Subgroups — #33056556

Order 18688.

614 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620504	C₂	1	1	yes	maximal, characteristic
2	#22618908	C₂	1	2	yes	maximal, complemented
3	#22608844	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608840	C₂	1	1	yes	maximal, characteristic, complemented
5	#22608836	C₂	1	2	yes	maximal, complemented
6	#22608756	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608741	C₂	1	2	yes	maximal, complemented
8	#22608691	C₂	1	1	yes	maximal, characteristic, complemented
9	#22608390	C₂	1	2	yes	maximal, complemented
10	#322×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#1579	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989124	C₂²	1	2	yes	—
13	#17988988	C₄	1	1	yes	characteristic
14	#17988972	C₄	1	1	yes	characteristic
15	#17988943	C₄	1	2	yes	—
16	#17988825	C₄	1	1	yes	characteristic
17	#17988824	C₄	1	1	yes	characteristic
18	#17988819	C₄	1	2	yes	—
19	#17988718	C₂²	1	2	yes	—
20	C₄×#11924015	C₂²	1	2	yes	—
21	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
22	#17988664	C₂²	1	2	yes	—
23	#17988657	—	2	2	no	—
24	C₈×D₂₉₂	—	2	2	no	—
25	C₈×D₂₉₂	C₂²	1	2	yes	—
26	#17988647	—	2	2	no	—
27	#17988643	C₂²	1	2	yes	—
28	#17988541	—	2	2	no	—
29	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
32	#17988339	C₂²	1	2	yes	—
33	#17988338	C₂²	1	1	yes	characteristic
34	Q₈⋊#10596831	C₂²	1	2	yes	—
35	#17988316	C₂²	1	2	yes	—
36	#17988314	C₂²	1	2	yes	—
37	#17988309	C₂²	1	1	yes	characteristic
38	#17988309	C₂²	1	1	yes	characteristic
39	#17988308	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₇₃	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
42	#334×C₇₃	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₇₃	C₂²	1	2	yes	—
44	#322×C₇₃	C₂²	1	1	yes	characteristic
45	C₂×#11924015	C₂×C₄	1	2	yes	—
46	C₂×D₅₈₄	C₂×C₄	1	1	yes	characteristic
47	C₂×#11924011	—	2	2	no	—
48	C₂×C₈×D₇₃	—	2	1	no	—
49	C₂×C₈×D₇₃	—	2	2	no	—
50	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₈×D₇₃	D₄	1	2	yes	—
53	#14556586	—	2	2	no	—
54	#14556582	C₂×C₄	1	2	yes	—
55	#14556580	C₂×C₄	1	2	yes	—
56	#14556576	—	2	2	no	—
57	#14556567	C₂×C₄	1	2	yes	—
58	#14556566	C₂×C₄	1	2	yes	—
59	C₄×D₂₉₂	—	2	2	no	—
60	C₄×D₂₉₂	C₂³	1	2	yes	—
61	#14556548	—	2	2	no	—
62	C₄²×D₇₃	C₂³	1	1	yes	characteristic
63	#14556525	C₂×C₄	1	1	yes	characteristic
64	#14556522	C₂×C₄	1	2	yes	—
65	C₄⋊#10596831	—	2	2	no	—
66	C₄⋊#10596831	C₂³	1	2	yes	—
67	#14556515	C₂×C₄	1	2	yes	—
68	#14556513	C₂×C₄	1	2	yes	—
69	#14556511	C₂×C₄	1	2	yes	—
70	#14556503	—	2	2	no	—
71	#14556503	—	2	2	no	—
72	#14556503	—	2	2	no	—
73	#14556502	—	2	2	no	—
74	#14556501	—	2	2	no	—
75	C₄×#10596831	C₂³	1	1	yes	characteristic
76	C₄×#10596831	C₂³	1	1	yes	characteristic
77	C₄×#10596831	C₂³	1	1	yes	characteristic
78	C₈×Dic₇₃	—	2	1	no	—
79	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₇₃	D₄	1	2	yes	—
82	C₂×Q₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
