Subgroups — C₂³×C₄×C₁₆×C₄₇ #72201361

Order 24064.

6276 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₁₆×C₄₇	C₂	1	2	yes	maximal
2	C₂³×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄×C₁₆×C₄₇	C₂	1	28	yes	maximal, direct product factor
4	C₂⁴×C₈×C₄₇	C₄	1	1	yes	characteristic
5	C₂⁴×C₈×C₄₇	C₂²	1	1	yes	characteristic
6	C₂³×C₄²×C₄₇	C₄	1	1	yes	characteristic
7	C₂³×C₁₆×C₄₇	C₄	1	32	yes	direct product factor
8	C₂³×C₁₆×C₄₇	C₂²	1	28	yes	—
9	C₂²×C₄×C₈×C₄₇	C₄	1	14	yes	—
10	C₂²×C₄×C₈×C₄₇	C₂²	1	14	yes	—
11	C₂×C₄×C₁₆×C₄₇	C₂²	1	112	yes	direct product factor
12	C₂⁴×C₄×C₄₇	C₈	1	2	yes	—
13	C₂⁴×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₈×C₄₇	C₈	1	16	yes	—
15	C₂³×C₈×C₄₇	C₂×C₄	1	7	yes	—
16	C₂³×C₈×C₄₇	C₂×C₄	1	14	yes	—
17	C₂³×C₈×C₄₇	C₂×C₄	1	16	yes	—
18	C₂³×C₈×C₄₇	C₂³	1	7	yes	—
19	C₂²×C₄²×C₄₇	C₈	1	14	yes	—
20	C₂²×C₄²×C₄₇	C₂×C₄	1	14	yes	—
21	C₂²×C₁₆×C₄₇	C₂×C₄	1	224	yes	direct product factor
22	C₂²×C₁₆×C₄₇	C₂³	1	56	yes	—
23	C₂×C₄×C₈×C₄₇	C₂×C₄	1	84	yes	—
24	C₂×C₄×C₈×C₄₇	C₂³	1	28	yes	—
25	C₄×C₁₆×C₄₇	C₂³	1	64	yes	direct product factor
26	C₂⁵×C₄₇	C₂×C₈	1	1	yes	characteristic
27	C₂³×C₄×C₄₇	C₁₆	1	32	yes	direct product factor
28	C₂³×C₄×C₄₇	C₄²	1	8	yes	—
29	C₂³×C₄×C₄₇	C₂×C₈	1	7	yes	—
30	C₂³×C₄×C₄₇	C₂×C₈	1	8	yes	—
31	C₂³×C₄×C₄₇	C₂×C₈	1	28	yes	—
32	C₂³×C₄×C₄₇	C₂²×C₄	1	7	yes	—
33	C₂²×C₈×C₄₇	C₄²	1	56	yes	—
34	C₂²×C₈×C₄₇	C₂×C₈	1	112	yes	—
35	C₂²×C₈×C₄₇	C₂²×C₄	1	21	yes	—
36	C₂²×C₈×C₄₇	C₂²×C₄	1	28	yes	—
37	C₂²×C₈×C₄₇	C₂²×C₄	1	56	yes	—
38	C₂²×C₈×C₄₇	C₂⁴	1	7	yes	—
39	C₂×C₄²×C₄₇	C₂×C₈	1	84	yes	—
40	C₂×C₄²×C₄₇	C₂²×C₄	1	28	yes	—
41	C₂×C₁₆×C₄₇	C₂²×C₄	1	224	yes	direct product factor
42	C₂×C₁₆×C₄₇	C₂⁴	1	16	yes	—
43	C₄×C₈×C₄₇	C₂²×C₄	1	56	yes	—
44	C₄×C₈×C₄₇	C₂⁴	1	8	yes	—
45	C₂⁴×C₄₇	C₄×C₈	1	8	yes	—
46	C₂⁴×C₄₇	C₂×C₁₆	1	16	yes	—
47	C₂⁴×C₄₇	C₂²×C₈	1	7	yes	—
48	C₂²×C₄×C₄₇	C₄×C₈	1	56	yes	—
49	C₂²×C₄×C₄₇	C₂×C₁₆	1	224	yes	direct product factor
50	C₂²×C₄×C₄₇	C₂×C₄²	1	28	yes	—
51	C₂²×C₄×C₄₇	C₂²×C₈	1	21	yes	—
52	C₂²×C₄×C₄₇	C₂²×C₈	1	28	yes	—
53	C₂²×C₄×C₄₇	C₂²×C₈	1	56	yes	—
54	C₂²×C₄×C₄₇	C₂³×C₄	1	7	yes	—
