Subgroups — C₂×C₃²×C₅×C₉×C₂₇ #72199337

Order 21870.

6944 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₅×C₉×C₂₇	C₂	1	1	yes	maximal, characteristic, direct product factor, Hall
2	C₂×C₃²×C₅×C₉²	C₃	1	1	yes	maximal, characteristic
3	C₂×C₃³×C₅×C₂₇	C₃	1	3	yes	maximal
4	C₂×C₃×C₅×C₉×C₂₇	C₃	1	36	yes	maximal, direct product factor
5	C₂×C₃²×C₉×C₂₇	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂×C₃³×C₅×C₉	C₉	1	3	yes	—
7	C₂×C₃³×C₅×C₉	C₃²	1	1	yes	characteristic
8	C₂×C₃²×C₅×C₂₇	C₉	1	81	yes	direct product factor
9	C₂×C₃²×C₅×C₂₇	C₃²	1	36	yes	—
10	C₂×C₃×C₅×C₉²	C₉	1	24	yes	—
11	C₂×C₃×C₅×C₉²	C₃²	1	12	yes	—
12	C₂×C₅×C₉×C₂₇	C₃²	1	81	yes	direct product factor
13	C₃³×C₅×C₂₇	C₂×C₃	1	3	yes	—
14	C₃²×C₅×C₉²	C₂×C₃	1	1	yes	characteristic
15	C₃×C₅×C₉×C₂₇	C₂×C₃	1	36	yes	direct product factor
16	C₃²×C₉×C₂₇	C₂×C₅	1	1	yes	characteristic, direct product factor, Sylow
17	C₂×C₃³×C₂₇	C₃×C₅	1	3	yes	—
18	C₂×C₃²×C₉²	C₃×C₅	1	1	yes	characteristic
19	C₂×C₃×C₉×C₂₇	C₃×C₅	1	36	yes	direct product factor
20	C₃³×C₅×C₉	C₂×C₉	1	3	yes	—
21	C₃³×C₅×C₉	C₂×C₃²	1	1	yes	characteristic
22	C₃²×C₅×C₂₇	C₂×C₉	1	81	yes	direct product factor
23	C₃²×C₅×C₂₇	C₂×C₃²	1	36	yes	—
24	C₃×C₅×C₉²	C₂×C₉	1	24	yes	—
25	C₃×C₅×C₉²	C₂×C₃²	1	12	yes	—
26	C₅×C₉×C₂₇	C₂×C₃²	1	81	yes	direct product factor
27	C₂×C₃⁴×C₅	C₃×C₉	1	1	yes	characteristic
28	C₂×C₃²×C₅×C₉	C₂₇	1	81	yes	direct product factor
29	C₂×C₃²×C₅×C₉	C₃×C₉	1	8	yes	—
30	C₂×C₃²×C₅×C₉	C₃×C₉	1	27	yes	—
31	C₂×C₃²×C₅×C₉	C₃×C₉	1	36	yes	—
32	C₂×C₃²×C₅×C₉	C₃³	1	4	yes	—
33	C₂×C₃×C₅×C₂₇	C₃×C₉	1	324	yes	direct product factor
34	C₂×C₃×C₅×C₂₇	C₃³	1	27	yes	—
35	C₂×C₅×C₉²	C₃×C₉	1	72	yes	—
36	C₂×C₅×C₉²	C₃³	1	9	yes	—
37	C₃³×C₂₇	C₂×C₃×C₅	1	3	yes	—
38	C₃²×C₉²	C₂×C₃×C₅	1	1	yes	characteristic
39	C₃×C₉×C₂₇	C₂×C₃×C₅	1	36	yes	direct product factor
