Subgroups — C₂⁵×C₃×C₉×C₂₇ #72200708

Order 23328.

75548 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₉×C₂₇	C₂	1	31	yes	maximal, direct product factor
2	C₂⁵×C₃²×C₂₇	C₃	1	3	yes	maximal
3	C₂⁵×C₃×C₉²	C₃	1	1	yes	maximal, characteristic
4	C₂⁵×C₉×C₂₇	C₃	1	9	yes	maximal, direct product factor
5	C₂³×C₃×C₉×C₂₇	C₂²	1	155	yes	direct product factor
6	C₂⁴×C₃²×C₂₇	C₂×C₃	1	93	yes	—
7	C₂⁴×C₃×C₉²	C₂×C₃	1	31	yes	—
8	C₂⁴×C₉×C₂₇	C₂×C₃	1	279	yes	direct product factor
9	C₂²×C₃×C₉×C₂₇	C₂³	1	155	yes	direct product factor
10	C₂⁵×C₃²×C₉	C₉	1	3	yes	—
11	C₂⁵×C₃²×C₉	C₃²	1	1	yes	characteristic
12	C₂⁵×C₉²	C₉	1	6	yes	—
13	C₂⁵×C₉²	C₃²	1	3	yes	—
14	C₂⁵×C₃×C₂₇	C₉	1	27	yes	direct product factor
15	C₂⁵×C₃×C₂₇	C₃²	1	9	yes	—
16	C₂³×C₃²×C₂₇	C₂²×C₃	1	465	yes	—
17	C₂³×C₃×C₉²	C₂²×C₃	1	155	yes	—
18	C₂³×C₉×C₂₇	C₂²×C₃	1	1395	yes	direct product factor
19	C₂×C₃×C₉×C₂₇	C₂⁴	1	31	yes	direct product factor
20	C₂⁴×C₃²×C₉	C₂×C₉	1	93	yes	—
21	C₂⁴×C₃²×C₉	C₂×C₃²	1	31	yes	—
22	C₂⁴×C₃×C₂₇	C₂×C₉	1	837	yes	direct product factor
23	C₂⁴×C₃×C₂₇	C₂×C₃²	1	279	yes	—
24	C₂⁴×C₉²	C₂×C₉	1	186	yes	—
25	C₂⁴×C₉²	C₂×C₃²	1	93	yes	—
26	C₂²×C₃²×C₂₇	C₂³×C₃	1	465	yes	—
27	C₂²×C₃×C₉²	C₂³×C₃	1	155	yes	—
28	C₂²×C₉×C₂₇	C₂³×C₃	1	1395	yes	direct product factor
29	C₂⁵×C₃³	C₃×C₉	1	1	yes	characteristic
30	C₂⁵×C₃×C₉	C₂₇	1	27	yes	direct product factor
31	C₂⁵×C₃×C₉	C₃×C₉	1	2	yes	—
32	C₂⁵×C₃×C₉	C₃×C₉	1	9	yes	—
33	C₂⁵×C₃×C₉	C₃×C₉	1	9	yes	—
34	C₂⁵×C₃×C₉	C₃³	1	1	yes	characteristic
35	C₂⁵×C₂₇	C₃×C₉	1	27	yes	direct product factor
36	C₃×C₉×C₂₇	C₂⁵	1	1	yes	characteristic, direct product factor, Sylow
37	C₂³×C₃²×C₉	C₂²×C₉	1	465	yes	—
38	C₂³×C₃²×C₉	C₂²×C₃²	1	155	yes	—
39	C₂³×C₃×C₂₇	C₂²×C₉	1	4185	yes	direct product factor
40	C₂³×C₃×C₂₇	C₂²×C₃²	1	1395	yes	—
41	C₂³×C₉²	C₂²×C₉	1	930	yes	—
42	C₂³×C₉²	C₂²×C₃²	1	465	yes	—
43	C₂×C₃²×C₂₇	C₂⁴×C₃	1	93	yes	—
44	C₂×C₃×C₉²	C₂⁴×C₃	1	31	yes	—
45	C₂×C₉×C₂₇	C₂⁴×C₃	1	279	yes	direct product factor
46	C₂⁴×C₃³	C₂×C₃×C₉	1	31	yes	—
47	C₂⁴×C₃×C₉	C₂×C₂₇	1	837	yes	direct product factor
48	C₂⁴×C₃×C₉	C₂×C₃×C₉	1	279	yes	—
49	C₂⁴×C₃×C₉	C₂×C₃×C₉	1	279	yes	—
50	C₂⁴×C₃×C₉	C₂×C₃×C₉	1	62	yes	—
