Subgroups — C₂⁴×C₂₇×C₃₂ #72192014

Order 13824.

6408 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	C₂⁴×C₁₆×C₂₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₂₇×C₃₂	C₂	1	30	yes	maximal, direct product factor
3	C₂⁴×C₉×C₃₂	C₃	1	1	yes	maximal, characteristic
4	C₂⁴×C₈×C₂₇	C₄	1	1	yes	characteristic
5	C₂³×C₁₆×C₂₇	C₄	1	15	yes	—
6	C₂³×C₁₆×C₂₇	C₂²	1	15	yes	—
7	C₂²×C₂₇×C₃₂	C₂²	1	140	yes	direct product factor
8	C₂⁴×C₉×C₁₆	C₂×C₃	1	1	yes	characteristic
9	C₂³×C₉×C₃₂	C₂×C₃	1	30	yes	—
10	C₂⁴×C₃×C₃₂	C₉	1	1	yes	characteristic
11	C₂⁴×C₄×C₂₇	C₈	1	1	yes	characteristic
12	C₂³×C₈×C₂₇	C₈	1	15	yes	—
13	C₂³×C₈×C₂₇	C₂×C₄	1	15	yes	—
14	C₂²×C₁₆×C₂₇	C₂×C₄	1	105	yes	—
15	C₂²×C₁₆×C₂₇	C₂³	1	35	yes	—
16	C₂×C₂₇×C₃₂	C₂³	1	120	yes	direct product factor
17	C₂⁴×C₈×C₉	C₃×C₄	1	1	yes	characteristic
18	C₂³×C₉×C₁₆	C₃×C₄	1	15	yes	—
19	C₂³×C₉×C₁₆	C₂²×C₃	1	15	yes	—
20	C₂²×C₉×C₃₂	C₂²×C₃	1	140	yes	—
21	C₂⁵×C₂₇	C₁₆	1	1	yes	characteristic
22	C₂³×C₄×C₂₇	C₁₆	1	15	yes	—
23	C₂³×C₄×C₂₇	C₂×C₈	1	15	yes	—
24	C₂²×C₈×C₂₇	C₂×C₈	1	105	yes	—
25	C₂²×C₈×C₂₇	C₂²×C₄	1	35	yes	—
26	C₂×C₁₆×C₂₇	C₂²×C₄	1	105	yes	—
27	C₂×C₁₆×C₂₇	C₂⁴	1	15	yes	—
28	C₂₇×C₃₂	C₂⁴	1	16	yes	direct product factor
29	C₂⁴×C₃×C₁₆	C₂×C₉	1	1	yes	characteristic
30	C₂³×C₃×C₃₂	C₂×C₉	1	30	yes	—
31	C₂⁴×C₄×C₉	C₃×C₈	1	1	yes	characteristic
32	C₂³×C₈×C₉	C₃×C₈	1	15	yes	—
33	C₂³×C₈×C₉	C₂×C₃×C₄	1	15	yes	—
34	C₂²×C₉×C₁₆	C₂×C₃×C₄	1	105	yes	—
35	C₂²×C₉×C₁₆	C₂³×C₃	1	35	yes	—
36	C₂×C₉×C₃₂	C₂³×C₃	1	120	yes	—
37	C₂⁴×C₃₂	C₂₇	1	1	yes	characteristic, direct product factor, Sylow
38	C₂⁴×C₂₇	C₃₂	1	16	yes	direct product factor
39	C₂⁴×C₂₇	C₂×C₁₆	1	15	yes	—
40	C₂²×C₄×C₂₇	C₂×C₁₆	1	105	yes	—
41	C₂²×C₄×C₂₇	C₂²×C₈	1	35	yes	—
42	C₂×C₈×C₂₇	C₂²×C₈	1	105	yes	—
43	C₂×C₈×C₂₇	C₂³×C₄	1	15	yes	—
44	C₁₆×C₂₇	C₂³×C₄	1	15	yes	—
45	C₁₆×C₂₇	C₂⁵	1	1	yes	characteristic
