Subgroups — #1708×D₇₃ #33073635

Order 18688.

866 conjugacy classes of subgroups.

The subgroup lattice has 372 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	#492×D₇₃	C₂	1	1	yes	maximal, characteristic
3	#487×D₇₃	C₂	1	2	yes	maximal, complemented
4	#486×D₇₃	C₂	1	2	yes	maximal
5	#22609534	C₂	1	2	yes	maximal, complemented
6	#22609533	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609532	C₂	1	1	yes	maximal, characteristic, complemented
8	#22609531	C₂	1	2	yes	maximal, complemented
9	#22609530	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈²×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#1708	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×SD₁₆×D₇₃	—	2	2	no	—
13	C₄⋊Q₈×D₇₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×D₇₃	C₂²	1	1	yes	characteristic
15	#127×D₇₃	C₂²	1	1	yes	characteristic
16	#17988985	C₂²	1	1	yes	characteristic, complemented
17	#17988984	—	2	2	no	—
18	#17988983	C₂²	1	2	yes	complemented
19	#17988961	C₂²	1	1	yes	characteristic
20	#17988960	C₂²	1	1	yes	characteristic
21	#17988959	C₂²	1	1	yes	characteristic
22	#17988958	—	2	2	no	—
23	#17988939	C₂²	1	2	yes	complemented
24	#17988938	C₂²	1	2	yes	complemented
25	#17988937	C₂²	1	2	yes	complemented
26	#17988936	—	2	4	no	—
27	#17988935	C₂²	1	2	yes	—
28	#17988934	C₂²	1	2	yes	—
29	#17988933	C₂²	1	2	yes	—
30	#17988932	—	2	4	no	—
31	#106×D₇₃	—	2	2	no	—
32	#103×D₇₃	—	2	4	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	C₈×#10596831	C₂²	1	1	yes	characteristic, complemented
37	C₈×#10596831	C₂²	1	2	yes	complemented
38	#498×C₇₃	C₂²	1	1	yes	characteristic, complemented
39	#492×C₇₃	C₂²	1	1	yes	characteristic
40	#487×C₇₃	C₂²	1	2	yes	complemented
41	#486×C₇₃	C₂²	1	2	yes	—
42	C₈²×C₇₃	C₂²	1	1	yes	characteristic, complemented
43	C₂×Q₈×D₇₃	—	2	2	no	—
44	C₂×Q₈×D₇₃	—	2	1	no	—
45	C₂×D₄×D₇₃	—	2	2	no	—
46	C₂×D₄×D₇₃	—	2	1	no	—
47	C₂×#11924017	—	2	2	no	—
48	C₂×#11924016	—	2	2	no	—
49	C₂×#11924015	—	2	2	no	—
50	SD₁₆×D₇₃	—	4	8	no	—
51	C₂×C₈×D₇₃	D₄	1	4	yes	—
52	C₂×C₈×D₇₃	D₄	1	2	yes	—
53	#14556591	C₂³	1	2	yes	—
54	#14556591	C₂³	1	1	yes	characteristic
55	#14556590	—	2	2	no	—
56	#14556590	—	2	2	no	—
57	#14556590	—	2	2	no	—
58	#14556589	—	2	2	no	—
59	#14556589	—	2	1	no	—
60	#14556588	C₂³	1	1	yes	characteristic
61	#14556587	—	2	2	no	—
62	#14556587	—	2	1	no	—
63	C₄⋊C₄×D₇₃	—	2	2	no	—
64	C₄⋊C₄×D₇₃	—	2	2	no	—
65	C₄⋊C₄×D₇₃	—	2	2	no	—
66	C₄²×D₇₃	C₂³	1	1	yes	characteristic
67	#14556523	—	2	2	no	—
68	#14556522	—	2	4	no	—
69	#14556516	—	2	4	no	—
70	#14556515	—	2	4	no	—
71	#14556514	—	2	4	no	—
72	#14556513	—	2	4	no	—
73	#14556512	—	2	4	no	—