84	#107×C₇₃	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₈×C₇₃	—	2	2	no	—
86	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
87	#103×C₇₃	C₂×C₄	1	2	yes	—
88	C₄×C₈×C₇₃	—	2	2	no	—
89	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
92	C₂×D₂₉₂	—	2	2	no	—
93	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₇₃	—	2	2	no	—
95	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
98	#11924015	—	2	4	no	—
99	D₅₈₄	—	2	2	no	—
100	#11924011	—	4	4	no	—
101	C₂×#10596831	—	2	2	no	—
102	C₂×#10596831	—	2	1	no	—
103	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
104	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
105	C₂×#10596831	C₂×D₄	1	2	yes	—
106	C₈×D₇₃	—	2	2	no	—
107	C₈×D₇₃	—	2	2	no	—
108	C₈×D₇₃	—	4	2	no	—
109	C₈×D₇₃	—	4	4	no	—
110	C₈×D₇₃	—	4	2	no	—
111	C₄⋊Dic₇₃	—	2	2	no	—
112	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
113	#11924007	—	2	2	no	—
114	#11924007	—	2	2	no	—
115	#11924005	—	2	2	no	—
116	#11924005	—	2	2	no	—
117	#11924005	—	2	2	no	—
118	#11924005	—	2	2	no	—
119	#11924005	—	2	2	no	—
120	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
121	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
122	C₄×Dic₇₃	#41	1	1	yes	characteristic
123	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
124	Q₁₆×C₇₃	—	2	2	no	—
125	C₂×C₈×C₇₃	—	2	2	no	—
126	C₂×C₈×C₇₃	—	2	1	no	—
127	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
130	C₄⋊C₄×C₇₃	—	2	2	no	—
131	C₄⋊C₄×C₇₃	C₂²×C₄	1	2	yes	—
132	C₄²×C₇₃	—	2	2	no	—
133	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
134	C₂²×D₇₃	—	2	2	no	—
135	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
136	Q₈×C₇₃	—	2	2	no	—
137	Q₈×C₇₃	—	2	2	no	—
138	C₂×C₄×C₇₃	—	2	2	no	—
139	C₂×C₄×C₇₃	—	2	2	no	—
140	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
141	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
143	C₂×Dic₇₃	—	2	2	no	—
144	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
146	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
147	D₂₉₂	—	2	2	no	—
148	D₂₉₂	—	2	2	no	—
149	D₂₉₂	—	4	4	no	—
150	C₄×D₇₃	—	2	2	no	—
151	C₄×D₇₃	—	2	1	no	—
152	C₄×D₇₃	—	2	1	no	—
153	C₄×D₇₃	—	2	2	no	—
154	C₄×D₇₃	—	4	2	no	—
155	C₄×D₇₃	—	4	2	no	—
156	C₈×C₇₃	—	2	1	no	—
157	C₈×C₇₃	—	2	2	no	—
158	C₈×C₇₃	—	4	1	no	—
159	C₈×C₇₃	—	4	2	no	—
160	C₈×C₇₃	#131	1	2	yes	—
161	#10596831	—	2	1	no	—
162	#10596831	—	2	2	no	—
163	#10596831	—	4	1	no	—
164	#10596831	—	4	2	no	—
165	#10596831	#131	1	2	yes	—
166	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂×D₇₃	—	2	1	no	—
168	C₂×D₇₃	—	2	1	no	—
169	C₂×D₇₃	—	4	2	no	—
170	C₂×D₇₃	—	4	2	no	—
171	C₂×D₇₃	—	4	2	no	—
172	C₂×D₇₃	#444	1	2	yes	—
173	C₄×C₇₃	—	2	1	no	—
174	C₄×C₇₃	—	4	2	no	—
175	C₄×C₇₃	—	4	2	no	—
176	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
177	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
178	C₄×C₇₃	#575	1	2	yes	—
179	Dic₇₃	—	2	1	no	—
180	Dic₇₃	—	2	1	no	—
181	Dic₇₃	—	2	1	no	—
182	Dic₇₃	—	4	2	no	—
183	#17963	—	73	1	no	maximal, Sylow
