Group #27141909 — subgroups

Subgroups — #27141909

Order 12416.

212 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂×#15690745	C₂	1	2	yes	maximal
3	#20260758	C₂	1	2	yes	maximal
4	C₂×#15690720	C₂	1	1	yes	maximal, characteristic
5	C₂×#108×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095495	C₂²	1	2	yes	—
7	C₂²×#11715895	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
9	#15690746	—	2	2	no	—
10	#15690745	—	2	4	no	—
11	#15690739	—	2	4	no	—
12	#15690720	C₄	1	4	yes	—
13	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
15	#108×C₉₇	—	2	2	no	—
16	C₂²×Dic₉₇	Q₈	1	2	yes	—
17	#13095495	—	2	2	no	—
18	#13095495	—	2	2	no	—
19	C₂×#11715895	—	2	2	no	—
20	C₂×#11715895	C₂×C₄	1	2	yes	—
21	C₂×#11715895	D₄	1	2	yes	—
22	C₂×#11715895	Q₈	1	2	yes	—
23	C₄×Dic₉₇	C₂×C₄	1	2	yes	—
24	C₄×Dic₉₇	D₄	1	2	yes	—
25	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₉₇	—	2	2	no	—
27	M₁₆×C₉₇	—	2	2	no	—
28	C₂×C₈×C₉₇	—	2	2	no	—
29	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₉₇	D₄	1	2	yes	—
31	C₂×C₈×C₉₇	Q₈	1	2	yes	—
32	C₂³×C₉₇	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₉₇	C₂×Q₈	1	1	yes	characteristic
39	C₂×Dic₉₇	—	4	2	no	—
40	C₂×Dic₉₇	C₄⋊C₄	1	4	yes	—
41	C₈×C₉₇	—	4	2	no	—
42	C₈×C₉₇	C₄⋊C₄	1	4	yes	—
43	#11715895	—	4	2	no	—
44	#11715895	C₄⋊C₄	1	4	yes	—
45	C₂²×C₉₇	—	2	2	no	—
46	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
48	C₂²×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
49	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₉₇	#127	1	1	yes	characteristic
52	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₉₇	—	2	4	no	—
54	C₂×C₉₇	—	2	1	no	—
55	C₂×C₉₇	—	2	1	no	—
56	C₂×C₉₇	#384	1	1	yes	characteristic
57	C₂×C₉₇	#430	1	2	yes	—
58	#1849	—	97	1	no	maximal, Sylow
59	C₉₇	#1849	1	1	yes	characteristic, complemented, Sylow
60	C₂×#108	—	97	2	no	—
61	C₂×#108	—	97	1	no	—
62	C₂×#108	D₉₇	1	1	yes	characteristic
63	#423	—	97	2	no	—
64	C₂×#97	—	97	1	no	—
65	C₂×M₁₆	—	97	2	no	—
66	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
67	C₂²×C₈	—	97	1	no	—
68	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
69	C₂×C₄²	—	97	1	no	—
70	#108	—	2	2	no	—
71	#108	—	194	4	no	—
72	#108	—	194	2	no	—
73	C₄⋊C₈	—	194	4	no	—
74	#97	—	97	4	no	—
75	C₂²×C₄	—	97	2	no	—
76	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
77	M₁₆	—	2	2	no	—
78	M₁₆	—	2	2	no	—
79	M₁₆	—	194	2	no	—
80	M₁₆	—	194	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	97	2	no	—
83	C₂×C₈	—	97	2	no	—
84	C₂×C₈	—	97	2	no	—
85	C₂×C₈	—	194	2	no	—
86	C₂×C₈	Dic₁₉₄	1	2	yes	—
87	C₂×C₈	C₄×D₉₇	1	2	yes	—
88	C₂×C₈	C₂²⋊D₉₇	1	2	yes	—
89	C₄²	—	97	2	no	—
90	C₄²	—	97	2	no	—
91	C₂³	Q₈×D₉₇	1	1	yes	characteristic
92	C₂×C₄	—	97	4	no	—
93	C₂×C₄	—	388	2	no	—
94	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
95	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
96	C₂×C₄	D₄×D₉₇	1	1	yes	characteristic
97	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
98	C₂×C₄	Q₈×D₉₇	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₁₉₄	1	1	yes	characteristic
100	C₈	—	4	2	no	—
101	C₈	—	97	4	no	—
102	C₈	—	388	2	no	—
103	C₈	#13095490	1	4	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	C₄⋊C₄×D₉₇	1	1	yes	characteristic
106	C₂²	#15690808	1	1	yes	characteristic
107	C₂²	#15690809	1	1	yes	characteristic
108	C₄	—	194	4	no	—
109	C₄	C₄⋊C₄×D₉₇	1	1	yes	characteristic
110	C₄	C₂×#13095490	1	1	yes	characteristic
111	C₄	#15690806	1	1	yes	characteristic
112	C₄	#15690809	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	#20260604	1	1	yes	characteristic
116	C₂	#20260782	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690746

