Subgroups — C₂×C₄×C₁₆×C₂₇ #72183116

Order 3456.

432 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	C₂²×C₁₆×C₂₇	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₂₇	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₆×C₂₇	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₄×C₉×C₁₆	C₃	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₂₇	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₂₇	C₂²	1	1	yes	characteristic
7	C₂×C₄²×C₂₇	C₄	1	1	yes	characteristic
8	C₂×C₁₆×C₂₇	C₄	1	8	yes	direct product factor
9	C₂×C₁₆×C₂₇	C₂²	1	4	yes	—
10	C₄×C₈×C₂₇	C₄	1	2	yes	—
11	C₄×C₈×C₂₇	C₂²	1	2	yes	—
12	C₂²×C₉×C₁₆	C₂×C₃	1	2	yes	—
13	C₂×C₄×C₈×C₉	C₂×C₃	1	1	yes	characteristic
14	C₄×C₉×C₁₆	C₂×C₃	1	4	yes	—
15	C₂²×C₄×C₂₇	C₈	1	2	yes	—
16	C₂²×C₄×C₂₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₂₇	C₈	1	4	yes	—
18	C₂×C₈×C₂₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₂₇	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₂₇	C₂×C₄	1	4	yes	—
21	C₂×C₈×C₂₇	C₂³	1	1	yes	characteristic
22	C₄²×C₂₇	C₈	1	2	yes	—
23	C₄²×C₂₇	C₂×C₄	1	2	yes	—
24	C₁₆×C₂₇	C₂×C₄	1	8	yes	direct product factor
25	C₂×C₃×C₄×C₁₆	C₉	1	1	yes	characteristic
26	C₂²×C₈×C₉	C₃×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
28	C₂×C₄²×C₉	C₃×C₄	1	1	yes	characteristic
29	C₂×C₉×C₁₆	C₃×C₄	1	8	yes	—
30	C₂×C₉×C₁₆	C₂²×C₃	1	4	yes	—
31	C₄×C₈×C₉	C₃×C₄	1	2	yes	—
32	C₄×C₈×C₉	C₂²×C₃	1	2	yes	—
33	C₂³×C₂₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₂₇	C₁₆	1	8	yes	direct product factor
35	C₂×C₄×C₂₇	C₄²	1	2	yes	—
36	C₂×C₄×C₂₇	C₂×C₈	1	1	yes	characteristic
37	C₂×C₄×C₂₇	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₂₇	C₂×C₈	1	4	yes	—
39	C₂×C₄×C₂₇	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₂₇	C₄²	1	2	yes	—
41	C₈×C₂₇	C₂×C₈	1	4	yes	—
42	C₈×C₂₇	C₂²×C₄	1	2	yes	—
43	C₂²×C₃×C₁₆	C₂×C₉	1	2	yes	—
44	C₂×C₃×C₄×C₈	C₂×C₉	1	1	yes	characteristic
45	C₃×C₄×C₁₆	C₂×C₉	1	4	yes	—
46	C₂²×C₄×C₉	C₃×C₈	1	2	yes	—
47	C₂²×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
48	C₂×C₈×C₉	C₃×C₈	1	4	yes	—
49	C₂×C₈×C₉	C₂×C₃×C₄	1	1	yes	characteristic
50	C₂×C₈×C₉	C₂×C₃×C₄	1	2	yes	—
51	C₂×C₈×C₉	C₂×C₃×C₄	1	4	yes	—
52	C₂×C₈×C₉	C₂³×C₃	1	1	yes	characteristic
53	C₄²×C₉	C₃×C₈	1	2	yes	—
54	C₄²×C₉	C₂×C₃×C₄	1	2	yes	—
55	C₉×C₁₆	C₂×C₃×C₄	1	8	yes	—
56	C₂×C₄×C₁₆	C₂₇	1	1	yes	characteristic, direct product factor, Sylow
57	C₂²×C₂₇	C₄×C₈	1	2	yes	—
58	C₂²×C₂₇	C₂×C₁₆	1	4	yes	—
