Subgroups — #31918478

Order 18176.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590614	C₂	1	2	yes	maximal
2	#22586542	C₂	1	1	yes	maximal, characteristic
3	#22586541	C₂	1	2	yes	maximal
4	C₃₂×Dic₇₁	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1424	C₂	1	1	yes	maximal, characteristic
6	#17986403	—	2	2	no	—
7	C₂×C₇₁⋊C₃₂	C₂²	1	1	yes	characteristic
8	#17985825	C₂²	1	2	yes	—
9	#17985824	—	2	2	no	—
10	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
12	#366×C₇₁	C₂²	1	2	yes	—
13	#14556127	—	4	2	no	—
14	C₂×C₇₁⋊C₁₆	D₄	1	1	yes	characteristic
15	#14556062	—	2	2	no	—
16	#14556061	—	4	2	no	—
17	C₈×Dic₇₁	D₄	1	1	yes	characteristic
18	C₇₁⋊C₃₂	Q₈	1	2	yes	complemented
19	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
20	#107×C₇₁	—	2	2	no	—
21	C₃₂×C₇₁	Q₈	1	2	yes	complemented
22	C₂×Dic₁₄₂	—	8	2	no	—
23	C₂×C₇₁⋊C₈	D₈	1	1	yes	characteristic
24	C₄⋊Dic₇₁	—	4	2	no	—
25	Dic₇₁⋊C₄	—	8	2	no	—
26	C₄×Dic₇₁	D₈	1	1	yes	characteristic
27	C₇₁⋊C₁₆	Q₁₆	1	2	yes	—
28	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₇₁	—	4	2	no	—
30	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
31	C₁₆×C₇₁	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₇₁	—	8	2	no	—
33	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
34	C₂×Dic₇₁	—	8	2	no	—
35	C₂×Dic₇₁	D₁₆	1	2	yes	—
36	Dic₁₄₂	—	16	4	no	—
37	C₈×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
38	C₈×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
39	C₇₁⋊C₈	Q₃₂	1	2	yes	—
40	C₂²×C₇₁	C₂×D₁₆	1	1	yes	characteristic
41	C₄×C₇₁	—	16	2	no	—
42	C₄×C₇₁	#500	1	1	yes	characteristic
43	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
44	Dic₇₁	—	16	2	no	—
45	Dic₇₁	SD₆₄	1	4	yes	—
46	#13733	—	71	1	no	maximal, Sylow
47	C₂×C₇₁	#2252	1	1	yes	characteristic
48	C₂×C₇₁	C₂×SD₆₄	1	2	yes	—
49	#2252	—	71	2	no	—
50	#1424	—	71	1	no	—
51	#1424	—	71	2	no	—
52	#1424	D₇₁	1	1	yes	characteristic
53	C₄×C₃₂	—	71	1	no	—
54	C₇₁	#13733	1	1	yes	characteristic, complemented, Sylow
55	#500	—	142	2	no	—
56	C₂×C₃₂	—	71	1	no	—
57	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
58	#366	—	71	2	no	—
59	#366	—	142	2	no	—
60	#366	C₂×D₇₁	1	2	yes	—
61	C₄×C₁₆	—	71	1	no	—
62	C₄⋊Q₈	—	284	2	no	—
63	C₂×C₁₆	—	71	1	no	—
64	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
65	#107	—	2	2	no	—
66	#107	—	284	2	no	—
67	#107	—	142	2	no	—
68	C₄×C₈	—	71	1	no	—
69	C₃₂	—	71	2	no	—
70	C₃₂	Dic₁₄₂	1	2	yes	complemented
71	C₂×Q₈	—	568	2	no	—
72	C₂×C₈	—	71	1	no	—
73	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	284	2	no	—
76	C₄⋊C₄	—	568	2	no	—
77	C₄²	—	71	1	no	—
78	C₁₆	—	71	2	no	—
79	C₁₆	Q₈×D₇₁	1	1	yes	characteristic
80	C₁₆	C₂×Dic₁₄₂	1	1	yes	characteristic
81	Q₈	—	1136	4	no	—
82	C₂×C₄	—	8	2	no	—
83	C₂×C₄	—	568	2	no	—
84	C₂×C₄	—	71	2	no	—
85	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
86	C₈	—	71	2	no	—
87	C₈	#14556127	1	1	yes	characteristic
88	C₈	Q₁₆×D₇₁	1	1	yes	characteristic
89	C₂²	D₁₆×D₇₁	1	1	yes	characteristic
90	C₄	—	16	2	no	—
91	C₄	—	71	4	no	—
92	C₄	—	1136	2	no	—
93	C₄	#17986403	1	1	yes	characteristic
94	C₄	Q₃₂×D₇₁	1	1	yes	characteristic
95	C₂	#22590614	1	1	yes	characteristic
96	C₂	SD₆₄×D₇₁	1	2	yes	—

	subgroup	G/H
center	C₂²	D₁₆×D₇₁
commutator, cosocle	C₂×C₁₆×C₇₁	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₁
Fitting	C₇₁×#1424	C₂
socle, prime radical	C₂²×C₇₁	C₂×D₁₆
hypercenter	#1424	D₇₁
hypocenter	C₇₁	#13733
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1424

Complements

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

N	H ≅ G/N	type
18. C₇₁⋊C₃₂	Q₈	semidirect
21. C₃₂×C₇₁	Q₈	semidirect
54. C₇₁	#13733	semidirect
70. C₃₂	Dic₁₄₂	semidirect

Maximal subgroups

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

subgroup	index
#22590614	2
#22586542	2
#22586541	2
C₃₂×Dic₇₁	2
C₇₁×#1424	2
#13733	71