C₂

yes

maximal, characteristic

C₂×#15690745

C₂

yes

maximal

#20260758

C₂

yes

maximal

C₂×#15690720

C₂

yes

maximal, characteristic

C₂×#108×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

C₂²×#11715895

C₂²

yes

characteristic

C₂×C₄×Dic₉₇

C₂²

yes

—

—

—

C₄

yes

—

C₂×M₁₆×C₉₇

C₂²

yes

—

C₂²×C₈×C₉₇

C₂²

yes

characteristic

#108×C₉₇

—

C₂²×Dic₉₇

Q₈

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#11715895

D₄

yes

—

C₂×#11715895

Q₈

yes

—

C₄×Dic₉₇

C₂×C₄

yes

—

C₄×Dic₉₇

D₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

M₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

D₄

yes

—

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×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

characteristic

C₂×Dic₉₇

—

C₂×Dic₉₇

C₄⋊C₄

yes

—

C₈×C₉₇

—

C₈×C₉₇

C₄⋊C₄

yes

—

#11715895

—

#11715895

C₄⋊C₄

yes

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

C₄⋊Q₈

yes

characteristic

C₂²×C₉₇

C₄⋊Q₈

yes

characteristic

C₄×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₉₇

#127

yes

characteristic

C₄×C₉₇

C₄⋊Q₈

yes

characteristic

Dic₉₇

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#384

yes

characteristic

C₂×C₉₇

#430

yes

—

#1849

—

maximal, Sylow

C₉₇

#1849

yes

characteristic, complemented, Sylow

C₂×#108

—

C₂×#108

—

C₂×#108

D₉₇

yes

characteristic

#423

—

C₂×#97

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₉₇

yes

characteristic

C₂×C₄²

—

#108

—

#108

—

194

—

#108

—

194

—

C₄⋊C₈

—

194

—

#97

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

Dic₁₉₄

yes

—

C₂×C₈

C₄×D₉₇

yes

—

C₂×C₈

C₂²⋊D₉₇

yes

—

C₄²

—

C₄²

—

C₂³

Q₈×D₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

388

—

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

C₂×C₄

D₄×D₉₇

yes

characteristic

C₂×C₄

D₄×D₉₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

C₂×C₄

Q₈×D₉₇

yes

characteristic

C₂×C₄

C₂×Dic₁₉₄

yes

characteristic

100

C₈

—

101

C₈

—

102

C₈

—

388

—

103

C₈

#13095490

yes

—

104

C₂²

—

105

C₂²

C₄⋊C₄×D₉₇

yes

characteristic

106

C₂²

#15690808

yes

characteristic

107

C₂²

#15690809

yes

characteristic

108

C₄

—

194

—

109

C₄

C₄⋊C₄×D₉₇

yes

characteristic

110

C₄

C₂×#13095490

yes

characteristic

111

C₄

#15690806

yes

characteristic

112

C₄

#15690809

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

#20260604

yes

characteristic

116

C₂

#20260782

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₉₇
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂×#108×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₄⋊C₄
prime radical	C₂³×C₉₇	C₂×Q₈
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×#108	D₉₇
hypocenter	C₉₇	#1849
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₉₇

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂×#108×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₄⋊C₄

prime radical

C₂³×C₉₇

C₂×Q₈

cosocle

C₂²×C₄×C₉₇

C₂³

D₉₇

C₉₇

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
59. C₉₇	#1849	semidirect

H ≅ G/N

type

59. C₉₇

#1849

semidirect

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

subgroup	index
C₂×#15690746	2
C₂×#15690745	2
#20260758	2
C₂×#15690720	2
C₂×#108×C₉₇	2
#1849	97

subgroup

index

C₂×#108×C₉₇

Subgroups — #27141909

Conjugacy classes of subgroups

Complements

Maximal subgroups