Subgroups — #1715×D₇₃ #33073701

Order 18688.

1030 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	#498×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#493×D₇₃	C₂	1	1	yes	maximal, characteristic
3	#492×D₇₃	C₂	1	2	yes	maximal, complemented
4	#486×D₇₃	C₂	1	2	yes	maximal, complemented
5	#22609561	C₂	1	2	yes	maximal, complemented
6	#22609560	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609559	C₂	1	1	yes	maximal, characteristic, complemented
8	#22609558	C₂	1	2	yes	maximal, complemented
9	#22609557	C₂	1	1	yes	maximal, characteristic, complemented
10	#322×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#1715	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×SD₁₆×D₇₃	—	2	4	no	—
13	C₂×D₈×D₇₃	—	2	2	no	—
14	C₄⋊Q₈×D₇₃	C₂²	1	2	yes	—
15	#127×D₇₃	C₂²	1	1	yes	characteristic
16	#127×D₇₃	C₂²	1	1	yes	characteristic
17	#17988985	C₂²	1	1	yes	characteristic, complemented
18	#17988984	—	2	2	no	—
19	#17988983	C₂²	1	2	yes	complemented
20	#17988964	C₂²	1	1	yes	characteristic
21	#17988963	C₂²	1	1	yes	characteristic
22	#17988963	C₂²	1	1	yes	characteristic
23	#17988962	—	2	2	no	—
24	#17988961	C₂²	1	2	yes	complemented
25	#17988960	C₂²	1	2	yes	complemented
26	#17988959	C₂²	1	2	yes	complemented
27	#17988958	—	2	4	no	—
28	#17988935	C₂²	1	2	yes	complemented
29	#17988934	C₂²	1	2	yes	complemented
30	#17988933	C₂²	1	2	yes	complemented
31	#17988932	—	2	4	no	—
32	#106×D₇₃	—	2	2	no	—
33	#102×D₇₃	—	2	4	no	—
34	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
35	C₄×C₈×D₇₃	C₂²	1	2	yes	—
36	#17988309	C₂²	1	2	yes	complemented
37	#17988308	C₂²	1	1	yes	characteristic, complemented
38	#498×C₇₃	C₂²	1	1	yes	characteristic, complemented
39	#493×C₇₃	C₂²	1	1	yes	characteristic
40	#492×C₇₃	C₂²	1	2	yes	complemented
41	#486×C₇₃	C₂²	1	2	yes	complemented
42	#322×C₇₃	C₂²	1	1	yes	characteristic, complemented
43	C₂×Q₈×D₇₃	—	2	2	no	—
44	C₂×D₄×D₇₃	—	2	2	no	—
45	C₂×D₄×D₇₃	—	2	1	no	—
46	C₂×D₄×D₇₃	—	2	1	no	—
47	C₂×D₄×D₇₃	—	2	2	no	—
48	C₂×#11924017	—	2	4	no	—
49	C₂×#11924016	—	2	4	no	—
50	C₂×#11924015	—	2	4	no	—
51	SD₁₆×D₇₃	—	4	16	no	—
52	C₂×D₅₈₄	—	2	2	no	—
53	C₂×#11924013	—	2	2	no	—
54	C₂×#11924013	—	2	2	no	—
55	D₈×D₇₃	—	4	8	no	—
56	C₂×C₈×D₇₃	D₄	1	4	yes	—
57	C₂×C₈×D₇₃	D₄	1	2	yes	—
58	#14556591	C₂³	1	2	yes	—
59	#14556590	—	2	2	no	—
60	#14556590	—	2	2	no	—
61	#14556589	—	2	2	no	—
62	#14556588	C₂³	1	1	yes	characteristic
63	#14556588	C₂³	1	1	yes	characteristic
64	#14556587	—	2	2	no	—
65	#14556587	—	2	2	no	—
66	#14556587	—	2	1	no	—
67	#14556587	—	2	1	no	—
68	C₄⋊C₄×D₇₃	—	2	2	no	—
69	C₄⋊C₄×D₇₃	—	2	2	no	—
