Subgroups — #31918506

Order 18176.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590649	C₂	1	2	yes	maximal, complemented
2	Dic₂₂₇₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#22586556	C₂	1	2	yes	maximal, complemented
4	D₄₅₄₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₆₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986431	—	2	2	no	—
7	Dic₁₁₃₆	C₂²	1	2	yes	—
8	#17985838	—	2	2	no	—
9	D₂₂₇₂	C₂²	1	2	yes	—
10	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
11	C₆₄×C₇₁	C₂²	1	1	yes	characteristic
12	C₆₄×C₇₁	C₂²	1	1	yes	characteristic
13	#14556150	—	4	2	no	—
14	Dic₅₆₈	—	2	2	no	—
15	#14556073	—	4	2	no	—
16	D₁₁₃₆	—	2	2	no	—
17	C₂×C₁₆×C₇₁	D₄	1	1	yes	characteristic
18	C₃₂×C₇₁	D₄	1	1	yes	characteristic
19	C₃₂×C₇₁	C₂³	1	1	yes	characteristic
20	#11765865	—	8	2	no	—
21	Dic₂₈₄	—	4	2	no	—
22	#11765854	—	8	2	no	—
23	D₅₆₈	—	4	2	no	—
24	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
25	C₁₆×C₇₁	D₈	1	1	yes	characteristic
26	C₁₆×C₇₁	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₇₁	D₁₆	1	1	yes	characteristic
28	C₂²⋊D₇₁	—	16	2	no	—
29	D₂₈₄	—	8	2	no	—
30	C₄×D₇₁	—	16	2	no	—
31	Dic₁₄₂	—	8	2	no	—
32	C₈×C₇₁	D₁₆	1	1	yes	characteristic
33	C₈×C₇₁	C₂×D₈	1	1	yes	characteristic
34	C₂²×C₇₁	D₃₂	1	1	yes	characteristic
35	C₂×D₇₁	—	16	2	no	—
36	C₄×C₇₁	D₃₂	1	1	yes	characteristic
37	C₄×C₇₁	C₂×D₁₆	1	1	yes	characteristic
38	Dic₇₁	—	16	2	no	—
39	#13741	—	71	1	no	maximal, Sylow
40	C₂×C₇₁	—	2	1	no	—
41	C₂×C₇₁	C₂×D₃₂	1	1	yes	characteristic
42	D₇₁	—	32	2	no	—
43	#2262	—	71	2	no	—
44	Q₁₂₈	—	71	1	no	—
45	SD₁₂₈	—	71	2	no	—
46	D₆₄	—	71	1	no	—
47	C₂×C₆₄	D₇₁	1	1	yes	characteristic, complemented
48	C₇₁	#13741	1	1	yes	characteristic, complemented, Sylow
49	#508	—	142	2	no	—
50	Q₆₄	—	71	2	no	—
51	SD₆₄	—	142	2	no	—
52	D₃₂	—	71	2	no	—
53	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
54	C₆₄	C₂×D₇₁	1	1	yes	characteristic
55	C₆₄	C₂×D₇₁	1	1	yes	characteristic
56	#135	—	284	2	no	—
57	Q₃₂	—	142	2	no	—
58	SD₃₂	—	284	2	no	—
59	D₁₆	—	142	2	no	—
60	C₂×C₁₆	D₂₈₄	1	1	yes	characteristic
61	C₃₂	D₂₈₄	1	1	yes	characteristic
62	C₃₂	C₂²×D₇₁	1	1	yes	characteristic
63	#41	—	568	2	no	—
64	Q₁₆	—	284	2	no	—
65	SD₁₆	—	568	2	no	—
66	D₈	—	284	2	no	—
67	C₂×C₈	D₅₆₈	1	1	yes	characteristic
68	C₁₆	D₅₆₈	1	1	yes	characteristic
69	C₁₆	C₂×D₂₈₄	1	1	yes	characteristic
70	Q₈	—	568	2	no	—
71	D₄	—	568	2	no	—
72	D₄	—	1136	2	no	—
73	C₂×C₄	—	1136	2	no	—
74	C₂×C₄	D₁₁₃₆	1	1	yes	characteristic
75	C₈	D₁₁₃₆	1	1	yes	characteristic
76	C₈	C₂×D₅₆₈	1	1	yes	characteristic
77	C₂²	—	1136	2	no	—
78	C₂²	D₂₂₇₂	1	1	yes	characteristic
79	C₄	—	1136	2	no	—
80	C₄	D₂₂₇₂	1	1	yes	characteristic
81	C₄	C₂×D₁₁₃₆	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	2272	2	no	—
84	C₂	C₂×D₂₂₇₂	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₂₂₇₂
commutator, cosocle	C₃₂×C₇₁	C₂³
Frattini	C₃₂	C₂²×D₇₁
Fitting	C₂×C₆₄×C₇₁	C₂
socle	C₂×C₇₁	C₂×D₃₂
hypercenter	C₂×C₆₄	D₇₁
hypocenter	C₇₁	#13741
LCS	G ▷ C₃₂×C₇₁ ▷ C₁₆×C₇₁ ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ 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
1. #22590649	C₂	semidirect
2. Dic₂₂₇₂	C₂	semidirect
3. #22586556	C₂	semidirect
4. D₄₅₄₄	C₂	semidirect
5. C₂×C₆₄×C₇₁	C₂	semidirect
47. C₂×C₆₄	D₇₁	semidirect
48. C₇₁	#13741	semidirect

Maximal subgroups

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

subgroup	index
#22590649	2
Dic₂₂₇₂	2
#22586556	2
D₄₅₄₄	2
C₂×C₆₄×C₇₁	2
#13741	71