74	#14556511	—	2	4	no	—
75	C₄×#10596831	C₂³	1	2	yes	—
76	C₄×#10596831	C₂³	1	1	yes	characteristic
77	C₈×Dic₇₃	D₄	1	4	yes	—
78	C₈×Dic₇₃	D₄	1	2	yes	—
79	C₂×SD₁₆×C₇₃	—	2	2	no	—
80	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
81	C₄⋊Q₈×C₇₃	C₂³	1	1	yes	characteristic
82	#127×C₇₃	C₂³	1	1	yes	characteristic
83	#106×C₇₃	—	2	2	no	—
84	#103×C₇₃	—	2	4	no	—
85	#102×C₇₃	—	2	4	no	—
86	C₄×C₈×C₇₃	C₂³	1	2	yes	—
87	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
88	C₂³×D₇₃	—	4	1	no	—
89	C₂×Dic₁₄₆	—	2	2	no	—
90	C₂×Dic₁₄₆	—	2	1	no	—
91	C₂×Dic₁₄₆	—	4	1	no	—
92	C₂×Dic₁₄₆	—	4	2	no	—
93	Q₈×D₇₃	—	4	4	no	—
94	Q₈×D₇₃	—	4	8	no	—
95	C₂×D₂₉₂	—	2	2	no	—
96	C₂×D₂₉₂	—	2	1	no	—
97	C₂×C₂²⋊D₇₃	—	4	1	no	—
98	C₂×C₂²⋊D₇₃	—	4	2	no	—
99	D₄×D₇₃	—	4	4	no	—
100	D₄×D₇₃	—	4	8	no	—
101	C₂×C₄×D₇₃	—	4	1	no	—
102	C₂×C₄×D₇₃	—	4	2	no	—
103	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
105	#11924017	—	4	4	no	—
106	#11924016	—	4	4	no	—
107	#11924015	—	4	4	no	—
108	C₂×#10596831	C₂×D₄	1	4	yes	—
109	C₂×#10596831	C₂×D₄	1	2	yes	—
110	C₈×D₇₃	—	2	8	no	—
111	C₈×D₇₃	SD₁₆	1	8	yes	—
112	C₄⋊Dic₇₃	—	2	2	no	—
113	C₄⋊Dic₇₃	—	2	2	no	—
114	C₄⋊Dic₇₃	—	2	2	no	—
115	#11924007	—	4	2	no	—
116	#11924007	—	4	2	no	—
117	#11924007	—	4	2	no	—
118	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
119	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
120	C₂×Q₈×C₇₃	—	2	2	no	—
121	C₂×Q₈×C₇₃	—	2	1	no	—
122	C₂×D₄×C₇₃	—	2	2	no	—
123	C₂×D₄×C₇₃	—	2	1	no	—
124	SD₁₆×C₇₃	—	4	4	no	—
125	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
126	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
127	C₄⋊C₄×C₇₃	—	2	2	no	—
128	C₄⋊C₄×C₇₃	—	2	2	no	—
129	C₄⋊C₄×C₇₃	—	2	2	no	—
130	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
131	C₂³×C₇₃	—	4	1	no	—
132	C₂²×D₇₃	—	4	1	no	—
133	C₂²×D₇₃	—	8	2	no	—
134	C₂²×D₇₃	—	8	4	no	—
135	C₂²×D₇₃	#127	1	1	yes	characteristic
136	Q₈×C₇₃	—	4	2	no	—
137	Q₈×C₇₃	—	4	4	no	—
138	D₄×C₇₃	—	4	2	no	—
139	D₄×C₇₃	—	4	4	no	—
140	C₂×C₄×C₇₃	—	4	1	no	—
141	C₂×C₄×C₇₃	—	4	2	no	—
142	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
144	C₂²⋊D₇₃	—	8	2	no	—
145	C₂²⋊D₇₃	—	8	4	no	—
146	C₂×Dic₇₃	—	4	1	no	—
147	C₂×Dic₇₃	—	4	2	no	—
148	C₂×Dic₇₃	#127	1	1	yes	characteristic
149	C₂×Dic₇₃	#127	1	2	yes	—
150	D₂₉₂	—	4	2	no	—
151	D₂₉₂	—	4	4	no	—
152	C₄×D₇₃	—	8	2	no	—
153	C₄×D₇₃	—	8	4	no	—
154	C₄×D₇₃	C₂×SD₁₆	1	4	yes	—
155	C₄×D₇₃	#135	1	8	yes	—
156	Dic₁₄₆	—	4	2	no	—
157	Dic₁₄₆	—	4	4	no	—
158	Dic₁₄₆	—	8	2	no	—
159	Dic₁₄₆	—	8	4	no	—
160	C₈×C₇₃	—	2	4	no	—
161	C₈×C₇₃	C₂×SD₁₆	1	4	yes	—
162	#10596831	—	2	4	no	—
163	#10596831	C₂×SD₁₆	1	4	yes	—
164	C₂²×C₇₃	—	8	1	no	—
165	C₂²×C₇₃	—	8	2	no	—
