Subgroups — #31912285

Order 18176.

350 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22589945	C₂	1	3	yes	maximal
2	#22589937	C₂	1	3	yes	maximal
3	C₇₁×#1747	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556068	C₂²	1	3	yes	—
5	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
6	#17985777	C₄	1	12	yes	complemented
7	#17985775	C₄	1	6	yes	complemented
8	C₄×C₇₁⋊C₁₆	C₄	1	6	yes	complemented
9	#405×C₇₁	C₂²	1	3	yes	—
10	#338×C₇₁	C₄	1	1	yes	characteristic
11	H(3,ℤ/4)×C₇₁	C₄	1	3	yes	—
12	#14556068	C₂×C₄	1	12	yes	—
13	C₂×C₇₁⋊C₁₆	C₂×C₄	1	3	yes	—
14	C₂×C₇₁⋊C₁₆	C₂×C₄	1	3	yes	—
15	C₂×C₇₁⋊C₁₆	C₂×C₄	1	6	yes	—
16	C₂×M₁₆×C₇₁	C₂×C₄	1	3	yes	—
17	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
18	#117×C₇₁	C₂×C₄	1	3	yes	—
19	#97×C₇₁	C₂×C₄	1	6	yes	—
20	C₄×C₈×C₇₁	C₂×C₄	1	6	yes	—
21	C₇₁⋊C₁₆	C₄²	1	1	yes	characteristic
22	C₇₁⋊C₁₆	C₄²	1	3	yes	—
23	C₇₁⋊C₁₆	C₄²	1	12	yes	—
24	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
25	M₁₆×C₇₁	—	2	6	no	—
26	C₂×C₈×C₇₁	C₄²	1	6	yes	—
27	C₂×C₈×C₇₁	C₂²×C₄	1	3	yes	—
28	C₂×C₈×C₇₁	C₂²×C₄	1	3	yes	—
29	C₄⋊C₄×C₇₁	—	2	3	no	—
30	C₂²⋊C₄×C₇₁	—	2	3	no	—
31	C₄²×C₇₁	C₄²	1	6	yes	—
32	C₂³×C₇₁	#131	1	1	yes	characteristic
33	C₂×C₄×C₇₁	—	2	6	no	—
34	C₂×C₄×C₇₁	C₂×C₄²	1	3	yes	—
35	C₂×C₄×C₇₁	#131	1	3	yes	—
36	C₈×C₇₁	—	2	6	no	—
37	C₈×C₇₁	C₂×C₄²	1	3	yes	—
38	C₈×C₇₁	C₂×C₄²	1	1	yes	characteristic
39	C₂²×C₇₁	—	4	1	no	—
40	C₂²×C₇₁	#405	1	3	yes	—
41	C₄×C₇₁	—	4	6	no	—
42	C₄×C₇₁	C₄³	1	1	yes	characteristic
43	C₄×C₇₁	#405	1	3	yes	—
44	#12542	—	71	1	no	maximal, Sylow
45	C₂×C₇₁	—	2	3	no	—
46	C₂×C₇₁	#1726	1	1	yes	characteristic
47	#2110	—	71	3	no	—
48	#2108	—	71	3	no	—
49	#1747	D₇₁	1	1	yes	characteristic
50	C₇₁	#12542	1	1	yes	characteristic, complemented, Sylow
51	C₂×M₃₂	—	71	3	no	—
52	C₂²×C₁₆	—	71	1	no	—
53	#405	C₂×D₇₁	1	3	yes	—
54	#347	—	71	12	no	—
55	#346	—	71	6	no	—
56	C₄×C₁₆	—	71	6	no	—
57	#338	Dic₇₁	1	1	yes	characteristic
58	H(3,ℤ/4)	Dic₇₁	1	3	yes	—
59	C₂×M₁₆	C₂×Dic₇₁	1	3	yes	—
60	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
61	#117	C₂×Dic₇₁	1	3	yes	—
62	M₃₂	—	71	12	no	—
63	C₂×C₁₆	—	71	3	no	—
64	C₂×C₁₆	—	71	6	no	—
65	C₂×C₁₆	—	71	3	no	—
66	#97	C₄×D₇₁	1	6	yes	—
67	C₄×C₈	C₄×D₇₁	1	6	yes	—
68	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
69	M₁₆	—	2	6	no	—
70	C₂×C₈	C₄×Dic₇₁	1	6	yes	—
71	C₂×C₈	C₂×C₄×D₇₁	1	3	yes	—
72	C₂×C₈	C₂×C₄×D₇₁	1	3	yes	—
73	C₄⋊C₄	—	2	3	no	—
74	C₂²⋊C₄	—	2	3	no	—
75	C₄²	C₄×Dic₇₁	1	6	yes	—
76	C₁₆	—	71	12	no	—
77	C₁₆	—	71	1	no	—
78	C₁₆	—	71	3	no	—
79	C₂³	#14556135	1	1	yes	characteristic
80	C₂×C₄	—	2	6	no	—
81	C₂×C₄	C₂×C₄×Dic₇₁	1	3	yes	—
82	C₂×C₄	#14556135	1	3	yes	—
83	C₈	—	2	6	no	—
84	C₈	C₄²×D₇₁	1	3	yes	—
85	C₈	C₄²×D₇₁	1	1	yes	characteristic
86	C₂²	—	4	1	no	—
87	C₂²	#17985955	1	3	yes	—
88	C₄	—	4	6	no	—
89	C₄	C₄²×Dic₇₁	1	1	yes	characteristic
90	C₄	#17985955	1	3	yes	—
91	C₂	—	2	3	no	—
92	C₂	#22588027	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄²×D₇₁
commutator	C₄×C₇₁	C₄³
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₇₁×#1747	C₂
socle	C₂×C₇₁	#1726
prime radical	C₂³×C₇₁	#131
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	#1747	D₇₁
hypocenter	C₇₁	#12542
LCS	G ▷ C₄×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ #1747

Complements

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

N	H ≅ G/N	type
6. #17985777	C₄	semidirect
7. #17985775	C₄	semidirect
8. C₄×C₇₁⋊C₁₆	C₄	semidirect
50. C₇₁	#12542	semidirect

Maximal subgroups

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

subgroup	index
#22589945	2
#22589937	2
C₇₁×#1747	2
#12542	71