Group #27142527 — subgroups

Subgroups — #27142527

Order 12416.

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

normal subgroups only

This group has 184 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	#20260784	C₂	1	2	yes	maximal
2	C₂×#15690746	C₂	1	1	yes	maximal, characteristic
3	#20260690	C₂	1	1	yes	maximal, characteristic
4	#20260507	C₂	1	2	yes	maximal
5	C₂×#108×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095495	C₂²	1	2	yes	—
7	C₂×#13095495	C₂²	1	1	yes	characteristic
8	#15690769	C₂²	1	1	yes	characteristic
9	#15690746	—	2	4	no	—
10	#15690746	—	2	1	no	—
11	#15690746	—	2	1	no	—
12	#15690720	—	2	1	no	—
13	C₈×Dic₉₇	—	2	1	no	—
14	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
15	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
16	#108×C₉₇	C₄	1	4	yes	complemented
17	#13095495	—	2	1	no	—
18	#13095495	—	2	2	no	—
19	#13095495	—	2	2	no	—
20	#13095495	D₄	1	2	yes	—
21	C₂×#11715895	—	2	1	no	—
22	C₂×#11715895	—	2	2	no	—
23	C₄⋊Dic₉₇	—	2	1	no	—
24	C₂²⋊Dic₉₇	Q₈	1	2	yes	—
25	C₄×Dic₉₇	—	2	1	no	—
26	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₉₇	—	2	2	no	—
28	M₁₆×C₉₇	C₂×C₄	1	4	yes	—
29	C₂×C₈×C₉₇	—	2	2	no	—
30	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
32	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
33	C₂×C₈×C₉₇	Q₈	1	2	yes	—
34	C₂³×C₉₇	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
40	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
41	C₂×Dic₉₇	—	2	2	no	—
42	C₈×C₉₇	—	2	4	no	—
43	C₈×C₉₇	C₄⋊C₄	1	2	yes	—
44	C₈×C₉₇	C₄⋊C₄	1	2	yes	—
45	#11715895	—	2	2	no	—
46	#11715895	—	4	2	no	—
47	C₂²×C₉₇	—	4	1	no	—
48	C₂²×C₉₇	C₄×Q₈	1	1	yes	characteristic
49	C₂²×C₉₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₉₇	C₄×D₄	1	1	yes	characteristic
53	C₄×C₉₇	#121	1	1	yes	characteristic
54	C₄×C₉₇	#125	1	1	yes	characteristic
55	Dic₉₇	—	4	2	no	—
56	C₂×C₉₇	—	2	1	no	—
57	C₂×C₉₇	—	2	1	no	—
58	C₂×C₉₇	—	2	1	no	—
59	C₂×C₉₇	#389	1	1	yes	characteristic
60	#1953	—	97	1	no	maximal, Sylow
61	C₉₇	#1953	1	1	yes	characteristic, complemented, Sylow
62	#430	—	97	2	no	—
63	C₂×#108	—	97	1	no	—
64	C₂×#108	D₉₇	1	1	yes	characteristic
65	#405	—	97	1	no	—
66	#344	—	97	2	no	—
67	C₂×M₁₆	—	97	1	no	—
68	C₂×M₁₆	—	97	2	no	—
69	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
70	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
71	#117	—	97	1	no	—
72	#108	—	194	4	no	—
73	#108	—	194	1	no	—
74	#108	—	194	1	no	—
75	#108	Dic₉₇	1	4	yes	complemented
76	#97	—	194	1	no	—
77	C₄×C₈	—	194	1	no	—
78	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
79	M₁₆	—	2	2	no	—
80	M₁₆	—	97	2	no	—
81	M₁₆	—	194	2	no	—
82	M₁₆	—	194	2	no	—
83	M₁₆	—	194	1	no	—
84	M₁₆	C₂×Dic₉₇	1	4	yes	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	194	2	no	—
87	C₂×C₈	—	194	1	no	—
88	C₂×C₈	Dic₁₉₄	1	2	yes	—
89	C₂×C₈	D₃₈₈	1	1	yes	characteristic
90	C₂×C₈	D₃₈₈	1	1	yes	characteristic
91	C₂×C₈	C₂×Dic₉₇	1	2	yes	—
92	C₄⋊C₄	—	194	1	no	—
93	C₂²⋊C₄	—	97	2	no	—
94	C₄²	—	194	1	no	—
95	C₂³	Q₈×D₉₇	1	1	yes	characteristic
96	C₂×C₄	—	194	2	no	—
97	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
98	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
99	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
100	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
101	C₂×C₄	#13095510	1	1	yes	characteristic
102	C₂×C₄	#13095512	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₈	—	388	2	no	—
105	C₈	—	194	2	no	—
106	C₈	C₄⋊Dic₉₇	1	2	yes	—
107	C₈	C₄⋊Dic₉₇	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	Q₈×Dic₉₇	1	1	yes	characteristic
110	C₂²	#15690790	1	1	yes	characteristic
111	C₂²	#15690809	1	1	yes	characteristic
112	C₄	—	388	2	no	—
113	C₄	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
114	C₄	D₄×Dic₉₇	1	1	yes	characteristic
115	C₄	#15690785	1	1	yes	characteristic
116	C₄	#15690801	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#20260634	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20260784

