Subgroups — #30743513

Order 17152.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22558343	C₂	1	1	yes	maximal, characteristic
2	#22558283	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1372	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₇⋊C₃₂	C₄	1	1	yes	characteristic
5	C₂×C₆₇⋊C₃₂	C₄	1	1	yes	characteristic
6	#16863149	—	2	1	no	—
7	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
8	#335×C₆₇	—	2	1	no	—
9	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
10	C₆₇⋊C₃₂	—	2	1	no	—
11	C₆₇⋊C₃₂	C₈	1	2	yes	complemented
12	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
13	C₂×C₁₆×C₆₇	Q₈	1	1	yes	characteristic
14	C₄⋊C₈×C₆₇	—	4	1	no	—
15	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
16	C₂×C₆₇⋊C₈	—	8	1	no	—
17	C₂×C₈×C₆₇	—	8	1	no	—
18	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
19	C₂×C₈×C₆₇	Q₁₆	1	1	yes	characteristic
20	C₄²×C₆₇	D₈	1	1	yes	characteristic
21	C₁₆×C₆₇	—	2	1	no	—
22	C₁₆×C₆₇	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₆₇	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₆₇	#107	1	1	yes	characteristic
25	C₂×C₄×C₆₇	SD₃₂	1	1	yes	characteristic
26	C₂×C₄×C₆₇	SD₃₂	1	1	yes	characteristic
27	C₈×C₆₇	—	2	1	no	—
28	C₈×C₆₇	—	16	1	no	—
29	C₈×C₆₇	C₄⋊C₈	1	1	yes	characteristic
30	C₈×C₆₇	#108	1	1	yes	characteristic
31	C₆₇⋊C₈	—	16	1	no	—
32	C₂²×C₆₇	#367	1	1	yes	characteristic
33	C₄×C₆₇	—	2	1	no	—
34	C₄×C₆₇	—	2	1	no	—
35	C₄×C₆₇	#335	1	1	yes	characteristic
36	C₄×C₆₇	#368	1	1	yes	characteristic
37	#7463	—	67	1	no	maximal, Sylow
38	C₂×C₆₇	#1372	1	1	yes	characteristic
39	C₂×C₆₇	#1426	1	1	yes	characteristic
40	C₂×C₆₇	#1426	1	1	yes	characteristic
41	#1397	—	67	1	no	—
42	#1372	—	67	1	no	—
43	#1372	D₆₇	1	1	yes	characteristic
44	C₆₇	#7463	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₃₂	—	67	1	no	—
46	C₂×C₃₂	—	67	1	no	—
47	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
48	#335	—	2	1	no	—
49	#335	—	134	1	no	—
50	C₂×C₁₆	Dic₁₃₄	1	1	yes	characteristic
51	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
52	C₄⋊C₈	—	4	1	no	—
53	C₄⋊C₈	—	268	1	no	—
54	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
55	C₃₂	—	67	2	no	—
56	C₃₂	—	134	1	no	—
57	C₂×C₈	—	8	1	no	—
58	C₂×C₈	—	536	1	no	—
59	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
60	C₂×C₈	#11761618	1	1	yes	characteristic
61	C₄²	#11761613	1	1	yes	characteristic
62	C₁₆	—	2	1	no	—
63	C₁₆	C₈×D₆₇	1	1	yes	characteristic
64	C₁₆	#11761611	1	1	yes	characteristic
65	C₂×C₄	#14523976	1	1	yes	characteristic
66	C₂×C₄	#14523986	1	1	yes	characteristic
67	C₂×C₄	#14523987	1	1	yes	characteristic
68	C₈	—	2	1	no	—
69	C₈	—	16	1	no	—
70	C₈	—	1072	1	no	—
71	C₈	Dic₆₇⋊C₈	1	1	yes	characteristic
72	C₈	#14523978	1	1	yes	characteristic
73	C₂²	#16863223	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	—	2	1	no	—
76	C₄	#16863148	1	1	yes	characteristic
77	C₄	#16863225	1	1	yes	characteristic
78	C₂	#22558282	1	1	yes	characteristic
79	C₂	#22558414	1	1	yes	characteristic
80	C₂	#22558416	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863223
commutator	C₁₆×C₆₇	C₂×C₈
Frattini	C₄×C₁₆	C₂×D₆₇
Fitting	C₆₇×#1372	C₂
socle, prime radical	C₂²×C₆₇	#367
cosocle	C₄×C₁₆×C₆₇	C₂²
hypercenter	#1372	D₆₇
hypocenter	C₆₇	#7463
LCS	G ▷ C₁₆×C₆₇ ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1372

Complements

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

N	H ≅ G/N	type
11. C₆₇⋊C₃₂	C₈	semidirect
44. C₆₇	#7463	semidirect

Maximal subgroups

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

subgroup	index
#22558343	2
#22558283	2
C₆₇×#1372	2
#7463	67