70	C₄²×D₇₃	C₂³	1	1	yes	characteristic
71	#14556523	—	2	2	no	—
72	#14556522	—	2	4	no	—
73	#14556513	—	2	4	no	—
74	#14556512	—	2	4	no	—
75	#14556511	—	2	4	no	—
76	C₄×#10596831	C₂³	1	2	yes	—
77	C₄×#10596831	C₂³	1	1	yes	characteristic
78	C₈×Dic₇₃	D₄	1	4	yes	—
79	C₈×Dic₇₃	D₄	1	2	yes	—
80	C₂×SD₁₆×C₇₃	—	2	4	no	—
81	C₂×D₈×C₇₃	—	2	2	no	—
82	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
83	#127×C₇₃	C₂³	1	1	yes	characteristic
84	#127×C₇₃	C₂³	1	1	yes	characteristic
85	#106×C₇₃	—	2	2	no	—
86	#102×C₇₃	—	2	4	no	—
87	C₄×C₈×C₇₃	C₂³	1	2	yes	—
88	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
89	C₂³×D₇₃	—	4	1	no	—
90	C₂³×D₇₃	—	4	1	no	—
91	C₂×Dic₁₄₆	—	2	2	no	—
92	C₂×Dic₁₄₆	—	4	2	no	—
93	Q₈×D₇₃	—	4	8	no	—
94	C₂×D₂₉₂	—	2	2	no	—
95	C₂×D₂₉₂	—	2	1	no	—
96	C₂×D₂₉₂	—	2	1	no	—
97	C₂×D₂₉₂	—	2	2	no	—
98	C₂×C₂²⋊D₇₃	—	4	1	no	—
99	C₂×C₂²⋊D₇₃	—	4	1	no	—
100	C₂×C₂²⋊D₇₃	—	4	2	no	—
101	C₂×C₂²⋊D₇₃	—	4	2	no	—
102	D₄×D₇₃	—	4	4	no	—
103	D₄×D₇₃	—	4	4	no	—
104	D₄×D₇₃	—	4	8	no	—
105	D₄×D₇₃	—	4	8	no	—
106	C₂×C₄×D₇₃	—	4	2	no	—
107	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
109	#11924017	—	4	8	no	—
110	#11924016	—	4	8	no	—
111	#11924015	—	4	8	no	—
112	D₅₈₄	—	4	4	no	—
113	#11924013	—	4	4	no	—
114	#11924013	—	4	4	no	—
115	C₂×#10596831	C₂×D₄	1	4	yes	—
116	C₂×#10596831	C₂×D₄	1	2	yes	—
117	C₈×D₇₃	—	2	8	no	—
118	C₈×D₇₃	SD₁₆	1	8	yes	complemented
119	C₄⋊Dic₇₃	—	2	2	no	—
120	C₄⋊Dic₇₃	—	2	2	no	—
121	#11924007	—	4	2	no	—
122	#11924007	—	4	2	no	—
123	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
124	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
125	C₂×Q₈×C₇₃	—	2	2	no	—
126	C₂×D₄×C₇₃	—	2	2	no	—
127	C₂×D₄×C₇₃	—	2	1	no	—
128	C₂×D₄×C₇₃	—	2	1	no	—
129	C₂×D₄×C₇₃	—	2	2	no	—
130	SD₁₆×C₇₃	—	4	8	no	—
131	D₈×C₇₃	—	4	4	no	—
132	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
133	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
134	C₄⋊C₄×C₇₃	—	2	2	no	—
135	C₄⋊C₄×C₇₃	—	2	2	no	—
136	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
137	C₂³×C₇₃	—	4	1	no	—
138	C₂³×C₇₃	—	4	1	no	—
139	C₂²×D₇₃	—	4	1	no	—
140	C₂²×D₇₃	—	4	1	no	—
141	C₂²×D₇₃	—	8	2	no	—
142	C₂²×D₇₃	—	8	2	no	—
143	C₂²×D₇₃	—	8	4	no	—
144	C₂²×D₇₃	—	8	4	no	—
145	C₂²×D₇₃	#127	1	1	yes	characteristic
146	Q₈×C₇₃	—	4	4	no	—
147	D₄×C₇₃	—	4	2	no	—
148	D₄×C₇₃	—	4	2	no	—
149	D₄×C₇₃	—	4	4	no	—
150	D₄×C₇₃	—	4	4	no	—
151	C₂×C₄×C₇₃	—	4	2	no	—
152	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
154	C₂²⋊D₇₃	—	8	2	no	—
155	C₂²⋊D₇₃	—	8	2	no	—
156	C₂²⋊D₇₃	—	8	4	no	—
157	C₂²⋊D₇₃	—	8	4	no	—
158	C₂×Dic₇₃	—	4	2	no	—
159	C₂×Dic₇₃	#127	1	1	yes	characteristic
