Subgroups — C₄²×C₅²×C₈² #72202829

Order 25600.

52776 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄³×C₅²×C₈	C₂	1	3	yes	maximal
2	C₂×C₄×C₅²×C₈²	C₂	1	12	yes	maximal
3	C₄⁴×C₅²	C₂²	1	1	yes	characteristic
4	C₂×C₄²×C₅²×C₈	C₄	1	24	yes	—
5	C₂×C₄²×C₅²×C₈	C₂²	1	18	yes	—
6	C₂²×C₅²×C₈²	C₂²	1	16	yes	—
7	C₄×C₅²×C₈²	C₄	1	96	yes	direct product factor
8	C₄²×C₅×C₈²	C₅	1	6	yes	maximal, direct product factor
9	C₂×C₄³×C₅²	C₂×C₄	1	12	yes	—
10	C₂×C₄³×C₅²	C₂³	1	3	yes	—
11	C₂²×C₄×C₅²×C₈	C₂×C₄	1	72	yes	—
12	C₂²×C₄×C₅²×C₈	C₂³	1	12	yes	—
13	C₄²×C₅²×C₈	C₈	1	192	yes	direct product factor
14	C₄²×C₅²×C₈	C₂×C₄	1	144	yes	—
15	C₂×C₅²×C₈²	C₂×C₄	1	192	yes	—
16	C₄³×C₅×C₈	C₂×C₅	1	18	yes	—
17	C₂×C₄×C₅×C₈²	C₂×C₅	1	72	yes	—
18	C₄²×C₈²	C₅²	1	1	yes	characteristic, direct product factor, Sylow
19	C₂³×C₅²×C₈	C₂²×C₄	1	24	yes	—
20	C₂²×C₄²×C₅²	C₄²	1	16	yes	—
21	C₂²×C₄²×C₅²	C₂²×C₄	1	18	yes	—
22	C₂²×C₄²×C₅²	C₂⁴	1	1	yes	characteristic
23	C₂×C₄×C₅²×C₈	C₄²	1	288	yes	—
24	C₂×C₄×C₅²×C₈	C₂×C₈	1	576	yes	—
25	C₂×C₄×C₅²×C₈	C₂²×C₄	1	144	yes	—
26	C₄³×C₅²	C₂×C₈	1	96	yes	—
27	C₄³×C₅²	C₂²×C₄	1	24	yes	—
28	C₅²×C₈²	C₄²	1	256	yes	direct product factor
29	C₂×C₄²×C₅×C₈	C₄×C₅	1	144	yes	—
30	C₂×C₄²×C₅×C₈	C₂²×C₅	1	108	yes	—
31	C₂²×C₅×C₈²	C₂²×C₅	1	96	yes	—
32	C₄⁴×C₅	C₂²×C₅	1	6	yes	—
33	C₄×C₅×C₈²	C₄×C₅	1	576	yes	direct product factor
34	C₂³×C₄×C₅²	C₂×C₄²	1	12	yes	—
35	C₂³×C₄×C₅²	C₂³×C₄	1	3	yes	—
36	C₂²×C₅²×C₈	C₂×C₄²	1	144	yes	—
37	C₂²×C₅²×C₈	C₂²×C₈	1	192	yes	—
38	C₂×C₄²×C₅²	C₄×C₈	1	192	yes	—
39	C₂×C₄²×C₅²	C₂×C₄²	1	72	yes	—
40	C₂×C₄²×C₅²	C₂²×C₈	1	144	yes	—
41	C₂×C₄²×C₅²	C₂³×C₄	1	12	yes	—
42	C₄×C₅²×C₈	C₄×C₈	1	1152	yes	direct product factor
43	C₄×C₅²×C₈	C₂×C₄²	1	192	yes	—
44	C₂²×C₄×C₅×C₈	C₂×C₄×C₅	1	432	yes	—
45	C₂²×C₄×C₅×C₈	C₂³×C₅	1	72	yes	—
46	C₂×C₄³×C₅	C₂×C₄×C₅	1	72	yes	—
47	C₂×C₄³×C₅	C₂³×C₅	1	18	yes	—
48	C₄²×C₅×C₈	C₅×C₈	1	1152	yes	direct product factor
49	C₄²×C₅×C₈	C₂×C₄×C₅	1	864	yes	—
50	C₂×C₅×C₈²	C₂×C₄×C₅	1	1152	yes	—
51	C₄³×C₈	C₂×C₅²	1	3	yes	—