C₂

yes

maximal

C₂×#15690746

C₂

yes

maximal, characteristic

#20260690

C₂

yes

maximal, characteristic

#20260507

C₂

yes

maximal

C₂×#108×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

C₂×#13095495

C₂²

yes

characteristic

#15690769

C₂²

yes

—

—

—

—

C₈×Dic₉₇

—

C₂×M₁₆×C₉₇

C₂²

yes

—

C₂²×C₈×C₉₇

C₂²

yes

characteristic

#108×C₉₇

C₄

yes

—

—

—

D₄

yes

—

C₂×#11715895

—

C₂×#11715895

—

C₄⋊Dic₉₇

—

C₂²⋊Dic₉₇

Q₈

yes

—

C₄×Dic₉₇

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

M₁₆×C₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

D₄

yes

characteristic

C₂×C₈×C₉₇

D₄

yes

characteristic

C₂×C₈×C₉₇

Q₈

yes

—

C₂³×C₉₇

C₂×Q₈

yes

characteristic

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

C₂×C₄×C₉₇

yes

C₂×C₄×C₉₇

yes

C₂×Dic₉₇

—

C₈×C₉₇

—

C₈×C₉₇

C₄⋊C₄

yes

—

C₈×C₉₇

C₄⋊C₄

yes

—

#11715895

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

C₄×Q₈

yes

characteristic

C₂²×C₉₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₉₇

C₄⋊Q₈

yes

characteristic

C₄×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₉₇

C₄×D₄

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

yes

—

C₉₇

yes

characteristic, complemented, Sylow

#430

—

C₂×#108

—

C₂×#108

D₉₇

yes

characteristic

#405

—

#344

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

—

C₂²×C₈

C₂×D₉₇

yes

characteristic

#117

—

#108

—

194

—

#108

—

194

—

#108

—

194

—

#108

Dic₉₇

yes

complemented

#97

—

194

—

C₄×C₈

—

194

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

194

—

M₁₆

—

194

—

M₁₆

—

194

—

M₁₆

C₂×Dic₉₇

yes

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

Dic₁₉₄

yes

—

C₂×C₈

D₃₈₈

yes

characteristic

C₂×C₈

D₃₈₈

yes

characteristic

C₂×C₈

C₂×Dic₉₇

yes

—

C₄⋊C₄

—

194

—

C₂²⋊C₄

—

C₄²

—

194

—

C₂³

Q₈×D₉₇

yes

characteristic

C₂×C₄

—

194

—

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

C₂×C₄

D₄×D₉₇

yes

characteristic

C₂×C₄

C₂×D₃₈₈

yes

characteristic

100

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

101

C₂×C₄

#13095510

yes

characteristic

102

C₂×C₄

#13095512

yes

characteristic

103

C₈

—

104

C₈

—

388

—

105

C₈

—

194

—

106

C₈

C₄⋊Dic₉₇

yes

—

107

C₈

C₄⋊Dic₉₇

yes

—

108

C₂²

—

109

C₂²

Q₈×Dic₉₇

yes

characteristic

110

C₂²

#15690790

yes

characteristic

111

C₂²

#15690809

yes

characteristic

112

C₄

—

388

—

113

C₄

C₂×C₄⋊Dic₉₇

yes

characteristic

114

C₄

D₄×Dic₉₇

yes

characteristic

115

C₄

#15690785

yes

characteristic

116

C₄

#15690801

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#20260634

yes

characteristic

	subgroup	G/H
center	C₄	#15690785
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×#108×C₉₇	C₂
socle	C₂×C₉₇	#389
prime radical	C₂³×C₉₇	C₂×Q₈
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×#108	D₉₇
hypocenter	C₉₇	#1953
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#15690785

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂×#108×C₉₇

C₂

socle

C₂×C₉₇

#389

prime radical

C₂³×C₉₇

C₂×Q₈

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
16. #108×C₉₇	C₄	semidirect
61. C₉₇	#1953	semidirect
75. #108	Dic₉₇	semidirect

H ≅ G/N

type

16. #108×C₉₇

C₄

semidirect

61. C₉₇

#1953

semidirect

75. #108

Dic₉₇

semidirect

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

subgroup	index
#20260784	2
C₂×#15690746	2
#20260690	2
#20260507	2
C₂×#108×C₉₇	2
#1953	97

subgroup

index

C₂×#108×C₉₇

Subgroups — #27142527

Conjugacy classes of subgroups

Complements

Maximal subgroups