160	C₂×Dic₇₃	#127	1	2	yes	—
161	D₂₉₂	—	4	2	no	—
162	D₂₉₂	—	4	2	no	—
163	D₂₉₂	—	4	4	no	—
164	D₂₉₂	—	4	4	no	—
165	C₄×D₇₃	—	8	4	no	—
166	C₄×D₇₃	C₂×SD₁₆	1	4	yes	—
167	C₄×D₇₃	AGL(1,ℤ/8)	1	8	yes	—
168	Dic₁₄₆	—	4	4	no	—
169	Dic₁₄₆	—	8	4	no	—
170	C₈×C₇₃	—	2	4	no	—
171	C₈×C₇₃	C₂×SD₁₆	1	4	yes	complemented
172	#10596831	—	2	4	no	—
173	#10596831	C₂×SD₁₆	1	4	yes	complemented
174	C₂²×C₇₃	—	8	1	no	—
175	C₂²×C₇₃	—	8	1	no	—
176	C₂²×C₇₃	—	8	2	no	—
177	C₂²×C₇₃	—	8	2	no	—
178	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
179	C₂×D₇₃	—	8	1	no	—
180	C₂×D₇₃	—	8	1	no	—
181	C₂×D₇₃	—	8	2	no	—
182	C₂×D₇₃	—	8	2	no	—
183	C₂×D₇₃	—	16	4	no	—
184	C₂×D₇₃	—	16	4	no	—
185	C₂×D₇₃	#492	1	2	yes	—
186	C₂×D₇₃	#496	1	4	yes	—
187	C₄×C₇₃	—	8	2	no	—
188	C₄×C₇₃	C₂²×SD₁₆	1	2	yes	—
189	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
190	Dic₇₃	—	8	2	no	—
191	Dic₇₃	#492	1	2	yes	—
192	Dic₇₃	#496	1	4	yes	—
193	C₂×#1715	—	73	1	no	maximal, Sylow
194	C₂×C₇₃	—	16	1	no	—
195	C₂×C₇₃	—	16	1	no	—
196	C₂×C₇₃	C₂×#492	1	1	yes	characteristic
197	C₂×C₇₃	C₂×#496	1	2	yes	—
198	D₇₃	—	16	1	no	—
199	D₇₃	—	16	1	no	—
200	D₇₃	#1715	1	4	yes	direct product factor
201	C₂×#498	—	73	1	no	—
202	C₂×#493	—	73	1	no	—
203	C₂×#492	—	73	2	no	—
204	C₂×#486	—	73	2	no	—
205	#1715	—	73	1	no	—
206	#1715	—	73	1	no	—
207	#1715	—	73	2	no	—
208	#1715	—	73	2	no	—
209	#1715	—	73	1	no	—
210	#1715	D₇₃	1	1	yes	characteristic, direct product factor
211	C₂×#322	—	73	1	no	—
212	C₇₃	C₂×#1715	1	1	yes	characteristic, complemented, Sylow
213	C₂²×SD₁₆	—	146	4	no	—
214	C₂²×D₈	—	146	2	no	—
215	C₂×C₄⋊Q₈	—	73	2	no	—
216	C₂×#127	—	73	1	no	—
217	C₂×#127	—	73	1	no	—
218	#498	—	73	2	no	—
219	#498	—	73	1	no	—
220	#498	—	146	2	no	—
221	#498	C₂×D₇₃	1	1	yes	characteristic, complemented
222	#493	—	73	1	no	—
223	#493	—	73	1	no	—
224	#493	—	73	1	no	—
225	#493	—	146	2	no	—
226	#493	C₂×D₇₃	1	1	yes	characteristic
227	#492	—	73	2	no	—
228	#492	—	73	2	no	—
229	#492	—	73	2	no	—
230	#492	—	146	4	no	—
231	#492	C₂×D₇₃	1	2	yes	complemented
232	#486	—	73	2	no	—
233	#486	—	73	2	no	—
234	#486	—	73	2	no	—
235	#486	—	146	4	no	—
236	#486	C₂×D₇₃	1	2	yes	complemented
237	C₂×#106	—	146	2	no	—
238	C₂×#102	—	146	4	no	—
239	C₂×C₄×C₈	—	73	2	no	—
240	C₂×C₄×C₈	—	73	1	no	—
241	#322	—	73	1	no	—
242	#322	—	73	2	no	—
243	#322	C₂×D₇₃	1	1	yes	characteristic, complemented
244	C₂²×Q₈	—	146	2	no	—
245	C₂²×D₄	—	146	1	no	—
246	C₂²×D₄	—	146	1	no	—
247	C₂²×D₄	—	146	2	no	—
248	C₂²×D₄	—	146	2	no	—
249	C₂×SD₁₆	—	2	4	no	—
250	C₂×SD₁₆	—	292	16	no	—
251	C₂×SD₁₆	—	146	4	no	—
252	C₂×SD₁₆	—	146	4	no	—
253	C₂×SD₁₆	—	146	4	no	—
254	C₂×D₈	—	2	2	no	—
