Subgroups — #544×C₂₇₁ #31773573

Order 17344.

206 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈×C₂₇₁	C₂	1	4	yes	maximal, complemented
2	#125×C₂₇₁	C₂	1	4	yes	maximal, complemented
3	C₂²⋊Q₈×C₂₇₁	C₂	1	4	yes	maximal
4	#117×C₂₇₁	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₄×C₂₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈×C₂₇₁	C₂²	1	4	yes	—
7	C₂²×C₄×C₂₇₁	C₂²	1	2	yes	—
8	C₂²×C₄×C₂₇₁	C₂²	1	1	yes	characteristic
9	C₄⋊C₄×C₂₇₁	C₂²	1	4	yes	—
10	C₄⋊C₄×C₂₇₁	C₂²	1	2	yes	—
11	C₄⋊C₄×C₂₇₁	C₂²	1	2	yes	—
12	C₄⋊C₄×C₂₇₁	C₂²	1	4	yes	—
13	C₄⋊C₄×C₂₇₁	C₂²	1	8	yes	—
14	C₂²⋊C₄×C₂₇₁	C₂²	1	4	yes	—
15	C₄²×C₂₇₁	C₂²	1	4	yes	—
16	C₂³×C₂₇₁	C₂³	1	1	yes	characteristic
17	Q₈×C₂₇₁	—	4	4	no	—
18	C₂×C₄×C₂₇₁	—	4	2	no	—
19	C₂×C₄×C₂₇₁	Q₈	1	4	yes	complemented
20	C₂×C₄×C₂₇₁	C₂³	1	4	yes	—
21	C₂×C₄×C₂₇₁	C₂³	1	4	yes	—
22	C₂×C₄×C₂₇₁	C₂³	1	4	yes	—
23	C₂×C₄×C₂₇₁	C₂³	1	2	yes	—
24	C₂²×C₂₇₁	—	2	1	no	—
25	C₂²×C₂₇₁	—	2	1	no	—
26	C₂²×C₂₇₁	C₂×Q₈	1	2	yes	—
27	C₂²×C₂₇₁	C₂⁴	1	1	yes	characteristic
28	C₄×C₂₇₁	—	2	4	no	—
29	C₄×C₂₇₁	—	2	4	no	—
30	C₄×C₂₇₁	—	2	4	no	—
31	C₄×C₂₇₁	C₂×Q₈	1	4	yes	—
32	C₂×C₂₇₁	—	2	2	no	—
33	C₂×C₂₇₁	C₂²×Q₈	1	1	yes	characteristic
34	C₂×C₂₇₁	H(5,𝔽₂)	1	1	yes	characteristic
35	C₂×C₂₇₁	2₋¹⁺⁴	1	1	yes	characteristic
36	C₂₇₁	#544	1	1	yes	characteristic, direct product factor, Sylow
37	#544	C₂₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
38	C₄⋊Q₈	C₂×C₂₇₁	1	4	yes	complemented
39	#125	C₂×C₂₇₁	1	4	yes	complemented
40	C₂²⋊Q₈	C₂×C₂₇₁	1	4	yes	—
41	#117	C₂×C₂₇₁	1	2	yes	—
42	C₂×C₄⋊C₄	C₂×C₂₇₁	1	1	yes	characteristic
43	C₂×Q₈	C₂²×C₂₇₁	1	4	yes	—
44	C₂²×C₄	C₂²×C₂₇₁	1	2	yes	—
45	C₂²×C₄	C₂²×C₂₇₁	1	1	yes	characteristic
46	C₄⋊C₄	C₂²×C₂₇₁	1	4	yes	—
47	C₄⋊C₄	C₂²×C₂₇₁	1	2	yes	—
48	C₄⋊C₄	C₂²×C₂₇₁	1	2	yes	—
49	C₄⋊C₄	C₂²×C₂₇₁	1	4	yes	—
50	C₄⋊C₄	C₂²×C₂₇₁	1	8	yes	—
51	C₂²⋊C₄	C₂²×C₂₇₁	1	4	yes	—
52	C₄²	C₂²×C₂₇₁	1	4	yes	—
53	C₂³	C₂³×C₂₇₁	1	1	yes	characteristic
54	Q₈	—	4	4	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	Q₈×C₂₇₁	1	4	yes	complemented
57	C₂×C₄	C₂³×C₂₇₁	1	4	yes	—
58	C₂×C₄	C₂³×C₂₇₁	1	4	yes	—
59	C₂×C₄	C₂³×C₂₇₁	1	4	yes	—
60	C₂×C₄	C₂³×C₂₇₁	1	2	yes	—
61	C₂²	—	2	1	no	—
62	C₂²	—	2	1	no	—
63	C₂²	C₂×Q₈×C₂₇₁	1	2	yes	—
64	C₂²	C₂⁴×C₂₇₁	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	2	4	no	—
67	C₄	—	2	4	no	—
68	C₄	C₂×Q₈×C₂₇₁	1	4	yes	—
69	C₂	—	2	2	no	—
70	C₂	C₂²×Q₈×C₂₇₁	1	1	yes	characteristic
71	C₂	H(5,𝔽₂)×C₂₇₁	1	1	yes	characteristic
72	C₂	2₋¹⁺⁴×C₂₇₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₂₇₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₂₇₁
prime radical	C₂³×C₂₇₁	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄⋊Q₈×C₂₇₁	C₂	semidirect
2. #125×C₂₇₁	C₂	semidirect
19. C₂×C₄×C₂₇₁	Q₈	semidirect
36. C₂₇₁	#544	direct (N × H)
37. #544	C₂₇₁	direct (N × H)
38. C₄⋊Q₈	C₂×C₂₇₁	semidirect
39. #125	C₂×C₂₇₁	semidirect
56. C₂×C₄	Q₈×C₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊Q₈×C₂₇₁	2
#125×C₂₇₁	2
C₂²⋊Q₈×C₂₇₁	2
#117×C₂₇₁	2
C₂×C₄⋊C₄×C₂₇₁	2
#544	271

Subgroups — #544×C271 #31773573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #544×C₂₇₁ #31773573