Group #41014374 — subgroups

Subgroups — #41014374

Order 27008.

284 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 97 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	#27213703	C₂	1	1	yes	maximal, characteristic
2	C₄²×Dic₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#27212907	C₂	1	4	yes	maximal
4	#423×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	#20269977	C₄	1	2	yes	complemented
6	#20269969	C₄	1	2	yes	complemented
7	#20269956	—	2	2	no	—
8	C₄×Dic₄₂₂	—	2	2	no	—
9	#20269936	C₂²	1	2	yes	—
10	C₂×C₄×Dic₂₁₁	—	2	2	no	—
11	C₂×C₄×Dic₂₁₁	C₂²	1	2	yes	—
12	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₂₁₁	—	2	2	no	—
15	C₂×Dic₄₂₂	—	2	2	no	—
16	C₂²×Dic₂₁₁	—	2	2	no	—
17	C₄⋊Dic₂₁₁	—	2	2	no	—
18	C₄⋊Dic₂₁₁	C₂×C₄	1	4	yes	—
19	Dic₂₁₁⋊C₄	—	2	4	no	—
20	C₂²⋊Dic₂₁₁	—	2	2	no	—
21	C₄×Dic₂₁₁	—	2	8	no	—
22	C₄×Dic₂₁₁	—	2	2	no	—
23	C₄×Dic₂₁₁	—	2	2	no	—
24	C₄×Dic₂₁₁	Q₈	1	4	yes	complemented
25	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
26	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
27	M₁₆×C₂₁₁	—	2	4	no	—
28	C₂×C₈×C₂₁₁	—	2	2	no	—
29	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₂₁₁	D₄	1	2	yes	—
32	Dic₄₂₂	—	4	4	no	—
33	C₂×Dic₂₁₁	—	2	8	no	—
34	C₂×Dic₂₁₁	—	2	4	no	—
35	C₂×Dic₂₁₁	—	2	4	no	—
36	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₁₁	—	2	4	no	—
38	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₂₁₁	#41	1	2	yes	—
44	C₈×C₂₁₁	—	4	2	no	—
45	C₂²×C₂₁₁	—	2	2	no	—
46	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
47	C₂²×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
48	C₄×C₂₁₁	—	2	2	no	—
49	C₄×C₂₁₁	#104	1	4	yes	—
50	C₄×C₂₁₁	C₄×Q₈	1	2	yes	—
51	C₄×C₂₁₁	#124	1	1	yes	characteristic
52	C₄×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₂₁₁	—	2	8	no	—
54	Dic₂₁₁	—	4	4	no	—
55	C₂×C₂₁₁	—	2	2	no	—
56	C₂×C₂₁₁	#391	1	1	yes	characteristic
57	C₂×C₂₁₁	C₂×#104	1	2	yes	—
58	C₂₁₁	#1994	1	1	yes	characteristic, complemented, Sylow
59	#1994	—	211	1	no	maximal, Sylow
60	#533	—	211	1	no	—
61	#423	D₂₁₁	1	1	yes	characteristic
62	C₄³	—	211	1	no	—
63	#339	—	211	4	no	—
64	C₂×M₁₆	C₂×D₂₁₁	1	2	yes	—
65	C₄⋊Q₈	—	211	2	no	—
66	#125	—	211	2	no	—
67	C₂²⋊Q₈	—	422	2	no	—
68	C₄×Q₈	—	422	2	no	—
69	#117	—	211	2	no	—
70	C₂×C₄²	—	422	2	no	—
71	C₂×C₄²	—	211	2	no	—
72	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
73	C₄⋊C₈	—	2	2	no	—
74	C₂×Q₈	—	422	2	no	—
75	C₂²×C₄	—	422	2	no	—
76	C₂²×C₄	D₈₄₄	1	2	yes	—
77	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
78	M₁₆	—	2	4	no	—
79	C₂×C₈	—	2	2	no	—
80	C₄⋊C₄	—	422	4	no	—
81	C₄⋊C₄	—	422	2	no	—
82	C₄⋊C₄	—	211	4	no	—
83	C₂²⋊C₄	—	422	2	no	—
84	C₄²	—	422	8	no	—
85	C₄²	—	422	2	no	—
86	C₄²	—	422	2	no	—
87	C₄²	—	211	4	no	—
88	C₄²	C₄×D₂₁₁	1	1	yes	characteristic
89	C₄²	C₄×D₂₁₁	1	1	yes	characteristic
90	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
91	C₂³	C₂×D₈₄₄	1	1	yes	characteristic
92	Q₈	—	844	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	422	4	no	—
95	C₂×C₄	—	422	8	no	—
96	C₂×C₄	—	422	4	no	—
97	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
98	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
99	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
101	C₂×C₄	Q₈×D₂₁₁	1	2	yes	—
102	C₂×C₄	#16809961	1	2	yes	—
103	C₈	—	4	2	no	—
104	C₂²	—	2	2	no	—
105	C₂²	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
106	C₂²	#20269957	1	2	yes	—
107	C₄	—	2	2	no	—
108	C₄	—	844	4	no	—
109	C₄	—	422	8	no	—
110	C₄	#20269903	1	4	yes	—
111	C₄	#20269942	1	2	yes	—
112	C₄	#20269964	1	1	yes	characteristic
113	C₄	#20269975	1	1	yes	characteristic
114	C₂	—	2	2	no	—
115	C₂	#27213051	1	1	yes	characteristic
116	C₂	C₂×#20269903	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27213703

