Subgroups — #33094332

Order 18688.

618 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22613239	C₂	1	1	yes	maximal, characteristic
2	#22610655	C₂	1	1	yes	maximal, characteristic
3	#22610652	C₂	1	1	yes	maximal, characteristic
4	#22610498	C₂	1	2	yes	maximal
5	#22610494	C₂	1	2	yes	maximal
6	#22610050	C₂	1	1	yes	maximal, characteristic
7	#22609807	C₂	1	2	yes	maximal
8	#22609803	C₂	1	2	yes	maximal
9	#22609798	C₂	1	1	yes	maximal, characteristic
10	#22609617	C₂	1	1	yes	maximal, characteristic
11	C₇₃×#1770	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
13	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
14	C₂×#14556520	C₂²	1	2	yes	—
15	C₂×#14556502	C₂²	1	1	yes	characteristic
16	C₂×#14556501	C₂²	1	1	yes	characteristic
17	C₂×#14556501	C₂²	1	1	yes	characteristic
18	C₂×#14556501	C₂²	1	2	yes	—
19	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
21	#17988529	C₄	1	2	yes	—
22	#17988526	C₄	1	2	yes	—
23	#17988520	C₄	1	2	yes	—
24	#17988519	C₄	1	2	yes	—
25	C₄×#11924007	C₂²	1	2	yes	—
26	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
27	C₄×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
28	#17988433	C₂²	1	1	yes	characteristic
29	#17988342	C₂²	1	2	yes	—
30	#17988341	C₂²	1	2	yes	—
31	#17988341	C₂²	1	2	yes	—
32	#17988341	C₂²	1	2	yes	—
33	#17988341	C₂²	1	2	yes	—
34	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
35	C₂×#97×C₇₃	C₂²	1	2	yes	—
36	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
37	C₄×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
38	#336×C₇₃	C₂²	1	2	yes	—
39	C₂²×#10596831	C₂³	1	2	yes	—
40	C₂²×#10596831	C₂³	1	1	yes	characteristic
41	C₂²×#10596831	C₂³	1	1	yes	characteristic
42	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
45	C₂×#11924007	C₂×C₄	1	2	yes	—
46	C₂×#11924007	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
48	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
50	C₄⋊#10596831	—	2	4	no	—
51	C₄⋊#10596831	—	2	2	no	—
52	#14556520	—	2	4	no	—
53	#14556502	—	2	2	no	—
54	#14556501	—	2	2	no	—
55	#14556501	—	2	2	no	—
56	#14556501	—	2	4	no	—
57	C₈×Dic₇₃	—	2	2	no	—
58	C₈×Dic₇₃	—	2	2	no	—
59	#14556498	C₂×C₄	1	2	yes	—
60	#14556498	C₂×C₄	1	2	yes	—
61	#14556498	C₂×C₄	1	1	yes	characteristic
62	#14556498	C₂×C₄	1	2	yes	—
63	#14556498	C₂×C₄	1	1	yes	characteristic
64	#14556498	C₂×C₄	1	2	yes	—
65	#14556498	C₂×C₄	1	2	yes	—
66	#14556498	C₂×C₄	1	2	yes	—
67	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
68	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
71	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
72	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₈×C₇₃	—	2	2	no	—
74	#97×C₇₃	—	2	4	no	—
75	C₄×C₈×C₇₃	Q₈	1	4	yes	—
76	#95×C₇₃	C₂×C₄	1	1	yes	characteristic
77	#95×C₇₃	C₂×C₄	1	1	yes	characteristic
78	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
