Subgroups — #33056836

Order 18688.

584 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620491	C₂	1	1	yes	maximal, characteristic
2	#22619006	C₂	1	1	yes	maximal, characteristic
3	#22608809	C₂	1	2	yes	maximal, complemented
4	#22608786	C₂	1	2	yes	maximal, complemented
5	#22608774	C₂	1	2	yes	maximal, complemented
6	#22608765	C₂	1	2	yes	maximal, complemented
7	#22608692	C₂	1	1	yes	maximal, characteristic, complemented
8	#22608404	C₂	1	1	yes	maximal, characteristic, complemented
9	#22608399	C₂	1	1	yes	maximal, characteristic
10	#22608224	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2985	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989137	C₂²	1	1	yes	characteristic
13	#17988983	C₄	1	1	yes	characteristic
14	#17988961	C₄	1	1	yes	characteristic
15	#17988937	C₄	1	1	yes	characteristic
16	#17988935	C₄	1	1	yes	characteristic
17	#17988818	C₄	1	2	yes	—
18	#17988818	C₄	1	2	yes	—
19	C₄×#11924015	C₂²	1	2	yes	—
20	C₄×#11924015	C₂²	1	2	yes	—
21	#17988666	C₂²	1	2	yes	—
22	#17988657	—	2	2	no	—
23	C₈×D₂₉₂	C₂²	1	2	yes	—
24	#17988644	C₂²	1	1	yes	characteristic
25	#17988640	C₂²	1	1	yes	characteristic
26	#17988552	—	2	2	no	—
27	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
28	#17988339	C₂²	1	2	yes	—
29	#17988337	C₂²	1	2	yes	—
30	Q₈⋊#10596831	C₂²	1	2	yes	—
31	Q₈⋊#10596831	C₂²	1	2	yes	—
32	#17988314	C₂²	1	2	yes	—
33	#17988314	C₂²	1	2	yes	—
34	#17988309	C₂²	1	1	yes	characteristic
35	#17988309	C₂²	1	1	yes	characteristic
36	#17988308	C₂²	1	1	yes	characteristic
37	#17988308	C₂²	1	1	yes	characteristic
38	#533×C₇₃	C₂²	1	1	yes	characteristic
39	#446×C₇₃	—	2	2	no	—
40	#446×C₇₃	C₂²	1	2	yes	—
41	C₈×Q₈×C₇₃	C₂²	1	2	yes	—
42	#433×C₇₃	—	2	2	no	—
43	#432×C₇₃	C₂²	1	1	yes	characteristic
44	#431×C₇₃	C₂²	1	1	yes	characteristic
45	C₂×#11924028	—	2	1	no	—
46	C₂×#11924015	C₂×C₄	1	2	yes	—
47	C₂×#11924015	C₂×C₄	1	2	yes	—
48	C₂×#11924011	—	2	2	no	—
49	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
50	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₇₃	—	2	2	no	—
52	#14556591	C₂×C₄	1	1	yes	characteristic
53	#14556588	C₂×C₄	1	1	yes	characteristic
54	#14556579	—	2	1	no	—
55	#14556565	—	2	1	no	—
56	#14556565	C₂×C₄	1	2	yes	—
57	C₄×D₂₉₂	—	2	1	no	—
58	C₄×D₂₉₂	C₂³	1	2	yes	—
59	#14556522	C₂×C₄	1	2	yes	—
60	#14556522	C₂×C₄	1	2	yes	—
61	C₄⋊#10596831	—	2	1	no	—
62	C₄⋊#10596831	C₂³	1	2	yes	—
63	#14556515	C₂×C₄	1	2	yes	—
64	#14556515	C₂×C₄	1	2	yes	—
65	#14556511	C₂×C₄	1	2	yes	—
66	#14556511	C₂×C₄	1	2	yes	—
67	C₂²⋊#10596831	—	2	1	no	—
68	C₂²⋊#10596831	—	2	1	no	—
69	#14556503	—	2	2	no	—
70	#14556503	—	2	2	no	—
71	#14556502	—	2	1	no	—
72	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
73	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
74	C₄×#10596831	C₂³	1	1	yes	characteristic
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	C₄⋊Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
79	C₄⋊Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
80	#125×C₇₃	—	2	1	no	—
81	C₂²⋊Q₈×C₇₃	—	2	1	no	—
82	C₂²⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
83	C₄×Q₈×C₇₃	—	2	1	no	—
84	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
85	#117×C₇₃	—	2	1	no	—
86	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₇₃	—	2	2	no	—
88	C₄⋊C₈×C₇₃	—	2	2	no	—
