Subgroups — #46834555

Order 34688.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄₃₃₆	C₂	1	1	yes	maximal, characteristic
2	#31773727	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₂₁₆₈	—	2	1	no	—
5	D₄₃₃₆	—	2	1	no	—
6	D₄₃₃₆	C₄	1	2	yes	—
7	#22576096	—	2	1	no	—
8	C₂×C₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
9	C₃₂×C₂₇₁	—	2	1	no	—
10	C₂×D₁₀₈₄	—	4	1	no	—
11	D₂₁₆₈	—	2	2	no	—
12	D₂₁₆₈	—	4	1	no	—
13	#16864638	—	4	1	no	—
14	C₂×C₈×C₂₇₁	D₄	1	1	yes	characteristic
15	C₁₆×C₂₇₁	C₂×C₄	1	1	yes	characteristic
16	C₁₆×C₂₇₁	D₄	1	1	yes	characteristic
17	C₂²×D₂₇₁	—	8	1	no	—
18	D₁₀₈₄	—	4	2	no	—
19	D₁₀₈₄	—	8	1	no	—
20	C₂₇₁⋊C₈	—	8	1	no	—
21	C₂×C₄×C₂₇₁	D₈	1	1	yes	characteristic
22	C₈×C₂₇₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₈×C₂₇₁	SD₁₆	1	1	yes	characteristic
24	C₂²×C₂₇₁	SD₃₂	1	1	yes	characteristic
25	C₂×D₂₇₁	—	8	2	no	—
26	C₂×D₂₇₁	—	16	2	no	—
27	C₄×C₂₇₁	#102	1	1	yes	characteristic
28	C₄×C₂₇₁	D₁₆	1	1	yes	characteristic
29	C₂×C₂₇₁	—	2	1	no	—
30	C₂×C₂₇₁	#357	1	1	yes	characteristic
31	D₂₇₁	—	16	2	no	—
32	C₂₇₁	#1419	1	1	yes	characteristic, complemented, Sylow
33	#1419	—	271	1	no	maximal, Sylow
34	C₂×D₁₆	—	271	1	no	—
35	M₆₄	D₂₇₁	1	1	yes	characteristic, complemented
36	#368	—	271	1	no	—
37	C₂×D₈	—	542	1	no	—
38	D₁₆	—	271	2	no	—
39	D₁₆	—	542	1	no	—
40	C₂×C₁₆	C₂×D₂₇₁	1	1	yes	characteristic
41	#108	—	542	1	no	—
42	C₃₂	—	2	1	no	—
43	C₂×D₄	—	1084	1	no	—
44	D₈	—	542	2	no	—
45	D₈	—	1084	1	no	—
46	M₁₆	—	1084	1	no	—
47	C₂×C₈	D₁₀₈₄	1	1	yes	characteristic
48	C₁₆	C₄×D₂₇₁	1	1	yes	characteristic
49	C₁₆	C₂²⋊D₂₇₁	1	1	yes	characteristic
50	C₂³	—	2168	1	no	—
51	D₄	—	1084	2	no	—
52	D₄	—	2168	1	no	—
53	C₂×C₄	D₂₁₆₈	1	1	yes	characteristic
54	C₈	—	2168	1	no	—
55	C₈	(C₂×D₂₇₁)⋊C₄	1	1	yes	characteristic
56	C₈	#16864643	1	1	yes	characteristic
57	C₂²	—	2168	2	no	—
58	C₂²	—	4336	2	no	—
59	C₂²	#22576105	1	1	yes	characteristic
60	C₄	#22576082	1	1	yes	characteristic
61	C₄	D₄₃₃₆	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	4336	2	no	—
64	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	C₂×D₄₃₃₆	C₂
cosocle	C₂×C₁₆×C₂₇₁	C₂²
hypercenter	M₆₄	D₂₇₁
hypocenter	C₂₇₁	#1419
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. #31773727	C₂	semidirect
3. M₆₄×C₂₇₁	C₂	semidirect
32. C₂₇₁	#1419	semidirect
35. M₆₄	D₂₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄₃₃₆	2
#31773727	2
M₆₄×C₂₇₁	2
#1419	271