255	C₂×D₈	—	292	8	no	—
256	C₂×D₈	—	146	2	no	—
257	C₂×D₈	—	146	2	no	—
258	C₂×D₈	—	146	2	no	—
259	C₂²×C₈	—	73	2	no	—
260	C₂²×C₈	—	73	4	no	—
261	C₄⋊Q₈	—	73	2	no	—
262	C₄⋊Q₈	—	146	2	no	—
263	C₄⋊Q₈	—	146	2	no	—
264	C₄⋊Q₈	—	146	2	no	—
265	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
266	#127	—	73	1	no	—
267	#127	—	73	1	no	—
268	#127	—	146	2	no	—
269	#127	—	146	2	no	—
270	#127	—	146	1	no	—
271	#127	—	146	1	no	—
272	#127	C₂²×D₇₃	1	1	yes	characteristic
273	#127	C₂²×D₇₃	1	1	yes	characteristic
274	C₂×C₄⋊C₄	—	146	2	no	—
275	C₂×C₄⋊C₄	—	146	2	no	—
276	C₂×C₄²	—	73	1	no	—
277	#106	—	2	2	no	—
278	#106	—	146	4	no	—
279	#106	—	146	2	no	—
280	#102	—	2	4	no	—
281	#102	—	146	4	no	—
282	#102	—	146	4	no	—
283	#102	—	146	4	no	—
284	C₄×C₈	—	73	4	no	—
285	C₄×C₈	—	73	2	no	—
286	C₄×C₈	—	73	2	no	—
287	C₄×C₈	—	73	1	no	—
288	C₄×C₈	C₂²×D₇₃	1	2	yes	—
289	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
290	C₂⁴	—	292	1	no	—
291	C₂⁴	—	292	1	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	292	2	no	—
294	C₂×Q₈	—	292	8	no	—
295	C₂×Q₈	—	146	2	no	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	—	2	1	no	—
298	C₂×D₄	—	2	1	no	—
299	C₂×D₄	—	2	2	no	—
300	C₂×D₄	—	292	4	no	—
301	C₂×D₄	—	292	4	no	—
302	C₂×D₄	—	292	8	no	—
303	C₂×D₄	—	292	8	no	—
304	C₂×D₄	—	292	1	no	—
305	C₂×D₄	—	292	1	no	—
306	C₂×D₄	—	292	2	no	—
307	C₂×D₄	—	292	2	no	—
308	C₂×D₄	—	146	2	no	—
309	C₂×D₄	—	146	1	no	—
310	C₂×D₄	—	146	1	no	—
311	C₂×D₄	—	146	2	no	—
312	C₂²×C₄	—	292	2	no	—
313	C₂²×C₄	—	73	1	no	—
314	C₂²×C₄	—	73	2	no	—
315	SD₁₆	—	4	8	no	—
316	SD₁₆	—	292	8	no	—
317	SD₁₆	—	292	8	no	—
318	SD₁₆	—	292	8	no	—
319	D₈	—	4	4	no	—
320	D₈	—	292	4	no	—
321	D₈	—	292	4	no	—
322	D₈	—	292	4	no	—
323	C₂×C₈	—	73	2	no	—
324	C₂×C₈	—	73	8	no	—
325	C₂×C₈	—	73	4	no	—
326	C₂×C₈	—	146	8	no	—
327	C₂×C₈	D₄×D₇₃	1	4	yes	—
328	C₂×C₈	D₄×D₇₃	1	2	yes	—
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	292	2	no	—
332	C₄⋊C₄	—	292	2	no	—
333	C₄⋊C₄	—	146	2	no	—
334	C₄⋊C₄	—	146	2	no	—
335	C₄²	—	73	2	no	—
336	C₄²	—	73	1	no	—
337	C₄²	C₂³×D₇₃	1	1	yes	characteristic
338	C₂³	—	4	1	no	—
339	C₂³	—	4	1	no	—
340	C₂³	—	292	1	no	—
341	C₂³	—	292	1	no	—
342	C₂³	—	584	2	no	—
343	C₂³	—	584	2	no	—
344	C₂³	—	584	4	no	—
345	C₂³	—	584	4	no	—
346	C₂³	—	73	1	no	—
347	Q₈	—	4	4	no	—
348	Q₈	—	292	4	no	—
349	Q₈	—	584	4	no	—
350	D₄	—	4	2	no	—
351	D₄	—	4	2	no	—
352	D₄	—	4	4	no	—
353	D₄	—	4	4	no	—
354	D₄	—	292	2	no	—
355	D₄	—	292	2	no	—
356	D₄	—	292	4	no	—
357	D₄	—	292	4	no	—
358	D₄	—	584	2	no	—
359	D₄	—	584	2	no	—
360	D₄	—	584	4	no	—
361	D₄	—	584	4	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	292	2	no	—
