Subgroups — #31918492

Order 18176.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22590633	C₂	1	1	yes	maximal, characteristic
2	#22586550	C₂	1	2	yes	maximal, complemented
3	C₂×C₇₁⋊C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₄×D₇₁	C₂	1	2	yes	maximal, complemented
5	C₇₁×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986415	C₄	1	1	yes	characteristic
7	#17985833	C₄	1	1	yes	characteristic
8	#17985832	C₄	1	2	yes	—
9	C₃₂×D₇₁	C₂²	1	2	yes	—
10	C₇₁⋊C₆₄	C₂²	1	1	yes	characteristic
11	C₇₁⋊C₆₄	C₂²	1	1	yes	characteristic
12	C₂×C₃₂×C₇₁	C₂²	1	1	yes	characteristic
13	C₆₄×C₇₁	C₂²	1	2	yes	—
14	#14556134	C₈	1	1	yes	characteristic
15	#14556068	C₈	1	1	yes	characteristic
16	#14556067	C₈	1	2	yes	—
17	C₁₆×D₇₁	C₂×C₄	1	2	yes	—
18	C₇₁⋊C₃₂	C₂×C₄	1	2	yes	—
19	C₂×C₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₃₂×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₃₂×C₇₁	C₂³	1	1	yes	characteristic
22	#11765865	C₁₆	1	1	yes	characteristic
23	#11765849	C₁₆	1	1	yes	characteristic
24	#11765848	C₁₆	1	2	yes	—
25	C₈×D₇₁	C₂×C₈	1	2	yes	—
26	C₇₁⋊C₁₆	C₂×C₈	1	2	yes	—
27	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
28	C₁₆×C₇₁	C₂×C₈	1	1	yes	characteristic
29	C₁₆×C₇₁	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
31	C₂²⋊D₇₁	C₃₂	1	2	yes	—
32	D₂₈₄	C₃₂	1	1	yes	characteristic
33	C₄×D₇₁	C₂×C₁₆	1	2	yes	—
34	Dic₁₄₂	C₃₂	1	1	yes	characteristic
35	C₈×C₇₁	C₂×C₁₆	1	1	yes	characteristic
36	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
37	C₇₁⋊C₈	C₂×C₁₆	1	2	yes	—
38	C₂²×C₇₁	C₂×C₃₂	1	1	yes	characteristic
39	C₂×D₇₁	C₂×C₃₂	1	2	yes	—
40	C₄×C₇₁	C₂×C₃₂	1	1	yes	characteristic
41	C₄×C₇₁	C₂²×C₁₆	1	1	yes	characteristic
42	Dic₇₁	C₂×C₃₂	1	2	yes	—
43	#13737	—	71	1	no	maximal, Sylow
44	C₂×C₇₁	—	2	1	no	—
45	C₂×C₇₁	C₂²×C₃₂	1	1	yes	characteristic
46	D₇₁	—	2	2	no	—
47	#2258	—	71	1	no	—
48	M₁₂₈	—	71	2	no	—
49	M₁₂₈	D₇₁	1	1	yes	characteristic, complemented
50	C₂×C₆₄	—	71	1	no	—
51	C₂×C₆₄	—	71	2	no	—
52	C₇₁	#13737	1	1	yes	characteristic, complemented, Sylow
53	#504	—	71	1	no	—
54	M₆₄	—	71	1	no	—
55	M₆₄	—	71	2	no	—
56	C₂×C₃₂	—	71	2	no	—
57	C₂×C₃₂	C₂×D₇₁	1	1	yes	characteristic
58	C₆₄	—	71	1	no	—
59	C₆₄	—	71	1	no	—
60	C₆₄	C₂×D₇₁	1	2	yes	—
61	#131	—	71	1	no	—
62	M₃₂	—	71	1	no	—
63	M₃₂	—	71	2	no	—
64	C₂×C₁₆	—	71	2	no	—
65	C₂×C₁₆	C₄×D₇₁	1	1	yes	characteristic
66	C₃₂	—	71	2	no	—
67	C₃₂	C₄×D₇₁	1	1	yes	characteristic
68	C₃₂	C₂²×D₇₁	1	1	yes	characteristic
69	#41	—	71	1	no	—
70	M₁₆	—	71	1	no	—
71	M₁₆	—	71	2	no	—
72	C₂×C₈	—	71	2	no	—
73	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
74	C₁₆	—	71	2	no	—
75	C₁₆	C₈×D₇₁	1	1	yes	characteristic
76	C₁₆	C₂×C₄×D₇₁	1	1	yes	characteristic
77	Q₈	—	71	1	no	—
78	D₄	—	71	2	no	—
79	D₄	—	71	1	no	—
80	C₂×C₄	—	71	2	no	—
81	C₂×C₄	C₁₆×D₇₁	1	1	yes	characteristic
82	C₈	—	71	2	no	—
83	C₈	C₁₆×D₇₁	1	1	yes	characteristic
84	C₈	C₂×C₈×D₇₁	1	1	yes	characteristic
85	C₂²	—	71	2	no	—
86	C₂²	C₃₂×D₇₁	1	1	yes	characteristic
87	C₄	—	71	2	no	—
88	C₄	C₃₂×D₇₁	1	1	yes	characteristic
89	C₄	C₂×C₁₆×D₇₁	1	1	yes	characteristic
90	C₂	—	2	1	no	—
91	C₂	—	142	2	no	—
92	C₂	C₂×C₃₂×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃₂	C₂²×D₇₁
commutator, socle	C₂×C₇₁	C₂²×C₃₂
Fitting	C₇₁×M₁₂₈	C₂
prime radical	#11765865	C₁₆
cosocle	C₃₂×C₇₁	C₂³
hypercenter	M₁₂₈	D₇₁
hypocenter	C₇₁	#13737
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₃₂ ◁ M₁₂₈

Complements

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

N	H ≅ G/N	type
2. #22586550	C₂	semidirect
3. C₂×C₇₁⋊C₆₄	C₂	semidirect
4. C₆₄×D₇₁	C₂	semidirect
5. C₇₁×M₁₂₈	C₂	semidirect
49. M₁₂₈	D₇₁	semidirect
52. C₇₁	#13737	semidirect

Maximal subgroups

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

subgroup	index
#22590633	2
#22586550	2
C₂×C₇₁⋊C₆₄	2
C₆₄×D₇₁	2
C₇₁×M₁₂₈	2
#13737	71