Subgroups — #31885343

Order 18176.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₂₇₂	C₂	1	1	yes	maximal, characteristic
2	#22586544	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₁₃₆	—	2	1	no	—
5	D₂₂₇₂	—	2	1	no	—
6	D₂₂₇₂	C₄	1	2	yes	—
7	#17985829	—	2	1	no	—
8	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
9	C₆₄×C₇₁	—	2	1	no	—
10	C₂×D₅₆₈	—	4	1	no	—
11	D₁₁₃₆	—	2	2	no	—
12	D₁₁₃₆	—	4	1	no	—
13	#14556064	—	4	1	no	—
14	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
15	C₃₂×C₇₁	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₇₁	D₄	1	1	yes	characteristic
17	C₂×D₂₈₄	—	8	1	no	—
18	D₅₆₈	—	4	2	no	—
19	D₅₆₈	—	8	1	no	—
20	#11765849	—	8	1	no	—
21	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
22	C₁₆×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₁₆×C₇₁	SD₁₆	1	1	yes	characteristic
24	C₂²×D₇₁	—	16	1	no	—
25	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
26	D₂₈₄	—	8	2	no	—
27	D₂₈₄	—	16	1	no	—
28	C₈×C₇₁	#102	1	1	yes	characteristic
29	C₈×C₇₁	SD₃₂	1	1	yes	characteristic
30	C₇₁⋊C₈	—	16	1	no	—
31	C₂²×C₇₁	SD₆₄	1	1	yes	characteristic
32	C₂×D₇₁	—	16	2	no	—
33	C₂×D₇₁	—	32	2	no	—
34	C₄×C₇₁	#357	1	1	yes	characteristic
35	C₄×C₇₁	D₃₂	1	1	yes	characteristic
36	#7541	—	71	1	no	maximal, Sylow
37	C₂×C₇₁	—	2	1	no	—
38	C₂×C₇₁	#1415	1	1	yes	characteristic
39	D₇₁	—	32	2	no	—
40	C₂×D₃₂	—	71	1	no	—
41	M₁₂₈	D₇₁	1	1	yes	characteristic, complemented
42	#1425	—	71	1	no	—
43	C₇₁	#7541	1	1	yes	characteristic, complemented, Sylow
44	C₂×D₁₆	—	142	1	no	—
45	D₃₂	—	71	2	no	—
46	D₃₂	—	142	1	no	—
47	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
48	#368	—	142	1	no	—
49	C₆₄	—	2	1	no	—
50	C₂×D₈	—	284	1	no	—
51	D₁₆	—	284	1	no	—
52	D₁₆	—	142	2	no	—
53	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
54	#108	—	284	1	no	—
55	C₃₂	C₄×D₇₁	1	1	yes	characteristic
56	C₃₂	C₂²⋊D₇₁	1	1	yes	characteristic
57	C₂×D₄	—	568	1	no	—
58	D₈	—	284	2	no	—
59	D₈	—	568	1	no	—
60	M₁₆	—	568	1	no	—
61	C₂×C₈	D₅₆₈	1	1	yes	characteristic
62	C₁₆	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
63	C₁₆	#11765854	1	1	yes	characteristic
64	C₂³	—	1136	1	no	—
65	D₄	—	568	2	no	—
66	D₄	—	1136	1	no	—
67	C₂×C₄	D₁₁₃₆	1	1	yes	characteristic
68	C₈	—	1136	1	no	—
69	C₈	#14556050	1	1	yes	characteristic
70	C₈	#14556073	1	1	yes	characteristic
71	C₂²	—	1136	2	no	—
72	C₂²	—	2272	2	no	—
73	C₂²	#17985838	1	1	yes	characteristic
74	C₄	#17985801	1	1	yes	characteristic
75	C₄	D₂₂₇₂	1	1	yes	characteristic
76	C₂	—	2	1	no	—
77	C₂	—	2272	2	no	—
78	C₂	#22586519	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22586519
commutator	C₃₂×C₇₁	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₇₁
Fitting	C₇₁×M₁₂₈	C₂
socle	C₂×C₇₁	#1415
prime radical	C₂×D₂₂₇₂	C₂
cosocle	C₂×C₃₂×C₇₁	C₂²
hypercenter	M₁₂₈	D₇₁
hypocenter	C₇₁	#7541
LCS	G ▷ C₃₂×C₇₁ ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
2. #22586544	C₂	semidirect
3. C₇₁×M₁₂₈	C₂	semidirect
41. M₁₂₈	D₇₁	semidirect
43. C₇₁	#7541	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₂₇₂	2
#22586544	2
C₇₁×M₁₂₈	2
#7541	71