Subgroups — C₂²×Q₆₄×C₆₇ #30713228

Order 17152.

462 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₃₂×C₆₇	C₂	1	2	yes	maximal
2	C₂×Q₆₄×C₆₇	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×Q₁₆×C₆₇	—	2	2	no	—
5	C₂×Q₃₂×C₆₇	—	2	6	no	—
6	C₂×Q₃₂×C₆₇	C₂²	1	12	yes	—
7	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
8	Q₆₄×C₆₇	C₂²	1	16	yes	direct product factor
9	C₂×C₃₂×C₆₇	C₂²	1	6	yes	—
10	C₂²×Q₈×C₆₇	—	4	2	no	—
11	C₂×Q₁₆×C₆₇	—	2	12	no	—
12	C₂×Q₁₆×C₆₇	—	4	6	no	—
13	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
14	Q₃₂×C₆₇	—	2	12	no	—
15	Q₃₂×C₆₇	C₂³	1	8	yes	—
16	C₂×C₁₆×C₆₇	D₄	1	3	yes	—
17	C₂×C₁₆×C₆₇	C₂³	1	3	yes	—
18	C₃₂×C₆₇	C₂³	1	4	yes	—
19	C₂×Q₈×C₆₇	—	4	12	no	—
20	C₂×Q₈×C₆₇	—	8	6	no	—
21	C₂²×C₄×C₆₇	—	8	2	no	—
22	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
23	Q₁₆×C₆₇	—	2	8	no	—
24	Q₁₆×C₆₇	—	4	12	no	—
25	C₂×C₈×C₆₇	D₈	1	3	yes	—
26	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
27	C₁₆×C₆₇	C₂×D₄	1	3	yes	—
28	C₁₆×C₆₇	C₂⁴	1	1	yes	characteristic
29	C₂³×C₆₇	D₁₆	1	1	yes	characteristic
30	Q₈×C₆₇	—	4	8	no	—
31	Q₈×C₆₇	—	8	12	no	—
32	C₂×C₄×C₆₇	—	8	12	no	—
33	C₂×C₄×C₆₇	D₁₆	1	3	yes	—
34	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
35	C₈×C₆₇	C₂×D₈	1	3	yes	—
36	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
37	C₂²×C₆₇	Q₆₄	1	4	yes	direct product factor
38	C₂²×C₆₇	C₂×D₁₆	1	3	yes	—
39	C₄×C₆₇	—	8	8	no	—
40	C₄×C₆₇	C₂×D₁₆	1	3	yes	—
41	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
42	C₂²×Q₆₄	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂×C₆₇	C₂×Q₆₄	1	6	yes	direct product factor
44	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
45	C₂²×Q₃₂	C₂×C₆₇	1	2	yes	—
46	C₂×Q₆₄	C₂×C₆₇	1	12	yes	direct product factor
47	C₂²×C₃₂	C₂×C₆₇	1	1	yes	characteristic
48	C₆₇	C₂²×Q₆₄	1	1	yes	characteristic, direct product factor, Sylow
49	C₂²×Q₁₆	—	2	2	no	—
50	C₂×Q₃₂	—	2	6	no	—
51	C₂×Q₃₂	C₂²×C₆₇	1	12	yes	—
52	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
53	Q₆₄	C₂²×C₆₇	1	16	yes	direct product factor
54	C₂×C₃₂	C₂²×C₆₇	1	6	yes	—
55	C₂²×Q₈	—	4	2	no	—
56	C₂×Q₁₆	—	2	12	no	—
57	C₂×Q₁₆	—	4	6	no	—
58	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
59	Q₃₂	—	2	12	no	—
60	Q₃₂	C₂³×C₆₇	1	8	yes	—
61	C₂×C₁₆	D₄×C₆₇	1	3	yes	—
62	C₂×C₁₆	C₂³×C₆₇	1	3	yes	—
63	C₃₂	C₂³×C₆₇	1	4	yes	—
64	C₂×Q₈	—	4	12	no	—
65	C₂×Q₈	—	8	6	no	—
66	C₂²×C₄	—	8	2	no	—
67	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
68	Q₁₆	—	2	8	no	—
69	Q₁₆	—	4	12	no	—
70	C₂×C₈	D₈×C₆₇	1	3	yes	—
71	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
72	C₁₆	C₂×D₄×C₆₇	1	3	yes	—
73	C₁₆	C₂⁴×C₆₇	1	1	yes	characteristic
74	C₂³	D₁₆×C₆₇	1	1	yes	characteristic
75	Q₈	—	4	8	no	—
76	Q₈	—	8	12	no	—
77	C₂×C₄	—	8	12	no	—
78	C₂×C₄	D₁₆×C₆₇	1	3	yes	—
79	C₂×C₄	C₂×D₈×C₆₇	1	3	yes	—
80	C₈	C₂×D₈×C₆₇	1	3	yes	—
81	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
82	C₂²	Q₆₄×C₆₇	1	4	yes	direct product factor
83	C₂²	C₂×D₁₆×C₆₇	1	3	yes	—
84	C₄	—	8	8	no	—
85	C₄	C₂×D₁₆×C₆₇	1	3	yes	—
86	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
87	C₂	C₂×Q₆₄×C₆₇	1	6	yes	direct product factor
88	C₂	C₂²×D₁₆×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁴×C₆₇
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂²×C₈×C₆₇ ◁ C₂²×C₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×Q₆₄×C₆₇	C₂	direct (N × H)
8. Q₆₄×C₆₇	C₂²	direct (N × H)
37. C₂²×C₆₇	Q₆₄	direct (N × H)
42. C₂²×Q₆₄	C₆₇	direct (N × H)
43. C₂×C₆₇	C₂×Q₆₄	direct (N × H)
46. C₂×Q₆₄	C₂×C₆₇	direct (N × H)
48. C₆₇	C₂²×Q₆₄	direct (N × H)
53. Q₆₄	C₂²×C₆₇	direct (N × H)
82. C₂²	Q₆₄×C₆₇	direct (N × H)
87. C₂	C₂×Q₆₄×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₃₂×C₆₇	2
C₂×Q₆₄×C₆₇	2
C₂²×C₃₂×C₆₇	2
C₂²×Q₆₄	67

Subgroups — C22×Q64×C67 #30713228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₆₄×C₆₇ #30713228