Subgroups — #2219×D₇₃ #33240432

Order 18688.

682 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₃₂×D₇₃	C₂	1	1	yes	maximal, characteristic
3	#467×D₇₃	C₂	1	2	yes	maximal
4	#22612040	C₂	1	1	yes	maximal, characteristic, complemented
5	#22612039	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612038	C₂	1	2	yes	maximal, complemented
7	#22612037	C₂	1	2	yes	maximal, complemented
8	#22612036	C₂	1	1	yes	maximal, characteristic, complemented
9	#367×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#358×D₇₃	C₂	1	2	yes	maximal, complemented
11	C₇₃×#2219	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₃	C₂²	1	2	yes	—
13	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
14	C₂²⋊Q₈×D₇₃	—	2	2	no	—
15	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic, complemented
16	C₂×#14556537	C₂²	1	2	yes	complemented
17	Q₃₂×D₇₃	—	2	4	no	—
18	C₂×#14556531	C₂²	1	1	yes	characteristic
19	C₂×#14556530	C₂²	1	1	yes	characteristic
20	C₂×#14556530	C₂²	1	1	yes	characteristic
21	M₃₂×D₇₃	—	2	4	no	—
22	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
23	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
24	#17988829	C₂²	1	2	yes	—
25	#17988828	C₂²	1	2	yes	—
26	#17988827	—	2	2	no	—
27	#17988826	—	2	2	no	—
28	#17988825	C₂²	1	2	yes	—
29	#107×D₇₃	C₂²	1	2	yes	—
30	#103×D₇₃	—	2	2	no	—
31	#17988415	C₂²	1	1	yes	characteristic, complemented
32	#17988414	C₂²	1	2	yes	complemented
33	#17988392	C₂²	1	2	yes	complemented
34	#17988391	C₂²	1	2	yes	complemented
35	#17988390	C₂²	1	2	yes	complemented
36	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic, complemented
37	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
38	#467×C₇₃	C₂²	1	2	yes	—
39	#367×C₇₃	C₂²	1	1	yes	characteristic, complemented
40	#358×C₇₃	C₂²	1	2	yes	complemented
41	C₂×Q₈×D₇₃	—	2	2	no	—
42	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
43	C₂×Dic₂₉₂	C₂³	1	2	yes	—
44	C₂×#11924018	—	2	2	no	—
45	Q₁₆×D₇₃	—	2	4	no	—
46	Q₁₆×D₇₃	—	2	4	no	—
47	C₂²×#10596831	D₄	1	1	yes	characteristic
48	C₂×C₈×D₇₃	—	2	2	no	—
49	C₂×C₈×D₇₃	—	2	2	no	—
50	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
51	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
52	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
53	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
54	#14556571	—	2	2	no	—
55	#14556570	—	2	2	no	—
56	#14556569	—	4	2	no	—
57	#14556568	—	4	2	no	—
58	#14556567	—	2	2	no	—
59	C₄⋊C₄×D₇₃	—	2	2	no	—
60	C₄⋊C₄×D₇₃	—	4	2	no	—
61	C₂²⋊C₄×D₇₃	—	4	2	no	—
62	Dic₅₈₄	—	2	2	no	—