51	C₂⁴×C₃×C₉	C₂×C₃³	1	31	yes	—
52	C₂⁴×C₂₇	C₂×C₃×C₉	1	837	yes	direct product factor
53	C₂²×C₃²×C₉	C₂³×C₉	1	465	yes	—
54	C₂²×C₃²×C₉	C₂³×C₃²	1	155	yes	—
55	C₂²×C₃×C₂₇	C₂³×C₉	1	4185	yes	direct product factor
56	C₂²×C₃×C₂₇	C₂³×C₃²	1	1395	yes	—
57	C₂²×C₉²	C₂³×C₉	1	930	yes	—
58	C₂²×C₉²	C₂³×C₃²	1	465	yes	—
59	C₂⁵×C₃²	C₉²	1	3	yes	—
60	C₂⁵×C₃²	C₃×C₂₇	1	9	yes	—
61	C₂⁵×C₃²	C₃²×C₉	1	1	yes	characteristic
62	C₂⁵×C₉	C₉²	1	6	yes	—
63	C₂⁵×C₉	C₃×C₂₇	1	27	yes	direct product factor
64	C₂⁵×C₉	C₃²×C₉	1	3	yes	—
65	C₃²×C₂₇	C₂⁵×C₃	1	3	yes	—
66	C₃×C₉²	C₂⁵×C₃	1	1	yes	characteristic
67	C₉×C₂₇	C₂⁵×C₃	1	9	yes	direct product factor
68	C₂³×C₃³	C₂²×C₃×C₉	1	155	yes	—
69	C₂³×C₃×C₉	C₂²×C₂₇	1	4185	yes	direct product factor
70	C₂³×C₃×C₉	C₂²×C₃×C₉	1	310	yes	—
71	C₂³×C₃×C₉	C₂²×C₃×C₉	1	1395	yes	—
72	C₂³×C₃×C₉	C₂²×C₃×C₉	1	1395	yes	—
73	C₂³×C₃×C₉	C₂²×C₃³	1	155	yes	—
74	C₂³×C₂₇	C₂²×C₃×C₉	1	4185	yes	direct product factor
75	C₂×C₃²×C₉	C₂⁴×C₉	1	93	yes	—
76	C₂×C₃²×C₉	C₂⁴×C₃²	1	31	yes	—
77	C₂×C₃×C₂₇	C₂⁴×C₉	1	837	yes	direct product factor
78	C₂×C₃×C₂₇	C₂⁴×C₃²	1	279	yes	—
79	C₂×C₉²	C₂⁴×C₉	1	186	yes	—
80	C₂×C₉²	C₂⁴×C₃²	1	93	yes	—
81	C₂⁴×C₃²	C₂×C₉²	1	93	yes	—
82	C₂⁴×C₃²	C₂×C₃×C₂₇	1	279	yes	—
83	C₂⁴×C₃²	C₂×C₃²×C₉	1	31	yes	—
84	C₂⁴×C₉	C₂×C₉²	1	186	yes	—
85	C₂⁴×C₉	C₂×C₃×C₂₇	1	837	yes	direct product factor
86	C₂⁴×C₉	C₂×C₃²×C₉	1	93	yes	—
87	C₂²×C₃³	C₂³×C₃×C₉	1	155	yes	—
88	C₂²×C₃×C₉	C₂³×C₂₇	1	4185	yes	direct product factor
89	C₂²×C₃×C₉	C₂³×C₃×C₉	1	310	yes	—
90	C₂²×C₃×C₉	C₂³×C₃×C₉	1	1395	yes	—
91	C₂²×C₃×C₉	C₂³×C₃×C₉	1	1395	yes	—
92	C₂²×C₃×C₉	C₂³×C₃³	1	155	yes	—
93	C₂²×C₂₇	C₂³×C₃×C₉	1	4185	yes	direct product factor
94	C₂⁵×C₃	C₉×C₂₇	1	9	yes	direct product factor
95	C₂⁵×C₃	C₃×C₉²	1	1	yes	characteristic
96	C₂⁵×C₃	C₃²×C₂₇	1	3	yes	—
97	C₃²×C₉	C₂⁵×C₉	1	3	yes	—
98	C₃²×C₉	C₂⁵×C₃²	1	1	yes	characteristic
99	C₃×C₂₇	C₂⁵×C₉	1	27	yes	direct product factor
100	C₃×C₂₇	C₂⁵×C₃²	1	9	yes	—
101	C₉²	C₂⁵×C₉	1	6	yes	—
102	C₉²	C₂⁵×C₃²	1	3	yes	—
103	C₂³×C₃²	C₂²×C₉²	1	465	yes	—
104	C₂³×C₃²	C₂²×C₃×C₂₇	1	1395	yes	—
105	C₂³×C₃²	C₂²×C₃²×C₉	1	155	yes	—
106	C₂³×C₉	C₂²×C₉²	1	930	yes	—
107	C₂³×C₉	C₂²×C₃×C₂₇	1	4185	yes	direct product factor