166	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
167	C₂×D₇₃	—	8	1	no	—
168	C₂×D₇₃	—	8	2	no	—
169	C₂×D₇₃	—	16	4	no	—
170	C₂×D₇₃	#492	1	2	yes	—
171	C₂×D₇₃	#495	1	4	yes	—
172	C₄×C₇₃	—	8	1	no	—
173	C₄×C₇₃	—	8	2	no	—
174	C₄×C₇₃	C₂²×SD₁₆	1	2	yes	—
175	C₄×C₇₃	C₂×#135	1	4	yes	—
176	Dic₇₃	—	8	1	no	—
177	Dic₇₃	—	8	2	no	—
178	Dic₇₃	#492	1	2	yes	—
179	Dic₇₃	#495	1	4	yes	—
180	C₂×#1708	—	73	1	no	maximal, Sylow
181	C₂×C₇₃	—	16	1	no	—
182	C₂×C₇₃	C₂×#492	1	1	yes	characteristic
183	C₂×C₇₃	C₂×#495	1	2	yes	—
184	D₇₃	—	16	1	no	—
185	D₇₃	#1708	1	4	yes	direct product factor
186	C₂×#498	—	73	1	no	—
187	C₂×#492	—	73	1	no	—
188	C₂×#487	—	73	2	no	—
189	C₂×#486	—	73	2	no	—
190	#1708	—	73	1	no	—
191	#1708	—	73	1	no	—
192	#1708	—	73	2	no	—
193	#1708	—	73	2	no	—
194	#1708	—	73	1	no	—
195	#1708	D₇₃	1	1	yes	characteristic, direct product factor
196	C₂×C₈²	—	73	1	no	—
197	C₇₃	C₂×#1708	1	1	yes	characteristic, complemented, Sylow
198	C₂²×SD₁₆	—	146	2	no	—
199	C₂×C₄⋊Q₈	—	73	2	no	—
200	C₂×C₄⋊Q₈	—	73	1	no	—
201	C₂×#127	—	73	1	no	—
202	#498	—	73	2	no	—
203	#498	—	73	1	no	—
204	#498	—	146	2	no	—
205	#498	C₂×D₇₃	1	1	yes	characteristic, complemented
206	#492	—	73	1	no	—
207	#492	—	73	1	no	—
208	#492	—	73	1	no	—
209	#492	—	146	2	no	—
210	#492	C₂×D₇₃	1	1	yes	characteristic
211	#487	—	73	2	no	—
212	#487	—	73	2	no	—
213	#487	—	73	2	no	—
214	#487	—	146	4	no	—
215	#487	C₂×D₇₃	1	2	yes	complemented
216	#486	—	73	2	no	—
217	#486	—	73	2	no	—
218	#486	—	73	2	no	—
219	#486	—	146	4	no	—
220	#486	C₂×D₇₃	1	2	yes	—
221	C₂×#106	—	146	2	no	—
222	C₂×#103	—	146	4	no	—
223	C₂×#102	—	146	4	no	—
224	C₂×C₄×C₈	—	73	2	no	—
225	C₂×C₄×C₈	—	73	1	no	—
226	C₈²	—	73	1	no	—
227	C₈²	—	73	2	no	—
228	C₈²	C₂×D₇₃	1	1	yes	characteristic, complemented
229	C₂²×Q₈	—	146	1	no	—
230	C₂²×Q₈	—	146	2	no	—
231	C₂²×D₄	—	146	1	no	—
232	C₂²×D₄	—	146	2	no	—
233	C₂×SD₁₆	—	2	2	no	—
234	C₂×SD₁₆	—	292	8	no	—
235	C₂×SD₁₆	—	146	2	no	—
236	C₂×SD₁₆	—	146	2	no	—
237	C₂×SD₁₆	—	146	2	no	—
238	C₂²×C₈	—	73	2	no	—
239	C₂²×C₈	—	73	4	no	—
240	C₄⋊Q₈	—	73	2	no	—
241	C₄⋊Q₈	—	73	1	no	—
242	C₄⋊Q₈	—	146	2	no	—
243	C₄⋊Q₈	—	146	2	no	—
244	C₄⋊Q₈	—	146	2	no	—
245	C₄⋊Q₈	—	146	2	no	—
246	C₄⋊Q₈	—	146	1	no	—
247	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
248	C₄⋊Q₈	C₂²×D₇₃	1	1	yes	characteristic
249	#127	—	73	1	no	—
250	#127	—	146	2	no	—
251	#127	—	146	1	no	—
252	#127	C₂²×D₇₃	1	1	yes	characteristic
253	C₂×C₄⋊C₄	—	146	2	no	—
254	C₂×C₄⋊C₄	—	146	2	no	—
255	C₂×C₄⋊C₄	—	146	2	no	—
256	C₂×C₄²	—	73	1	no	—
257	#106	—	2	2	no	—
258	#106	—	146	4	no	—
259	#106	—	146	2	no	—
260	#103	—	2	4	no	—
261	#103	—	146	4	no	—
