Subgroups — #46838673

Order 34688.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31774329	C₂	1	2	yes	maximal, complemented
2	Dic₄₃₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#31773736	C₂	1	2	yes	maximal, complemented
4	D₈₆₇₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₂×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22576182	—	2	2	no	—
7	Dic₂₁₆₈	C₂²	1	2	yes	—
8	#22576105	—	2	2	no	—
9	D₄₃₃₆	C₂²	1	2	yes	—
10	C₂×C₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
11	C₃₂×C₂₇₁	C₂²	1	1	yes	characteristic
12	C₃₂×C₂₇₁	C₂²	1	1	yes	characteristic
13	#16864654	—	4	2	no	—
14	Dic₁₀₈₄	—	2	2	no	—
15	#16864643	—	4	2	no	—
16	D₂₁₆₈	—	2	2	no	—
17	C₂×C₈×C₂₇₁	D₄	1	1	yes	characteristic
18	C₁₆×C₂₇₁	D₄	1	1	yes	characteristic
19	C₁₆×C₂₇₁	C₂³	1	1	yes	characteristic
20	Dic₅₄₂	—	4	2	no	—
21	D₁₀₈₄	—	4	2	no	—
22	C₂²⋊D₂₇₁	—	8	2	no	—
23	C₄×D₂₇₁	—	8	2	no	—
24	C₂×C₄×C₂₇₁	D₈	1	1	yes	characteristic
25	C₈×C₂₇₁	D₈	1	1	yes	characteristic
26	C₈×C₂₇₁	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₂₇₁	D₁₆	1	1	yes	characteristic
28	C₂×D₂₇₁	—	8	2	no	—
29	C₄×C₂₇₁	D₁₆	1	1	yes	characteristic
30	C₄×C₂₇₁	C₂×D₈	1	1	yes	characteristic
31	Dic₂₇₁	—	8	2	no	—
32	C₂×C₂₇₁	—	2	1	no	—
33	C₂×C₂₇₁	C₂×D₁₆	1	1	yes	characteristic
34	D₂₇₁	—	16	2	no	—
35	C₂₇₁	#2262	1	1	yes	characteristic, complemented, Sylow
36	#2262	—	271	1	no	maximal, Sylow
37	#508	—	271	2	no	—
38	Q₆₄	—	271	1	no	—
39	SD₆₄	—	271	2	no	—
40	D₃₂	—	271	1	no	—
41	C₂×C₃₂	D₂₇₁	1	1	yes	characteristic, complemented
42	#135	—	542	2	no	—
43	Q₃₂	—	271	2	no	—
44	SD₃₂	—	542	2	no	—
45	D₁₆	—	271	2	no	—
46	C₂×C₁₆	C₂×D₂₇₁	1	1	yes	characteristic
47	C₃₂	C₂×D₂₇₁	1	1	yes	characteristic
48	C₃₂	C₂×D₂₇₁	1	1	yes	characteristic
49	#41	—	1084	2	no	—
50	Q₁₆	—	542	2	no	—
51	SD₁₆	—	1084	2	no	—
52	D₈	—	542	2	no	—
53	C₂×C₈	D₁₀₈₄	1	1	yes	characteristic
54	C₁₆	D₁₀₈₄	1	1	yes	characteristic
55	C₁₆	C₂²×D₂₇₁	1	1	yes	characteristic
56	Q₈	—	1084	2	no	—
57	D₄	—	1084	2	no	—
58	D₄	—	2168	2	no	—
59	C₂×C₄	—	2168	2	no	—
60	C₂×C₄	D₂₁₆₈	1	1	yes	characteristic
61	C₈	D₂₁₆₈	1	1	yes	characteristic
62	C₈	C₂×D₁₀₈₄	1	1	yes	characteristic
63	C₂²	—	2168	2	no	—
64	C₂²	D₄₃₃₆	1	1	yes	characteristic
65	C₄	—	2168	2	no	—
66	C₄	D₄₃₃₆	1	1	yes	characteristic
67	C₄	C₂×D₂₁₆₈	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4336	2	no	—
70	C₂	C₂×D₄₃₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄₃₃₆
commutator, cosocle	C₁₆×C₂₇₁	C₂³
Frattini	C₁₆	C₂²×D₂₇₁
Fitting	C₂×C₃₂×C₂₇₁	C₂
socle	C₂×C₂₇₁	C₂×D₁₆
hypercenter	C₂×C₃₂	D₂₇₁
hypocenter	C₂₇₁	#2262
LCS	G ▷ C₁₆×C₂₇₁ ▷ C₈×C₂₇₁ ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
1. #31774329	C₂	semidirect
2. Dic₄₃₃₆	C₂	semidirect
3. #31773736	C₂	semidirect
4. D₈₆₇₂	C₂	semidirect
5. C₂×C₃₂×C₂₇₁	C₂	semidirect
35. C₂₇₁	#2262	semidirect
41. C₂×C₃₂	D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
#31774329	2
Dic₄₃₃₆	2
#31773736	2
D₈₆₇₂	2
C₂×C₃₂×C₂₇₁	2
#2262	271