Subgroups — #33069830

Order 18688.

602 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620515	C₂	1	1	yes	maximal, characteristic
2	#22619731	C₂	1	2	yes	maximal
3	C₈×Q₈×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#22609388	C₂	1	2	yes	maximal, complemented
5	#22609387	C₂	1	2	yes	maximal, complemented
6	#22609336	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609335	C₂	1	1	yes	maximal, characteristic, complemented
8	#22609298	C₂	1	1	yes	maximal, characteristic, complemented
9	#22608688	C₂	1	1	yes	maximal, characteristic
10	#22608620	C₂	1	2	yes	maximal
11	C₇₃×#1670	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	#17988937	C₂²	1	2	yes	—
16	#17988897	—	2	2	no	—
17	#17988896	—	2	2	no	—
18	#17988883	C₂²	1	2	yes	—
19	#17988882	C₂²	1	2	yes	—
20	#17988869	C₂²	1	2	yes	—
21	#17988825	—	2	1	no	—
22	#17988825	—	2	1	no	—
23	Q₈×#10596831	C₂²	1	1	yes	characteristic
24	C₈×Dic₁₄₆	—	2	1	no	—
25	C₈×Dic₁₄₆	C₂²	1	1	yes	characteristic
26	#17988711	—	2	1	no	—
27	#17988711	C₂²	1	1	yes	characteristic
28	Q₁₆×Dic₇₃	—	2	1	no	—
29	Q₁₆×Dic₇₃	—	2	1	no	—
30	#17988626	—	2	2	no	—
31	#17988609	C₂²	1	2	yes	—
32	C₄⋊C₈×D₇₃	—	2	1	no	—
33	C₄⋊C₈×D₇₃	C₂²	1	1	yes	characteristic
34	#17988594	—	2	2	no	—
35	C₄×C₈×D₇₃	—	2	1	no	—
36	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
37	#17988340	C₂²	1	1	yes	characteristic
38	#17988339	C₂²	1	2	yes	—
39	#17988335	C₂²	1	2	yes	—
40	#17988335	C₂²	1	2	yes	—
41	#17988333	C₂²	1	2	yes	—
42	#494×C₇₃	C₂²	1	1	yes	characteristic
43	#475×C₇₃	C₂²	1	2	yes	—
44	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
45	#335×C₇₃	C₂²	1	1	yes	characteristic
46	#332×C₇₃	C₂²	1	2	yes	—
47	C₂×Q₈×D₇₃	D₄	1	1	yes	characteristic
48	C₂×#11924011	—	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	C₄⋊#10596831	—	2	1	no	—
74	C₄⋊#10596831	C₂³	1	2	yes	—
75	C₄⋊#10596831	C₂³	1	1	yes	characteristic
76	#14556520	—	2	1	no	—
77	#14556520	—	2	2	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	#14556515	—	2	2	no	—
84	#14556514	—	2	2	no	—
85	C₄×#10596831	—	2	1	no	—
86	C₄×#10596831	C₂³	1	1	yes	characteristic
87	C₈×Dic₇₃	—	2	1	no	—
88	C₈×Dic₇₃	—	2	1	no	—
89	C₈×Dic₇₃	D₄	1	1	yes	characteristic
90	C₈×Dic₇₃	D₄	1	1	yes	characteristic
91	C₂×Q₁₆×C₇₃	—	2	1	no	—
92	C₂×Q₁₆×C₇₃	—	2	1	no	—
93	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
94	C₄×Q₈×C₇₃	C₂³	1	1	yes	characteristic
95	#106×C₇₃	—	2	2	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	C₄×C₈×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	—	2	2	no	—
106	Q₈×D₇₃	Q₁₆	1	4	yes	complemented
107	C₂×C₄×D₇₃	—	2	1	no	—
108	C₂×C₄×D₇₃	—	2	1	no	—
109	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
112	#11924011	—	4	4	no	—
113	C₂×#10596831	—	2	1	no	—
114	C₂×#10596831	—	2	2	no	—
115	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
116	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
117	C₈×D₇₃	—	2	2	no	—
118	C₈×D₇₃	—	4	1	no	—
119	C₈×D₇₃	—	4	2	no	—
