Subgroups — #41013539

Order 27008.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213170	C₂	1	2	yes	maximal
2	#27212911	C₂	1	4	yes	maximal
3	C₂×#97×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#16809946	C₂²	1	4	yes	—
5	C₄⋊(C₂₁₁⋊C₈)	—	2	4	no	—
6	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
8	#97×C₂₁₁	C₄	1	4	yes	—
9	#16809946	—	2	8	no	—
10	C₂×C₂₁₁⋊C₈	—	2	4	no	—
11	C₂²×C₄×C₂₁₁	Q₈	1	2	yes	—
12	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
13	C₂×C₈×C₂₁₁	—	2	4	no	—
14	C₂×C₈×C₂₁₁	C₂×C₄	1	4	yes	—
15	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
16	C₄²×C₂₁₁	D₄	1	2	yes	—
17	C₂₁₁⋊C₈	—	4	4	no	—
18	C₂³×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
19	C₂×C₄×C₂₁₁	—	4	2	no	—
20	C₂×C₄×C₂₁₁	C₄⋊C₄	1	4	yes	—
21	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
22	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
23	C₂×C₄×C₂₁₁	#41	1	4	yes	—
24	C₈×C₂₁₁	—	2	4	no	—
25	C₂²×C₂₁₁	—	2	2	no	—
26	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
27	C₂²×C₂₁₁	#125	1	2	yes	—
28	C₄×C₂₁₁	—	2	4	no	—
29	C₄×C₂₁₁	#117	1	2	yes	—
30	C₄×C₂₁₁	#124	1	2	yes	—
31	C₂×C₂₁₁	—	2	2	no	—
32	C₂×C₂₁₁	#382	1	1	yes	characteristic
33	C₂×C₂₁₁	#430	1	2	yes	—
34	C₂₁₁	#1838	1	1	yes	characteristic, complemented, Sylow
35	#1838	—	211	1	no	maximal, Sylow
36	#423	—	211	2	no	—
37	C₂×#97	D₂₁₁	1	1	yes	characteristic
38	#341	—	211	4	no	—
39	C₂×M₁₆	—	211	4	no	—
40	C₂²×C₈	C₂×D₂₁₁	1	2	yes	—
41	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
42	C₄⋊C₈	—	422	4	no	—
43	#97	Dic₂₁₁	1	4	yes	—
44	C₂²×C₄	Dic₄₂₂	1	2	yes	—
45	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
46	M₁₆	—	422	8	no	—
47	C₂×C₈	—	2	4	no	—
48	C₂×C₈	—	422	4	no	—
49	C₂×C₈	C₂×Dic₂₁₁	1	4	yes	—
50	C₄²	C₂×Dic₂₁₁	1	2	yes	—
51	C₄²	D₈₄₄	1	2	yes	—
52	C₂³	C₂×Dic₄₂₂	1	1	yes	characteristic
53	C₂×C₄	—	4	2	no	—
54	C₂×C₄	C₄⋊Dic₂₁₁	1	4	yes	—
55	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
56	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
57	C₂×C₄	#16809962	1	4	yes	—
58	C₈	—	2	4	no	—
59	C₈	—	844	4	no	—
60	C₂²	—	2	2	no	—
61	C₂²	C₂×C₄⋊Dic₂₁₁	1	1	yes	characteristic
62	C₂²	#20269969	1	2	yes	—
63	C₄	—	2	4	no	—
64	C₄	#20269936	1	2	yes	—
65	C₄	#20269965	1	2	yes	—
66	C₂	—	2	2	no	—
67	C₂	#27213012	1	1	yes	characteristic
68	C₂	#27213195	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₈₄₄
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	C₂×#97×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₄⋊C₄
prime radical	C₂³×C₂₁₁	C₂×Q₈
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	C₂×#97	D₂₁₁
hypocenter	C₂₁₁	#1838
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
34. C₂₁₁	#1838	semidirect

Maximal subgroups

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

subgroup	index
#27213170	2
#27212911	2
C₂×#97×C₂₁₁	2
#1838	211