Subgroups — #33068623

Order 18688.

586 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620442	C₂	1	1	yes	maximal, characteristic
2	#22619969	C₂	1	2	yes	maximal
3	#446×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#22609460	C₂	1	1	yes	maximal, characteristic, complemented
5	#22609459	C₂	1	1	yes	maximal, characteristic, complemented
6	#22609446	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609436	C₂	1	2	yes	maximal, complemented
8	#22609435	C₂	1	2	yes	maximal, complemented
9	#22608676	C₂	1	1	yes	maximal, characteristic
10	#22608587	C₂	1	2	yes	maximal
11	C₇₃×#1694	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989142	C₂²	1	2	yes	—
13	C₄×Q₈×D₇₃	C₂²	1	1	yes	characteristic
14	#17988992	C₂²	1	1	yes	characteristic
15	#17988988	—	2	1	no	—
16	#17988988	—	2	1	no	—
17	#17988983	C₂²	1	2	yes	—
18	#17988921	—	2	2	no	—
19	#17988883	C₂²	1	2	yes	—
20	#17988882	C₂²	1	2	yes	—
21	#17988794	C₂²	1	2	yes	—
22	#17988741	—	2	2	no	—
23	#17988718	C₂²	1	1	yes	characteristic
24	#17988717	C₂²	1	1	yes	characteristic
25	#17988716	—	2	1	no	—
26	#17988715	—	2	1	no	—
27	#17988714	C₂²	1	1	yes	characteristic
28	Q₁₆×Dic₇₃	—	2	2	no	—
29	#17988679	—	2	2	no	—
30	#17988639	C₂²	1	2	yes	—
31	#17988624	—	2	1	no	—
32	#17988624	—	2	1	no	—
33	C₄⋊C₈×D₇₃	—	2	1	no	—
34	C₄⋊C₈×D₇₃	C₂²	1	1	yes	characteristic
35	#97×D₇₃	—	2	1	no	—
36	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
37	#17988339	C₂²	1	2	yes	—
38	#17988338	C₂²	1	1	yes	characteristic
39	#17988335	C₂²	1	2	yes	—
40	#17988335	C₂²	1	2	yes	—
41	#17988333	C₂²	1	2	yes	—
42	#500×C₇₃	C₂²	1	1	yes	characteristic
43	#462×C₇₃	C₂²	1	2	yes	—
44	#446×C₇₃	C₂²	1	1	yes	characteristic
45	#334×C₇₃	C₂²	1	1	yes	characteristic
46	#332×C₇₃	C₂²	1	2	yes	—
47	C₂×Q₈×D₇₃	D₄	1	1	yes	characteristic
48	C₂×#11924018	—	2	2	no	—
49	C₂×C₈×D₇₃	—	2	1	no	—
50	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
51	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
52	#14556591	C₂³	1	2	yes	—
53	#14556582	—	2	2	no	—
54	#14556571	—	2	2	no	—
55	#14556567	—	2	2	no	—
56	Q₈×Dic₇₃	—	2	2	no	—
57	Q₈×Dic₇₃	D₄	1	1	yes	characteristic
58	C₄×Dic₁₄₆	—	2	1	no	—
59	C₄×Dic₁₄₆	C₂³	1	1	yes	characteristic
60	#14556556	—	2	1	no	—
61	#14556556	—	2	2	no	—
62	#14556556	D₄	1	1	yes	characteristic
63	#14556549	—	2	2	no	—
64	C₄⋊C₄×D₇₃	—	2	1	no	—
65	C₄⋊C₄×D₇₃	D₄	1	1	yes	characteristic
66	C₄²×D₇₃	—	2	1	no	—
67	C₄²×D₇₃	C₂³	1	1	yes	characteristic
68	#14556525	—	2	2	no	—
69	#14556525	—	2	1	no	—
