Subgroups — #30743503

Order 17152.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊(C₆₇⋊C₃₂)	C₂	1	1	yes	maximal, characteristic
2	C₁₆×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₆₇×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₂×C₆₇⋊C₃₂	—	2	1	no	—
5	C₄×C₆₇⋊C₁₆	C₄	1	1	yes	characteristic
6	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
7	C₂×C₃₂×C₆₇	—	2	1	no	—
8	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₆₇⋊C₁₆	—	2	1	no	—
10	C₄×C₆₇⋊C₈	C₈	1	1	yes	characteristic
11	C₄×C₆₇⋊C₈	C₈	1	1	yes	characteristic
12	C₆₇⋊C₃₂	—	4	1	no	—
13	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₆₇	Q₈	1	1	yes	characteristic
15	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
16	C₃₂×C₆₇	—	4	1	no	—
17	C₂×C₆₇⋊C₈	—	2	1	no	—
18	C₂×C₆₇⋊C₈	C₁₆	1	1	yes	characteristic
19	C₂×C₆₇⋊C₈	C₁₆	1	1	yes	characteristic
20	C₆₇⋊C₁₆	—	2	2	no	—
21	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
22	C₂×C₈×C₆₇	M₁₆	1	1	yes	characteristic
23	C₄²×C₆₇	C₂×C₈	1	1	yes	characteristic
24	C₁₆×C₆₇	—	2	1	no	—
25	C₁₆×C₆₇	D₈	1	1	yes	characteristic
26	C₁₆×C₆₇	Q₁₆	1	1	yes	characteristic
27	C₂×C₄×C₆₇	C₄⋊C₈	1	1	yes	characteristic
28	C₂×C₄×C₆₇	C₂×C₁₆	1	1	yes	characteristic
29	C₂×C₄×C₆₇	M₃₂	1	1	yes	characteristic
30	C₈×C₆₇	—	2	1	no	—
31	C₈×C₆₇	#107	1	1	yes	characteristic
32	C₈×C₆₇	#108	1	1	yes	characteristic
33	C₆₇⋊C₈	—	2	2	no	—
34	C₆₇⋊C₈	—	2	1	no	—
35	C₆₇⋊C₈	C₃₂	1	2	yes	complemented
36	C₂²×C₆₇	C₄⋊C₁₆	1	1	yes	characteristic
37	C₄×C₆₇	—	2	1	no	—
38	C₄×C₆₇	#335	1	1	yes	characteristic
39	C₄×C₆₇	#364	1	1	yes	characteristic
40	C₄×C₆₇	C₂×C₃₂	1	1	yes	characteristic
41	C₄×C₆₇	M₆₄	1	1	yes	characteristic
42	#7458	—	67	1	no	maximal, Sylow
43	C₂×C₆₇	#1370	1	1	yes	characteristic
44	C₂×C₆₇	C₄⋊C₃₂	1	1	yes	characteristic
45	C₂×C₆₇	#1422	1	1	yes	characteristic
46	C₄⋊C₃₂	—	67	1	no	—
47	C₄⋊C₃₂	D₆₇	1	1	yes	characteristic
48	C₈×C₁₆	—	67	1	no	—
49	C₆₇	#7458	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃₂	—	2	1	no	—
51	C₂×C₃₂	—	134	1	no	—
52	C₄×C₁₆	—	67	1	no	—
53	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
54	C₈²	—	67	1	no	—
55	C₂×C₁₆	—	134	1	no	—
56	C₂×C₁₆	Dic₁₃₄	1	1	yes	characteristic
57	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
58	C₄×C₈	—	67	1	no	—
59	C₄×C₈	—	67	1	no	—
60	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
61	C₃₂	—	4	1	no	—
62	C₃₂	—	268	1	no	—
63	C₂×C₈	—	67	1	no	—
64	C₂×C₈	—	67	1	no	—
65	C₂×C₈	—	134	1	no	—
66	C₂×C₈	Dic₆₇⋊C₄	1	1	yes	characteristic
67	C₂×C₈	#11761611	1	1	yes	characteristic
68	C₄²	C₈×D₆₇	1	1	yes	characteristic
69	C₁₆	—	2	1	no	—
70	C₁₆	—	134	2	no	—
71	C₁₆	#11761613	1	1	yes	characteristic
72	C₁₆	#11761618	1	1	yes	characteristic
73	C₂×C₄	Dic₆₇⋊C₈	1	1	yes	characteristic
74	C₂×C₄	C₁₆×D₆₇	1	1	yes	characteristic
75	C₂×C₄	#14523982	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	67	2	no	—
78	C₈	—	134	2	no	—
79	C₈	—	134	1	no	—
80	C₈	#14523976	1	1	yes	characteristic
81	C₈	#14523978	1	1	yes	characteristic
82	C₂²	Dic₆₇⋊C₁₆	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	#16863148	1	1	yes	characteristic
85	C₄	#16863216	1	1	yes	characteristic
86	C₄	C₃₂×D₆₇	1	1	yes	characteristic
87	C₄	#16863229	1	1	yes	characteristic
88	C₂	#22558278	1	1	yes	characteristic
89	C₂	Dic₆₇⋊C₃₂	1	1	yes	characteristic
90	C₂	#22558406	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	C₂²⋊D₆₇
commutator	C₄×C₆₇	C₂×C₃₂
Frattini	C₄×C₁₆	C₂×D₆₇
Fitting	C₆₇×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₆₇	C₄⋊C₁₆
cosocle	C₄×C₁₆×C₆₇	C₂²
hypercenter	C₄⋊C₃₂	D₆₇
hypocenter	C₆₇	#7458
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₁₆ ◁ C₄⋊C₃₂

Complements

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

N	H ≅ G/N	type
35. C₆₇⋊C₈	C₃₂	semidirect
49. C₆₇	#7458	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊(C₆₇⋊C₃₂)	2
C₁₆×C₆₇⋊C₈	2
C₆₇×C₄⋊C₃₂	2
#7458	67