63	#14556537	—	2	4	no	—
64	#14556531	—	2	2	no	—
65	#14556530	—	2	2	no	—
66	#14556530	—	2	2	no	—
67	C₂×#11924003	C₂³	1	1	yes	characteristic
68	C₂×#11924003	C₂³	1	1	yes	characteristic
69	C₁₆×D₇₃	—	2	2	no	—
70	C₁₆×D₇₃	D₄	1	4	yes	—
71	#14556525	C₂³	1	2	yes	—
72	#14556524	—	2	2	no	—
73	#14556516	—	2	2	no	—
74	#14556515	—	2	2	no	—
75	#14556514	—	2	2	no	—
76	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
77	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
78	C₂²⋊Q₈×C₇₃	—	2	2	no	—
79	Q₃₂×C₇₃	—	2	2	no	—
80	M₃₂×C₇₃	—	2	2	no	—
81	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
83	#107×C₇₃	C₂³	1	2	yes	—
84	#103×C₇₃	—	2	2	no	—
85	C₂³×D₇₃	D₈	1	1	yes	characteristic
86	C₂×Dic₁₄₆	—	2	2	no	—
87	C₂×Dic₁₄₆	—	4	2	no	—
88	Q₈×D₇₃	—	4	4	no	—
89	Q₈×D₇₃	—	4	4	no	—
90	C₂²×Dic₇₃	D₈	1	1	yes	characteristic
91	C₂×C₄×D₇₃	—	2	2	no	—
92	C₂×C₄×D₇₃	—	2	2	no	—
93	C₂×C₄×D₇₃	—	4	2	no	—
94	C₂×C₄×D₇₃	—	4	2	no	—
95	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
96	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
97	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
99	Dic₂₉₂	—	2	2	no	—
100	Dic₂₉₂	—	2	2	no	—
101	#11924018	—	4	4	no	—
102	C₂×#10596831	—	2	2	no	—
103	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
104	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
105	C₈×D₇₃	—	2	2	no	—
106	C₈×D₇₃	—	2	1	no	—
107	C₈×D₇₃	—	2	1	no	—
108	C₈×D₇₃	—	2	2	no	—
109	C₈×D₇₃	C₂×D₄	1	2	yes	—
110	C₈×D₇₃	#41	1	2	yes	—
111	C₄⋊Dic₇₃	—	2	2	no	—
112	C₄⋊Dic₇₃	—	4	2	no	—
113	#11924007	—	4	2	no	—
114	#11924007	—	4	2	no	—
115	#11924007	—	4	2	no	—
116	C₂²⋊Dic₇₃	—	4	2	no	—
117	#11924005	—	4	2	no	—
118	#11924005	—	4	2	no	—
119	#11924003	—	2	1	no	—
120	#11924003	C₂×D₄	1	2	yes	—
121	C₂×Q₈×C₇₃	—	2	2	no	—
122	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
123	Q₁₆×C₇₃	—	2	2	no	—
124	Q₁₆×C₇₃	—	2	2	no	—
125	C₂×C₈×C₇₃	—	2	1	no	—
126	C₂×C₈×C₇₃	—	2	1	no	—
127	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
129	C₄⋊C₄×C₇₃	—	2	2	no	—
130	C₄⋊C₄×C₇₃	—	4	2	no	—
131	C₂²⋊C₄×C₇₃	—	4	2	no	—
132	C₁₆×C₇₃	—	2	1	no	—
133	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
134	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
135	C₂²×D₇₃	—	2	2	no	—
136	C₂²×D₇₃	—	2	4	no	—
137	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
138	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
139	Q₈×C₇₃	—	4	2	no	—
140	Q₈×C₇₃	—	4	2	no	—
141	C₂×C₄×C₇₃	—	2	1	no	—
142	C₂×C₄×C₇₃	—	2	1	no	—
143	C₂×C₄×C₇₃	—	4	2	no	—
144	C₂×C₄×C₇₃	—	4	2	no	—
