Subgroups — #1412×C₂₁₁ #41009032

Order 27008.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#490×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	#353×C₂₁₁	C₂	1	2	yes	maximal
3	#127×C₂₁₁	C₂²	1	1	yes	characteristic
4	#125×C₂₁₁	C₄	1	1	yes	characteristic, complemented
5	#102×C₂₁₁	—	2	2	no	—
6	#99×C₂₁₁	—	2	2	no	—
7	#97×C₂₁₁	C₄	1	1	yes	characteristic, complemented
8	C₂×D₄×C₂₁₁	—	4	1	no	—
9	C₂×D₄×C₂₁₁	D₄	1	2	yes	—
10	C₂×C₈×C₂₁₁	—	2	1	no	—
11	C₄⋊C₄×C₂₁₁	—	2	1	no	—
12	C₄⋊C₄×C₂₁₁	—	4	1	no	—
13	C₂²⋊C₄×C₂₁₁	—	4	2	no	—
14	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
15	C₂³×C₂₁₁	—	2	2	no	—
16	D₄×C₂₁₁	—	4	2	no	—
17	D₄×C₂₁₁	—	8	2	no	—
18	C₂×C₄×C₂₁₁	—	2	1	no	—
19	C₂×C₄×C₂₁₁	—	4	1	no	—
20	C₂×C₄×C₂₁₁	—	8	2	no	—
21	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
22	C₈×C₂₁₁	—	4	1	no	—
23	C₂²×C₂₁₁	—	4	2	no	—
24	C₂²×C₂₁₁	—	8	2	no	—
25	C₂²×C₂₁₁	#99	1	1	yes	characteristic
26	C₄×C₂₁₁	—	2	1	no	—
27	C₄×C₂₁₁	—	4	1	no	—
28	C₄×C₂₁₁	—	8	1	no	—
29	C₄×C₂₁₁	—	16	2	no	—
30	C₂×C₂₁₁	—	2	1	no	—
31	C₂×C₂₁₁	—	8	2	no	—
32	C₂×C₂₁₁	#354	1	1	yes	characteristic
33	C₂₁₁	#1412	1	1	yes	characteristic, direct product factor, Sylow
34	#1412	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	#490	C₂×C₂₁₁	1	1	yes	characteristic
36	#353	C₂×C₂₁₁	1	2	yes	—
37	#127	C₂²×C₂₁₁	1	1	yes	characteristic
38	#125	C₄×C₂₁₁	1	1	yes	characteristic, complemented
39	#102	—	2	2	no	—
40	#99	—	2	2	no	—
41	#97	C₄×C₂₁₁	1	1	yes	characteristic, complemented
42	C₂×D₄	—	4	1	no	—
43	C₂×D₄	D₄×C₂₁₁	1	2	yes	—
44	C₂×C₈	—	2	1	no	—
45	C₄⋊C₄	—	2	1	no	—
46	C₄⋊C₄	—	4	1	no	—
47	C₂²⋊C₄	—	4	2	no	—
48	C₄²	C₂×C₄×C₂₁₁	1	1	yes	characteristic
49	C₂³	—	2	2	no	—
50	D₄	—	4	2	no	—
51	D₄	—	8	2	no	—
52	C₂×C₄	—	2	1	no	—
53	C₂×C₄	—	4	1	no	—
54	C₂×C₄	—	8	2	no	—
55	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
56	C₈	—	4	1	no	—
57	C₂²	—	4	2	no	—
58	C₂²	—	8	2	no	—
59	C₂²	#99×C₂₁₁	1	1	yes	characteristic
60	C₄	—	2	1	no	—
61	C₄	—	4	1	no	—
62	C₄	—	8	1	no	—
63	C₄	—	16	2	no	—
64	C₂	—	2	1	no	—
65	C₂	—	8	2	no	—
66	C₂	#354×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₁₁	#354
commutator	C₄²	C₂×C₄×C₂₁₁
Frattini, cosocle	#127	C₂²×C₂₁₁
prime radical	#127×C₂₁₁	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₁₁ ◁ C₂²×C₂₁₁ ◁ C₂×C₄×C₂₁₁ ◁ #127×C₂₁₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. #125×C₂₁₁	C₄	semidirect
7. #97×C₂₁₁	C₄	semidirect
33. C₂₁₁	#1412	direct (N × H)
34. #1412	C₂₁₁	direct (N × H)
38. #125	C₄×C₂₁₁	semidirect
41. #97	C₄×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#490×C₂₁₁	2
#353×C₂₁₁	2
#1412	211

Subgroups — #1412×C211 #41009032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1412×C₂₁₁ #41009032