59	C₂²×C₂₇	C₂²×C₈	1	1	yes	characteristic
60	C₄×C₂₇	C₄×C₈	1	2	yes	—
61	C₄×C₂₇	C₂×C₁₆	1	8	yes	direct product factor
62	C₄×C₂₇	C₂×C₄²	1	1	yes	characteristic
63	C₄×C₂₇	C₂²×C₈	1	1	yes	characteristic
64	C₂²×C₃×C₈	C₄×C₉	1	1	yes	characteristic
65	C₂²×C₃×C₈	C₂²×C₉	1	1	yes	characteristic
66	C₂×C₃×C₄²	C₄×C₉	1	1	yes	characteristic
67	C₂×C₃×C₁₆	C₄×C₉	1	8	yes	—
68	C₂×C₃×C₁₆	C₂²×C₉	1	4	yes	—
69	C₃×C₄×C₈	C₄×C₉	1	2	yes	—
70	C₃×C₄×C₈	C₂²×C₉	1	2	yes	—
71	C₂³×C₉	C₂×C₃×C₈	1	1	yes	characteristic
72	C₂×C₄×C₉	C₃×C₁₆	1	8	yes	—
73	C₂×C₄×C₉	C₃×C₄²	1	2	yes	—
74	C₂×C₄×C₉	C₂×C₃×C₈	1	1	yes	characteristic
75	C₂×C₄×C₉	C₂×C₃×C₈	1	2	yes	—
76	C₂×C₄×C₉	C₂×C₃×C₈	1	4	yes	—
77	C₂×C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
78	C₈×C₉	C₃×C₄²	1	2	yes	—
79	C₈×C₉	C₂×C₃×C₈	1	4	yes	—
80	C₈×C₉	C₂²×C₃×C₄	1	2	yes	—
81	C₂²×C₁₆	C₂×C₂₇	1	2	yes	—
82	C₂×C₄×C₈	C₂×C₂₇	1	1	yes	characteristic
83	C₄×C₁₆	C₂×C₂₇	1	4	yes	direct product factor
84	C₂×C₂₇	C₄×C₁₆	1	4	yes	direct product factor
85	C₂×C₂₇	C₂×C₄×C₈	1	1	yes	characteristic
86	C₂×C₂₇	C₂²×C₁₆	1	2	yes	—
87	C₂²×C₃×C₄	C₈×C₉	1	2	yes	—
88	C₂²×C₃×C₄	C₂×C₄×C₉	1	1	yes	characteristic
89	C₂×C₃×C₈	C₈×C₉	1	4	yes	—
90	C₂×C₃×C₈	C₂×C₄×C₉	1	1	yes	characteristic
91	C₂×C₃×C₈	C₂×C₄×C₉	1	2	yes	—
92	C₂×C₃×C₈	C₂×C₄×C₉	1	4	yes	—
93	C₂×C₃×C₈	C₂³×C₉	1	1	yes	characteristic
94	C₃×C₄²	C₈×C₉	1	2	yes	—
95	C₃×C₄²	C₂×C₄×C₉	1	2	yes	—
96	C₃×C₁₆	C₂×C₄×C₉	1	8	yes	—
97	C₂²×C₉	C₃×C₄×C₈	1	2	yes	—
98	C₂²×C₉	C₂×C₃×C₁₆	1	4	yes	—
99	C₂²×C₉	C₂²×C₃×C₈	1	1	yes	characteristic
100	C₄×C₉	C₃×C₄×C₈	1	2	yes	—
101	C₄×C₉	C₂×C₃×C₁₆	1	8	yes	—
102	C₄×C₉	C₂×C₃×C₄²	1	1	yes	characteristic
103	C₄×C₉	C₂²×C₃×C₈	1	1	yes	characteristic
104	C₂²×C₈	C₄×C₂₇	1	1	yes	characteristic
105	C₂²×C₈	C₂²×C₂₇	1	1	yes	characteristic
106	C₂×C₄²	C₄×C₂₇	1	1	yes	characteristic
107	C₂×C₁₆	C₄×C₂₇	1	8	yes	direct product factor
108	C₂×C₁₆	C₂²×C₂₇	1	4	yes	—
109	C₄×C₈	C₄×C₂₇	1	2	yes	—
110	C₄×C₈	C₂²×C₂₇	1	2	yes	—
111	C₂₇	C₂×C₄×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
112	C₂³×C₃	C₂×C₈×C₉	1	1	yes	characteristic
113	C₂×C₃×C₄	C₉×C₁₆	1	8	yes	—
114	C₂×C₃×C₄	C₄²×C₉	1	2	yes	—
115	C₂×C₃×C₄	C₂×C₈×C₉	1	1	yes	characteristic
116	C₂×C₃×C₄	C₂×C₈×C₉	1	2	yes	—
117	C₂×C₃×C₄	C₂×C₈×C₉	1	4	yes	—
118	C₂×C₃×C₄	C₂²×C₄×C₉	1	1	yes	characteristic
119	C₃×C₈	C₄²×C₉	1	2	yes	—
120	C₃×C₈	C₂×C₈×C₉	1	4	yes	—
121	C₃×C₈	C₂²×C₄×C₉	1	2	yes	—
122	C₂×C₉	C₃×C₄×C₁₆	1	4	yes	—
123	C₂×C₉	C₂×C₃×C₄×C₈	1	1	yes	characteristic