89	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
90	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
91	C₂²⋊C₈×C₇₃	—	2	1	no	—
92	C₂²⋊C₈×C₇₃	D₄	1	2	yes	—
93	#97×C₇₃	—	2	2	no	—
94	#97×C₇₃	—	2	1	no	—
95	C₄×C₈×C₇₃	—	2	2	no	—
96	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
97	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
98	#11924028	—	8	2	no	—
99	C₂×Dic₁₄₆	—	2	1	no	—
100	C₂×D₂₉₂	—	2	1	no	—
101	C₂×D₂₉₂	—	2	1	no	—
102	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
103	C₂×C₂²⋊D₇₃	—	2	2	no	—
104	C₂×C₄×D₇₃	—	2	2	no	—
105	#11924015	—	4	2	no	—
106	#11924015	—	4	2	no	—
107	#11924011	—	8	2	no	—
108	C₂×#10596831	—	2	2	no	—
109	C₂×#10596831	—	2	1	no	—
110	C₂×#10596831	—	2	1	no	—
111	C₂×#10596831	—	4	1	no	—
112	C₂×#10596831	—	4	1	no	—
113	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
114	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
115	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
116	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
117	C₈×D₇₃	—	8	2	no	—
118	C₄⋊Dic₇₃	—	2	1	no	—
119	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
120	#11924005	—	2	2	no	—
121	#11924005	—	2	2	no	—
122	C₂×Q₈×C₇₃	—	2	1	no	—
123	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
124	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
125	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
126	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
127	C₂×C₈×C₇₃	—	2	2	no	—
128	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
129	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
130	C₄⋊C₄×C₇₃	—	2	1	no	—
131	C₄⋊C₄×C₇₃	—	2	1	no	—
132	C₄⋊C₄×C₇₃	—	2	1	no	—
133	C₄⋊C₄×C₇₃	—	2	2	no	—
134	C₄⋊C₄×C₇₃	—	2	2	no	—
135	C₄⋊C₄×C₇₃	C₂²×C₄	1	2	yes	—
136	C₂²⋊C₄×C₇₃	—	2	2	no	—
137	C₄²×C₇₃	—	2	2	no	—
138	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
139	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
140	C₄²×C₇₃	#41	1	1	yes	characteristic
141	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
142	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
143	C₂²×D₇₃	—	2	2	no	—
144	Q₈×C₇₃	—	2	2	no	—
145	Q₈×C₇₃	—	2	2	no	—
146	Q₈×C₇₃	—	8	1	no	—
147	C₂×C₄×C₇₃	—	2	1	no	—
148	C₂×C₄×C₇₃	—	2	1	no	—
149	C₂×C₄×C₇₃	—	2	2	no	—
150	C₂×C₄×C₇₃	—	2	2	no	—
151	C₂×C₄×C₇₃	—	2	2	no	—
152	C₂×C₄×C₇₃	—	4	1	no	—
153	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
155	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
156	C₂×C₄×C₇₃	C₂³×C₄	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	C₂²⋊D₇₃	—	8	2	no	—
160	C₂×Dic₇₃	—	2	2	no	—
161	D₂₉₂	—	2	2	no	—
162	D₂₉₂	—	2	2	no	—
163	D₂₉₂	—	8	1	no	—
164	D₂₉₂	—	8	1	no	—
165	C₄×D₇₃	—	4	2	no	—
166	C₄×D₇₃	—	4	2	no	—
167	Dic₁₄₆	—	8	1	no	—
168	C₈×C₇₃	—	2	2	no	—
169	C₈×C₇₃	—	2	2	no	—
170	C₈×C₇₃	—	4	2	no	—
171	#10596831	—	2	1	no	—
172	#10596831	—	2	1	no	—
173	#10596831	—	2	1	no	—
174	#10596831	—	2	1	no	—
175	#10596831	—	4	1	no	—
176	#10596831	—	4	1	no	—
177	#10596831	—	4	2	no	—
178	C₂²×C₇₃	—	2	1	no	—
179	C₂²×C₇₃	—	2	1	no	—
180	C₂²×C₇₃	—	2	1	no	—
181	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
182	C₂×D₇₃	—	4	2	no	—
183	C₂×D₇₃	—	4	2	no	—
184	C₂×D₇₃	—	4	2	no	—
185	C₄×C₇₃	—	2	1	no	—
186	C₄×C₇₃	—	2	1	no	—
187	C₄×C₇₃	—	2	1	no	—
188	C₄×C₇₃	—	2	1	no	—
189	C₄×C₇₃	—	4	2	no	—
