Subgroups — #30743486

Order 17152.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562484	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(C₆₇⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1371	C₂	1	1	yes	maximal, characteristic
4	#16863801	—	2	1	no	—
5	C₂×C₆₇⋊C₃₂	—	2	1	no	—
6	#16863222	—	2	1	no	—
7	#16863222	C₄	1	2	yes	—
8	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	#335×C₆₇	—	2	1	no	—
10	#14524043	—	4	1	no	—
11	#14523977	—	2	2	no	—
12	#14523977	—	4	1	no	—
13	C₆₇⋊C₃₂	—	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	—	4	1	no	—
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	—	8	1	no	—
22	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
24	C₄²×C₆₇	D₈	1	1	yes	characteristic
25	C₁₆×C₆₇	—	2	1	no	—
26	C₁₆×C₆₇	SD₁₆	1	1	yes	characteristic
27	C₁₆×C₆₇	Q₁₆	1	1	yes	characteristic
28	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
29	C₂×C₄×C₆₇	D₁₆	1	1	yes	characteristic
30	C₂×C₄×C₆₇	SD₃₂	1	1	yes	characteristic
31	C₂×Dic₆₇	—	8	2	no	—
32	Dic₁₃₄	—	16	2	no	—
33	C₈×C₆₇	—	2	1	no	—
34	C₈×C₆₇	—	16	1	no	—
35	C₈×C₆₇	#101	1	1	yes	characteristic
36	C₈×C₆₇	#103	1	1	yes	characteristic
37	C₂²×C₆₇	#357	1	1	yes	characteristic
38	C₄×C₆₇	—	2	1	no	—
39	C₄×C₆₇	#332	1	1	yes	characteristic
40	C₄×C₆₇	#362	1	1	yes	characteristic
41	C₄×C₆₇	SD₆₄	1	1	yes	characteristic
42	C₄×C₆₇	Q₆₄	1	1	yes	characteristic
43	Dic₆₇	—	16	2	no	—
44	#7451	—	67	1	no	maximal, Sylow
45	C₂×C₆₇	#1364	1	1	yes	characteristic
46	C₂×C₆₇	#1416	1	1	yes	characteristic
47	C₂×C₆₇	#1419	1	1	yes	characteristic
48	#2252	—	67	1	no	—
49	C₄⋊C₃₂	—	67	1	no	—
50	#1371	D₆₇	1	1	yes	characteristic
51	C₆₇	#7451	1	1	yes	characteristic, complemented, Sylow
52	#500	—	134	1	no	—
53	C₂×C₃₂	—	134	1	no	—
54	#366	—	67	2	no	—
55	#366	—	134	1	no	—
56	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
57	#335	—	2	1	no	—
58	C₄⋊Q₈	—	268	1	no	—
59	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
60	C₂×C₁₆	D₂₆₈	1	1	yes	characteristic
61	#107	—	268	1	no	—
62	#107	—	134	2	no	—
63	C₄⋊C₈	—	4	1	no	—
64	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
65	C₃₂	—	268	1	no	—
66	C₂×Q₈	—	536	1	no	—
67	C₂×C₈	—	8	1	no	—
68	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
69	C₂×C₈	#11761615	1	1	yes	characteristic
70	C₄⋊C₄	—	268	2	no	—
71	C₄⋊C₄	—	536	1	no	—
72	C₄²	#11761613	1	1	yes	characteristic
73	C₁₆	—	2	1	no	—
74	C₁₆	#11761617	1	1	yes	characteristic
75	C₁₆	Dic₂₆₈	1	1	yes	characteristic
76	Q₈	—	1072	2	no	—
77	C₂×C₄	—	536	2	no	—
78	C₂×C₄	#14523963	1	1	yes	characteristic
79	C₂×C₄	#14523984	1	1	yes	characteristic
80	C₂×C₄	#14523986	1	1	yes	characteristic
81	C₈	—	2	1	no	—
82	C₈	—	16	1	no	—
83	C₈	#14523961	1	1	yes	characteristic
84	C₈	#14523968	1	1	yes	characteristic
85	C₂²	#16863196	1	1	yes	characteristic
86	C₄	—	2	1	no	—
87	C₄	—	1072	2	no	—
88	C₄	#16863143	1	1	yes	characteristic
89	C₄	#16863211	1	1	yes	characteristic
90	C₄	#16863234	1	1	yes	characteristic
91	C₄	#16863236	1	1	yes	characteristic
92	C₂	#22558265	1	1	yes	characteristic
93	C₂	#22558389	1	1	yes	characteristic
94	C₂	#22558398	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16863196
commutator	C₂×C₁₆×C₆₇	C₂×C₄
Frattini	C₄×C₁₆	C₂×D₆₇
Fitting	C₆₇×#1371	C₂
socle, prime radical	C₂²×C₆₇	#357
cosocle	C₄×C₁₆×C₆₇	C₂²
hypercenter	#1371	D₆₇
hypocenter	C₆₇	#7451
LCS	G ▷ C₂×C₁₆×C₆₇ ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1371

Complements

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

N	H ≅ G/N	type
51. C₆₇	#7451	semidirect

Maximal subgroups

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

subgroup	index
#22562484	2
C₄⋊(C₆₇⋊C₃₂)	2
C₆₇×#1371	2
#7451	67