364	C₂×C₄	—	584	4	no	—
365	C₂×C₄	—	73	1	no	—
366	C₂×C₄	—	73	2	no	—
367	C₂×C₄	—	73	4	no	—
368	C₂×C₄	—	73	8	no	—
369	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
370	C₂×C₄	C₂×D₄×D₇₃	1	2	yes	—
371	C₈	—	2	4	no	—
372	C₈	—	73	4	no	—
373	C₈	—	146	4	no	—
374	C₈	SD₁₆×D₇₃	1	4	yes	complemented
375	C₂²	—	8	1	no	—
376	C₂²	—	8	1	no	—
377	C₂²	—	8	2	no	—
378	C₂²	—	8	2	no	—
379	C₂²	—	584	1	no	—
380	C₂²	—	584	1	no	—
381	C₂²	—	584	2	no	—
382	C₂²	—	584	2	no	—
383	C₂²	—	73	2	no	—
384	C₂²	—	73	4	no	—
385	C₂²	—	1168	4	no	—
386	C₂²	—	1168	4	no	—
387	C₂²	#127×D₇₃	1	1	yes	characteristic
388	C₄	—	8	2	no	—
389	C₄	—	584	2	no	—
390	C₄	—	73	2	no	—
391	C₄	—	73	4	no	—
392	C₄	C₂×SD₁₆×D₇₃	1	2	yes	—
393	C₄	AGL(1,ℤ/8)×D₇₃	1	4	yes	—
394	C₂	—	16	1	no	—
395	C₂	—	16	1	no	—
396	C₂	—	73	4	no	—
397	C₂	—	1168	1	no	—
398	C₂	—	1168	1	no	—
399	C₂	#492×D₇₃	1	1	yes	characteristic
400	C₂	#496×D₇₃	1	2	yes	—

	subgroup	G/H
center	C₂²	#127×D₇₃
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1715	C₂
socle	C₂²×C₇₃	C₂×#127
prime radical	#493×D₇₃	C₂
hypercenter	#1715	D₇₃
hypocenter	C₇₃	C₂×#1715
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1715

Complements

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

N	H ≅ G/N	type
1. #498×D₇₃	C₂	semidirect
3. #492×D₇₃	C₂	semidirect
4. #486×D₇₃	C₂	semidirect
5. #22609561	C₂	semidirect
6. #22609560	C₂	semidirect
7. #22609559	C₂	semidirect
8. #22609558	C₂	semidirect
9. #22609557	C₂	semidirect
10. #322×D₇₃	C₂	semidirect
11. C₇₃×#1715	C₂	semidirect
17. #17988985	C₂²	semidirect
19. #17988983	C₂²	semidirect
24. #17988961	C₂²	semidirect
25. #17988960	C₂²	semidirect
26. #17988959	C₂²	semidirect
28. #17988935	C₂²	semidirect
29. #17988934	C₂²	semidirect
30. #17988933	C₂²	semidirect
36. #17988309	C₂²	semidirect
37. #17988308	C₂²	semidirect
38. #498×C₇₃	C₂²	semidirect
40. #492×C₇₃	C₂²	semidirect
41. #486×C₇₃	C₂²	semidirect
42. #322×C₇₃	C₂²	semidirect
118. C₈×D₇₃	SD₁₆	semidirect
171. C₈×C₇₃	C₂×SD₁₆	semidirect
173. #10596831	C₂×SD₁₆	semidirect
200. D₇₃	#1715	direct (N × H)
210. #1715	D₇₃	direct (N × H)
212. C₇₃	C₂×#1715	semidirect
221. #498	C₂×D₇₃	semidirect
231. #492	C₂×D₇₃	semidirect
236. #486	C₂×D₇₃	semidirect
243. #322	C₂×D₇₃	semidirect
374. C₈	SD₁₆×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#498×D₇₃	2
#493×D₇₃	2
#492×D₇₃	2
#486×D₇₃	2
#22609561	2
#22609560	2
#22609559	2
#22609558	2
#22609557	2
#322×D₇₃	2
C₇₃×#1715	2
C₂×#1715	73

Subgroups — #1715×D73 #33073701

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1715×D₇₃ #33073701