Group #27141778 — subgroups

Subgroups — #27141778

Order 12416.

272 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 109 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	2	yes	maximal, complemented
2	#20260699	C₂	1	2	yes	maximal
3	#20260500	C₂	1	2	yes	maximal, complemented
4	C₂²×M₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#13095495	C₂²	1	4	yes	—
6	#15690746	—	2	4	no	—
7	C₂²⋊#11715895	—	2	4	no	—
8	C₂³×C₄×C₉₇	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×C₉₇	—	2	4	no	—
10	C₂×M₁₆×C₉₇	C₄	1	4	yes	—
11	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₉₇	C₂²	1	1	yes	characteristic
13	#13095495	—	2	4	no	—
14	#13095495	—	2	4	no	—
15	C₂×#11715895	—	2	4	no	—
16	C₂⁴×C₉₇	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₉₇	—	2	2	no	—
18	C₂²×C₄×C₉₇	—	2	2	no	—
19	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₉₇	D₄	1	2	yes	—
21	C₂²×C₄×C₉₇	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
23	M₁₆×C₉₇	—	2	8	no	—
24	C₂×C₈×C₉₇	—	2	2	no	—
25	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₉₇	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₉₇	D₄	1	4	yes	—
28	C₂³×C₉₇	—	2	2	no	—
29	C₂³×C₉₇	—	4	2	no	—
30	C₂³×C₉₇	C₄⋊C₄	1	2	yes	—
31	C₂³×C₉₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₉₇	—	2	2	no	—
33	C₂×C₄×C₉₇	—	2	2	no	—
34	C₂×C₄×C₉₇	—	2	2	no	—
35	C₂×C₄×C₉₇	—	2	4	no	—
36	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₉₇	#41	1	2	yes	—
41	C₈×C₉₇	—	2	2	no	—
42	C₈×C₉₇	C₂²⋊C₄	1	4	yes	—
43	#11715895	—	4	4	no	—
44	C₂²×C₉₇	—	2	2	no	—
45	C₂²×C₉₇	—	2	2	no	—
46	C₂²×C₉₇	—	2	4	no	—
47	C₂²×C₉₇	—	4	4	no	—
48	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₉₇	C₂²⋊Q₈	1	2	yes	—
50	C₄×C₉₇	—	2	2	no	—
51	C₄×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₄×C₉₇	#117	1	1	yes	characteristic
53	C₄×C₉₇	#121	1	2	yes	—
54	C₂×C₉₇	—	2	2	no	—
55	C₂×C₉₇	—	4	2	no	—
56	C₂×C₉₇	#380	1	1	yes	characteristic
57	C₂×C₉₇	#430	1	2	yes	—
58	#1811	—	97	1	no	maximal, Sylow
59	C₉₇	#1811	1	1	yes	characteristic, complemented, Sylow
60	C₂²×M₁₆	D₉₇	1	1	yes	characteristic
61	C₂×#108	—	97	2	no	—
62	#407	—	97	2	no	—
63	#341	—	97	2	no	—
64	C₂³×C₄	C₂×D₉₇	1	1	yes	characteristic
65	C₂×M₁₆	—	2	4	no	—
66	C₂×M₁₆	—	97	4	no	—
67	C₂×M₁₆	Dic₉₇	1	4	yes	—
68	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
69	C₂²×C₈	C₂×D₉₇	1	1	yes	characteristic
70	#108	—	194	4	no	—
71	C₂²⋊C₈	—	194	4	no	—
72	C₂⁴	Dic₁₉₄	1	1	yes	characteristic
73	C₂²×C₄	—	2	2	no	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	Dic₁₉₄	1	1	yes	characteristic
76	C₂²×C₄	D₃₈₈	1	2	yes	—
77	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
78	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
79	M₁₆	—	2	8	no	—
80	M₁₆	—	194	4	no	—
81	M₁₆	—	194	4	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	194	4	no	—
84	C₂×C₈	C₂×Dic₉₇	1	2	yes	—
85	C₂×C₈	C₂×Dic₉₇	1	2	yes	—
86	C₂×C₈	C₂²⋊D₉₇	1	4	yes	—
87	C₂³	—	2	2	no	—
88	C₂³	—	4	2	no	—
89	C₂³	C₄⋊Dic₉₇	1	2	yes	—
90	C₂³	C₂×Dic₁₉₄	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	C₄⋊Dic₉₇	1	2	yes	—
96	C₂×C₄	C₂²×Dic₉₇	1	1	yes	characteristic
97	C₂×C₄	C₂×C₂²⋊D₉₇	1	2	yes	—
98	C₂×C₄	C₂×D₃₈₈	1	1	yes	characteristic
99	C₂×C₄	#13095511	1	2	yes	—
100	C₈	—	2	2	no	—
101	C₈	—	388	4	no	—
102	C₈	C₂²⋊Dic₉₇	1	4	yes	—
103	C₂²	—	2	2	no	—
104	C₂²	—	2	4	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	4	4	no	—
107	C₂²	C₂×C₄⋊Dic₉₇	1	1	yes	characteristic
108	C₂²	#15690789	1	2	yes	—
109	C₄	—	2	2	no	—
110	C₄	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
111	C₄	#15690769	1	1	yes	characteristic
112	C₄	#15690784	1	2	yes	—
113	C₂	—	2	2	no	—
114	C₂	—	4	2	no	—
115	C₂	#20260595	1	1	yes	characteristic
116	C₂	#20260784	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690746

