Subgroups — #31884191

Order 18176.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22586410	C₂	1	1	yes	maximal, characteristic
2	C₈×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1370	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₇₁⋊C₁₆)	—	2	1	no	—
5	C₄×C₇₁⋊C₁₆	C₄	1	1	yes	characteristic
6	C₄×C₇₁⋊C₁₆	C₄	1	1	yes	characteristic
7	C₄⋊C₁₆×C₇₁	—	2	1	no	—
8	C₈²×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×C₇₁⋊C₁₆	—	2	1	no	—
10	C₂×C₇₁⋊C₁₆	—	4	1	no	—
11	C₂×C₇₁⋊C₁₆	C₈	1	1	yes	characteristic
12	C₂×C₇₁⋊C₁₆	C₈	1	1	yes	characteristic
13	C₂×C₁₆×C₇₁	—	4	1	no	—
14	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
15	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
16	C₄×C₈×C₇₁	Q₈	1	1	yes	characteristic
17	C₇₁⋊C₁₆	—	2	2	no	—
18	C₇₁⋊C₁₆	—	2	1	no	—
19	C₇₁⋊C₁₆	—	8	1	no	—
20	C₇₁⋊C₁₆	C₁₆	1	2	yes	complemented
21	C₂×C₈×C₇₁	—	2	1	no	—
22	C₂×C₈×C₇₁	C₂×C₈	1	1	yes	characteristic
23	C₂×C₈×C₇₁	M₁₆	1	1	yes	characteristic
24	C₂×C₈×C₇₁	D₈	1	1	yes	characteristic
25	C₂×C₈×C₇₁	Q₁₆	1	1	yes	characteristic
26	C₄²×C₇₁	C₄⋊C₄	1	1	yes	characteristic
27	C₁₆×C₇₁	—	8	1	no	—
28	C₂×C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
29	C₂×C₄×C₇₁	#107	1	1	yes	characteristic
30	C₂×C₄×C₇₁	#108	1	1	yes	characteristic
31	C₈×C₇₁	—	2	2	no	—
32	C₈×C₇₁	—	2	1	no	—
33	C₈×C₇₁	—	2	1	no	—
34	C₈×C₇₁	C₂×C₁₆	1	1	yes	characteristic
35	C₈×C₇₁	M₃₂	1	1	yes	characteristic
36	C₈×C₇₁	D₁₆	1	1	yes	characteristic
37	C₈×C₇₁	Q₃₂	1	1	yes	characteristic
38	C₂²×C₇₁	#335	1	1	yes	characteristic
39	C₄×C₇₁	—	2	1	no	—
40	C₄×C₇₁	C₄⋊C₁₆	1	1	yes	characteristic
41	C₄×C₇₁	#364	1	1	yes	characteristic
42	C₄×C₇₁	#366	1	1	yes	characteristic
43	C₄×C₇₁	#368	1	1	yes	characteristic
44	#7100	—	71	1	no	maximal, Sylow
45	C₂×C₇₁	#1370	1	1	yes	characteristic
46	C₂×C₇₁	#1371	1	1	yes	characteristic
47	C₂×C₇₁	#1373	1	1	yes	characteristic
48	#1370	—	71	1	no	—
49	#1370	D₇₁	1	1	yes	characteristic
50	C₈×C₁₆	—	71	1	no	—
51	C₇₁	#7100	1	1	yes	characteristic, complemented, Sylow
52	C₄⋊C₁₆	—	2	1	no	—
53	C₄⋊C₁₆	—	142	1	no	—
54	C₄×C₁₆	—	71	1	no	—
55	C₄×C₁₆	—	71	1	no	—
56	C₈²	C₂×D₇₁	1	1	yes	characteristic
57	C₂×C₁₆	—	4	1	no	—
58	C₂×C₁₆	—	284	1	no	—
59	C₂×C₁₆	—	71	1	no	—
60	C₂×C₁₆	—	71	1	no	—
61	C₂×C₁₆	—	142	1	no	—
62	C₄×C₈	Dic₁₄₂	1	1	yes	characteristic
63	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
64	C₄×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
65	C₂×C₈	—	2	1	no	—
66	C₂×C₈	C₈×D₇₁	1	1	yes	characteristic
67	C₂×C₈	#11765848	1	1	yes	characteristic
68	C₂×C₈	#11765850	1	1	yes	characteristic
69	C₂×C₈	#11765855	1	1	yes	characteristic
70	C₄²	Dic₇₁⋊C₄	1	1	yes	characteristic
71	C₁₆	—	8	1	no	—
72	C₁₆	—	568	1	no	—
73	C₁₆	—	71	2	no	—
74	C₁₆	—	142	2	no	—
75	C₁₆	—	142	1	no	—
76	C₂×C₄	Dic₇₁⋊C₈	1	1	yes	characteristic
77	C₂×C₄	#14556061	1	1	yes	characteristic
78	C₂×C₄	#14556063	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	—	2	1	no	—
81	C₈	—	2	1	no	—
82	C₈	C₁₆×D₇₁	1	1	yes	characteristic
83	C₈	#14556067	1	1	yes	characteristic
84	C₈	#14556069	1	1	yes	characteristic
85	C₈	#14556074	1	1	yes	characteristic
86	C₂²	#17985751	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	Dic₇₁⋊C₁₆	1	1	yes	characteristic
89	C₄	#17985819	1	1	yes	characteristic
90	C₄	#17985824	1	1	yes	characteristic
91	C₄	#17985828	1	1	yes	characteristic
92	C₂	#22586409	1	1	yes	characteristic
93	C₂	#22586411	1	1	yes	characteristic
94	C₂	#22586415	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	#11765850
commutator	C₈×C₇₁	C₂×C₁₆
Frattini	C₈²	C₂×D₇₁
Fitting	C₇₁×#1370	C₂
socle, prime radical	C₂²×C₇₁	#335
cosocle	C₈²×C₇₁	C₂²
hypercenter	#1370	D₇₁
hypocenter	C₇₁	#7100
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈² ◁ #1370

Complements

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

N	H ≅ G/N	type
20. C₇₁⋊C₁₆	C₁₆	semidirect
51. C₇₁	#7100	semidirect

Maximal subgroups

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

subgroup	index
#22586410	2
C₈×C₇₁⋊C₁₆	2
C₇₁×#1370	2
#7100	71