79	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
80	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
81	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
82	C₂×#10596831	—	2	1	no	—
83	C₂×#10596831	—	2	1	no	—
84	C₂×#10596831	—	2	1	no	—
85	C₂×#10596831	—	2	1	no	—
86	C₂×#10596831	—	2	2	no	—
87	C₂×#10596831	—	2	2	no	—
88	C₂×#10596831	—	2	2	no	—
89	C₂×#10596831	#41	1	2	yes	—
90	C₂×#10596831	#41	1	2	yes	—
91	C₄⋊Dic₇₃	—	4	2	no	—
92	C₄⋊Dic₇₃	—	4	2	no	—
93	C₄⋊Dic₇₃	—	4	2	no	—
94	#11924007	—	4	2	no	—
95	#11924007	—	4	2	no	—
96	C₄×Dic₇₃	—	2	1	no	—
97	C₄×Dic₇₃	—	2	1	no	—
98	C₄×Dic₇₃	—	2	1	no	—
99	C₄×Dic₇₃	—	2	1	no	—
100	C₄×Dic₇₃	—	2	2	no	—
101	C₄×Dic₇₃	—	2	2	no	—
102	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
107	C₂×C₈×C₇₃	—	2	1	no	—
108	C₂×C₈×C₇₃	—	2	1	no	—
109	C₂×C₈×C₇₃	—	2	2	no	—
110	C₂×C₈×C₇₃	—	2	2	no	—
111	C₂×C₈×C₇₃	—	2	2	no	—
112	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
113	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
114	C₂×C₈×C₇₃	#41	1	2	yes	—
115	C₂×C₈×C₇₃	#41	1	2	yes	—
116	C₄⋊C₄×C₇₃	—	4	2	no	—
117	C₄²×C₇₃	—	2	2	no	—
118	C₄²×C₇₃	—	2	2	no	—
119	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
120	C₄²×C₇₃	#41	1	2	yes	—
121	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₇₃	—	2	2	no	—
123	C₂×C₄×C₇₃	—	2	2	no	—
124	C₂×C₄×C₇₃	—	2	2	no	—
125	C₂×C₄×C₇₃	—	2	2	no	—
126	C₂×C₄×C₇₃	—	2	2	no	—
127	C₂×C₄×C₇₃	—	2	2	no	—
128	C₂×C₄×C₇₃	—	2	2	no	—
129	C₂×C₄×C₇₃	—	2	2	no	—
130	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
132	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
135	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
136	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₇₃	—	2	2	no	—
139	C₂×Dic₇₃	—	2	2	no	—
140	C₂×Dic₇₃	—	2	2	no	—
141	C₂×Dic₇₃	—	2	2	no	—
142	C₂×Dic₇₃	—	2	2	no	—
143	C₂×Dic₇₃	—	2	2	no	—
144	C₂×Dic₇₃	—	2	2	no	—
145	C₂×Dic₇₃	—	2	2	no	—
146	C₂×Dic₇₃	—	2	2	no	—
147	C₂×Dic₇₃	—	2	2	no	—
148	C₂×Dic₇₃	—	2	2	no	—
149	C₂×Dic₇₃	—	2	2	no	—
150	C₈×C₇₃	—	2	2	no	—
151	C₈×C₇₃	—	4	2	no	—
152	C₈×C₇₃	#125	1	4	yes	—
153	#10596831	—	2	2	no	—
154	#10596831	—	2	2	no	—
155	#10596831	—	4	2	no	—
156	C₂²×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
157	C₂²×C₇₃	C₄×#41	1	1	yes	characteristic
158	C₂²×C₇₃	C₄×#41	1	1	yes	characteristic
159	C₂²×C₇₃	#568	1	2	yes	—
160	C₂²×C₇₃	#568	1	2	yes	—
161	C₄×C₇₃	—	2	2	no	—
162	C₄×C₇₃	—	2	2	no	—
163	C₄×C₇₃	—	4	2	no	—
164	C₄×C₇₃	—	4	2	no	—
165	C₄×C₇₃	C₂×#125	1	1	yes	characteristic
166	C₄×C₇₃	#529	1	1	yes	characteristic
167	C₄×C₇₃	#529	1	1	yes	characteristic
168	C₄×C₇₃	#533	1	1	yes	characteristic
169	Dic₇₃	—	4	2	no	—
170	Dic₇₃	—	4	2	no	—
171	Dic₇₃	—	4	2	no	—
172	Dic₇₃	—	4	2	no	—
173	#21677	—	73	1	no	maximal, Sylow
174	C₂×C₇₃	C₄×#125	1	1	yes	characteristic
175	C₂×C₇₃	#2961	1	1	yes	characteristic
176	C₂×C₇₃	#2961	1	1	yes	characteristic
