Group #41014219 — subgroups

Subgroups — #41014219

Order 27008.

308 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 113 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×#20269901	C₂	1	4	yes	maximal
2	C₂×#20269888	C₂	1	1	yes	maximal, characteristic
3	#27213012	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
7	#20269901	—	2	8	no	—
8	#20269888	—	2	2	no	—
9	C₂₁₁⋊#95	—	2	2	no	—
10	C₂²×Q₈×C₂₁₁	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₂₁₁	—	2	2	no	—
12	C₄⋊Q₈×C₂₁₁	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₂₁₁	—	2	2	no	—
14	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₂₁₁	—	2	2	no	—
16	C₂×C₂₁₁⋊C₈	—	2	2	no	—
17	C₂×C₂₁₁⋊C₈	—	2	2	no	—
18	C₂×C₂₁₁⋊C₈	D₄	1	4	yes	—
19	C₄⋊Dic₂₁₁	—	2	4	no	—
20	C₂×Q₈×C₂₁₁	—	2	2	no	—
21	C₂×Q₈×C₂₁₁	—	2	4	no	—
22	C₂×Q₈×C₂₁₁	—	2	4	no	—
23	C₂×Q₈×C₂₁₁	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₁₁	—	2	2	no	—
25	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
26	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₂₁₁	—	2	8	no	—
28	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
29	C₄²×C₂₁₁	D₄	1	2	yes	—
30	C₂×Dic₂₁₁	—	4	2	no	—
31	C₂×Dic₂₁₁	—	4	4	no	—
32	C₂₁₁⋊C₈	—	2	4	no	—
33	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₂₁₁	—	2	8	no	—
35	Q₈×C₂₁₁	—	4	4	no	—
36	C₂×C₄×C₂₁₁	—	2	4	no	—
37	C₂×C₄×C₂₁₁	—	2	4	no	—
38	C₂×C₄×C₂₁₁	—	4	2	no	—
39	C₂×C₄×C₂₁₁	—	4	2	no	—
40	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₁₁	#41	1	2	yes	—
45	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₁₁	#124	1	1	yes	characteristic
47	C₂²×C₂₁₁	#127	1	1	yes	characteristic
48	C₂²×C₂₁₁	#137	1	4	yes	—
49	C₄×C₂₁₁	—	2	4	no	—
50	C₄×C₂₁₁	—	4	4	no	—
51	C₄×C₂₁₁	C₄×D₄	1	2	yes	—
52	C₄×C₂₁₁	#121	1	2	yes	—
53	Dic₂₁₁	—	8	2	no	—
54	C₂×C₂₁₁	#390	1	1	yes	characteristic
55	C₂×C₂₁₁	#419	1	2	yes	—
56	C₂×C₂₁₁	#491	1	2	yes	—
57	C₂×C₂₁₁	#497	1	2	yes	—
58	C₂₁₁	#1960	1	1	yes	characteristic, complemented, Sylow
59	#1960	—	211	1	no	maximal, Sylow
60	C₂×C₄⋊Q₈	D₂₁₁	1	1	yes	characteristic
61	C₂×#103	—	211	4	no	—
62	C₂×#97	—	211	1	no	—
63	#382	—	211	1	no	—
64	C₂²×Q₈	C₂×D₂₁₁	1	2	yes	—
65	C₂²×C₈	—	211	2	no	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	Dic₂₁₁	1	4	yes	complemented
68	C₂×C₄⋊C₄	—	2	2	no	—
69	C₂×C₄⋊C₄	—	211	2	no	—
70	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
71	#103	—	422	8	no	—
72	#97	—	422	2	no	—
73	#95	—	422	2	no	—
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	C₂×Dic₂₁₁	1	4	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	—	422	2	no	—
80	C₂²×C₄	C₂²⋊D₂₁₁	1	2	yes	—
81	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
82	C₂×C₈	—	422	2	no	—
83	C₂×C₈	—	422	2	no	—
84	C₂×C₈	—	211	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	422	4	no	—
87	C₄²	C₂×Dic₂₁₁	1	2	yes	—
88	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
89	C₂³	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
90	Q₈	—	2	8	no	—
91	Q₈	—	4	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	844	2	no	—
97	C₂×C₄	—	844	4	no	—
98	C₂×C₄	C₂²⋊Dic₂₁₁	1	4	yes	—
99	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
100	C₂×C₄	D₄×D₂₁₁	1	2	yes	—
101	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
102	C₂×C₄	#16809961	1	2	yes	—
103	C₈	—	422	4	no	—
104	C₂²	C₂×C₂²⋊Dic₂₁₁	1	1	yes	characteristic
105	C₂²	#20269964	1	1	yes	characteristic
106	C₂²	#20269973	1	1	yes	characteristic
107	C₂²	#20270008	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	4	4	no	—
110	C₄	—	1688	2	no	—
111	C₄	D₄×Dic₂₁₁	1	2	yes	—
112	C₄	#20269948	1	2	yes	—
113	C₂	#27213049	1	1	yes	characteristic
114	C₂	#27213155	1	2	yes	—
115	C₂	#27213514	1	2	yes	—
116	C₂	#27213538	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269901

