Subgroups — C₃×C₈×C₁₀₂₄ #72201816

Order 24576.

284 conjugacy classes of subgroups.

The subgroup lattice has 160 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₈×C₅₁₂	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₁₀₂₄	C₂	1	2	yes	maximal
3	C₈×C₁₀₂₄	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₃×C₈×C₂₅₆	C₄	1	1	yes	characteristic
5	C₃×C₄×C₅₁₂	C₄	1	1	yes	characteristic
6	C₃×C₄×C₅₁₂	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₁₀₂₄	C₄	1	4	yes	—
8	C₈×C₅₁₂	C₂×C₃	1	1	yes	characteristic
9	C₄×C₁₀₂₄	C₂×C₃	1	2	yes	—
10	C₃×C₈×C₁₂₈	C₈	1	1	yes	characteristic
11	C₃×C₄×C₂₅₆	C₈	1	1	yes	characteristic
12	C₃×C₄×C₂₅₆	C₂×C₄	1	1	yes	characteristic
13	C₂×C₃×C₅₁₂	C₈	1	2	yes	—
14	C₂×C₃×C₅₁₂	C₂×C₄	1	2	yes	—
15	C₃×C₁₀₂₄	C₈	1	8	yes	direct product factor
16	C₈×C₂₅₆	C₃×C₄	1	1	yes	characteristic
17	C₄×C₅₁₂	C₃×C₄	1	1	yes	characteristic
18	C₄×C₅₁₂	C₂²×C₃	1	1	yes	characteristic
19	C₂×C₁₀₂₄	C₃×C₄	1	4	yes	—
20	C₃×C₈×C₆₄	C₁₆	1	1	yes	characteristic
21	C₃×C₄×C₁₂₈	C₁₆	1	1	yes	characteristic
22	C₃×C₄×C₁₂₈	C₂×C₈	1	1	yes	characteristic
23	C₂×C₃×C₂₅₆	C₁₆	1	2	yes	—
24	C₂×C₃×C₂₅₆	C₄²	1	1	yes	characteristic
25	C₂×C₃×C₂₅₆	C₂×C₈	1	1	yes	characteristic
26	C₃×C₅₁₂	C₁₆	1	4	yes	—
27	C₃×C₅₁₂	C₂×C₈	1	4	yes	—
28	C₈×C₁₂₈	C₃×C₈	1	1	yes	characteristic
29	C₄×C₂₅₆	C₃×C₈	1	1	yes	characteristic
30	C₄×C₂₅₆	C₂×C₃×C₄	1	1	yes	characteristic
31	C₂×C₅₁₂	C₃×C₈	1	2	yes	—
32	C₂×C₅₁₂	C₂×C₃×C₄	1	2	yes	—
33	C₁₀₂₄	C₃×C₈	1	8	yes	direct product factor
34	C₂×C₃×C₁₂₈	C₃₂	1	2	yes	—
35	C₂×C₃×C₁₂₈	C₄×C₈	1	1	yes	characteristic
36	C₂×C₃×C₁₂₈	C₂×C₁₆	1	1	yes	characteristic
37	C₃×C₄×C₆₄	C₃₂	1	1	yes	characteristic
38	C₃×C₄×C₆₄	C₂×C₁₆	1	1	yes	characteristic
39	C₃×C₈×C₃₂	C₃₂	1	1	yes	characteristic
40	C₃×C₂₅₆	C₃₂	1	4	yes	—
41	C₃×C₂₅₆	C₄×C₈	1	2	yes	—
42	C₃×C₂₅₆	C₂×C₁₆	1	2	yes	—
43	C₂×C₂₅₆	C₃×C₁₆	1	2	yes	—
44	C₂×C₂₅₆	C₃×C₄²	1	1	yes	characteristic
45	C₂×C₂₅₆	C₂×C₃×C₈	1	1	yes	characteristic
46	C₄×C₁₂₈	C₃×C₁₆	1	1	yes	characteristic
47	C₄×C₁₂₈	C₂×C₃×C₈	1	1	yes	characteristic
48	C₈×C₆₄	C₃×C₁₆	1	1	yes	characteristic
49	C₅₁₂	C₃×C₁₆	1	4	yes	—
50	C₅₁₂	C₂×C₃×C₈	1	4	yes	—
51	C₂×C₃×C₆₄	C₆₄	1	2	yes	—
52	C₂×C₃×C₆₄	C₄×C₁₆	1	1	yes	characteristic
53	C₂×C₃×C₆₄	C₂×C₃₂	1	1	yes	characteristic
54	C₃×C₄×C₃₂	C₆₄	1	1	yes	characteristic
