Subgroups — C₂³×M₁₆×D₇₃ #33999739

Order 18688.

13310 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₄×D₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂³×#11924012	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂³×#11924011	C₂	1	2	yes	maximal, complemented
4	C₂²×M₁₆×D₇₃	C₂	1	56	yes	maximal, direct product factor
5	C₂³×C₈×D₇₃	C₂	1	2	yes	maximal, complemented
6	C₂³×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₇₃	C₄	1	1	yes	characteristic
8	C₂⁴×Dic₇₃	C₄	1	1	yes	characteristic
9	C₂³×C₄×D₇₃	C₄	1	28	yes	—
10	C₂³×C₄×D₇₃	C₄	1	1	yes	characteristic
11	C₂³×C₄×D₇₃	C₄	1	1	yes	characteristic
12	C₂³×C₄×D₇₃	C₂²	1	1	yes	characteristic
13	C₂³×C₄×D₇₃	C₂²	1	28	yes	—
14	C₂³×C₄×D₇₃	C₂²	1	1	yes	characteristic
15	C₂²×#11924012	C₂²	1	28	yes	complemented
16	C₂²×#11924011	C₂²	1	56	yes	complemented
17	C₂×M₁₆×D₇₃	C₂²	1	448	yes	direct product factor
18	C₂³×#10596831	C₂²	1	2	yes	complemented
19	C₂²×C₈×D₇₃	C₂²	1	56	yes	complemented
20	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
21	C₂²×M₁₆×C₇₃	C₂²	1	28	yes	complemented
22	C₂³×C₈×C₇₃	C₂²	1	2	yes	complemented
23	C₂⁴×D₇₃	—	2	16	no	—
24	C₂⁴×D₇₃	C₂×C₄	1	28	yes	—
25	C₂⁴×D₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂⁴×D₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×Dic₇₃	C₂×C₄	1	28	yes	—
28	C₂³×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×Dic₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×D₇₃	C₂×C₄	1	336	yes	—
31	C₂²×C₄×D₇₃	C₂×C₄	1	28	yes	—
32	C₂²×C₄×D₇₃	C₂×C₄	1	14	yes	—
33	C₂²×C₄×D₇₃	C₂×C₄	1	14	yes	—
34	C₂²×C₄×D₇₃	C₂×C₄	1	28	yes	—
35	C₂²×C₄×D₇₃	C₂³	1	14	yes	—
36	C₂²×C₄×D₇₃	C₂³	1	112	yes	—
37	C₂²×C₄×D₇₃	C₂³	1	14	yes	—
38	C₂×#11924012	C₂³	1	112	yes	complemented
39	C₂×#11924011	C₂³	1	224	yes	complemented
40	M₁₆×D₇₃	C₂³	1	512	yes	direct product factor
41	C₂²×#10596831	C₂³	1	28	yes	complemented
42	C₂×C₈×D₇₃	C₂³	1	224	yes	complemented
43	C₂⁵×C₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₇₃	C₂×C₄	1	14	yes	—
45	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂³×C₄×C₇₃	C₂³	1	14	yes	—