C₂

yes

maximal, complemented

#20260699

C₂

yes

maximal

#20260500

C₂

yes

maximal, complemented

C₂²×M₁₆×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

#15690746

—

C₂²⋊#11715895

—

C₂³×C₄×C₉₇

C₂²

yes

characteristic

C₂×M₁₆×C₉₇

—

C₂×M₁₆×C₉₇

C₄

yes

—

C₂²×C₈×C₉₇

C₂²

yes

characteristic

C₂²×C₈×C₉₇

C₂²

yes

—

—

—

C₂⁴×C₉₇

Q₈

yes

characteristic

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

Q₈

yes

characteristic

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

D₄

yes

—

C₂³×C₉₇

—

C₂³×C₉₇

—

C₂³×C₉₇

C₄⋊C₄

yes

—

C₂³×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₈×C₉₇

—

C₈×C₉₇

C₂²⋊C₄

yes

—

#11715895

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

C₂²⋊Q₈

yes

—

C₄×C₉₇

—

C₄×C₉₇

C₂×C₂²⋊C₄

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#380

yes

characteristic

C₂×C₉₇

#430

yes

—

#1811

—

maximal, Sylow

C₉₇

#1811

yes

characteristic, complemented, Sylow

C₂²×M₁₆

D₉₇

yes

—

—

—

C₂³×C₄

C₂×D₉₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

Dic₉₇

yes

—

C₂²×C₈

C₂×D₉₇

yes

characteristic

C₂²×C₈

C₂×D₉₇

yes

characteristic

#108

—

194

—

C₂²⋊C₈

—

194

—

C₂⁴

Dic₁₉₄

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₉₄

yes

characteristic

C₂²×C₄

D₃₈₈

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

194

—

M₁₆

—

194

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

C₂×Dic₉₇

yes

—

C₂×C₈

C₂×Dic₉₇

yes

—

C₂×C₈

C₂²⋊D₉₇

yes

—

C₂³

—

C₂³

—

C₂³

C₄⋊Dic₉₇

yes

—

C₂³

C₂×Dic₁₉₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₉₇

yes

—

C₂×C₄

C₂²×Dic₉₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₉₇

yes

—

C₂×C₄

C₂×D₃₈₈

yes

characteristic

C₂×C₄

#13095511

yes

—

100

C₈

—

101

C₈

—

388

—

102

C₈

C₂²⋊Dic₉₇

yes

—

103

C₂²

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×C₄⋊Dic₉₇

yes

characteristic

108

C₂²

#15690789

yes

—

109

C₄

—

110

C₄

C₂×C₂²⋊Dic₉₇

yes

characteristic

111

C₄

#15690769

yes

characteristic

112

C₄

#15690784

yes

—

113

C₂

—

114

C₂

—

115

C₂

#20260595

yes

characteristic

116

C₂

#20260784

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₃₈₈
commutator	C₂×C₄×C₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₂²×M₁₆×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₉₇	Q₈
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂²×M₁₆	D₉₇
hypocenter	C₉₇	#1811
LCS	G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×D₃₈₈

commutator

C₂×C₄×C₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₂²×M₁₆×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₄⋊C₄

prime radical

C₂⁴×C₉₇

Q₈

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₂²×M₁₆

D₉₇

hypocenter

C₉₇

#1811

LCS

G ▷ C₂×C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×M₁₆

N	H ≅ G/N	type
1. C₂×#15690746	C₂	semidirect
3. #20260500	C₂	semidirect
59. C₉₇	#1811	semidirect

H ≅ G/N

type

1. C₂×#15690746

C₂

semidirect

3. #20260500

C₂

semidirect

59. C₉₇

#1811

semidirect

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

subgroup	index
C₂×#15690746	2
#20260699	2
#20260500	2
C₂²×M₁₆×C₉₇	2
#1811	97

subgroup

index

C₂²×M₁₆×C₉₇

Subgroups — #27141778

Conjugacy classes of subgroups

Complements

Maximal subgroups