Subgroups — #31885331

Order 18176.

98 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	#22586540	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₁₃₆	—	2	1	no	—
5	D₂₂₇₂	—	2	1	no	—
6	D₂₂₇₂	C₄	1	2	yes	complemented
7	#17985825	—	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	#14556062	—	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	C₄⋊Dic₇₁	—	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	C₂×Dic₇₁	—	16	1	no	—
27	D₂₈₄	—	8	2	no	—
28	D₂₈₄	—	16	1	no	—
29	C₈×C₇₁	#102	1	1	yes	characteristic
30	C₈×C₇₁	SD₃₂	1	1	yes	characteristic
31	C₂²×C₇₁	D₃₂	1	1	yes	characteristic
32	C₂×D₇₁	—	16	2	no	—
33	C₂×D₇₁	—	32	1	no	—
34	C₂×D₇₁	—	32	1	no	—
35	C₄×C₇₁	#357	1	1	yes	characteristic
36	C₄×C₇₁	SD₆₄	1	1	yes	characteristic
37	Dic₇₁	—	32	1	no	—
38	#7537	—	71	1	no	maximal, Sylow
39	C₂×C₇₁	#1415	1	1	yes	characteristic
40	C₂×C₇₁	D₆₄	1	1	yes	characteristic
41	C₂×C₇₁	SD₁₂₈	1	1	yes	characteristic
42	D₇₁	—	32	2	no	—
43	C₂×D₃₂	—	71	1	no	—
44	C₂×C₆₄	D₇₁	1	1	yes	characteristic, complemented
45	#1423	—	71	1	no	—
46	C₇₁	#7537	1	1	yes	characteristic, complemented, Sylow
47	C₂×D₁₆	—	142	1	no	—
48	D₃₂	—	71	2	no	—
49	D₃₂	—	142	1	no	—
50	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
51	#366	—	142	1	no	—
52	C₆₄	—	2	1	no	—
53	C₂×D₈	—	284	1	no	—
54	D₁₆	—	284	1	no	—
55	D₁₆	—	142	2	no	—
56	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
57	#107	—	284	1	no	—
58	C₃₂	C₄×D₇₁	1	1	yes	characteristic
59	C₃₂	C₂²⋊D₇₁	1	1	yes	characteristic
60	C₂×D₄	—	568	1	no	—
61	D₈	—	284	2	no	—
62	D₈	—	568	1	no	—
63	C₂×C₈	D₅₆₈	1	1	yes	characteristic
64	C₄⋊C₄	—	568	1	no	—
65	C₁₆	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
66	C₁₆	#11765854	1	1	yes	characteristic
67	C₂³	—	1136	1	no	—
68	D₄	—	568	2	no	—
69	D₄	—	1136	1	no	—
70	C₂×C₄	—	1136	1	no	—
71	C₂×C₄	D₁₁₃₆	1	1	yes	characteristic
72	C₈	#14556050	1	1	yes	characteristic
73	C₈	#14556073	1	1	yes	characteristic
74	C₂²	—	1136	2	no	—
75	C₂²	—	2272	1	no	—
76	C₂²	—	2272	1	no	—
77	C₂²	D₂₂₇₂	1	1	yes	characteristic
78	C₄	—	2272	1	no	—
79	C₄	#17985801	1	1	yes	characteristic
80	C₄	#17985838	1	1	yes	characteristic
81	C₂	—	2272	2	no	—
82	C₂	#22586519	1	1	yes	characteristic
83	C₂	D₄₅₄₄	1	1	yes	characteristic
84	C₂	#22586556	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₂₂₇₂
commutator	C₃₂×C₇₁	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₇₁
Fitting	C₂×C₆₄×C₇₁	C₂
socle	C₂²×C₇₁	D₃₂
prime radical	C₂×D₂₂₇₂	C₂
cosocle	C₂×C₃₂×C₇₁	C₂²
hypercenter	C₂×C₆₄	D₇₁
hypocenter	C₇₁	#7537
LCS	G ▷ C₃₂×C₇₁ ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
2. #22586540	C₂	semidirect
3. C₂×C₆₄×C₇₁	C₂	semidirect
6. D₂₂₇₂	C₄	semidirect
44. C₂×C₆₄	D₇₁	semidirect
46. C₇₁	#7537	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
#22586540	2
C₂×C₆₄×C₇₁	2
#7537	71