Group #41014379 — subgroups

Subgroups — #41014379

Order 27008.

244 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 93 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	#27213170	C₂	1	1	yes	maximal, characteristic
3	#27212906	C₂	1	4	yes	maximal
4	#376×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#16809946	C₂²	1	2	yes	—
6	#20269977	C₄	1	2	yes	complemented
7	#20269969	C₄	1	2	yes	complemented
8	#20269956	—	2	2	no	—
9	C₄×Dic₄₂₂	—	2	2	no	—
10	#20269936	C₂²	1	2	yes	—
11	C₄⋊(C₂₁₁⋊C₈)	—	2	2	no	—
12	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
14	#95×C₂₁₁	—	2	2	no	—
15	C₂×Dic₄₂₂	—	2	2	no	—
16	#16809946	—	2	4	no	—
17	C₂×C₂₁₁⋊C₈	—	2	2	no	—
18	C₄⋊Dic₂₁₁	—	2	2	no	—
19	C₄⋊Dic₂₁₁	C₂×C₄	1	4	yes	—
20	Dic₂₁₁⋊C₄	—	2	4	no	—
21	C₂²⋊Dic₂₁₁	—	2	2	no	—
22	C₄×Dic₂₁₁	—	2	2	no	—
23	C₂²×C₄×C₂₁₁	—	2	2	no	—
24	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
25	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
26	C₄²×C₂₁₁	—	2	2	no	—
27	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
28	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₂₁₁	D₄	1	2	yes	—
30	C₄²×C₂₁₁	Q₈	1	4	yes	complemented
31	Dic₄₂₂	—	4	4	no	—
32	C₂×Dic₂₁₁	—	2	4	no	—
33	C₂₁₁⋊C₈	—	4	2	no	—
34	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₂₁₁	—	2	4	no	—
36	C₂×C₄×C₂₁₁	—	4	2	no	—
37	C₂×C₄×C₂₁₁	—	4	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	C₂²×C₂₁₁	—	2	2	no	—
45	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₂₁₁	—	2	2	no	—
48	C₄×C₂₁₁	—	2	2	no	—
49	C₄×C₂₁₁	—	4	4	no	—
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	—	4	4	no	—
54	C₂×C₂₁₁	—	2	2	no	—
55	C₂×C₂₁₁	#391	1	1	yes	characteristic
56	C₂×C₂₁₁	#421	1	2	yes	—
57	C₂₁₁	#1995	1	1	yes	characteristic, complemented, Sylow
58	#1995	—	211	1	no	maximal, Sylow
59	#533	—	211	1	no	—
60	#423	—	211	1	no	—
61	#376	D₂₁₁	1	1	yes	characteristic
62	#339	—	211	4	no	—
63	C₂×M₁₆	—	211	2	no	—
64	C₄⋊Q₈	—	211	2	no	—
65	#125	—	211	2	no	—
66	C₂²⋊Q₈	—	422	2	no	—
67	C₄×Q₈	—	422	2	no	—
68	#117	—	211	2	no	—
69	C₂×C₄²	C₂×D₂₁₁	1	2	yes	—
70	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
71	C₄⋊C₈	—	422	2	no	—
72	#95	—	2	2	no	—
73	C₂×Q₈	—	422	2	no	—
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	C₂²⋊D₂₁₁	1	2	yes	—
76	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
77	M₁₆	—	422	4	no	—
78	C₂×C₈	—	422	2	no	—
79	C₄⋊C₄	—	422	4	no	—
80	C₄⋊C₄	—	422	2	no	—
81	C₄⋊C₄	—	211	4	no	—
82	C₂²⋊C₄	—	422	2	no	—
83	C₄²	—	2	2	no	—
84	C₄²	—	422	2	no	—
85	C₄²	C₄×D₂₁₁	1	1	yes	characteristic
86	C₄²	C₄×D₂₁₁	1	1	yes	characteristic
87	C₄²	D₈₄₄	1	2	yes	—
88	C₄²	Dic₄₂₂	1	4	yes	complemented
89	C₂³	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
90	Q₈	—	844	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	4	4	no	—
94	C₂×C₄	—	422	4	no	—
95	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
96	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
97	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
98	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₄₂₂	1	2	yes	—
100	C₂×C₄	#16809962	1	2	yes	—
101	C₈	—	844	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
104	C₂²	#20269956	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	4	4	no	—
108	C₄	—	844	4	no	—
109	C₄	C₄×Dic₄₂₂	1	2	yes	—
110	C₄	#20269965	1	1	yes	characteristic
111	C₄	#20269977	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#27213052	1	1	yes	characteristic
114	C₂	#27213164	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27213703

