Subgroups — C₂³×C₈×C₉×C₂₅ #72192562

Order 14400.

1521 conjugacy classes of subgroups.

The subgroup lattice has 171 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₄×C₉×C₂₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×C₉×C₂₅	C₂	1	14	yes	maximal, direct product factor
3	C₂³×C₃×C₈×C₂₅	C₃	1	1	yes	maximal, characteristic
4	C₂⁴×C₉×C₂₅	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₉×C₂₅	C₄	1	7	yes	—
6	C₂²×C₄×C₉×C₂₅	C₂²	1	7	yes	—
7	C₂×C₈×C₉×C₂₅	C₂²	1	28	yes	direct product factor
8	C₂³×C₅×C₈×C₉	C₅	1	1	yes	maximal, characteristic
9	C₂³×C₃×C₄×C₂₅	C₂×C₃	1	1	yes	characteristic
10	C₂²×C₃×C₈×C₂₅	C₂×C₃	1	14	yes	—
11	C₂³×C₉×C₂₅	C₈	1	8	yes	direct product factor
12	C₂³×C₉×C₂₅	C₂×C₄	1	7	yes	—
13	C₂×C₄×C₉×C₂₅	C₂×C₄	1	21	yes	—
14	C₂×C₄×C₉×C₂₅	C₂³	1	7	yes	—
15	C₈×C₉×C₂₅	C₂³	1	8	yes	direct product factor
16	C₂³×C₈×C₂₅	C₉	1	1	yes	characteristic, direct product factor, Hall
17	C₂³×C₄×C₅×C₉	C₂×C₅	1	1	yes	characteristic
18	C₂²×C₅×C₈×C₉	C₂×C₅	1	14	yes	—
19	C₂⁴×C₃×C₂₅	C₃×C₄	1	1	yes	characteristic
20	C₂²×C₃×C₄×C₂₅	C₃×C₄	1	7	yes	—
21	C₂²×C₃×C₄×C₂₅	C₂²×C₃	1	7	yes	—
22	C₂×C₃×C₈×C₂₅	C₂²×C₃	1	28	yes	—
23	C₂³×C₃×C₅×C₈	C₃×C₅	1	1	yes	characteristic
24	C₂²×C₉×C₂₅	C₂×C₈	1	28	yes	direct product factor
25	C₂²×C₉×C₂₅	C₂²×C₄	1	7	yes	—
26	C₄×C₉×C₂₅	C₂²×C₄	1	7	yes	—
27	C₄×C₉×C₂₅	C₂⁴	1	1	yes	characteristic
28	C₂³×C₄×C₂₅	C₂×C₉	1	1	yes	characteristic
29	C₂²×C₈×C₂₅	C₂×C₉	1	14	yes	direct product factor
30	C₂⁴×C₅×C₉	C₄×C₅	1	1	yes	characteristic
31	C₂²×C₄×C₅×C₉	C₄×C₅	1	7	yes	—
32	C₂²×C₄×C₅×C₉	C₂²×C₅	1	7	yes	—
33	C₂×C₅×C₈×C₉	C₂²×C₅	1	28	yes	—
34	C₂³×C₃×C₂₅	C₃×C₈	1	8	yes	—
35	C₂³×C₃×C₂₅	C₂×C₃×C₄	1	7	yes	—
36	C₂×C₃×C₄×C₂₅	C₂×C₃×C₄	1	21	yes	—
37	C₂×C₃×C₄×C₂₅	C₂³×C₃	1	7	yes	—
38	C₃×C₈×C₂₅	C₂³×C₃	1	8	yes	—
39	C₂³×C₈×C₉	C₂₅	1	1	yes	characteristic, direct product factor, Hall
40	C₂³×C₃×C₄×C₅	C₂×C₃×C₅	1	1	yes	characteristic
41	C₂²×C₃×C₅×C₈	C₂×C₃×C₅	1	14	yes	—
42	C₂×C₉×C₂₅	C₂²×C₈	1	14	yes	direct product factor
43	C₂×C₉×C₂₅	C₂³×C₄	1	1	yes	characteristic
44	C₂⁴×C₂₅	C₄×C₉	1	1	yes	characteristic
45	C₂²×C₄×C₂₅	C₄×C₉	1	7	yes	—
46	C₂²×C₄×C₂₅	C₂²×C₉	1	7	yes	—
47	C₂×C₈×C₂₅	C₂²×C₉	1	28	yes	direct product factor