55	C₂×C₈×C₄₇	C₂×C₄²	1	84	yes	—
56	C₂×C₈×C₄₇	C₂²×C₈	1	112	yes	—
57	C₂×C₈×C₄₇	C₂³×C₄	1	7	yes	—
58	C₂×C₈×C₄₇	C₂³×C₄	1	8	yes	—
59	C₂×C₈×C₄₇	C₂³×C₄	1	28	yes	—
60	C₂×C₈×C₄₇	C₂⁵	1	1	yes	characteristic
61	C₄²×C₄₇	C₂²×C₈	1	56	yes	—
62	C₄²×C₄₇	C₂³×C₄	1	8	yes	—
63	C₁₆×C₄₇	C₂³×C₄	1	32	yes	direct product factor
64	C₂³×C₄×C₁₆	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂³×C₄₇	C₄×C₁₆	1	64	yes	direct product factor
66	C₂³×C₄₇	C₂×C₄×C₈	1	28	yes	—
67	C₂³×C₄₇	C₂²×C₁₆	1	56	yes	—
68	C₂³×C₄₇	C₂³×C₈	1	7	yes	—
69	C₂×C₄×C₄₇	C₂×C₄×C₈	1	84	yes	—
70	C₂×C₄×C₄₇	C₂²×C₁₆	1	224	yes	direct product factor
71	C₂×C₄×C₄₇	C₂²×C₄²	1	14	yes	—
72	C₂×C₄×C₄₇	C₂³×C₈	1	7	yes	—
73	C₂×C₄×C₄₇	C₂³×C₈	1	14	yes	—
74	C₂×C₄×C₄₇	C₂³×C₈	1	16	yes	—
75	C₂×C₄×C₄₇	C₂⁴×C₄	1	1	yes	characteristic
76	C₈×C₄₇	C₂²×C₄²	1	14	yes	—
77	C₈×C₄₇	C₂³×C₈	1	16	yes	—
78	C₈×C₄₇	C₂⁴×C₄	1	2	yes	—
79	C₂⁴×C₁₆	C₂×C₄₇	1	2	yes	—
80	C₂³×C₄×C₈	C₂×C₄₇	1	1	yes	characteristic
81	C₂²×C₄×C₁₆	C₂×C₄₇	1	28	yes	direct product factor
82	C₂²×C₄₇	C₂×C₄×C₁₆	1	112	yes	direct product factor
83	C₂²×C₄₇	C₂²×C₄×C₈	1	14	yes	—
84	C₂²×C₄₇	C₂³×C₁₆	1	28	yes	—
85	C₂²×C₄₇	C₂⁴×C₈	1	1	yes	characteristic
86	C₄×C₄₇	C₂²×C₄×C₈	1	14	yes	—
87	C₄×C₄₇	C₂³×C₁₆	1	32	yes	direct product factor
88	C₄×C₄₇	C₂³×C₄²	1	1	yes	characteristic
89	C₄×C₄₇	C₂⁴×C₈	1	1	yes	characteristic
90	C₂⁴×C₈	C₄×C₄₇	1	1	yes	characteristic
91	C₂⁴×C₈	C₂²×C₄₇	1	1	yes	characteristic
92	C₂³×C₄²	C₄×C₄₇	1	1	yes	characteristic
93	C₂³×C₁₆	C₄×C₄₇	1	32	yes	direct product factor
94	C₂³×C₁₆	C₂²×C₄₇	1	28	yes	—
95	C₂²×C₄×C₈	C₄×C₄₇	1	14	yes	—
96	C₂²×C₄×C₈	C₂²×C₄₇	1	14	yes	—
97	C₂×C₄×C₁₆	C₂²×C₄₇	1	112	yes	direct product factor
98	C₂×C₄₇	C₂²×C₄×C₁₆	1	28	yes	direct product factor
99	C₂×C₄₇	C₂³×C₄×C₈	1	1	yes	characteristic
100	C₂×C₄₇	C₂⁴×C₁₆	1	2	yes	—
101	C₂⁴×C₄	C₈×C₄₇	1	2	yes	—
102	C₂⁴×C₄	C₂×C₄×C₄₇	1	1	yes	characteristic
103	C₂³×C₈	C₈×C₄₇	1	16	yes	—
104	C₂³×C₈	C₂×C₄×C₄₇	1	7	yes	—
105	C₂³×C₈	C₂×C₄×C₄₇	1	14	yes	—
106	C₂³×C₈	C₂×C₄×C₄₇	1	16	yes	—
107	C₂³×C₈	C₂³×C₄₇	1	7	yes	—
108	C₂²×C₄²	C₈×C₄₇	1	14	yes	—
