Subgroups — #31885103

Order 18176.

108 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	#22586412	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#2252	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇₁⋊C₃₂	—	2	1	no	—
5	#17985752	—	2	1	no	—
6	#500×C₇₁	—	2	1	no	—
7	#366×C₇₁	—	2	1	no	—
8	#366×C₇₁	C₄	1	2	yes	—
9	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
10	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
11	C₇₁⋊C₃₂	—	4	1	no	—
12	C₄⋊Q₈×C₇₁	—	4	1	no	—
13	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
15	#107×C₇₁	—	2	2	no	—
16	#107×C₇₁	—	4	1	no	—
17	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
18	C₂×C₇₁⋊C₈	—	8	1	no	—
19	C₂×Q₈×C₇₁	—	8	1	no	—
20	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
21	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
22	C₄⋊C₄×C₇₁	—	4	2	no	—
23	C₄⋊C₄×C₇₁	—	8	1	no	—
24	C₄²×C₇₁	D₈	1	1	yes	characteristic
25	C₁₆×C₇₁	—	2	1	no	—
26	C₁₆×C₇₁	SD₁₆	1	1	yes	characteristic
27	C₁₆×C₇₁	Q₁₆	1	1	yes	characteristic
28	Q₈×C₇₁	—	16	1	no	—
29	Q₈×C₇₁	—	16	1	no	—
30	C₂×C₄×C₇₁	—	8	2	no	—
31	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
32	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
33	C₂×C₄×C₇₁	SD₃₂	1	1	yes	characteristic
34	C₈×C₇₁	—	2	1	no	—
35	C₈×C₇₁	#101	1	1	yes	characteristic
36	C₈×C₇₁	#103	1	1	yes	characteristic
37	C₇₁⋊C₈	—	16	1	no	—
38	C₂²×C₇₁	#357	1	1	yes	characteristic
39	C₄×C₇₁	—	2	1	no	—
40	C₄×C₇₁	—	16	2	no	—
41	C₄×C₇₁	#332	1	1	yes	characteristic
42	C₄×C₇₁	#362	1	1	yes	characteristic
43	C₄×C₇₁	D₃₂	1	1	yes	characteristic
44	C₄×C₇₁	SD₆₄	1	1	yes	characteristic
45	#7450	—	71	1	no	maximal, Sylow
46	C₂×C₇₁	#1364	1	1	yes	characteristic
47	C₂×C₇₁	#1415	1	1	yes	characteristic
48	C₂×C₇₁	#1420	1	1	yes	characteristic
49	#2252	D₇₁	1	1	yes	characteristic
50	C₄⋊C₃₂	—	71	1	no	—
51	#1371	—	71	1	no	—
52	C₇₁	#7450	1	1	yes	characteristic, complemented, Sylow
53	#500	—	2	1	no	—
54	C₂×C₃₂	—	142	1	no	—
55	#366	—	2	1	no	—
56	#366	Dic₇₁	1	2	yes	—
57	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
58	#335	—	142	1	no	—
59	C₄⋊Q₈	—	4	1	no	—
60	C₂×C₁₆	C₂×Dic₇₁	1	1	yes	characteristic
61	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
62	#107	—	2	2	no	—
63	#107	—	4	1	no	—
64	C₄⋊C₈	—	284	1	no	—
65	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
66	C₃₂	—	284	1	no	—
67	C₂×Q₈	—	8	1	no	—
68	C₂×C₈	—	568	1	no	—
69	C₂×C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
70	C₂×C₈	#11765853	1	1	yes	characteristic
71	C₄⋊C₄	—	4	2	no	—
72	C₄⋊C₄	—	8	1	no	—
73	C₄²	#11765850	1	1	yes	characteristic
74	C₁₆	—	2	1	no	—
75	C₁₆	#11765852	1	1	yes	characteristic
76	C₁₆	#11765855	1	1	yes	characteristic
77	Q₈	—	16	1	no	—
78	Q₈	—	16	1	no	—
79	C₂×C₄	—	8	2	no	—
80	C₂×C₄	#14556049	1	1	yes	characteristic
81	C₂×C₄	#14556069	1	1	yes	characteristic
82	C₂×C₄	#14556072	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	1136	1	no	—
85	C₈	#14556047	1	1	yes	characteristic
86	C₈	#14556052	1	1	yes	characteristic
87	C₂²	#17985800	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	16	2	no	—
90	C₄	#17985745	1	1	yes	characteristic
91	C₄	#17985815	1	1	yes	characteristic
92	C₄	#17985834	1	1	yes	characteristic
93	C₄	#17985837	1	1	yes	characteristic
94	C₂	#22586395	1	1	yes	characteristic
95	C₂	#22586518	1	1	yes	characteristic
96	C₂	#22586533	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17985800
commutator	C₂×C₁₆×C₇₁	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₇₁
Fitting	C₇₁×#2252	C₂
socle, prime radical	C₂²×C₇₁	#357
cosocle	C₄×C₁₆×C₇₁	C₂²
hypercenter	#2252	D₇₁
hypocenter	C₇₁	#7450
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2252

Complements

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

N	H ≅ G/N	type
52. C₇₁	#7450	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
#22586412	2
C₇₁×#2252	2
#7450	71