Subgroups — #31912159

Order 18176.

246 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590567	C₂	1	4	yes	maximal
2	#22587950	C₂	1	1	yes	maximal, characteristic
3	#22586257	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1713	C₂	1	1	yes	maximal, characteristic
5	#17986404	C₂²	1	2	yes	—
6	#17986350	—	2	2	no	—
7	#17985803	—	2	8	no	—
8	C₄×C₇₁⋊C₁₆	C₂²	1	2	yes	—
9	#500×C₇₁	—	2	2	no	—
10	#494×C₇₁	C₂²	1	2	yes	—
11	C₈²×C₇₁	C₂²	1	1	yes	characteristic
12	#14556128	—	2	2	no	—
13	C₂×C₇₁⋊C₁₆	D₄	1	4	yes	—
14	#14556062	—	2	4	no	—
15	#14556053	—	4	4	no	—
16	C₂×Q₁₆×C₇₁	—	2	4	no	—
17	C₄⋊Q₈×C₇₁	—	2	2	no	—
18	#107×C₇₁	—	4	4	no	—
19	C₄×C₈×C₇₁	D₄	1	2	yes	—
20	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
21	C₂×Dic₁₄₂	—	4	2	no	—
22	C₄⋊Dic₇₁	—	4	2	no	—
23	C₄⋊Dic₇₁	—	4	2	no	—
24	C₇₁⋊C₁₆	—	2	4	no	—
25	C₂×Q₈×C₇₁	—	4	2	no	—
26	Q₁₆×C₇₁	—	4	8	no	—
27	C₂×C₈×C₇₁	D₈	1	4	yes	—
28	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
29	C₄⋊C₄×C₇₁	—	4	4	no	—
30	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
31	Q₈×C₇₁	—	8	4	no	—
32	C₂×C₄×C₇₁	—	8	2	no	—
33	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
35	C₂×Dic₇₁	—	8	2	no	—
36	Dic₁₄₂	—	8	4	no	—
37	C₈×C₇₁	—	2	4	no	—
38	C₈×C₇₁	#135	1	4	yes	—
39	C₂²×C₇₁	#493	1	1	yes	characteristic
40	C₄×C₇₁	—	16	2	no	—
41	C₄×C₇₁	#495	1	2	yes	—
42	C₄×C₇₁	#510	1	4	yes	—
43	Dic₇₁	—	16	2	no	—
44	#12514	—	71	1	no	maximal, Sylow
45	C₂×C₇₁	#1709	1	1	yes	characteristic
46	C₂×C₇₁	#2251	1	2	yes	—
47	#2241	—	71	4	no	—
48	#1713	D₇₁	1	1	yes	characteristic
49	#1710	—	71	1	no	—
50	#1311	—	71	1	no	—
51	C₇₁	#12514	1	1	yes	characteristic, complemented, Sylow
52	#500	—	2	2	no	—
53	#500	—	71	2	no	—
54	#494	C₂×D₇₁	1	2	yes	—
55	#487	—	142	2	no	—
56	#358	—	142	8	no	—
57	C₄×C₁₆	—	71	2	no	—
58	C₈²	C₂×D₇₁	1	1	yes	characteristic
59	C₂×Q₁₆	—	2	4	no	—
60	C₄⋊Q₈	—	2	2	no	—
61	C₄⋊Q₈	—	142	2	no	—
62	C₂×C₁₆	—	71	4	no	—
63	#107	—	4	4	no	—
64	#107	—	142	4	no	—
65	#103	—	284	4	no	—
66	C₄×C₈	C₂²⋊D₇₁	1	2	yes	—
67	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
68	C₂×Q₈	—	4	2	no	—
69	C₂×Q₈	—	284	2	no	—
70	Q₁₆	—	4	8	no	—
71	C₂×C₈	#11765850	1	4	yes	—
72	C₂×C₈	D₄×D₇₁	1	2	yes	—
73	C₄⋊C₄	—	4	4	no	—
74	C₄⋊C₄	—	284	2	no	—
75	C₄⋊C₄	—	284	2	no	—
76	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
77	C₁₆	—	142	4	no	—
78	Q₈	—	8	4	no	—
79	Q₈	—	568	4	no	—
80	C₂×C₄	—	8	2	no	—
81	C₂×C₄	—	568	2	no	—
82	C₂×C₄	#14556124	1	1	yes	characteristic
83	C₂×C₄	C₂×#11765850	1	2	yes	—
84	C₈	—	2	4	no	—
85	C₈	#14556147	1	4	yes	—
86	C₂²	#17986375	1	1	yes	characteristic
87	C₄	—	16	2	no	—
88	C₄	—	1136	2	no	—
89	C₄	#17986382	1	2	yes	—
90	C₄	#17986440	1	4	yes	—
91	C₂	#22587945	1	1	yes	characteristic
92	C₂	#22590609	1	2	yes	—

	subgroup	G/H
center	C₂²	#17986375
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1713	C₂
socle, prime radical	C₂²×C₇₁	#493
hypercenter	#1713	D₇₁
hypocenter	C₇₁	#12514
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1713

Complements

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

N	H ≅ G/N	type
51. C₇₁	#12514	semidirect

Maximal subgroups

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

subgroup	index
#22590567	2
#22587950	2
#22586257	2
C₇₁×#1713	2
#12514	71