262	#103	—	146	4	no	—
263	#103	—	146	4	no	—
264	#102	—	2	4	no	—
265	#102	—	146	4	no	—
266	#102	—	146	4	no	—
267	#102	—	146	4	no	—
268	C₄×C₈	—	73	4	no	—
269	C₄×C₈	—	73	2	no	—
270	C₄×C₈	—	73	2	no	—
271	C₄×C₈	—	73	1	no	—
272	C₄×C₈	C₂²×D₇₃	1	2	yes	—
273	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
274	C₂⁴	—	292	1	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	—	2	1	no	—
277	C₂×Q₈	—	292	2	no	—
278	C₂×Q₈	—	292	1	no	—
279	C₂×Q₈	—	292	4	no	—
280	C₂×Q₈	—	292	8	no	—
281	C₂×Q₈	—	146	2	no	—
282	C₂×Q₈	—	146	1	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	292	4	no	—
286	C₂×D₄	—	292	8	no	—
287	C₂×D₄	—	292	1	no	—
288	C₂×D₄	—	292	2	no	—
289	C₂×D₄	—	146	2	no	—
290	C₂×D₄	—	146	1	no	—
291	C₂²×C₄	—	292	2	no	—
292	C₂²×C₄	—	292	1	no	—
293	C₂²×C₄	—	73	1	no	—
294	C₂²×C₄	—	73	2	no	—
295	SD₁₆	—	4	4	no	—
296	SD₁₆	—	292	4	no	—
297	SD₁₆	—	292	4	no	—
298	SD₁₆	—	292	4	no	—
299	C₂×C₈	—	73	2	no	—
300	C₂×C₈	—	73	8	no	—
301	C₂×C₈	—	73	4	no	—
302	C₂×C₈	—	146	8	no	—
303	C₂×C₈	D₄×D₇₃	1	4	yes	—
304	C₂×C₈	D₄×D₇₃	1	2	yes	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	2	2	no	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	292	2	no	—
309	C₄⋊C₄	—	292	2	no	—
310	C₄⋊C₄	—	292	2	no	—
311	C₄⋊C₄	—	146	2	no	—
312	C₄⋊C₄	—	146	2	no	—
313	C₄⋊C₄	—	146	2	no	—
314	C₄²	—	73	1	no	—
315	C₄²	—	73	2	no	—
316	C₄²	C₂³×D₇₃	1	1	yes	characteristic
317	C₂³	—	4	1	no	—
318	C₂³	—	292	1	no	—
319	C₂³	—	584	2	no	—
320	C₂³	—	584	4	no	—
321	C₂³	—	73	1	no	—
322	Q₈	—	4	2	no	—
323	Q₈	—	4	4	no	—
324	Q₈	—	292	2	no	—
325	Q₈	—	292	4	no	—
326	Q₈	—	584	2	no	—
327	Q₈	—	584	4	no	—
328	D₄	—	4	2	no	—
329	D₄	—	4	4	no	—
330	D₄	—	292	2	no	—
331	D₄	—	292	4	no	—
332	D₄	—	584	2	no	—
333	D₄	—	584	4	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	292	1	no	—
337	C₂×C₄	—	292	2	no	—
338	C₂×C₄	—	584	2	no	—
339	C₂×C₄	—	584	4	no	—
340	C₂×C₄	—	73	1	no	—
341	C₂×C₄	—	73	2	no	—
342	C₂×C₄	—	73	4	no	—
343	C₂×C₄	—	73	8	no	—
344	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
345	C₂×C₄	C₂×D₄×D₇₃	1	2	yes	—
346	C₈	—	2	4	no	—
347	C₈	—	73	4	no	—
348	C₈	—	146	4	no	—
349	C₈	SD₁₆×D₇₃	1	4	yes	—
350	C₂²	—	8	1	no	—
351	C₂²	—	8	2	no	—
352	C₂²	—	584	1	no	—
353	C₂²	—	584	2	no	—
354	C₂²	—	73	2	no	—
355	C₂²	—	73	4	no	—
356	C₂²	—	1168	4	no	—
357	C₂²	#127×D₇₃	1	1	yes	characteristic
358	C₄	—	8	1	no	—
359	C₄	—	8	2	no	—
360	C₄	—	584	1	no	—
361	C₄	—	584	2	no	—
362	C₄	—	73	2	no	—
363	C₄	—	73	4	no	—
364	C₄	C₂×SD₁₆×D₇₃	1	2	yes	—
365	C₄	#135×D₇₃	1	4	yes	—
366	C₂	—	16	1	no	—
367	C₂	—	73	4	no	—
368	C₂	—	1168	1	no	—
369	C₂	#492×D₇₃	1	1	yes	characteristic
370	C₂	#495×D₇₃	1	2	yes	—