70	#14556525	—	2	1	no	—
71	#14556524	—	2	2	no	—
72	#14556522	—	2	2	no	—
73	#14556522	—	2	2	no	—
74	C₄⋊#10596831	—	2	1	no	—
75	C₄⋊#10596831	C₂³	1	2	yes	—
76	C₄⋊#10596831	C₂³	1	1	yes	characteristic
77	#14556520	—	2	1	no	—
78	#14556520	—	2	1	no	—
79	#14556520	—	2	1	no	—
80	#14556516	—	2	2	no	—
81	#14556515	—	2	2	no	—
82	#14556515	—	2	2	no	—
83	#14556503	—	2	2	no	—
84	#14556502	—	2	1	no	—
85	#14556501	—	2	1	no	—
86	#14556501	—	2	1	no	—
87	C₄×#10596831	C₂³	1	1	yes	characteristic
88	C₈×Dic₇₃	—	2	2	no	—
89	C₈×Dic₇₃	D₄	1	1	yes	characteristic
90	C₈×Dic₇₃	D₄	1	1	yes	characteristic
91	C₂×Q₁₆×C₇₃	—	2	2	no	—
92	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
93	C₄×Q₈×C₇₃	C₂³	1	1	yes	characteristic
94	#107×C₇₃	—	2	1	no	—
95	#107×C₇₃	—	2	1	no	—
96	C₄⋊C₈×C₇₃	—	2	1	no	—
97	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
98	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
99	#103×C₇₃	—	2	2	no	—
100	#97×C₇₃	—	2	1	no	—
101	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
102	C₂×Dic₁₄₆	—	2	1	no	—
103	C₂×Dic₁₄₆	—	2	2	no	—
104	C₂×Dic₁₄₆	C₂×D₄	1	1	yes	characteristic
105	Q₈×D₇₃	—	4	2	no	—
106	C₂×C₄×D₇₃	—	2	1	no	—
107	C₂×C₄×D₇₃	—	2	1	no	—
108	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
111	#11924018	—	4	4	no	—
112	C₂×#10596831	—	2	2	no	—
113	C₂×#10596831	—	2	1	no	—
114	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
115	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
116	C₈×D₇₃	—	2	2	no	—
117	C₈×D₇₃	—	4	2	no	—
118	C₈×D₇₃	Q₁₆	1	4	yes	complemented
119	C₄⋊Dic₇₃	—	2	1	no	—
120	C₄⋊Dic₇₃	—	2	2	no	—
121	C₄⋊Dic₇₃	—	2	2	no	—
122	C₄⋊Dic₇₃	—	4	2	no	—
123	C₄⋊Dic₇₃	C₂×D₄	1	1	yes	characteristic
124	#11924007	—	2	1	no	—
125	#11924007	—	2	1	no	—
126	#11924007	—	2	1	no	—
127	#11924007	—	4	2	no	—
128	#11924005	—	4	2	no	—
129	C₄×Dic₇₃	—	2	1	no	—
130	C₄×Dic₇₃	—	2	1	no	—
131	C₄×Dic₇₃	—	2	1	no	—
132	C₄×Dic₇₃	—	4	2	no	—
133	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
134	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
135	C₄×Dic₇₃	#41	1	1	yes	characteristic
136	C₂×Q₈×C₇₃	—	2	2	no	—
137	C₂×Q₈×C₇₃	C₂×D₄	1	1	yes	characteristic
138	Q₁₆×C₇₃	—	4	4	no	—
139	C₂×C₈×C₇₃	—	2	1	no	—
140	C₂×C₈×C₇₃	—	2	2	no	—
141	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
143	C₄⋊C₄×C₇₃	—	2	1	no	—
144	C₄⋊C₄×C₇₃	—	2	2	no	—
145	C₄⋊C₄×C₇₃	—	2	2	no	—
146	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
147	C₄²×C₇₃	—	2	1	no	—
148	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
