Subgroups — #41011309

Order 27008.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213203	C₂	1	2	yes	maximal
2	#423×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₂₁₁⋊C₈)	—	2	2	no	—
4	C₂²⋊(C₂₁₁⋊C₈)	C₄	1	4	yes	—
5	#20269888	—	2	2	no	—
6	C₂×M₁₆×C₂₁₁	C₄	1	2	yes	—
7	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
8	C₄⋊C₈×C₂₁₁	—	2	2	no	—
9	C₂×C₂₁₁⋊C₈	—	2	4	no	—
10	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
11	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
12	M₁₆×C₂₁₁	—	4	2	no	—
13	C₂×C₈×C₂₁₁	—	2	2	no	—
14	C₄²×C₂₁₁	D₄	1	2	yes	—
15	C₄²×C₂₁₁	D₄	1	1	yes	characteristic
16	C₄²×C₂₁₁	Q₈	1	1	yes	characteristic
17	C₂₁₁⋊C₈	—	4	4	no	—
18	C₂³×C₂₁₁	C₄²	1	1	yes	characteristic
19	C₂×C₄×C₂₁₁	—	2	1	no	—
20	C₂×C₄×C₂₁₁	—	2	1	no	—
21	C₂×C₄×C₂₁₁	—	4	2	no	—
22	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
23	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₂₁₁	C₄⋊C₄	1	1	yes	characteristic
26	C₈×C₂₁₁	—	4	2	no	—
27	C₂²×C₂₁₁	—	2	1	no	—
28	C₂²×C₂₁₁	—	2	2	no	—
29	C₂²×C₂₁₁	#95	1	1	yes	characteristic
30	C₄×C₂₁₁	—	2	1	no	—
31	C₄×C₂₁₁	—	2	2	no	—
32	C₄×C₂₁₁	—	2	2	no	—
33	C₄×C₂₁₁	#100	1	1	yes	characteristic
34	C₄×C₂₁₁	#101	1	1	yes	characteristic
35	C₂×C₂₁₁	—	4	1	no	—
36	C₂×C₂₁₁	#343	1	1	yes	characteristic
37	C₂×C₂₁₁	#344	1	2	yes	—
38	C₂₁₁	#1307	1	1	yes	characteristic, complemented, Sylow
39	#1307	—	211	1	no	maximal, Sylow
40	#432	—	211	2	no	—
41	#423	D₂₁₁	1	1	yes	characteristic
42	C₂×M₁₆	Dic₂₁₁	1	2	yes	—
43	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
44	C₄⋊C₈	—	2	2	no	—
45	C₄⋊C₈	—	422	2	no	—
46	C₂²⋊C₈	—	211	4	no	—
47	#97	—	422	2	no	—
48	C₂²×C₄	C₄×D₂₁₁	1	2	yes	—
49	C₂²×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
50	M₁₆	—	4	2	no	—
51	C₂×C₈	—	2	2	no	—
52	C₂×C₈	—	422	4	no	—
53	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
54	C₄²	D₈₄₄	1	1	yes	characteristic
55	C₄²	Dic₄₂₂	1	1	yes	characteristic
56	C₂³	C₄×Dic₂₁₁	1	1	yes	characteristic
57	C₂×C₄	—	2	1	no	—
58	C₂×C₄	—	2	1	no	—
59	C₂×C₄	—	4	2	no	—
60	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	2	yes	—
61	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
62	C₂×C₄	Dic₂₁₁⋊C₄	1	2	yes	—
63	C₂×C₄	C₄⋊Dic₂₁₁	1	1	yes	characteristic
64	C₈	—	4	2	no	—
65	C₈	—	844	4	no	—
66	C₂²	—	2	1	no	—
67	C₂²	—	2	2	no	—
68	C₂²	C₂₁₁⋊#95	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	2	2	no	—
71	C₄	—	2	2	no	—
72	C₄	#20269894	1	1	yes	characteristic
73	C₄	#20269896	1	1	yes	characteristic
74	C₂	—	4	1	no	—
75	C₂	#27212915	1	1	yes	characteristic
76	C₂	#27212918	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂₁₁⋊#95
commutator, prime radical	C₂³×C₂₁₁	C₄²
Frattini	C₂×C₄²	C₂×D₂₁₁
Fitting	#423×C₂₁₁	C₂
socle	C₂²×C₂₁₁	#95
cosocle	C₂×C₄²×C₂₁₁	C₂²
hypercenter	#423	D₂₁₁
hypocenter	C₂₁₁	#1307
LCS	G ▷ C₂³×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
38. C₂₁₁	#1307	semidirect

Maximal subgroups

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

subgroup	index
#27213203	2
#423×C₂₁₁	2
#1307	211