Subgroups — C₇₃×#19940 #32929344

Order 18688.

556 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₃×#3369	C₂	1	2	yes	maximal
3	C₇₃×#3342	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#1710	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1704	C₂	1	2	yes	maximal, complemented
6	C₇₃×#1675	C₂	1	1	yes	maximal, characteristic
7	C₇₃×#1674	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₃×#1645	C₂	1	2	yes	maximal, complemented
9	C₇₃×#1638	C₂	1	2	yes	maximal
10	C₇₃×#1591	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
12	Q₈²×C₇₃	C₂²	1	2	yes	—
13	#556×C₇₃	C₂²	1	2	yes	—
14	#500×C₇₃	—	2	1	no	—
15	#500×C₇₃	C₂²	1	1	yes	characteristic
16	#500×C₇₃	C₂²	1	1	yes	characteristic
17	#489×C₇₃	—	2	2	no	—
18	#487×C₇₃	C₂²	1	2	yes	—
19	#487×C₇₃	C₂²	1	2	yes	—
20	#485×C₇₃	—	2	2	no	—
21	#477×C₇₃	—	2	2	no	—
22	#477×C₇₃	C₂²	1	2	yes	—
23	#476×C₇₃	C₂²	1	2	yes	—
24	#475×C₇₃	—	2	2	no	—
25	#475×C₇₃	C₂²	1	2	yes	—
26	#467×C₇₃	—	2	1	no	—
27	#464×C₇₃	—	2	2	no	—
28	#462×C₇₃	C₂²	1	2	yes	—
29	#446×C₇₃	C₂²	1	2	yes	—
30	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
31	#441×C₇₃	—	2	2	no	—
32	C₄×Q₁₆×C₇₃	—	2	1	no	—
33	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
34	C₄×SD₁₆×C₇₃	C₂²	1	1	yes	characteristic
35	C₈×D₄×C₇₃	C₂²	1	1	yes	characteristic
36	#427×C₇₃	—	2	1	no	—
37	#335×C₇₃	C₂²	1	1	yes	characteristic
38	#334×C₇₃	C₂²	1	1	yes	characteristic
39	#333×C₇₃	C₂²	1	2	yes	—
40	#332×C₇₃	C₂²	1	2	yes	—
41	Q₈⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈×C₇₃	C₂²	1	2	yes	—
43	D₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
44	C₈²×C₇₃	C₂²	1	1	yes	characteristic
45	C₂×Q₁₆×C₇₃	—	2	2	no	—
46	C₂×Q₁₆×C₇₃	D₄	1	1	yes	characteristic
47	C₂×SD₁₆×C₇₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₇₃	—	2	1	no	—
49	C₄⋊Q₈×C₇₃	—	2	2	no	—
50	C₄⋊Q₈×C₇₃	—	2	2	no	—
51	C₄⋊Q₈×C₇₃	—	4	2	no	—
52	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
53	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
54	#126×C₇₃	—	4	2	no	—
55	#124×C₇₃	—	2	2	no	—
56	C₂²⋊Q₈×C₇₃	—	2	2	no	—
57	C₄×Q₈×C₇₃	—	2	2	no	—
58	C₄×Q₈×C₇₃	—	2	2	no	—
59	C₄×Q₈×C₇₃	—	2	2	no	—
60	C₄×Q₈×C₇₃	C₂³	1	1	yes	characteristic
61	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
62	C₄×D₄×C₇₃	C₂³	1	1	yes	characteristic
63	#117×C₇₃	—	2	2	no	—
64	#107×C₇₃	—	2	1	no	—