149	C₂²×D₇₃	#121	1	1	yes	characteristic
150	Q₈×C₇₃	—	2	2	no	—
151	Q₈×C₇₃	—	4	2	no	—
152	Q₈×C₇₃	—	4	2	no	—
153	C₂×C₄×C₇₃	—	2	1	no	—
154	C₂×C₄×C₇₃	—	2	1	no	—
155	C₂×C₄×C₇₃	—	4	2	no	—
156	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
159	C₂×Dic₇₃	—	2	1	no	—
160	C₂×Dic₇₃	—	2	1	no	—
161	C₂×Dic₇₃	—	4	2	no	—
162	C₂×Dic₇₃	#121	1	1	yes	characteristic
163	C₂×Dic₇₃	#121	1	1	yes	characteristic
164	C₂×Dic₇₃	#121	1	1	yes	characteristic
165	C₄×D₇₃	—	2	2	no	—
166	C₄×D₇₃	—	4	1	no	—
167	C₄×D₇₃	—	4	2	no	—
168	C₄×D₇₃	—	4	2	no	—
169	C₄×D₇₃	C₂×Q₁₆	1	2	yes	—
170	C₄×D₇₃	#137	1	2	yes	—
171	Dic₁₄₆	—	2	2	no	—
172	Dic₁₄₆	—	4	2	no	—
173	Dic₁₄₆	—	8	2	no	—
174	C₈×C₇₃	—	2	1	no	—
175	C₈×C₇₃	—	4	1	no	—
176	C₈×C₇₃	—	4	2	no	—
177	C₈×C₇₃	C₂×Q₁₆	1	2	yes	—
178	#10596831	—	2	1	no	—
179	#10596831	—	4	1	no	—
180	#10596831	—	4	2	no	—
181	#10596831	C₂×Q₁₆	1	2	yes	—
182	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
183	C₂×D₇₃	—	2	1	no	—
184	C₂×D₇₃	—	2	1	no	—
185	C₂×D₇₃	#462	1	2	yes	—
186	C₄×C₇₃	—	2	1	no	—
187	C₄×C₇₃	—	4	1	no	—
188	C₄×C₇₃	—	4	1	no	—
189	C₄×C₇₃	—	8	2	no	—
190	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
191	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
192	C₄×C₇₃	#575	1	2	yes	—
193	Dic₇₃	—	2	1	no	—
194	Dic₇₃	—	2	1	no	—
195	Dic₇₃	—	4	1	no	—
196	Dic₇₃	—	4	1	no	—
197	Dic₇₃	—	8	2	no	—
198	Dic₇₃	#462	1	2	yes	—
199	#19150	—	73	1	no	maximal, Sylow
200	C₂×C₇₃	C₂×#462	1	1	yes	characteristic
201	C₂×C₇₃	#3054	1	1	yes	characteristic
202	C₂×C₇₃	#3067	1	1	yes	characteristic
203	D₇₃	—	2	2	no	—
204	#3160	—	73	1	no	—
205	#3104	—	73	2	no	—
206	C₂×#446	—	73	1	no	—
207	#1696	—	73	1	no	—
208	#1696	—	73	1	no	—
209	#1694	—	73	1	no	—
210	#1694	D₇₃	1	1	yes	characteristic, complemented
211	#1692	—	73	2	no	—
212	#1692	—	73	2	no	—
213	#1564	—	73	1	no	—
214	#1542	—	73	2	no	—
215	C₇₃	#19150	1	1	yes	characteristic, complemented, Sylow
216	#533	—	73	2	no	—
217	C₂×C₄×Q₈	—	73	1	no	—
218	#500	—	73	1	no	—
219	#500	C₂×D₇₃	1	1	yes	characteristic
220	#499	—	146	1	no	—
221	#499	—	146	1	no	—
222	#498	—	73	2	no	—
223	#484	—	146	2	no	—
224	#477	—	73	2	no	—
225	#477	—	73	2	no	—
226	#462	—	73	2	no	—
227	#462	C₂×D₇₃	1	2	yes	—
228	#451	—	146	2	no	—
229	#446	—	73	1	no	—
230	#446	—	73	1	no	—
231	#446	—	73	1	no	—
232	#446	—	146	1	no	—
233	#446	—	146	1	no	—
234	#446	C₂×D₇₃	1	1	yes	characteristic
235	C₄×Q₁₆	—	146	2	no	—
236	C₄×Q₁₆	—	146	2	no	—
237	#431	—	73	2	no	—
238	#427	—	146	1	no	—
239	#427	—	146	1	no	—
240	C₂×C₄⋊C₈	—	73	1	no	—
