Subgroups — #30743492

Order 17152.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562491	C₂	1	1	yes	maximal, characteristic
2	#22558407	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1373	C₂	1	1	yes	maximal, characteristic
4	#16863801	—	2	1	no	—
5	#16863230	—	2	1	no	—
6	#16863224	—	2	1	no	—
7	#16863224	C₄	1	2	yes	—
8	#364×C₆₇	—	2	1	no	—
9	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	#14524043	—	4	1	no	—
11	#14523977	—	2	2	no	—
12	#14523977	—	4	1	no	—
13	C₆₇⋊C₃₂	—	4	1	no	—
14	M₃₂×C₆₇	—	4	1	no	—
15	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
17	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
18	C₂×Dic₁₃₄	—	8	1	no	—
19	C₄⋊Dic₆₇	—	4	2	no	—
20	C₄⋊Dic₆₇	—	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	—	8	1	no	—
26	C₁₆×C₆₇	SD₁₆	1	1	yes	characteristic
27	C₁₆×C₆₇	Q₁₆	1	1	yes	characteristic
28	C₂×C₄×C₆₇	—	2	1	no	—
29	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
30	C₂×Dic₆₇	—	8	2	no	—
31	Dic₁₃₄	—	16	2	no	—
32	C₈×C₆₇	#101	1	1	yes	characteristic
33	C₈×C₆₇	#103	1	1	yes	characteristic
34	C₈×C₆₇	SD₃₂	1	1	yes	characteristic
35	C₈×C₆₇	Q₃₂	1	1	yes	characteristic
36	C₂²×C₆₇	#361	1	1	yes	characteristic
37	C₄×C₆₇	—	2	2	no	—
38	C₄×C₆₇	#332	1	1	yes	characteristic
39	C₄×C₆₇	#358	1	1	yes	characteristic
40	Dic₆₇	—	16	2	no	—
41	#7453	—	67	1	no	maximal, Sylow
42	C₂×C₆₇	—	2	1	no	—
43	C₂×C₆₇	#1365	1	1	yes	characteristic
44	#2254	—	67	1	no	—
45	#1422	—	67	1	no	—
46	#1373	D₆₇	1	1	yes	characteristic
47	C₆₇	#7453	1	1	yes	characteristic, complemented, Sylow
48	#500	—	134	1	no	—
49	M₆₄	—	134	1	no	—
50	#367	—	67	2	no	—
51	#367	—	134	1	no	—
52	#364	—	2	1	no	—
53	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
54	C₄⋊Q₈	—	268	1	no	—
55	M₃₂	—	4	1	no	—
56	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
57	C₂×C₁₆	D₂₆₈	1	1	yes	characteristic
58	#107	—	268	1	no	—
59	#107	—	134	2	no	—
60	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
61	C₃₂	—	268	1	no	—
62	C₂×Q₈	—	536	1	no	—
63	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
64	C₂×C₈	#11761613	1	1	yes	characteristic
65	C₄⋊C₄	—	268	2	no	—
66	C₄⋊C₄	—	536	1	no	—
67	C₄²	#11761615	1	1	yes	characteristic
68	C₁₆	—	2	1	no	—
69	C₁₆	—	8	1	no	—
70	C₁₆	#11761617	1	1	yes	characteristic
71	C₁₆	Dic₂₆₈	1	1	yes	characteristic
72	Q₈	—	1072	2	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	536	2	no	—
75	C₂×C₄	#14523963	1	1	yes	characteristic
76	C₈	#14523961	1	1	yes	characteristic
77	C₈	#14523968	1	1	yes	characteristic
78	C₈	#14523987	1	1	yes	characteristic
79	C₈	#14523989	1	1	yes	characteristic
80	C₂²	#16863208	1	1	yes	characteristic
81	C₄	—	2	2	no	—
82	C₄	—	1072	2	no	—
83	C₄	#16863143	1	1	yes	characteristic
84	C₄	#16863199	1	1	yes	characteristic
85	C₂	—	2	1	no	—
86	C₂	#22558268	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#22558268
commutator	C₂×C₁₆×C₆₇	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₆₇
Fitting	C₆₇×#1373	C₂
socle	C₂×C₆₇	#1365
prime radical	C₂²×C₆₇	#361
cosocle	C₄×C₁₆×C₆₇	C₂²
hypercenter	#1373	D₆₇
hypocenter	C₆₇	#7453
LCS	G ▷ C₂×C₁₆×C₆₇ ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1373

Complements

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

N	H ≅ G/N	type
47. C₆₇	#7453	semidirect

Maximal subgroups

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

subgroup	index
#22562491	2
#22558407	2
C₆₇×#1373	2
#7453	67