	subgroup	G/H
center	C₂²	#127×D₇₃
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1708	C₂
socle	C₂²×C₇₃	C₂×#127
prime radical	#127×D₇₃	C₂²
hypercenter	#1708	D₇₃
hypocenter	C₇₃	C₂×#1708
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1708

Complements

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

N	H ≅ G/N	type
1. #498×D₇₃	C₂	semidirect
3. #487×D₇₃	C₂	semidirect
5. #22609534	C₂	semidirect
6. #22609533	C₂	semidirect
7. #22609532	C₂	semidirect
8. #22609531	C₂	semidirect
9. #22609530	C₂	semidirect
10. C₈²×D₇₃	C₂	semidirect
11. C₇₃×#1708	C₂	semidirect
16. #17988985	C₂²	semidirect
18. #17988983	C₂²	semidirect
23. #17988939	C₂²	semidirect
24. #17988938	C₂²	semidirect
25. #17988937	C₂²	semidirect
36. C₈×#10596831	C₂²	semidirect
37. C₈×#10596831	C₂²	semidirect
38. #498×C₇₃	C₂²	semidirect
40. #487×C₇₃	C₂²	semidirect
42. C₈²×C₇₃	C₂²	semidirect
185. D₇₃	#1708	direct (N × H)
195. #1708	D₇₃	direct (N × H)
197. C₇₃	C₂×#1708	semidirect
205. #498	C₂×D₇₃	semidirect
215. #487	C₂×D₇₃	semidirect
228. C₈²	C₂×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
#492×D₇₃	2
#487×D₇₃	2
#486×D₇₃	2
#22609534	2
#22609533	2
#22609532	2
#22609531	2
#22609530	2
C₈²×D₇₃	2
C₇₃×#1708	2
C₂×#1708	73

Subgroups — #1708×D73 #33073635

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1708×D₇₃ #33073635