Subgroups — #31885105

Order 18176.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590615	C₂	1	1	yes	maximal, characteristic
2	#22586412	C₂	1	1	yes	maximal, characteristic
3	C₇₁×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	#17986404	—	2	1	no	—
5	#17985825	—	2	1	no	—
6	#17985825	C₄	1	2	yes	—
7	#17985752	—	2	1	no	—
8	C₂×C₃₂×C₇₁	—	2	1	no	—
9	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
10	#14556128	—	4	1	no	—
11	#14556062	—	2	2	no	—
12	#14556062	—	4	1	no	—
13	C₄⋊(C₇₁⋊C₈)	—	4	1	no	—
14	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
15	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
16	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
17	C₃₂×C₇₁	—	4	1	no	—
18	C₂×Dic₁₄₂	—	8	1	no	—
19	C₂×C₇₁⋊C₈	—	8	1	no	—
20	C₄⋊Dic₇₁	—	4	2	no	—
21	C₄⋊Dic₇₁	—	8	1	no	—
22	C₂×C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₈×C₇₁	SD₁₆	1	1	yes	characteristic
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	C₂×C₄×C₇₁	#102	1	1	yes	characteristic
29	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
30	C₂×C₄×C₇₁	SD₃₂	1	1	yes	characteristic
31	C₂×Dic₇₁	—	8	2	no	—
32	Dic₁₄₂	—	16	2	no	—
33	C₈×C₇₁	—	2	1	no	—
34	C₈×C₇₁	#101	1	1	yes	characteristic
35	C₈×C₇₁	#103	1	1	yes	characteristic
36	C₇₁⋊C₈	—	16	1	no	—
37	C₂²×C₇₁	#357	1	1	yes	characteristic
38	C₄×C₇₁	—	2	1	no	—
39	C₄×C₇₁	#332	1	1	yes	characteristic
40	C₄×C₇₁	#362	1	1	yes	characteristic
41	C₄×C₇₁	SD₆₄	1	1	yes	characteristic
42	C₄×C₇₁	Q₆₄	1	1	yes	characteristic
43	Dic₇₁	—	16	2	no	—
44	#7451	—	71	1	no	maximal, Sylow
45	C₂×C₇₁	#1364	1	1	yes	characteristic
46	C₂×C₇₁	#1416	1	1	yes	characteristic
47	C₂×C₇₁	#1419	1	1	yes	characteristic
48	#2252	—	71	1	no	—
49	C₄⋊C₃₂	D₇₁	1	1	yes	characteristic
50	#1371	—	71	1	no	—
51	C₇₁	#7451	1	1	yes	characteristic, complemented, Sylow
52	#500	—	142	1	no	—
53	C₂×C₃₂	—	2	1	no	—
54	#366	—	71	2	no	—
55	#366	—	142	1	no	—
56	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
57	#335	—	142	1	no	—
58	C₄⋊Q₈	—	284	1	no	—
59	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
60	C₂×C₁₆	C₂²⋊D₇₁	1	1	yes	characteristic
61	#107	—	284	1	no	—
62	#107	—	142	2	no	—
63	C₄⋊C₈	—	284	1	no	—
64	C₄×C₈	D₂₈₄	1	1	yes	characteristic
65	C₃₂	—	4	1	no	—
66	C₂×Q₈	—	568	1	no	—
67	C₂×C₈	—	568	1	no	—
68	C₂×C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
69	C₂×C₈	#11765854	1	1	yes	characteristic
70	C₄⋊C₄	—	284	2	no	—
71	C₄⋊C₄	—	568	1	no	—
72	C₄²	D₅₆₈	1	1	yes	characteristic
73	C₁₆	—	2	1	no	—
74	C₁₆	#11765853	1	1	yes	characteristic
75	C₁₆	#11765855	1	1	yes	characteristic
76	Q₈	—	1136	2	no	—
77	C₂×C₄	—	568	2	no	—
78	C₂×C₄	#14556050	1	1	yes	characteristic
79	C₂×C₄	D₁₁₃₆	1	1	yes	characteristic
80	C₂×C₄	#14556073	1	1	yes	characteristic
81	C₈	—	2	1	no	—
82	C₈	—	1136	1	no	—
83	C₈	#14556046	1	1	yes	characteristic
84	C₈	#14556051	1	1	yes	characteristic
85	C₂²	#17985801	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	1136	2	no	—
88	C₄	#17985744	1	1	yes	characteristic
89	C₄	#17985816	1	1	yes	characteristic
90	C₄	#17985838	1	1	yes	characteristic
91	C₄	Dic₁₁₃₆	1	1	yes	characteristic
92	C₂	#22586394	1	1	yes	characteristic
93	C₂	#22586522	1	1	yes	characteristic
94	C₂	#22586531	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17985801
commutator	C₂×C₁₆×C₇₁	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₇₁
Fitting	C₇₁×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₇₁	#357
cosocle	C₄×C₁₆×C₇₁	C₂²
hypercenter	C₄⋊C₃₂	D₇₁
hypocenter	C₇₁	#7451
LCS	G ▷ C₂×C₁₆×C₇₁ ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

Complements

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

N	H ≅ G/N	type
51. C₇₁	#7451	semidirect

Maximal subgroups

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

subgroup	index
#22590615	2
#22586412	2
C₇₁×C₄⋊C₃₂	2
#7451	71