65	#107×C₇₃	—	2	1	no	—
66	#107×C₇₃	—	2	1	no	—
67	#107×C₇₃	—	2	1	no	—
68	#107×C₇₃	—	2	2	no	—
69	#107×C₇₃	D₄	1	1	yes	characteristic
70	#106×C₇₃	—	2	2	no	—
71	#106×C₇₃	—	2	2	no	—
72	#106×C₇₃	D₄	1	1	yes	characteristic
73	C₄⋊C₈×C₇₃	—	2	1	no	—
74	C₄⋊C₈×C₇₃	—	2	2	no	—
75	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
78	#103×C₇₃	—	2	2	no	—
79	#103×C₇₃	—	2	2	no	—
80	#103×C₇₃	—	2	2	no	—
81	#103×C₇₃	—	2	2	no	—
82	#103×C₇₃	—	2	2	no	—
83	#103×C₇₃	—	4	2	no	—
84	#103×C₇₃	D₄	1	1	yes	characteristic
85	#103×C₇₃	D₄	1	2	yes	—
86	#102×C₇₃	—	2	2	no	—
87	#102×C₇₃	D₄	1	1	yes	characteristic
88	C₂²⋊C₈×C₇₃	—	2	1	no	—
89	#97×C₇₃	—	2	2	no	—
90	C₄×C₈×C₇₃	—	2	1	no	—
91	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₇₃	—	2	2	no	—
95	C₂×Q₈×C₇₃	—	2	2	no	—
96	C₂×Q₈×C₇₃	—	4	2	no	—
97	C₂×Q₈×C₇₃	C₂×D₄	1	2	yes	—
98	C₂×Q₈×C₇₃	C₂×D₄	1	1	yes	characteristic
99	C₂×D₄×C₇₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₇₃	—	2	1	no	—
101	Q₁₆×C₇₃	—	2	2	no	—
102	Q₁₆×C₇₃	—	4	4	no	—
103	SD₁₆×C₇₃	—	2	2	no	—
104	C₂×C₈×C₇₃	—	2	1	no	—
105	C₂×C₈×C₇₃	—	2	2	no	—
106	C₂×C₈×C₇₃	—	2	1	no	—
107	C₂×C₈×C₇₃	—	2	1	no	—
108	C₂×C₈×C₇₃	—	4	2	no	—
109	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
113	C₄⋊C₄×C₇₃	—	2	2	no	—
114	C₄⋊C₄×C₇₃	—	2	1	no	—
115	C₄⋊C₄×C₇₃	—	2	2	no	—
116	C₄⋊C₄×C₇₃	—	2	2	no	—
117	C₄⋊C₄×C₇₃	—	2	2	no	—
118	C₄⋊C₄×C₇₃	—	2	2	no	—
119	C₄⋊C₄×C₇₃	—	4	2	no	—
120	C₄⋊C₄×C₇₃	—	4	2	no	—
121	C₄⋊C₄×C₇₃	—	4	2	no	—
122	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
123	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
124	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
125	C₂²⋊C₄×C₇₃	—	2	1	no	—
126	C₂²⋊C₄×C₇₃	—	4	2	no	—
127	C₂²⋊C₄×C₇₃	—	4	2	no	—
128	C₄²×C₇₃	—	2	1	no	—
129	C₄²×C₇₃	—	2	2	no	—
130	C₄²×C₇₃	—	4	2	no	—
131	C₄²×C₇₃	—	4	2	no	—
132	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
133	C₂³×C₇₃	—	2	1	no	—
134	Q₈×C₇₃	—	2	1	no	—
135	Q₈×C₇₃	—	2	2	no	—
136	Q₈×C₇₃	—	2	2	no	—
137	Q₈×C₇₃	—	4	2	no	—
138	Q₈×C₇₃	—	4	2	no	—
139	Q₈×C₇₃	—	8	2	no	—
140	Q₈×C₇₃	—	16	2	no	—
141	Q₈×C₇₃	#135	1	2	yes	—
142	D₄×C₇₃	—	2	1	no	—
143	D₄×C₇₃	—	2	1	no	—
144	C₂×C₄×C₇₃	—	2	1	no	—
145	C₂×C₄×C₇₃	—	2	2	no	—
146	C₂×C₄×C₇₃	—	2	2	no	—
147	C₂×C₄×C₇₃	—	2	1	no	—
148	C₂×C₄×C₇₃	—	2	1	no	—
149	C₂×C₄×C₇₃	—	4	1	no	—
150	C₂×C₄×C₇₃	—	4	1	no	—
151	C₂×C₄×C₇₃	—	4	2	no	—
