Subgroups — #46838679

Order 34688.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#31774329	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄₃₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#31773736	C₂	1	2	yes	maximal, complemented
4	D₈₆₇₂	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22576182	—	2	1	no	—
7	C₂×D₂₁₆₈	—	2	1	no	—
8	Dic₂₁₆₈	C₂²	1	1	yes	characteristic, complemented
9	#22576105	—	2	1	no	—
10	D₄₃₃₆	—	2	1	no	—
11	D₄₃₃₆	C₂²	1	2	yes	—
12	D₄₃₃₆	C₂²	1	1	yes	characteristic, complemented
13	C₂×C₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
14	C₃₂×C₂₇₁	C₂²	1	2	yes	complemented
15	#16864654	—	4	1	no	—
16	C₂×D₁₀₈₄	—	4	1	no	—
17	Dic₁₀₈₄	—	2	1	no	—
18	#16864643	—	4	1	no	—
19	D₂₁₆₈	—	2	2	no	—
20	D₂₁₆₈	—	2	1	no	—
21	D₂₁₆₈	—	4	1	no	—
22	C₂×C₈×C₂₇₁	D₄	1	1	yes	characteristic
23	C₁₆×C₂₇₁	D₄	1	1	yes	characteristic
24	C₁₆×C₂₇₁	C₂³	1	1	yes	characteristic
25	C₂²×D₂₇₁	—	8	1	no	—
26	Dic₅₄₂	—	4	1	no	—
27	D₁₀₈₄	—	4	2	no	—
28	D₁₀₈₄	—	4	1	no	—
29	D₁₀₈₄	—	8	1	no	—
30	C₂²⋊D₂₇₁	—	8	1	no	—
31	C₄×D₂₇₁	—	8	1	no	—
32	C₂×C₄×C₂₇₁	D₈	1	1	yes	characteristic
33	C₈×C₂₇₁	D₈	1	1	yes	characteristic
34	C₈×C₂₇₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₂₇₁	D₁₆	1	1	yes	characteristic
36	C₂×D₂₇₁	—	8	1	no	—
37	C₂×D₂₇₁	—	8	2	no	—
38	C₂×D₂₇₁	—	16	2	no	—
39	C₄×C₂₇₁	D₁₆	1	1	yes	characteristic
40	C₄×C₂₇₁	C₂×D₈	1	1	yes	characteristic
41	Dic₂₇₁	—	8	1	no	—
42	C₂×C₂₇₁	—	2	1	no	—
43	C₂×C₂₇₁	C₂×D₁₆	1	1	yes	characteristic
44	D₂₇₁	—	16	1	no	—
45	D₂₇₁	—	16	2	no	—
46	C₂₇₁	#2263	1	1	yes	characteristic, complemented, Sylow
47	#2263	—	271	1	no	maximal, Sylow
48	#508	—	271	1	no	—
49	C₂×D₁₆	—	271	1	no	—
50	SD₆₄	—	271	2	no	—
51	D₃₂	—	271	2	no	—
52	M₆₄	D₂₇₁	1	1	yes	characteristic, complemented
53	#135	—	542	1	no	—
54	C₂×D₈	—	542	1	no	—
55	Q₃₂	—	271	1	no	—
56	SD₃₂	—	542	1	no	—
57	D₁₆	—	271	1	no	—
58	D₁₆	—	271	2	no	—
59	D₁₆	—	542	1	no	—
60	C₂×C₁₆	C₂×D₂₇₁	1	1	yes	characteristic
61	C₃₂	C₂×D₂₇₁	1	2	yes	complemented
62	#41	—	1084	1	no	—
63	C₂×D₄	—	1084	1	no	—
64	Q₁₆	—	542	1	no	—
65	SD₁₆	—	1084	1	no	—
66	D₈	—	542	2	no	—
67	D₈	—	542	1	no	—
68	D₈	—	1084	1	no	—
69	C₂×C₈	D₁₀₈₄	1	1	yes	characteristic
70	C₁₆	D₁₀₈₄	1	1	yes	characteristic
71	C₁₆	C₂²×D₂₇₁	1	1	yes	characteristic
72	C₂³	—	2168	1	no	—
73	Q₈	—	1084	1	no	—
74	D₄	—	1084	2	no	—
75	D₄	—	1084	1	no	—
76	D₄	—	2168	1	no	—
77	D₄	—	2168	1	no	—
78	C₂×C₄	—	2168	1	no	—
79	C₂×C₄	D₂₁₆₈	1	1	yes	characteristic
80	C₈	D₂₁₆₈	1	1	yes	characteristic
81	C₈	C₂×D₁₀₈₄	1	1	yes	characteristic
82	C₂²	—	2168	1	no	—
83	C₂²	—	2168	2	no	—
84	C₂²	—	4336	2	no	—
85	C₂²	D₄₃₃₆	1	1	yes	characteristic
86	C₄	—	2168	1	no	—
87	C₄	D₄₃₃₆	1	1	yes	characteristic
88	C₄	C₂×D₂₁₆₈	1	1	yes	characteristic
89	C₂	—	2	1	no	—
90	C₂	—	4336	2	no	—
91	C₂	—	4336	1	no	—
92	C₂	C₂×D₄₃₃₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄₃₃₆
commutator, cosocle	C₁₆×C₂₇₁	C₂³
Frattini	C₁₆	C₂²×D₂₇₁
Fitting	M₆₄×C₂₇₁	C₂
socle	C₂×C₂₇₁	C₂×D₁₆
hypercenter	M₆₄	D₂₇₁
hypocenter	C₂₇₁	#2263
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
1. #31774329	C₂	semidirect
2. C₂×D₄₃₃₆	C₂	semidirect
3. #31773736	C₂	semidirect
4. D₈₆₇₂	C₂	semidirect
5. M₆₄×C₂₇₁	C₂	semidirect
8. Dic₂₁₆₈	C₂²	semidirect
12. D₄₃₃₆	C₂²	semidirect
14. C₃₂×C₂₇₁	C₂²	semidirect
46. C₂₇₁	#2263	semidirect
52. M₆₄	D₂₇₁	semidirect
61. C₃₂	C₂×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
C₂×D₄₃₃₆	2
#31773736	2
D₈₆₇₂	2
M₆₄×C₂₇₁	2
#2263	271