Subgroups — C₄²×C₆₇⋊C₁₆ #30770650

Order 17152.

558 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₆₇⋊C₁₆	C₂	1	6	yes	maximal
2	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₆₇⋊C₁₆	C₂²	1	4	yes	—
4	C₄×C₆₇⋊C₁₆	C₄	1	24	yes	direct product factor
5	C₂×C₄×C₈×C₆₇	C₄	1	3	yes	—
6	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
7	C₄³×C₆₇	C₄	1	1	yes	characteristic
8	C₂×C₆₇⋊C₁₆	C₂×C₄	1	24	yes	—
9	C₂²×C₈×C₆₇	C₂×C₄	1	3	yes	—
10	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
11	C₂×C₄²×C₆₇	C₈	1	4	yes	—
12	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
13	C₄×C₈×C₆₇	C₈	1	12	yes	—
14	C₄×C₈×C₆₇	C₂×C₄	1	12	yes	—
15	C₆₇⋊C₁₆	C₄²	1	16	yes	direct product factor
16	C₂²×C₄×C₆₇	C₂×C₈	1	6	yes	—
17	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₈×C₆₇	C₄²	1	6	yes	—
19	C₂×C₈×C₆₇	C₂×C₈	1	12	yes	—
20	C₂×C₈×C₆₇	C₂²×C₄	1	6	yes	—
21	C₄²×C₆₇	C₁₆	1	16	yes	complemented
22	C₄²×C₆₇	C₄²	1	6	yes	—
23	C₄²×C₆₇	C₂×C₈	1	6	yes	—
24	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
25	C₂×C₄×C₆₇	C₄×C₈	1	12	yes	—
26	C₂×C₄×C₆₇	C₂×C₁₆	1	24	yes	—
27	C₂×C₄×C₆₇	C₂×C₄²	1	3	yes	—
28	C₂×C₄×C₆₇	C₂²×C₈	1	3	yes	—
29	C₈×C₆₇	C₄×C₈	1	12	yes	—
30	C₈×C₆₇	C₂×C₄²	1	4	yes	—
31	C₂²×C₆₇	C₂×C₄×C₈	1	3	yes	—
32	C₂²×C₆₇	C₂²×C₁₆	1	4	yes	—
33	C₄×C₆₇	C₄×C₁₆	1	24	yes	complemented
34	C₄×C₆₇	C₄³	1	1	yes	characteristic
35	C₄×C₆₇	C₂×C₄×C₈	1	3	yes	—
36	C₄²×C₁₆	—	67	1	no	maximal, Sylow
37	C₂×C₆₇	C₄²×C₈	1	1	yes	characteristic
38	C₂×C₆₇	C₂×C₄×C₁₆	1	6	yes	—
39	C₂×C₄×C₁₆	—	67	6	no	—
40	C₄²×C₈	D₆₇	1	1	yes	characteristic
41	C₆₇	C₄²×C₁₆	1	1	yes	characteristic, complemented, Sylow
42	C₂²×C₁₆	—	67	4	no	—
43	C₂×C₄×C₈	Dic₆₇	1	3	yes	—
44	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
45	C₄³	Dic₆₇	1	1	yes	characteristic
46	C₄×C₁₆	—	67	24	no	—
47	C₂²×C₈	C₂×Dic₆₇	1	3	yes	—
48	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
49	C₂×C₄²	C₆₇⋊C₈	1	4	yes	—
50	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
51	C₂×C₁₆	—	67	24	no	—
52	C₄×C₈	C₆₇⋊C₈	1	12	yes	—
53	C₄×C₈	C₄×D₆₇	1	12	yes	—
54	C₂²×C₄	C₂×C₆₇⋊C₈	1	6	yes	—
55	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
56	C₂×C₈	C₄×Dic₆₇	1	6	yes	—
57	C₂×C₈	C₂×C₆₇⋊C₈	1	12	yes	—
58	C₂×C₈	C₂×C₄×D₆₇	1	6	yes	—
59	C₄²	C₆₇⋊C₁₆	1	16	yes	direct product factor
60	C₄²	C₄×Dic₆₇	1	6	yes	—
61	C₄²	C₂×C₆₇⋊C₈	1	6	yes	—
62	C₁₆	—	67	16	no	—
63	C₂³	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
64	C₂×C₄	C₄×C₆₇⋊C₈	1	12	yes	—
65	C₂×C₄	C₂×C₆₇⋊C₁₆	1	24	yes	—
66	C₂×C₄	C₂×C₄×Dic₆₇	1	3	yes	—
67	C₂×C₄	C₂²×C₆₇⋊C₈	1	3	yes	—
68	C₈	C₄×C₆₇⋊C₈	1	12	yes	—
69	C₈	C₄²×D₆₇	1	4	yes	—
70	C₂²	C₂×C₄×C₆₇⋊C₈	1	3	yes	—
71	C₂²	C₂²×C₆₇⋊C₁₆	1	4	yes	—
72	C₄	C₄×C₆₇⋊C₁₆	1	24	yes	direct product factor
73	C₄	C₄²×Dic₆₇	1	1	yes	characteristic
74	C₄	C₂×C₄×C₆₇⋊C₈	1	3	yes	—
75	C₂	C₄²×C₆₇⋊C₈	1	1	yes	characteristic
76	C₂	C₂×C₄×C₆₇⋊C₁₆	1	6	yes	—

	subgroup	G/H
center, hypercenter	C₄²×C₈	D₆₇
commutator, hypocenter	C₆₇	C₄²×C₁₆
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₄²×C₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂²×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
LCS	G ▷ C₆₇
UCS	C₁ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
4. C₄×C₆₇⋊C₁₆	C₄	direct (N × H)
15. C₆₇⋊C₁₆	C₄²	direct (N × H)
21. C₄²×C₆₇	C₁₆	semidirect
33. C₄×C₆₇	C₄×C₁₆	semidirect
41. C₆₇	C₄²×C₁₆	semidirect
59. C₄²	C₆₇⋊C₁₆	direct (N × H)
72. C₄	C₄×C₆₇⋊C₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₆₇⋊C₁₆	2
C₄²×C₈×C₆₇	2
C₄²×C₁₆	67

Subgroups — C42×C67⋊C16 #30770650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₆₇⋊C₁₆ #30770650