Subgroups — C₂⁴×Dic₂₈₄ #32911234

Order 18176.

11166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Dic₁₄₂	C₂	1	2	yes	maximal
2	C₂³×Dic₂₈₄	C₂	1	60	yes	maximal, direct product factor
3	C₂⁴×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂³×Dic₁₄₂	—	2	30	no	—
5	C₂³×Dic₁₄₂	C₂²	1	60	yes	—
6	C₂⁴×Dic₇₁	—	2	2	no	—
7	C₂²×Dic₂₈₄	C₂²	1	560	yes	direct product factor
8	C₂⁴×C₄×C₇₁	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₇₁	C₂²	1	30	yes	—
10	C₂²×Dic₁₄₂	—	2	420	no	—
11	C₂²×Dic₁₄₂	C₂³	1	280	yes	—
12	C₂³×Dic₇₁	—	2	60	no	—
13	C₂×Dic₂₈₄	C₂³	1	960	yes	direct product factor
14	C₂⁵×C₇₁	D₄	1	1	yes	characteristic
15	C₂³×C₄×C₇₁	D₄	1	15	yes	—
16	C₂³×C₄×C₇₁	C₂³	1	15	yes	—
17	C₂²×C₈×C₇₁	C₂³	1	140	yes	—
18	C₂×Dic₁₄₂	—	2	840	no	—
19	C₂×Dic₁₄₂	C₂⁴	1	240	yes	—
20	C₂²×Dic₇₁	—	2	280	no	—
21	Dic₂₈₄	C₂⁴	1	256	yes	direct product factor
22	C₂⁴×C₇₁	Q₁₆	1	16	yes	complemented
23	C₂⁴×C₇₁	C₂×D₄	1	15	yes	—
24	C₂²×C₄×C₇₁	C₂×D₄	1	105	yes	—
25	C₂²×C₄×C₇₁	C₂⁴	1	35	yes	—
26	C₂×C₈×C₇₁	C₂⁴	1	120	yes	—
27	C₂³×C₇₁	C₂×Q₁₆	1	120	yes	complemented
28	C₂³×C₇₁	C₂²×D₄	1	35	yes	—
29	C₂×C₄×C₇₁	C₂²×D₄	1	105	yes	—
30	C₂×C₄×C₇₁	C₂⁵	1	15	yes	—
31	C₂×Dic₇₁	—	2	240	no	—
32	Dic₁₄₂	—	2	240	no	—
33	Dic₁₄₂	C₂⁵	1	32	yes	—
34	C₈×C₇₁	C₂⁵	1	16	yes	—
35	C₂²×C₇₁	C₂²×Q₁₆	1	140	yes	complemented
36	C₂²×C₇₁	C₂³×D₄	1	15	yes	—
37	C₄×C₇₁	C₂³×D₄	1	15	yes	—
38	C₄×C₇₁	C₂⁶	1	1	yes	characteristic
39	Dic₇₁	—	2	32	no	—
40	C₂⁴×Q₁₆	—	71	1	no	maximal, Sylow
41	C₂×C₇₁	C₂³×Q₁₆	1	30	yes	complemented
42	C₂×C₇₁	C₂⁴×D₄	1	1	yes	characteristic
43	C₂⁴×Q₈	—	71	2	no	—
44	C₂³×Q₁₆	—	71	60	no	—
45	C₂⁴×C₈	D₇₁	1	1	yes	characteristic
46	C₇₁	C₂⁴×Q₁₆	1	1	yes	characteristic, complemented, Sylow
47	C₂³×Q₈	—	71	60	no	—
48	C₂³×Q₈	—	142	30	no	—
49	C₂⁴×C₄	—	142	2	no	—
50	C₂⁴×C₄	C₂×D₇₁	1	1	yes	characteristic
51	C₂²×Q₁₆	—	71	560	no	—
52	C₂³×C₈	C₂×D₇₁	1	30	yes	—
53	C₂⁵	D₂₈₄	1	1	yes	characteristic
54	C₂²×Q₈	—	71	280	no	—
55	C₂²×Q₈	—	142	420	no	—
56	C₂³×C₄	—	142	60	no	—
57	C₂³×C₄	D₂₈₄	1	15	yes	—
58	C₂³×C₄	C₂²×D₇₁	1	15	yes	—
59	C₂×Q₁₆	—	71	960	no	—
60	C₂²×C₈	C₂²×D₇₁	1	140	yes	—
61	C₂⁴	Dic₂₈₄	1	16	yes	direct product factor
62	C₂⁴	C₂×D₂₈₄	1	15	yes	—
63	C₂×Q₈	—	71	240	no	—
64	C₂×Q₈	—	142	840	no	—
65	C₂²×C₄	—	142	280	no	—
66	C₂²×C₄	C₂×D₂₈₄	1	105	yes	—
67	C₂²×C₄	C₂³×D₇₁	1	35	yes	—
68	Q₁₆	—	71	256	no	—
69	C₂×C₈	C₂³×D₇₁	1	120	yes	—
70	C₂³	C₂×Dic₂₈₄	1	120	yes	direct product factor
71	C₂³	C₂²×D₂₈₄	1	35	yes	—
72	Q₈	—	71	32	no	—
73	Q₈	—	142	240	no	—
74	C₂×C₄	—	142	240	no	—
75	C₂×C₄	C₂²×D₂₈₄	1	105	yes	—
76	C₂×C₄	C₂⁴×D₇₁	1	15	yes	—
77	C₈	C₂⁴×D₇₁	1	16	yes	—
78	C₂²	C₂²×Dic₂₈₄	1	140	yes	direct product factor
79	C₂²	C₂³×D₂₈₄	1	15	yes	—
80	C₄	—	142	32	no	—
81	C₄	C₂³×D₂₈₄	1	15	yes	—
82	C₄	C₂⁵×D₇₁	1	1	yes	characteristic
83	C₂	C₂³×Dic₂₈₄	1	30	yes	direct product factor
84	C₂	C₂⁴×D₂₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	D₂₈₄
commutator, cosocle	C₄×C₇₁	C₂⁶
Frattini	C₄	C₂⁵×D₇₁
Fitting	C₂⁴×C₈×C₇₁	C₂
socle, prime radical	C₂⁵×C₇₁	D₄
hypercenter	C₂⁴×C₈	D₇₁
hypocenter	C₇₁	C₂⁴×Q₁₆
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂⁵ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈

Complements

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

N	H ≅ G/N	type
2. C₂³×Dic₂₈₄	C₂	direct (N × H)
7. C₂²×Dic₂₈₄	C₂²	direct (N × H)
13. C₂×Dic₂₈₄	C₂³	direct (N × H)
21. Dic₂₈₄	C₂⁴	direct (N × H)
22. C₂⁴×C₇₁	Q₁₆	semidirect
27. C₂³×C₇₁	C₂×Q₁₆	semidirect
35. C₂²×C₇₁	C₂²×Q₁₆	semidirect
41. C₂×C₇₁	C₂³×Q₁₆	semidirect
46. C₇₁	C₂⁴×Q₁₆	semidirect
61. C₂⁴	Dic₂₈₄	direct (N × H)
70. C₂³	C₂×Dic₂₈₄	direct (N × H)
78. C₂²	C₂²×Dic₂₈₄	direct (N × H)
83. C₂	C₂³×Dic₂₈₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Dic₁₄₂	2
C₂³×Dic₂₈₄	2
C₂⁴×C₈×C₇₁	2
C₂⁴×Q₁₆	71

Subgroups — C24×Dic284 #32911234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×Dic₂₈₄ #32911234