Subgroups — C₇₃×#23453 #32932857

Order 18688.

1282 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#125×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#417×C₇₃	C₂	1	2	yes	maximal
3	C₇₃×#2297	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#2026	C₂	1	4	yes	maximal, complemented
5	C₇₃×#2025	C₂	1	4	yes	maximal, complemented
6	C₇₃×#1736	C₂	1	2	yes	maximal, complemented
7	C₇₃×#1728	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₇₃	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
10	C₂×#125×C₇₃	—	2	8	no	—
11	C₂×#125×C₇₃	—	2	2	no	—
12	C₂×#125×C₇₃	C₂²	1	4	yes	—
13	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₇₃	—	2	2	no	—
15	C₂²×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
17	#417×C₇₃	—	2	8	no	—
18	C₂×#103×C₇₃	C₂²	1	4	yes	—
19	C₂×#102×C₇₃	C₂²	1	4	yes	—
20	#391×C₇₃	C₂²	1	2	yes	—
21	#390×C₇₃	C₂²	1	2	yes	—
22	#386×C₇₃	—	2	2	no	—
23	C₄×C₂²⋊C₄×C₇₃	—	2	2	no	—
24	#340×C₇₃	C₂²	1	8	yes	—
25	#336×C₇₃	C₂²	1	4	yes	—
26	C₂×#41×C₇₃	—	2	2	no	—
27	C₂×#41×C₇₃	—	2	4	no	—
28	C₂×#41×C₇₃	—	2	4	no	—
29	C₂×#41×C₇₃	Q₈	1	4	yes	—
30	C₂²×Q₈×C₇₃	C₂³	1	1	yes	characteristic
31	C₂²×D₄×C₇₃	—	2	1	no	—
32	C₂²×D₄×C₇₃	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₇₃	—	2	2	no	—
34	C₂³×C₄×C₇₃	—	2	1	no	—
35	C₂³×C₄×C₇₃	D₄	1	2	yes	—
36	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₇₃	—	2	4	no	—
38	C₂×M₁₆×C₇₃	—	2	4	no	—
39	C₂×M₁₆×C₇₃	—	2	4	no	—
40	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
41	#125×C₇₃	—	2	16	no	—
42	#125×C₇₃	—	2	16	no	—
43	#117×C₇₃	—	2	4	no	—
44	C₂×C₄⋊C₄×C₇₃	—	2	16	no	—
45	C₂×C₄⋊C₄×C₇₃	—	2	8	no	—
46	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
48	C₂×C₄⋊C₄×C₇₃	D₄	1	8	yes	—
49	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
51	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
52	C₂×C₂²⋊C₄×C₇₃	—	2	4	no	—
53	C₂×C₄²×C₇₃	—	2	4	no	—
54	C₂×C₄²×C₇₃	—	2	1	no	—
55	C₂×C₄²×C₇₃	D₄	1	2	yes	—
56	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
57	#103×C₇₃	—	2	8	no	—
58	#102×C₇₃	—	2	8	no	—
59	#95×C₇₃	—	2	4	no	—
60	C₂⁴×C₇₃	—	2	1	no	—
61	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
62	#41×C₇₃	—	2	16	no	—
63	#41×C₇₃	—	4	8	no	—
64	C₂×Q₈×C₇₃	—	2	1	no	—
65	C₂×Q₈×C₇₃	—	2	2	no	—
66	C₂×Q₈×C₇₃	—	2	2	no	—
67	C₂×Q₈×C₇₃	C₂×Q₈	1	2	yes	—
68	C₂×D₄×C₇₃	—	2	4	no	—
69	C₂×D₄×C₇₃	—	2	2	no	—
70	C₂×D₄×C₇₃	—	2	2	no	—
71	C₂×D₄×C₇₃	—	2	1	no	—
72	C₂×D₄×C₇₃	—	4	2	no	—
73	C₂×D₄×C₇₃	—	4	2	no	—
74	C₂×D₄×C₇₃	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₇₃	—	2	4	no	—
76	C₂²×C₄×C₇₃	—	2	8	no	—
77	C₂²×C₄×C₇₃	—	2	1	no	—
78	C₂²×C₄×C₇₃	—	2	4	no	—
79	C₂²×C₄×C₇₃	—	2	4	no	—
