Subgroups — #30770572

Order 17152.

238 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562491	C₂	1	2	yes	maximal
2	#22562484	C₂	1	2	yes	maximal
3	#22559819	C₂	1	2	yes	maximal
4	C₈×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#16863801	C₂²	1	4	yes	—
6	#16863747	—	2	4	no	—
7	#16863224	—	2	4	no	—
8	#16863222	—	2	4	no	—
9	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₈²×C₆₇	C₂²	1	1	yes	characteristic
12	#14524043	—	2	4	no	—
13	#14523977	—	2	4	no	—
14	#14523977	—	2	4	no	—
15	#14523968	—	4	8	no	—
16	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
17	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
18	C₄×C₈×C₆₇	D₄	1	2	yes	—
19	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
20	C₂×Dic₁₃₄	—	4	4	no	—
21	C₄⋊Dic₆₇	—	4	4	no	—
22	C₄⋊Dic₆₇	—	4	4	no	—
23	C₂×C₈×C₆₇	D₈	1	4	yes	—
24	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
25	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
26	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
27	C₁₆×C₆₇	Q₁₆	1	4	yes	—
28	C₁₆×C₆₇	Q₁₆	1	4	yes	—
29	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
30	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
31	C₂×Dic₆₇	—	8	4	no	—
32	Dic₁₃₄	—	8	8	no	—
33	C₈×C₆₇	—	2	4	no	—
34	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
35	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
36	C₂²×C₆₇	#493	1	1	yes	characteristic
37	C₄×C₆₇	#494	1	1	yes	characteristic
38	C₄×C₆₇	#494	1	1	yes	characteristic
39	C₄×C₆₇	#508	1	4	yes	—
40	Dic₆₇	—	16	4	no	—
41	#12522	—	67	1	no	maximal, Sylow
42	C₂×C₆₇	#1713	1	1	yes	characteristic
43	C₂×C₆₇	#2249	1	2	yes	—
44	#2254	—	67	2	no	—
45	#2252	—	67	2	no	—
46	#1710	—	67	2	no	—
47	C₈×C₁₆	D₆₇	1	1	yes	characteristic
48	C₆₇	#12522	1	1	yes	characteristic, complemented, Sylow
49	#500	—	67	4	no	—
50	#487	—	134	4	no	—
51	#367	—	134	4	no	—
52	#366	—	134	4	no	—
53	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
54	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
55	C₈²	C₂×D₆₇	1	1	yes	characteristic
56	C₄⋊Q₈	—	134	4	no	—
57	C₂×C₁₆	D₂₆₈	1	2	yes	—
58	C₂×C₁₆	D₂₆₈	1	2	yes	—
59	#107	—	134	4	no	—
60	#107	—	134	4	no	—
61	#103	—	268	8	no	—
62	C₄×C₈	D₂₆₈	1	2	yes	—
63	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
64	C₂×Q₈	—	268	4	no	—
65	C₂×C₈	D₅₃₆	1	4	yes	—
66	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
67	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
68	C₄⋊C₄	—	268	4	no	—
69	C₄⋊C₄	—	268	4	no	—
70	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
71	C₁₆	Dic₂₆₈	1	4	yes	—
72	C₁₆	Dic₂₆₈	1	4	yes	—
73	Q₈	—	536	8	no	—
74	C₂×C₄	—	536	4	no	—
75	C₂×C₄	#14524040	1	1	yes	characteristic
76	C₂×C₄	C₂×D₅₃₆	1	2	yes	—
77	C₈	—	2	4	no	—
78	C₈	C₂×Dic₂₆₈	1	2	yes	—
79	C₈	C₂×Dic₂₆₈	1	2	yes	—
80	C₂²	#16863773	1	1	yes	characteristic
81	C₄	—	1072	4	no	—
82	C₄	#16863776	1	1	yes	characteristic
83	C₄	#16863776	1	1	yes	characteristic
84	C₄	#16863828	1	4	yes	—
85	C₂	#22559827	1	1	yes	characteristic
86	C₂	#22562470	1	2	yes	—

	subgroup	G/H
center	C₂²	#16863773
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₈×C₁₆×C₆₇	C₂
socle, prime radical	C₂²×C₆₇	#493
hypercenter	C₈×C₁₆	D₆₇
hypocenter	C₆₇	#12522
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
48. C₆₇	#12522	semidirect

Maximal subgroups

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

subgroup	index
#22562491	2
#22562484	2
#22559819	2
C₈×C₁₆×C₆₇	2
#12522	67