47	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₇₃	C₂³	1	112	yes	complemented
49	C₂²×C₈×C₇₃	C₂³	1	28	yes	complemented
50	C₂³×D₇₃	—	2	8	no	—
51	C₂³×D₇₃	—	2	224	no	—
52	C₂³×D₇₃	M₁₆	1	16	yes	direct product factor
53	C₂³×D₇₃	C₂²×C₄	1	14	yes	—
54	C₂³×D₇₃	C₂²×C₄	1	112	yes	—
55	C₂³×D₇₃	C₂²×C₄	1	14	yes	—
56	C₂²×Dic₇₃	C₂²×C₄	1	112	yes	—
57	C₂²×Dic₇₃	C₂²×C₄	1	14	yes	—
58	C₂²×Dic₇₃	C₂²×C₄	1	14	yes	—
59	C₂×C₄×D₇₃	C₂²×C₄	1	112	yes	—
60	C₂×C₄×D₇₃	C₂²×C₄	1	448	yes	—
61	C₂×C₄×D₇₃	C₂²×C₄	1	84	yes	—
62	C₂×C₄×D₇₃	C₂²×C₄	1	84	yes	—
63	C₂×C₄×D₇₃	C₂²×C₄	1	112	yes	—
64	C₂×C₄×D₇₃	C₂⁴	1	28	yes	—
65	C₂×C₄×D₇₃	C₂⁴	1	64	yes	—
66	C₂×C₄×D₇₃	C₂⁴	1	28	yes	—
67	#11924012	C₂⁴	1	64	yes	complemented
68	#11924011	C₂⁴	1	128	yes	complemented
69	C₂×#10596831	C₂⁴	1	56	yes	complemented
70	C₈×D₇₃	C₂⁴	1	128	yes	complemented
71	C₂⁴×C₇₃	—	2	8	no	—
72	C₂⁴×C₇₃	C₂²×C₄	1	14	yes	—
73	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₃	C₂²×C₄	1	84	yes	—
75	C₂²×C₄×C₇₃	C₂²×C₄	1	14	yes	—
76	C₂²×C₄×C₇₃	C₂²×C₄	1	7	yes	—
77	C₂²×C₄×C₇₃	C₂⁴	1	7	yes	—
78	C₂²×C₄×C₇₃	C₂⁴	1	28	yes	—
79	M₁₆×C₇₃	C₂⁴	1	64	yes	complemented
80	C₂×C₈×C₇₃	C₂⁴	1	56	yes	complemented
81	C₂³×C₇₃	—	2	56	no	—
82	C₂³×C₇₃	C₂×M₁₆	1	8	yes	complemented
83	C₂³×C₇₃	C₂³×C₄	1	28	yes	—
84	C₂³×C₇₃	C₂³×C₄	1	7	yes	—
85	C₂²×D₇₃	—	2	56	no	—
86	C₂²×D₇₃	—	2	448	no	—
87	C₂²×D₇₃	C₂×M₁₆	1	112	yes	direct product factor
88	C₂²×D₇₃	C₂³×C₄	1	28	yes	—
89	C₂²×D₇₃	C₂³×C₄	1	64	yes	—
90	C₂²×D₇₃	C₂³×C₄	1	28	yes	—
91	C₂×C₄×C₇₃	C₂³×C₄	1	56	yes	—
92	C₂×C₄×C₇₃	C₂³×C₄	1	21	yes	—
93	C₂×C₄×C₇₃	C₂³×C₄	1	28	yes	—
94	C₂×C₄×C₇₃	C₂⁵	1	7	yes	—
95	C₂×C₄×C₇₃	C₂⁵	1	8	yes	—
96	C₂×Dic₇₃	C₂³×C₄	1	64	yes	—
97	C₂×Dic₇₃	C₂³×C₄	1	28	yes	—
98	C₂×Dic₇₃	C₂³×C₄	1	28	yes	—
99	C₄×D₇₃	C₂³×C₄	1	64	yes	—
100	C₄×D₇₃	C₂³×C₄	1	56	yes	—
101	C₄×D₇₃	C₂³×C₄	1	56	yes	—
102	C₄×D₇₃	C₂³×C₄	1	64	yes	—
103	C₄×D₇₃	C₂⁵	1	8	yes	—
104	C₄×D₇₃	C₂⁵	1	8	yes	—
105	C₈×C₇₃	C₂⁵	1	16	yes	complemented
106	#10596831	C₂⁵	1	16	yes	complemented
107	C₂²×C₇₃	—	2	56	no	—
108	C₂²×C₇₃	C₂²×M₁₆	1	28	yes	complemented
109	C₂²×C₇₃	C₂⁴×C₄	1	8	yes	—
110	C₂²×C₇₃	C₂⁴×C₄	1	7	yes	—
