Group #41013255 — subgroups

Subgroups — #41013255

Order 27008.

260 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 141 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⋊(C₂₁₁⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#20269888	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#27212901	C₂	1	2	yes	maximal
5	#376×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
8	C₄⋊(C₂₁₁⋊C₈)	C₄	1	8	yes	complemented
9	#20269888	—	2	2	no	—
10	C₄×C₂₁₁⋊C₈	—	2	2	no	—
11	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
14	#95×C₂₁₁	C₄	1	4	yes	—
15	C₂×C₂₁₁⋊C₈	—	2	2	no	—
16	C₂×C₂₁₁⋊C₈	—	2	2	no	—
17	C₂×C₂₁₁⋊C₈	—	2	2	no	—
18	C₂×C₂₁₁⋊C₈	—	2	2	no	—
19	C₂×C₂₁₁⋊C₈	C₂×C₄	1	4	yes	—
20	C₂×C₂₁₁⋊C₈	C₂×C₄	1	4	yes	—
21	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
25	C₄²×C₂₁₁	—	2	2	no	—
26	C₄²×C₂₁₁	C₂×C₄	1	4	yes	—
27	C₄²×C₂₁₁	D₄	1	2	yes	—
28	C₄²×C₂₁₁	Q₈	1	2	yes	—
29	C₂₁₁⋊C₈	—	2	4	no	—
30	C₂₁₁⋊C₈	—	2	4	no	—
31	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₁₁	—	2	4	no	—
33	C₂×C₄×C₂₁₁	—	2	4	no	—
34	C₂×C₄×C₂₁₁	—	2	2	no	—
35	C₂×C₄×C₂₁₁	—	2	4	no	—
36	C₂×C₄×C₂₁₁	C₄²	1	4	yes	—
37	C₂×C₄×C₂₁₁	C₄⋊C₄	1	4	yes	—
38	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₂₁₁	#41	1	2	yes	—
42	C₂²×C₂₁₁	C₂×C₄²	1	1	yes	characteristic
43	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₂₁₁	#117	1	1	yes	characteristic
45	C₂²×C₂₁₁	#131	1	4	yes	—
46	C₄×C₂₁₁	—	2	4	no	—
47	C₄×C₂₁₁	—	2	4	no	—
48	C₄×C₂₁₁	—	4	2	no	—
49	C₄×C₂₁₁	C₄×D₄	1	2	yes	—
50	C₄×C₂₁₁	C₄×Q₈	1	2	yes	—
51	C₂×C₂₁₁	C₄×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₂₁₁	#405	1	2	yes	—
53	C₂×C₂₁₁	#424	1	2	yes	—
54	C₂×C₂₁₁	#433	1	2	yes	—
55	C₂₁₁	#1768	1	1	yes	characteristic, complemented, Sylow
56	#1768	—	211	1	no	maximal, Sylow
57	C₂×C₄⋊C₈	—	211	2	no	—
58	C₂×#97	—	211	1	no	—
59	C₂×C₄×C₈	—	211	1	no	—
60	#376	D₂₁₁	1	1	yes	characteristic
61	#336	—	211	2	no	—
62	C₂²×C₈	—	211	2	no	—
63	C₂²×C₈	—	211	2	no	—
64	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
66	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
67	C₄⋊C₈	—	211	8	no	—
68	#97	—	422	2	no	—
69	C₄×C₈	—	422	2	no	—
70	#95	Dic₂₁₁	1	4	yes	—
71	C₂²×C₄	C₂×Dic₂₁₁	1	2	yes	—
72	C₂²×C₄	C₂×Dic₂₁₁	1	2	yes	—
73	C₂²×C₄	C₂×Dic₂₁₁	1	2	yes	—
74	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
75	C₂×C₈	—	422	2	no	—
76	C₂×C₈	—	422	2	no	—
77	C₂×C₈	—	422	2	no	—
78	C₂×C₈	—	422	2	no	—
79	C₂×C₈	—	211	4	no	—
80	C₂×C₈	—	211	4	no	—
81	C₄²	—	2	2	no	—
82	C₄²	C₄×D₂₁₁	1	4	yes	—
83	C₄²	D₈₄₄	1	2	yes	—
84	C₄²	Dic₄₂₂	1	2	yes	—
85	C₂³	C₂²×Dic₂₁₁	1	1	yes	characteristic
86	C₂×C₄	—	2	4	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	4	no	—
90	C₂×C₄	C₄×Dic₂₁₁	1	4	yes	—
91	C₂×C₄	C₄⋊Dic₂₁₁	1	4	yes	—
92	C₂×C₄	C₂×C₄×D₂₁₁	1	2	yes	—
93	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
95	C₂×C₄	#16809962	1	2	yes	—
96	C₈	—	422	4	no	—
97	C₈	—	422	4	no	—
98	C₂²	C₂×C₄×Dic₂₁₁	1	1	yes	characteristic
99	C₂²	C₂×C₄⋊Dic₂₁₁	1	1	yes	characteristic
100	C₂²	#20269936	1	1	yes	characteristic
101	C₂²	#20269984	1	4	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	2	4	no	—
104	C₄	—	4	2	no	—
105	C₄	C₄×D₈₄₄	1	2	yes	—
106	C₄	C₄×Dic₄₂₂	1	2	yes	—
107	C₂	C₄×C₄⋊Dic₂₁₁	1	1	yes	characteristic
108	C₂	#27213102	1	2	yes	—
109	C₂	#27213173	1	2	yes	—
110	C₂	#27213207	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₁₁⋊C₈)