46	C₂⁴×C₃×C₈	C₄×C₉	1	1	yes	characteristic
47	C₂³×C₃×C₁₆	C₄×C₉	1	15	yes	—
48	C₂³×C₃×C₁₆	C₂²×C₉	1	15	yes	—
49	C₂²×C₃×C₃₂	C₂²×C₉	1	140	yes	—
50	C₂⁵×C₉	C₃×C₁₆	1	1	yes	characteristic
51	C₂³×C₄×C₉	C₃×C₁₆	1	15	yes	—
52	C₂³×C₄×C₉	C₂×C₃×C₈	1	15	yes	—
53	C₂²×C₈×C₉	C₂×C₃×C₈	1	105	yes	—
54	C₂²×C₈×C₉	C₂²×C₃×C₄	1	35	yes	—
55	C₂×C₉×C₁₆	C₂²×C₃×C₄	1	105	yes	—
56	C₂×C₉×C₁₆	C₂⁴×C₃	1	15	yes	—
57	C₉×C₃₂	C₂⁴×C₃	1	16	yes	—
58	C₂⁴×C₁₆	C₂×C₂₇	1	1	yes	characteristic
59	C₂³×C₃₂	C₂×C₂₇	1	30	yes	direct product factor
60	C₂³×C₂₇	C₂×C₃₂	1	120	yes	direct product factor
61	C₂³×C₂₇	C₂²×C₁₆	1	35	yes	—
62	C₂×C₄×C₂₇	C₂²×C₁₆	1	105	yes	—
63	C₂×C₄×C₂₇	C₂³×C₈	1	15	yes	—
64	C₈×C₂₇	C₂³×C₈	1	15	yes	—
65	C₈×C₂₇	C₂⁴×C₄	1	1	yes	characteristic
66	C₂⁴×C₃×C₄	C₈×C₉	1	1	yes	characteristic
67	C₂³×C₃×C₈	C₈×C₉	1	15	yes	—
68	C₂³×C₃×C₈	C₂×C₄×C₉	1	15	yes	—
69	C₂²×C₃×C₁₆	C₂×C₄×C₉	1	105	yes	—
70	C₂²×C₃×C₁₆	C₂³×C₉	1	35	yes	—
71	C₂×C₃×C₃₂	C₂³×C₉	1	120	yes	—
72	C₂⁴×C₉	C₃×C₃₂	1	16	yes	—
73	C₂⁴×C₉	C₂×C₃×C₁₆	1	15	yes	—
74	C₂²×C₄×C₉	C₂×C₃×C₁₆	1	105	yes	—
75	C₂²×C₄×C₉	C₂²×C₃×C₈	1	35	yes	—
76	C₂×C₈×C₉	C₂²×C₃×C₈	1	105	yes	—
77	C₂×C₈×C₉	C₂³×C₃×C₄	1	15	yes	—
78	C₉×C₁₆	C₂³×C₃×C₄	1	15	yes	—
79	C₉×C₁₆	C₂⁵×C₃	1	1	yes	characteristic
80	C₂⁴×C₈	C₄×C₂₇	1	1	yes	characteristic
81	C₂³×C₁₆	C₄×C₂₇	1	15	yes	—
82	C₂³×C₁₆	C₂²×C₂₇	1	15	yes	—
83	C₂²×C₃₂	C₂²×C₂₇	1	140	yes	direct product factor
84	C₂²×C₂₇	C₂²×C₃₂	1	140	yes	direct product factor
85	C₂²×C₂₇	C₂³×C₁₆	1	15	yes	—
86	C₄×C₂₇	C₂³×C₁₆	1	15	yes	—
87	C₄×C₂₇	C₂⁴×C₈	1	1	yes	characteristic
88	C₂⁵×C₃	C₉×C₁₆	1	1	yes	characteristic
89	C₂³×C₃×C₄	C₉×C₁₆	1	15	yes	—
90	C₂³×C₃×C₄	C₂×C₈×C₉	1	15	yes	—
91	C₂²×C₃×C₈	C₂×C₈×C₉	1	105	yes	—
92	C₂²×C₃×C₈	C₂²×C₄×C₉	1	35	yes	—
93	C₂×C₃×C₁₆	C₂²×C₄×C₉	1	105	yes	—
94	C₂×C₃×C₁₆	C₂⁴×C₉	1	15	yes	—
95	C₃×C₃₂	C₂⁴×C₉	1	16	yes	—
96	C₂³×C₉	C₂×C₃×C₃₂	1	120	yes	—
