Subgroups — #2110×D₇₃ #33238639

Order 18688.

738 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₇₃	C₂	1	2	yes	maximal
2	#405×D₇₃	C₂	1	1	yes	maximal, characteristic
3	#22611536	C₂	1	1	yes	maximal, characteristic, complemented
4	#22611535	C₂	1	2	yes	maximal, complemented
5	#22611534	C₂	1	4	yes	maximal, complemented
6	#347×D₇₃	C₂	1	4	yes	maximal, complemented
7	C₇₃×#2110	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×M₁₆×D₇₃	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
10	#117×D₇₃	C₄	1	1	yes	characteristic
11	C₂×#14556531	C₂²	1	2	yes	—
12	C₂×#14556530	C₄	1	4	yes	complemented
13	M₃₂×D₇₃	C₄	1	4	yes	complemented
14	M₃₂×D₇₃	C₄	1	4	yes	complemented
15	M₃₂×D₇₃	C₂²	1	4	yes	—
16	M₃₂×D₇₃	C₂²	1	4	yes	—
17	C₂×C₁₆×D₇₃	C₄	1	4	yes	complemented
18	#17988545	C₄	1	1	yes	characteristic
19	#17988544	C₂²	1	2	yes	—
20	#17988543	C₄	1	1	yes	characteristic
21	#17988542	C₂²	1	1	yes	characteristic
22	#17988541	C₄	1	2	yes	—
23	#97×D₇₃	C₂²	1	2	yes	—
24	C₄×C₈×D₇₃	C₄	1	2	yes	—
25	#17988365	C₂²	1	4	yes	complemented
26	#17988364	C₂²	1	8	yes	complemented
27	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
28	#405×C₇₃	C₂²	1	1	yes	characteristic
29	#347×C₇₃	C₂²	1	4	yes	complemented
30	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
32	C₂×#11924011	C₂×C₄	1	2	yes	—
33	M₁₆×D₇₃	—	2	2	no	—
34	M₁₆×D₇₃	—	2	2	no	—
35	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
43	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
44	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
45	#14556550	C₂×C₄	1	1	yes	characteristic
46	#14556549	—	2	1	no	—
47	#14556548	C₂×C₄	1	2	yes	—
48	#14556547	—	2	1	no	—
49	C₄⋊C₄×D₇₃	—	2	1	no	—
50	C₂²⋊C₄×D₇₃	—	2	1	no	—
51	C₄²×D₇₃	C₂×C₄	1	2	yes	—
52	#14556531	C₂×C₄	1	4	yes	complemented
53	#14556531	C₂³	1	4	yes	—
54	#14556530	C₂×C₄	1	8	yes	—
55	#14556530	C₂×C₄	1	8	yes	complemented
56	C₂×#11924003	C₂×C₄	1	4	yes	complemented
57	C₁₆×D₇₃	C₂×C₄	1	8	yes	—
58	C₁₆×D₇₃	C₂×C₄	1	8	yes	complemented
59	#14556502	C₂×C₄	1	2	yes	—
60	#14556501	C₂×C₄	1	2	yes	—
61	#14556501	C₂³	1	2	yes	—
62	C₄×#10596831	C₂×C₄	1	2	yes	—
63	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
64	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
65	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
67	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
68	M₃₂×C₇₃	C₂×C₄	1	4	yes	complemented
69	M₃₂×C₇₃	C₂³	1	4	yes	—
70	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	complemented
71	#97×C₇₃	C₂³	1	2	yes	—
72	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
73	C₂³×D₇₃	M₁₆	1	1	yes	characteristic
74	C₂²×Dic₇₃	M₁₆	1	1	yes	characteristic
75	C₂×C₄×D₇₃	—	2	2	no	—
76	C₂×C₄×D₇₃	C₄²	1	2	yes	—
77	C₂×C₄×D₇₃	C₄²	1	2	yes	—
78	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
79	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
80	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
81	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