52	C₂×C₄×C₈²	C₂×C₅²	1	12	yes	—
53	C₂⁴×C₅²	C₂²×C₄²	1	1	yes	characteristic
54	C₂²×C₄×C₅²	C₄³	1	24	yes	—
55	C₂²×C₄×C₅²	C₂×C₄×C₈	1	144	yes	—
56	C₂²×C₄×C₅²	C₂²×C₄²	1	18	yes	—
57	C₂²×C₄×C₅²	C₂³×C₈	1	24	yes	—
58	C₂×C₅²×C₈	C₄³	1	96	yes	—
59	C₂×C₅²×C₈	C₂×C₄×C₈	1	576	yes	—
60	C₄²×C₅²	C₈²	1	256	yes	direct product factor
61	C₄²×C₅²	C₂×C₄×C₈	1	288	yes	—
62	C₄²×C₅²	C₂²×C₄²	1	16	yes	—
63	C₂³×C₅×C₈	C₂²×C₄×C₅	1	144	yes	—
64	C₂²×C₄²×C₅	C₄²×C₅	1	96	yes	—
65	C₂²×C₄²×C₅	C₂²×C₄×C₅	1	108	yes	—
66	C₂²×C₄²×C₅	C₂⁴×C₅	1	6	yes	—
67	C₂×C₄×C₅×C₈	C₄²×C₅	1	1728	yes	—
68	C₂×C₄×C₅×C₈	C₂×C₅×C₈	1	3456	yes	—
69	C₂×C₄×C₅×C₈	C₂²×C₄×C₅	1	864	yes	—
70	C₄³×C₅	C₂×C₅×C₈	1	576	yes	—
71	C₄³×C₅	C₂²×C₄×C₅	1	144	yes	—
72	C₅×C₈²	C₄²×C₅	1	1536	yes	direct product factor
73	C₂×C₄²×C₈	C₄×C₅²	1	24	yes	—
74	C₂×C₄²×C₈	C₂²×C₅²	1	18	yes	—
75	C₂²×C₈²	C₂²×C₅²	1	16	yes	—
76	C₄⁴	C₂²×C₅²	1	1	yes	characteristic
77	C₄×C₈²	C₄×C₅²	1	96	yes	direct product factor
78	C₂³×C₅²	C₂×C₄³	1	3	yes	—
79	C₂³×C₅²	C₂²×C₄×C₈	1	12	yes	—
80	C₂×C₄×C₅²	C₂×C₈²	1	192	yes	—
81	C₂×C₄×C₅²	C₄²×C₈	1	144	yes	—
82	C₂×C₄×C₅²	C₂×C₄³	1	12	yes	—
83	C₂×C₄×C₅²	C₂²×C₄×C₈	1	72	yes	—
84	C₅²×C₈	C₄²×C₈	1	192	yes	direct product factor
85	C₂³×C₄×C₅	C₂×C₄²×C₅	1	72	yes	—
86	C₂³×C₄×C₅	C₂³×C₄×C₅	1	18	yes	—
87	C₂²×C₅×C₈	C₂×C₄²×C₅	1	864	yes	—
88	C₂²×C₅×C₈	C₂²×C₅×C₈	1	1152	yes	—
89	C₂×C₄²×C₅	C₄×C₅×C₈	1	1152	yes	—
90	C₂×C₄²×C₅	C₂×C₄²×C₅	1	432	yes	—
91	C₂×C₄²×C₅	C₂²×C₅×C₈	1	864	yes	—
92	C₂×C₄²×C₅	C₂³×C₄×C₅	1	72	yes	—
93	C₄×C₅×C₈	C₄×C₅×C₈	1	6912	yes	direct product factor
94	C₄×C₅×C₈	C₂×C₄²×C₅	1	1152	yes	—
95	C₂²×C₄×C₈	C₂×C₄×C₅²	1	72	yes	—
96	C₂²×C₄×C₈	C₂³×C₅²	1	12	yes	—
97	C₂×C₄³	C₂×C₄×C₅²	1	12	yes	—
98	C₂×C₄³	C₂³×C₅²	1	3	yes	—
99	C₄²×C₈	C₅²×C₈	1	192	yes	direct product factor
100	C₄²×C₈	C₂×C₄×C₅²	1	144	yes	—
101	C₂×C₈²	C₂×C₄×C₅²	1	192	yes	—
102	C₂²×C₅²	C₄⁴	1	1	yes	characteristic
103	C₂²×C₅²	C₂²×C₈²	1	16	yes	—
104	C₂²×C₅²	C₂×C₄²×C₈	1	18	yes	—
105	C₄×C₅²	C₄×C₈²	1	96	yes	direct product factor