109	C₂²×C₄²	C₂×C₄×C₄₇	1	14	yes	—
110	C₂²×C₁₆	C₂×C₄×C₄₇	1	224	yes	direct product factor
111	C₂²×C₁₆	C₂³×C₄₇	1	56	yes	—
112	C₂×C₄×C₈	C₂×C₄×C₄₇	1	84	yes	—
113	C₂×C₄×C₈	C₂³×C₄₇	1	28	yes	—
114	C₄×C₁₆	C₂³×C₄₇	1	64	yes	direct product factor
115	C₄₇	C₂³×C₄×C₁₆	1	1	yes	characteristic, direct product factor, Sylow
116	C₂⁵	C₂×C₈×C₄₇	1	1	yes	characteristic
117	C₂³×C₄	C₁₆×C₄₇	1	32	yes	direct product factor
118	C₂³×C₄	C₄²×C₄₇	1	8	yes	—
119	C₂³×C₄	C₂×C₈×C₄₇	1	7	yes	—
120	C₂³×C₄	C₂×C₈×C₄₇	1	8	yes	—
121	C₂³×C₄	C₂×C₈×C₄₇	1	28	yes	—
122	C₂³×C₄	C₂²×C₄×C₄₇	1	7	yes	—
123	C₂²×C₈	C₄²×C₄₇	1	56	yes	—
124	C₂²×C₈	C₂×C₈×C₄₇	1	112	yes	—
125	C₂²×C₈	C₂²×C₄×C₄₇	1	21	yes	—
126	C₂²×C₈	C₂²×C₄×C₄₇	1	28	yes	—
127	C₂²×C₈	C₂²×C₄×C₄₇	1	56	yes	—
128	C₂²×C₈	C₂⁴×C₄₇	1	7	yes	—
129	C₂×C₄²	C₂×C₈×C₄₇	1	84	yes	—
130	C₂×C₄²	C₂²×C₄×C₄₇	1	28	yes	—
131	C₂×C₁₆	C₂²×C₄×C₄₇	1	224	yes	direct product factor
132	C₂×C₁₆	C₂⁴×C₄₇	1	16	yes	—
133	C₄×C₈	C₂²×C₄×C₄₇	1	56	yes	—
134	C₄×C₈	C₂⁴×C₄₇	1	8	yes	—
135	C₂⁴	C₄×C₈×C₄₇	1	8	yes	—
136	C₂⁴	C₂×C₁₆×C₄₇	1	16	yes	—
137	C₂⁴	C₂²×C₈×C₄₇	1	7	yes	—
138	C₂²×C₄	C₄×C₈×C₄₇	1	56	yes	—
139	C₂²×C₄	C₂×C₁₆×C₄₇	1	224	yes	direct product factor
140	C₂²×C₄	C₂×C₄²×C₄₇	1	28	yes	—
141	C₂²×C₄	C₂²×C₈×C₄₇	1	21	yes	—
142	C₂²×C₄	C₂²×C₈×C₄₇	1	28	yes	—
143	C₂²×C₄	C₂²×C₈×C₄₇	1	56	yes	—
144	C₂²×C₄	C₂³×C₄×C₄₇	1	7	yes	—
145	C₂×C₈	C₂×C₄²×C₄₇	1	84	yes	—
146	C₂×C₈	C₂²×C₈×C₄₇	1	112	yes	—
147	C₂×C₈	C₂³×C₄×C₄₇	1	7	yes	—
148	C₂×C₈	C₂³×C₄×C₄₇	1	8	yes	—
149	C₂×C₈	C₂³×C₄×C₄₇	1	28	yes	—
150	C₂×C₈	C₂⁵×C₄₇	1	1	yes	characteristic
151	C₄²	C₂²×C₈×C₄₇	1	56	yes	—
152	C₄²	C₂³×C₄×C₄₇	1	8	yes	—
153	C₁₆	C₂³×C₄×C₄₇	1	32	yes	direct product factor
154	C₂³	C₄×C₁₆×C₄₇	1	64	yes	direct product factor
155	C₂³	C₂×C₄×C₈×C₄₇	1	28	yes	—
156	C₂³	C₂²×C₁₆×C₄₇	1	56	yes	—
157	C₂³	C₂³×C₈×C₄₇	1	7	yes	—
158	C₂×C₄	C₂×C₄×C₈×C₄₇	1	84	yes	—
159	C₂×C₄	C₂²×C₁₆×C₄₇	1	224	yes	direct product factor
160	C₂×C₄	C₂²×C₄²×C₄₇	1	14	yes	—
161	C₂×C₄	C₂³×C₈×C₄₇	1	7	yes	—
162	C₂×C₄	C₂³×C₈×C₄₇	1	14	yes	—
163	C₂×C₄	C₂³×C₈×C₄₇	1	16	yes	—
164	C₂×C₄	C₂⁴×C₄×C₄₇	1	1	yes	characteristic