177	C₂×C₇₃	#2967	1	2	yes	—
178	C₂×C₇₃	#2967	1	2	yes	—
179	#2402	—	73	1	no	—
180	#1940	—	73	1	no	—
181	#1940	—	73	1	no	—
182	#1917	—	73	2	no	—
183	#1917	—	73	2	no	—
184	#1832	—	73	1	no	—
185	#1771	—	73	2	no	—
186	#1771	—	73	2	no	—
187	#1770	—	73	1	no	—
188	#1770	D₇₃	1	1	yes	characteristic
189	#1726	—	73	1	no	—
190	C₇₃	#21677	1	1	yes	characteristic, complemented, Sylow
191	C₂×C₄⋊C₈	—	73	2	no	—
192	C₂×C₄⋊C₈	—	73	1	no	—
193	C₂×C₄⋊C₈	—	73	2	no	—
194	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
195	C₂×#97	—	73	1	no	—
196	C₂×#97	—	73	1	no	—
197	C₂×#97	—	73	2	no	—
198	C₂×#97	—	73	1	no	—
199	C₂×#97	C₂×D₇₃	1	2	yes	—
200	C₂×C₄×C₈	—	73	1	no	—
201	C₂×C₄×C₈	—	73	1	no	—
202	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
203	#400	—	73	2	no	—
204	#400	—	73	2	no	—
205	#398	—	73	2	no	—
206	#398	—	73	2	no	—
207	C₄×C₄⋊C₄	—	73	2	no	—
208	C₄×C₄⋊C₄	—	73	1	no	—
209	C₄×C₄⋊C₄	—	73	2	no	—
210	C₄×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
211	#376	—	73	1	no	—
212	#336	—	73	2	no	—
213	#336	—	73	2	no	—
214	#336	—	73	2	no	—
215	#336	—	73	2	no	—
216	#336	—	73	2	no	—
217	#336	C₂×D₇₃	1	2	yes	—
218	C₂²×C₈	—	73	1	no	—
219	C₂²×C₈	—	73	1	no	—
220	C₂²×C₈	—	73	2	no	—
221	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
222	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
223	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
224	C₂×C₄⋊C₄	—	73	2	no	—
225	C₂×C₄⋊C₄	—	73	2	no	—
226	C₂×C₄⋊C₄	—	73	2	no	—
227	C₂×C₄⋊C₄	—	73	2	no	—
228	C₂×C₄⋊C₄	—	73	2	no	—
229	C₂×C₄⋊C₄	C₄×D₇₃	1	2	yes	—
230	C₂×C₄²	—	73	1	no	—
231	C₂×C₄²	—	73	1	no	—
232	C₂×C₄²	—	73	2	no	—
233	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
234	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
235	C₄⋊C₈	—	2	2	no	—
236	C₄⋊C₈	—	146	4	no	—
237	C₄⋊C₈	—	146	4	no	—
238	C₄⋊C₈	—	146	2	no	—
239	#97	—	2	4	no	—
240	#97	—	146	2	no	—
241	#97	—	146	2	no	—
242	#97	—	146	4	no	—
243	#97	—	146	2	no	—
244	C₄×C₈	—	146	2	no	—
245	C₄×C₈	—	146	2	no	—
246	C₄×C₈	Dic₁₄₆	1	4	yes	—
247	#95	—	73	2	no	—
248	#95	—	73	2	no	—
249	#95	—	73	1	no	—
250	#95	—	73	2	no	—
251	#95	—	73	2	no	—
252	#95	—	73	2	no	—
253	#95	—	73	2	no	—
254	#95	—	73	1	no	—
255	#95	C₄×D₇₃	1	1	yes	characteristic
256	#95	C₄×D₇₃	1	1	yes	characteristic
257	C₂²×C₄	—	73	2	no	—
258	C₂²×C₄	—	73	2	no	—
259	C₂²×C₄	—	73	2	no	—
260	C₂²×C₄	—	73	2	no	—
261	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
262	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
263	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
265	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
266	C₂×C₈	—	2	1	no	—
267	C₂×C₈	—	2	1	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	73	2	no	—
272	C₂×C₈	—	73	2	no	—
273	C₂×C₈	—	146	2	no	—
274	C₂×C₈	—	146	2	no	—
275	C₂×C₈	—	146	2	no	—
276	C₂×C₈	—	146	1	no	—
277	C₂×C₈	—	146	1	no	—