190	C₄×C₇₃	—	4	2	no	—
191	C₄×C₇₃	#568	1	1	yes	characteristic
192	C₄×C₇₃	#568	1	1	yes	characteristic
193	C₄×C₇₃	#577	1	2	yes	—
194	Dic₇₃	—	4	2	no	—
195	#17990	—	73	1	no	maximal, Sylow
196	C₂×C₇₃	—	4	1	no	—
197	C₂×C₇₃	#2932	1	1	yes	characteristic
198	C₂×C₇₃	#2949	1	1	yes	characteristic
199	C₂×C₇₃	#2957	1	1	yes	characteristic
200	D₇₃	—	8	2	no	—
201	#3167	—	73	1	no	—
202	#2985	D₇₃	1	1	yes	characteristic, complemented
203	#2981	—	73	1	no	—
204	#1588	—	73	2	no	—
205	#1585	—	73	2	no	—
206	#1583	—	73	2	no	—
207	#1582	—	73	2	no	—
208	#1568	—	73	1	no	—
209	#1497	—	73	1	no	—
210	#1496	—	73	1	no	—
211	#1452	—	73	1	no	—
212	C₇₃	#17990	1	1	yes	characteristic, complemented, Sylow
213	#533	C₂×D₇₃	1	1	yes	characteristic
214	#532	—	73	1	no	—
215	#498	—	73	1	no	—
216	#492	—	73	1	no	—
217	#487	—	73	1	no	—
218	#486	—	73	1	no	—
219	#465	—	73	2	no	—
220	#465	—	73	2	no	—
221	#446	—	2	2	no	—
222	#446	C₂×D₇₃	1	2	yes	—
223	C₈×Q₈	C₂×D₇₃	1	2	yes	—
224	C₄×SD₁₆	—	73	2	no	—
225	C₄×SD₁₆	—	73	2	no	—
226	#436	—	73	2	no	—
227	#435	—	146	2	no	—
228	C₈×D₄	—	73	2	no	—
229	#433	—	2	2	no	—
230	#432	—	73	1	no	—
231	#432	C₂×D₇₃	1	1	yes	characteristic
232	#431	—	73	1	no	—
233	#431	C₂×D₇₃	1	1	yes	characteristic
234	#408	—	146	2	no	—
235	C₂×C₄×C₈	—	73	1	no	—
236	#335	—	73	2	no	—
237	#334	—	73	2	no	—
238	Q₈⋊C₈	—	73	2	no	—
239	Q₈⋊C₈	—	73	2	no	—
240	D₄⋊C₈	—	73	2	no	—
241	D₄⋊C₈	—	73	2	no	—
242	#322	—	73	1	no	—
243	#322	—	73	1	no	—
244	#322	—	73	1	no	—
245	#322	—	73	1	no	—
246	C₂×#41	—	146	1	no	—
247	C₂×SD₁₆	—	73	2	no	—
248	C₂×SD₁₆	—	73	2	no	—
249	C₂×M₁₆	—	146	2	no	—
250	C₂²×C₈	—	73	1	no	—
251	C₂²×C₈	—	73	1	no	—
252	C₂²×C₈	—	146	2	no	—
253	C₄⋊Q₈	—	73	1	no	—
254	C₄⋊Q₈	C₄×D₇₃	1	1	yes	characteristic
255	C₄⋊Q₈	C₄×D₇₃	1	1	yes	characteristic
256	#127	—	73	1	no	—
257	#125	—	2	1	no	—
258	#124	—	146	1	no	—
259	C₂²⋊Q₈	—	2	1	no	—
260	C₂²⋊Q₈	C₄×D₇₃	1	2	yes	—
261	#121	—	73	2	no	—
262	#121	—	146	1	no	—
263	C₄×Q₈	—	2	1	no	—
264	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
265	C₄×D₄	—	73	2	no	—
266	C₄×D₄	—	146	1	no	—
267	#117	—	2	1	no	—
268	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
269	#106	—	73	2	no	—
270	#106	—	73	2	no	—
271	C₄⋊C₈	—	2	2	no	—
272	C₄⋊C₈	—	2	2	no	—
273	C₄⋊C₈	—	73	2	no	—
274	C₄⋊C₈	—	146	1	no	—
275	C₄⋊C₈	C₂²⋊D₇₃	1	2	yes	—
276	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
277	#103	—	73	2	no	—
278	#103	—	73	2	no	—
279	#102	—	73	2	no	—
280	#102	—	73	2	no	—
281	C₂²⋊C₈	—	2	1	no	—
282	C₂²⋊C₈	—	146	1	no	—
283	C₂²⋊C₈	—	146	1	no	—
284	C₂²⋊C₈	—	146	2	no	—
285	C₂²⋊C₈	—	146	2	no	—
286	C₂²⋊C₈	C₂²⋊D₇₃	1	2	yes	—
287	#97	—	2	2	no	—
288	#97	—	2	1	no	—
289	#97	—	146	1	no	—
290	C₄×C₈	—	2	2	no	—
291	C₄×C₈	—	73	1	no	—
292	C₄×C₈	—	73	1	no	—
293	C₄×C₈	—	73	1	no	—
294	C₄×C₈	—	73	1	no	—
295	C₄×C₈	—	73	1	no	—
296	C₄×C₈	—	73	1	no	—
297	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
298	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
299	#41	—	584	2	no	—
300	C₂×Q₈	—	2	1	no	—
301	C₂×Q₈	—	146	1	no	—
302	C₂×Q₈	C₂×C₄×D₇₃	1	2	yes	—
303	C₂×D₄	—	73	2	no	—
304	C₂×D₄	—	146	2	no	—
305	C₂×D₄	—	146	1	no	—
306	C₂×D₄	—	146	1	no	—
307	C₂²×C₄	—	146	2	no	—
308	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
309	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
310	C₂²×C₄	#11924028	1	1	yes	characteristic