111	C₂×D₇₃	—	2	56	no	—
112	C₂×D₇₃	—	2	128	no	—
113	C₂×D₇₃	C₂²×M₁₆	1	112	yes	direct product factor
114	C₂×D₇₃	C₂⁴×C₄	1	8	yes	—
115	C₂×D₇₃	C₂⁴×C₄	1	8	yes	—
116	C₄×C₇₃	C₂⁴×C₄	1	7	yes	—
117	C₄×C₇₃	C₂⁴×C₄	1	8	yes	—
118	C₄×C₇₃	C₂⁶	1	1	yes	characteristic
119	Dic₇₃	C₂⁴×C₄	1	8	yes	—
120	Dic₇₃	C₂⁴×C₄	1	8	yes	—
121	C₂⁴×M₁₆	—	73	1	no	maximal, Sylow
122	C₂×C₇₃	—	2	8	no	—
123	C₂×C₇₃	C₂³×M₁₆	1	14	yes	complemented
124	C₂×C₇₃	C₂⁵×C₄	1	1	yes	characteristic
125	D₇₃	—	2	8	no	—
126	D₇₃	C₂³×M₁₆	1	16	yes	direct product factor
127	C₂⁵×C₄	—	73	1	no	—
128	C₂³×M₁₆	—	73	1	no	—
129	C₂³×M₁₆	—	73	56	no	—
130	C₂³×M₁₆	—	73	2	no	—
131	C₂³×M₁₆	D₇₃	1	1	yes	characteristic, direct product factor
132	C₂⁴×C₈	—	73	2	no	—
133	C₇₃	C₂⁴×M₁₆	1	1	yes	characteristic, complemented, Sylow
134	C₂⁶	—	73	1	no	—
135	C₂⁴×C₄	—	73	1	no	—
136	C₂⁴×C₄	—	73	1	no	—
137	C₂⁴×C₄	—	73	28	no	—
138	C₂⁴×C₄	—	73	1	no	—
139	C₂⁴×C₄	—	73	28	no	—
140	C₂⁴×C₄	—	73	1	no	—
141	C₂⁴×C₄	—	73	1	no	—
142	C₂⁴×C₄	C₂×D₇₃	1	1	yes	characteristic
143	C₂²×M₁₆	—	73	28	no	—
144	C₂²×M₁₆	—	73	448	no	—
145	C₂²×M₁₆	—	73	56	no	—
146	C₂²×M₁₆	C₂×D₇₃	1	28	yes	direct product factor
147	C₂³×C₈	—	73	2	no	—
148	C₂³×C₈	—	73	56	no	—
149	C₂³×C₈	C₂×D₇₃	1	2	yes	complemented
150	C₂⁵	—	73	1	no	—
151	C₂⁵	—	73	28	no	—
152	C₂⁵	—	73	1	no	—
153	C₂⁵	—	146	16	no	—
154	C₂⁵	C₄×D₇₃	1	1	yes	characteristic
155	C₂³×C₄	—	73	28	no	—
156	C₂³×C₄	—	73	1	no	—
157	C₂³×C₄	—	73	1	no	—
158	C₂³×C₄	—	73	28	no	—
159	C₂³×C₄	—	73	336	no	—
160	C₂³×C₄	—	73	28	no	—
161	C₂³×C₄	—	73	14	no	—
162	C₂³×C₄	—	73	14	no	—
163	C₂³×C₄	—	73	14	no	—
164	C₂³×C₄	—	73	112	no	—
165	C₂³×C₄	—	73	14	no	—
166	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
167	C₂³×C₄	C₄×D₇₃	1	14	yes	—
168	C₂³×C₄	C₂²×D₇₃	1	14	yes	—
169	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
170	C₂×M₁₆	—	73	112	no	—
171	C₂×M₁₆	—	73	512	no	—
172	C₂×M₁₆	—	73	224	no	—
173	C₂×M₁₆	C₂²×D₇₃	1	112	yes	direct product factor
174	C₂²×C₈	—	73	28	no	—
175	C₂²×C₈	—	73	224	no	—
176	C₂²×C₈	C₂²×D₇₃	1	28	yes	complemented
177	C₂⁴	—	2	8	no	—
178	C₂⁴	—	73	16	no	—