55	C₃×C₄×C₃₂	C₂×C₃₂	1	1	yes	characteristic
56	C₃×C₈×C₁₆	C₆₄	1	1	yes	characteristic
57	C₃×C₁₂₈	C₆₄	1	4	yes	—
58	C₃×C₁₂₈	C₈²	1	1	yes	characteristic
59	C₃×C₁₂₈	C₄×C₁₆	1	1	yes	characteristic
60	C₃×C₁₂₈	C₂×C₃₂	1	2	yes	—
61	C₂×C₁₂₈	C₃×C₃₂	1	2	yes	—
62	C₂×C₁₂₈	C₃×C₄×C₈	1	1	yes	characteristic
63	C₂×C₁₂₈	C₂×C₃×C₁₆	1	1	yes	characteristic
64	C₄×C₆₄	C₃×C₃₂	1	1	yes	characteristic
65	C₄×C₆₄	C₂×C₃×C₁₆	1	1	yes	characteristic
66	C₈×C₃₂	C₃×C₃₂	1	1	yes	characteristic
67	C₂₅₆	C₃×C₃₂	1	4	yes	—
68	C₂₅₆	C₃×C₄×C₈	1	2	yes	—
69	C₂₅₆	C₂×C₃×C₁₆	1	2	yes	—
70	C₂×C₃×C₃₂	C₁₂₈	1	2	yes	—
71	C₂×C₃×C₃₂	C₄×C₃₂	1	1	yes	characteristic
72	C₂×C₃×C₃₂	C₂×C₆₄	1	1	yes	characteristic
73	C₃×C₄×C₁₆	C₁₂₈	1	1	yes	characteristic
74	C₃×C₄×C₁₆	C₂×C₆₄	1	1	yes	characteristic
75	C₃×C₈²	C₁₂₈	1	1	yes	characteristic
76	C₃×C₆₄	C₁₂₈	1	4	yes	—
77	C₃×C₆₄	C₈×C₁₆	1	1	yes	characteristic
78	C₃×C₆₄	C₄×C₃₂	1	1	yes	characteristic
79	C₃×C₆₄	C₂×C₆₄	1	2	yes	—
80	C₂×C₆₄	C₃×C₆₄	1	2	yes	—
81	C₂×C₆₄	C₃×C₄×C₁₆	1	1	yes	characteristic
82	C₂×C₆₄	C₂×C₃×C₃₂	1	1	yes	characteristic
83	C₄×C₃₂	C₃×C₆₄	1	1	yes	characteristic
84	C₄×C₃₂	C₂×C₃×C₃₂	1	1	yes	characteristic
85	C₈×C₁₆	C₃×C₆₄	1	1	yes	characteristic
86	C₁₂₈	C₃×C₆₄	1	4	yes	—
87	C₁₂₈	C₃×C₈²	1	1	yes	characteristic
88	C₁₂₈	C₃×C₄×C₁₆	1	1	yes	characteristic
89	C₁₂₈	C₂×C₃×C₃₂	1	2	yes	—
90	C₂×C₃×C₁₆	C₂₅₆	1	2	yes	—
91	C₂×C₃×C₁₆	C₄×C₆₄	1	1	yes	characteristic
92	C₂×C₃×C₁₆	C₂×C₁₂₈	1	1	yes	characteristic
93	C₃×C₄×C₈	C₂₅₆	1	2	yes	—
94	C₃×C₄×C₈	C₂×C₁₂₈	1	1	yes	characteristic
95	C₃×C₃₂	C₂₅₆	1	4	yes	—
96	C₃×C₃₂	C₈×C₃₂	1	1	yes	characteristic
97	C₃×C₃₂	C₄×C₆₄	1	1	yes	characteristic
98	C₃×C₃₂	C₂×C₁₂₈	1	2	yes	—
99	C₂×C₃₂	C₃×C₁₂₈	1	2	yes	—
100	C₂×C₃₂	C₃×C₄×C₃₂	1	1	yes	characteristic
101	C₂×C₃₂	C₂×C₃×C₆₄	1	1	yes	characteristic
102	C₄×C₁₆	C₃×C₁₂₈	1	1	yes	characteristic
103	C₄×C₁₆	C₂×C₃×C₆₄	1	1	yes	characteristic
104	C₈²	C₃×C₁₂₈	1	1	yes	characteristic
105	C₆₄	C₃×C₁₂₈	1	4	yes	—
106	C₆₄	C₃×C₈×C₁₆	1	1	yes	characteristic
107	C₆₄	C₃×C₄×C₃₂	1	1	yes	characteristic
108	C₆₄	C₂×C₃×C₆₄	1	2	yes	—
109	C₂×C₃×C₈	C₅₁₂	1	4	yes	—
110	C₂×C₃×C₈	C₄×C₁₂₈	1	1	yes	characteristic
111	C₂×C₃×C₈	C₂×C₂₅₆	1	1	yes	characteristic
112	C₃×C₄²	C₂×C₂₅₆	1	1	yes	characteristic
113	C₃×C₁₆	C₅₁₂	1	4	yes	—
114	C₃×C₁₆	C₈×C₆₄	1	1	yes	characteristic