82	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
84	#11924012	—	2	2	no	—
85	#11924011	—	2	4	no	—
86	C₂×#10596831	C₄²	1	2	yes	—
87	C₂×#10596831	C₄²	1	2	yes	—
88	C₂×#10596831	C₂²×C₄	1	2	yes	—
89	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
90	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
91	C₈×D₇₃	—	2	2	no	—
92	C₈×D₇₃	—	2	2	no	—
93	C₈×D₇₃	C₄²	1	1	yes	characteristic
94	C₈×D₇₃	C₄²	1	1	yes	characteristic
95	C₈×D₇₃	C₄²	1	2	yes	—
96	C₈×D₇₃	C₄²	1	1	yes	characteristic
97	C₈×D₇₃	C₄²	1	1	yes	characteristic
98	C₈×D₇₃	C₄²	1	2	yes	—
99	C₈×D₇₃	C₂²×C₄	1	2	yes	—
100	C₈×D₇₃	C₂²×C₄	1	2	yes	—
101	C₈×D₇₃	C₂²×C₄	1	2	yes	—
102	C₈×D₇₃	C₂²×C₄	1	2	yes	—
103	C₄⋊Dic₇₃	—	2	1	no	—
104	#11924007	—	2	2	no	—
105	C₂²⋊Dic₇₃	—	2	1	no	—
106	#11924005	—	2	2	no	—
107	C₄×Dic₇₃	—	2	2	no	—
108	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
109	#11924003	C₂²×C₄	1	8	yes	—
110	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
111	M₁₆×C₇₃	—	2	2	no	—
112	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
113	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
116	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₇₃	—	2	1	no	—
118	C₂²⋊C₄×C₇₃	—	2	1	no	—
119	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
120	C₁₆×C₇₃	C₂²×C₄	1	8	yes	—
121	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
122	C₂²×D₇₃	—	4	2	no	—
123	C₂²×D₇₃	#97	1	2	yes	—
124	C₂²×D₇₃	#97	1	2	yes	—
125	C₂²×D₇₃	C₂×M₁₆	1	1	yes	characteristic
126	C₂²×D₇₃	C₂×M₁₆	1	1	yes	characteristic
127	C₂×C₄×C₇₃	—	2	2	no	—
128	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
130	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
131	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
132	C₂×Dic₇₃	—	2	2	no	—
133	C₂×Dic₇₃	#97	1	2	yes	—
134	C₂×Dic₇₃	#97	1	2	yes	—
135	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
136	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
137	C₄×D₇₃	—	4	4	no	—
138	C₄×D₇₃	#97	1	2	yes	—
139	C₄×D₇₃	#97	1	2	yes	—
140	C₄×D₇₃	#97	1	2	yes	—
141	C₄×D₇₃	#97	1	2	yes	—
142	C₄×D₇₃	C₂×C₄²	1	2	yes	—
143	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
144	C₄×D₇₃	C₂×C₄²	1	1	yes	characteristic
145	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
146	C₄×D₇₃	C₂×M₁₆	1	1	yes	characteristic
147	C₄×D₇₃	C₂×M₁₆	1	2	yes	—
148	C₈×C₇₃	—	2	2	no	—
149	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
150	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
151	C₈×C₇₃	C₂³×C₄	1	2	yes	—
152	#10596831	—	2	2	no	—
153	#10596831	C₂×C₄²	1	2	yes	—
154	#10596831	C₂×C₄²	1	1	yes	characteristic
155	#10596831	C₂×C₄²	1	1	yes	characteristic
156	C₂²×C₇₃	—	4	1	no	—
157	C₂²×C₇₃	C₂×#97	1	2	yes	—
158	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
159	C₂×D₇₃	—	2	2	no	—
160	C₂×D₇₃	—	2	4	no	—
161	C₂×D₇₃	—	4	2	no	—
162	C₂×D₇₃	—	4	1	no	—
163	C₂×D₇₃	C₂×#97	1	1	yes	characteristic
164	C₂×D₇₃	C₂×#97	1	2	yes	—
165	C₂×D₇₃	C₂×#97	1	1	yes	characteristic