152	C₂×C₄×C₇₃	—	4	2	no	—
153	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
156	C₈×C₇₃	—	2	1	no	—
157	C₈×C₇₃	—	2	1	no	—
158	C₈×C₇₃	—	2	1	no	—
159	C₈×C₇₃	—	4	1	no	—
160	C₈×C₇₃	—	4	2	no	—
161	C₈×C₇₃	—	4	1	no	—
162	C₈×C₇₃	—	4	1	no	—
163	C₈×C₇₃	#137	1	2	yes	—
164	C₂²×C₇₃	—	4	1	no	—
165	C₂²×C₇₃	—	4	1	no	—
166	C₂²×C₇₃	—	4	1	no	—
167	C₂²×C₇₃	D₄²	1	1	yes	characteristic
168	C₄×C₇₃	—	2	2	no	—
169	C₄×C₇₃	—	2	1	no	—
170	C₄×C₇₃	—	4	1	no	—
171	C₄×C₇₃	—	4	2	no	—
172	C₄×C₇₃	—	4	2	no	—
173	C₄×C₇₃	—	4	1	no	—
174	C₄×C₇₃	—	8	2	no	—
175	C₄×C₇₃	—	8	2	no	—
176	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
177	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
178	C₄×C₇₃	#578	1	1	yes	characteristic
179	C₄×C₇₃	#578	1	1	yes	characteristic
180	#19940	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂×C₇₃	—	8	1	no	—
182	C₂×C₇₃	#3276	1	1	yes	characteristic
183	C₂×C₇₃	#3285	1	1	yes	characteristic
184	C₂×C₇₃	#3290	1	1	yes	characteristic
185	Q₈×Q₁₆	C₂×C₇₃	1	1	yes	characteristic, complemented
186	#3369	C₂×C₇₃	1	2	yes	—
187	#3342	C₂×C₇₃	1	1	yes	characteristic
188	#1710	C₂×C₇₃	1	1	yes	characteristic, complemented
189	#1704	C₂×C₇₃	1	2	yes	complemented
190	#1675	C₂×C₇₃	1	1	yes	characteristic
191	#1674	C₂×C₇₃	1	1	yes	characteristic, complemented
192	#1645	C₂×C₇₃	1	2	yes	complemented
193	#1638	C₂×C₇₃	1	2	yes	—
194	#1591	C₂×C₇₃	1	1	yes	characteristic, complemented
195	C₈×SD₁₆	C₂×C₇₃	1	1	yes	characteristic
196	C₇₃	#19940	1	1	yes	characteristic, direct product factor, Sylow
197	Q₈²	C₂²×C₇₃	1	2	yes	—
198	#556	C₂²×C₇₃	1	2	yes	—
199	#500	—	2	1	no	—
200	#500	C₂²×C₇₃	1	1	yes	characteristic
201	#500	C₂²×C₇₃	1	1	yes	characteristic
202	#489	—	2	2	no	—
203	#487	C₂²×C₇₃	1	2	yes	—
204	#487	C₂²×C₇₃	1	2	yes	—
205	#485	—	2	2	no	—
206	#477	—	2	2	no	—
207	#477	C₂²×C₇₃	1	2	yes	—
208	#476	C₂²×C₇₃	1	2	yes	—
209	#475	—	2	2	no	—
210	#475	C₂²×C₇₃	1	2	yes	—
211	#467	—	2	1	no	—
212	#464	—	2	2	no	—
213	#462	C₂²×C₇₃	1	2	yes	—
214	#446	C₂²×C₇₃	1	2	yes	—
215	C₈×Q₈	C₂²×C₇₃	1	1	yes	characteristic
216	#441	—	2	2	no	—
217	C₄×Q₁₆	—	2	1	no	—
218	C₄×Q₁₆	C₂²×C₇₃	1	1	yes	characteristic
219	C₄×SD₁₆	C₂²×C₇₃	1	1	yes	characteristic
220	C₈×D₄	C₂²×C₇₃	1	1	yes	characteristic
221	#427	—	2	1	no	—
222	#335	C₂²×C₇₃	1	1	yes	characteristic
223	#334	C₂²×C₇₃	1	1	yes	characteristic
224	#333	C₂²×C₇₃	1	2	yes	—
