Group #41013301 — subgroups

Subgroups — C₄×#20269912 #41013301

Order 27008.

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×#20269912	C₂	1	2	yes	maximal
2	C₄×#16809946	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#27212910	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#16809946	C₂²	1	2	yes	—
7	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
9	#20269912	C₄	1	8	yes	direct product factor
10	#20269888	—	2	1	no	—
11	C₄×C₂₁₁⋊C₈	—	2	1	no	—
12	C₄×C₂₁₁⋊C₈	C₄	1	4	yes	—
13	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
15	#97×C₂₁₁	—	2	1	no	—
16	C₄×C₈×C₂₁₁	—	2	1	no	—
17	#16809946	—	2	2	no	—
18	#16809946	C₂×C₄	1	4	yes	—
19	C₂×C₂₁₁⋊C₈	—	2	2	no	—
20	C₂×C₂₁₁⋊C₈	—	2	2	no	—
21	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₂₁₁⋊C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₁₁	Q₈	1	2	yes	—
25	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₁₁	—	2	2	no	—
27	M₁₆×C₂₁₁	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₂₁₁	—	2	2	no	—
29	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
30	C₄²×C₂₁₁	D₄	1	1	yes	characteristic
31	C₄²×C₂₁₁	D₄	1	1	yes	characteristic
32	C₂₁₁⋊C₈	—	2	2	no	—
33	C₂₁₁⋊C₈	—	2	4	no	—
34	C₂₁₁⋊C₈	C₄²	1	4	yes	—
35	C₂³×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₂₁₁	—	2	4	no	—
37	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₁₁	#41	1	2	yes	—
43	C₈×C₂₁₁	—	2	4	no	—
44	C₂²×C₂₁₁	—	2	2	no	—
45	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₁₁	C₄×Q₈	1	2	yes	—
47	C₄×C₂₁₁	—	2	2	no	—
48	C₄×C₂₁₁	#108	1	4	yes	complemented
49	C₄×C₂₁₁	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₂₁₁	#117	1	1	yes	characteristic
51	C₄×C₂₁₁	C₄×D₄	1	2	yes	—
52	C₂×C₂₁₁	—	2	2	no	—
53	C₂×C₂₁₁	C₄×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₂₁₁	C₂×#108	1	2	yes	—
55	C₂₁₁	C₄×#108	1	1	yes	characteristic, complemented, Sylow
56	C₄×#108	—	211	1	no	maximal, Sylow
57	C₂×#108	—	211	2	no	—
58	C₄×M₁₆	—	211	1	no	—
59	C₄×M₁₆	D₂₁₁	1	1	yes	characteristic
60	C₂×C₄×C₈	—	211	1	no	—
61	#341	—	211	2	no	—
62	C₂×M₁₆	—	211	2	no	—
63	C₂×M₁₆	C₂×D₂₁₁	1	2	yes	—
64	C₂²×C₈	—	211	1	no	—
65	C₂²×C₈	—	211	1	no	—
66	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
67	#108	—	211	8	no	—
68	#97	—	2	1	no	—
69	#97	—	422	1	no	—
70	C₄×C₈	—	2	1	no	—
71	C₄×C₈	—	422	1	no	—
72	C₄×C₈	—	211	4	no	—
73	C₂²×C₄	Dic₄₂₂	1	2	yes	—
74	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	422	2	no	—
77	M₁₆	—	211	4	no	—
78	M₁₆	C₄×D₂₁₁	1	4	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	422	2	no	—
81	C₂×C₈	—	422	2	no	—
82	C₂×C₈	—	211	2	no	—
83	C₂×C₈	—	211	2	no	—
84	C₂×C₈	—	211	4	no	—
85	C₄²	C₄×D₂₁₁	1	2	yes	—
86	C₄²	C₂²⋊D₂₁₁	1	1	yes	characteristic
87	C₄²	C₂²⋊D₂₁₁	1	1	yes	characteristic
88	C₂³	C₂×Dic₄₂₂	1	1	yes	characteristic
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	Dic₂₁₁⋊C₄	1	2	yes	—
91	C₂×C₄	Dic₂₁₁⋊C₄	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₂₁₁	1	2	yes	—
94	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
95	C₂×C₄	#16809962	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	422	2	no	—
98	C₈	—	422	4	no	—
99	C₈	—	211	4	no	—
100	C₂²	—	2	2	no	—
101	C₂²	C₂×Dic₂₁₁⋊C₄	1	1	yes	characteristic
102	C₂²	C₄×Dic₄₂₂	1	2	yes	—
103	C₄	—	2	2	no	—
104	C₄	#20269912	1	4	yes	direct product factor
105	C₄	C₄²×D₂₁₁	1	1	yes	characteristic
106	C₄	#20269934	1	1	yes	characteristic
107	C₄	C₄×C₂²⋊D₂₁₁	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×Dic₂₁₁⋊C₄	1	1	yes	characteristic
110	C₂	C₂×#20269912	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269912

