Group #41009316 — subgroups

Subgroups — #1696×C₂₁₁ #41009316

Order 27008.

140 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 68 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	#500×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#477×C₂₁₁	C₂	1	2	yes	maximal
3	#446×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#334×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	#333×C₂₁₁	C₂	1	2	yes	maximal
6	C₄⋊Q₈×C₂₁₁	C₂²	1	2	yes	—
7	C₄×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
8	#107×C₂₁₁	—	2	1	no	—
9	#107×C₂₁₁	—	2	1	no	—
10	#107×C₂₁₁	—	2	2	no	—
11	C₄⋊C₈×C₂₁₁	—	2	1	no	—
12	C₄⋊C₈×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₂₁₁	C₂²	1	2	yes	—
14	#103×C₂₁₁	—	2	2	no	—
15	#97×C₂₁₁	—	2	1	no	—
16	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₂₁₁	—	2	2	no	—
18	C₂×Q₈×C₂₁₁	D₄	1	1	yes	characteristic
19	C₂×C₈×C₂₁₁	—	2	1	no	—
20	C₂×C₈×C₂₁₁	—	2	2	no	—
21	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
22	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
23	C₄⋊C₄×C₂₁₁	—	2	1	no	—
24	C₄⋊C₄×C₂₁₁	—	2	2	no	—
25	C₄⋊C₄×C₂₁₁	—	2	2	no	—
26	C₄⋊C₄×C₂₁₁	D₄	1	1	yes	characteristic
27	C₄²×C₂₁₁	—	2	1	no	—
28	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
29	Q₈×C₂₁₁	—	2	2	no	—
30	Q₈×C₂₁₁	—	8	2	no	—
31	C₂×C₄×C₂₁₁	—	2	1	no	—
32	C₂×C₄×C₂₁₁	—	2	1	no	—
33	C₂×C₄×C₂₁₁	—	4	2	no	—
34	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₁₁	#41	1	1	yes	characteristic
37	C₈×C₂₁₁	—	2	1	no	—
38	C₈×C₂₁₁	—	4	1	no	—
39	C₈×C₂₁₁	—	4	2	no	—
40	C₈×C₂₁₁	Q₁₆	1	2	yes	—
41	C₂²×C₂₁₁	#121	1	1	yes	characteristic
42	C₄×C₂₁₁	—	2	1	no	—
43	C₄×C₂₁₁	—	4	1	no	—
44	C₄×C₂₁₁	—	4	1	no	—
45	C₄×C₂₁₁	—	8	2	no	—
46	C₄×C₂₁₁	C₂×Q₁₆	1	1	yes	characteristic
47	C₄×C₂₁₁	#137	1	1	yes	characteristic
48	C₄×C₂₁₁	#137	1	2	yes	—
49	C₂×C₂₁₁	#462	1	1	yes	characteristic
50	C₂×C₂₁₁	#470	1	1	yes	characteristic
51	C₂×C₂₁₁	#472	1	1	yes	characteristic
52	C₂₁₁	#1696	1	1	yes	characteristic, direct product factor, Sylow
53	#1696	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	#500	C₂×C₂₁₁	1	1	yes	characteristic
55	#477	C₂×C₂₁₁	1	2	yes	—
56	#446	C₂×C₂₁₁	1	1	yes	characteristic
57	#334	C₂×C₂₁₁	1	1	yes	characteristic
58	#333	C₂×C₂₁₁	1	2	yes	—
59	C₄⋊Q₈	C₂²×C₂₁₁	1	2	yes	—
60	C₄×Q₈	C₂²×C₂₁₁	1	1	yes	characteristic
61	#107	—	2	1	no	—
62	#107	—	2	1	no	—
63	#107	—	2	2	no	—
64	C₄⋊C₈	—	2	1	no	—
65	C₄⋊C₈	C₂²×C₂₁₁	1	1	yes	characteristic
66	C₄⋊C₈	C₂²×C₂₁₁	1	2	yes	—
67	#103	—	2	2	no	—
68	#97	—	2	1	no	—
69	C₄×C₈	C₂²×C₂₁₁	1	1	yes	characteristic
70	C₂×Q₈	—	2	2	no	—
71	C₂×Q₈	D₄×C₂₁₁	1	1	yes	characteristic
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	D₄×C₂₁₁	1	1	yes	characteristic
75	C₂×C₈	D₄×C₂₁₁	1	1	yes	characteristic
76	C₄⋊C₄	—	2	1	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	2	2	no	—
79	C₄⋊C₄	D₄×C₂₁₁	1	1	yes	characteristic
80	C₄²	—	2	1	no	—
81	C₄²	C₂³×C₂₁₁	1	1	yes	characteristic
82	Q₈	—	2	2	no	—
83	Q₈	—	8	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
88	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
89	C₂×C₄	#41×C₂₁₁	1	1	yes	characteristic
90	C₈	—	2	1	no	—
91	C₈	—	4	1	no	—
92	C₈	—	4	2	no	—
93	C₈	Q₁₆×C₂₁₁	1	2	yes	—
94	C₂²	#121×C₂₁₁	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	4	1	no	—
97	C₄	—	4	1	no	—
98	C₄	—	8	2	no	—
99	C₄	C₂×Q₁₆×C₂₁₁	1	1	yes	characteristic
100	C₄	#137×C₂₁₁	1	1	yes	characteristic
101	C₄	#137×C₂₁₁	1	2	yes	—
102	C₂	#462×C₂₁₁	1	1	yes	characteristic
103	C₂	#470×C₂₁₁	1	1	yes	characteristic
104	C₂	#472×C₂₁₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#500×C₂₁₁

