Group #27141817 — subgroups

Subgroups — #27141817

Order 12416.

272 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 85 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×#15690769	C₂	1	1	yes	maximal, characteristic
2	#20260699	C₂	1	1	yes	maximal, characteristic
3	#20260503	C₂	1	4	yes	maximal, complemented
4	#407×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095495	C₂²	1	2	yes	—
6	#15690769	—	2	2	no	—
7	#15690769	—	2	2	no	—
8	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
9	C₂×C₂²⋊Dic₉₇	C₄	1	2	yes	—
10	C₂×C₄×Dic₉₇	C₄	1	2	yes	—
11	C₂²⋊#11715895	—	2	2	no	—
12	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
14	C₂²⋊C₈×C₉₇	—	2	2	no	—
15	C₂²×Dic₉₇	C₂×C₄	1	4	yes	—
16	#13095495	—	2	4	no	—
17	C₂×#11715895	—	2	2	no	—
18	C₄⋊Dic₉₇	—	2	2	no	—
19	C₄⋊Dic₉₇	—	2	2	no	—
20	C₂²⋊Dic₉₇	—	2	4	no	—
21	C₄×Dic₉₇	—	2	4	no	—
22	C₂⁴×C₉₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₉₇	—	2	4	no	—
24	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₉₇	D₄	1	2	yes	—
26	C₂²×C₄×C₉₇	D₄	1	2	yes	—
27	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
28	M₁₆×C₉₇	—	2	4	no	—
29	C₂×C₈×C₉₇	—	2	2	no	—
30	C₂³×C₉₇	—	2	2	no	—
31	C₂³×C₉₇	—	4	2	no	—
32	C₂³×C₉₇	C₂²⋊C₄	1	2	yes	—
33	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₉₇	—	2	2	no	—
35	C₂×C₄×C₉₇	—	2	4	no	—
36	C₂×C₄×C₉₇	—	2	4	no	—
37	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₉₇	#41	1	4	yes	—
41	C₂×Dic₉₇	—	2	4	no	—
42	C₂×Dic₉₇	—	4	4	no	—
43	C₈×C₉₇	—	4	2	no	—
44	#11715895	—	4	2	no	—
45	C₂²×C₉₇	—	2	2	no	—
46	C₂²×C₉₇	—	2	2	no	—
47	C₂²×C₉₇	—	2	4	no	—
48	C₂²×C₉₇	—	4	4	no	—
49	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₉₇	#117	1	2	yes	—
51	C₄×C₉₇	—	2	2	no	—
52	C₄×C₉₇	#123	1	2	yes	—
53	C₄×C₉₇	#123	1	2	yes	—
54	Dic₉₇	—	4	4	no	—
55	C₂×C₉₇	—	2	2	no	—
56	C₂×C₉₇	—	4	2	no	—
57	C₂×C₉₇	#381	1	1	yes	characteristic
58	C₂×C₉₇	#413	1	2	yes	—
59	#1822	—	97	1	no	maximal, Sylow
60	C₉₇	#1822	1	1	yes	characteristic, complemented, Sylow
61	C₂×#117	—	97	1	no	—
62	#407	—	97	1	no	—
63	#407	D₉₇	1	1	yes	characteristic
64	#343	—	97	4	no	—
65	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
66	C₂×M₁₆	—	97	2	no	—
67	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
68	#117	—	194	2	no	—
69	#117	—	194	2	no	—
70	C₂×C₄⋊C₄	—	97	2	no	—
71	C₂×C₂²⋊C₄	—	97	2	no	—
72	C₂×C₄²	—	97	2	no	—
73	C₂²⋊C₈	—	2	2	no	—
74	C₂²⋊C₈	—	194	2	no	—
75	C₂⁴	C₄×D₉₇	1	1	yes	characteristic
76	C₂²×C₄	—	2	4	no	—
77	C₂²×C₄	—	97	4	no	—
78	C₂²×C₄	C₄×D₉₇	1	1	yes	characteristic
79	C₂²×C₄	D₃₈₈	1	2	yes	—
80	C₂²×C₄	C₂²⋊D₉₇	1	2	yes	—
81	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
82	M₁₆	—	2	4	no	—
83	M₁₆	—	194	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	194	2	no	—
86	C₄⋊C₄	—	194	2	no	—
87	C₄⋊C₄	—	194	2	no	—
88	C₂²⋊C₄	—	194	4	no	—
89	C₄²	—	194	4	no	—
90	C₂³	—	2	2	no	—
91	C₂³	—	4	2	no	—
92	C₂³	#13095488	1	2	yes	—
93	C₂³	C₂×C₄×D₉₇	1	1	yes	characteristic
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	388	4	no	—
98	C₂×C₄	—	194	4	no	—
99	C₂×C₄	#13095488	1	2	yes	—
100	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
101	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
102	C₂×C₄	#13095510	1	4	yes	—
103	C₈	—	4	2	no	—
104	C₈	—	388	2	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	2	4	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	4	4	no	—
109	C₂²	C₂×#13095488	1	1	yes	characteristic
110	C₂²	#15690766	1	2	yes	—
111	C₄	—	2	2	no	—
112	C₄	—	388	4	no	—
113	C₄	#15690791	1	2	yes	—
114	C₄	#15690794	1	2	yes	—
115	C₂	—	2	2	no	—
116	C₂	—	4	2	no	—
117	C₂	#20260596	1	1	yes	characteristic
118	C₂	#20260717	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690769