40	C₂×C₃³×C₉	C₅×C₉	1	3	yes	—
41	C₂×C₃³×C₉	C₃²×C₅	1	1	yes	characteristic
42	C₂×C₃²×C₂₇	C₅×C₉	1	81	yes	direct product factor
43	C₂×C₃²×C₂₇	C₃²×C₅	1	36	yes	—
44	C₂×C₃×C₉²	C₅×C₉	1	24	yes	—
45	C₂×C₃×C₉²	C₃²×C₅	1	12	yes	—
46	C₂×C₉×C₂₇	C₃²×C₅	1	81	yes	direct product factor
47	C₃⁴×C₅	C₂×C₃×C₉	1	1	yes	characteristic
48	C₃²×C₅×C₉	C₂×C₂₇	1	81	yes	direct product factor
49	C₃²×C₅×C₉	C₂×C₃×C₉	1	8	yes	—
50	C₃²×C₅×C₉	C₂×C₃×C₉	1	27	yes	—
51	C₃²×C₅×C₉	C₂×C₃×C₉	1	36	yes	—
52	C₃²×C₅×C₉	C₂×C₃³	1	4	yes	—
53	C₃×C₅×C₂₇	C₂×C₃×C₉	1	324	yes	direct product factor
54	C₃×C₅×C₂₇	C₂×C₃³	1	27	yes	—
55	C₅×C₉²	C₂×C₃×C₉	1	72	yes	—
56	C₅×C₉²	C₂×C₃³	1	9	yes	—
57	C₂×C₃³×C₅	C₉²	1	9	yes	—
58	C₂×C₃³×C₅	C₃×C₂₇	1	27	yes	—
59	C₂×C₃³×C₅	C₃²×C₉	1	4	yes	—
60	C₂×C₃×C₅×C₉	C₉²	1	72	yes	—
61	C₂×C₃×C₅×C₉	C₃×C₂₇	1	324	yes	direct product factor
62	C₂×C₃×C₅×C₉	C₃²×C₉	1	8	yes	—
63	C₂×C₃×C₅×C₉	C₃²×C₉	1	27	yes	—
64	C₂×C₃×C₅×C₉	C₃²×C₉	1	36	yes	—
65	C₂×C₃×C₅×C₉	C₃⁴	1	1	yes	characteristic
66	C₂×C₅×C₂₇	C₃²×C₉	1	81	yes	direct product factor
67	C₃³×C₉	C₂×C₅×C₉	1	3	yes	—
68	C₃³×C₉	C₂×C₃²×C₅	1	1	yes	characteristic
69	C₃²×C₂₇	C₂×C₅×C₉	1	81	yes	direct product factor
70	C₃²×C₂₇	C₂×C₃²×C₅	1	36	yes	—
71	C₃×C₉²	C₂×C₅×C₉	1	24	yes	—
72	C₃×C₉²	C₂×C₃²×C₅	1	12	yes	—
73	C₉×C₂₇	C₂×C₃²×C₅	1	81	yes	direct product factor
74	C₂×C₃⁴	C₃×C₅×C₉	1	1	yes	characteristic
75	C₂×C₃²×C₉	C₅×C₂₇	1	81	yes	direct product factor
76	C₂×C₃²×C₉	C₃×C₅×C₉	1	8	yes	—
77	C₂×C₃²×C₉	C₃×C₅×C₉	1	27	yes	—
78	C₂×C₃²×C₉	C₃×C₅×C₉	1	36	yes	—
79	C₂×C₃²×C₉	C₃³×C₅	1	4	yes	—
80	C₂×C₃×C₂₇	C₃×C₅×C₉	1	324	yes	direct product factor
81	C₂×C₃×C₂₇	C₃³×C₅	1	27	yes	—
82	C₂×C₉²	C₃×C₅×C₉	1	72	yes	—
83	C₂×C₉²	C₃³×C₅	1	9	yes	—
84	C₃³×C₅	C₂×C₉²	1	9	yes	—
85	C₃³×C₅	C₂×C₃×C₂₇	1	27	yes	—
86	C₃³×C₅	C₂×C₃²×C₉	1	4	yes	—
87	C₃×C₅×C₉	C₂×C₉²	1	72	yes	—