106	C₄×C₅²	C₂×C₄²×C₈	1	24	yes	—
107	C₂⁴×C₅	C₂²×C₄²×C₅	1	6	yes	—
108	C₂²×C₄×C₅	C₄³×C₅	1	144	yes	—
109	C₂²×C₄×C₅	C₂×C₄×C₅×C₈	1	864	yes	—
110	C₂²×C₄×C₅	C₂²×C₄²×C₅	1	108	yes	—
111	C₂²×C₄×C₅	C₂³×C₅×C₈	1	144	yes	—
112	C₂×C₅×C₈	C₄³×C₅	1	576	yes	—
113	C₂×C₅×C₈	C₂×C₄×C₅×C₈	1	3456	yes	—
114	C₄²×C₅	C₅×C₈²	1	1536	yes	direct product factor
115	C₄²×C₅	C₂×C₄×C₅×C₈	1	1728	yes	—
116	C₄²×C₅	C₂²×C₄²×C₅	1	96	yes	—
117	C₂³×C₈	C₂²×C₄×C₅²	1	24	yes	—
118	C₂²×C₄²	C₄²×C₅²	1	16	yes	—
119	C₂²×C₄²	C₂²×C₄×C₅²	1	18	yes	—
120	C₂²×C₄²	C₂⁴×C₅²	1	1	yes	characteristic
121	C₂×C₄×C₈	C₄²×C₅²	1	288	yes	—
122	C₂×C₄×C₈	C₂×C₅²×C₈	1	576	yes	—
123	C₂×C₄×C₈	C₂²×C₄×C₅²	1	144	yes	—
124	C₄³	C₂×C₅²×C₈	1	96	yes	—
125	C₄³	C₂²×C₄×C₅²	1	24	yes	—
126	C₈²	C₄²×C₅²	1	256	yes	direct product factor
127	C₂×C₅²	C₂×C₄×C₈²	1	12	yes	—
128	C₂×C₅²	C₄³×C₈	1	3	yes	—
129	C₂³×C₅	C₂×C₄³×C₅	1	18	yes	—
130	C₂³×C₅	C₂²×C₄×C₅×C₈	1	72	yes	—
131	C₂×C₄×C₅	C₂×C₅×C₈²	1	1152	yes	—
132	C₂×C₄×C₅	C₄²×C₅×C₈	1	864	yes	—
133	C₂×C₄×C₅	C₂×C₄³×C₅	1	72	yes	—
134	C₂×C₄×C₅	C₂²×C₄×C₅×C₈	1	432	yes	—
135	C₅×C₈	C₄²×C₅×C₈	1	1152	yes	direct product factor
136	C₂³×C₄	C₂×C₄²×C₅²	1	12	yes	—
137	C₂³×C₄	C₂³×C₄×C₅²	1	3	yes	—
138	C₂²×C₈	C₂×C₄²×C₅²	1	144	yes	—
139	C₂²×C₈	C₂²×C₅²×C₈	1	192	yes	—
140	C₂×C₄²	C₄×C₅²×C₈	1	192	yes	—
141	C₂×C₄²	C₂×C₄²×C₅²	1	72	yes	—
142	C₂×C₄²	C₂²×C₅²×C₈	1	144	yes	—
143	C₂×C₄²	C₂³×C₄×C₅²	1	12	yes	—
144	C₄×C₈	C₄×C₅²×C₈	1	1152	yes	direct product factor
145	C₄×C₈	C₂×C₄²×C₅²	1	192	yes	—
146	C₅²	C₄²×C₈²	1	1	yes	characteristic, direct product factor, Sylow
147	C₂²×C₅	C₄⁴×C₅	1	6	yes	—
148	C₂²×C₅	C₂²×C₅×C₈²	1	96	yes	—
149	C₂²×C₅	C₂×C₄²×C₅×C₈	1	108	yes	—
150	C₄×C₅	C₄×C₅×C₈²	1	576	yes	direct product factor
151	C₄×C₅	C₂×C₄²×C₅×C₈	1	144	yes	—
152	C₂⁴	C₂²×C₄²×C₅²	1	1	yes	characteristic
153	C₂²×C₄	C₄³×C₅²	1	24	yes	—
154	C₂²×C₄	C₂×C₄×C₅²×C₈	1	144	yes	—
155	C₂²×C₄	C₂²×C₄²×C₅²	1	18	yes	—
156	C₂²×C₄	C₂³×C₅²×C₈	1	24	yes	—
157	C₂×C₈	C₄³×C₅²	1	96	yes	—
158	C₂×C₈	C₂×C₄×C₅²×C₈	1	576	yes	—
