Subgroups — C₂²×M₆₄×C₆₇ #30713223

Order 17152.

502 conjugacy classes of subgroups.

The subgroup lattice has 114 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×M₆₄×C₆₇	C₂	1	12	yes	maximal, direct product factor
3	C₂²×C₃₂×C₆₇	C₂	1	2	yes	maximal, complemented
4	C₂³×C₈×C₆₇	C₄	1	1	yes	characteristic
5	C₂²×C₁₆×C₆₇	C₄	1	1	yes	characteristic
6	C₂²×C₁₆×C₆₇	C₄	1	6	yes	—
7	C₂²×C₁₆×C₆₇	C₂²	1	6	yes	—
8	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	M₆₄×C₆₇	C₂²	1	16	yes	direct product factor
10	C₂×C₃₂×C₆₇	C₂²	1	12	yes	complemented
11	C₂³×C₄×C₆₇	C₈	1	1	yes	characteristic
12	C₂²×C₈×C₆₇	C₈	1	1	yes	characteristic
13	C₂²×C₈×C₆₇	C₈	1	6	yes	—
14	C₂²×C₈×C₆₇	C₂×C₄	1	6	yes	—
15	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆×C₆₇	C₂×C₄	1	6	yes	—
17	C₂×C₁₆×C₆₇	C₂×C₄	1	12	yes	—
18	C₂×C₁₆×C₆₇	C₂×C₄	1	3	yes	—
19	C₂×C₁₆×C₆₇	C₂³	1	4	yes	—
20	C₂×C₁₆×C₆₇	C₂³	1	3	yes	—
21	C₃₂×C₆₇	C₂³	1	8	yes	complemented
22	C₂⁴×C₆₇	C₁₆	1	1	yes	characteristic
23	C₂²×C₄×C₆₇	C₁₆	1	6	yes	—
24	C₂²×C₄×C₆₇	C₁₆	1	1	yes	characteristic
25	C₂²×C₄×C₆₇	C₂×C₈	1	6	yes	—
26	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₆₇	C₂×C₈	1	12	yes	—
28	C₂×C₈×C₆₇	C₂×C₈	1	6	yes	—
29	C₂×C₈×C₆₇	C₂×C₈	1	3	yes	—
30	C₂×C₈×C₆₇	C₂²×C₄	1	4	yes	—
31	C₂×C₈×C₆₇	C₂²×C₄	1	3	yes	—
32	C₁₆×C₆₇	C₂²×C₄	1	4	yes	—
33	C₁₆×C₆₇	C₂²×C₄	1	3	yes	—
34	C₁₆×C₆₇	C₂⁴	1	1	yes	characteristic
35	C₂³×C₆₇	—	2	4	no	—
36	C₂³×C₆₇	C₂×C₁₆	1	6	yes	—
37	C₂³×C₆₇	C₂×C₁₆	1	1	yes	characteristic
38	C₂×C₄×C₆₇	C₂×C₁₆	1	12	yes	—
39	C₂×C₄×C₆₇	C₂×C₁₆	1	3	yes	—
40	C₂×C₄×C₆₇	C₂×C₁₆	1	6	yes	—
41	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
42	C₂×C₄×C₆₇	C₂²×C₈	1	3	yes	—
43	C₈×C₆₇	C₂²×C₈	1	4	yes	—
44	C₈×C₆₇	C₂²×C₈	1	3	yes	—
45	C₈×C₆₇	C₂³×C₄	1	1	yes	characteristic
46	C₂²×C₆₇	—	2	12	no	—
47	C₂²×C₆₇	M₆₄	1	4	yes	direct product factor
48	C₂²×C₆₇	C₂²×C₁₆	1	4	yes	—
49	C₂²×C₆₇	C₂²×C₁₆	1	3	yes	—
50	C₄×C₆₇	C₂²×C₁₆	1	3	yes	—
51	C₄×C₆₇	C₂²×C₁₆	1	4	yes	—
52	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
53	C₂²×M₆₄	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂×C₆₇	—	2	4	no	—
55	C₂×C₆₇	C₂×M₆₄	1	6	yes	direct product factor
56	C₂×C₆₇	C₂³×C₁₆	1	1	yes	characteristic
57	C₂³×C₁₆	C₂×C₆₇	1	1	yes	characteristic
58	C₂×M₆₄	C₂×C₆₇	1	12	yes	direct product factor
59	C₂²×C₃₂	C₂×C₆₇	1	2	yes	complemented
60	C₆₇	C₂²×M₆₄	1	1	yes	characteristic, direct product factor, Sylow
61	C₂³×C₈	C₄×C₆₇	1	1	yes	characteristic
62	C₂²×C₁₆	C₄×C₆₇	1	1	yes	characteristic
