Subgroups — #31885362

Order 18176.

44 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₄×D₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁⋊C₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×C₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₇₁	C₄	1	1	yes	characteristic
5	C₇₁⋊C₆₄	C₄	1	1	yes	characteristic
6	C₆₄×C₇₁	C₂²	1	1	yes	characteristic
7	C₁₆×D₇₁	C₈	1	1	yes	characteristic
8	C₇₁⋊C₃₂	C₈	1	1	yes	characteristic
9	C₃₂×C₇₁	C₂×C₄	1	1	yes	characteristic
10	C₈×D₇₁	C₁₆	1	1	yes	characteristic
11	C₇₁⋊C₁₆	C₁₆	1	1	yes	characteristic
12	C₁₆×C₇₁	C₂×C₈	1	1	yes	characteristic
13	C₄×D₇₁	C₃₂	1	1	yes	characteristic
14	C₈×C₇₁	C₂×C₁₆	1	1	yes	characteristic
15	C₇₁⋊C₈	C₃₂	1	1	yes	characteristic
16	C₂×D₇₁	C₆₄	1	1	yes	characteristic
17	C₄×C₇₁	C₂×C₃₂	1	1	yes	characteristic
18	Dic₇₁	C₆₄	1	1	yes	characteristic
19	M₂₅₆	—	71	1	no	maximal, Sylow
20	C₂×C₇₁	C₂×C₆₄	1	1	yes	characteristic
21	D₇₁	—	2	1	no	—
22	C₂×C₆₄	—	71	1	no	—
23	C₁₂₈	—	71	1	no	—
24	C₁₂₈	D₇₁	1	1	yes	characteristic, complemented
25	C₇₁	M₂₅₆	1	1	yes	characteristic, complemented, Sylow
26	C₂×C₃₂	—	71	1	no	—
27	C₆₄	—	71	1	no	—
28	C₆₄	C₂×D₇₁	1	1	yes	characteristic
29	C₂×C₁₆	—	71	1	no	—
30	C₃₂	—	71	1	no	—
31	C₃₂	C₄×D₇₁	1	1	yes	characteristic
32	C₂×C₈	—	71	1	no	—
33	C₁₆	—	71	1	no	—
34	C₁₆	C₈×D₇₁	1	1	yes	characteristic
35	C₂×C₄	—	71	1	no	—
36	C₈	—	71	1	no	—
37	C₈	C₁₆×D₇₁	1	1	yes	characteristic
38	C₂²	—	71	1	no	—
39	C₄	—	71	1	no	—
40	C₄	C₃₂×D₇₁	1	1	yes	characteristic
41	C₂	—	142	1	no	—
42	C₂	C₆₄×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₆₄	C₂×D₇₁
commutator, socle	C₂×C₇₁	C₂×C₆₄
Fitting	C₇₁×C₁₂₈	C₂
prime radical	C₂×D₇₁	C₆₄
cosocle	C₆₄×C₇₁	C₂²
hypercenter	C₁₂₈	D₇₁
hypocenter	C₇₁	M₂₅₆
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₆₄ ◁ C₁₂₈

Complements

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

N	H ≅ G/N	type
2. C₇₁⋊C₁₂₈	C₂	semidirect
3. C₇₁×C₁₂₈	C₂	semidirect
24. C₁₂₈	D₇₁	semidirect
25. C₇₁	M₂₅₆	semidirect

Maximal subgroups

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

subgroup	index
C₆₄×D₇₁	2
C₇₁⋊C₁₂₈	2
C₇₁×C₁₂₈	2
M₂₅₆	71