179	C₂⁴	—	73	14	no	—
180	C₂⁴	—	73	112	no	—
181	C₂⁴	—	73	14	no	—
182	C₂⁴	—	146	224	no	—
183	C₂⁴	—	146	8	no	—
184	C₂⁴	C₂×C₄×D₇₃	1	14	yes	—
185	C₂⁴	C₂×C₄×D₇₃	1	1	yes	characteristic
186	C₂²×C₄	—	73	112	no	—
187	C₂²×C₄	—	73	14	no	—
188	C₂²×C₄	—	73	14	no	—
189	C₂²×C₄	—	73	84	no	—
190	C₂²×C₄	—	73	112	no	—
191	C₂²×C₄	—	73	448	no	—
192	C₂²×C₄	—	73	112	no	—
193	C₂²×C₄	—	73	84	no	—
194	C₂²×C₄	—	73	28	no	—
195	C₂²×C₄	—	73	64	no	—
196	C₂²×C₄	—	73	28	no	—
197	C₂²×C₄	C₂×C₄×D₇₃	1	84	yes	—
198	C₂²×C₄	C₂×C₄×D₇₃	1	14	yes	—
199	C₂²×C₄	C₂×C₄×D₇₃	1	7	yes	—
200	C₂²×C₄	C₂³×D₇₃	1	7	yes	—
201	C₂²×C₄	C₂³×D₇₃	1	28	yes	—
202	M₁₆	—	73	64	no	—
203	M₁₆	—	73	128	no	—
204	M₁₆	C₂³×D₇₃	1	64	yes	direct product factor
205	C₂×C₈	—	73	56	no	—
206	C₂×C₈	—	73	128	no	—
207	C₂×C₈	C₂³×D₇₃	1	56	yes	complemented
208	C₂³	—	2	56	no	—
209	C₂³	—	73	28	no	—
210	C₂³	—	73	112	no	—
211	C₂³	—	73	64	no	—
212	C₂³	—	73	28	no	—
213	C₂³	—	146	448	no	—
214	C₂³	—	146	56	no	—
215	C₂³	M₁₆×D₇₃	1	8	yes	direct product factor
216	C₂³	C₂²×C₄×D₇₃	1	28	yes	—
217	C₂³	C₂²×C₄×D₇₃	1	7	yes	—
218	C₂×C₄	—	73	64	no	—
219	C₂×C₄	—	73	28	no	—
220	C₂×C₄	—	73	28	no	—
221	C₂×C₄	—	73	56	no	—
222	C₂×C₄	—	73	64	no	—
223	C₂×C₄	—	73	64	no	—
224	C₂×C₄	—	73	56	no	—
225	C₂×C₄	—	73	8	no	—
226	C₂×C₄	—	73	8	no	—
227	C₂×C₄	C₂²×C₄×D₇₃	1	56	yes	—
228	C₂×C₄	C₂²×C₄×D₇₃	1	28	yes	—
229	C₂×C₄	C₂²×C₄×D₇₃	1	21	yes	—
230	C₂×C₄	C₂⁴×D₇₃	1	7	yes	—
231	C₂×C₄	C₂⁴×D₇₃	1	8	yes	—
232	C₈	—	73	16	no	—
233	C₈	C₂⁴×D₇₃	1	16	yes	complemented
234	C₂²	—	2	56	no	—
235	C₂²	—	73	112	no	—
236	C₂²	—	73	8	no	—
237	C₂²	—	73	8	no	—
238	C₂²	—	146	128	no	—
239	C₂²	—	146	56	no	—
240	C₂²	C₂×M₁₆×D₇₃	1	28	yes	direct product factor
241	C₂²	C₂³×C₄×D₇₃	1	8	yes	—
242	C₂²	C₂³×C₄×D₇₃	1	7	yes	—
243	C₄	—	73	8	no	—
244	C₄	—	73	8	no	—
245	C₄	C₂³×C₄×D₇₃	1	7	yes	—
246	C₄	C₂³×C₄×D₇₃	1	8	yes	—
247	C₄	C₂⁵×D₇₃	1	1	yes	characteristic
248	C₂	—	2	8	no	—
249	C₂	—	73	16	no	—
250	C₂	—	146	8	no	—
251	C₂	C₂²×M₁₆×D₇₃	1	14	yes	direct product factor
252	C₂	C₂⁴×C₄×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂²×D₇₃
