Subgroups — #30743502

Order 17152.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22558345	C₂	1	1	yes	maximal, characteristic
2	#22558275	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1369	C₂	1	1	yes	maximal, characteristic
4	#16863230	C₄	1	1	yes	characteristic
5	#16863230	C₄	1	1	yes	characteristic
6	#16863147	—	2	1	no	—
7	#364×C₆₇	—	2	1	no	—
8	#346×C₆₇	C₂²	1	1	yes	characteristic
9	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
10	C₆₇⋊C₃₂	—	2	1	no	—
11	C₆₇⋊C₃₂	C₈	1	2	yes	complemented
12	M₃₂×C₆₇	—	4	1	no	—
13	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
14	C₂×C₁₆×C₆₇	Q₈	1	1	yes	characteristic
15	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
16	C₂×C₆₇⋊C₈	—	8	1	no	—
17	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
19	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
20	C₁₆×C₆₇	—	2	1	no	—
21	C₁₆×C₆₇	—	8	1	no	—
22	C₁₆×C₆₇	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₆₇	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₆₇	—	2	1	no	—
25	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
26	C₈×C₆₇	—	2	1	no	—
27	C₈×C₆₇	C₄⋊C₈	1	1	yes	characteristic
28	C₈×C₆₇	#108	1	1	yes	characteristic
29	C₆₇⋊C₈	—	16	1	no	—
30	C₂²×C₆₇	#365	1	1	yes	characteristic
31	C₄×C₆₇	—	4	1	no	—
32	C₄×C₆₇	#334	1	1	yes	characteristic
33	C₄×C₆₇	#365	1	1	yes	characteristic
34	#7457	—	67	1	no	maximal, Sylow
35	C₂×C₆₇	—	2	1	no	—
36	C₂×C₆₇	#1368	1	1	yes	characteristic
37	#1398	—	67	1	no	—
38	#1369	D₆₇	1	1	yes	characteristic
39	#1368	—	67	1	no	—
40	C₆₇	#7457	1	1	yes	characteristic, complemented, Sylow
41	M₆₄	—	67	1	no	—
42	M₆₄	—	67	1	no	—
43	#364	—	2	1	no	—
44	#346	C₂×D₆₇	1	1	yes	characteristic
45	#334	—	134	1	no	—
46	M₃₂	—	4	1	no	—
47	C₂×C₁₆	Dic₁₃₄	1	1	yes	characteristic
48	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
49	C₄⋊C₈	—	268	1	no	—
50	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
51	C₃₂	—	67	2	no	—
52	C₃₂	—	134	1	no	—
53	C₂×C₈	—	536	1	no	—
54	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
55	C₂×C₈	#11761616	1	1	yes	characteristic
56	C₄²	#11761615	1	1	yes	characteristic
57	C₁₆	—	2	1	no	—
58	C₁₆	—	8	1	no	—
59	C₁₆	C₈×D₆₇	1	1	yes	characteristic
60	C₁₆	#11761611	1	1	yes	characteristic
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	#14523974	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	1072	1	no	—
65	C₈	Dic₆₇⋊C₈	1	1	yes	characteristic
66	C₈	#14523978	1	1	yes	characteristic
67	C₂²	#16863220	1	1	yes	characteristic
68	C₄	—	4	1	no	—
69	C₄	#16863146	1	1	yes	characteristic
70	C₄	#16863218	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	#22558274	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22558274
commutator	C₁₆×C₆₇	C₂×C₈
Frattini	#346	C₂×D₆₇
Fitting	C₆₇×#1369	C₂
socle	C₂×C₆₇	#1368
prime radical	C₂²×C₆₇	#365
cosocle	#346×C₆₇	C₂²
hypercenter	#1369	D₆₇
hypocenter	C₆₇	#7457
LCS	G ▷ C₁₆×C₆₇ ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346 ◁ #1369

Complements

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

N	H ≅ G/N	type
11. C₆₇⋊C₃₂	C₈	semidirect
40. C₆₇	#7457	semidirect

Maximal subgroups

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

subgroup	index
#22558345	2
#22558275	2
C₆₇×#1369	2
#7457	67