241	C₂×C₄⋊C₈	—	146	1	no	—
242	C₂×#97	—	146	1	no	—
243	C₂×C₄×C₈	—	73	1	no	—
244	#335	—	73	2	no	—
245	#334	—	73	1	no	—
246	#334	C₂×D₇₃	1	1	yes	characteristic
247	#333	—	73	2	no	—
248	#333	—	73	2	no	—
249	#332	—	73	2	no	—
250	#332	C₂×D₇₃	1	2	yes	—
251	C₂²×Q₈	—	73	1	no	—
252	C₂×Q₁₆	—	2	2	no	—
253	C₂×Q₁₆	—	146	2	no	—
254	C₂²×C₈	—	73	1	no	—
255	C₂²×C₈	—	73	1	no	—
256	C₂²×C₈	—	146	1	no	—
257	C₄⋊Q₈	—	73	2	no	—
258	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
259	#125	—	146	2	no	—
260	C₂²⋊Q₈	—	146	2	no	—
261	C₂²⋊Q₈	—	146	2	no	—
262	C₄×Q₈	—	73	1	no	—
263	C₄×Q₈	—	73	1	no	—
264	C₄×Q₈	—	73	1	no	—
265	C₄×Q₈	—	146	1	no	—
266	C₄×Q₈	—	146	2	no	—
267	C₄×Q₈	—	146	2	no	—
268	C₄×Q₈	—	146	1	no	—
269	C₄×Q₈	C₂²×D₇₃	1	1	yes	characteristic
270	#117	—	146	2	no	—
271	C₂×C₄⋊C₄	—	73	1	no	—
272	C₂×C₄⋊C₄	—	146	1	no	—
273	C₂×C₄²	—	73	1	no	—
274	C₂×C₄²	—	146	1	no	—
275	#107	—	2	1	no	—
276	#107	—	2	1	no	—
277	#107	—	146	2	no	—
278	#107	—	146	1	no	—
279	#107	—	146	1	no	—
280	#107	—	146	2	no	—
281	#106	—	146	2	no	—
282	#106	—	146	2	no	—
283	C₄⋊C₈	—	2	1	no	—
284	C₄⋊C₈	—	73	1	no	—
285	C₄⋊C₈	—	73	2	no	—
286	C₄⋊C₈	—	146	1	no	—
287	C₄⋊C₈	—	146	1	no	—
288	C₄⋊C₈	—	146	1	no	—
289	C₄⋊C₈	—	146	1	no	—
290	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
291	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
292	#103	—	2	2	no	—
293	#103	—	146	2	no	—
294	#103	—	146	2	no	—
295	#103	—	146	2	no	—
296	C₂²⋊C₈	—	146	2	no	—
297	#97	—	2	1	no	—
298	#97	—	146	1	no	—
299	#97	—	146	1	no	—
300	#97	—	146	1	no	—
301	C₄×C₈	—	73	1	no	—
302	C₄×C₈	—	73	1	no	—
303	C₄×C₈	—	73	1	no	—
304	C₄×C₈	—	146	2	no	—
305	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
306	C₂×Q₈	—	2	2	no	—
307	C₂×Q₈	—	292	2	no	—
308	C₂×Q₈	—	73	1	no	—
309	C₂×Q₈	—	146	1	no	—
310	C₂×Q₈	—	146	2	no	—
311	C₂×Q₈	D₄×D₇₃	1	1	yes	characteristic
312	C₂²×C₄	—	73	1	no	—
313	C₂²×C₄	—	73	1	no	—
314	C₂²×C₄	—	73	1	no	—
315	C₂²×C₄	—	146	1	no	—
316	C₂²×C₄	—	146	1	no	—
317	Q₁₆	—	4	4	no	—
318	Q₁₆	—	292	4	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	292	2	no	—
322	C₂×C₈	—	73	1	no	—
323	C₂×C₈	—	73	4	no	—
324	C₂×C₈	—	73	1	no	—
325	C₂×C₈	—	146	2	no	—
326	C₂×C₈	—	146	2	no	—
327	C₂×C₈	—	146	1	no	—
328	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
329	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	292	2	no	—
334	C₄⋊C₄	—	292	2	no	—
335	C₄⋊C₄	—	73	1	no	—
336	C₄⋊C₄	—	146	1	no	—
337	C₄⋊C₄	—	146	1	no	—
338	C₄⋊C₄	—	146	1	no	—
339	C₄⋊C₄	—	146	2	no	—