commutator	C₂×C₇₃	C₂⁵×C₄
Frattini	C₄	C₂⁵×D₇₃
Fitting	C₂³×M₁₆×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂²×C₄
prime radical	C₂⁵×D₇₃	C₄
cosocle	C₄×C₇₃	C₂⁶
hypercenter	C₂³×M₁₆	D₇₃
hypocenter	C₇₃	C₂⁴×M₁₆
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×M₁₆

Complements

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

N	H ≅ G/N	type
2. C₂³×#11924012	C₂	semidirect
3. C₂³×#11924011	C₂	semidirect
4. C₂²×M₁₆×D₇₃	C₂	direct (N × H)
5. C₂³×C₈×D₇₃	C₂	semidirect
6. C₂³×M₁₆×C₇₃	C₂	semidirect
15. C₂²×#11924012	C₂²	semidirect
16. C₂²×#11924011	C₂²	semidirect
17. C₂×M₁₆×D₇₃	C₂²	direct (N × H)
18. C₂³×#10596831	C₂²	semidirect
19. C₂²×C₈×D₇₃	C₂²	semidirect
21. C₂²×M₁₆×C₇₃	C₂²	semidirect
22. C₂³×C₈×C₇₃	C₂²	semidirect
38. C₂×#11924012	C₂³	semidirect
39. C₂×#11924011	C₂³	semidirect
40. M₁₆×D₇₃	C₂³	direct (N × H)
41. C₂²×#10596831	C₂³	semidirect
42. C₂×C₈×D₇₃	C₂³	semidirect
48. C₂×M₁₆×C₇₃	C₂³	semidirect
49. C₂²×C₈×C₇₃	C₂³	semidirect
52. C₂³×D₇₃	M₁₆	direct (N × H)
67. #11924012	C₂⁴	semidirect
68. #11924011	C₂⁴	semidirect
69. C₂×#10596831	C₂⁴	semidirect
70. C₈×D₇₃	C₂⁴	semidirect
79. M₁₆×C₇₃	C₂⁴	semidirect
80. C₂×C₈×C₇₃	C₂⁴	semidirect
82. C₂³×C₇₃	C₂×M₁₆	semidirect
87. C₂²×D₇₃	C₂×M₁₆	direct (N × H)
105. C₈×C₇₃	C₂⁵	semidirect
106. #10596831	C₂⁵	semidirect
108. C₂²×C₇₃	C₂²×M₁₆	semidirect
113. C₂×D₇₃	C₂²×M₁₆	direct (N × H)
123. C₂×C₇₃	C₂³×M₁₆	semidirect
126. D₇₃	C₂³×M₁₆	direct (N × H)
131. C₂³×M₁₆	D₇₃	direct (N × H)
133. C₇₃	C₂⁴×M₁₆	semidirect
146. C₂²×M₁₆	C₂×D₇₃	direct (N × H)
149. C₂³×C₈	C₂×D₇₃	semidirect
173. C₂×M₁₆	C₂²×D₇₃	direct (N × H)
176. C₂²×C₈	C₂²×D₇₃	semidirect
204. M₁₆	C₂³×D₇₃	direct (N × H)
207. C₂×C₈	C₂³×D₇₃	semidirect
215. C₂³	M₁₆×D₇₃	direct (N × H)
233. C₈	C₂⁴×D₇₃	semidirect
240. C₂²	C₂×M₁₆×D₇₃	direct (N × H)
251. C₂	C₂²×M₁₆×D₇₃	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×C₄×D₇₃	2
C₂³×#11924012	2
C₂³×#11924011	2
C₂²×M₁₆×D₇₃	2
C₂³×C₈×D₇₃	2
C₂³×M₁₆×C₇₃	2
C₂⁴×M₁₆	73

Subgroups — C23×M16×D73 #33999739

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×M₁₆×D₇₃ #33999739