120	C₄⋊Dic₇₃	—	2	1	no	—
121	C₄⋊Dic₇₃	—	2	2	no	—
122	C₄⋊Dic₇₃	—	2	2	no	—
123	C₄⋊Dic₇₃	—	4	2	no	—
124	C₄⋊Dic₇₃	C₂×D₄	1	1	yes	characteristic
125	#11924007	—	2	1	no	—
126	#11924007	—	2	1	no	—
127	#11924007	—	2	1	no	—
128	#11924007	—	4	2	no	—
129	#11924005	—	4	2	no	—
130	C₄×Dic₇₃	—	2	1	no	—
131	C₄×Dic₇₃	—	2	1	no	—
132	C₄×Dic₇₃	—	2	1	no	—
133	C₄×Dic₇₃	—	4	2	no	—
134	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
135	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
136	C₄×Dic₇₃	#41	1	1	yes	characteristic
137	C₂×Q₈×C₇₃	—	2	2	no	—
138	C₂×Q₈×C₇₃	C₂×D₄	1	1	yes	characteristic
139	Q₁₆×C₇₃	—	4	2	no	—
140	Q₁₆×C₇₃	—	4	2	no	—
141	C₂×C₈×C₇₃	—	2	1	no	—
142	C₂×C₈×C₇₃	—	2	2	no	—
143	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
145	C₄⋊C₄×C₇₃	—	2	1	no	—
146	C₄⋊C₄×C₇₃	—	2	2	no	—
147	C₄⋊C₄×C₇₃	—	2	2	no	—
148	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₇₃	—	2	1	no	—
150	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
151	C₂²×D₇₃	#121	1	1	yes	characteristic
152	Q₈×C₇₃	—	2	1	no	—
153	Q₈×C₇₃	—	4	2	no	—
154	Q₈×C₇₃	—	4	2	no	—
155	Q₈×C₇₃	C₂×Q₁₆	1	2	yes	—
156	C₂×C₄×C₇₃	—	2	1	no	—
157	C₂×C₄×C₇₃	—	2	1	no	—
158	C₂×C₄×C₇₃	—	4	2	no	—
159	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
162	C₂×Dic₇₃	—	2	1	no	—
163	C₂×Dic₇₃	—	2	1	no	—
164	C₂×Dic₇₃	—	4	2	no	—
165	C₂×Dic₇₃	#121	1	1	yes	characteristic
166	C₂×Dic₇₃	#121	1	1	yes	characteristic
167	C₂×Dic₇₃	#121	1	1	yes	characteristic
168	C₄×D₇₃	—	2	4	no	—
169	C₄×D₇₃	—	2	1	no	—
170	C₄×D₇₃	—	2	1	no	—
171	C₄×D₇₃	—	4	1	no	—
172	C₄×D₇₃	—	4	2	no	—
173	C₄×D₇₃	C₂×Q₁₆	1	2	yes	—
174	C₄×D₇₃	#135	1	2	yes	—
175	Dic₁₄₆	—	2	1	no	—
176	Dic₁₄₆	—	2	4	no	—
177	Dic₁₄₆	—	8	2	no	—
178	Dic₁₄₆	C₂×Q₁₆	1	2	yes	—
179	C₈×C₇₃	—	2	1	no	—
180	C₈×C₇₃	—	4	2	no	—
181	C₈×C₇₃	—	4	1	no	—
182	C₈×C₇₃	#137	1	2	yes	—
183	#10596831	—	2	1	no	—
184	#10596831	—	2	1	no	—
185	#10596831	—	4	2	no	—
186	#10596831	—	4	1	no	—
187	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
188	C₂×D₇₃	—	2	1	no	—
189	C₂×D₇₃	—	2	1	no	—
190	C₂×D₇₃	#467	1	2	yes	—
191	C₄×C₇₃	—	2	2	no	—
192	C₄×C₇₃	—	2	1	no	—
193	C₄×C₇₃	—	4	1	no	—
194	C₄×C₇₃	—	8	2	no	—
195	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
196	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
197	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
198	C₄×C₇₃	#575	1	1	yes	characteristic
199	Dic₇₃	—	2	1	no	—
200	Dic₇₃	—	2	2	no	—
201	Dic₇₃	—	2	1	no	—
202	Dic₇₃	—	4	1	no	—
203	Dic₇₃	—	8	2	no	—
204	Dic₇₃	#467	1	2	yes	—
205	#19262	—	73	1	no	maximal, Sylow
206	C₂×C₇₃	#3037	1	1	yes	characteristic
207	C₂×C₇₃	C₂×#467	1	1	yes	characteristic
208	C₂×C₇₃	#3067	1	1	yes	characteristic
209	D₇₃	—	2	2	no	—
210	#3170	—	73	1	no	—
211	#3078	—	73	2	no	—
212	C₂×C₈×Q₈	—	73	1	no	—
213	#1684	—	73	2	no	—
214	#1684	—	73	2	no	—
215	#1676	—	73	1	no	—
216	#1676	—	73	1	no	—
217	#1670	—	73	1	no	—
218	#1670	D₇₃	1	1	yes	characteristic, complemented
219	#1567	—	73	1	no	—
220	#1550	—	73	2	no	—