145	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×Dic₇₃	—	2	2	no	—
148	C₂×Dic₇₃	—	4	2	no	—
149	C₂×Dic₇₃	—	4	2	no	—
150	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
151	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
152	C₄×D₇₃	—	2	2	no	—
153	C₄×D₇₃	—	2	1	no	—
154	C₄×D₇₃	—	2	1	no	—
155	C₄×D₇₃	—	2	2	no	—
156	C₄×D₇₃	—	8	4	no	—
157	C₄×D₇₃	—	8	4	no	—
158	C₄×D₇₃	#121	1	2	yes	—
159	C₄×D₇₃	#135	1	2	yes	—
160	Dic₁₄₆	—	4	2	no	—
161	Dic₁₄₆	—	4	2	no	—
162	Dic₁₄₆	—	8	4	no	—
163	C₈×C₇₃	—	2	1	no	—
164	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
165	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
166	#10596831	—	2	1	no	—
167	#10596831	#121	1	2	yes	—
168	C₂²×C₇₃	—	2	2	no	—
169	C₂²×C₇₃	—	2	1	no	—
170	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
171	C₂×D₇₃	—	2	2	no	—
172	C₂×D₇₃	—	2	1	no	—
173	C₂×D₇₃	—	4	4	no	—
174	C₂×D₇₃	#466	1	2	yes	—
175	C₂×D₇₃	#510	1	4	yes	—
176	C₄×C₇₃	—	2	1	no	—
177	C₄×C₇₃	—	8	2	no	—
178	C₄×C₇₃	—	8	2	no	—
179	C₄×C₇₃	C₂×#121	1	1	yes	characteristic
180	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
181	Dic₇₃	—	2	1	no	—
182	Dic₇₃	—	8	2	no	—
183	Dic₇₃	—	8	2	no	—
184	Dic₇₃	#466	1	2	yes	—
185	C₂×#2219	—	73	1	no	maximal, Sylow
186	C₂×C₇₃	—	4	1	no	—
187	C₂×C₇₃	C₂×#466	1	1	yes	characteristic
188	C₂×C₇₃	C₂×#510	1	2	yes	—
189	D₇₃	—	4	1	no	—
190	D₇₃	#2219	1	4	yes	direct product factor
191	C₂²×Q₃₂	—	73	1	no	—
192	C₂²×M₃₂	—	73	1	no	—
193	C₂×#467	—	73	2	no	—
194	#2219	—	73	1	no	—
195	#2219	—	73	1	no	—
196	#2219	—	73	2	no	—
197	#2219	—	73	1	no	—
198	#2219	—	73	2	no	—
199	#2219	D₇₃	1	1	yes	characteristic, direct product factor
200	C₂×#367	—	73	1	no	—
201	C₂×#358	—	73	2	no	—
202	C₇₃	C₂×#2219	1	1	yes	characteristic, complemented, Sylow
203	C₂²×Q₁₆	—	73	2	no	—
204	C₂³×C₈	—	73	1	no	—
205	C₂×C₂²⋊Q₈	—	146	2	no	—
206	C₂×Q₃₂	—	73	2	no	—
207	C₂×Q₃₂	—	73	1	no	—
208	C₂×Q₃₂	—	146	4	no	—
209	C₂×Q₃₂	C₂×D₇₃	1	1	yes	characteristic, complemented
210	C₂×M₃₂	—	73	1	no	—
211	C₂×M₃₂	—	73	1	no	—
212	C₂×M₃₂	—	73	1	no	—
213	C₂×M₃₂	—	146	4	no	—
214	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
215	C₂²×C₁₆	—	73	1	no	—
216	C₂²×C₁₆	—	73	1	no	—
217	#467	—	73	2	no	—
218	#467	—	73	2	no	—
219	#467	—	73	2	no	—
220	#467	—	146	2	no	—
221	#467	—	146	2	no	—
222	#467	C₂×D₇₃	1	2	yes	—
223	C₂×#107	—	73	2	no	—
224	C₂×#103	—	146	2	no	—
225	#367	—	73	1	no	—
226	#367	—	73	2	no	—
227	#367	C₂×D₇₃	1	1	yes	characteristic, complemented
228	#358	—	73	2	no	—
229	#358	—	73	2	no	—
230	#358	—	73	2	no	—
231	#358	C₂×D₇₃	1	2	yes	complemented
