Group #41014392 — subgroups

Subgroups — #41014392

Order 27008.

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

normal subgroups only

This group has 228 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	#27213170	C₂	1	1	yes	maximal, characteristic
2	C₄×#16809946	C₂	1	1	yes	maximal, characteristic
3	#27212915	C₂	1	4	yes	maximal
4	C₂×#125×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#16809946	C₂²	1	2	yes	—
6	C₂×#16809946	C₂²	1	2	yes	—
7	C₄⋊(C₂₁₁⋊C₈)	—	2	2	no	—
8	#20269888	—	2	1	no	—
9	C₄×C₂₁₁⋊C₈	—	2	1	no	—
10	#125×C₂₁₁	—	2	1	no	—
11	#125×C₂₁₁	—	2	1	no	—
12	#125×C₂₁₁	—	2	2	no	—
13	C₂×C₄⋊C₄×C₂₁₁	C₄	1	4	yes	—
14	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
16	#16809946	—	2	2	no	—
17	#16809946	—	2	4	no	—
18	#16809946	Q₈	1	4	yes	—
19	C₂×C₂₁₁⋊C₈	—	2	2	no	—
20	C₂×C₂₁₁⋊C₈	—	2	2	no	—
21	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₂₁₁	—	2	4	no	—
25	C₄⋊C₄×C₂₁₁	—	2	4	no	—
26	C₄⋊C₄×C₂₁₁	—	2	2	no	—
27	C₄⋊C₄×C₂₁₁	—	2	2	no	—
28	C₄²×C₂₁₁	D₄	1	1	yes	characteristic
29	C₄²×C₂₁₁	D₄	1	1	yes	characteristic
30	C₄²×C₂₁₁	D₄	1	2	yes	—
31	C₂₁₁⋊C₈	—	2	4	no	—
32	C₂₁₁⋊C₈	—	4	2	no	—
33	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₁₁	—	2	4	no	—
35	C₂×C₄×C₂₁₁	—	4	2	no	—
36	C₂×C₄×C₂₁₁	—	4	4	no	—
37	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	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×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₂₁₁	#41	1	2	yes	—
43	C₂²×C₂₁₁	—	2	2	no	—
44	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂²×C₂₁₁	C₄×Q₈	1	2	yes	—
46	C₄×C₂₁₁	—	2	2	no	—
47	C₄×C₂₁₁	—	2	2	no	—
48	C₄×C₂₁₁	—	4	4	no	—
49	C₄×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
50	C₄×C₂₁₁	#124	1	1	yes	characteristic
51	C₄×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₂₁₁	—	2	2	no	—
53	C₂×C₂₁₁	#391	1	1	yes	characteristic
54	C₂×C₂₁₁	#413	1	2	yes	—
55	C₂₁₁	#1998	1	1	yes	characteristic, complemented, Sylow
56	#1998	—	211	1	no	maximal, Sylow
57	C₂×#125	D₂₁₁	1	1	yes	characteristic
58	#423	—	211	1	no	—
59	C₄×M₁₆	—	211	1	no	—
60	#343	—	211	4	no	—
61	C₂×M₁₆	—	211	2	no	—
62	C₂×M₁₆	—	211	2	no	—
63	#125	—	2	1	no	—
64	#125	—	2	1	no	—
65	#125	—	2	2	no	—
66	C₂×C₄⋊C₄	Dic₂₁₁	1	4	yes	—
67	C₂×C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
68	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
69	C₄⋊C₈	—	422	2	no	—
70	#97	—	422	1	no	—
71	C₄×C₈	—	422	1	no	—
72	C₂²×C₄	C₂×Dic₂₁₁	1	4	yes	—
73	C₂²×C₄	C₂×Dic₂₁₁	1	2	yes	—
74	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
75	M₁₆	—	422	4	no	—
76	M₁₆	—	422	2	no	—
77	M₁₆	—	211	4	no	—
78	C₂×C₈	—	422	2	no	—
79	C₂×C₈	—	422	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	4	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄²	C₂²⋊D₂₁₁	1	1	yes	characteristic
85	C₄²	C₂²⋊D₂₁₁	1	1	yes	characteristic
86	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
87	C₂³	C₂²×Dic₂₁₁	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	4	no	—
91	C₂×C₄	C₂²⋊Dic₂₁₁	1	2	yes	—
92	C₂×C₄	C₂²⋊Dic₂₁₁	1	2	yes	—
93	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
94	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
95	C₂×C₄	Q₈×D₂₁₁	1	2	yes	—
96	C₂×C₄	#16809963	1	2	yes	—
97	C₈	—	844	2	no	—
98	C₈	—	422	4	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₂×C₂²⋊Dic₂₁₁	1	1	yes	characteristic
101	C₂²	Q₈×Dic₂₁₁	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	2	no	—
104	C₄	—	4	4	no	—
105	C₄	#20269953	1	2	yes	—
106	C₄	#20269966	1	1	yes	characteristic
107	C₄	#20269975	1	1	yes	characteristic
108	C₂	—	2	2	no	—
109	C₂	#27213053	1	1	yes	characteristic
110	C₂	#27213130	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27213170