221	C₇₃	#19262	1	1	yes	characteristic, complemented, Sylow
222	#533	—	73	2	no	—
223	C₂×C₄×Q₈	—	73	1	no	—
224	#500	—	73	1	no	—
225	#494	C₂×D₇₃	1	1	yes	characteristic
226	#487	—	73	2	no	—
227	#479	—	146	2	no	—
228	#479	—	146	2	no	—
229	#477	—	73	2	no	—
230	#477	—	73	2	no	—
231	#475	—	73	2	no	—
232	#475	C₂×D₇₃	1	2	yes	—
233	#467	—	146	1	no	—
234	#467	—	146	1	no	—
235	C₈×Q₈	—	73	1	no	—
236	C₈×Q₈	—	73	1	no	—
237	C₈×Q₈	—	73	1	no	—
238	C₈×Q₈	—	146	1	no	—
239	C₈×Q₈	—	146	1	no	—
240	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
241	C₄×Q₁₆	—	146	1	no	—
242	C₄×Q₁₆	—	146	1	no	—
243	#428	—	146	2	no	—
244	#423	—	73	2	no	—
245	C₂×C₄⋊C₈	—	73	1	no	—
246	C₂×C₄⋊C₈	—	146	1	no	—
247	#419	—	146	2	no	—
248	C₂×C₄×C₈	—	73	1	no	—
249	C₂×C₄×C₈	—	146	1	no	—
250	#335	—	73	2	no	—
251	#335	—	73	1	no	—
252	#335	C₂×D₇₃	1	1	yes	characteristic
253	#333	—	73	2	no	—
254	#333	—	73	2	no	—
255	#332	—	73	2	no	—
256	#332	C₂×D₇₃	1	2	yes	—
257	C₂²×Q₈	—	73	1	no	—
258	C₂×Q₁₆	—	2	1	no	—
259	C₂×Q₁₆	—	2	1	no	—
260	C₂×M₁₆	—	146	2	no	—
261	C₂²×C₈	—	73	1	no	—
262	C₂²×C₈	—	73	1	no	—
263	C₂²×C₈	—	146	1	no	—
264	C₄⋊Q₈	—	73	2	no	—
265	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
266	#125	—	146	2	no	—
267	C₂²⋊Q₈	—	146	2	no	—
268	C₂²⋊Q₈	—	146	2	no	—
269	C₄×Q₈	—	73	1	no	—
270	C₄×Q₈	—	73	1	no	—
271	C₄×Q₈	—	73	1	no	—
272	C₄×Q₈	—	146	1	no	—
273	C₄×Q₈	—	146	2	no	—
274	C₄×Q₈	—	146	2	no	—
275	C₄×Q₈	—	146	1	no	—
276	C₄×Q₈	C₂²×D₇₃	1	1	yes	characteristic
277	#117	—	146	2	no	—
278	C₂×C₄⋊C₄	—	73	1	no	—
279	C₂×C₄⋊C₄	—	146	1	no	—
280	C₂×C₄²	—	73	1	no	—
281	C₂×C₄²	—	146	1	no	—
282	#107	—	146	2	no	—
283	#107	—	146	1	no	—
284	#107	—	146	1	no	—
285	#107	—	146	2	no	—
286	#106	—	2	2	no	—
287	#106	—	146	2	no	—
288	C₄⋊C₈	—	2	1	no	—
289	C₄⋊C₈	—	73	1	no	—
290	C₄⋊C₈	—	73	2	no	—
291	C₄⋊C₈	—	146	1	no	—
292	C₄⋊C₈	—	146	1	no	—
293	C₄⋊C₈	—	146	2	no	—
294	C₄⋊C₈	—	146	1	no	—
295	C₄⋊C₈	—	146	1	no	—
296	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
297	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
298	#103	—	2	2	no	—
299	#103	—	146	2	no	—
300	#103	—	146	2	no	—
301	#103	—	146	2	no	—
302	#103	—	146	2	no	—
303	#103	—	146	2	no	—
304	C₄×C₈	—	2	1	no	—
305	C₄×C₈	—	73	1	no	—
306	C₄×C₈	—	73	1	no	—
307	C₄×C₈	—	73	1	no	—
308	C₄×C₈	—	146	1	no	—
309	C₄×C₈	—	146	1	no	—
310	C₄×C₈	—	146	1	no	—
311	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
312	C₂×Q₈	—	2	2	no	—
313	C₂×Q₈	—	73	4	no	—
314	C₂×Q₈	—	73	1	no	—
315	C₂×Q₈	—	146	1	no	—
316	C₂×Q₈	—	146	2	no	—
317	C₂×Q₈	—	146	2	no	—
318	C₂×Q₈	D₄×D₇₃	1	1	yes	characteristic
319	C₂²×C₄	—	73	1	no	—
320	C₂²×C₄	—	73	1	no	—
321	C₂²×C₄	—	73	1	no	—
322	C₂²×C₄	—	146	1	no	—
323	C₂²×C₄	—	146	1	no	—
324	Q₁₆	—	4	2	no	—
325	Q₁₆	—	4	2	no	—
326	M₁₆	—	292	4	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	292	1	no	—
330	C₂×C₈	—	292	2	no	—
331	C₂×C₈	—	73	1	no	—
332	C₂×C₈	—	73	1	no	—
333	C₂×C₈	—	146	1	no	—
334	C₂×C₈	—	146	2	no	—
335	C₂×C₈	—	146	2	no	—