166	C₄×C₇₃	—	4	2	no	—
167	C₄×C₇₃	C₂×#97	1	2	yes	—
168	C₄×C₇₃	C₂²×C₄²	1	1	yes	characteristic
169	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
170	Dic₇₃	—	4	2	no	—
171	Dic₇₃	C₂×#97	1	1	yes	characteristic
172	Dic₇₃	C₂×#97	1	1	yes	characteristic
173	Dic₇₃	C₂×#97	1	2	yes	—
174	C₂×#2110	—	73	1	no	maximal, Sylow
175	C₂×C₇₃	—	2	1	no	—
176	C₂×C₇₃	—	2	2	no	—
177	C₂×C₇₃	C₂²×#97	1	1	yes	characteristic
178	D₇₃	—	2	1	no	—
179	D₇₃	—	2	2	no	—
180	D₇₃	#2110	1	2	yes	direct product factor
181	C₂²×M₃₂	—	73	2	no	—
182	C₂×#405	—	73	1	no	—
183	#2110	—	73	2	no	—
184	#2110	—	73	1	no	—
185	#2110	—	73	4	no	—
186	#2110	D₇₃	1	1	yes	characteristic, direct product factor
187	C₂×#347	—	73	4	no	—
188	C₇₃	C₂×#2110	1	1	yes	characteristic, complemented, Sylow
189	C₂²×M₁₆	—	73	1	no	—
190	C₂³×C₈	—	73	1	no	—
191	C₂×#117	—	73	1	no	—
192	C₂×M₃₂	—	73	2	no	—
193	C₂×M₃₂	—	73	4	no	—
194	C₂×M₃₂	—	73	4	no	—
195	C₂×M₃₂	—	73	4	no	—
196	C₂×M₃₂	—	73	4	no	—
197	C₂×M₃₂	—	73	4	no	—
198	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
199	C₂²×C₁₆	—	73	4	no	—
200	#405	—	73	1	no	—
201	#405	—	73	1	no	—
202	#405	—	73	1	no	—
203	#405	—	73	2	no	—
204	#405	—	73	2	no	—
205	#405	C₂×D₇₃	1	1	yes	characteristic
206	C₂×#97	—	73	2	no	—
207	C₂×C₄×C₈	—	73	2	no	—
208	#347	—	73	8	no	—
209	#347	—	73	4	no	—
210	#347	C₂×D₇₃	1	4	yes	complemented
211	C₂³×C₄	—	73	1	no	—
212	C₂×M₁₆	—	73	1	no	—
213	C₂×M₁₆	—	73	2	no	—
214	C₂×M₁₆	—	146	2	no	—
215	C₂×M₁₆	—	146	2	no	—
216	C₂×M₁₆	C₄×D₇₃	1	1	yes	characteristic
217	C₂²×C₈	—	73	1	no	—
218	C₂²×C₈	—	73	1	no	—
219	C₂²×C₈	—	73	1	no	—
220	C₂²×C₈	—	73	2	no	—
221	C₂²×C₈	—	73	2	no	—
222	C₂²×C₈	—	73	2	no	—
223	C₂²×C₈	—	73	2	no	—
224	C₂²×C₈	—	73	1	no	—
225	C₂²×C₈	—	73	1	no	—
226	C₂²×C₈	—	73	2	no	—
227	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
228	#117	—	73	1	no	—
229	#117	—	73	2	no	—
230	#117	—	146	1	no	—
231	#117	—	146	1	no	—
232	#117	C₄×D₇₃	1	1	yes	characteristic
233	C₂×C₄⋊C₄	—	146	1	no	—
234	C₂×C₂²⋊C₄	—	146	1	no	—
235	C₂×C₄²	—	73	2	no	—
236	M₃₂	—	73	4	no	—
237	M₃₂	—	73	4	no	—
238	M₃₂	—	73	8	no	—
239	M₃₂	—	73	8	no	—
240	M₃₂	C₄×D₇₃	1	4	yes	complemented
241	M₃₂	C₂²×D₇₃	1	4	yes	—
242	C₂×C₁₆	—	73	4	no	—
243	C₂×C₁₆	—	73	8	no	—
244	C₂×C₁₆	—	73	8	no	—
245	C₂×C₁₆	C₄×D₇₃	1	4	yes	complemented
246	#97	—	73	2	no	—
247	#97	—	73	2	no	—
248	#97	—	73	2	no	—
249	#97	C₂²×D₇₃	1	2	yes	—
250	C₄×C₈	—	73	2	no	—
251	C₄×C₈	—	73	2	no	—
252	C₄×C₈	—	73	2	no	—
253	C₄×C₈	C₄×D₇₃	1	2	yes	—
254	C₂⁴	—	73	1	no	—
255	C₂²×C₄	—	73	1	no	—
256	C₂²×C₄	—	73	2	no	—
257	C₂²×C₄	—	73	2	no	—
258	C₂²×C₄	—	73	1	no	—
259	C₂²×C₄	—	73	1	no	—
260	C₂²×C₄	—	73	2	no	—
261	C₂²×C₄	—	73	2	no	—
262	C₂²×C₄	—	73	1	no	—
263	C₂²×C₄	—	73	1	no	—
264	C₂²×C₄	—	146	2	no	—
265	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
266	M₁₆	—	2	2	no	—
267	M₁₆	—	146	4	no	—
268	M₁₆	—	146	2	no	—
269	C₂×C₈	—	73	2	no	—
270	C₂×C₈	—	73	2	no	—
271	C₂×C₈	—	73	1	no	—
272	C₂×C₈	—	73	1	no	—
273	C₂×C₈	—	73	1	no	—
274	C₂×C₈	—	73	1	no	—