63	C₂²×C₁₆	C₄×C₆₇	1	6	yes	—
64	C₂²×C₁₆	C₂²×C₆₇	1	6	yes	—
65	C₂²×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
66	M₆₄	C₂²×C₆₇	1	16	yes	direct product factor
67	C₂×C₃₂	C₂²×C₆₇	1	12	yes	complemented
68	C₂³×C₄	C₈×C₆₇	1	1	yes	characteristic
69	C₂²×C₈	C₈×C₆₇	1	6	yes	—
70	C₂²×C₈	C₈×C₆₇	1	1	yes	characteristic
71	C₂²×C₈	C₂×C₄×C₆₇	1	6	yes	—
72	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
73	C₂×C₁₆	C₂×C₄×C₆₇	1	12	yes	—
74	C₂×C₁₆	C₂×C₄×C₆₇	1	6	yes	—
75	C₂×C₁₆	C₂×C₄×C₆₇	1	3	yes	—
76	C₂×C₁₆	C₂³×C₆₇	1	3	yes	—
77	C₂×C₁₆	C₂³×C₆₇	1	4	yes	—
78	C₃₂	C₂³×C₆₇	1	8	yes	complemented
79	C₂⁴	C₁₆×C₆₇	1	1	yes	characteristic
80	C₂²×C₄	C₁₆×C₆₇	1	6	yes	—
81	C₂²×C₄	C₁₆×C₆₇	1	1	yes	characteristic
82	C₂²×C₄	C₂×C₈×C₆₇	1	6	yes	—
83	C₂²×C₄	C₂×C₈×C₆₇	1	1	yes	characteristic
84	C₂×C₈	C₂×C₈×C₆₇	1	12	yes	—
85	C₂×C₈	C₂×C₈×C₆₇	1	6	yes	—
86	C₂×C₈	C₂×C₈×C₆₇	1	3	yes	—
87	C₂×C₈	C₂²×C₄×C₆₇	1	3	yes	—
88	C₂×C₈	C₂²×C₄×C₆₇	1	4	yes	—
89	C₁₆	C₂²×C₄×C₆₇	1	3	yes	—
90	C₁₆	C₂²×C₄×C₆₇	1	4	yes	—
91	C₁₆	C₂⁴×C₆₇	1	1	yes	characteristic
92	C₂³	—	2	4	no	—
93	C₂³	C₂×C₁₆×C₆₇	1	6	yes	—
94	C₂³	C₂×C₁₆×C₆₇	1	1	yes	characteristic
95	C₂×C₄	C₂×C₁₆×C₆₇	1	12	yes	—
96	C₂×C₄	C₂×C₁₆×C₆₇	1	3	yes	—
97	C₂×C₄	C₂×C₁₆×C₆₇	1	6	yes	—
98	C₂×C₄	C₂²×C₈×C₆₇	1	4	yes	—
99	C₂×C₄	C₂²×C₈×C₆₇	1	3	yes	—
100	C₈	C₂²×C₈×C₆₇	1	3	yes	—
101	C₈	C₂²×C₈×C₆₇	1	4	yes	—
102	C₈	C₂³×C₄×C₆₇	1	1	yes	characteristic
103	C₂²	—	2	12	no	—
104	C₂²	M₆₄×C₆₇	1	4	yes	direct product factor
105	C₂²	C₂²×C₁₆×C₆₇	1	4	yes	—
106	C₂²	C₂²×C₁₆×C₆₇	1	3	yes	—
107	C₄	C₂²×C₁₆×C₆₇	1	3	yes	—
108	C₄	C₂²×C₁₆×C₆₇	1	4	yes	—
109	C₄	C₂³×C₈×C₆₇	1	1	yes	characteristic
110	C₂	—	2	4	no	—
111	C₂	C₂×M₆₄×C₆₇	1	6	yes	direct product factor
112	C₂	C₂³×C₁₆×C₆₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×M₆₄×C₆₇	C₂	direct (N × H)
3. C₂²×C₃₂×C₆₇	C₂	semidirect
9. M₆₄×C₆₇	C₂²	direct (N × H)
10. C₂×C₃₂×C₆₇	C₂²	semidirect
21. C₃₂×C₆₇	C₂³	semidirect
47. C₂²×C₆₇	M₆₄	direct (N × H)
53. C₂²×M₆₄	C₆₇	direct (N × H)
55. C₂×C₆₇	C₂×M₆₄	direct (N × H)
58. C₂×M₆₄	C₂×C₆₇	direct (N × H)
59. C₂²×C₃₂	C₂×C₆₇	semidirect
60. C₆₇	C₂²×M₆₄	direct (N × H)
66. M₆₄	C₂²×C₆₇	direct (N × H)
67. C₂×C₃₂	C₂²×C₆₇	semidirect
78. C₃₂	C₂³×C₆₇	semidirect
104. C₂²	M₆₄×C₆₇	direct (N × H)
111. C₂	C₂×M₆₄×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
C₂³×C₁₆×C₆₇	2
C₂×M₆₄×C₆₇	2
C₂²×C₃₂×C₆₇	2
C₂²×M₆₄	67

Subgroups — C22×M64×C67 #30713223

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×M₆₄×C₆₇ #30713223