Subgroups — C₆₇×#7457 #30686708

Order 17152.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#1398	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1369	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1368	C₂	1	1	yes	maximal, characteristic
4	M₆₄×C₆₇	C₄	1	1	yes	characteristic
5	M₆₄×C₆₇	C₄	1	1	yes	characteristic
6	#364×C₆₇	—	2	1	no	—
7	#346×C₆₇	C₂²	1	1	yes	characteristic
8	#334×C₆₇	—	2	1	no	—
9	M₃₂×C₆₇	—	4	1	no	—
10	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
11	C₂×C₁₆×C₆₇	Q₈	1	1	yes	characteristic
12	C₄⋊C₈×C₆₇	—	4	1	no	—
13	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
14	C₃₂×C₆₇	—	2	1	no	—
15	C₃₂×C₆₇	C₈	1	2	yes	complemented
16	C₂×C₈×C₆₇	—	8	1	no	—
17	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
18	C₂×C₈×C₆₇	SD₁₆	1	1	yes	characteristic
19	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
20	C₁₆×C₆₇	—	2	1	no	—
21	C₁₆×C₆₇	—	8	1	no	—
22	C₁₆×C₆₇	C₂×C₈	1	1	yes	characteristic
23	C₁₆×C₆₇	M₁₆	1	1	yes	characteristic
24	C₂×C₄×C₆₇	—	2	1	no	—
25	C₂×C₄×C₆₇	#106	1	1	yes	characteristic
26	C₈×C₆₇	—	2	1	no	—
27	C₈×C₆₇	—	16	1	no	—
28	C₈×C₆₇	C₄⋊C₈	1	1	yes	characteristic
29	C₈×C₆₇	#108	1	1	yes	characteristic
30	C₂²×C₆₇	#365	1	1	yes	characteristic
31	C₄×C₆₇	—	4	1	no	—
32	C₄×C₆₇	#334	1	1	yes	characteristic
33	C₄×C₆₇	#365	1	1	yes	characteristic
34	#7457	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂×C₆₇	—	2	1	no	—
36	C₂×C₆₇	#1368	1	1	yes	characteristic
37	#1398	C₂×C₆₇	1	1	yes	characteristic
38	#1369	C₂×C₆₇	1	1	yes	characteristic
39	#1368	C₂×C₆₇	1	1	yes	characteristic
40	C₆₇	#7457	1	1	yes	characteristic, direct product factor, Sylow
41	M₆₄	C₄×C₆₇	1	1	yes	characteristic
42	M₆₄	C₄×C₆₇	1	1	yes	characteristic
43	#364	—	2	1	no	—
44	#346	C₂²×C₆₇	1	1	yes	characteristic
45	#334	—	2	1	no	—
46	M₃₂	—	4	1	no	—
47	C₂×C₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
48	C₂×C₁₆	Q₈×C₆₇	1	1	yes	characteristic
49	C₄⋊C₈	—	4	1	no	—
50	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
51	C₃₂	—	2	1	no	—
52	C₃₂	C₈×C₆₇	1	2	yes	complemented
53	C₂×C₈	—	8	1	no	—
54	C₂×C₈	C₄⋊C₄×C₆₇	1	1	yes	characteristic
55	C₂×C₈	SD₁₆×C₆₇	1	1	yes	characteristic
56	C₄²	SD₁₆×C₆₇	1	1	yes	characteristic
57	C₁₆	—	2	1	no	—
58	C₁₆	—	8	1	no	—
59	C₁₆	C₂×C₈×C₆₇	1	1	yes	characteristic
60	C₁₆	M₁₆×C₆₇	1	1	yes	characteristic
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	#106×C₆₇	1	1	yes	characteristic
63	C₈	—	2	1	no	—
64	C₈	—	16	1	no	—
65	C₈	C₄⋊C₈×C₆₇	1	1	yes	characteristic
66	C₈	#108×C₆₇	1	1	yes	characteristic
67	C₂²	#365×C₆₇	1	1	yes	characteristic
68	C₄	—	4	1	no	—
69	C₄	#334×C₆₇	1	1	yes	characteristic
70	C₄	#365×C₆₇	1	1	yes	characteristic
71	C₂	—	2	1	no	—
72	C₂	C₆₇×#1368	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	#1368
commutator	C₁₆	C₂×C₈×C₆₇
Frattini, cosocle	#346	C₂²×C₆₇
prime radical	C₂²×C₆₇	#365
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂×C₄×C₆₇ ◁ C₄×C₈×C₆₇ ◁ #346×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₆₇	C₈	semidirect
34. #7457	C₆₇	direct (N × H)
40. C₆₇	#7457	direct (N × H)
52. C₃₂	C₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#1398	2
C₆₇×#1369	2
C₆₇×#1368	2
#7457	67

Subgroups — C67×#7457 #30686708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#7457 #30686708