88	C₃×C₅×C₉	C₂×C₃×C₂₇	1	324	yes	direct product factor
89	C₃×C₅×C₉	C₂×C₃²×C₉	1	8	yes	—
90	C₃×C₅×C₉	C₂×C₃²×C₉	1	27	yes	—
91	C₃×C₅×C₉	C₂×C₃²×C₉	1	36	yes	—
92	C₃×C₅×C₉	C₂×C₃⁴	1	1	yes	characteristic
93	C₅×C₂₇	C₂×C₃²×C₉	1	81	yes	direct product factor
94	C₂×C₃²×C₅	C₉×C₂₇	1	81	yes	direct product factor
95	C₂×C₃²×C₅	C₃×C₉²	1	12	yes	—
96	C₂×C₃²×C₅	C₃²×C₂₇	1	36	yes	—
97	C₂×C₃²×C₅	C₃³×C₉	1	1	yes	characteristic
98	C₂×C₅×C₉	C₃×C₉²	1	24	yes	—
99	C₂×C₅×C₉	C₃²×C₂₇	1	81	yes	direct product factor
100	C₂×C₅×C₉	C₃³×C₉	1	3	yes	—
101	C₃⁴	C₂×C₃×C₅×C₉	1	1	yes	characteristic
102	C₃²×C₉	C₂×C₅×C₂₇	1	81	yes	direct product factor
103	C₃²×C₉	C₂×C₃×C₅×C₉	1	8	yes	—
104	C₃²×C₉	C₂×C₃×C₅×C₉	1	27	yes	—
105	C₃²×C₉	C₂×C₃×C₅×C₉	1	36	yes	—
106	C₃²×C₉	C₂×C₃³×C₅	1	4	yes	—
107	C₃×C₂₇	C₂×C₃×C₅×C₉	1	324	yes	direct product factor
108	C₃×C₂₇	C₂×C₃³×C₅	1	27	yes	—
109	C₉²	C₂×C₃×C₅×C₉	1	72	yes	—
110	C₉²	C₂×C₃³×C₅	1	9	yes	—
111	C₂×C₃³	C₅×C₉²	1	9	yes	—
112	C₂×C₃³	C₃×C₅×C₂₇	1	27	yes	—
113	C₂×C₃³	C₃²×C₅×C₉	1	4	yes	—
114	C₂×C₃×C₉	C₅×C₉²	1	72	yes	—
115	C₂×C₃×C₉	C₃×C₅×C₂₇	1	324	yes	direct product factor
116	C₂×C₃×C₉	C₃²×C₅×C₉	1	8	yes	—
117	C₂×C₃×C₉	C₃²×C₅×C₉	1	27	yes	—
118	C₂×C₃×C₉	C₃²×C₅×C₉	1	36	yes	—
119	C₂×C₃×C₉	C₃⁴×C₅	1	1	yes	characteristic
120	C₂×C₂₇	C₃²×C₅×C₉	1	81	yes	direct product factor
121	C₃²×C₅	C₂×C₉×C₂₇	1	81	yes	direct product factor
122	C₃²×C₅	C₂×C₃×C₉²	1	12	yes	—
123	C₃²×C₅	C₂×C₃²×C₂₇	1	36	yes	—
124	C₃²×C₅	C₂×C₃³×C₉	1	1	yes	characteristic
125	C₅×C₉	C₂×C₃×C₉²	1	24	yes	—
126	C₅×C₉	C₂×C₃²×C₂₇	1	81	yes	direct product factor
127	C₅×C₉	C₂×C₃³×C₉	1	3	yes	—
128	C₂×C₃×C₅	C₃×C₉×C₂₇	1	36	yes	direct product factor
129	C₂×C₃×C₅	C₃²×C₉²	1	1	yes	characteristic
130	C₂×C₃×C₅	C₃³×C₂₇	1	3	yes	—
131	C₃³	C₂×C₅×C₉²	1	9	yes	—
132	C₃³	C₂×C₃×C₅×C₂₇	1	27	yes	—
133	C₃³	C₂×C₃²×C₅×C₉	1	4	yes	—
134	C₃×C₉	C₂×C₅×C₉²	1	72	yes	—