80	C₂²×C₄×C₇₃	—	2	2	no	—
81	C₂²×C₄×C₇₃	—	4	4	no	—
82	C₂²×C₄×C₇₃	—	4	8	no	—
83	C₂²×C₄×C₇₃	—	4	2	no	—
84	C₂²×C₄×C₇₃	—	4	2	no	—
85	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
88	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₇₃	#41	1	4	yes	—
90	C₂²×C₄×C₇₃	#41	1	2	yes	—
91	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
92	M₁₆×C₇₃	—	4	8	no	—
93	C₂×C₈×C₇₃	—	2	4	no	—
94	C₂×C₈×C₇₃	—	2	4	no	—
95	C₂×C₈×C₇₃	—	2	4	no	—
96	C₄⋊C₄×C₇₃	—	2	32	no	—
97	C₄⋊C₄×C₇₃	—	2	8	no	—
98	C₄⋊C₄×C₇₃	—	2	4	no	—
99	C₄⋊C₄×C₇₃	—	2	4	no	—
100	C₄⋊C₄×C₇₃	C₂×D₄	1	8	yes	—
101	C₂²⋊C₄×C₇₃	—	4	8	no	—
102	C₂²⋊C₄×C₇₃	—	4	2	no	—
103	C₄²×C₇₃	—	2	4	no	—
104	C₄²×C₇₃	—	2	4	no	—
105	C₄²×C₇₃	—	4	2	no	—
106	C₂³×C₇₃	—	2	4	no	—
107	C₂³×C₇₃	—	2	2	no	—
108	C₂³×C₇₃	—	4	1	no	—
109	C₂³×C₇₃	—	4	1	no	—
110	C₂³×C₇₃	—	4	4	no	—
111	C₂³×C₇₃	#120	1	2	yes	—
112	C₂³×C₇₃	C₂²⋊Q₈	1	2	yes	—
113	C₂³×C₇₃	#124	1	2	yes	—
114	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
115	Q₈×C₇₃	—	2	4	no	—
116	Q₈×C₇₃	—	4	2	no	—
117	D₄×C₇₃	—	2	4	no	—
118	D₄×C₇₃	—	4	8	no	—
119	D₄×C₇₃	—	4	2	no	—
120	C₂×C₄×C₇₃	—	2	4	no	—
121	C₂×C₄×C₇₃	—	2	8	no	—
122	C₂×C₄×C₇₃	—	2	2	no	—
123	C₂×C₄×C₇₃	—	2	4	no	—
124	C₂×C₄×C₇₃	—	2	4	no	—
125	C₂×C₄×C₇₃	—	4	8	no	—
126	C₂×C₄×C₇₃	—	4	16	no	—
127	C₂×C₄×C₇₃	—	4	1	no	—
128	C₂×C₄×C₇₃	—	4	1	no	—
129	C₂×C₄×C₇₃	—	4	8	no	—
130	C₂×C₄×C₇₃	—	4	8	no	—
131	C₂×C₄×C₇₃	—	4	2	no	—
132	C₂×C₄×C₇₃	—	4	4	no	—
133	C₂×C₄×C₇₃	#120	1	2	yes	—
134	C₂×C₄×C₇₃	C₂²⋊Q₈	1	8	yes	—
135	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
136	C₂×C₄×C₇₃	#124	1	8	yes	—
137	C₂×C₄×C₇₃	#124	1	2	yes	—
138	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
139	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
141	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
142	C₈×C₇₃	—	4	4	no	—
143	C₂²×C₇₃	—	2	4	no	—
144	C₂²×C₇₃	—	2	4	no	—
145	C₂²×C₇₃	—	2	4	no	—
146	C₂²×C₇₃	—	4	4	no	—
147	C₂²×C₇₃	—	4	2	no	—
148	C₂²×C₇₃	—	8	2	no	—
149	C₂²×C₇₃	—	8	2	no	—
150	C₂²×C₇₃	#393	1	4	yes	—
151	C₂²×C₇₃	C₂×#120	1	1	yes	characteristic
152	C₂²×C₇₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
153	C₂²×C₇₃	C₂×#124	1	1	yes	characteristic
154	C₂²×C₇₃	#575	1	4	yes	—
155	C₄×C₇₃	—	4	2	no	—
156	C₄×C₇₃	—	4	4	no	—
157	C₄×C₇₃	—	4	8	no	—
158	C₄×C₇₃	—	8	2	no	—
159	C₄×C₇₃	#393	1	4	yes	—
160	C₄×C₇₃	C₂×C₂²⋊Q₈	1	2	yes	—
161	C₄×C₇₃	C₂×#124	1	2	yes	—
162	#23453	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂×C₇₃	—	2	4	no	—
164	C₂×C₇₃	—	8	2	no	—
165	C₂×C₇₃	C₂×#393	1	1	yes	characteristic
166	C₂×C₇₃	#3005	1	2	yes	—
167	C₂×C₇₃	#3080	1	2	yes	—
168	C₂×C₇₃	#3142	1	2	yes	—