115	C₃×C₁₆	C₄×C₁₂₈	1	1	yes	characteristic
116	C₃×C₁₆	C₂×C₂₅₆	1	2	yes	—
117	C₂×C₁₆	C₃×C₂₅₆	1	2	yes	—
118	C₂×C₁₆	C₃×C₄×C₆₄	1	1	yes	characteristic
119	C₂×C₁₆	C₂×C₃×C₁₂₈	1	1	yes	characteristic
120	C₄×C₈	C₃×C₂₅₆	1	2	yes	—
121	C₄×C₈	C₂×C₃×C₁₂₈	1	1	yes	characteristic
122	C₃₂	C₃×C₂₅₆	1	4	yes	—
123	C₃₂	C₃×C₈×C₃₂	1	1	yes	characteristic
124	C₃₂	C₃×C₄×C₆₄	1	1	yes	characteristic
125	C₃₂	C₂×C₃×C₁₂₈	1	2	yes	—
126	C₂×C₃×C₄	C₂×C₅₁₂	1	2	yes	—
127	C₂×C₃×C₄	C₄×C₂₅₆	1	1	yes	characteristic
128	C₃×C₈	C₁₀₂₄	1	8	yes	direct product factor
129	C₃×C₈	C₂×C₅₁₂	1	2	yes	—
130	C₃×C₈	C₄×C₂₅₆	1	1	yes	characteristic
131	C₃×C₈	C₈×C₁₂₈	1	1	yes	characteristic
132	C₂×C₈	C₃×C₅₁₂	1	4	yes	—
133	C₂×C₈	C₂×C₃×C₂₅₆	1	1	yes	characteristic
134	C₂×C₈	C₃×C₄×C₁₂₈	1	1	yes	characteristic
135	C₄²	C₂×C₃×C₂₅₆	1	1	yes	characteristic
136	C₁₆	C₃×C₅₁₂	1	4	yes	—
137	C₁₆	C₂×C₃×C₂₅₆	1	2	yes	—
138	C₁₆	C₃×C₄×C₁₂₈	1	1	yes	characteristic
139	C₁₆	C₃×C₈×C₆₄	1	1	yes	characteristic
140	C₂²×C₃	C₄×C₅₁₂	1	1	yes	characteristic
141	C₃×C₄	C₂×C₁₀₂₄	1	4	yes	—
142	C₃×C₄	C₄×C₅₁₂	1	1	yes	characteristic
143	C₃×C₄	C₈×C₂₅₆	1	1	yes	characteristic
144	C₂×C₄	C₂×C₃×C₅₁₂	1	2	yes	—
145	C₂×C₄	C₃×C₄×C₂₅₆	1	1	yes	characteristic
146	C₈	C₃×C₁₀₂₄	1	8	yes	direct product factor
147	C₈	C₂×C₃×C₅₁₂	1	2	yes	—
148	C₈	C₃×C₄×C₂₅₆	1	1	yes	characteristic
149	C₈	C₃×C₈×C₁₂₈	1	1	yes	characteristic
150	C₂×C₃	C₄×C₁₀₂₄	1	2	yes	—
151	C₂×C₃	C₈×C₅₁₂	1	1	yes	characteristic
152	C₂²	C₃×C₄×C₅₁₂	1	1	yes	characteristic
153	C₄	C₂×C₃×C₁₀₂₄	1	4	yes	—
154	C₄	C₃×C₄×C₅₁₂	1	1	yes	characteristic
155	C₄	C₃×C₈×C₂₅₆	1	1	yes	characteristic
156	C₃	C₈×C₁₀₂₄	1	1	yes	characteristic, direct product factor, Sylow
157	C₂	C₃×C₄×C₁₀₂₄	1	2	yes	—
158	C₂	C₃×C₈×C₅₁₂	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₄×C₅₁₂	C₂²×C₃
socle, prime radical	C₂²×C₃	C₄×C₅₁₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈×C₁₀₂₄	C₃	direct (N × H)
15. C₃×C₁₀₂₄	C₈	direct (N × H)
33. C₁₀₂₄	C₃×C₈	direct (N × H)
128. C₃×C₈	C₁₀₂₄	direct (N × H)
146. C₈	C₃×C₁₀₂₄	direct (N × H)
156. C₃	C₈×C₁₀₂₄	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₈×C₅₁₂	2
C₃×C₄×C₁₀₂₄	2
C₈×C₁₀₂₄	3

Subgroups — C3×C8×C1024 #72201816

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₈×C₁₀₂₄ #72201816