48	C₂³×C₅×C₉	C₅×C₈	1	8	yes	—
49	C₂³×C₅×C₉	C₂×C₄×C₅	1	7	yes	—
50	C₂×C₄×C₅×C₉	C₂×C₄×C₅	1	21	yes	—
51	C₂×C₄×C₅×C₉	C₂³×C₅	1	7	yes	—
52	C₅×C₈×C₉	C₂³×C₅	1	8	yes	—
53	C₂³×C₅×C₈	C₅×C₉	1	1	yes	characteristic
54	C₂²×C₃×C₂₅	C₂×C₃×C₈	1	28	yes	—
55	C₂²×C₃×C₂₅	C₂²×C₃×C₄	1	7	yes	—
56	C₃×C₄×C₂₅	C₂²×C₃×C₄	1	7	yes	—
57	C₃×C₄×C₂₅	C₂⁴×C₃	1	1	yes	characteristic
58	C₂³×C₄×C₉	C₂×C₂₅	1	1	yes	characteristic
59	C₂²×C₈×C₉	C₂×C₂₅	1	14	yes	direct product factor
60	C₂⁴×C₃×C₅	C₃×C₄×C₅	1	1	yes	characteristic
61	C₂²×C₃×C₄×C₅	C₃×C₄×C₅	1	7	yes	—
62	C₂²×C₃×C₄×C₅	C₂²×C₃×C₅	1	7	yes	—
63	C₂×C₃×C₅×C₈	C₂²×C₃×C₅	1	28	yes	—
64	C₉×C₂₅	C₂³×C₈	1	1	yes	characteristic, direct product factor, Hall
65	C₂³×C₂₅	C₈×C₉	1	8	yes	direct product factor
66	C₂³×C₂₅	C₂×C₄×C₉	1	7	yes	—
67	C₂×C₄×C₂₅	C₂×C₄×C₉	1	21	yes	—
68	C₂×C₄×C₂₅	C₂³×C₉	1	7	yes	—
69	C₈×C₂₅	C₂³×C₉	1	8	yes	direct product factor
70	C₂³×C₃×C₈	C₃×C₂₅	1	1	yes	characteristic
71	C₂²×C₅×C₉	C₂×C₅×C₈	1	28	yes	—
72	C₂²×C₅×C₉	C₂²×C₄×C₅	1	7	yes	—
73	C₄×C₅×C₉	C₂²×C₄×C₅	1	7	yes	—
74	C₄×C₅×C₉	C₂⁴×C₅	1	1	yes	characteristic
75	C₂³×C₄×C₅	C₂×C₅×C₉	1	1	yes	characteristic
76	C₂²×C₅×C₈	C₂×C₅×C₉	1	14	yes	—
77	C₂×C₃×C₂₅	C₂²×C₃×C₈	1	14	yes	—
78	C₂×C₃×C₂₅	C₂³×C₃×C₄	1	1	yes	characteristic
79	C₂⁴×C₉	C₄×C₂₅	1	1	yes	characteristic
80	C₂²×C₄×C₉	C₄×C₂₅	1	7	yes	—
81	C₂²×C₄×C₉	C₂²×C₂₅	1	7	yes	—
82	C₂×C₈×C₉	C₂²×C₂₅	1	28	yes	direct product factor
83	C₂³×C₃×C₅	C₃×C₅×C₈	1	8	yes	—
84	C₂³×C₃×C₅	C₂×C₃×C₄×C₅	1	7	yes	—
85	C₂×C₃×C₄×C₅	C₂×C₃×C₄×C₅	1	21	yes	—
86	C₂×C₃×C₄×C₅	C₂³×C₃×C₅	1	7	yes	—
87	C₃×C₅×C₈	C₂³×C₃×C₅	1	8	yes	—
88	C₂²×C₂₅	C₂×C₈×C₉	1	28	yes	direct product factor
89	C₂²×C₂₅	C₂²×C₄×C₉	1	7	yes	—
90	C₄×C₂₅	C₂²×C₄×C₉	1	7	yes	—
91	C₄×C₂₅	C₂⁴×C₉	1	1	yes	characteristic
92	C₂³×C₃×C₄	C₂×C₃×C₂₅	1	1	yes	characteristic
93	C₂²×C₃×C₈	C₂×C₃×C₂₅	1	14	yes	—
94	C₂×C₅×C₉	C₂²×C₅×C₈	1	14	yes	—
95	C₂×C₅×C₉	C₂³×C₄×C₅	1	1	yes	characteristic
96	C₂⁴×C₅	C₄×C₅×C₉	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₄×C₅×C₉	1	7	yes	—
98	C₂²×C₄×C₅	C₂²×C₅×C₉	1	7	yes	—
99	C₂×C₅×C₈	C₂²×C₅×C₉	1	28	yes	—
100	C₃×C₂₅	C₂³×C₃×C₈	1	1	yes	characteristic
101	C₂³×C₉	C₈×C₂₅	1	8	yes	direct product factor
