Subgroups — C₂×#494×C₂₁₁ #41010765

Order 27008.

552 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₁₆×C₂₁₁	C₂	1	4	yes	maximal
2	C₂×C₄⋊Q₈×C₂₁₁	C₂	1	2	yes	maximal
3	#494×C₂₁₁	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₂₁₁	C₂²	1	4	yes	—
6	C₂×Q₁₆×C₂₁₁	—	2	16	no	—
7	C₂×Q₁₆×C₂₁₁	C₂²	1	16	yes	—
8	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₂₁₁	—	2	4	no	—
10	C₄⋊Q₈×C₂₁₁	C₂²	1	8	yes	—
11	C₂×C₄⋊C₄×C₂₁₁	—	2	4	no	—
12	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₂₁₁	C₂²	1	4	yes	—
14	C₂×Q₈×C₂₁₁	—	2	4	no	—
15	C₂×Q₈×C₂₁₁	—	2	16	no	—
16	C₂×Q₈×C₂₁₁	C₂³	1	8	yes	—
17	C₂²×C₄×C₂₁₁	—	2	4	no	—
18	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
19	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
20	Q₁₆×C₂₁₁	—	2	32	no	—
21	C₂×C₈×C₂₁₁	D₄	1	8	yes	—
22	C₂×C₈×C₂₁₁	C₂³	1	4	yes	—
23	C₄⋊C₄×C₂₁₁	—	2	16	no	—
24	C₄²×C₂₁₁	D₄	1	2	yes	—
25	C₄²×C₂₁₁	C₂³	1	2	yes	—
26	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
27	Q₈×C₂₁₁	—	2	16	no	—
28	Q₈×C₂₁₁	—	4	8	no	—
29	C₂×C₄×C₂₁₁	—	2	8	no	—
30	C₂×C₄×C₂₁₁	—	4	8	no	—
31	C₂×C₄×C₂₁₁	Q₁₆	1	8	yes	complemented
32	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₁₁	C₂×D₄	1	4	yes	—
34	C₂×C₄×C₂₁₁	C₂×D₄	1	4	yes	—
35	C₂×C₄×C₂₁₁	C₂⁴	1	1	yes	characteristic
36	C₈×C₂₁₁	C₂×D₄	1	8	yes	—
37	C₂²×C₂₁₁	#127	1	2	yes	—
38	C₂²×C₂₁₁	C₂×Q₁₆	1	4	yes	—
39	C₂²×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
40	C₄×C₂₁₁	—	4	8	no	—
41	C₄×C₂₁₁	#127	1	2	yes	—
42	C₄×C₂₁₁	C₂×Q₁₆	1	8	yes	complemented
43	C₄×C₂₁₁	C₂²×D₄	1	2	yes	—
44	C₂×C₂₁₁	#494	1	4	yes	direct product factor
45	C₂×C₂₁₁	C₂×#127	1	1	yes	characteristic
46	C₂×C₂₁₁	C₂²×Q₁₆	1	2	yes	—
47	C₂₁₁	C₂×#494	1	1	yes	characteristic, direct product factor, Sylow
48	C₂×#494	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂²×Q₁₆	C₂×C₂₁₁	1	4	yes	—
50	C₂×C₄⋊Q₈	C₂×C₂₁₁	1	2	yes	—
51	#494	C₂×C₂₁₁	1	8	yes	direct product factor
52	C₂×C₄×C₈	C₂×C₂₁₁	1	1	yes	characteristic
53	C₂²×Q₈	C₂²×C₂₁₁	1	4	yes	—
54	C₂×Q₁₆	—	2	16	no	—
55	C₂×Q₁₆	C₂²×C₂₁₁	1	16	yes	—
56	C₂²×C₈	C₂²×C₂₁₁	1	2	yes	—
57	C₄⋊Q₈	—	2	4	no	—
58	C₄⋊Q₈	C₂²×C₂₁₁	1	8	yes	—
59	C₂×C₄⋊C₄	—	2	4	no	—
60	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
61	C₄×C₈	C₂²×C₂₁₁	1	4	yes	—
62	C₂×Q₈	—	2	4	no	—
63	C₂×Q₈	—	2	16	no	—
64	C₂×Q₈	C₂³×C₂₁₁	1	8	yes	—
65	C₂²×C₄	—	2	4	no	—
66	C₂²×C₄	D₄×C₂₁₁	1	2	yes	—
67	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
68	Q₁₆	—	2	32	no	—
69	C₂×C₈	D₄×C₂₁₁	1	8	yes	—
70	C₂×C₈	C₂³×C₂₁₁	1	4	yes	—
71	C₄⋊C₄	—	2	16	no	—
72	C₄²	D₄×C₂₁₁	1	2	yes	—
73	C₄²	C₂³×C₂₁₁	1	2	yes	—
74	C₂³	C₂×D₄×C₂₁₁	1	1	yes	characteristic
75	Q₈	—	2	16	no	—
76	Q₈	—	4	8	no	—
77	C₂×C₄	—	2	8	no	—
78	C₂×C₄	—	4	8	no	—
79	C₂×C₄	Q₁₆×C₂₁₁	1	8	yes	complemented
80	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
81	C₂×C₄	C₂×D₄×C₂₁₁	1	4	yes	—
82	C₂×C₄	C₂×D₄×C₂₁₁	1	4	yes	—
83	C₂×C₄	C₂⁴×C₂₁₁	1	1	yes	characteristic
84	C₈	C₂×D₄×C₂₁₁	1	8	yes	—
85	C₂²	#127×C₂₁₁	1	2	yes	—
86	C₂²	C₂×Q₁₆×C₂₁₁	1	4	yes	—
87	C₂²	C₂²×D₄×C₂₁₁	1	1	yes	characteristic
88	C₄	—	4	8	no	—
89	C₄	#127×C₂₁₁	1	2	yes	—
90	C₄	C₂×Q₁₆×C₂₁₁	1	8	yes	complemented
91	C₄	C₂²×D₄×C₂₁₁	1	2	yes	—
92	C₂	#494×C₂₁₁	1	4	yes	direct product factor
93	C₂	C₂×#127×C₂₁₁	1	1	yes	characteristic
94	C₂	C₂²×Q₁₆×C₂₁₁	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₁₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₂₁₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₁₁ ◁ C₂×C₄²×C₂₁₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. #494×C₂₁₁	C₂	direct (N × H)
31. C₂×C₄×C₂₁₁	Q₁₆	semidirect
42. C₄×C₂₁₁	C₂×Q₁₆	semidirect
44. C₂×C₂₁₁	#494	direct (N × H)
47. C₂₁₁	C₂×#494	direct (N × H)
48. C₂×#494	C₂₁₁	direct (N × H)
51. #494	C₂×C₂₁₁	direct (N × H)
79. C₂×C₄	Q₁₆×C₂₁₁	semidirect
90. C₄	C₂×Q₁₆×C₂₁₁	semidirect
92. C₂	#494×C₂₁₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₁₆×C₂₁₁	2
C₂×C₄⋊Q₈×C₂₁₁	2
#494×C₂₁₁	2
C₂×C₄×C₈×C₂₁₁	2
C₂×#494	211

Subgroups — C2×#494×C211 #41010765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#494×C₂₁₁ #41010765