336	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
337	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	292	2	no	—
342	C₄⋊C₄	—	292	2	no	—
343	C₄⋊C₄	—	73	1	no	—
344	C₄⋊C₄	—	146	1	no	—
345	C₄⋊C₄	—	146	1	no	—
346	C₄⋊C₄	—	146	1	no	—
347	C₄⋊C₄	—	146	2	no	—
348	C₄⋊C₄	—	146	2	no	—
349	C₄⋊C₄	—	146	1	no	—
350	C₄⋊C₄	D₄×D₇₃	1	1	yes	characteristic
351	C₂²⋊C₄	—	292	2	no	—
352	C₄²	—	2	1	no	—
353	C₄²	—	292	2	no	—
354	C₄²	—	73	1	no	—
355	C₄²	—	73	1	no	—
356	C₄²	—	73	1	no	—
357	C₄²	—	146	1	no	—
358	C₄²	—	146	1	no	—
359	C₄²	—	146	1	no	—
360	C₄²	C₂³×D₇₃	1	1	yes	characteristic
361	C₂³	—	73	1	no	—
362	Q₈	—	2	1	no	—
363	Q₈	—	4	2	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	584	2	no	—
366	Q₈	—	73	2	no	—
367	Q₈	—	146	4	no	—
368	Q₈	—	146	1	no	—
369	Q₈	Q₁₆×D₇₃	1	2	yes	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	292	1	no	—
374	C₂×C₄	—	292	2	no	—
375	C₂×C₄	—	292	2	no	—
376	C₂×C₄	—	73	1	no	—
377	C₂×C₄	—	73	2	no	—
378	C₂×C₄	—	73	2	no	—
379	C₂×C₄	—	73	1	no	—
380	C₂×C₄	—	73	1	no	—
381	C₂×C₄	—	146	4	no	—
382	C₂×C₄	—	146	1	no	—
383	C₂×C₄	—	146	1	no	—
384	C₂×C₄	—	146	1	no	—
385	C₂×C₄	—	146	1	no	—
386	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
387	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
388	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
389	C₈	—	2	1	no	—
390	C₈	—	4	1	no	—
391	C₈	—	4	2	no	—
392	C₈	—	292	1	no	—
393	C₈	—	292	2	no	—
394	C₈	—	146	1	no	—
395	C₈	—	146	1	no	—
396	C₈	#14556620	1	2	yes	—
397	C₂²	—	73	2	no	—
398	C₂²	—	146	1	no	—
399	C₂²	—	146	1	no	—
400	C₂²	#121×D₇₃	1	1	yes	characteristic
401	C₄	—	2	1	no	—
402	C₄	—	2	2	no	—
403	C₄	—	4	1	no	—
404	C₄	—	8	2	no	—
405	C₄	—	292	1	no	—
406	C₄	—	584	2	no	—
407	C₄	—	73	2	no	—
408	C₄	—	146	1	no	—
409	C₄	—	146	1	no	—
410	C₄	—	146	2	no	—
411	C₄	C₂×Q₁₆×D₇₃	1	1	yes	characteristic
412	C₄	#135×D₇₃	1	1	yes	characteristic
413	C₄	C₂×#14556620	1	1	yes	characteristic
414	C₄	#17989383	1	1	yes	characteristic
415	C₂	—	146	2	no	—
416	C₂	#22619398	1	1	yes	characteristic
417	C₂	#467×D₇₃	1	1	yes	characteristic
418	C₂	#22619653	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#121×D₇₃
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1670	C₂
socle	C₂²×C₇₃	C₂×#121
prime radical	C₂²×D₇₃	#121
hypercenter	#1670	D₇₃
hypocenter	C₇₃	#19262
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1670

Complements

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

N	H ≅ G/N	type
4. #22609388	C₂	semidirect
5. #22609387	C₂	semidirect
6. #22609336	C₂	semidirect
7. #22609335	C₂	semidirect
8. #22609298	C₂	semidirect
11. C₇₃×#1670	C₂	semidirect
106. Q₈×D₇₃	Q₁₆	semidirect
218. #1670	D₇₃	semidirect
221. C₇₃	#19262	semidirect

Maximal subgroups

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

subgroup	index
#22620515	2
#22619731	2
C₈×Q₈×D₇₃	2
#22609388	2
#22609387	2
#22609336	2
#22609335	2
#22609298	2
#22608688	2
#22608620	2
C₇₃×#1670	2
#19262	73