102	C₂³×C₉	C₂×C₄×C₂₅	1	7	yes	—
103	C₂×C₄×C₉	C₂×C₄×C₂₅	1	21	yes	—
104	C₂×C₄×C₉	C₂³×C₂₅	1	7	yes	—
105	C₈×C₉	C₂³×C₂₅	1	8	yes	direct product factor
106	C₂³×C₈	C₉×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
107	C₂²×C₃×C₅	C₂×C₃×C₅×C₈	1	28	yes	—
108	C₂²×C₃×C₅	C₂²×C₃×C₄×C₅	1	7	yes	—
109	C₃×C₄×C₅	C₂²×C₃×C₄×C₅	1	7	yes	—
110	C₃×C₄×C₅	C₂⁴×C₃×C₅	1	1	yes	characteristic
111	C₂×C₂₅	C₂²×C₈×C₉	1	14	yes	direct product factor
112	C₂×C₂₅	C₂³×C₄×C₉	1	1	yes	characteristic
113	C₂⁴×C₃	C₃×C₄×C₂₅	1	1	yes	characteristic
114	C₂²×C₃×C₄	C₃×C₄×C₂₅	1	7	yes	—
115	C₂²×C₃×C₄	C₂²×C₃×C₂₅	1	7	yes	—
116	C₂×C₃×C₈	C₂²×C₃×C₂₅	1	28	yes	—
117	C₅×C₉	C₂³×C₅×C₈	1	1	yes	characteristic
118	C₂³×C₅	C₅×C₈×C₉	1	8	yes	—
119	C₂³×C₅	C₂×C₄×C₅×C₉	1	7	yes	—
120	C₂×C₄×C₅	C₂×C₄×C₅×C₉	1	21	yes	—
121	C₂×C₄×C₅	C₂³×C₅×C₉	1	7	yes	—
122	C₅×C₈	C₂³×C₅×C₉	1	8	yes	—
123	C₂²×C₉	C₂×C₈×C₂₅	1	28	yes	direct product factor
124	C₂²×C₉	C₂²×C₄×C₂₅	1	7	yes	—
125	C₄×C₉	C₂²×C₄×C₂₅	1	7	yes	—
126	C₄×C₉	C₂⁴×C₂₅	1	1	yes	characteristic
127	C₂³×C₄	C₂×C₉×C₂₅	1	1	yes	characteristic
128	C₂²×C₈	C₂×C₉×C₂₅	1	14	yes	direct product factor
129	C₂×C₃×C₅	C₂²×C₃×C₅×C₈	1	14	yes	—
130	C₂×C₃×C₅	C₂³×C₃×C₄×C₅	1	1	yes	characteristic
131	C₂₅	C₂³×C₈×C₉	1	1	yes	characteristic, direct product factor, Sylow
132	C₂³×C₃	C₃×C₈×C₂₅	1	8	yes	—
133	C₂³×C₃	C₂×C₃×C₄×C₂₅	1	7	yes	—
134	C₂×C₃×C₄	C₂×C₃×C₄×C₂₅	1	21	yes	—
135	C₂×C₃×C₄	C₂³×C₃×C₂₅	1	7	yes	—
136	C₃×C₈	C₂³×C₃×C₂₅	1	8	yes	—
137	C₂²×C₅	C₂×C₅×C₈×C₉	1	28	yes	—
138	C₂²×C₅	C₂²×C₄×C₅×C₉	1	7	yes	—
139	C₄×C₅	C₂²×C₄×C₅×C₉	1	7	yes	—
140	C₄×C₅	C₂⁴×C₅×C₉	1	1	yes	characteristic
141	C₂×C₉	C₂²×C₈×C₂₅	1	14	yes	direct product factor
142	C₂×C₉	C₂³×C₄×C₂₅	1	1	yes	characteristic
143	C₂⁴	C₄×C₉×C₂₅	1	1	yes	characteristic
144	C₂²×C₄	C₄×C₉×C₂₅	1	7	yes	—
145	C₂²×C₄	C₂²×C₉×C₂₅	1	7	yes	—
146	C₂×C₈	C₂²×C₉×C₂₅	1	28	yes	direct product factor
147	C₃×C₅	C₂³×C₃×C₅×C₈	1	1	yes	characteristic
148	C₂²×C₃	C₂×C₃×C₈×C₂₅	1	28	yes	—
149	C₂²×C₃	C₂²×C₃×C₄×C₂₅	1	7	yes	—
150	C₃×C₄	C₂²×C₃×C₄×C₂₅	1	7	yes	—
151	C₃×C₄	C₂⁴×C₃×C₂₅	1	1	yes	characteristic
152	C₂×C₅	C₂²×C₅×C₈×C₉	1	14	yes	—
153	C₂×C₅	C₂³×C₄×C₅×C₉	1	1	yes	characteristic