C₂

yes

maximal

C₂×#20269888

C₂

yes

maximal, characteristic

#27213012

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

#20269901

—

#20269888

—

C₂₁₁⋊#95

—

C₂²×Q₈×C₂₁₁

C₂²

yes

—

C₄⋊Q₈×C₂₁₁

—

C₄⋊Q₈×C₂₁₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₁₁

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₂²×Dic₂₁₁

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

D₄

yes

—

C₄⋊Dic₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

D₄

yes

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

Q₈×C₂₁₁

—

Q₈×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₄×D₄

yes

—

C₄×C₂₁₁

#121

yes

—

Dic₂₁₁

—

C₂×C₂₁₁

#390

yes

characteristic

C₂×C₂₁₁

#419

yes

—

C₂×C₂₁₁

#491

yes

—

C₂×C₂₁₁

#497

yes

—

C₂₁₁

#1960

yes

characteristic, complemented, Sylow

#1960

—

211

maximal, Sylow

C₂×C₄⋊Q₈

D₂₁₁

yes

characteristic

C₂×#103

—

211

—

C₂×#97

—

211

—

#382

—

211

—

C₂²×Q₈

C₂×D₂₁₁

yes

—

C₂²×C₈

—

211

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₂₁₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

#103

—

422

—

#97

—

422

—

#95

—

422

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₁₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

422

—

C₂²×C₄

C₂²⋊D₂₁₁

yes

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

211

—

C₄⋊C₄

—

C₄⋊C₄

—

422

—

C₄²

C₂×Dic₂₁₁

yes

—

C₄²

C₂²⋊D₂₁₁

yes

—

C₂³

C₂×C₂²⋊D₂₁₁

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

844

—

C₂×C₄

—

844

—

C₂×C₄

C₂²⋊Dic₂₁₁

yes

—

C₂×C₄

C₂²×Dic₂₁₁

yes

characteristic

100

C₂×C₄

D₄×D₂₁₁

yes

—

101

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

102

C₂×C₄

#16809961

yes

—

103

C₈

—

422

—

104

C₂²

C₂×C₂²⋊Dic₂₁₁

yes

characteristic

105

C₂²

#20269964

yes

characteristic

106

C₂²

#20269973

yes

characteristic

107

C₂²

#20270008

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

1688

—

111

C₄

D₄×Dic₂₁₁

yes

—

112

C₄

#20269948

yes

—

113

C₂

#27213049

yes

characteristic

114

C₂

#27213155

yes

—

115

C₂

#27213514

yes

—

116

C₂

#27213538

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂×C₄⋊Q₈×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂×C₄⋊Q₈	D₂₁₁
hypocenter	C₂₁₁	#1960
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₁₁

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

C₂×C₄⋊Q₈×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂×D₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

hypercenter

C₂×C₄⋊Q₈

D₂₁₁

hypocenter

C₂₁₁

#1960

LCS

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₂₁₁	C₄	semidirect
58. C₂₁₁	#1960	semidirect
67. C₄⋊Q₈	Dic₂₁₁	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₂₁₁

C₄

semidirect

58. C₂₁₁

#1960

semidirect

67. C₄⋊Q₈

Dic₂₁₁

semidirect

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

subgroup	index
C₂×#20269901	2
C₂×#20269888	2
#27213012	2
C₂×C₄⋊Q₈×C₂₁₁	2
#1960	211

subgroup

index

C₂×C₄⋊Q₈×C₂₁₁

211

Subgroups — #41014219

Conjugacy classes of subgroups

Complements

Maximal subgroups