184	C₂×C₇₃	#2931	1	1	yes	characteristic
185	C₂×C₇₃	#2943	1	1	yes	characteristic
186	C₂×C₇₃	C₂×#444	1	1	yes	characteristic
187	D₇₃	—	2	2	no	—
188	D₇₃	—	8	2	no	—
189	#3168	—	73	1	no	—
190	#2966	—	73	2	no	—
191	#1594	—	73	1	no	—
192	#1593	—	73	1	no	—
193	#1592	—	73	2	no	—
194	#1580	—	73	1	no	—
195	#1579	D₇₃	1	1	yes	characteristic, complemented
196	SD₁₆⋊C₈	—	73	2	no	—
197	#1568	—	73	1	no	—
198	#1494	—	73	2	no	—
199	C₂×#322	—	73	1	no	—
200	C₇₃	#17963	1	1	yes	characteristic, complemented, Sylow
201	#529	—	73	2	no	—
202	#499	—	73	1	no	—
203	#495	—	73	1	no	—
204	#488	—	73	2	no	—
205	#467	—	73	1	no	—
206	#466	—	73	1	no	—
207	#465	—	73	2	no	—
208	#446	—	73	2	no	—
209	C₈×Q₈	C₂×D₇₃	1	2	yes	—
210	C₄×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
211	C₄×SD₁₆	—	73	2	no	—
212	C₄×D₈	—	73	1	no	—
213	#436	—	73	2	no	—
214	#435	—	146	2	no	—
215	C₈×D₄	—	73	2	no	—
216	C₈×D₄	—	146	2	no	—
217	#433	—	146	2	no	—
218	#432	—	73	2	no	—
219	#405	—	146	2	no	—
220	C₂×C₄×C₈	—	73	1	no	—
221	C₂×C₄×C₈	—	73	1	no	—
222	C₂×C₄×C₈	—	73	1	no	—
223	#335	—	73	2	no	—
224	#334	—	73	1	no	—
225	#334	C₂×D₇₃	1	1	yes	characteristic
226	Q₈⋊C₈	—	73	2	no	—
227	Q₈⋊C₈	C₂×D₇₃	1	2	yes	—
228	D₄⋊C₈	—	73	2	no	—
229	D₄⋊C₈	—	73	2	no	—
230	#322	—	73	1	no	—
231	#322	—	73	1	no	—
232	#322	—	73	1	no	—
233	#322	C₂×D₇₃	1	1	yes	characteristic
234	C₂×Q₁₆	C₄×D₇₃	1	1	yes	characteristic
235	C₂×SD₁₆	—	73	2	no	—
236	C₂×D₈	—	73	1	no	—
237	C₂×M₁₆	—	146	2	no	—
238	C₂²×C₈	—	73	2	no	—
239	C₂²×C₈	—	73	1	no	—
240	C₂²×C₈	—	73	1	no	—
241	C₂²×C₈	—	146	1	no	—
242	C₂²×C₈	—	146	2	no	—
243	#126	—	146	2	no	—
244	#125	—	73	2	no	—
245	#124	—	73	2	no	—
246	#123	—	146	2	no	—
247	C₂²⋊Q₈	—	73	2	no	—
248	#121	—	73	2	no	—
249	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
250	C₄×D₄	—	73	2	no	—
251	C₄×D₄	—	146	2	no	—
252	#117	—	146	2	no	—
253	C₂×C₄²	—	73	1	no	—
254	#107	—	73	1	no	—
255	#107	C₄×D₇₃	1	1	yes	characteristic
256	#106	—	73	2	no	—
257	C₄⋊C₈	—	2	2	no	—
258	C₄⋊C₈	—	73	2	no	—
259	C₄⋊C₈	—	146	2	no	—
260	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
261	#103	—	73	2	no	—
262	#103	C₄×D₇₃	1	2	yes	—
263	#102	—	73	2	no	—
264	#102	—	73	2	no	—
265	C₂²⋊C₈	—	146	2	no	—
266	C₂²⋊C₈	—	146	2	no	—
267	C₂²⋊C₈	—	146	2	no	—
268	#97	—	146	2	no	—
269	#97	—	146	2	no	—
270	C₄×C₈	—	2	2	no	—
271	C₄×C₈	—	73	1	no	—
272	C₄×C₈	—	73	1	no	—
273	C₄×C₈	—	73	2	no	—
274	C₄×C₈	—	73	1	no	—
275	C₄×C₈	—	73	1	no	—
276	C₄×C₈	—	73	1	no	—
277	C₄×C₈	—	146	1	no	—
278	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
281	C₂×Q₈	C₂×C₄×D₇₃	1	2	yes	—
282	C₂×D₄	—	73	2	no	—
283	C₂×D₄	—	146	2	no	—
284	C₂²×C₄	—	73	1	no	—
285	C₂²×C₄	—	73	1	no	—
286	C₂²×C₄	—	73	1	no	—
287	C₂²×C₄	—	146	2	no	—
288	Q₁₆	—	2	2	no	—
289	SD₁₆	—	146	4	no	—
290	D₈	—	146	2	no	—
291	M₁₆	—	292	4	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	292	4	no	—
295	C₂×C₈	—	292	2	no	—
296	C₂×C₈	—	292	2	no	—
297	C₂×C₈	—	73	1	no	—
298	C₂×C₈	—	73	1	no	—
299	C₂×C₈	—	73	2	no	—
