Subgroups — #31913854

Order 18176.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22586418	C₂	1	6	yes	maximal
2	C₇₁×#1726	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₇₁⋊C₁₆	C₂²	1	4	yes	—
4	C₂×#97×C₇₁	C₄	1	3	yes	—
5	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
6	#376×C₇₁	C₄	1	1	yes	characteristic
7	C₂×C₇₁⋊C₁₆	—	2	12	no	—
8	C₂²×C₈×C₇₁	C₂×C₄	1	3	yes	—
9	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
10	C₂×C₄²×C₇₁	C₂×C₄	1	3	yes	—
11	#97×C₇₁	—	2	6	no	—
12	C₄×C₈×C₇₁	—	2	6	no	—
13	#95×C₇₁	C₈	1	4	yes	—
14	C₇₁⋊C₁₆	—	4	4	no	—
15	C₂²×C₄×C₇₁	C₂×C₈	1	6	yes	—
16	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
17	C₂×C₈×C₇₁	—	2	3	no	—
18	C₂×C₈×C₇₁	—	2	3	no	—
19	C₂×C₈×C₇₁	M₁₆	1	6	yes	—
20	C₂×C₈×C₇₁	#41	1	6	yes	—
21	C₄²×C₇₁	—	2	3	no	—
22	C₄²×C₇₁	—	2	3	no	—
23	C₂³×C₇₁	C₂²×C₈	1	1	yes	characteristic
24	C₂×C₄×C₇₁	—	2	6	no	—
25	C₂×C₄×C₇₁	—	2	12	no	—
26	C₂×C₄×C₇₁	#117	1	3	yes	—
27	C₂×C₄×C₇₁	C₂×M₁₆	1	3	yes	—
28	C₈×C₇₁	—	2	6	no	—
29	C₈×C₇₁	#126	1	4	yes	—
30	C₂²×C₇₁	#431	1	3	yes	—
31	C₂²×C₇₁	#504	1	4	yes	—
32	C₄×C₇₁	—	4	6	no	—
33	C₄×C₇₁	#383	1	1	yes	characteristic
34	C₄×C₇₁	#432	1	3	yes	—
35	#12895	—	71	1	no	maximal, Sylow
36	C₂×C₇₁	#1839	1	1	yes	characteristic
37	C₂×C₇₁	#2164	1	6	yes	—
38	#1726	D₇₁	1	1	yes	characteristic
39	#1374	—	71	6	no	—
40	C₇₁	#12895	1	1	yes	characteristic, complemented, Sylow
41	C₂²×C₁₆	—	71	4	no	—
42	C₂×#97	Dic₇₁	1	3	yes	—
43	C₂×C₄×C₈	C₂×D₇₁	1	3	yes	—
44	#376	Dic₇₁	1	1	yes	characteristic
45	C₂²×C₈	C₂×Dic₇₁	1	3	yes	—
46	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
47	C₂×C₄²	C₂×Dic₇₁	1	3	yes	—
48	C₂×C₁₆	—	142	12	no	—
49	#97	—	2	6	no	—
50	C₄×C₈	—	2	6	no	—
51	#95	C₇₁⋊C₈	1	4	yes	—
52	C₂²×C₄	C₂×C₇₁⋊C₈	1	6	yes	—
53	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
54	C₂×C₈	—	2	3	no	—
55	C₂×C₈	—	2	3	no	—
56	C₂×C₈	#11765849	1	6	yes	—
57	C₂×C₈	#11765865	1	6	yes	—
58	C₄²	—	2	3	no	—
59	C₄²	—	2	3	no	—
60	C₁₆	—	284	4	no	—
61	C₂³	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
62	C₂×C₄	—	2	6	no	—
63	C₂×C₄	—	2	12	no	—
64	C₂×C₄	#14556087	1	3	yes	—
65	C₂×C₄	C₂×#11765849	1	3	yes	—
66	C₈	—	2	6	no	—
67	C₈	#14556122	1	4	yes	—
68	C₂²	#17986052	1	3	yes	—
69	C₂²	#17986416	1	4	yes	—
70	C₄	—	4	6	no	—
71	C₄	#17985867	1	1	yes	characteristic
72	C₄	#17986056	1	3	yes	—
73	C₂	#22588483	1	1	yes	characteristic
74	C₂	#22590147	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₁
commutator, socle, prime radical	C₂³×C₇₁	C₂²×C₈
Fitting	C₇₁×#1726	C₂
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1726	D₇₁
hypocenter	C₇₁	#12895
LCS	G ▷ C₂³×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
40. C₇₁	#12895	semidirect

Maximal subgroups

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

subgroup	index
#22586418	2
C₇₁×#1726	2
#12895	71