Subgroups — #1746×C₂₁₁ #41009366

Order 27008.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#97×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
3	#376×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#336×C₂₁₁	C₂	1	4	yes	maximal
5	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
6	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₁₁	C₂²	1	2	yes	—
8	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
9	#97×C₂₁₁	C₄	1	4	yes	complemented
10	C₄×C₈×C₂₁₁	C₄	1	4	yes	complemented
11	#95×C₂₁₁	C₄	1	4	yes	—
12	C₂²×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
13	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
15	C₂×C₈×C₂₁₁	—	2	4	no	—
16	C₂×C₈×C₂₁₁	—	2	4	no	—
17	C₂×C₈×C₂₁₁	C₂×C₄	1	4	yes	—
18	C₂×C₈×C₂₁₁	C₂×C₄	1	4	yes	—
19	C₄²×C₂₁₁	—	2	4	no	—
20	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
21	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
22	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₁₁	—	2	4	no	—
24	C₂×C₄×C₂₁₁	—	2	4	no	—
25	C₂×C₄×C₂₁₁	—	2	8	no	—
26	C₂×C₄×C₂₁₁	C₄²	1	4	yes	—
27	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₂₁₁	#41	1	4	yes	—
30	C₈×C₂₁₁	—	2	4	no	—
31	C₈×C₂₁₁	—	2	4	no	—
32	C₂²×C₂₁₁	C₂×C₄²	1	1	yes	characteristic
33	C₂²×C₂₁₁	#117	1	2	yes	—
34	C₂²×C₂₁₁	#131	1	4	yes	—
35	C₄×C₂₁₁	—	2	4	no	—
36	C₄×C₂₁₁	—	4	4	no	—
37	C₄×C₂₁₁	#117	1	2	yes	—
38	C₄×C₂₁₁	#117	1	2	yes	—
39	C₂×C₂₁₁	#376	1	1	yes	characteristic
40	C₂×C₂₁₁	#405	1	2	yes	—
41	C₂×C₂₁₁	#433	1	4	yes	—
42	C₂₁₁	#1746	1	1	yes	characteristic, direct product factor, Sylow
43	#1746	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂×#97	C₂×C₂₁₁	1	1	yes	characteristic
45	C₂×C₄×C₈	C₂×C₂₁₁	1	1	yes	characteristic
46	#376	C₂×C₂₁₁	1	1	yes	characteristic
47	#336	C₂×C₂₁₁	1	4	yes	—
48	C₂²×C₈	C₂²×C₂₁₁	1	2	yes	—
49	C₂²×C₈	C₂²×C₂₁₁	1	2	yes	—
50	C₂×C₄²	C₂²×C₂₁₁	1	2	yes	—
51	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
52	#97	C₄×C₂₁₁	1	4	yes	complemented
53	C₄×C₈	C₄×C₂₁₁	1	4	yes	complemented
54	#95	C₄×C₂₁₁	1	4	yes	—
55	C₂²×C₄	C₂×C₄×C₂₁₁	1	4	yes	—
56	C₂²×C₄	C₂×C₄×C₂₁₁	1	2	yes	—
57	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
58	C₂×C₈	—	2	4	no	—
59	C₂×C₈	—	2	4	no	—
60	C₂×C₈	C₂×C₄×C₂₁₁	1	4	yes	—
61	C₂×C₈	C₂×C₄×C₂₁₁	1	4	yes	—
62	C₄²	—	2	4	no	—
63	C₄²	C₂×C₄×C₂₁₁	1	2	yes	—
64	C₄²	C₂×C₄×C₂₁₁	1	2	yes	—
65	C₂³	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	—	2	4	no	—
68	C₂×C₄	—	2	8	no	—
69	C₂×C₄	C₄²×C₂₁₁	1	4	yes	—
70	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
71	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
72	C₂×C₄	#41×C₂₁₁	1	4	yes	—
73	C₈	—	2	4	no	—
74	C₈	—	2	4	no	—
75	C₂²	C₂×C₄²×C₂₁₁	1	1	yes	characteristic
76	C₂²	#117×C₂₁₁	1	2	yes	—
77	C₂²	#131×C₂₁₁	1	4	yes	—
78	C₄	—	2	4	no	—
79	C₄	—	4	4	no	—
80	C₄	#117×C₂₁₁	1	2	yes	—
81	C₄	#117×C₂₁₁	1	2	yes	—
82	C₂	#376×C₂₁₁	1	1	yes	characteristic
83	C₂	#405×C₂₁₁	1	2	yes	—
84	C₂	#433×C₂₁₁	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₂₁₁	C₂³
commutator	C₂²	C₂×C₄²×C₂₁₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₁₁
socle, prime radical	C₂³×C₂₁₁	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
9. #97×C₂₁₁	C₄	semidirect
10. C₄×C₈×C₂₁₁	C₄	semidirect
42. C₂₁₁	#1746	direct (N × H)
43. #1746	C₂₁₁	direct (N × H)
52. #97	C₄×C₂₁₁	semidirect
53. C₄×C₈	C₄×C₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#97×C₂₁₁	2
C₂×C₄×C₈×C₂₁₁	2
#376×C₂₁₁	2
#336×C₂₁₁	2
#1746	211

Subgroups — #1746×C211 #41009366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1746×C₂₁₁ #41009366