300	C₂×C₈	—	146	2	no	—
301	C₂×C₈	—	146	2	no	—
302	C₂×C₈	—	146	1	no	—
303	C₂×C₈	—	146	2	no	—
304	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
305	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
306	C₂×C₈	D₄×D₇₃	1	2	yes	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	73	2	no	—
309	C₄⋊C₄	—	146	2	no	—
310	C₄⋊C₄	—	146	2	no	—
311	C₄⋊C₄	—	146	2	no	—
312	C₄⋊C₄	C₂×C₄×D₇₃	1	2	yes	—
313	C₂²⋊C₄	—	146	2	no	—
314	C₂²⋊C₄	—	146	2	no	—
315	C₂²⋊C₄	—	146	2	no	—
316	C₂²⋊C₄	—	146	2	no	—
317	C₂²⋊C₄	—	146	2	no	—
318	C₄²	—	2	2	no	—
319	C₄²	—	73	1	no	—
320	C₄²	—	73	1	no	—
321	C₄²	—	73	1	no	—
322	C₄²	C₂³×D₇₃	1	1	yes	characteristic
323	C₂³	—	73	1	no	—
324	C₂³	—	146	2	no	—
325	Q₈	—	2	2	no	—
326	Q₈	—	2	2	no	—
327	D₄	—	292	4	no	—
328	D₄	—	146	2	no	—
329	D₄	—	146	2	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	2	no	—
332	C₂×C₄	—	292	2	no	—
333	C₂×C₄	—	292	2	no	—
334	C₂×C₄	—	73	1	no	—
335	C₂×C₄	—	73	1	no	—
336	C₂×C₄	—	73	1	no	—
337	C₂×C₄	—	146	2	no	—
338	C₂×C₄	—	146	2	no	—
339	C₂×C₄	—	146	2	no	—
340	C₂×C₄	—	146	1	no	—
341	C₂×C₄	—	146	1	no	—
342	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
343	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
344	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
345	C₈	—	2	2	no	—
346	C₈	—	2	1	no	—
347	C₈	—	4	1	no	—
348	C₈	—	4	2	no	—
349	C₈	—	292	1	no	—
350	C₈	—	292	2	no	—
351	C₈	—	73	2	no	—
352	C₈	—	146	2	no	—
353	C₈	—	146	1	no	—
354	C₈	#14556597	1	2	yes	—
355	C₂²	—	292	2	no	—
356	C₂²	—	292	2	no	—
357	C₂²	—	292	2	no	—
358	C₂²	—	73	2	no	—
359	C₂²	—	146	1	no	—
360	C₂²	—	146	1	no	—
361	C₂²	C₄×D₄×D₇₃	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	4	2	no	—
364	C₄	—	4	2	no	—
365	C₄	—	292	2	no	—
366	C₄	—	146	1	no	—
367	C₄	—	146	1	no	—
368	C₄	—	146	1	no	—
369	C₄	#131×D₇₃	1	1	yes	characteristic
370	C₄	C₂×#14556597	1	1	yes	characteristic
371	C₄	#17989386	1	2	yes	—
372	C₂	—	584	2	no	—
373	C₂	—	146	2	no	—
374	C₂	#22618638	1	1	yes	characteristic
375	C₂	#22618730	1	1	yes	characteristic
376	C₂	#444×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1579	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#17988824	C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1579	D₇₃
hypocenter	C₇₃	#17963
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

Complements

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

N	H ≅ G/N	type
2. #22618908	C₂	semidirect
3. #22608844	C₂	semidirect
4. #22608840	C₂	semidirect
5. #22608836	C₂	semidirect
6. #22608756	C₂	semidirect
7. #22608741	C₂	semidirect
8. #22608691	C₂	semidirect
9. #22608390	C₂	semidirect
10. #322×D₇₃	C₂	semidirect
11. C₇₃×#1579	C₂	semidirect
195. #1579	D₇₃	semidirect
200. C₇₃	#17963	semidirect

Maximal subgroups

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

subgroup	index
#22620504	2
#22618908	2
#22608844	2
#22608840	2
#22608836	2
#22608756	2
#22608741	2
#22608691	2
#22608390	2
#322×D₇₃	2
C₇₃×#1579	2
#17963	73