Subgroups — #41013538

Order 27008.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213170	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269888	C₂	1	1	yes	maximal, characteristic
3	#27212910	C₂	1	4	yes	maximal
4	#423×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#16809946	C₂²	1	2	yes	—
6	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
7	C₄⋊(C₂₁₁⋊C₈)	—	2	2	no	—
8	#20269888	C₄	1	4	yes	—
9	C₂×M₁₆×C₂₁₁	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₂₁₁	—	2	2	no	—
12	#16809946	—	2	4	no	—
13	C₂×C₂₁₁⋊C₈	—	2	2	no	—
14	C₂×C₂₁₁⋊C₈	—	2	2	no	—
15	C₂×C₂₁₁⋊C₈	—	2	2	no	—
16	C₂×C₂₁₁⋊C₈	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₂₁₁	Q₈	1	2	yes	—
18	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
19	M₁₆×C₂₁₁	—	2	4	no	—
20	C₂×C₈×C₂₁₁	—	2	2	no	—
21	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
22	C₄²×C₂₁₁	D₄	1	2	yes	—
23	C₂₁₁⋊C₈	—	2	4	no	—
24	C₂₁₁⋊C₈	—	4	2	no	—
25	C₂³×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₄×C₂₁₁	—	4	2	no	—
27	C₂×C₄×C₂₁₁	C₄⋊C₄	1	4	yes	—
28	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₂₁₁	#41	1	2	yes	—
31	C₂×C₄×C₂₁₁	#41	1	2	yes	—
32	C₈×C₂₁₁	—	4	2	no	—
33	C₂²×C₂₁₁	—	2	2	no	—
34	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₂₁₁	#125	1	2	yes	—
36	C₄×C₂₁₁	—	2	4	no	—
37	C₄×C₂₁₁	#117	1	2	yes	—
38	C₄×C₂₁₁	#124	1	1	yes	characteristic
39	C₄×C₂₁₁	#124	1	1	yes	characteristic
40	C₂×C₂₁₁	—	2	2	no	—
41	C₂×C₂₁₁	#382	1	1	yes	characteristic
42	C₂×C₂₁₁	#430	1	2	yes	—
43	C₂₁₁	#1838	1	1	yes	characteristic, complemented, Sylow
44	#1838	—	211	1	no	maximal, Sylow
45	#423	—	211	1	no	—
46	#423	D₂₁₁	1	1	yes	characteristic
47	C₂×#97	—	211	1	no	—
48	#341	—	211	4	no	—
49	C₂×M₁₆	—	211	2	no	—
50	C₂×M₁₆	C₂×D₂₁₁	1	2	yes	—
51	C₂²×C₈	—	211	2	no	—
52	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
53	C₄⋊C₈	—	2	2	no	—
54	C₄⋊C₈	—	422	2	no	—
55	#97	—	211	4	no	—
56	C₂²×C₄	Dic₄₂₂	1	2	yes	—
57	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
58	M₁₆	—	2	4	no	—
59	M₁₆	—	422	4	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	422	2	no	—
62	C₂×C₈	—	422	2	no	—
63	C₂×C₈	—	422	2	no	—
64	C₂×C₈	—	211	4	no	—
65	C₄²	C₄×D₂₁₁	1	2	yes	—
66	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
67	C₂³	C₂×Dic₄₂₂	1	1	yes	characteristic
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	Dic₂₁₁⋊C₄	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
71	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
72	C₂×C₄	#16809961	1	2	yes	—
73	C₂×C₄	#16809963	1	2	yes	—
74	C₈	—	4	2	no	—
75	C₈	—	844	2	no	—
76	C₈	—	422	4	no	—
77	C₂²	—	2	2	no	—
78	C₂²	C₂×Dic₂₁₁⋊C₄	1	1	yes	characteristic
79	C₂²	#20269968	1	2	yes	—
80	C₄	—	2	4	no	—
81	C₄	#20269935	1	2	yes	—
82	C₄	#20269964	1	1	yes	characteristic
83	C₄	#20269966	1	1	yes	characteristic
84	C₂	—	2	2	no	—
85	C₂	#27213011	1	1	yes	characteristic
86	C₂	#27213194	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#423×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₄⋊C₄
prime radical	C₂³×C₂₁₁	C₂×Q₈
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#423	D₂₁₁
hypocenter	C₂₁₁	#1838
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
43. 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
C₂×#20269888	2
#27212910	2
#423×C₂₁₁	2
#1838	211