340	C₄⋊C₄	—	146	2	no	—
341	C₄⋊C₄	—	146	1	no	—
342	C₄⋊C₄	D₄×D₇₃	1	1	yes	characteristic
343	C₂²⋊C₄	—	292	2	no	—
344	C₄²	—	2	1	no	—
345	C₄²	—	292	2	no	—
346	C₄²	—	73	1	no	—
347	C₄²	—	73	1	no	—
348	C₄²	—	73	1	no	—
349	C₄²	—	146	1	no	—
350	C₄²	—	146	1	no	—
351	C₄²	—	146	1	no	—
352	C₄²	C₂³×D₇₃	1	1	yes	characteristic
353	C₂³	—	73	1	no	—
354	Q₈	—	2	2	no	—
355	Q₈	—	4	2	no	—
356	Q₈	—	4	2	no	—
357	Q₈	—	292	2	no	—
358	Q₈	—	584	2	no	—
359	Q₈	—	146	2	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	292	1	no	—
364	C₂×C₄	—	292	2	no	—
365	C₂×C₄	—	292	2	no	—
366	C₂×C₄	—	292	2	no	—
367	C₂×C₄	—	73	1	no	—
368	C₂×C₄	—	73	2	no	—
369	C₂×C₄	—	73	2	no	—
370	C₂×C₄	—	73	1	no	—
371	C₂×C₄	—	73	1	no	—
372	C₂×C₄	—	146	1	no	—
373	C₂×C₄	—	146	1	no	—
374	C₂×C₄	—	146	2	no	—
375	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
376	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
377	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
378	C₈	—	2	1	no	—
379	C₈	—	4	1	no	—
380	C₈	—	4	2	no	—
381	C₈	—	292	1	no	—
382	C₈	—	292	2	no	—
383	C₈	—	73	2	no	—
384	C₈	—	146	1	no	—
385	C₈	Q₁₆×D₇₃	1	2	yes	—
386	C₂²	—	73	2	no	—
387	C₂²	—	146	1	no	—
388	C₂²	—	146	1	no	—
389	C₂²	#121×D₇₃	1	1	yes	characteristic
390	C₄	—	2	1	no	—
391	C₄	—	4	1	no	—
392	C₄	—	4	1	no	—
393	C₄	—	8	2	no	—
394	C₄	—	292	1	no	—
395	C₄	—	292	1	no	—
396	C₄	—	584	2	no	—
397	C₄	—	73	2	no	—
398	C₄	—	146	1	no	—
399	C₄	—	146	1	no	—
400	C₄	C₂×Q₁₆×D₇₃	1	1	yes	characteristic
401	C₄	#137×D₇₃	1	1	yes	characteristic
402	C₄	#17989383	1	2	yes	—
403	C₂	—	146	2	no	—
404	C₂	#462×D₇₃	1	1	yes	characteristic
405	C₂	#22619538	1	1	yes	characteristic
406	C₂	#22619653	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₇₃
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1694	C₂
socle	C₂²×C₇₃	C₂×#121
prime radical	C₂²×D₇₃	#121
hypercenter	#1694	D₇₃
hypocenter	C₇₃	#19150
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1694

Complements

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

N	H ≅ G/N	type
4. #22609460	C₂	semidirect
5. #22609459	C₂	semidirect
6. #22609446	C₂	semidirect
7. #22609436	C₂	semidirect
8. #22609435	C₂	semidirect
11. C₇₃×#1694	C₂	semidirect
118. C₈×D₇₃	Q₁₆	semidirect
210. #1694	D₇₃	semidirect
215. C₇₃	#19150	semidirect

Maximal subgroups

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

subgroup	index
#22620442	2
#22619969	2
#446×D₇₃	2
#22609460	2
#22609459	2
#22609446	2
#22609436	2
#22609435	2
#22608676	2
#22608587	2
C₇₃×#1694	2
#19150	73