311	SD₁₆	—	292	2	no	—
312	SD₁₆	—	292	2	no	—
313	M₁₆	—	584	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	292	1	no	—
316	C₂×C₈	—	292	1	no	—
317	C₂×C₈	—	584	2	no	—
318	C₂×C₈	—	73	1	no	—
319	C₂×C₈	—	73	1	no	—
320	C₂×C₈	—	73	1	no	—
321	C₂×C₈	—	73	1	no	—
322	C₂×C₈	—	146	2	no	—
323	C₂×C₈	—	146	1	no	—
324	C₂×C₈	—	146	1	no	—
325	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
326	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	—	2	1	no	—
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	73	2	no	—
333	C₄⋊C₄	—	146	1	no	—
334	C₄⋊C₄	C₂×C₄×D₇₃	1	2	yes	—
335	C₂²⋊C₄	—	2	2	no	—
336	C₂²⋊C₄	—	146	2	no	—
337	C₂²⋊C₄	—	146	2	no	—
338	C₄²	—	2	2	no	—
339	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
340	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
341	C₄²	#11924028	1	1	yes	characteristic
342	C₄²	C₂³×D₇₃	1	1	yes	characteristic
343	C₂³	—	146	2	no	—
344	C₂³	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
345	Q₈	—	2	2	no	—
346	Q₈	—	2	2	no	—
347	Q₈	—	8	1	no	—
348	Q₈	—	584	1	no	—
349	D₄	—	584	1	no	—
350	D₄	—	584	1	no	—
351	D₄	—	584	2	no	—
352	D₄	—	146	2	no	—
353	D₄	—	146	2	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	292	2	no	—
361	C₂×C₄	—	292	2	no	—
362	C₂×C₄	—	146	2	no	—
363	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
364	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
365	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
366	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
367	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
368	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
369	C₈	—	2	2	no	—
370	C₈	—	2	2	no	—
371	C₈	—	4	2	no	—
372	C₈	—	292	1	no	—
373	C₈	—	292	1	no	—
374	C₈	—	292	2	no	—
375	C₈	—	146	1	no	—
376	C₈	—	146	1	no	—
377	C₈	—	146	1	no	—
378	C₈	—	146	1	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	1	no	—
381	C₂²	—	2	1	no	—
382	C₂²	—	292	2	no	—
383	C₂²	—	292	2	no	—
384	C₂²	—	292	2	no	—
385	C₂²	C₂×C₄×C₂²⋊D₇₃	1	1	yes	characteristic
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	2	1	no	—
389	C₄	—	2	1	no	—
390	C₄	—	4	2	no	—
391	C₄	—	4	2	no	—
392	C₄	—	292	2	no	—
393	C₄	#17989352	1	1	yes	characteristic
394	C₄	#17989352	1	1	yes	characteristic
395	C₄	#17989395	1	2	yes	—
396	C₂	—	4	1	no	—
397	C₂	—	584	2	no	—
398	C₂	#22618647	1	1	yes	characteristic
399	C₂	#22618790	1	1	yes	characteristic
400	C₂	#22618846	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#2985	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#17989137	C₂²
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#2985	D₇₃
hypocenter	C₇₃	#17990
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #2985

Complements

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

N	H ≅ G/N	type
3. #22608809	C₂	semidirect
4. #22608786	C₂	semidirect
5. #22608774	C₂	semidirect
6. #22608765	C₂	semidirect
7. #22608692	C₂	semidirect
8. #22608404	C₂	semidirect
10. #22608224	C₂	semidirect
11. C₇₃×#2985	C₂	semidirect
202. #2985	D₇₃	semidirect
212. C₇₃	#17990	semidirect

Maximal subgroups

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

subgroup	index
#22620491	2
#22619006	2
#22608809	2
#22608786	2
#22608774	2
#22608765	2
#22608692	2
#22608404	2
#22608399	2
#22608224	2
C₇₃×#2985	2
#17990	73