225	#332	C₂²×C₇₃	1	2	yes	—
226	Q₈⋊C₈	C₂²×C₇₃	1	1	yes	characteristic
227	Q₈⋊C₈	C₂²×C₇₃	1	2	yes	—
228	D₄⋊C₈	C₂²×C₇₃	1	1	yes	characteristic
229	C₈²	C₂²×C₇₃	1	1	yes	characteristic
230	C₂×Q₁₆	—	2	2	no	—
231	C₂×Q₁₆	D₄×C₇₃	1	1	yes	characteristic
232	C₂×SD₁₆	D₄×C₇₃	1	1	yes	characteristic
233	C₂²×C₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	2	no	—
235	C₄⋊Q₈	—	2	2	no	—
236	C₄⋊Q₈	—	4	2	no	—
237	C₄⋊Q₈	C₂³×C₇₃	1	2	yes	—
238	C₄⋊Q₈	C₂³×C₇₃	1	2	yes	—
239	#126	—	4	2	no	—
240	#124	—	2	2	no	—
241	C₂²⋊Q₈	—	2	2	no	—
242	C₄×Q₈	—	2	2	no	—
243	C₄×Q₈	—	2	2	no	—
244	C₄×Q₈	—	2	2	no	—
245	C₄×Q₈	C₂³×C₇₃	1	2	yes	—
246	C₄×Q₈	C₂³×C₇₃	1	1	yes	characteristic
247	C₄×D₄	C₂³×C₇₃	1	1	yes	characteristic
248	#117	—	2	2	no	—
249	#107	—	2	1	no	—
250	#107	—	2	1	no	—
251	#107	—	2	1	no	—
252	#107	—	2	1	no	—
253	#107	—	2	2	no	—
254	#107	D₄×C₇₃	1	1	yes	characteristic
255	#106	—	2	2	no	—
256	#106	—	2	2	no	—
257	#106	D₄×C₇₃	1	1	yes	characteristic
258	C₄⋊C₈	—	2	1	no	—
259	C₄⋊C₈	—	2	2	no	—
260	C₄⋊C₈	C₂³×C₇₃	1	1	yes	characteristic
261	C₄⋊C₈	C₂³×C₇₃	1	2	yes	—
262	C₄⋊C₈	C₂³×C₇₃	1	1	yes	characteristic
263	#103	—	2	2	no	—
264	#103	—	2	2	no	—
265	#103	—	2	2	no	—
266	#103	—	2	2	no	—
267	#103	—	2	2	no	—
268	#103	—	4	2	no	—
269	#103	D₄×C₇₃	1	1	yes	characteristic
270	#103	D₄×C₇₃	1	2	yes	—
271	#102	—	2	2	no	—
272	#102	D₄×C₇₃	1	1	yes	characteristic
273	C₂²⋊C₈	—	2	1	no	—
274	#97	—	2	2	no	—
275	C₄×C₈	—	2	1	no	—
276	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
277	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
278	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
279	C₂×Q₈	—	2	2	no	—
280	C₂×Q₈	—	2	2	no	—
281	C₂×Q₈	—	4	2	no	—
282	C₂×Q₈	C₂×D₄×C₇₃	1	2	yes	—
283	C₂×Q₈	C₂×D₄×C₇₃	1	1	yes	characteristic
284	C₂×D₄	C₂×D₄×C₇₃	1	1	yes	characteristic
285	C₂²×C₄	—	2	1	no	—
286	Q₁₆	—	2	2	no	—
287	Q₁₆	—	4	4	no	—
288	SD₁₆	—	2	2	no	—
289	C₂×C₈	—	2	1	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	1	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	4	2	no	—
294	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
295	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
296	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
297	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	4	2	no	—
305	C₄⋊C₄	—	4	2	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
