Subgroups — #2263×C₂₁₁ #41009883

Order 27008.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#508×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₁₆×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
3	SD₆₄×C₂₁₁	C₂	1	2	yes	maximal, complemented
4	D₃₂×C₂₁₁	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#135×C₂₁₁	—	2	1	no	—
7	C₂×D₈×C₂₁₁	—	2	1	no	—
8	Q₃₂×C₂₁₁	C₂²	1	1	yes	characteristic, complemented
9	SD₃₂×C₂₁₁	—	2	1	no	—
10	D₁₆×C₂₁₁	—	2	1	no	—
11	D₁₆×C₂₁₁	C₂²	1	2	yes	—
12	D₁₆×C₂₁₁	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₁₆×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₃₂×C₂₁₁	C₂²	1	2	yes	complemented
15	#41×C₂₁₁	—	4	1	no	—
16	C₂×D₄×C₂₁₁	—	4	1	no	—
17	Q₁₆×C₂₁₁	—	2	1	no	—
18	SD₁₆×C₂₁₁	—	4	1	no	—
19	D₈×C₂₁₁	—	2	2	no	—
20	D₈×C₂₁₁	—	2	1	no	—
21	D₈×C₂₁₁	—	4	1	no	—
22	C₂×C₈×C₂₁₁	D₄	1	1	yes	characteristic
23	C₁₆×C₂₁₁	D₄	1	1	yes	characteristic
24	C₁₆×C₂₁₁	C₂³	1	1	yes	characteristic
25	C₂³×C₂₁₁	—	8	1	no	—
26	Q₈×C₂₁₁	—	4	1	no	—
27	D₄×C₂₁₁	—	4	2	no	—
28	D₄×C₂₁₁	—	4	1	no	—
29	D₄×C₂₁₁	—	8	1	no	—
30	D₄×C₂₁₁	—	8	1	no	—
31	C₂×C₄×C₂₁₁	—	8	1	no	—
32	C₂×C₄×C₂₁₁	D₈	1	1	yes	characteristic
33	C₈×C₂₁₁	D₈	1	1	yes	characteristic
34	C₈×C₂₁₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₂₁₁	—	8	1	no	—
36	C₂²×C₂₁₁	—	8	2	no	—
37	C₂²×C₂₁₁	—	16	2	no	—
38	C₂²×C₂₁₁	D₁₆	1	1	yes	characteristic
39	C₄×C₂₁₁	—	8	1	no	—
40	C₄×C₂₁₁	D₁₆	1	1	yes	characteristic
41	C₄×C₂₁₁	C₂×D₈	1	1	yes	characteristic
42	C₂×C₂₁₁	—	2	1	no	—
43	C₂×C₂₁₁	—	16	1	no	—
44	C₂×C₂₁₁	—	16	2	no	—
45	C₂×C₂₁₁	C₂×D₁₆	1	1	yes	characteristic
46	C₂₁₁	#2263	1	1	yes	characteristic, direct product factor, Sylow
47	#2263	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	#508	C₂×C₂₁₁	1	1	yes	characteristic, complemented
49	C₂×D₁₆	C₂×C₂₁₁	1	1	yes	characteristic, complemented
50	SD₆₄	C₂×C₂₁₁	1	2	yes	complemented
51	D₃₂	C₂×C₂₁₁	1	2	yes	complemented
52	M₆₄	C₂×C₂₁₁	1	1	yes	characteristic, complemented
53	#135	—	2	1	no	—
54	C₂×D₈	—	2	1	no	—
55	Q₃₂	C₂²×C₂₁₁	1	1	yes	characteristic, complemented
56	SD₃₂	—	2	1	no	—
57	D₁₆	—	2	1	no	—
58	D₁₆	C₂²×C₂₁₁	1	2	yes	—
59	D₁₆	C₂²×C₂₁₁	1	1	yes	characteristic, complemented
60	C₂×C₁₆	C₂²×C₂₁₁	1	1	yes	characteristic
61	C₃₂	C₂²×C₂₁₁	1	2	yes	complemented
62	#41	—	4	1	no	—
63	C₂×D₄	—	4	1	no	—
64	Q₁₆	—	2	1	no	—
65	SD₁₆	—	4	1	no	—
66	D₈	—	2	2	no	—
67	D₈	—	2	1	no	—
68	D₈	—	4	1	no	—
69	C₂×C₈	D₄×C₂₁₁	1	1	yes	characteristic
70	C₁₆	D₄×C₂₁₁	1	1	yes	characteristic
71	C₁₆	C₂³×C₂₁₁	1	1	yes	characteristic
72	C₂³	—	8	1	no	—
73	Q₈	—	4	1	no	—
74	D₄	—	4	2	no	—
75	D₄	—	4	1	no	—
76	D₄	—	8	1	no	—
77	D₄	—	8	1	no	—
78	C₂×C₄	—	8	1	no	—
79	C₂×C₄	D₈×C₂₁₁	1	1	yes	characteristic
80	C₈	D₈×C₂₁₁	1	1	yes	characteristic
81	C₈	C₂×D₄×C₂₁₁	1	1	yes	characteristic
82	C₂²	—	8	1	no	—
83	C₂²	—	8	2	no	—
84	C₂²	—	16	2	no	—
85	C₂²	D₁₆×C₂₁₁	1	1	yes	characteristic
86	C₄	—	8	1	no	—
87	C₄	D₁₆×C₂₁₁	1	1	yes	characteristic
88	C₄	C₂×D₈×C₂₁₁	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	16	2	no	—
91	C₂	—	16	1	no	—
92	C₂	C₂×D₁₆×C₂₁₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₂₁₁	C₂×D₁₆
commutator, Frattini, cosocle	C₁₆	C₂³×C₂₁₁
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₂₁₁ ◁ C₂×C₄×C₂₁₁ ◁ C₂×C₈×C₂₁₁ ◁ C₂×C₁₆×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. #508×C₂₁₁	C₂	semidirect
2. C₂×D₁₆×C₂₁₁	C₂	semidirect
3. SD₆₄×C₂₁₁	C₂	semidirect
4. D₃₂×C₂₁₁	C₂	semidirect
5. M₆₄×C₂₁₁	C₂	semidirect
8. Q₃₂×C₂₁₁	C₂²	semidirect
12. D₁₆×C₂₁₁	C₂²	semidirect
14. C₃₂×C₂₁₁	C₂²	semidirect
46. C₂₁₁	#2263	direct (N × H)
47. #2263	C₂₁₁	direct (N × H)
48. #508	C₂×C₂₁₁	semidirect
49. C₂×D₁₆	C₂×C₂₁₁	semidirect
50. SD₆₄	C₂×C₂₁₁	semidirect
51. D₃₂	C₂×C₂₁₁	semidirect
52. M₆₄	C₂×C₂₁₁	semidirect
55. Q₃₂	C₂²×C₂₁₁	semidirect
59. D₁₆	C₂²×C₂₁₁	semidirect
61. C₃₂	C₂²×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
#508×C₂₁₁	2
C₂×D₁₆×C₂₁₁	2
SD₆₄×C₂₁₁	2
D₃₂×C₂₁₁	2
M₆₄×C₂₁₁	2
#2263	211

Subgroups — #2263×C211 #41009883

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2263×C₂₁₁ #41009883