169	C₂²×#125	C₂×C₇₃	1	1	yes	characteristic, complemented
170	C₂×#417	C₂×C₇₃	1	2	yes	—
171	#2297	C₂×C₇₃	1	1	yes	characteristic
172	#2026	C₂×C₇₃	1	4	yes	complemented
173	#2025	C₂×C₇₃	1	4	yes	complemented
174	#1736	C₂×C₇₃	1	2	yes	complemented
175	#1728	C₂×C₇₃	1	1	yes	characteristic, complemented
176	C₇₃	#23453	1	1	yes	characteristic, direct product factor, Sylow
177	C₂²×#41	C₂²×C₇₃	1	1	yes	characteristic
178	C₂²×M₁₆	C₂²×C₇₃	1	2	yes	—
179	C₂×#125	—	2	8	no	—
180	C₂×#125	—	2	2	no	—
181	C₂×#125	C₂²×C₇₃	1	4	yes	—
182	C₂×#117	C₂²×C₇₃	1	1	yes	characteristic
183	C₂²×C₄⋊C₄	—	2	2	no	—
184	C₂²×C₄⋊C₄	C₂²×C₇₃	1	2	yes	—
185	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
186	#417	—	2	8	no	—
187	C₂×#103	C₂²×C₇₃	1	4	yes	—
188	C₂×#102	C₂²×C₇₃	1	4	yes	—
189	#391	C₂²×C₇₃	1	2	yes	—
190	#390	C₂²×C₇₃	1	2	yes	—
191	#386	—	2	2	no	—
192	C₄×C₂²⋊C₄	—	2	2	no	—
193	#340	C₂²×C₇₃	1	8	yes	—
194	#336	C₂²×C₇₃	1	4	yes	—
195	C₂×#41	—	2	2	no	—
196	C₂×#41	—	2	4	no	—
197	C₂×#41	—	2	4	no	—
198	C₂×#41	Q₈×C₇₃	1	4	yes	—
199	C₂²×Q₈	C₂³×C₇₃	1	1	yes	characteristic
200	C₂²×D₄	—	2	1	no	—
201	C₂²×D₄	C₂³×C₇₃	1	1	yes	characteristic
202	C₂³×C₄	—	2	2	no	—
203	C₂³×C₄	—	2	1	no	—
204	C₂³×C₄	D₄×C₇₃	1	2	yes	—
205	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
206	C₂×M₁₆	—	2	4	no	—
207	C₂×M₁₆	—	2	4	no	—
208	C₂×M₁₆	—	2	4	no	—
209	C₂²×C₈	C₂³×C₇₃	1	4	yes	—
210	#125	—	2	16	no	—
211	#125	—	2	16	no	—
212	#117	—	2	4	no	—
213	C₂×C₄⋊C₄	—	2	16	no	—
214	C₂×C₄⋊C₄	—	2	8	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	4	no	—
217	C₂×C₄⋊C₄	D₄×C₇₃	1	8	yes	—
218	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
219	C₂×C₄⋊C₄	C₂³×C₇₃	1	4	yes	—
220	C₂×C₂²⋊C₄	—	2	1	no	—
221	C₂×C₂²⋊C₄	—	2	4	no	—
222	C₂×C₄²	—	2	4	no	—
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	D₄×C₇₃	1	2	yes	—
225	C₂×C₄²	C₂³×C₇₃	1	2	yes	—
226	#103	—	2	8	no	—
227	#102	—	2	8	no	—
228	#95	—	2	4	no	—
229	C₂⁴	—	2	1	no	—
230	C₂⁴	C₂×D₄×C₇₃	1	1	yes	characteristic
231	#41	—	2	16	no	—
232	#41	—	4	8	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	C₂×Q₈×C₇₃	1	2	yes	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	C₂×Q₈×C₇₃	1	2	yes	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	8	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	4	4	no	—
251	C₂²×C₄	—	4	8	no	—
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
255	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
257	C₂²×C₄	C₂×Q₈×C₇₃	1	2	yes	—
258	C₂²×C₄	#41×C₇₃	1	4	yes	—
259	C₂²×C₄	#41×C₇₃	1	2	yes	—
260	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
261	M₁₆	—	4	8	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	—	2	4	no	—
