Subgroups — #46834552

Order 34688.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31774329	C₂	1	1	yes	maximal, characteristic
2	#31773725	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#22576182	—	2	1	no	—
5	Dic₂₁₆₈	C₄	1	1	yes	characteristic, complemented
6	#22576105	—	2	1	no	—
7	D₄₃₃₆	C₄	1	1	yes	characteristic, complemented
8	#22576094	—	2	1	no	—
9	C₂×C₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
10	C₃₂×C₂₇₁	—	2	1	no	—
11	#16864654	—	4	1	no	—
12	Dic₁₀₈₄	—	2	1	no	—
13	#16864643	—	4	1	no	—
14	D₂₁₆₈	—	2	1	no	—
15	C₄⋊Dic₂₇₁	—	4	1	no	—
16	C₂×C₈×C₂₇₁	D₄	1	1	yes	characteristic
17	C₁₆×C₂₇₁	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₂₇₁	D₄	1	1	yes	characteristic
19	Dic₅₄₂	—	4	1	no	—
20	D₁₀₈₄	—	4	1	no	—
21	C₂²⋊D₂₇₁	—	8	1	no	—
22	C₂×Dic₂₇₁	—	8	1	no	—
23	C₄×D₂₇₁	—	8	1	no	—
24	C₂×C₄×C₂₇₁	D₈	1	1	yes	characteristic
25	C₈×C₂₇₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₂₇₁	SD₁₆	1	1	yes	characteristic
27	C₂²×C₂₇₁	D₁₆	1	1	yes	characteristic
28	C₂×D₂₇₁	—	8	1	no	—
29	C₄×C₂₇₁	#102	1	1	yes	characteristic
30	C₄×C₂₇₁	SD₃₂	1	1	yes	characteristic
31	Dic₂₇₁	—	8	1	no	—
32	Dic₂₇₁	—	16	1	no	—
33	C₂×C₂₇₁	—	2	1	no	—
34	C₂×C₂₇₁	#357	1	1	yes	characteristic
35	D₂₇₁	—	16	1	no	—
36	C₂₇₁	#1418	1	1	yes	characteristic, complemented, Sylow
37	#1418	—	271	1	no	maximal, Sylow
38	#508	—	271	1	no	—
39	M₆₄	D₂₇₁	1	1	yes	characteristic, complemented
40	#367	—	271	1	no	—
41	#135	—	542	1	no	—
42	Q₃₂	—	271	1	no	—
43	SD₃₂	—	542	1	no	—
44	D₁₆	—	271	1	no	—
45	C₂×C₁₆	C₂×D₂₇₁	1	1	yes	characteristic
46	#107	—	542	1	no	—
47	C₃₂	—	2	1	no	—
48	#41	—	1084	1	no	—
49	Q₁₆	—	542	1	no	—
50	SD₁₆	—	1084	1	no	—
51	D₈	—	542	1	no	—
52	C₂×C₈	D₁₀₈₄	1	1	yes	characteristic
53	C₄⋊C₄	—	1084	1	no	—
54	C₁₆	C₄×D₂₇₁	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₂₇₁	1	1	yes	characteristic
56	Q₈	—	1084	1	no	—
57	D₄	—	1084	1	no	—
58	D₄	—	2168	1	no	—
59	C₂×C₄	—	2168	1	no	—
60	C₂×C₄	—	2168	1	no	—
61	C₂×C₄	D₂₁₆₈	1	1	yes	characteristic
62	C₈	(C₂×D₂₇₁)⋊C₄	1	1	yes	characteristic
63	C₈	#16864643	1	1	yes	characteristic
64	C₂²	—	2168	1	no	—
65	C₂²	D₄₃₃₆	1	1	yes	characteristic
66	C₄	—	2168	1	no	—
67	C₄	—	4336	1	no	—
68	C₄	#22576082	1	1	yes	characteristic
69	C₄	#22576105	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	4336	1	no	—
72	C₂	#31773699	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#31773699
commutator	C₁₆×C₂₇₁	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₂₇₁
Fitting	M₆₄×C₂₇₁	C₂
socle	C₂×C₂₇₁	#357
prime radical	#31774329	C₂
cosocle	C₂×C₁₆×C₂₇₁	C₂²
hypercenter	M₆₄	D₂₇₁
hypocenter	C₂₇₁	#1418
LCS	G ▷ C₁₆×C₂₇₁ ▷ C₈×C₂₇₁ ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #31773725	C₂	semidirect
3. M₆₄×C₂₇₁	C₂	semidirect
5. Dic₂₁₆₈	C₄	semidirect
7. D₄₃₃₆	C₄	semidirect
36. C₂₇₁	#1418	semidirect
39. M₆₄	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
#31773725	2
M₆₄×C₂₇₁	2
#1418	271