C₂

yes

maximal, characteristic

C₄²×Dic₂₁₁

C₂

yes

maximal, characteristic

#27212907

C₂

yes

maximal

#423×C₂₁₁

C₂

yes

maximal, characteristic

#20269977

C₄

yes

complemented

#20269969

C₄

yes

complemented

#20269956

—

C₄×Dic₄₂₂

—

#20269936

C₂²

yes

—

C₂×C₄×Dic₂₁₁

—

C₂×C₄×Dic₂₁₁

C₂²

yes

—

C₂×M₁₆×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₁₁

—

C₂×Dic₄₂₂

—

C₂²×Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

C₂×C₄

yes

—

Dic₂₁₁⋊C₄

—

C₂²⋊Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₄×Dic₂₁₁

Q₈

yes

complemented

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

M₁₆×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

characteristic

C₄²×C₂₁₁

C₂×C₄

yes

characteristic

C₄²×C₂₁₁

D₄

yes

—

Dic₄₂₂

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×Q₈

yes

—

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₁₁

C₂²⋊Q₈

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#104

yes

—

C₄×C₂₁₁

C₄×Q₈

yes

—

C₄×C₂₁₁

#124

yes

characteristic

C₄×C₂₁₁

C₄⋊Q₈

yes

characteristic

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#391

yes

characteristic

C₂×C₂₁₁

C₂×#104

yes

—

C₂₁₁

#1994

yes

characteristic, complemented, Sylow

#1994

—

211

maximal, Sylow

#533

—

211

—

#423

D₂₁₁

yes

characteristic

C₄³

—

211

—

#339

—

211

—

C₂×M₁₆

C₂×D₂₁₁

yes

—

C₄⋊Q₈

—

211

—

#125

—

211

—

C₂²⋊Q₈

—

422

—

C₄×Q₈

—

422

—

#117

—

211

—

C₂×C₄²

—

422

—

C₂×C₄²

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₄⋊C₈

—

C₂×Q₈

—

422

—

C₂²×C₄

—

422

—

C₂²×C₄

D₈₄₄

yes

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

211

—

C₂²⋊C₄

—

422

—

C₄²

—

422

—

C₄²

—

422

—

C₄²

—

422

—

C₄²

—

211

—

C₄²

C₄×D₂₁₁

yes

characteristic

C₄²

C₄×D₂₁₁

yes

characteristic

C₄²

C₂²⋊D₂₁₁

yes

—

C₂³

C₂×D₈₄₄

yes

characteristic

Q₈

—

844

—

C₂×C₄

—

C₂×C₄

—

422

—

C₂×C₄

—

422

—

C₂×C₄

—

422

—

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

—

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

101

C₂×C₄

Q₈×D₂₁₁

yes

—

102

C₂×C₄

#16809961

yes

—

103

C₈

—

104

C₂²

—

105

C₂²

C₂×(C₂×D₂₁₁)⋊C₄

yes

characteristic

106

C₂²

#20269957

yes

—

107

C₄

—

108

C₄

—

844

—

109

C₄

—

422

—

110

C₄

#20269903

yes

—

111

C₄

#20269942

yes

—

112

C₄

#20269964

yes

characteristic

113

C₄

#20269975

yes

characteristic

114

C₂

—

115

C₂

#27213051

yes

characteristic

116

C₂

C₂×#20269903

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#423×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₁₁	C₂×D₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#423	D₂₁₁
hypocenter	C₂₁₁	#1994
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₁₁

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#423×C₂₁₁

C₂

socle

C₂²×C₂₁₁

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₁₁

C₂×D₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
5. #20269977	C₄	semidirect
6. #20269969	C₄	semidirect
24. C₄×Dic₂₁₁	Q₈	semidirect
58. C₂₁₁	#1994	semidirect

H ≅ G/N

type

5. #20269977

C₄

semidirect

6. #20269969

C₄

semidirect

24. C₄×Dic₂₁₁

Q₈

semidirect

58. C₂₁₁

#1994

semidirect

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

subgroup	index
#27213703	2
C₄²×Dic₂₁₁	2
#27212907	2
#423×C₂₁₁	2
#1994	211

subgroup

index

#27213703

C₄²×Dic₂₁₁

#27212907

#423×C₂₁₁

#1994

211

Subgroups — #41014374

Conjugacy classes of subgroups

Complements

Maximal subgroups