278	C₂×C₈	—	146	1	no	—
279	C₂×C₈	—	146	1	no	—
280	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
281	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
282	C₂×C₈	#11924027	1	2	yes	—
283	C₂×C₈	#11924029	1	2	yes	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	292	2	no	—
286	C₄⋊C₄	—	292	2	no	—
287	C₄⋊C₄	—	292	2	no	—
288	C₄⋊C₄	—	292	2	no	—
289	C₄⋊C₄	—	292	2	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	2	no	—
292	C₄²	—	146	1	no	—
293	C₄²	—	146	1	no	—
294	C₄²	—	146	2	no	—
295	C₄²	—	146	1	no	—
296	C₄²	—	146	1	no	—
297	C₄²	—	146	2	no	—
298	C₄²	C₂×Dic₁₄₆	1	2	yes	—
299	C₄²	#11924028	1	2	yes	—
300	C₂³	C₂²×C₄×D₇₃	1	1	yes	characteristic
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	146	2	no	—
310	C₂×C₄	—	146	2	no	—
311	C₂×C₄	—	146	2	no	—
312	C₂×C₄	—	146	2	no	—
313	C₂×C₄	—	146	2	no	—
314	C₂×C₄	—	146	2	no	—
315	C₂×C₄	—	146	2	no	—
316	C₂×C₄	—	146	2	no	—
317	C₂×C₄	—	146	2	no	—
318	C₂×C₄	—	146	2	no	—
319	C₂×C₄	—	146	2	no	—
320	C₂×C₄	—	146	2	no	—
321	C₂×C₄	C₄×Dic₁₄₆	1	2	yes	—
322	C₂×C₄	C₄×Dic₁₄₆	1	2	yes	—
323	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
324	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
325	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
326	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
327	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
328	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	4	2	no	—
331	C₈	—	292	2	no	—
332	C₈	—	146	2	no	—
333	C₈	—	146	2	no	—
334	C₈	#14556582	1	4	yes	—
335	C₂²	C₂×C₄×Dic₁₄₆	1	1	yes	characteristic
336	C₂²	C₄×#11924027	1	1	yes	characteristic
337	C₂²	C₄×#11924029	1	1	yes	characteristic
338	C₂²	#17989352	1	2	yes	—
339	C₂²	#17989353	1	2	yes	—
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	4	2	no	—
343	C₄	—	4	2	no	—
344	C₄	—	292	2	no	—
345	C₄	—	292	2	no	—
346	C₄	—	292	2	no	—
347	C₄	—	292	2	no	—
348	C₄	C₂×#14556582	1	1	yes	characteristic
349	C₄	#17989119	1	1	yes	characteristic
350	C₄	#17989124	1	1	yes	characteristic
351	C₄	#17989140	1	1	yes	characteristic
352	C₂	C₄×#14556582	1	1	yes	characteristic
353	C₂	#22618869	1	1	yes	characteristic
354	C₂	#22618871	1	1	yes	characteristic
355	C₂	#22618915	1	2	yes	—
356	C₂	#22618917	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₃
commutator, socle, prime radical	C₂³×C₇₃	C₂³×C₄
Fitting	C₇₃×#1770	C₂
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1770	D₇₃
hypocenter	C₇₃	#21677
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1770

Complements

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

N	H ≅ G/N	type
190. C₇₃	#21677	semidirect

Maximal subgroups

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

subgroup	index
#22613239	2
#22610655	2
#22610652	2
#22610498	2
#22610494	2
#22610050	2
#22609807	2
#22609803	2
#22609798	2
#22609617	2
C₇₃×#1770	2
#21677	73