Group #27141639 — subgroups

Subgroups — C₄×#15690745 #27141639

Order 12416.

228 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 121 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×#15690745	C₂	1	2	yes	maximal
2	C₄×#13095495	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#11715895	C₂	1	1	yes	maximal, characteristic
4	#20260499	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#13095495	C₂²	1	2	yes	—
7	C₂²×#11715895	C₂²	1	1	yes	characteristic
8	C₂²×#11715895	C₂²	1	1	yes	characteristic
9	#15690745	C₄	1	8	yes	direct product factor
10	#15690721	—	2	1	no	—
11	C₄×#11715895	—	2	1	no	—
12	C₄×#11715895	C₄	1	4	yes	—
13	C₂×M₁₆×C₉₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	#97×C₉₇	—	2	1	no	—
16	C₄×C₈×C₉₇	—	2	1	no	—
17	#13095495	—	2	2	no	—
18	#13095495	C₂×C₄	1	4	yes	—
19	C₂×#11715895	—	2	2	no	—
20	C₂×#11715895	—	2	2	no	—
21	C₂×#11715895	C₂×C₄	1	2	yes	—
22	C₂×#11715895	C₂×C₄	1	2	yes	—
23	C₂×#11715895	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₉₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₉₇	—	2	2	no	—
27	M₁₆×C₉₇	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₉₇	—	2	2	no	—
29	C₄²×C₉₇	C₂×C₄	1	2	yes	—
30	C₄²×C₉₇	D₄	1	1	yes	characteristic
31	C₄²×C₉₇	D₄	1	1	yes	characteristic
32	C₂³×C₉₇	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₉₇	—	2	4	no	—
34	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₉₇	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₉₇	#41	1	2	yes	—
40	C₈×C₉₇	—	2	4	no	—
41	#11715895	—	2	2	no	—
42	#11715895	—	2	4	no	—
43	#11715895	C₄²	1	4	yes	—
44	C₂²×C₉₇	—	2	2	no	—
45	C₂²×C₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₉₇	C₄×Q₈	1	2	yes	—
47	C₄×C₉₇	—	2	2	no	—
48	C₄×C₉₇	#108	1	4	yes	complemented
49	C₄×C₉₇	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₉₇	#117	1	1	yes	characteristic
51	C₄×C₉₇	C₄×D₄	1	2	yes	—
52	C₂×C₉₇	—	2	2	no	—
53	C₂×C₉₇	C₄×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₉₇	C₂×#108	1	2	yes	—
55	C₄×#108	—	97	1	no	maximal, Sylow
56	C₉₇	C₄×#108	1	1	yes	characteristic, complemented, Sylow
57	C₂×#108	—	97	2	no	—
58	C₄×M₁₆	—	97	1	no	—
59	C₄×M₁₆	D₉₇	1	1	yes	characteristic
60	C₂×C₄×C₈	—	97	1	no	—
61	#341	—	97	2	no	—
62	C₂×M₁₆	—	97	2	no	—
63	C₂×M₁₆	C₂×D₉₇	1	2	yes	—
64	C₂²×C₈	—	97	1	no	—
65	C₂²×C₈	—	97	1	no	—
66	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
67	#108	—	97	8	no	—
68	#97	—	2	1	no	—
69	#97	—	194	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	—	97	4	no	—
72	C₄×C₈	—	194	1	no	—
73	C₂²×C₄	Dic₁₉₄	1	2	yes	—
74	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	97	4	no	—
77	M₁₆	—	194	2	no	—
78	M₁₆	C₄×D₉₇	1	4	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	97	2	no	—
81	C₂×C₈	—	97	4	no	—
82	C₂×C₈	—	97	2	no	—
83	C₂×C₈	—	194	2	no	—
84	C₂×C₈	—	194	2	no	—
85	C₄²	C₄×D₉₇	1	2	yes	—
86	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
87	C₄²	C₂²⋊D₉₇	1	1	yes	characteristic
88	C₂³	C₂×Dic₁₉₄	1	1	yes	characteristic
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	#13095490	1	2	yes	—
91	C₂×C₄	#13095490	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₉₇	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₉₇	1	2	yes	—
94	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
95	C₂×C₄	#13095511	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	97	4	no	—
98	C₈	—	194	2	no	—
99	C₈	—	194	4	no	—
100	C₂²	—	2	2	no	—
101	C₂²	C₂×#13095490	1	1	yes	characteristic
102	C₂²	C₄×Dic₁₉₄	1	2	yes	—
103	C₄	—	2	2	no	—
104	C₄	#15690745	1	4	yes	direct product factor
105	C₄	C₄²×D₉₇	1	1	yes	characteristic
106	C₄	#15690767	1	1	yes	characteristic
107	C₄	C₄×C₂²⋊D₉₇	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×#13095490	1	1	yes	characteristic
110	C₂	C₂×#15690745	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690745

