Subgroups — #30743493

Order 17152.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562488	C₂	1	1	yes	maximal, characteristic
2	#22558285	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1422	C₂	1	1	yes	maximal, characteristic
4	#16863798	—	2	1	no	—
5	#16863224	C₄	1	2	yes	—
6	#16863222	—	2	1	no	—
7	#16863217	—	2	1	no	—
8	M₆₄×C₆₇	—	2	1	no	—
9	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	#14524035	—	4	1	no	—
11	#14523983	—	4	1	no	—
12	#14523977	—	2	2	no	—
13	#14523975	—	4	1	no	—
14	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
15	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
16	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
17	C₃₂×C₆₇	—	4	1	no	—
18	C₄⋊Dic₆₇	—	4	2	no	—
19	Dic₆₇⋊C₄	—	8	2	no	—
20	C₆₇⋊C₁₆	—	8	1	no	—
21	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
23	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
24	C₁₆×C₆₇	—	2	1	no	—
25	C₁₆×C₆₇	SD₁₆	1	1	yes	characteristic
26	C₁₆×C₆₇	Q₁₆	1	1	yes	characteristic
27	C₂×C₄×C₆₇	—	2	1	no	—
28	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
29	C₂×Dic₆₇	—	8	2	no	—
30	C₈×C₆₇	#101	1	1	yes	characteristic
31	C₈×C₆₇	#103	1	1	yes	characteristic
32	C₈×C₆₇	SD₃₂	1	1	yes	characteristic
33	C₈×C₆₇	Q₃₂	1	1	yes	characteristic
34	C₂²×C₆₇	#361	1	1	yes	characteristic
35	C₄×C₆₇	—	4	1	no	—
36	C₄×C₆₇	#332	1	1	yes	characteristic
37	C₄×C₆₇	#358	1	1	yes	characteristic
38	Dic₆₇	—	16	2	no	—
39	#7454	—	67	1	no	maximal, Sylow
40	C₂×C₆₇	—	2	1	no	—
41	C₂×C₆₇	#1365	1	1	yes	characteristic
42	#2253	—	67	1	no	—
43	#1422	D₆₇	1	1	yes	characteristic
44	#1373	—	67	1	no	—
45	C₆₇	#7454	1	1	yes	characteristic, complemented, Sylow
46	#499	—	134	1	no	—
47	M₆₄	—	2	1	no	—
48	#367	—	67	2	no	—
49	#366	—	134	1	no	—
50	#364	—	134	1	no	—
51	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
52	#125	—	268	1	no	—
53	M₃₂	—	268	1	no	—
54	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
55	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
56	#107	—	134	2	no	—
57	#106	—	268	1	no	—
58	C₄×C₈	D₂₆₈	1	1	yes	characteristic
59	C₃₂	—	4	1	no	—
60	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
61	C₂×C₈	D₅₃₆	1	1	yes	characteristic
62	C₄⋊C₄	—	268	2	no	—
63	C₄⋊C₄	—	536	2	no	—
64	C₄²	#11761617	1	1	yes	characteristic
65	C₁₆	—	2	1	no	—
66	C₁₆	—	536	1	no	—
67	C₁₆	#11761616	1	1	yes	characteristic
68	C₁₆	#11761618	1	1	yes	characteristic
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	536	2	no	—
71	C₂×C₄	#14523965	1	1	yes	characteristic
72	C₈	#14523961	1	1	yes	characteristic
73	C₈	#14523966	1	1	yes	characteristic
74	C₈	#14523988	1	1	yes	characteristic
75	C₈	Dic₅₃₆	1	1	yes	characteristic
76	C₂²	#16863210	1	1	yes	characteristic
77	C₄	—	4	1	no	—
78	C₄	—	1072	2	no	—
79	C₄	#16863141	1	1	yes	characteristic
80	C₄	#16863201	1	1	yes	characteristic
81	C₂	—	2	1	no	—
82	C₂	#22558266	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22558266
commutator	C₂×C₁₆×C₆₇	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₆₇
Fitting	C₆₇×#1422	C₂
socle	C₂×C₆₇	#1365
prime radical	C₂²×C₆₇	#361
cosocle	C₄×C₁₆×C₆₇	C₂²
hypercenter	#1422	D₆₇
hypocenter	C₆₇	#7454
LCS	G ▷ C₂×C₁₆×C₆₇ ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1422

Complements

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

N	H ≅ G/N	type
45. C₆₇	#7454	semidirect

Maximal subgroups

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

subgroup	index
#22562488	2
#22558285	2
C₆₇×#1422	2
#7454	67