C₂

yes

maximal

C₄×#16809946

C₂

yes

maximal, characteristic

C₂×C₄×C₂₁₁⋊C₈

C₂

yes

maximal, characteristic

#27212910

C₂

yes

maximal

C₄×M₁₆×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

C₂²

yes

—

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

characteristic

#20269912

C₄

yes

direct product factor

#20269888

—

C₄×C₂₁₁⋊C₈

—

C₄×C₂₁₁⋊C₈

C₄

yes

—

C₂×M₁₆×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

#97×C₂₁₁

—

C₄×C₈×C₂₁₁

—

#16809946

—

#16809946

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

Q₈

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

M₁₆×C₂₁₁

—

M₁₆×C₂₁₁

C₂×C₄

yes

—

C₂×C₈×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

D₄

yes

characteristic

C₄²×C₂₁₁

D₄

yes

characteristic

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

C₄²

yes

—

C₂³×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂²×C₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₁₁

C₄×Q₈

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#108

yes

complemented

C₄×C₂₁₁

C₂×C₄²

yes

characteristic

C₄×C₂₁₁

#117

yes

characteristic

C₄×C₂₁₁

C₄×D₄

yes

—

C₂×C₂₁₁

—

C₂×C₂₁₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₁₁

C₂×#108

yes

—

C₂₁₁

C₄×#108

yes

characteristic, complemented, Sylow

C₄×#108

—

211

maximal, Sylow

C₂×#108

—

211

—

C₄×M₁₆

—

211

—

C₄×M₁₆

D₂₁₁

yes

characteristic

C₂×C₄×C₈

—

211

—

#341

—

211

—

C₂×M₁₆

—

211

—

C₂×M₁₆

C₂×D₂₁₁

yes

—

C₂²×C₈

—

211

—

C₂²×C₈

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

#108

—

211

—

#97

—

#97

—

422

—

C₄×C₈

—

C₄×C₈

—

422

—

C₄×C₈

—

211

—

C₂²×C₄

Dic₄₂₂

yes

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

M₁₆

—

422

—

M₁₆

—

211

—

M₁₆

C₄×D₂₁₁

yes

—

C₂×C₈

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

211

—

C₂×C₈

—

211

—

C₂×C₈

—

211

—

C₄²

C₄×D₂₁₁

yes

—

C₄²

C₂²⋊D₂₁₁

yes

characteristic

C₄²

C₂²⋊D₂₁₁

yes

characteristic

C₂³

C₂×Dic₄₂₂

yes

characteristic

C₂×C₄

—

C₂×C₄

Dic₂₁₁⋊C₄

yes

—

C₂×C₄

Dic₂₁₁⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

C₂×C₄

C₂×C₄×D₂₁₁

yes

—

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

C₂×C₄

#16809962

yes

—

C₈

—

C₈

—

422

—

C₈

—

422

—

C₈

—

211

—

100

C₂²

—

101

C₂²

C₂×Dic₂₁₁⋊C₄

yes

characteristic

102

C₂²

C₄×Dic₄₂₂

yes

—

103

C₄

—

104

C₄

#20269912

yes

direct product factor

105

C₄

C₄²×D₂₁₁

yes

characteristic

106

C₄

#20269934

yes

characteristic

107

C₄

C₄×C₂²⋊D₂₁₁

yes

—

108

C₂

—

109

C₂

C₄×Dic₂₁₁⋊C₄

yes

characteristic

110

C₂

C₂×#20269912

yes

—

	subgroup	G/H
center	C₄²	C₂²⋊D₂₁₁
commutator	C₄×C₂₁₁	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₄×M₁₆×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₄⋊C₄
prime radical	C₂³×C₂₁₁	C₂×Q₈
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₄×M₁₆	D₂₁₁
hypocenter	C₂₁₁	C₄×#108
LCS	G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₄²

C₂²⋊D₂₁₁

commutator

C₄×C₂₁₁

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

C₄×M₁₆×C₂₁₁

C₂

socle

C₂²×C₂₁₁

C₂×C₄⋊C₄

prime radical

C₂³×C₂₁₁

C₂×Q₈

cosocle

C₂²×C₄×C₂₁₁

C₂³

hypercenter

C₄×M₁₆

D₂₁₁

hypocenter

C₂₁₁

C₄×#108

LCS

G ▷ C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
9. #20269912	C₄	direct (N × H)
48. C₄×C₂₁₁	#108	semidirect
55. C₂₁₁	C₄×#108	semidirect
104. C₄	#20269912	direct (N × H)

H ≅ G/N

type

9. #20269912

C₄

direct (N × H)

48. C₄×C₂₁₁

#108

semidirect

55. C₂₁₁

C₄×#108

semidirect

104. C₄

#20269912

direct (N × H)

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

subgroup	index
C₂×#20269912	2
C₄×#16809946	2
C₂×C₄×C₂₁₁⋊C₈	2
#27212910	2
C₄×M₁₆×C₂₁₁	2
C₄×#108	211

subgroup

index

C₂×C₄×C₂₁₁⋊C₈

C₄×M₁₆×C₂₁₁

211

Subgroups — C₄×#20269912 #41013301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#20269912 #41013301

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#20269912 #41013301