Group #13136172 — subgroups

Subgroups — C₄²×C₂²⋊C₂₇ #13136172

Order 1728.

339 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 105 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×C₄×C₂²⋊C₂₇	C₂	1	3	yes	maximal
2	C₂²×C₄²×C₉	C₃	1	1	yes	maximal, characteristic
3	C₂²×C₂²⋊C₂₇	C₂²	1	1	yes	characteristic
4	C₄×C₂²⋊C₂₇	C₄	1	6	yes	direct product factor
5	C₄²×C₂₇	—	4	1	no	maximal
6	C₂³×C₄×C₉	C₂×C₃	1	3	yes	—
7	C₂×C₄²×C₉	—	3	1	no	—
8	C₂×C₄²×C₉	—	3	3	no	—
9	C₂×C₂²⋊C₂₇	C₂×C₄	1	3	yes	—
10	C₂×C₄×C₂₇	—	4	3	no	—
11	C₂²×C₃×C₄²	C₉	1	1	yes	characteristic
12	C₂⁴×C₉	C₂²×C₃	1	1	yes	characteristic
13	C₂²×C₄×C₉	—	3	3	no	—
14	C₂²×C₄×C₉	—	3	3	no	—
15	C₂²×C₄×C₉	—	3	6	no	—
16	C₂²×C₄×C₉	C₃×C₄	1	6	yes	—
17	C₄²×C₉	—	3	2	no	—
18	C₄²×C₉	—	3	3	no	—
19	C₄²×C₉	A₄	1	1	yes	characteristic
20	C₂²×C₂₇	—	4	1	no	—
21	C₂²⋊C₂₇	C₄²	1	1	yes	characteristic, direct product factor
22	C₄×C₂₇	—	4	6	no	—
23	C₂³×C₃×C₄	C₂×C₉	1	3	yes	—
24	C₂×C₃×C₄²	—	3	1	no	—
25	C₂×C₃×C₄²	—	3	3	no	—
26	C₂³×C₉	—	3	3	no	—
27	C₂³×C₉	—	3	1	no	—
28	C₂³×C₉	C₂×C₃×C₄	1	3	yes	—
29	C₂×C₄×C₉	—	3	6	no	—
30	C₂×C₄×C₉	—	3	3	no	—
31	C₂×C₄×C₉	—	3	6	no	—
32	C₂×C₄×C₉	—	3	6	no	—
33	C₂×C₄×C₉	—	3	6	no	—
34	C₂×C₄×C₉	C₂×A₄	1	3	yes	—
35	C₂²×C₄²	C₂₇	1	1	yes	characteristic, complemented, Sylow
36	C₂×C₂₇	—	4	3	no	—
37	C₂⁴×C₃	C₂²×C₉	1	1	yes	characteristic
38	C₂²×C₃×C₄	—	3	3	no	—
39	C₂²×C₃×C₄	—	3	3	no	—
40	C₂²×C₃×C₄	—	3	6	no	—
41	C₂²×C₃×C₄	C₄×C₉	1	6	yes	—
42	C₃×C₄²	—	3	2	no	—
43	C₃×C₄²	—	3	3	no	—
44	C₃×C₄²	C₂²⋊C₉	1	1	yes	characteristic
45	C₂²×C₉	—	3	3	no	—
46	C₂²×C₉	—	3	2	no	—
47	C₂²×C₉	—	3	3	no	—
48	C₂²×C₉	—	3	3	no	—
49	C₂²×C₉	C₃×C₄²	1	1	yes	characteristic