135	C₃×C₉	C₂×C₃×C₅×C₂₇	1	324	yes	direct product factor
136	C₃×C₉	C₂×C₃²×C₅×C₉	1	8	yes	—
137	C₃×C₉	C₂×C₃²×C₅×C₉	1	27	yes	—
138	C₃×C₉	C₂×C₃²×C₅×C₉	1	36	yes	—
139	C₃×C₉	C₂×C₃⁴×C₅	1	1	yes	characteristic
140	C₂₇	C₂×C₃²×C₅×C₉	1	81	yes	direct product factor
141	C₂×C₃²	C₅×C₉×C₂₇	1	81	yes	direct product factor
142	C₂×C₃²	C₃×C₅×C₉²	1	12	yes	—
143	C₂×C₃²	C₃²×C₅×C₂₇	1	36	yes	—
144	C₂×C₃²	C₃³×C₅×C₉	1	1	yes	characteristic
145	C₂×C₉	C₃×C₅×C₉²	1	24	yes	—
146	C₂×C₉	C₃²×C₅×C₂₇	1	81	yes	direct product factor
147	C₂×C₉	C₃³×C₅×C₉	1	3	yes	—
148	C₃×C₅	C₂×C₃×C₉×C₂₇	1	36	yes	direct product factor
149	C₃×C₅	C₂×C₃²×C₉²	1	1	yes	characteristic
150	C₃×C₅	C₂×C₃³×C₂₇	1	3	yes	—
151	C₂×C₅	C₃²×C₉×C₂₇	1	1	yes	characteristic, direct product factor, Hall
152	C₃²	C₂×C₅×C₉×C₂₇	1	81	yes	direct product factor
153	C₃²	C₂×C₃×C₅×C₉²	1	12	yes	—
154	C₃²	C₂×C₃²×C₅×C₂₇	1	36	yes	—
155	C₃²	C₂×C₃³×C₅×C₉	1	1	yes	characteristic
156	C₉	C₂×C₃×C₅×C₉²	1	24	yes	—
157	C₉	C₂×C₃²×C₅×C₂₇	1	81	yes	direct product factor
158	C₉	C₂×C₃³×C₅×C₉	1	3	yes	—
159	C₂×C₃	C₃×C₅×C₉×C₂₇	1	36	yes	direct product factor
160	C₂×C₃	C₃²×C₅×C₉²	1	1	yes	characteristic
161	C₂×C₃	C₃³×C₅×C₂₇	1	3	yes	—
162	C₅	C₂×C₃²×C₉×C₂₇	1	1	yes	characteristic, direct product factor, Sylow
163	C₃	C₂×C₃×C₅×C₉×C₂₇	1	36	yes	direct product factor
164	C₃	C₂×C₃³×C₅×C₂₇	1	3	yes	—
165	C₃	C₂×C₃²×C₅×C₉²	1	1	yes	characteristic
166	C₂	C₃²×C₅×C₉×C₂₇	1	1	yes	characteristic, direct product factor, Sylow

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₃×C₉	C₂×C₃⁴×C₅
socle, prime radical	C₂×C₃⁴×C₅	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. C₃²×C₅×C₉×C₂₇	C₂	direct (N × H)
4. C₂×C₃×C₅×C₉×C₂₇	C₃	direct (N × H)
5. C₂×C₃²×C₉×C₂₇	C₅	direct (N × H)
8. C₂×C₃²×C₅×C₂₇	C₉	direct (N × H)
12. C₂×C₅×C₉×C₂₇	C₃²	direct (N × H)