C₂

yes

maximal, characteristic

#27213170

C₂

yes

maximal, characteristic

#27212906

C₂

yes

maximal

#376×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

C₂²

yes

—

#20269977

C₄

yes

complemented

#20269969

C₄

yes

complemented

#20269956

—

C₄×Dic₄₂₂

—

#20269936

C₂²

yes

—

C₄⋊(C₂₁₁⋊C₈)

—

C₂×C₄²×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

#95×C₂₁₁

—

C₂×Dic₄₂₂

—

#16809946

—

C₂×C₂₁₁⋊C₈

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

C₂×C₄

yes

—

Dic₂₁₁⋊C₄

—

C₂²⋊Dic₂₁₁

—

C₄×Dic₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₄²×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

characteristic

C₄²×C₂₁₁

C₂×C₄

yes

characteristic

C₄²×C₂₁₁

D₄

yes

—

C₄²×C₂₁₁

Q₈

yes

complemented

Dic₄₂₂

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

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

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₁₁

C₂²⋊Q₈

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₄×Q₈

yes

—

C₄×C₂₁₁

#124

yes

characteristic

C₄×C₂₁₁

C₄⋊Q₈

yes

characteristic

Dic₂₁₁

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#391

yes

characteristic

C₂×C₂₁₁

#421

yes

—

C₂₁₁

#1995

yes

characteristic, complemented, Sylow

#1995

—

211

maximal, Sylow

#533

—

211

—

#423

—

211

—

#376

D₂₁₁

yes

characteristic

#339

—

211

—

C₂×M₁₆

—

211

—

C₄⋊Q₈

—

211

—

#125

—

211

—

C₂²⋊Q₈

—

422

—

C₄×Q₈

—

422

—

#117

—

211

—

C₂×C₄²

C₂×D₂₁₁

yes

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₄⋊C₈

—

422

—

#95

—

C₂×Q₈

—

422

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₁₁

yes

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

422

—

C₂×C₈

—

422

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

211

—

C₂²⋊C₄

—

422

—

C₄²

—

C₄²

—

422

—

C₄²

C₄×D₂₁₁

yes

characteristic

C₄²

C₄×D₂₁₁

yes

characteristic

C₄²

D₈₄₄

yes

—

C₄²

Dic₄₂₂

yes

complemented

C₂³

C₂×C₂²⋊D₂₁₁

yes

characteristic

Q₈

—

844

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

422

—

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

—

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

C₂×C₄

C₂×D₈₄₄

yes

characteristic

C₂×C₄

C₂×Dic₄₂₂

yes

—

100

C₂×C₄

#16809962

yes

—

101

C₈

—

844

—

102

C₂²

—

103

C₂²

C₂×(C₂×D₂₁₁)⋊C₄

yes

characteristic

104

C₂²

#20269956

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

108

C₄

—

844

—

109

C₄

C₄×Dic₄₂₂

yes

—

110

C₄

#20269965

yes

characteristic

111

C₄

#20269977

yes

characteristic

112

C₂

—

113

C₂

#27213052

yes

characteristic

114

C₂

#27213164

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₈₄₄
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#376×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₁₁	C₂×D₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#376	D₂₁₁
hypocenter	C₂₁₁	#1995
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×D₈₄₄

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#376×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 ◁ #376

N	H ≅ G/N	type
6. #20269977	C₄	semidirect
7. #20269969	C₄	semidirect
30. C₄²×C₂₁₁	Q₈	semidirect
57. C₂₁₁	#1995	semidirect
88. C₄²	Dic₄₂₂	semidirect

H ≅ G/N

type

6. #20269977

C₄

semidirect

7. #20269969

C₄

semidirect

30. C₄²×C₂₁₁

Q₈

semidirect

57. C₂₁₁

#1995

semidirect

88. C₄²

Dic₄₂₂

semidirect

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

subgroup	index
#27213703	2
#27213170	2
#27212906	2
#376×C₂₁₁	2
#1995	211

subgroup

index

#376×C₂₁₁

211

Subgroups — #41014379

Conjugacy classes of subgroups

Complements

Maximal subgroups