159	C₄²	C₅²×C₈²	1	256	yes	direct product factor
160	C₄²	C₂×C₄×C₅²×C₈	1	288	yes	—
161	C₄²	C₂²×C₄²×C₅²	1	16	yes	—
162	C₂×C₅	C₂×C₄×C₅×C₈²	1	72	yes	—
163	C₂×C₅	C₄³×C₅×C₈	1	18	yes	—
164	C₂³	C₂²×C₄×C₅²×C₈	1	12	yes	—
165	C₂³	C₂×C₄³×C₅²	1	3	yes	—
166	C₂×C₄	C₂×C₅²×C₈²	1	192	yes	—
167	C₂×C₄	C₄²×C₅²×C₈	1	144	yes	—
168	C₂×C₄	C₂²×C₄×C₅²×C₈	1	72	yes	—
169	C₂×C₄	C₂×C₄³×C₅²	1	12	yes	—
170	C₈	C₄²×C₅²×C₈	1	192	yes	direct product factor
171	C₅	C₄²×C₅×C₈²	1	6	yes	direct product factor
172	C₂²	C₂²×C₅²×C₈²	1	16	yes	—
173	C₂²	C₂×C₄²×C₅²×C₈	1	18	yes	—
174	C₂²	C₄⁴×C₅²	1	1	yes	characteristic
175	C₄	C₄×C₅²×C₈²	1	96	yes	direct product factor
176	C₄	C₂×C₄²×C₅²×C₈	1	24	yes	—
177	C₂	C₂×C₄×C₅²×C₈²	1	12	yes	—
178	C₂	C₄³×C₅²×C₈	1	3	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂²×C₄²	C₂⁴×C₅²
socle, prime radical	C₂⁴×C₅²	C₂²×C₄²
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
7. C₄×C₅²×C₈²	C₄	direct (N × H)
8. C₄²×C₅×C₈²	C₅	direct (N × H)
13. C₄²×C₅²×C₈	C₈	direct (N × H)
18. C₄²×C₈²	C₅²	direct (N × H)
28. C₅²×C₈²	C₄²	direct (N × H)
33. C₄×C₅×C₈²	C₄×C₅	direct (N × H)
42. C₄×C₅²×C₈	C₄×C₈	direct (N × H)
48. C₄²×C₅×C₈	C₅×C₈	direct (N × H)
60. C₄²×C₅²	C₈²	direct (N × H)
72. C₅×C₈²	C₄²×C₅	direct (N × H)
77. C₄×C₈²	C₄×C₅²	direct (N × H)
84. C₅²×C₈	C₄²×C₈	direct (N × H)
93. C₄×C₅×C₈	C₄×C₅×C₈	direct (N × H)
99. C₄²×C₈	C₅²×C₈	direct (N × H)
105. C₄×C₅²	C₄×C₈²	direct (N × H)
114. C₄²×C₅	C₅×C₈²	direct (N × H)
126. C₈²	C₄²×C₅²	direct (N × H)
135. C₅×C₈	C₄²×C₅×C₈	direct (N × H)
144. C₄×C₈	C₄×C₅²×C₈	direct (N × H)
146. C₅²	C₄²×C₈²	direct (N × H)
150. C₄×C₅	C₄×C₅×C₈²	direct (N × H)
159. C₄²	C₅²×C₈²	direct (N × H)
170. C₈	C₄²×C₅²×C₈	direct (N × H)
171. C₅	C₄²×C₅×C₈²	direct (N × H)
175. C₄	C₄×C₅²×C₈²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄³×C₅²×C₈	2
C₂×C₄×C₅²×C₈²	2
C₄²×C₅×C₈²	5

Subgroups — C42×C52×C82 #72202829

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄²×C₅²×C₈² #72202829