C₂

yes

maximal

C₄×#13095495

C₂

yes

maximal, characteristic

C₂×C₄×#11715895

C₂

yes

maximal, characteristic

#20260499

C₂

yes

maximal

C₄×M₁₆×C₉₇

C₂

yes

maximal, characteristic

C₂×#13095495

C₂²

yes

—

C₂²×#11715895

C₂²

yes

characteristic

C₂²×#11715895

C₂²

yes

characteristic

#15690745

C₄

yes

direct product factor

#15690721

—

C₄×#11715895

—

C₄×#11715895

C₄

yes

—

C₂×M₁₆×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#97×C₉₇

—

C₄×C₈×C₉₇

—

#13095495

—

#13095495

C₂×C₄

yes

—

C₂×#11715895

—

C₂×#11715895

—

C₂×#11715895

C₂×C₄

yes

—

C₂×#11715895

C₂×C₄

yes

—

C₂×#11715895

C₂×C₄

yes

—

C₂²×C₄×C₉₇

Q₈

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

M₁₆×C₉₇

—

M₁₆×C₉₇

C₂×C₄

yes

—

C₂×C₈×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

D₄

yes

characteristic

C₂³×C₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

—

C₈×C₉₇

—

#11715895

—

#11715895

—

#11715895

C₄²

yes

—

C₂²×C₉₇

—

C₂²×C₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₉₇

C₄×Q₈

yes

—

C₄×C₉₇

—

C₄×C₉₇

#108

yes

complemented

C₄×C₉₇

C₂×C₄²

yes

characteristic

C₄×C₉₇

#117

yes

characteristic

C₄×C₉₇

C₄×D₄

yes

—

C₂×C₉₇

—

C₂×C₉₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₉₇

C₂×#108

yes

—

C₄×#108

—

maximal, Sylow

C₉₇

C₄×#108

yes

characteristic, complemented, Sylow

C₂×#108

—

C₄×M₁₆

—

C₄×M₁₆

D₉₇

yes

characteristic

C₂×C₄×C₈

—

#341

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₉₇

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×D₉₇

yes

—

—

—

194

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

194

—

C₂²×C₄

Dic₁₉₄

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

194

—

M₁₆

C₄×D₉₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄²

C₄×D₉₇

yes

—

C₄²

C₂²⋊D₉₇

yes

characteristic

C₄²

C₂²⋊D₉₇

yes

characteristic

C₂³

C₂×Dic₁₉₄

yes

characteristic

C₂×C₄

—

C₂×C₄

#13095490

yes

—

C₂×C₄

#13095490

yes

—

C₂×C₄

C₂×C₄×D₉₇

yes

characteristic

C₂×C₄

C₂×C₄×D₉₇

yes

—

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

C₂×C₄

#13095511

yes

—

C₈

—

C₈

—

C₈

—

194

—

C₈

—

194

—

100

C₂²

—

101

C₂²

C₂×#13095490

yes

characteristic

102

C₂²

C₄×Dic₁₉₄

yes

—

103

C₄

—

104

C₄

#15690745

yes

direct product factor

105

C₄

C₄²×D₉₇

yes

characteristic

106

C₄

#15690767

yes

characteristic

107

C₄

C₄×C₂²⋊D₉₇

yes

—

108

C₂

—

109

C₂

C₄×#13095490

yes

characteristic

110

C₂

C₂×#15690745

yes

—

	subgroup	G/H
center	C₄²	C₂²⋊D₉₇
commutator	C₄×C₉₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₉₇
Fitting	C₄×M₁₆×C₉₇	C₂
socle	C₂²×C₉₇	C₂×C₄⋊C₄
prime radical	C₂³×C₉₇	C₂×Q₈
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₄×M₁₆	D₉₇
hypocenter	C₉₇	C₄×#108
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₄²

C₂²⋊D₉₇

commutator

C₄×C₉₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₉₇

Fitting

C₄×M₁₆×C₉₇

C₂

socle

C₂²×C₉₇

C₂×C₄⋊C₄

prime radical

C₂³×C₉₇

C₂×Q₈

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₄×M₁₆

D₉₇

hypocenter

C₉₇

C₄×#108

LCS

G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
9. #15690745	C₄	direct (N × H)
48. C₄×C₉₇	#108	semidirect
56. C₉₇	C₄×#108	semidirect
104. C₄	#15690745	direct (N × H)

H ≅ G/N

type

9. #15690745

C₄

direct (N × H)

48. C₄×C₉₇

#108

semidirect

56. C₉₇

C₄×#108

semidirect

104. C₄

#15690745

direct (N × H)

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

subgroup	index
C₂×#15690745	2
C₄×#13095495	2
C₂×C₄×#11715895	2
#20260499	2
C₄×M₁₆×C₉₇	2
C₄×#108	97

subgroup

index

C₄×M₁₆×C₉₇

Subgroups — C₄×#15690745 #27141639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#15690745 #27141639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15690745 #27141639