Subgroups — C₆₇×#13731 #30692982

Order 17152.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2252	C₂	1	2	yes	maximal
2	C₆₇×#1423	C₂	1	4	yes	maximal
3	C₄×C₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#500×C₆₇	—	2	2	no	—
5	C₂×C₃₂×C₆₇	C₂²	1	2	yes	—
6	#366×C₆₇	—	2	2	no	—
7	#366×C₆₇	C₂²	1	4	yes	—
8	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	C₄⋊Q₈×C₆₇	—	4	2	no	—
10	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
11	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
12	#107×C₆₇	—	2	4	no	—
13	#107×C₆₇	—	4	2	no	—
14	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
15	C₃₂×C₆₇	Q₈	1	4	yes	complemented
16	C₂×Q₈×C₆₇	—	8	2	no	—
17	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
18	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
19	C₄⋊C₄×C₆₇	—	4	4	no	—
20	C₄⋊C₄×C₆₇	—	8	2	no	—
21	C₄²×C₆₇	D₈	1	1	yes	characteristic
22	C₁₆×C₆₇	Q₁₆	1	2	yes	—
23	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
24	Q₈×C₆₇	—	16	4	no	—
25	C₂×C₄×C₆₇	—	8	4	no	—
26	C₂×C₄×C₆₇	D₁₆	1	1	yes	characteristic
27	C₂×C₄×C₆₇	D₁₆	1	1	yes	characteristic
28	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
29	C₈×C₆₇	Q₃₂	1	2	yes	—
30	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
31	C₈×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
32	C₂²×C₆₇	C₂×D₁₆	1	1	yes	characteristic
33	C₄×C₆₇	—	16	4	no	—
34	C₄×C₆₇	D₃₂	1	2	yes	—
35	C₄×C₆₇	Q₆₄	1	2	yes	—
36	C₄×C₆₇	#500	1	1	yes	characteristic
37	C₄×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
38	#13731	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂×C₆₇	#2252	1	1	yes	characteristic
40	C₂×C₆₇	C₂×D₃₂	1	1	yes	characteristic
41	C₂×C₆₇	C₂×Q₆₄	1	1	yes	characteristic
42	#2252	C₂×C₆₇	1	2	yes	—
43	#1423	C₂×C₆₇	1	4	yes	—
44	C₄×C₃₂	C₂×C₆₇	1	1	yes	characteristic
45	C₆₇	#13731	1	1	yes	characteristic, direct product factor, Sylow
46	#500	—	2	2	no	—
47	C₂×C₃₂	C₂²×C₆₇	1	2	yes	—
48	#366	—	2	2	no	—
49	#366	C₂²×C₆₇	1	4	yes	—
50	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
51	C₄⋊Q₈	—	4	2	no	—
52	C₂×C₁₆	D₄×C₆₇	1	1	yes	characteristic
53	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
54	#107	—	2	4	no	—
55	#107	—	4	2	no	—
56	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
57	C₃₂	Q₈×C₆₇	1	4	yes	complemented
58	C₂×Q₈	—	8	2	no	—
59	C₂×C₈	D₈×C₆₇	1	1	yes	characteristic
60	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
61	C₄⋊C₄	—	4	4	no	—
62	C₄⋊C₄	—	8	2	no	—
63	C₄²	D₈×C₆₇	1	1	yes	characteristic
64	C₁₆	Q₁₆×C₆₇	1	2	yes	—
65	C₁₆	C₂×Q₈×C₆₇	1	2	yes	—
66	Q₈	—	16	4	no	—
67	C₂×C₄	—	8	4	no	—
68	C₂×C₄	D₁₆×C₆₇	1	1	yes	characteristic
69	C₂×C₄	D₁₆×C₆₇	1	1	yes	characteristic
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	—	16	4	no	—
76	C₄	D₃₂×C₆₇	1	2	yes	—
77	C₄	Q₆₄×C₆₇	1	2	yes	—
78	C₄	#500×C₆₇	1	1	yes	characteristic
79	C₄	C₂×Q₃₂×C₆₇	1	1	yes	characteristic
80	C₂	C₆₇×#2252	1	1	yes	characteristic
81	C₂	C₂×D₃₂×C₆₇	1	1	yes	characteristic
82	C₂	C₂×Q₆₄×C₆₇	1	1	yes	characteristic

	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
15. C₃₂×C₆₇	Q₈	semidirect
38. #13731	C₆₇	direct (N × H)
45. C₆₇	#13731	direct (N × H)
57. C₃₂	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2252	2
C₆₇×#1423	2
C₄×C₃₂×C₆₇	2
#13731	67

Subgroups — C67×#13731 #30692982

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#13731 #30692982