×C₉₇	—	4	2	no	—
50	C₂²×C₉₇	C₂²⋊C₈	1	2	yes	—
51	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
52	C₄×C₉₇	—	2	2	no	—
53	C₄×C₉₇	C₂²⋊C₈	1	2	yes	—
54	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₉₇	C₂×M₁₆	1	1	yes	characteristic
56	C₂×C₉₇	—	2	1	no	—
57	C₂×C₉₇	—	2	2	no	—
58	C₂×C₉₇	—	4	2	no	—
59	C₂×C₉₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
60	#2115	—	97	1	no	maximal, Sylow
61	C₉₇	#2115	1	1	yes	characteristic, complemented, Sylow
62	C₂×#131	D₉₇	1	1	yes	characteristic
63	C₂×M₃₂	—	97	2	no	—
64	#349	—	97	4	no	—
65	C₂×#41	Dic₉₇	1	1	yes	characteristic
66	#131	—	2	2	no	—
67	#131	—	2	2	no	—
68	C₂×M₁₆	Dic₉₇	1	1	yes	characteristic
69	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
70	C₂²×C₈	Dic₉₇	1	2	yes	—
71	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
72	M₃₂	—	97	4	no	—
73	M₃₂	—	194	2	no	—
74	C₂×C₁₆	—	194	2	no	—
75	#41	—	2	2	no	—
76	#41	—	2	2	no	—
77	C₂×Q₈	#11715895	1	1	yes	characteristic
78	C₂×D₄	#11715895	1	2	yes	—
79	C₂×D₄	#11715895	1	1	yes	characteristic
80	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
81	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
82	M₁₆	—	2	2	no	—
83	M₁₆	—	2	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	C₂×Dic₉₇	1	2	yes	—
87	C₂×C₈	C₂×Dic₉₇	1	1	yes	characteristic
88	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
89	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
90	C₂×C₈	C₂²×D₉₇	1	1	yes	characteristic
91	C₁₆	—	194	4	no	—
92	C₂³	C₂×#11715895	1	2	yes	—
93	C₂³	C₂×#11715895	1	1	yes	characteristic
94	Q₈	—	2	2	no	—
95	D₄	—	2	4	no	—
96	D₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₂²⋊Dic₉₇	1	2	yes	—
100	C₂×C₄	C₂×#11715895	1	2	yes	—
101	C₂×C₄	C₂×#11715895	1	1	yes	characteristic
102	C₂×C₄	#13095495	1	2	yes	—
103	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
104	C₈	—	2	2	no	—
105	C₈	C₂²⋊Dic₉₇	1	1	yes	characteristic
106	C₈	C₂²⋊Dic₉₇	1	1	yes	characteristic
107	C₈	C₂×C₂²⋊D₉₇	1	2	yes	—
108	C₂²	—	2	2	no	—
109	C₂²	—	4	1	no	—
110	C₂²	—	4	2	no	—
111	C₂²	C₂²⋊#11715895	1	2	yes	—
112	C₂²	C₂²×#11715895	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	C₂²⋊#11715895	1	2	yes	—
115	C₄	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
116	C₄	C₂×#13095495	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	2	no	—
119	C₂	—	4	2	no	—
120	C₂	C₂×C₂²⋊#11715895	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690750

