Subgroups — #31912193

Order 18176.

238 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590622	C₂	1	2	yes	maximal
2	#22590615	C₂	1	2	yes	maximal
3	#22587950	C₂	1	2	yes	maximal
4	C₈×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
5	#17986404	C₂²	1	4	yes	—
6	#17986350	—	2	4	no	—
7	#17985827	—	2	4	no	—
8	#17985825	—	2	4	no	—
9	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
11	C₈²×C₇₁	C₂²	1	1	yes	characteristic
12	#14556128	—	2	4	no	—
13	#14556062	—	2	4	no	—
14	#14556062	—	2	4	no	—
15	#14556053	—	4	8	no	—
16	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
17	C₂×C₁₆×C₇₁	D₄	1	2	yes	—
18	C₄×C₈×C₇₁	D₄	1	2	yes	—
19	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
20	C₂×Dic₁₄₂	—	4	4	no	—
21	C₄⋊Dic₇₁	—	4	4	no	—
22	C₄⋊Dic₇₁	—	4	4	no	—
23	C₂×C₈×C₇₁	D₈	1	4	yes	—
24	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
25	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
26	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
27	C₁₆×C₇₁	Q₁₆	1	4	yes	—
28	C₁₆×C₇₁	Q₁₆	1	4	yes	—
29	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
30	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
31	C₂×Dic₇₁	—	8	4	no	—
32	Dic₁₄₂	—	8	8	no	—
33	C₈×C₇₁	—	2	4	no	—
34	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
35	C₈×C₇₁	C₂×Q₁₆	1	2	yes	—
36	C₂²×C₇₁	#493	1	1	yes	characteristic
37	C₄×C₇₁	#494	1	1	yes	characteristic
38	C₄×C₇₁	#494	1	1	yes	characteristic
39	C₄×C₇₁	#508	1	4	yes	—
40	Dic₇₁	—	16	4	no	—
41	#12522	—	71	1	no	maximal, Sylow
42	C₂×C₇₁	#1713	1	1	yes	characteristic
43	C₂×C₇₁	#2249	1	2	yes	—
44	#2254	—	71	2	no	—
45	#2252	—	71	2	no	—
46	#1710	—	71	2	no	—
47	C₈×C₁₆	D₇₁	1	1	yes	characteristic
48	C₇₁	#12522	1	1	yes	characteristic, complemented, Sylow
49	#500	—	71	4	no	—
50	#487	—	142	4	no	—
51	#367	—	142	4	no	—
52	#366	—	142	4	no	—
53	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
54	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
55	C₈²	C₂×D₇₁	1	1	yes	characteristic
56	C₄⋊Q₈	—	142	4	no	—
57	C₂×C₁₆	D₂₈₄	1	2	yes	—
58	C₂×C₁₆	D₂₈₄	1	2	yes	—
59	#107	—	142	4	no	—
60	#107	—	142	4	no	—
61	#103	—	284	8	no	—
62	C₄×C₈	D₂₈₄	1	2	yes	—
63	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
64	C₂×Q₈	—	284	4	no	—
65	C₂×C₈	D₅₆₈	1	4	yes	—
66	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
67	C₂×C₈	C₂×D₂₈₄	1	1	yes	characteristic
68	C₄⋊C₄	—	284	4	no	—
69	C₄⋊C₄	—	284	4	no	—
70	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
71	C₁₆	Dic₂₈₄	1	4	yes	—
72	C₁₆	Dic₂₈₄	1	4	yes	—
73	Q₈	—	568	8	no	—
74	C₂×C₄	—	568	4	no	—
75	C₂×C₄	#14556125	1	1	yes	characteristic
76	C₂×C₄	C₂×D₅₆₈	1	2	yes	—
77	C₈	—	2	4	no	—
78	C₈	C₂×Dic₂₈₄	1	2	yes	—
79	C₈	C₂×Dic₂₈₄	1	2	yes	—
80	C₂²	#17986376	1	1	yes	characteristic
81	C₄	—	1136	4	no	—
82	C₄	#17986379	1	1	yes	characteristic
83	C₄	#17986379	1	1	yes	characteristic
84	C₄	#17986431	1	4	yes	—
85	C₂	#22587958	1	1	yes	characteristic
86	C₂	#22590601	1	2	yes	—

	subgroup	G/H
center	C₂²	#17986376
commutator, cosocle	C₄×C₈×C₇₁	C₂³
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₈×C₁₆×C₇₁	C₂
socle, prime radical	C₂²×C₇₁	#493
hypercenter	C₈×C₁₆	D₇₁
hypocenter	C₇₁	#12522
LCS	G ▷ C₄×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
48. C₇₁	#12522	semidirect

Maximal subgroups

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

subgroup	index
#22590622	2
#22590615	2
#22587950	2
C₈×C₁₆×C₇₁	2
#12522	71