Subgroups — C₂²×D₃₂×C₆₇ #30713226

Order 17152.

542 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×D₃₂×C₆₇	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₈×C₆₇	—	2	2	no	—
5	C₂×D₁₆×C₆₇	—	2	6	no	—
6	C₂×D₁₆×C₆₇	C₂²	1	12	yes	complemented
7	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
8	D₃₂×C₆₇	C₂²	1	16	yes	direct product factor
9	C₂×C₃₂×C₆₇	C₂²	1	6	yes	complemented
10	C₂²×D₄×C₆₇	—	4	2	no	—
11	C₂×D₈×C₆₇	—	2	12	no	—
12	C₂×D₈×C₆₇	—	4	6	no	—
13	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
14	D₁₆×C₆₇	—	2	12	no	—
15	D₁₆×C₆₇	C₂³	1	8	yes	complemented
16	C₂×C₁₆×C₆₇	D₄	1	3	yes	—
17	C₂×C₁₆×C₆₇	C₂³	1	3	yes	—
18	C₃₂×C₆₇	C₂³	1	4	yes	complemented
19	C₂⁴×C₆₇	—	8	2	no	—
20	C₂×D₄×C₆₇	—	4	12	no	—
21	C₂×D₄×C₆₇	—	8	6	no	—
22	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
23	D₈×C₆₇	—	2	8	no	—
24	D₈×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	—	8	12	no	—
30	C₂³×C₆₇	—	16	8	no	—
31	C₂³×C₆₇	D₁₆	1	1	yes	characteristic
32	D₄×C₆₇	—	4	8	no	—
33	D₄×C₆₇	—	8	12	no	—
34	C₂×C₄×C₆₇	D₁₆	1	3	yes	—
35	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
36	C₈×C₆₇	C₂×D₈	1	3	yes	—
37	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
38	C₂²×C₆₇	—	8	8	no	—
39	C₂²×C₆₇	—	16	24	no	—
40	C₂²×C₆₇	D₃₂	1	4	yes	direct product factor
41	C₂²×C₆₇	C₂×D₁₆	1	3	yes	—
42	C₄×C₆₇	C₂×D₁₆	1	3	yes	—
43	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
44	C₂²×D₃₂	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂×C₆₇	—	16	8	no	—
46	C₂×C₆₇	C₂×D₃₂	1	6	yes	direct product factor
47	C₂×C₆₇	C₂²×D₁₆	1	1	yes	characteristic
48	C₂²×D₁₆	C₂×C₆₇	1	2	yes	complemented
49	C₂×D₃₂	C₂×C₆₇	1	12	yes	direct product factor
50	C₂²×C₃₂	C₂×C₆₇	1	1	yes	characteristic, complemented
51	C₆₇	C₂²×D₃₂	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²×D₈	—	2	2	no	—
53	C₂×D₁₆	—	2	6	no	—
54	C₂×D₁₆	C₂²×C₆₇	1	12	yes	complemented
55	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
56	D₃₂	C₂²×C₆₇	1	16	yes	direct product factor
57	C₂×C₃₂	C₂²×C₆₇	1	6	yes	complemented
58	C₂²×D₄	—	4	2	no	—
59	C₂×D₈	—	2	12	no	—
60	C₂×D₈	—	4	6	no	—
61	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
62	D₁₆	—	2	12	no	—
63	D₁₆	C₂³×C₆₇	1	8	yes	complemented
64	C₂×C₁₆	D₄×C₆₇	1	3	yes	—
65	C₂×C₁₆	C₂³×C₆₇	1	3	yes	—
66	C₃₂	C₂³×C₆₇	1	4	yes	complemented
67	C₂⁴	—	8	2	no	—
68	C₂×D₄	—	4	12	no	—
69	C₂×D₄	—	8	6	no	—
70	C₂²×C₄	D₈×C₆₇	1	1	yes	characteristic
71	D₈	—	2	8	no	—
72	D₈	—	4	12	no	—
73	C₂×C₈	D₈×C₆₇	1	3	yes	—
74	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
75	C₁₆	C₂×D₄×C₆₇	1	3	yes	—
76	C₁₆	C₂⁴×C₆₇	1	1	yes	characteristic
77	C₂³	—	8	12	no	—
78	C₂³	—	16	8	no	—
79	C₂³	D₁₆×C₆₇	1	1	yes	characteristic
80	D₄	—	4	8	no	—
81	D₄	—	8	12	no	—
82	C₂×C₄	D₁₆×C₆₇	1	3	yes	—
83	C₂×C₄	C₂×D₈×C₆₇	1	3	yes	—
84	C₈	C₂×D₈×C₆₇	1	3	yes	—
85	C₈	C₂²×D₄×C₆₇	1	1	yes	characteristic
86	C₂²	—	8	8	no	—
87	C₂²	—	16	24	no	—
88	C₂²	D₃₂×C₆₇	1	4	yes	direct product factor
89	C₂²	C₂×D₁₆×C₆₇	1	3	yes	—
90	C₄	C₂×D₁₆×C₆₇	1	3	yes	—
91	C₄	C₂²×D₈×C₆₇	1	1	yes	characteristic
92	C₂	—	16	8	no	—
93	C₂	C₂×D₃₂×C₆₇	1	6	yes	direct product factor
94	C₂	C₂²×D₁₆×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	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
1. C₂²×D₁₆×C₆₇	C₂	semidirect
2. C₂×D₃₂×C₆₇	C₂	direct (N × H)
3. C₂²×C₃₂×C₆₇	C₂	semidirect
6. C₂×D₁₆×C₆₇	C₂²	semidirect
8. D₃₂×C₆₇	C₂²	direct (N × H)
9. C₂×C₃₂×C₆₇	C₂²	semidirect
15. D₁₆×C₆₇	C₂³	semidirect
18. C₃₂×C₆₇	C₂³	semidirect
40. C₂²×C₆₇	D₃₂	direct (N × H)
44. C₂²×D₃₂	C₆₇	direct (N × H)
46. C₂×C₆₇	C₂×D₃₂	direct (N × H)
48. C₂²×D₁₆	C₂×C₆₇	semidirect
49. C₂×D₃₂	C₂×C₆₇	direct (N × H)
50. C₂²×C₃₂	C₂×C₆₇	semidirect
51. C₆₇	C₂²×D₃₂	direct (N × H)
54. C₂×D₁₆	C₂²×C₆₇	semidirect
56. D₃₂	C₂²×C₆₇	direct (N × H)
57. C₂×C₃₂	C₂²×C₆₇	semidirect
63. D₁₆	C₂³×C₆₇	semidirect
66. C₃₂	C₂³×C₆₇	semidirect
88. C₂²	D₃₂×C₆₇	direct (N × H)
93. C₂	C₂×D₃₂×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×D₁₆×C₆₇	2
C₂×D₃₂×C₆₇	2
C₂²×C₃₂×C₆₇	2
C₂²×D₃₂	67

Subgroups — C22×D32×C67 #30713226

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₃₂×C₆₇ #30713226