154	C₉	C₂³×C₈×C₂₅	1	1	yes	characteristic, direct product factor, Sylow
155	C₂³	C₈×C₉×C₂₅	1	8	yes	direct product factor
156	C₂³	C₂×C₄×C₉×C₂₅	1	7	yes	—
157	C₂×C₄	C₂×C₄×C₉×C₂₅	1	21	yes	—
158	C₂×C₄	C₂³×C₉×C₂₅	1	7	yes	—
159	C₈	C₂³×C₉×C₂₅	1	8	yes	direct product factor
160	C₂×C₃	C₂²×C₃×C₈×C₂₅	1	14	yes	—
161	C₂×C₃	C₂³×C₃×C₄×C₂₅	1	1	yes	characteristic
162	C₅	C₂³×C₅×C₈×C₉	1	1	yes	characteristic
163	C₂²	C₂×C₈×C₉×C₂₅	1	28	yes	direct product factor
164	C₂²	C₂²×C₄×C₉×C₂₅	1	7	yes	—
165	C₄	C₂²×C₄×C₉×C₂₅	1	7	yes	—
166	C₄	C₂⁴×C₉×C₂₅	1	1	yes	characteristic
167	C₃	C₂³×C₃×C₈×C₂₅	1	1	yes	characteristic
168	C₂	C₂²×C₈×C₉×C₂₅	1	14	yes	direct product factor
169	C₂	C₂³×C₄×C₉×C₂₅	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₄×C₅	C₂⁴×C₃×C₅
socle, prime radical	C₂⁴×C₃×C₅	C₃×C₄×C₅
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₈×C₉×C₂₅	C₂	direct (N × H)
7. C₂×C₈×C₉×C₂₅	C₂²	direct (N × H)
11. C₂³×C₉×C₂₅	C₈	direct (N × H)
15. C₈×C₉×C₂₅	C₂³	direct (N × H)
16. C₂³×C₈×C₂₅	C₉	direct (N × H)
24. C₂²×C₉×C₂₅	C₂×C₈	direct (N × H)
29. C₂²×C₈×C₂₅	C₂×C₉	direct (N × H)
39. C₂³×C₈×C₉	C₂₅	direct (N × H)
42. C₂×C₉×C₂₅	C₂²×C₈	direct (N × H)
47. C₂×C₈×C₂₅	C₂²×C₉	direct (N × H)
59. C₂²×C₈×C₉	C₂×C₂₅	direct (N × H)
64. C₉×C₂₅	C₂³×C₈	direct (N × H)
65. C₂³×C₂₅	C₈×C₉	direct (N × H)
69. C₈×C₂₅	C₂³×C₉	direct (N × H)
82. C₂×C₈×C₉	C₂²×C₂₅	direct (N × H)
88. C₂²×C₂₅	C₂×C₈×C₉	direct (N × H)
101. C₂³×C₉	C₈×C₂₅	direct (N × H)
105. C₈×C₉	C₂³×C₂₅	direct (N × H)
106. C₂³×C₈	C₉×C₂₅	direct (N × H)
111. C₂×C₂₅	C₂²×C₈×C₉	direct (N × H)
123. C₂²×C₉	C₂×C₈×C₂₅	direct (N × H)
128. C₂²×C₈	C₂×C₉×C₂₅	direct (N × H)
131. C₂₅	C₂³×C₈×C₉	direct (N × H)
141. C₂×C₉	C₂²×C₈×C₂₅	direct (N × H)
146. C₂×C₈	C₂²×C₉×C₂₅	direct (N × H)
154. C₉	C₂³×C₈×C₂₅	direct (N × H)
155. C₂³	C₈×C₉×C₂₅	direct (N × H)
159. C₈	C₂³×C₉×C₂₅	direct (N × H)
163. C₂²	C₂×C₈×C₉×C₂₅	direct (N × H)
168. C₂	C₂²×C₈×C₉×C₂₅	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×C₄×C₉×C₂₅	2
C₂²×C₈×C₉×C₂₅	2
C₂³×C₃×C₈×C₂₅	3
C₂³×C₅×C₈×C₉	5

Subgroups — C23×C8×C9×C25 #72192562

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₈×C₉×C₂₅ #72192562