C₂

yes

maximal

C₂×#20269888

C₂

yes

maximal, characteristic

C₂×C₄×C₂₁₁⋊C₈

C₂

yes

maximal, characteristic

#27212901

C₂

yes

maximal

#376×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

—

C₂²×C₂₁₁⋊C₈

C₂²

yes

—

C₄⋊(C₂₁₁⋊C₈)

C₄

yes

complemented

#20269888

—

C₄×C₂₁₁⋊C₈

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

#95×C₂₁₁

C₄

yes

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₄²×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

D₄

yes

—

C₄²×C₂₁₁

Q₈

yes

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₄²

yes

—

C₂×C₄×C₂₁₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₂²×C₂₁₁

C₂×C₄²

yes

characteristic

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₄×D₄

yes

—

C₄×C₂₁₁

C₄×Q₈

yes

—

C₂×C₂₁₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₁₁

#405

yes

—

C₂×C₂₁₁

#424

yes

—

C₂×C₂₁₁

#433

yes

—

C₂₁₁

#1768

yes

characteristic, complemented, Sylow

#1768

—

211

maximal, Sylow

C₂×C₄⋊C₈

—

211

—

C₂×#97

—

211

—

C₂×C₄×C₈

—

211

—

#376

D₂₁₁

yes

characteristic

#336

—

211

—

C₂²×C₈

—

211

—

C₂²×C₈

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₄⋊C₈

—

211

—

#97

—

422

—

C₄×C₈

—

422

—

#95

Dic₂₁₁

yes

—

C₂²×C₄

C₂×Dic₂₁₁

yes

—

C₂²×C₄

C₂×Dic₂₁₁

yes

—

C₂²×C₄

C₂×Dic₂₁₁

yes

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

211

—

C₂×C₈

—

211

—

C₄²

—

C₄²

C₄×D₂₁₁

yes

—

C₄²

D₈₄₄

yes

—

C₄²

Dic₄₂₂

yes

—

C₂³

C₂²×Dic₂₁₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₂₁₁

yes

—

C₂×C₄

C₄⋊Dic₂₁₁

yes

—

C₂×C₄

C₂×C₄×D₂₁₁

yes

—

C₂×C₄

C₂×D₈₄₄

yes

characteristic

C₂×C₄

C₂×Dic₄₂₂

yes

characteristic

C₂×C₄

#16809962

yes

—

C₈

—

422

—

C₈

—

422

—

C₂²

C₂×C₄×Dic₂₁₁

yes

characteristic

C₂²

C₂×C₄⋊Dic₂₁₁

yes

characteristic

100

C₂²

#20269936

yes

characteristic

101

C₂²

#20269984

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₄×D₈₄₄

yes

—

106

C₄

C₄×Dic₄₂₂

yes

—

107

C₂

C₄×C₄⋊Dic₂₁₁

yes

characteristic

108

C₂

#27213102

yes

—

109

C₂

#27213173

yes

—

110

C₂

#27213207

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₁₁
commutator	C₂²×C₂₁₁	C₂×C₄²
Fitting	#376×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂²×C₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#376	D₂₁₁
hypocenter	C₂₁₁	#1768
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂²×C₄ ◁ #376

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₁₁

commutator

C₂²×C₂₁₁

C₂×C₄²

Fitting

#376×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂²×C₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂²×C₂₁₁ ▷ C₂₁₁

C₁ ◁ C₂²×C₄ ◁ #376

N	H ≅ G/N	type
8. C₄⋊(C₂₁₁⋊C₈)	C₄	semidirect
55. C₂₁₁	#1768	semidirect

H ≅ G/N

type

8. C₄⋊(C₂₁₁⋊C₈)

C₄

semidirect

55. C₂₁₁

#1768

semidirect

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

subgroup	index
C₂×C₄⋊(C₂₁₁⋊C₈)	2
C₂×#20269888	2
C₂×C₄×C₂₁₁⋊C₈	2
#27212901	2
#376×C₂₁₁	2
#1768	211

subgroup

index

C₂×C₄⋊(C₂₁₁⋊C₈)

C₂×#20269888

C₂×C₄×C₂₁₁⋊C₈

#27212901

#376×C₂₁₁

#1768

211

Subgroups — #41013255

Conjugacy classes of subgroups

Complements

Maximal subgroups