C₂

yes

maximal, characteristic

C₄×#16809946

C₂

yes

maximal, characteristic

#27212915

C₂

yes

maximal

C₂×#125×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#16809946

C₂²

yes

—

C₂×#16809946

C₂²

yes

—

C₄⋊(C₂₁₁⋊C₈)

—

#20269888

—

C₄×C₂₁₁⋊C₈

—

#125×C₂₁₁

—

#125×C₂₁₁

—

#125×C₂₁₁

—

C₂×C₄⋊C₄×C₂₁₁

C₄

yes

—

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

—

—

Q₈

yes

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

D₄

yes

characteristic

C₄²×C₂₁₁

D₄

yes

characteristic

C₄²×C₂₁₁

D₄

yes

—

C₂₁₁⋊C₈

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂²×C₄

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

C₂×D₄

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

C₄×Q₈

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂²⋊Q₈

yes

—

C₄×C₂₁₁

#124

yes

characteristic

C₄×C₂₁₁

C₄⋊Q₈

yes

characteristic

C₂×C₂₁₁

—

C₂×C₂₁₁

#391

yes

characteristic

C₂×C₂₁₁

#413

yes

—

C₂₁₁

#1998

yes

characteristic, complemented, Sylow

#1998

—

211

maximal, Sylow

C₂×#125

D₂₁₁

yes

characteristic

#423

—

211

—

C₄×M₁₆

—

211

—

#343

—

211

—

C₂×M₁₆

—

211

—

C₂×M₁₆

—

211

—

#125

—

#125

—

#125

—

C₂×C₄⋊C₄

Dic₂₁₁

yes

—

C₂×C₄⋊C₄

C₂×D₂₁₁

yes

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

C₄⋊C₈

—

422

—

#97

—

422

—

C₄×C₈

—

422

—

C₂²×C₄

C₂×Dic₂₁₁

yes

—

C₂²×C₄

C₂×Dic₂₁₁

yes

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

M₁₆

—

422

—

M₁₆

—

422

—

M₁₆

—

211

—

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₂₁₁

yes

characteristic

C₄²

C₂²⋊D₂₁₁

yes

characteristic

C₄²

C₂²⋊D₂₁₁

yes

—

C₂³

C₂²×Dic₂₁₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₂₁₁

yes

—

C₂×C₄

C₂²⋊Dic₂₁₁

yes

—

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₁₁

yes

characteristic

C₂×C₄

Q₈×D₂₁₁

yes

—

C₂×C₄

#16809963

yes

—

C₈

—

844

—

C₈

—

422

—

C₂²

—

100

C₂²

C₂×C₂²⋊Dic₂₁₁

yes

characteristic

101

C₂²

Q₈×Dic₂₁₁

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#20269953

yes

—

106

C₄

#20269966

yes

characteristic

107

C₄

#20269975

yes

characteristic

108

C₂

—

109

C₂

#27213053

yes

characteristic

110

C₂

#27213130

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₁₁
commutator, prime radical	C₂³×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂×#125×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂×#125	D₂₁₁
hypocenter	C₂₁₁	#1998
LCS	G ▷ C₂³×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₁₁

commutator, prime radical

C₂³×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

C₂×#125×C₂₁₁

C₂

socle

C₂²×C₂₁₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂³×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
55. C₂₁₁	#1998	semidirect

H ≅ G/N

type

55. C₂₁₁

#1998

semidirect

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

subgroup	index
#27213170	2
C₄×#16809946	2
#27212915	2
C₂×#125×C₂₁₁	2
#1998	211

subgroup

index

C₂×#125×C₂₁₁

211

Subgroups — #41014392

Conjugacy classes of subgroups

Complements

Maximal subgroups