Subgroups — #41011683

Order 27008.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213197	C₂	1	2	yes	maximal
2	C₄×C₈×D₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#27212867	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈×C₂₁₁⋊C₈	C₂	1	2	yes	maximal, complemented
5	#322×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₂₁₁	C₄	1	2	yes	—
7	C₄²×D₂₁₁	C₂²	1	1	yes	characteristic
8	Dic₂₁₁⋊C₈	C₄	1	4	yes	—
9	(C₂×D₂₁₁)⋊C₈	C₄	1	4	yes	—
10	C₄×C₂₁₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₄×C₂₁₁⋊C₈	C₂²	1	2	yes	—
12	C₈×Dic₂₁₁	C₄	1	2	yes	—
13	C₄×C₈×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₂₁₁	C₂²	1	2	yes	—
15	C₂×C₄×D₂₁₁	C₂×C₄	1	2	yes	—
16	C₂×C₄×D₂₁₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₂₁₁⋊C₈	C₂×C₄	1	4	yes	—
18	C₂×C₂₁₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₈×D₂₁₁	C₈	1	8	yes	complemented
20	C₄×Dic₂₁₁	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₂₁₁	C₂×C₄	1	2	yes	—
22	C₂×C₈×C₂₁₁	C₂×C₄	1	4	yes	—
23	C₂×C₈×C₂₁₁	C₂×C₄	1	2	yes	—
24	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
25	C₂²×D₂₁₁	C₄²	1	1	yes	characteristic
26	C₂×Dic₂₁₁	C₄²	1	2	yes	—
27	C₂×Dic₂₁₁	C₄²	1	1	yes	characteristic
28	C₄×D₂₁₁	—	2	4	no	—
29	C₄×D₂₁₁	C₂×C₈	1	4	yes	—
30	C₂₁₁⋊C₈	—	2	4	no	—
31	C₂₁₁⋊C₈	C₂×C₈	1	4	yes	—
32	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
34	C₈×C₂₁₁	—	2	4	no	—
35	C₈×C₂₁₁	C₂×C₈	1	4	yes	—
36	C₂²×C₂₁₁	C₂×C₄²	1	1	yes	characteristic
37	C₂×D₂₁₁	—	2	2	no	—
38	C₂×D₂₁₁	C₄×C₈	1	2	yes	—
39	C₄×C₂₁₁	C₂²×C₈	1	2	yes	—
40	C₄×C₂₁₁	#131	1	4	yes	—
41	Dic₂₁₁	—	2	2	no	—
42	Dic₂₁₁	C₄×C₈	1	2	yes	—
43	C₂×C₂₁₁	C₂×C₄×C₈	1	1	yes	characteristic
44	C₂×C₂₁₁	#405	1	2	yes	—
45	D₂₁₁	—	2	2	no	—
46	C₂₁₁	#1450	1	1	yes	characteristic, complemented, Sylow
47	#1450	—	211	1	no	maximal, Sylow
48	#431	—	211	2	no	—
49	C₂×C₄×C₈	—	211	1	no	—
50	#322	—	211	1	no	—
51	#322	D₂₁₁	1	1	yes	characteristic, complemented
52	C₈²	—	211	2	no	—
53	C₂²×C₈	—	211	2	no	—
54	C₂×C₄²	—	211	1	no	—
55	C₄⋊C₈	—	211	4	no	—
56	C₂²⋊C₈	—	211	4	no	—
57	C₄×C₈	—	211	2	no	—
58	C₄×C₈	—	211	1	no	—
59	C₄×C₈	—	211	2	no	—
60	C₄×C₈	C₂×D₂₁₁	1	2	yes	—
61	C₄×C₈	C₂×D₂₁₁	1	1	yes	characteristic
62	C₂²×C₄	—	211	2	no	—
63	C₂²×C₄	—	211	1	no	—
64	C₂×C₈	—	211	4	no	—
65	C₂×C₈	—	211	2	no	—
66	C₂×C₈	—	211	8	no	—
67	C₂×C₈	C₄×D₂₁₁	1	4	yes	—
68	C₂×C₈	C₄×D₂₁₁	1	2	yes	—
69	C₄²	—	211	1	no	—
70	C₄²	—	211	2	no	—
71	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
72	C₂³	—	211	1	no	—
73	C₂×C₄	—	422	4	no	—
74	C₂×C₄	—	211	2	no	—
75	C₂×C₄	—	211	1	no	—
76	C₂×C₄	—	211	4	no	—
77	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
78	C₂×C₄	C₂×C₄×D₂₁₁	1	2	yes	—
79	C₈	—	2	4	no	—
80	C₈	—	422	4	no	—
81	C₈	—	211	4	no	—
82	C₈	C₈×D₂₁₁	1	4	yes	—
83	C₂²	—	422	2	no	—
84	C₂²	—	211	2	no	—
85	C₂²	C₄²×D₂₁₁	1	1	yes	characteristic
86	C₄	—	422	2	no	—
87	C₄	—	211	2	no	—
88	C₄	C₂×C₈×D₂₁₁	1	2	yes	—
89	C₄	#20269985	1	4	yes	—
90	C₂	—	422	2	no	—
91	C₂	C₄×C₈×D₂₁₁	1	1	yes	characteristic
92	C₂	#27213103	1	2	yes	—

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₂₁₁
commutator	C₂×C₂₁₁	C₂×C₄×C₈
Fitting	#322×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₄²
prime radical	C₂²×D₂₁₁	C₄²
cosocle	C₄²×C₂₁₁	C₂³
hypercenter	#322	D₂₁₁
hypocenter	C₂₁₁	#1450
LCS	G ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
3. #27212867	C₂	semidirect
4. C₈×C₂₁₁⋊C₈	C₂	semidirect
5. #322×C₂₁₁	C₂	semidirect
19. C₈×D₂₁₁	C₈	semidirect
46. C₂₁₁	#1450	semidirect
51. #322	D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#27213197	2
C₄×C₈×D₂₁₁	2
#27212867	2
C₈×C₂₁₁⋊C₈	2
#322×C₂₁₁	2
#1450	211