Subgroups — #46838249

Order 34688.

144 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	#31774013	C₂	1	2	yes	maximal, complemented
3	#31773703	C₂	1	2	yes	maximal, complemented
4	#31773672	C₂	1	1	yes	maximal, characteristic, complemented
5	#368×C₂₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#22576182	C₂²	1	2	yes	—
7	#22576168	—	2	2	no	—
8	Dic₂₁₆₈	C₄	1	1	yes	characteristic, complemented
9	#22576105	C₄	1	2	yes	complemented
10	D₄₃₃₆	C₄	1	1	yes	characteristic, complemented
11	C₂×C₂₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	#22576086	—	2	2	no	—
13	C₈×Dic₂₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₂₇₁	C₂²	1	1	yes	characteristic
15	#108×C₂₇₁	C₂²	1	2	yes	—
16	#16864654	—	2	2	no	—
17	Dic₁₀₈₄	C₂×C₄	1	2	yes	—
18	#16864643	—	2	2	no	—
19	D₂₁₆₈	C₂×C₄	1	2	yes	—
20	#16864637	—	4	2	no	—
21	C₂×C₂₇₁⋊C₈	D₄	1	1	yes	characteristic
22	C₈×D₂₇₁	—	4	2	no	—
23	C₄×Dic₂₇₁	D₄	1	1	yes	characteristic
24	C₂₇₁⋊C₁₆	—	2	1	no	—
25	M₁₆×C₂₇₁	—	2	2	no	—
26	C₂×C₈×C₂₇₁	C₂³	1	1	yes	characteristic
27	C₁₆×C₂₇₁	C₂×C₄	1	1	yes	characteristic
28	C₁₆×C₂₇₁	C₂×C₄	1	1	yes	characteristic
29	Dic₅₄₂	—	2	2	no	—
30	D₁₀₈₄	—	2	2	no	—
31	C₂²⋊D₂₇₁	—	4	2	no	—
32	C₂×Dic₂₇₁	—	2	1	no	—
33	C₄×D₂₇₁	—	4	2	no	—
34	C₂₇₁⋊C₈	D₈	1	2	yes	—
35	C₂×C₄×C₂₇₁	C₂×D₄	1	1	yes	characteristic
36	C₈×C₂₇₁	—	4	2	no	—
37	C₈×C₂₇₁	C₂²×C₄	1	1	yes	characteristic
38	C₈×C₂₇₁	#41	1	1	yes	characteristic
39	C₂²×C₂₇₁	#135	1	1	yes	characteristic
40	C₂×D₂₇₁	—	4	2	no	—
41	C₄×C₂₇₁	C₄×D₄	1	1	yes	characteristic
42	C₄×C₂₇₁	C₂×D₈	1	1	yes	characteristic
43	Dic₂₇₁	—	4	1	no	—
44	Dic₂₇₁	—	4	2	no	—
45	C₂×C₂₇₁	—	2	1	no	—
46	C₂×C₂₇₁	C₄×D₈	1	1	yes	characteristic
47	D₂₇₁	—	8	2	no	—
48	C₂₇₁	#2179	1	1	yes	characteristic, complemented, Sylow
49	#2179	—	271	1	no	maximal, Sylow
50	#508	—	271	1	no	—
51	#443	—	271	2	no	—
52	#368	D₂₇₁	1	1	yes	characteristic, complemented
53	#359	—	271	2	no	—
54	#346	—	271	1	no	—
55	#135	—	271	2	no	—
56	#131	—	542	2	no	—
57	Q₃₂	—	271	1	no	—
58	SD₃₂	—	271	2	no	—
59	D₁₆	—	271	1	no	—
60	C₂×C₁₆	—	271	1	no	—
61	C₂×C₁₆	C₂×D₂₇₁	1	1	yes	characteristic
62	#108	C₂×D₂₇₁	1	2	yes	—
63	#104	—	542	2	no	—
64	C₄×C₈	—	271	1	no	—
65	#41	—	542	2	no	—
66	Q₁₆	—	271	2	no	—
67	SD₁₆	—	542	2	no	—
68	D₈	—	271	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	1084	2	no	—
71	C₂×C₈	—	271	1	no	—
72	C₂×C₈	—	1084	2	no	—
73	C₂×C₈	C₂²×D₂₇₁	1	1	yes	characteristic
74	C₄²	—	271	1	no	—
75	C₁₆	—	542	1	no	—
76	C₁₆	C₄×D₂₇₁	1	1	yes	characteristic
77	C₁₆	C₄×D₂₇₁	1	1	yes	characteristic
78	Q₈	—	542	2	no	—
79	D₄	—	542	2	no	—
80	D₄	—	1084	2	no	—
81	C₂×C₄	—	542	1	no	—
82	C₂×C₄	—	1084	2	no	—
83	C₂×C₄	D₄×D₂₇₁	1	1	yes	characteristic
84	C₈	—	4	2	no	—
85	C₈	—	271	2	no	—
86	C₈	C₂×C₄×D₂₇₁	1	1	yes	characteristic
87	C₈	#16864655	1	1	yes	characteristic
88	C₂²	—	1084	2	no	—
89	C₂²	#22576183	1	1	yes	characteristic
90	C₄	—	1084	1	no	—
91	C₄	—	1084	2	no	—
92	C₄	#22576122	1	1	yes	characteristic
93	C₄	D₈×D₂₇₁	1	1	yes	characteristic
94	C₂	—	2	1	no	—
95	C₂	—	2168	2	no	—
96	C₂	#31773979	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₈×D₂₇₁
commutator	C₈×C₂₇₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₂₇₁
Fitting	#368×C₂₇₁	C₂
socle	C₂×C₂₇₁	C₄×D₈
prime radical	#31774329	C₂
cosocle	C₂×C₈×C₂₇₁	C₂³
hypercenter	#368	D₂₇₁
hypocenter	C₂₇₁	#2179
LCS	G ▷ C₈×C₂₇₁ ▷ C₄×C₂₇₁ ▷ C₂×C₂₇₁ ▷ C₂₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

Complements

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

N	H ≅ G/N	type
2. #31774013	C₂	semidirect
3. #31773703	C₂	semidirect
4. #31773672	C₂	semidirect
5. #368×C₂₇₁	C₂	semidirect
8. Dic₂₁₆₈	C₄	semidirect
9. #22576105	C₄	semidirect
10. D₄₃₃₆	C₄	semidirect
48. C₂₇₁	#2179	semidirect
52. #368	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
#31774013	2
#31773703	2
#31773672	2
#368×C₂₇₁	2
#2179	271