C₂

yes

maximal, characteristic

#477×C₂₁₁

C₂

yes

maximal

#446×C₂₁₁

C₂

yes

maximal, characteristic

#334×C₂₁₁

C₂

yes

maximal, characteristic

#333×C₂₁₁

C₂

yes

maximal

C₄⋊Q₈×C₂₁₁

C₂²

yes

—

C₄×Q₈×C₂₁₁

C₂²

yes

characteristic

#107×C₂₁₁

—

#107×C₂₁₁

—

#107×C₂₁₁

—

C₄⋊C₈×C₂₁₁

—

C₄⋊C₈×C₂₁₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₁₁

C₂²

yes

—

#103×C₂₁₁

—

#97×C₂₁₁

—

C₄×C₈×C₂₁₁

C₂²

yes

characteristic

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

D₄

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

D₄

yes

characteristic

C₂×C₈×C₂₁₁

D₄

yes

characteristic

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

D₄

yes

characteristic

C₄²×C₂₁₁

—

C₄²×C₂₁₁

C₂³

yes

characteristic

Q₈×C₂₁₁

—

Q₈×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

characteristic

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₈×C₂₁₁

—

C₈×C₂₁₁

Q₁₆

yes

—

C₂²×C₂₁₁

#121

yes

characteristic

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×Q₁₆

yes

C₄×C₂₁₁

yes

C₄×C₂₁₁

yes

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, direct product factor, Sylow

#1696

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

#500

C₂×C₂₁₁

yes

characteristic

#477

C₂×C₂₁₁

yes

—

#446

C₂×C₂₁₁

yes

characteristic

#334

C₂×C₂₁₁

yes

characteristic

#333

C₂×C₂₁₁

yes

—

C₄⋊Q₈

C₂²×C₂₁₁

yes

—

C₄×Q₈

C₂²×C₂₁₁

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₂₁₁

yes

characteristic

C₄⋊C₈

C₂²×C₂₁₁

yes

—

#103

—

#97

—

C₄×C₈

C₂²×C₂₁₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

D₄×C₂₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₂₁₁

yes

characteristic

C₂×C₈

D₄×C₂₁₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₂₁₁

yes

characteristic

C₄²

—

C₄²

C₂³×C₂₁₁

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₂₁₁

yes

characteristic

C₂×C₄

C₂×D₄×C₂₁₁

yes

characteristic

C₂×C₄

#41×C₂₁₁

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

Q₁₆×C₂₁₁

yes

—

C₂²

#121×C₂₁₁

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

C₂×Q₁₆×C₂₁₁

yes

characteristic

100

C₄

#137×C₂₁₁

yes

characteristic

101

C₄

#137×C₂₁₁

yes

—

102

C₂

#462×C₂₁₁

yes

characteristic

103

C₂

#470×C₂₁₁

yes

characteristic

104

C₂

#472×C₂₁₁

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₁₁	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₂₁₁
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₁₁ ◁ C₄²×C₂₁₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₂₁₁

#121

commutator, Frattini, cosocle

C₄²

C₂³×C₂₁₁

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₁₁ ◁ C₄²×C₂₁₁ ◁ G

N	H ≅ G/N	type
52. C₂₁₁	#1696	direct (N × H)
53. #1696	C₂₁₁	direct (N × H)

H ≅ G/N

type

52. C₂₁₁

#1696

direct (N × H)

53. #1696

C₂₁₁

direct (N × H)

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

subgroup	index
#500×C₂₁₁	2
#477×C₂₁₁	2
#446×C₂₁₁	2
#334×C₂₁₁	2
#333×C₂₁₁	2
#1696	211

subgroup

index

#500×C₂₁₁

#477×C₂₁₁

#446×C₂₁₁

#334×C₂₁₁

#333×C₂₁₁

#1696

211

Subgroups — #1696×C₂₁₁ #41009316

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1696×C211 #41009316

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1696×C₂₁₁ #41009316