C₂

yes

maximal, characteristic

#20260699

C₂

yes

maximal, characteristic

#20260503

C₂

yes

maximal, complemented

#407×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

#15690769

—

#15690769

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×C₂²⋊Dic₉₇

C₄

yes

—

C₂×C₄×Dic₉₇

C₄

yes

—

C₂²⋊#11715895

—

C₂³×C₄×C₉₇

C₂²

yes

characteristic

C₂×M₁₆×C₉₇

C₂²

yes

—

C₂²⋊C₈×C₉₇

—

C₂²×Dic₉₇

C₂×C₄

yes

—

#13095495

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₄⋊Dic₉₇

—

C₂²⋊Dic₉₇

—

C₄×Dic₉₇

—

C₂⁴×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

C₂²⋊C₄

yes

—

C₂³×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₈×C₉₇

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

#117

yes

—

C₄×C₉₇

—

C₄×C₉₇

#123

yes

—

C₄×C₉₇

#123

yes

—

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#381

yes

characteristic

C₂×C₉₇

#413

yes

—

#1822

—

maximal, Sylow

C₉₇

#1822

yes

characteristic, complemented, Sylow

C₂×#117

—

#407

—

#407

D₉₇

yes

characteristic

#343

—

C₂³×C₄

C₂×D₉₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

—

#117

—

194

—

#117

—

194

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

—

194

—

C₂⁴

C₄×D₉₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₉₇

yes

characteristic

C₂²×C₄

D₃₈₈

yes

—

C₂²×C₄

C₂²⋊D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

194

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₂²⋊C₄

—

194

—

C₄²

—

194

—

C₂³

—

C₂³

—

C₂³

#13095488

yes

—

C₂³

C₂×C₄×D₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

388

—

C₂×C₄

—

194

—

C₂×C₄

#13095488

yes

—

100

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

101

C₂×C₄

C₂×D₃₈₈

yes

characteristic

102

C₂×C₄

#13095510

yes

—

103

C₈

—

104

C₈

—

388

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×#13095488

yes

characteristic

110

C₂²

#15690766

yes

—

111

C₄

—

112

C₄

—

388

—

113

C₄

#15690791

yes

—

114

C₄

#15690794

yes

—

115

C₂

—

116

C₂

—

117

C₂

#20260596

yes

characteristic

118

C₂

#20260717

yes

—

	subgroup	G/H
center	C₂²	C₂×#13095488
commutator	C₂³×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	#407×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₉₇	C₂×C₄
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	#407	D₉₇
hypocenter	C₉₇	#1822
LCS	G ▷ C₂³×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂×#13095488

commutator

C₂³×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

#407×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₉₇

C₂×C₄

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂³×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
3. #20260503	C₂	semidirect
60. C₉₇	#1822	semidirect

H ≅ G/N

type

3. #20260503

C₂

semidirect

60. C₉₇

#1822

semidirect

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

subgroup	index
C₂×#15690769	2
#20260699	2
#20260503	2
#407×C₉₇	2
#1822	97

subgroup

index

#407×C₉₇

Subgroups — #27141817

Conjugacy classes of subgroups

Complements

Maximal subgroups