108	C₂³×C₉	C₂²×C₃²×C₉	1	465	yes	—
109	C₂×C₃³	C₂⁴×C₃×C₉	1	31	yes	—
110	C₂×C₃×C₉	C₂⁴×C₂₇	1	837	yes	direct product factor
111	C₂×C₃×C₉	C₂⁴×C₃×C₉	1	279	yes	—
112	C₂×C₃×C₉	C₂⁴×C₃×C₉	1	279	yes	—
113	C₂×C₃×C₉	C₂⁴×C₃×C₉	1	62	yes	—
114	C₂×C₃×C₉	C₂⁴×C₃³	1	31	yes	—
115	C₂×C₂₇	C₂⁴×C₃×C₉	1	837	yes	direct product factor
116	C₂⁴×C₃	C₂×C₉×C₂₇	1	279	yes	direct product factor
117	C₂⁴×C₃	C₂×C₃×C₉²	1	31	yes	—
118	C₂⁴×C₃	C₂×C₃²×C₂₇	1	93	yes	—
119	C₂²×C₃²	C₂³×C₉²	1	465	yes	—
120	C₂²×C₃²	C₂³×C₃×C₂₇	1	1395	yes	—
121	C₂²×C₃²	C₂³×C₃²×C₉	1	155	yes	—
122	C₂²×C₉	C₂³×C₉²	1	930	yes	—
123	C₂²×C₉	C₂³×C₃×C₂₇	1	4185	yes	direct product factor
124	C₂²×C₉	C₂³×C₃²×C₉	1	465	yes	—
125	C₂⁵	C₃×C₉×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
126	C₃³	C₂⁵×C₃×C₉	1	1	yes	characteristic
127	C₃×C₉	C₂⁵×C₂₇	1	27	yes	direct product factor
128	C₃×C₉	C₂⁵×C₃×C₉	1	2	yes	—
129	C₃×C₉	C₂⁵×C₃×C₉	1	9	yes	—
130	C₃×C₉	C₂⁵×C₃×C₉	1	9	yes	—
131	C₃×C₉	C₂⁵×C₃³	1	1	yes	characteristic
132	C₂₇	C₂⁵×C₃×C₉	1	27	yes	direct product factor
133	C₂³×C₃	C₂²×C₉×C₂₇	1	1395	yes	direct product factor
134	C₂³×C₃	C₂²×C₃×C₉²	1	155	yes	—
135	C₂³×C₃	C₂²×C₃²×C₂₇	1	465	yes	—
136	C₂×C₃²	C₂⁴×C₉²	1	93	yes	—
137	C₂×C₃²	C₂⁴×C₃×C₂₇	1	279	yes	—
138	C₂×C₃²	C₂⁴×C₃²×C₉	1	31	yes	—
139	C₂×C₉	C₂⁴×C₉²	1	186	yes	—
140	C₂×C₉	C₂⁴×C₃×C₂₇	1	837	yes	direct product factor
141	C₂×C₉	C₂⁴×C₃²×C₉	1	93	yes	—
142	C₂⁴	C₂×C₃×C₉×C₂₇	1	31	yes	direct product factor
143	C₂²×C₃	C₂³×C₉×C₂₇	1	1395	yes	direct product factor
144	C₂²×C₃	C₂³×C₃×C₉²	1	155	yes	—
145	C₂²×C₃	C₂³×C₃²×C₂₇	1	465	yes	—
146	C₃²	C₂⁵×C₃×C₂₇	1	9	yes	—
147	C₃²	C₂⁵×C₉²	1	3	yes	—
148	C₃²	C₂⁵×C₃²×C₉	1	1	yes	characteristic
149	C₉	C₂⁵×C₃×C₂₇	1	27	yes	direct product factor
150	C₉	C₂⁵×C₉²	1	6	yes	—
151	C₉	C₂⁵×C₃²×C₉	1	3	yes	—
152	C₂³	C₂²×C₃×C₉×C₂₇	1	155	yes	direct product factor
153	C₂×C₃	C₂⁴×C₉×C₂₇	1	279	yes	direct product factor
154	C₂×C₃	C₂⁴×C₃×C₉²	1	31	yes	—
155	C₂×C₃	C₂⁴×C₃²×C₂₇	1	93	yes	—
156	C₂²	C₂³×C₃×C₉×C₂₇	1	155	yes	direct product factor
157	C₃	C₂⁵×C₉×C₂₇	1	9	yes	direct product factor
158	C₃	C₂⁵×C₃×C₉²	1	1	yes	characteristic
159	C₃	C₂⁵×C₃²×C₂₇	1	3	yes	—