15. C₃×C₅×C₉×C₂₇	C₂×C₃	direct (N × H)
16. C₃²×C₉×C₂₇	C₂×C₅	direct (N × H)
19. C₂×C₃×C₉×C₂₇	C₃×C₅	direct (N × H)
22. C₃²×C₅×C₂₇	C₂×C₉	direct (N × H)
26. C₅×C₉×C₂₇	C₂×C₃²	direct (N × H)
28. C₂×C₃²×C₅×C₉	C₂₇	direct (N × H)
33. C₂×C₃×C₅×C₂₇	C₃×C₉	direct (N × H)
39. C₃×C₉×C₂₇	C₂×C₃×C₅	direct (N × H)
42. C₂×C₃²×C₂₇	C₅×C₉	direct (N × H)
46. C₂×C₉×C₂₇	C₃²×C₅	direct (N × H)
48. C₃²×C₅×C₉	C₂×C₂₇	direct (N × H)
53. C₃×C₅×C₂₇	C₂×C₃×C₉	direct (N × H)
61. C₂×C₃×C₅×C₉	C₃×C₂₇	direct (N × H)
66. C₂×C₅×C₂₇	C₃²×C₉	direct (N × H)
69. C₃²×C₂₇	C₂×C₅×C₉	direct (N × H)
73. C₉×C₂₇	C₂×C₃²×C₅	direct (N × H)
75. C₂×C₃²×C₉	C₅×C₂₇	direct (N × H)
80. C₂×C₃×C₂₇	C₃×C₅×C₉	direct (N × H)
88. C₃×C₅×C₉	C₂×C₃×C₂₇	direct (N × H)
93. C₅×C₂₇	C₂×C₃²×C₉	direct (N × H)
94. C₂×C₃²×C₅	C₉×C₂₇	direct (N × H)
99. C₂×C₅×C₉	C₃²×C₂₇	direct (N × H)
102. C₃²×C₉	C₂×C₅×C₂₇	direct (N × H)
107. C₃×C₂₇	C₂×C₃×C₅×C₉	direct (N × H)
115. C₂×C₃×C₉	C₃×C₅×C₂₇	direct (N × H)
120. C₂×C₂₇	C₃²×C₅×C₉	direct (N × H)
121. C₃²×C₅	C₂×C₉×C₂₇	direct (N × H)
126. C₅×C₉	C₂×C₃²×C₂₇	direct (N × H)
128. C₂×C₃×C₅	C₃×C₉×C₂₇	direct (N × H)
135. C₃×C₉	C₂×C₃×C₅×C₂₇	direct (N × H)
140. C₂₇	C₂×C₃²×C₅×C₉	direct (N × H)
141. C₂×C₃²	C₅×C₉×C₂₇	direct (N × H)
146. C₂×C₉	C₃²×C₅×C₂₇	direct (N × H)
148. C₃×C₅	C₂×C₃×C₉×C₂₇	direct (N × H)
151. C₂×C₅	C₃²×C₉×C₂₇	direct (N × H)
152. C₃²	C₂×C₅×C₉×C₂₇	direct (N × H)
157. C₉	C₂×C₃²×C₅×C₂₇	direct (N × H)
159. C₂×C₃	C₃×C₅×C₉×C₂₇	direct (N × H)
162. C₅	C₂×C₃²×C₉×C₂₇	direct (N × H)
163. C₃	C₂×C₃×C₅×C₉×C₂₇	direct (N × H)
166. C₂	C₃²×C₅×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
C₃²×C₅×C₉×C₂₇	2
C₂×C₃²×C₅×C₉²	3
C₂×C₃³×C₅×C₂₇	3
C₂×C₃×C₅×C₉×C₂₇	3
C₂×C₃²×C₉×C₂₇	5

Subgroups — C2×C32×C5×C9×C27 #72199337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×C₅×C₉×C₂₇ #72199337