232	C₂²×Q₈	—	146	2	no	—
233	C₂³×C₄	—	73	1	no	—
234	C₂×Q₁₆	—	73	2	no	—
235	C₂×Q₁₆	—	146	4	no	—
236	C₂×Q₁₆	—	146	4	no	—
237	C₂×Q₁₆	—	146	2	no	—
238	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
239	C₂²×C₈	—	73	1	no	—
240	C₂²×C₈	—	73	1	no	—
241	C₂²×C₈	—	73	1	no	—
242	C₂²×C₈	—	73	1	no	—
243	C₂²×C₈	—	73	1	no	—
244	C₂²×C₈	—	146	2	no	—
245	C₂²×C₈	—	146	2	no	—
246	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
247	C₂²⋊Q₈	—	2	2	no	—
248	C₂²⋊Q₈	—	292	2	no	—
249	C₂²⋊Q₈	—	292	2	no	—
250	C₂²⋊Q₈	—	146	2	no	—
251	C₂²⋊Q₈	—	146	2	no	—
252	C₂²⋊Q₈	—	146	2	no	—
253	C₂×C₄⋊C₄	—	292	2	no	—
254	C₂×C₄⋊C₄	—	146	2	no	—
255	C₂×C₂²⋊C₄	—	292	2	no	—
256	Q₃₂	—	2	2	no	—
257	Q₃₂	—	146	4	no	—
258	Q₃₂	—	146	2	no	—
259	M₃₂	—	2	2	no	—
260	M₃₂	—	146	2	no	—
261	M₃₂	—	146	2	no	—
262	M₃₂	—	146	2	no	—
263	C₂×C₁₆	—	73	1	no	—
264	C₂×C₁₆	—	73	4	no	—
265	C₂×C₁₆	—	73	1	no	—
266	C₂×C₁₆	—	146	2	no	—
267	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
268	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
269	#107	—	73	2	no	—
270	#107	—	146	2	no	—
271	#107	C₂²×D₇₃	1	2	yes	—
272	#103	—	2	2	no	—
273	#103	—	146	2	no	—
274	#103	—	146	2	no	—
275	#103	—	146	2	no	—
276	C₂⁴	—	73	1	no	—
277	C₂×Q₈	—	2	2	no	—
278	C₂×Q₈	—	292	2	no	—
279	C₂×Q₈	—	292	4	no	—
280	C₂×Q₈	—	292	4	no	—
281	C₂×Q₈	—	146	2	no	—
282	C₂²×C₄	—	292	2	no	—
283	C₂²×C₄	—	292	2	no	—
284	C₂²×C₄	—	73	1	no	—
285	C₂²×C₄	—	73	1	no	—
286	C₂²×C₄	—	73	1	no	—
287	C₂²×C₄	—	73	1	no	—
288	C₂²×C₄	—	73	1	no	—
289	C₂²×C₄	—	146	2	no	—
290	C₂²×C₄	—	146	2	no	—
291	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
292	Q₁₆	—	2	2	no	—
293	Q₁₆	—	2	2	no	—
294	Q₁₆	—	292	4	no	—
295	Q₁₆	—	146	2	no	—
296	Q₁₆	—	146	2	no	—
297	C₂×C₈	—	2	1	no	—
298	C₂×C₈	—	2	1	no	—
299	C₂×C₈	—	73	1	no	—
300	C₂×C₈	—	73	2	no	—
301	C₂×C₈	—	73	2	no	—
302	C₂×C₈	—	73	1	no	—
303	C₂×C₈	—	146	2	no	—
304	C₂×C₈	—	146	2	no	—
305	C₂×C₈	—	146	2	no	—
306	C₂×C₈	—	146	1	no	—
307	C₂×C₈	—	146	1	no	—
308	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
309	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	4	2	no	—
312	C₄⋊C₄	—	292	2	no	—
313	C₄⋊C₄	—	292	2	no	—
314	C₄⋊C₄	—	292	2	no	—
315	C₄⋊C₄	—	292	2	no	—
316	C₄⋊C₄	—	146	2	no	—
317	C₂²⋊C₄	—	4	2	no	—
318	C₂²⋊C₄	—	292	2	no	—
319	C₂²⋊C₄	—	292	2	no	—
320	C₂²⋊C₄	—	292	2	no	—
321	C₁₆	—	2	1	no	—
322	C₁₆	—	73	2	no	—
323	C₁₆	—	146	1	no	—
324	C₁₆	D₄×D₇₃	1	2	yes	—
325	C₂³	—	73	1	no	—
326	C₂³	—	73	1	no	—
327	C₂³	—	146	2	no	—
328	C₂³	—	146	4	no	—
329	C₂³	D₈×D₇₃	1	1	yes	characteristic
