Subgroups — #31885132

Order 18176.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇₁⋊C₃₂	C₂	1	1	yes	maximal, characteristic
2	#22586416	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1373	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇₁⋊C₃₂	C₄	1	1	yes	characteristic
5	C₂×C₇₁⋊C₃₂	C₄	1	1	yes	characteristic
6	#17985820	—	2	1	no	—
7	#364×C₇₁	—	2	1	no	—
8	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
9	#14556068	—	4	1	no	—
10	C₇₁⋊C₃₂	—	2	1	no	—
11	C₇₁⋊C₃₂	C₈	1	2	yes	—
12	M₃₂×C₇₁	—	4	1	no	—
13	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₇₁	Q₈	1	1	yes	characteristic
15	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
16	C₇₁⋊C₁₆	—	8	1	no	—
17	C₂×C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
19	C₄²×C₇₁	Q₁₆	1	1	yes	characteristic
20	C₁₆×C₇₁	—	2	1	no	—
21	C₁₆×C₇₁	—	8	1	no	—
22	C₁₆×C₇₁	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₇₁	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₇₁	—	2	1	no	—
25	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
26	C₈×C₇₁	C₄⋊C₈	1	1	yes	characteristic
27	C₈×C₇₁	#108	1	1	yes	characteristic
28	C₈×C₇₁	D₁₆	1	1	yes	characteristic
29	C₈×C₇₁	Q₃₂	1	1	yes	characteristic
30	C₂²×C₇₁	#368	1	1	yes	characteristic
31	C₄×C₇₁	—	2	2	no	—
32	C₄×C₇₁	#335	1	1	yes	characteristic
33	C₄×C₇₁	#366	1	1	yes	characteristic
34	#7462	—	71	1	no	maximal, Sylow
35	C₂×C₇₁	—	2	1	no	—
36	C₂×C₇₁	#1371	1	1	yes	characteristic
37	C₄×C₃₂	—	71	1	no	—
38	#1373	—	71	1	no	—
39	#1373	D₇₁	1	1	yes	characteristic
40	C₇₁	#7462	1	1	yes	characteristic, complemented, Sylow
41	C₂×C₃₂	—	71	1	no	—
42	C₂×C₃₂	—	71	1	no	—
43	#364	—	2	1	no	—
44	#364	—	142	1	no	—
45	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
46	M₃₂	—	4	1	no	—
47	M₃₂	—	284	1	no	—
48	C₂×C₁₆	Dic₁₄₂	1	1	yes	characteristic
49	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
50	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
51	C₃₂	—	71	2	no	—
52	C₃₂	—	142	1	no	—
53	C₂×C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
54	C₂×C₈	#11765850	1	1	yes	characteristic
55	C₄²	#11765855	1	1	yes	characteristic
56	C₁₆	—	2	1	no	—
57	C₁₆	—	8	1	no	—
58	C₁₆	—	568	1	no	—
59	C₁₆	C₈×D₇₁	1	1	yes	characteristic
60	C₁₆	#11765848	1	1	yes	characteristic
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	#14556061	1	1	yes	characteristic
63	C₈	Dic₇₁⋊C₈	1	1	yes	characteristic
64	C₈	#14556063	1	1	yes	characteristic
65	C₈	#14556069	1	1	yes	characteristic
66	C₈	#14556074	1	1	yes	characteristic
67	C₂²	#17985828	1	1	yes	characteristic
68	C₄	—	2	2	no	—
69	C₄	#17985751	1	1	yes	characteristic
70	C₄	#17985824	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#22586411	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14556069
commutator	C₁₆×C₇₁	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₇₁
Fitting	C₇₁×#1373	C₂
socle	C₂×C₇₁	#1371
prime radical	C₂²×C₇₁	#368
cosocle	C₄×C₁₆×C₇₁	C₂²
hypercenter	#1373	D₇₁
hypocenter	C₇₁	#7462
LCS	G ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1373

Complements

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

N	H ≅ G/N	type
40. C₇₁	#7462	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₇₁⋊C₃₂	2
#22586416	2
C₇₁×#1373	2
#7462	71