Subgroups — C₆₇×#13732 #30692983

Order 17152.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2253	C₂	1	2	yes	maximal
2	C₆₇×#1424	C₂	1	2	yes	maximal
3	C₆₇×#1423	C₂	1	2	yes	maximal
4	C₄×C₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#499×C₆₇	—	2	2	no	—
6	C₂×C₃₂×C₆₇	C₂²	1	1	yes	characteristic
7	C₂×C₃₂×C₆₇	C₂²	1	1	yes	characteristic
8	#367×C₆₇	—	2	2	no	—
9	#366×C₆₇	C₂²	1	4	yes	—
10	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	#125×C₆₇	—	4	2	no	—
12	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
14	#107×C₆₇	—	2	4	no	—
15	#106×C₆₇	—	4	2	no	—
16	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
17	C₃₂×C₆₇	Q₈	1	2	yes	—
18	C₃₂×C₆₇	Q₈	1	2	yes	—
19	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
20	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
21	C₄⋊C₄×C₆₇	—	4	4	no	—
22	C₄⋊C₄×C₆₇	—	8	4	no	—
23	C₄²×C₆₇	D₈	1	1	yes	characteristic
24	C₁₆×C₆₇	Q₁₆	1	2	yes	—
25	C₁₆×C₆₇	C₂×Q₈	1	1	yes	characteristic
26	C₁₆×C₆₇	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₄×C₆₇	—	8	4	no	—
28	C₂×C₄×C₆₇	D₁₆	1	2	yes	—
29	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
30	C₈×C₆₇	Q₃₂	1	2	yes	—
31	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
32	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
33	C₂²×C₆₇	C₂×D₁₆	1	1	yes	characteristic
34	C₄×C₆₇	—	2	2	no	—
35	C₄×C₆₇	—	16	4	no	—
36	C₄×C₆₇	#500	1	1	yes	characteristic
37	C₄×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
38	#13732	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂×C₆₇	#2252	1	1	yes	characteristic
40	C₂×C₆₇	#2262	1	2	yes	—
41	#2253	C₂×C₆₇	1	2	yes	—
42	#1424	C₂×C₆₇	1	2	yes	—
43	#1423	C₂×C₆₇	1	2	yes	—
44	C₄×C₃₂	C₂×C₆₇	1	1	yes	characteristic
45	C₆₇	#13732	1	1	yes	characteristic, direct product factor, Sylow
46	#499	—	2	2	no	—
47	C₂×C₃₂	C₂²×C₆₇	1	1	yes	characteristic
48	C₂×C₃₂	C₂²×C₆₇	1	1	yes	characteristic
49	#367	—	2	2	no	—
50	#366	C₂²×C₆₇	1	4	yes	—
51	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
52	#125	—	4	2	no	—
53	C₂×C₁₆	D₄×C₆₇	1	1	yes	characteristic
54	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
55	#107	—	2	4	no	—
56	#106	—	4	2	no	—
57	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
58	C₃₂	Q₈×C₆₇	1	2	yes	—
59	C₃₂	Q₈×C₆₇	1	2	yes	—
60	C₂×C₈	D₈×C₆₇	1	1	yes	characteristic
61	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
62	C₄⋊C₄	—	4	4	no	—
63	C₄⋊C₄	—	8	4	no	—
64	C₄²	D₈×C₆₇	1	1	yes	characteristic
65	C₁₆	Q₁₆×C₆₇	1	2	yes	—
66	C₁₆	C₂×Q₈×C₆₇	1	1	yes	characteristic
67	C₁₆	C₂×Q₈×C₆₇	1	1	yes	characteristic
68	C₂×C₄	—	8	4	no	—
69	C₂×C₄	D₁₆×C₆₇	1	2	yes	—
70	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
71	C₈	Q₃₂×C₆₇	1	2	yes	—
72	C₈	C₄⋊Q₈×C₆₇	1	1	yes	characteristic
73	C₈	C₂×Q₁₆×C₆₇	1	1	yes	characteristic
74	C₂²	C₂×D₁₆×C₆₇	1	1	yes	characteristic
75	C₄	—	2	2	no	—
76	C₄	—	16	4	no	—
77	C₄	#500×C₆₇	1	1	yes	characteristic
78	C₄	C₂×Q₃₂×C₆₇	1	1	yes	characteristic
79	C₂	C₆₇×#2252	1	1	yes	characteristic
80	C₂	C₆₇×#2262	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²×C₆₇	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₆₇
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ C₄×C₈×C₆₇ ◁ C₄×C₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
38. #13732	C₆₇	direct (N × H)
45. C₆₇	#13732	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#2253	2
C₆₇×#1424	2
C₆₇×#1423	2
C₄×C₃₂×C₆₇	2
#13732	67

Subgroups — C67×#13732 #30692983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#13732 #30692983