265	C₄⋊C₄	—	2	32	no	—
266	C₄⋊C₄	—	2	8	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	C₂×D₄×C₇₃	1	8	yes	—
270	C₂²⋊C₄	—	4	8	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₄²	—	2	4	no	—
273	C₄²	—	2	4	no	—
274	C₄²	—	4	2	no	—
275	C₂³	—	2	4	no	—
276	C₂³	—	2	2	no	—
277	C₂³	—	4	1	no	—
278	C₂³	—	4	1	no	—
279	C₂³	—	4	4	no	—
280	C₂³	#120×C₇₃	1	2	yes	—
281	C₂³	C₂²⋊Q₈×C₇₃	1	2	yes	—
282	C₂³	#124×C₇₃	1	2	yes	—
283	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
284	Q₈	—	2	4	no	—
285	Q₈	—	4	2	no	—
286	D₄	—	2	4	no	—
287	D₄	—	4	8	no	—
288	D₄	—	4	2	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	8	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	4	8	no	—
295	C₂×C₄	—	4	16	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	8	no	—
299	C₂×C₄	—	4	8	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	#120×C₇₃	1	2	yes	—
303	C₂×C₄	C₂²⋊Q₈×C₇₃	1	2	yes	—
304	C₂×C₄	C₂²⋊Q₈×C₇₃	1	8	yes	—
305	C₂×C₄	#124×C₇₃	1	2	yes	—
306	C₂×C₄	#124×C₇₃	1	8	yes	—
307	C₂×C₄	C₂²×D₄×C₇₃	1	2	yes	—
308	C₂×C₄	C₂²×Q₈×C₇₃	1	1	yes	characteristic
309	C₂×C₄	C₂×#41×C₇₃	1	2	yes	—
310	C₂×C₄	C₂×#41×C₇₃	1	1	yes	characteristic
311	C₈	—	4	4	no	—
312	C₂²	—	2	4	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	2	4	no	—
315	C₂²	—	4	4	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	#393×C₇₃	1	4	yes	—
320	C₂²	C₂×#120×C₇₃	1	1	yes	characteristic
321	C₂²	C₂×C₂²⋊Q₈×C₇₃	1	1	yes	characteristic
322	C₂²	C₂×#124×C₇₃	1	1	yes	characteristic
323	C₂²	#575×C₇₃	1	4	yes	—
324	C₄	—	4	4	no	—
325	C₄	—	4	8	no	—
326	C₄	—	4	2	no	—
327	C₄	—	8	2	no	—
328	C₄	#393×C₇₃	1	4	yes	—
329	C₄	C₂×C₂²⋊Q₈×C₇₃	1	2	yes	—
330	C₄	C₂×#124×C₇₃	1	2	yes	—
331	C₂	—	2	4	no	—
332	C₂	—	8	2	no	—
333	C₂	C₂×#393×C₇₃	1	1	yes	characteristic
334	C₂	C₇₃×#3005	1	2	yes	—
335	C₂	C₇₃×#3080	1	2	yes	—
336	C₂	C₇₃×#3142	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₃
prime radical	C₂²×#41×C₇₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ C₂²×C₄²×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×#125×C₇₃	C₂	semidirect
4. C₇₃×#2026	C₂	semidirect
5. C₇₃×#2025	C₂	semidirect
6. C₇₃×#1736	C₂	semidirect
7. C₇₃×#1728	C₂	semidirect
162. #23453	C₇₃	direct (N × H)
169. C₂²×#125	C₂×C₇₃	semidirect
172. #2026	C₂×C₇₃	semidirect
173. #2025	C₂×C₇₃	semidirect
174. #1736	C₂×C₇₃	semidirect
175. #1728	C₂×C₇₃	semidirect
176. C₇₃	#23453	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×#125×C₇₃	2
C₂×#417×C₇₃	2
C₇₃×#2297	2
C₇₃×#2026	2
C₇₃×#2025	2
C₇₃×#1736	2
C₇₃×#1728	2
#23453	73

Subgroups — C73×#23453 #32932857

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#23453 #32932857