160	C₂	C₂⁴×C₃×C₉×C₂₇	1	31	yes	direct product factor

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₉	C₂⁵×C₃³
socle, prime radical	C₂⁵×C₃³	C₃×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃×C₉×C₂₇	C₂	direct (N × H)
4. C₂⁵×C₉×C₂₇	C₃	direct (N × H)
5. C₂³×C₃×C₉×C₂₇	C₂²	direct (N × H)
8. C₂⁴×C₉×C₂₇	C₂×C₃	direct (N × H)
9. C₂²×C₃×C₉×C₂₇	C₂³	direct (N × H)
14. C₂⁵×C₃×C₂₇	C₉	direct (N × H)
18. C₂³×C₉×C₂₇	C₂²×C₃	direct (N × H)
19. C₂×C₃×C₉×C₂₇	C₂⁴	direct (N × H)
22. C₂⁴×C₃×C₂₇	C₂×C₉	direct (N × H)
28. C₂²×C₉×C₂₇	C₂³×C₃	direct (N × H)
30. C₂⁵×C₃×C₉	C₂₇	direct (N × H)
35. C₂⁵×C₂₇	C₃×C₉	direct (N × H)
36. C₃×C₉×C₂₇	C₂⁵	direct (N × H)
39. C₂³×C₃×C₂₇	C₂²×C₉	direct (N × H)
45. C₂×C₉×C₂₇	C₂⁴×C₃	direct (N × H)
47. C₂⁴×C₃×C₉	C₂×C₂₇	direct (N × H)
52. C₂⁴×C₂₇	C₂×C₃×C₉	direct (N × H)
55. C₂²×C₃×C₂₇	C₂³×C₉	direct (N × H)
63. C₂⁵×C₉	C₃×C₂₇	direct (N × H)
67. C₉×C₂₇	C₂⁵×C₃	direct (N × H)
69. C₂³×C₃×C₉	C₂²×C₂₇	direct (N × H)
74. C₂³×C₂₇	C₂²×C₃×C₉	direct (N × H)
77. C₂×C₃×C₂₇	C₂⁴×C₉	direct (N × H)
85. C₂⁴×C₉	C₂×C₃×C₂₇	direct (N × H)
88. C₂²×C₃×C₉	C₂³×C₂₇	direct (N × H)
93. C₂²×C₂₇	C₂³×C₃×C₉	direct (N × H)
94. C₂⁵×C₃	C₉×C₂₇	direct (N × H)
99. C₃×C₂₇	C₂⁵×C₉	direct (N × H)
107. C₂³×C₉	C₂²×C₃×C₂₇	direct (N × H)
110. C₂×C₃×C₉	C₂⁴×C₂₇	direct (N × H)
115. C₂×C₂₇	C₂⁴×C₃×C₉	direct (N × H)
116. C₂⁴×C₃	C₂×C₉×C₂₇	direct (N × H)
123. C₂²×C₉	C₂³×C₃×C₂₇	direct (N × H)
125. C₂⁵	C₃×C₉×C₂₇	direct (N × H)
127. C₃×C₉	C₂⁵×C₂₇	direct (N × H)
132. C₂₇	C₂⁵×C₃×C₉	direct (N × H)
133. C₂³×C₃	C₂²×C₉×C₂₇	direct (N × H)
140. C₂×C₉	C₂⁴×C₃×C₂₇	direct (N × H)
142. C₂⁴	C₂×C₃×C₉×C₂₇	direct (N × H)
143. C₂²×C₃	C₂³×C₉×C₂₇	direct (N × H)
149. C₉	C₂⁵×C₃×C₂₇	direct (N × H)
152. C₂³	C₂²×C₃×C₉×C₂₇	direct (N × H)
153. C₂×C₃	C₂⁴×C₉×C₂₇	direct (N × H)
156. C₂²	C₂³×C₃×C₉×C₂₇	direct (N × H)
157. C₃	C₂⁵×C₉×C₂₇	direct (N × H)
160. C₂	C₂⁴×C₃×C₉×C₂₇	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×C₃×C₉×C₂₇	2
C₂⁵×C₃²×C₂₇	3
C₂⁵×C₃×C₉²	3
C₂⁵×C₉×C₂₇	3

Subgroups — C25×C3×C9×C27 #72200708

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁵×C₃×C₉×C₂₇ #72200708