165	C₈	C₂²×C₄²×C₄₇	1	14	yes	—
166	C₈	C₂³×C₈×C₄₇	1	16	yes	—
167	C₈	C₂⁴×C₄×C₄₇	1	2	yes	—
168	C₂²	C₂×C₄×C₁₆×C₄₇	1	112	yes	direct product factor
169	C₂²	C₂²×C₄×C₈×C₄₇	1	14	yes	—
170	C₂²	C₂³×C₁₆×C₄₇	1	28	yes	—
171	C₂²	C₂⁴×C₈×C₄₇	1	1	yes	characteristic
172	C₄	C₂²×C₄×C₈×C₄₇	1	14	yes	—
173	C₄	C₂³×C₁₆×C₄₇	1	32	yes	direct product factor
174	C₄	C₂³×C₄²×C₄₇	1	1	yes	characteristic
175	C₄	C₂⁴×C₈×C₄₇	1	1	yes	characteristic
176	C₂	C₂²×C₄×C₁₆×C₄₇	1	28	yes	direct product factor
177	C₂	C₂³×C₄×C₈×C₄₇	1	1	yes	characteristic
178	C₂	C₂⁴×C₁₆×C₄₇	1	2	yes	—

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₈	C₂⁵×C₄₇
socle, prime radical	C₂⁵×C₄₇	C₂×C₈
LCS	G ▷ C₁
UCS	C₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄×C₁₆×C₄₇	C₂	direct (N × H)
7. C₂³×C₁₆×C₄₇	C₄	direct (N × H)
11. C₂×C₄×C₁₆×C₄₇	C₂²	direct (N × H)
21. C₂²×C₁₆×C₄₇	C₂×C₄	direct (N × H)
25. C₄×C₁₆×C₄₇	C₂³	direct (N × H)
27. C₂³×C₄×C₄₇	C₁₆	direct (N × H)
41. C₂×C₁₆×C₄₇	C₂²×C₄	direct (N × H)
49. C₂²×C₄×C₄₇	C₂×C₁₆	direct (N × H)
63. C₁₆×C₄₇	C₂³×C₄	direct (N × H)
64. C₂³×C₄×C₁₆	C₄₇	direct (N × H)
65. C₂³×C₄₇	C₄×C₁₆	direct (N × H)
70. C₂×C₄×C₄₇	C₂²×C₁₆	direct (N × H)
81. C₂²×C₄×C₁₆	C₂×C₄₇	direct (N × H)
82. C₂²×C₄₇	C₂×C₄×C₁₆	direct (N × H)
87. C₄×C₄₇	C₂³×C₁₆	direct (N × H)
93. C₂³×C₁₆	C₄×C₄₇	direct (N × H)
97. C₂×C₄×C₁₆	C₂²×C₄₇	direct (N × H)
98. C₂×C₄₇	C₂²×C₄×C₁₆	direct (N × H)
110. C₂²×C₁₆	C₂×C₄×C₄₇	direct (N × H)
114. C₄×C₁₆	C₂³×C₄₇	direct (N × H)
115. C₄₇	C₂³×C₄×C₁₆	direct (N × H)
117. C₂³×C₄	C₁₆×C₄₇	direct (N × H)
131. C₂×C₁₆	C₂²×C₄×C₄₇	direct (N × H)
139. C₂²×C₄	C₂×C₁₆×C₄₇	direct (N × H)
153. C₁₆	C₂³×C₄×C₄₇	direct (N × H)
154. C₂³	C₄×C₁₆×C₄₇	direct (N × H)
159. C₂×C₄	C₂²×C₁₆×C₄₇	direct (N × H)
168. C₂²	C₂×C₄×C₁₆×C₄₇	direct (N × H)
173. C₄	C₂³×C₁₆×C₄₇	direct (N × H)
176. C₂	C₂²×C₄×C₁₆×C₄₇	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×C₄₇	2
C₂³×C₄×C₈×C₄₇	2
C₂²×C₄×C₁₆×C₄₇	2
C₂³×C₄×C₁₆	47

Subgroups — C23×C4×C16×C47 #72201361

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×C₁₆×C₄₇ #72201361