97	C₂³×C₉	C₂²×C₃×C₁₆	1	35	yes	—
98	C₂×C₄×C₉	C₂²×C₃×C₁₆	1	105	yes	—
99	C₂×C₄×C₉	C₂³×C₃×C₈	1	15	yes	—
100	C₈×C₉	C₂³×C₃×C₈	1	15	yes	—
101	C₈×C₉	C₂⁴×C₃×C₄	1	1	yes	characteristic
102	C₂⁴×C₄	C₈×C₂₇	1	1	yes	characteristic
103	C₂³×C₈	C₈×C₂₇	1	15	yes	—
104	C₂³×C₈	C₂×C₄×C₂₇	1	15	yes	—
105	C₂²×C₁₆	C₂×C₄×C₂₇	1	105	yes	—
106	C₂²×C₁₆	C₂³×C₂₇	1	35	yes	—
107	C₂×C₃₂	C₂³×C₂₇	1	120	yes	direct product factor
108	C₂×C₂₇	C₂³×C₃₂	1	30	yes	direct product factor
109	C₂×C₂₇	C₂⁴×C₁₆	1	1	yes	characteristic
110	C₂⁴×C₃	C₉×C₃₂	1	16	yes	—
111	C₂⁴×C₃	C₂×C₉×C₁₆	1	15	yes	—
112	C₂²×C₃×C₄	C₂×C₉×C₁₆	1	105	yes	—
113	C₂²×C₃×C₄	C₂²×C₈×C₉	1	35	yes	—
114	C₂×C₃×C₈	C₂²×C₈×C₉	1	105	yes	—
115	C₂×C₃×C₈	C₂³×C₄×C₉	1	15	yes	—
116	C₃×C₁₆	C₂³×C₄×C₉	1	15	yes	—
117	C₃×C₁₆	C₂⁵×C₉	1	1	yes	characteristic
118	C₂²×C₉	C₂²×C₃×C₃₂	1	140	yes	—
119	C₂²×C₉	C₂³×C₃×C₁₆	1	15	yes	—
120	C₄×C₉	C₂³×C₃×C₁₆	1	15	yes	—
121	C₄×C₉	C₂⁴×C₃×C₈	1	1	yes	characteristic
122	C₂⁵	C₁₆×C₂₇	1	1	yes	characteristic
123	C₂³×C₄	C₁₆×C₂₇	1	15	yes	—
124	C₂³×C₄	C₂×C₈×C₂₇	1	15	yes	—
125	C₂²×C₈	C₂×C₈×C₂₇	1	105	yes	—
126	C₂²×C₈	C₂²×C₄×C₂₇	1	35	yes	—
127	C₂×C₁₆	C₂²×C₄×C₂₇	1	105	yes	—
128	C₂×C₁₆	C₂⁴×C₂₇	1	15	yes	—
129	C₃₂	C₂⁴×C₂₇	1	16	yes	direct product factor
130	C₂₇	C₂⁴×C₃₂	1	1	yes	characteristic, direct product factor, Sylow
131	C₂³×C₃	C₂×C₉×C₃₂	1	120	yes	—
132	C₂³×C₃	C₂²×C₉×C₁₆	1	35	yes	—
133	C₂×C₃×C₄	C₂²×C₉×C₁₆	1	105	yes	—
134	C₂×C₃×C₄	C₂³×C₈×C₉	1	15	yes	—
135	C₃×C₈	C₂³×C₈×C₉	1	15	yes	—
136	C₃×C₈	C₂⁴×C₄×C₉	1	1	yes	characteristic
137	C₂×C₉	C₂³×C₃×C₃₂	1	30	yes	—
138	C₂×C₉	C₂⁴×C₃×C₁₆	1	1	yes	characteristic
139	C₂⁴	C₂₇×C₃₂	1	16	yes	direct product factor
140	C₂⁴	C₂×C₁₆×C₂₇	1	15	yes	—
141	C₂²×C₄	C₂×C₁₆×C₂₇	1	105	yes	—
142	C₂²×C₄	C₂²×C₈×C₂₇	1	35	yes	—
143	C₂×C₈	C₂²×C₈×C₂₇	1	105	yes	—
144	C₂×C₈	C₂³×C₄×C₂₇	1	15	yes	—