308	C₄⋊C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
309	C₄⋊C₄	C₂×D₄×C₇₃	1	2	yes	—
310	C₂²⋊C₄	—	2	1	no	—
311	C₂²⋊C₄	—	4	2	no	—
312	C₂²⋊C₄	—	4	2	no	—
313	C₄²	—	2	1	no	—
314	C₄²	—	2	2	no	—
315	C₄²	—	4	2	no	—
316	C₄²	—	4	2	no	—
317	C₄²	C₂⁴×C₇₃	1	1	yes	characteristic
318	C₂³	—	2	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	2	2	no	—
321	Q₈	—	2	2	no	—
322	Q₈	—	4	2	no	—
323	Q₈	—	4	2	no	—
324	Q₈	—	8	2	no	—
325	Q₈	—	16	2	no	—
326	Q₈	#135×C₇₃	1	2	yes	—
327	D₄	—	2	1	no	—
328	D₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	1	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	C₂²×D₄×C₇₃	1	1	yes	characteristic
339	C₂×C₄	C₂²×D₄×C₇₃	1	1	yes	characteristic
340	C₂×C₄	H(5,𝔽₂)×C₇₃	1	1	yes	characteristic
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	4	1	no	—
345	C₈	—	4	2	no	—
346	C₈	—	4	1	no	—
347	C₈	—	4	1	no	—
348	C₈	#137×C₇₃	1	2	yes	—
349	C₂²	—	4	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	D₄²×C₇₃	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	2	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	2	no	—
359	C₄	—	8	2	no	—
360	C₄	—	8	2	no	—
361	C₄	C₂×#135×C₇₃	1	1	yes	characteristic
362	C₄	C₂×#137×C₇₃	1	1	yes	characteristic
363	C₄	#578×C₇₃	1	1	yes	characteristic
364	C₄	#578×C₇₃	1	1	yes	characteristic
365	C₂	—	8	1	no	—
366	C₂	C₇₃×#3276	1	1	yes	characteristic
367	C₂	C₇₃×#3285	1	1	yes	characteristic
368	C₂	C₇₃×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇₃
prime radical	C₂×D₄×C₇₃	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₄²×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×Q₁₆×C₇₃	C₂	semidirect
4. C₇₃×#1710	C₂	semidirect
5. C₇₃×#1704	C₂	semidirect
7. C₇₃×#1674	C₂	semidirect
8. C₇₃×#1645	C₂	semidirect
10. C₇₃×#1591	C₂	semidirect
180. #19940	C₇₃	direct (N × H)
185. Q₈×Q₁₆	C₂×C₇₃	semidirect
188. #1710	C₂×C₇₃	semidirect
189. #1704	C₂×C₇₃	semidirect
191. #1674	C₂×C₇₃	semidirect
192. #1645	C₂×C₇₃	semidirect
194. #1591	C₂×C₇₃	semidirect
196. C₇₃	#19940	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
Q₈×Q₁₆×C₇₃	2
C₇₃×#3369	2
C₇₃×#3342	2
C₇₃×#1710	2
C₇₃×#1704	2
C₇₃×#1675	2
C₇₃×#1674	2
C₇₃×#1645	2
C₇₃×#1638	2
C₇₃×#1591	2
C₈×SD₁₆×C₇₃	2
#19940	73

Subgroups — C73×#19940 #32929344

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#19940 #32929344