50	C₂²×C₉	C₂²×A₄	1	1	yes	characteristic
51	C₄×C₉	—	3	6	no	—
52	C₄×C₉	C₄×A₄	1	6	yes	—
53	C₂³×C₄	C₂×C₂₇	1	3	yes	—
54	C₂×C₄²	—	3	1	no	—
55	C₂×C₄²	—	3	3	no	—
56	C₂₇	—	4	1	no	Sylow
57	C₂³×C₃	—	3	3	no	—
58	C₂³×C₃	—	3	1	no	—
59	C₂³×C₃	C₂×C₄×C₉	1	3	yes	—
60	C₂×C₃×C₄	—	3	6	no	—
61	C₂×C₃×C₄	—	3	3	no	—
62	C₂×C₃×C₄	—	3	6	no	—
63	C₂×C₃×C₄	—	3	6	no	—
64	C₂×C₃×C₄	—	3	6	no	—
65	C₂×C₃×C₄	C₂×C₂²⋊C₉	1	3	yes	—
66	C₂×C₉	—	3	3	no	—
67	C₂×C₉	—	3	1	no	—
68	C₂×C₉	C₂×C₄×A₄	1	3	yes	—
69	C₂⁴	C₂²×C₂₇	1	1	yes	characteristic
70	C₂²×C₄	—	3	3	no	—
71	C₂²×C₄	—	3	3	no	—
72	C₂²×C₄	—	3	6	no	—
73	C₂²×C₄	C₄×C₂₇	1	6	yes	complemented
74	C₄²	—	3	2	no	—
75	C₄²	—	3	3	no	—
76	C₄²	C₂²⋊C₂₇	1	1	yes	characteristic, direct product factor
77	C₂²×C₃	—	3	3	no	—
78	C₂²×C₃	—	3	2	no	—
79	C₂²×C₃	—	3	3	no	—
80	C₂²×C₃	—	3	3	no	—
81	C₂²×C₃	C₄²×C₉	1	1	yes	characteristic
82	C₂²×C₃	C₂²×C₂²⋊C₉	1	1	yes	characteristic
83	C₃×C₄	—	3	6	no	—
84	C₃×C₄	C₄×C₂²⋊C₉	1	6	yes	—
85	C₉	C₄²×A₄	1	1	yes	characteristic
86	C₂³	—	3	3	no	—
87	C₂³	—	3	1	no	—
88	C₂³	C₂×C₄×C₂₇	1	3	yes	—
89	C₂×C₄	—	3	6	no	—
90	C₂×C₄	—	3	3	no	—
91	C₂×C₄	—	3	6	no	—
92	C₂×C₄	—	3	6	no	—
93	C₂×C₄	—	3	6	no	—
94	C₂×C₄	C₂×C₂²⋊C₂₇	1	3	yes	—
95	C₂×C₃	—	3	1	no	—
96	C₂×C₃	—	3	3	no	—
97	C₂×C₃	C₂×C₄×C₂²⋊C₉	1	3	yes	—
98	C₂²	—	3	3	no	—
99	C₂²	—	3	2	no	—
100	C₂²	—	3	3	no	—
101	C₂²	—	3	3	no	—
102	C₂²	C₄²×C₂₇	1	1	yes	characteristic, complemented
103	C₂²	C₂²×C₂²⋊C₂₇	1	1	yes	characteristic
104	C₄	—	3	6	no	—
105	C₄	C₄×C₂²⋊C₂₇	1	6	yes	direct product factor
106	C₃	C₄²×C₂²⋊C₉	1	1	yes	characteristic
107	C₂	—	3	1	no	—
108	C₂	—	3	3	no	—
109	C₂	C₂×C₄×C₂²⋊C₂₇	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₂²⋊C₂₇