145	C₁₆	C₂³×C₄×C₂₇	1	15	yes	—
146	C₁₆	C₂⁵×C₂₇	1	1	yes	characteristic
147	C₂²×C₃	C₂²×C₉×C₃₂	1	140	yes	—
148	C₂²×C₃	C₂³×C₉×C₁₆	1	15	yes	—
149	C₃×C₄	C₂³×C₉×C₁₆	1	15	yes	—
150	C₃×C₄	C₂⁴×C₈×C₉	1	1	yes	characteristic
151	C₉	C₂⁴×C₃×C₃₂	1	1	yes	characteristic
152	C₂³	C₂×C₂₇×C₃₂	1	120	yes	direct product factor
153	C₂³	C₂²×C₁₆×C₂₇	1	35	yes	—
154	C₂×C₄	C₂²×C₁₆×C₂₇	1	105	yes	—
155	C₂×C₄	C₂³×C₈×C₂₇	1	15	yes	—
156	C₈	C₂³×C₈×C₂₇	1	15	yes	—
157	C₈	C₂⁴×C₄×C₂₇	1	1	yes	characteristic
158	C₂×C₃	C₂³×C₉×C₃₂	1	30	yes	—
159	C₂×C₃	C₂⁴×C₉×C₁₆	1	1	yes	characteristic
160	C₂²	C₂²×C₂₇×C₃₂	1	140	yes	direct product factor
161	C₂²	C₂³×C₁₆×C₂₇	1	15	yes	—
162	C₄	C₂³×C₁₆×C₂₇	1	15	yes	—
163	C₄	C₂⁴×C₈×C₂₇	1	1	yes	characteristic
164	C₃	C₂⁴×C₉×C₃₂	1	1	yes	characteristic
165	C₂	C₂³×C₂₇×C₃₂	1	30	yes	direct product factor
166	C₂	C₂⁴×C₁₆×C₂₇	1	1	yes	characteristic

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₉×C₁₆	C₂⁵×C₃
socle, prime radical	C₂⁵×C₃	C₉×C₁₆
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×C₂₇×C₃₂	C₂	direct (N × H)
7. C₂²×C₂₇×C₃₂	C₂²	direct (N × H)
16. C₂×C₂₇×C₃₂	C₂³	direct (N × H)
28. C₂₇×C₃₂	C₂⁴	direct (N × H)
37. C₂⁴×C₃₂	C₂₇	direct (N × H)
38. C₂⁴×C₂₇	C₃₂	direct (N × H)
59. C₂³×C₃₂	C₂×C₂₇	direct (N × H)
60. C₂³×C₂₇	C₂×C₃₂	direct (N × H)
83. C₂²×C₃₂	C₂²×C₂₇	direct (N × H)
84. C₂²×C₂₇	C₂²×C₃₂	direct (N × H)
107. C₂×C₃₂	C₂³×C₂₇	direct (N × H)
108. C₂×C₂₇	C₂³×C₃₂	direct (N × H)
129. C₃₂	C₂⁴×C₂₇	direct (N × H)
130. C₂₇	C₂⁴×C₃₂	direct (N × H)
139. C₂⁴	C₂₇×C₃₂	direct (N × H)
152. C₂³	C₂×C₂₇×C₃₂	direct (N × H)
160. C₂²	C₂²×C₂₇×C₃₂	direct (N × H)
165. C₂	C₂³×C₂₇×C₃₂	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×C₁₆×C₂₇	2
C₂³×C₂₇×C₃₂	2
C₂⁴×C₉×C₃₂	3

Subgroups — C24×C27×C32 #72192014

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂⁴×C₂₇×C₃₂ #72192014