C₂

yes

maximal, complemented

#20260517

C₂

yes

maximal, complemented

C₂×#131×C₉₇

C₂

yes

maximal, characteristic

#15690750

—

#15690750

C₂²

yes

—

C₂×#13095486

—

C₂×#41×C₉₇

C₄

yes

characteristic

#131×C₉₇

—

#131×C₉₇

—

C₂×M₁₆×C₉₇

C₄

yes

characteristic

C₂×M₁₆×C₉₇

C₂²

yes

—

C₂²×C₈×C₉₇

C₄

yes

—

C₂²×C₈×C₉₇

C₂²

yes

characteristic

#13095486

—

#41×C₉₇

—

#41×C₉₇

—

C₂×Q₈×C₉₇

C₈

yes

characteristic

C₂×D₄×C₉₇

C₈

yes

—

C₂×D₄×C₉₇

C₈

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

M₁₆×C₉₇

—

M₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

C₂×C₄

yes

characteristic

C₂×C₈×C₉₇

D₄

yes

—

C₂×C₈×C₉₇

D₄

yes

—

C₂×C₈×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

C₂×C₈

yes

—

C₂³×C₉₇

C₂×C₈

yes

characteristic

Q₈×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂×C₈

yes

—

C₂×C₄×C₉₇

C₂×C₈

yes

characteristic

C₂×C₄×C₉₇

M₁₆

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₈×C₉₇

—

C₈×C₉₇

C₂²⋊C₄

yes

characteristic

C₈×C₉₇

C₂²⋊C₄

yes

characteristic

C₈×C₉₇

C₂×D₄

yes

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂²⋊C₈

yes

—

C₂²×C₉₇

C₂²×C₈

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

C₂²⋊C₈

yes

—

C₄×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₉₇

C₂×M₁₆

yes

characteristic

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

C₂×C₂²⋊C₈

yes

—

C₉₇

yes

characteristic, complemented, Sylow

C₂×#131

D₉₇

yes

characteristic

C₂×M₃₂

—

#349

—

C₂×#41

Dic₉₇

yes

characteristic

#131

—

#131

—

C₂×M₁₆

Dic₉₇

yes

characteristic

C₂×M₁₆

C₂×D₉₇

yes

—

C₂²×C₈

Dic₉₇

yes

—

C₂²×C₈

C₂×D₉₇

yes

characteristic

M₃₂

—

M₃₂

—

194

—

C₂×C₁₆

—

194

—

#41

—

#41

—

C₂×Q₈

#11715895

yes

characteristic

C₂×D₄

#11715895

yes

—

C₂×D₄

#11715895

yes

characteristic

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₉₇

yes

—

C₂×C₈

C₂×Dic₉₇

yes

characteristic

C₂×C₈

C₂²⋊D₉₇

yes

—

C₂×C₈

C₂²⋊D₉₇

yes

—

C₂×C₈

C₂²×D₉₇

yes

characteristic

C₁₆

—

194

—

C₂³

C₂×#11715895

yes

—

C₂³

C₂×#11715895

yes

characteristic

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₉₇

yes

—

100

C₂×C₄

C₂×#11715895

yes

—

101

C₂×C₄

C₂×#11715895

yes

characteristic

102

C₂×C₄

#13095495

yes

—

103

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

104

C₈

—

105

C₈

C₂²⋊Dic₉₇

yes

characteristic

106

C₈

C₂²⋊Dic₉₇

yes

characteristic

107

C₈

C₂×C₂²⋊D₉₇

yes

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

C₂²⋊#11715895

yes

—

112

C₂²

C₂²×#11715895

yes

characteristic

113

C₄

—

114

C₄

C₂²⋊#11715895

yes

—

115

C₄

C₂×C₂²⋊Dic₉₇

yes

characteristic

116

C₄

C₂×#13095495

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

C₂×C₂²⋊#11715895

yes

characteristic

	subgroup	G/H
center	C₈	C₂²⋊Dic₉₇
commutator	C₂²×C₉₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₉₇
Fitting	C₂×#131×C₉₇	C₂
socle	C₂×C₉₇	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₉₇	C₄
cosocle	C₂×C₈×C₉₇	C₂³
hypercenter	C₂×#131	D₉₇
hypocenter	C₉₇	#2115
LCS	G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

subgroup

G/H

center

C₈

C₂²⋊Dic₉₇

commutator

C₂²×C₉₇

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₉₇

Fitting

C₂×#131×C₉₇

C₂

socle

C₂×C₉₇

C₂×C₂²⋊C₈

prime radical

C₂×#41×C₉₇

C₄

cosocle

C₂×C₈×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂²×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#131

N	H ≅ G/N	type
1. C₂×#15690750	C₂	semidirect
2. #20260517	C₂	semidirect
61. C₉₇	#2115	semidirect

H ≅ G/N

type

1. C₂×#15690750

C₂

semidirect

2. #20260517

C₂

semidirect

61. C₉₇

#2115

semidirect

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

subgroup	index
C₂×#15690750	2
#20260517	2
C₂×#131×C₉₇	2
#2115	97

subgroup

index

C₂×#15690750

#20260517

C₂×#131×C₉₇

#2115

Subgroups — #27143357

Conjugacy classes of subgroups

Complements

Maximal subgroups