330	Q₈	—	4	2	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	292	2	no	—
333	Q₈	—	292	2	no	—
334	Q₈	—	584	4	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	292	2	no	—
340	C₂×C₄	—	292	2	no	—
341	C₂×C₄	—	584	4	no	—
342	C₂×C₄	—	584	4	no	—
343	C₂×C₄	—	73	1	no	—
344	C₂×C₄	—	73	2	no	—
345	C₂×C₄	—	73	2	no	—
346	C₂×C₄	—	73	1	no	—
347	C₂×C₄	—	146	2	no	—
348	C₂×C₄	—	146	2	no	—
349	C₂×C₄	—	146	2	no	—
350	C₂×C₄	—	146	1	no	—
351	C₂×C₄	—	146	1	no	—
352	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
354	C₈	—	2	1	no	—
355	C₈	—	73	2	no	—
356	C₈	—	146	1	no	—
357	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
358	C₈	#41×D₇₃	1	1	yes	characteristic
359	C₂²	—	2	2	no	—
360	C₂²	—	2	1	no	—
361	C₂²	—	292	4	no	—
362	C₂²	—	73	4	no	—
363	C₂²	—	73	2	no	—
364	C₂²	—	146	2	no	—
365	C₂²	—	146	1	no	—
366	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	8	2	no	—
369	C₄	—	8	2	no	—
370	C₄	—	584	2	no	—
371	C₄	—	584	2	no	—
372	C₄	—	73	2	no	—
373	C₄	—	146	1	no	—
374	C₄	#121×D₇₃	1	1	yes	characteristic
375	C₄	#135×D₇₃	1	1	yes	characteristic
376	C₂	—	4	1	no	—
377	C₂	—	292	1	no	—
378	C₂	—	73	4	no	—
379	C₂	#466×D₇₃	1	1	yes	characteristic
380	C₂	#510×D₇₃	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2219	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂³×D₇₃	D₈
hypercenter	#2219	D₇₃
hypocenter	C₇₃	C₂×#2219
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2219

Complements

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

N	H ≅ G/N	type
1. C₂×Q₃₂×D₇₃	C₂	semidirect
4. #22612040	C₂	semidirect
5. #22612039	C₂	semidirect
6. #22612038	C₂	semidirect
7. #22612037	C₂	semidirect
8. #22612036	C₂	semidirect
9. #367×D₇₃	C₂	semidirect
10. #358×D₇₃	C₂	semidirect
11. C₇₃×#2219	C₂	semidirect
15. C₂×Dic₅₈₄	C₂²	semidirect
16. C₂×#14556537	C₂²	semidirect
31. #17988415	C₂²	semidirect
32. #17988414	C₂²	semidirect
33. #17988392	C₂²	semidirect
34. #17988391	C₂²	semidirect
35. #17988390	C₂²	semidirect
36. C₂×Q₃₂×C₇₃	C₂²	semidirect
39. #367×C₇₃	C₂²	semidirect
40. #358×C₇₃	C₂²	semidirect
190. D₇₃	#2219	direct (N × H)
199. #2219	D₇₃	direct (N × H)
202. C₇₃	C₂×#2219	semidirect
209. C₂×Q₃₂	C₂×D₇₃	semidirect
227. #367	C₂×D₇₃	semidirect
231. #358	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×D₇₃	2
C₂×M₃₂×D₇₃	2
#467×D₇₃	2
#22612040	2
#22612039	2
#22612038	2
#22612037	2
#22612036	2
#367×D₇₃	2
#358×D₇₃	2
C₇₃×#2219	2
C₂×#2219	73

Subgroups — #2219×D73 #33240432

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2219×D₇₃ #33240432