C₂

yes

maximal

C₂²×C₄²×C₉

C₃

yes

maximal, characteristic

C₂²×C₂²⋊C₂₇

C₂²

yes

characteristic

C₄×C₂²⋊C₂₇

C₄

yes

direct product factor

C₄²×C₂₇

—

maximal

C₂³×C₄×C₉

C₂×C₃

yes

—

C₂×C₄²×C₉

—

C₂×C₄²×C₉

—

C₂×C₂²⋊C₂₇

C₂×C₄

yes

—

C₂×C₄×C₂₇

—

C₂²×C₃×C₄²

C₉

yes

characteristic

C₂⁴×C₉

C₂²×C₃

yes

characteristic

C₂²×C₄×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₃×C₄

yes

—

C₄²×C₉

—

C₄²×C₉

—

C₄²×C₉

A₄

yes

characteristic

C₂²×C₂₇

—

C₂²⋊C₂₇

C₄²

yes

characteristic, direct product factor

C₄×C₂₇

—

C₂³×C₃×C₄

C₂×C₉

yes

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

—

C₂³×C₉

—

C₂³×C₉

—

C₂³×C₉

C₂×C₃×C₄

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×A₄

yes

—

C₂²×C₄²

C₂₇

yes

characteristic, complemented, Sylow

C₂×C₂₇

—

C₂⁴×C₃

C₂²×C₉

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₄×C₉

yes

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂²⋊C₉

yes

characteristic

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₃×C₄²

yes

characteristic

C₂²×C₉

C₂²×A₄

yes

characteristic

C₄×C₉

—

C₄×C₉

C₄×A₄

yes

—

C₂³×C₄

C₂×C₂₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂₇

—

Sylow

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₂×C₄×C₉

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₂²⋊C₉

yes

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₂×C₄×A₄

yes

—

C₂⁴

C₂²×C₂₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×C₂₇

yes

complemented

C₄²

—

C₄²

—

C₄²

C₂²⋊C₂₇

yes

characteristic, direct product factor

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₄²×C₉

yes

characteristic

C₂²×C₃

C₂²×C₂²⋊C₉

yes

characteristic

C₃×C₄

—

C₃×C₄

C₄×C₂²⋊C₉

yes

—

C₉

C₄²×A₄

yes

characteristic

C₂³

—

C₂³

—

C₂³

C₂×C₄×C₂₇

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₂₇

yes

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

C₂×C₄×C₂²⋊C₉

yes

—

C₂²

—

C₂²

—

100

C₂²

—

101

C₂²

—

102

C₂²

C₄²×C₂₇

yes

characteristic, complemented

103

C₂²

C₂²×C₂²⋊C₂₇

yes

characteristic

104

C₄

—

105

C₄

C₄×C₂²⋊C₂₇

yes

direct product factor

106

C₃

C₄²×C₂²⋊C₉

yes

characteristic

107

C₂

—

108

C₂

—

109

C₂

C₂×C₄×C₂²⋊C₂₇

yes

—

	subgroup	G/H
center, hypercenter	C₄²×C₉	A₄
commutator, hypocenter	C₂²	C₄²×C₂₇
Frattini	C₂²×C₉	C₂²×A₄
Fitting	C₂²×C₄²×C₉	C₃
socle, prime radical	C₂⁴×C₃	C₂²×C₉
cosocle	C₂⁴×C₉	C₂²×C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₄²×C₉

subgroup

G/H

center, hypercenter

C₄²×C₉

A₄

commutator, hypocenter

C₂²

C₄²×C₂₇

Frattini

C₂²×C₉

C₂²×A₄

Fitting

C₂²×C₄²×C₉

C₃

socle, prime radical

C₂⁴×C₃

C₂²×C₉

cosocle

C₂⁴×C₉

C₂²×C₃

LCS

G ▷ C₂²

UCS

C₁ ◁ C₄²×C₉

N	H ≅ G/N	type
4. C₄×C₂²⋊C₂₇	C₄	direct (N × H)
21. C₂²⋊C₂₇	C₄²	direct (N × H)
35. C₂²×C₄²	C₂₇	semidirect
73. C₂²×C₄	C₄×C₂₇	semidirect
76. C₄²	C₂²⋊C₂₇	direct (N × H)
102. C₂²	C₄²×C₂₇	semidirect
105. C₄	C₄×C₂²⋊C₂₇	direct (N × H)

H ≅ G/N

type

4. C₄×C₂²⋊C₂₇

C₄

direct (N × H)

21. C₂²⋊C₂₇

C₄²

direct (N × H)

35. C₂²×C₄²

C₂₇

semidirect

73. C₂²×C₄

C₄×C₂₇

semidirect

76. C₄²

C₂²⋊C₂₇

direct (N × H)

102. C₂²

C₄²×C₂₇

semidirect

105. C₄

C₄×C₂²⋊C₂₇

direct (N × H)

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

subgroup	index
C₂×C₄×C₂²⋊C₂₇	2
C₂²×C₄²×C₉	3
C₄²×C₂₇	4

subgroup

index

C₂×C₄×C₂²⋊C₂₇

C₂²×C₄²×C₉

C₄²×C₂₇

Subgroups — C₄²×C₂²⋊C₂₇ #13136172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C42×C22⋊C27 #13136172

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₂²⋊C₂₇ #13136172