124	C₂×C₉	C₂²×C₃×C₁₆	1	2	yes	—
125	C₂²×C₄	C₈×C₂₇	1	2	yes	—
126	C₂²×C₄	C₂×C₄×C₂₇	1	1	yes	characteristic
127	C₂×C₈	C₈×C₂₇	1	4	yes	—
128	C₂×C₈	C₂×C₄×C₂₇	1	1	yes	characteristic
129	C₂×C₈	C₂×C₄×C₂₇	1	2	yes	—
130	C₂×C₈	C₂×C₄×C₂₇	1	4	yes	—
131	C₂×C₈	C₂³×C₂₇	1	1	yes	characteristic
132	C₄²	C₈×C₂₇	1	2	yes	—
133	C₄²	C₂×C₄×C₂₇	1	2	yes	—
134	C₁₆	C₂×C₄×C₂₇	1	8	yes	direct product factor
135	C₂²×C₃	C₄×C₈×C₉	1	2	yes	—
136	C₂²×C₃	C₂×C₉×C₁₆	1	4	yes	—
137	C₂²×C₃	C₂²×C₈×C₉	1	1	yes	characteristic
138	C₃×C₄	C₄×C₈×C₉	1	2	yes	—
139	C₃×C₄	C₂×C₉×C₁₆	1	8	yes	—
140	C₃×C₄	C₂×C₄²×C₉	1	1	yes	characteristic
141	C₃×C₄	C₂²×C₈×C₉	1	1	yes	characteristic
142	C₉	C₂×C₃×C₄×C₁₆	1	1	yes	characteristic
143	C₂³	C₂×C₈×C₂₇	1	1	yes	characteristic
144	C₂×C₄	C₁₆×C₂₇	1	8	yes	direct product factor
145	C₂×C₄	C₄²×C₂₇	1	2	yes	—
146	C₂×C₄	C₂×C₈×C₂₇	1	1	yes	characteristic
147	C₂×C₄	C₂×C₈×C₂₇	1	2	yes	—
148	C₂×C₄	C₂×C₈×C₂₇	1	4	yes	—
149	C₂×C₄	C₂²×C₄×C₂₇	1	1	yes	characteristic
150	C₈	C₄²×C₂₇	1	2	yes	—
151	C₈	C₂×C₈×C₂₇	1	4	yes	—
152	C₈	C₂²×C₄×C₂₇	1	2	yes	—
153	C₂×C₃	C₄×C₉×C₁₆	1	4	yes	—
154	C₂×C₃	C₂×C₄×C₈×C₉	1	1	yes	characteristic
155	C₂×C₃	C₂²×C₉×C₁₆	1	2	yes	—
156	C₂²	C₄×C₈×C₂₇	1	2	yes	—
157	C₂²	C₂×C₁₆×C₂₇	1	4	yes	—
158	C₂²	C₂²×C₈×C₂₇	1	1	yes	characteristic
159	C₄	C₄×C₈×C₂₇	1	2	yes	—
160	C₄	C₂×C₁₆×C₂₇	1	8	yes	direct product factor
161	C₄	C₂×C₄²×C₂₇	1	1	yes	characteristic
162	C₄	C₂²×C₈×C₂₇	1	1	yes	characteristic
163	C₃	C₂×C₄×C₉×C₁₆	1	1	yes	characteristic
164	C₂	C₄×C₁₆×C₂₇	1	4	yes	direct product factor
165	C₂	C₂×C₄×C₈×C₂₇	1	1	yes	characteristic
166	C₂	C₂²×C₁₆×C₂₇	1	2	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₈×C₉	C₂³×C₃
socle, prime radical	C₂³×C₃	C₂×C₈×C₉
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₁₆×C₂₇	C₂	direct (N × H)
8. C₂×C₁₆×C₂₇	C₄	direct (N × H)
24. C₁₆×C₂₇	C₂×C₄	direct (N × H)
34. C₂×C₄×C₂₇	C₁₆	direct (N × H)
56. C₂×C₄×C₁₆	C₂₇	direct (N × H)
61. C₄×C₂₇	C₂×C₁₆	direct (N × H)
83. C₄×C₁₆	C₂×C₂₇	direct (N × H)
84. C₂×C₂₇	C₄×C₁₆	direct (N × H)
107. C₂×C₁₆	C₄×C₂₇	direct (N × H)
111. C₂₇	C₂×C₄×C₁₆	direct (N × H)
134. C₁₆	C₂×C₄×C₂₇	direct (N × H)
144. C₂×C₄	C₁₆×C₂₇	direct (N × H)
160. C₄	C₂×C₁₆×C₂₇	direct (N × H)
164. C₂	C₄×C₁₆×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
C₂²×C₁₆×C₂₇	2
C₂×C₄×C₈×C₂₇	2
C₄×C₁₆×C₂₇	2
C₂×C₄×C₉×C₁₆	3

Subgroups — C2×C4×C16×C27 #72183116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₁₆×C₂₇ #72183116