275	C₂×C₈	—	73	2	no	—
276	C₂×C₈	—	73	1	no	—
277	C₂×C₈	—	73	1	no	—
278	C₂×C₈	—	73	2	no	—
279	C₂×C₈	—	73	2	no	—
280	C₂×C₈	—	73	2	no	—
281	C₂×C₈	—	73	2	no	—
282	C₂×C₈	—	73	2	no	—
283	C₂×C₈	—	73	2	no	—
284	C₂×C₈	—	146	2	no	—
285	C₂×C₈	—	146	2	no	—
286	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
287	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
288	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
289	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
290	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	146	2	no	—
293	C₄⋊C₄	—	146	1	no	—
294	C₂²⋊C₄	—	2	1	no	—
295	C₂²⋊C₄	—	146	2	no	—
296	C₂²⋊C₄	—	146	1	no	—
297	C₄²	—	73	2	no	—
298	C₄²	—	146	2	no	—
299	C₄²	C₂×C₄×D₇₃	1	2	yes	—
300	C₁₆	—	73	8	no	—
301	C₁₆	C₂×C₄×D₇₃	1	8	yes	—
302	C₂³	—	292	2	no	—
303	C₂³	—	73	2	no	—
304	C₂³	—	73	1	no	—
305	C₂³	—	73	2	no	—
306	C₂³	—	73	1	no	—
307	C₂³	M₁₆×D₇₃	1	1	yes	characteristic
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	292	4	no	—
310	C₂×C₄	—	73	2	no	—
311	C₂×C₄	—	73	2	no	—
312	C₂×C₄	—	73	1	no	—
313	C₂×C₄	—	73	1	no	—
314	C₂×C₄	—	73	1	no	—
315	C₂×C₄	—	73	2	no	—
316	C₂×C₄	—	73	1	no	—
317	C₂×C₄	—	73	2	no	—
318	C₂×C₄	—	73	2	no	—
319	C₂×C₄	—	73	2	no	—
320	C₂×C₄	—	73	2	no	—
321	C₂×C₄	—	73	1	no	—
322	C₂×C₄	—	73	1	no	—
323	C₂×C₄	—	73	2	no	—
324	C₂×C₄	—	146	2	no	—
325	C₂×C₄	C₄²×D₇₃	1	2	yes	—
326	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
327	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
328	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	73	1	no	—
331	C₈	—	73	1	no	—
332	C₈	—	73	2	no	—
333	C₈	—	146	2	no	—
334	C₈	C₄²×D₇₃	1	1	yes	characteristic
335	C₈	C₄²×D₇₃	1	1	yes	characteristic
336	C₈	C₂²×C₄×D₇₃	1	2	yes	—
337	C₂²	—	4	1	no	—
338	C₂²	—	292	1	no	—
339	C₂²	—	292	2	no	—
340	C₂²	—	73	1	no	—
341	C₂²	—	73	1	no	—
342	C₂²	—	73	2	no	—
343	C₂²	—	146	2	no	—
344	C₂²	—	146	4	no	—
345	C₂²	#97×D₇₃	1	2	yes	—
346	C₂²	C₂×M₁₆×D₇₃	1	1	yes	characteristic
347	C₄	—	4	2	no	—
348	C₄	—	292	2	no	—
349	C₄	—	73	1	no	—
350	C₄	—	73	1	no	—
351	C₄	—	73	2	no	—
352	C₄	#97×D₇₃	1	2	yes	—
353	C₄	C₂×C₄²×D₇₃	1	1	yes	characteristic
354	C₄	C₂×M₁₆×D₇₃	1	1	yes	characteristic
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	73	2	no	—
358	C₂	—	146	1	no	—
359	C₂	—	146	2	no	—
360	C₂	C₂×#97×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄²×D₇₃
commutator	C₄×C₇₃	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2110	C₂
socle	C₂×C₇₃	C₂²×#97
prime radical	C₂³×D₇₃	M₁₆
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#2110	D₇₃
hypocenter	C₇₃	C₂×#2110
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ #2110

Complements

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

N	H ≅ G/N	type
3. #22611536	C₂	semidirect
4. #22611535	C₂	semidirect
5. #22611534	C₂	semidirect
6. #347×D₇₃	C₂	semidirect
7. C₇₃×#2110	C₂	semidirect
12. C₂×#14556530	C₄	semidirect
13. M₃₂×D₇₃	C₄	semidirect
14. M₃₂×D₇₃	C₄	semidirect
17. C₂×C₁₆×D₇₃	C₄	semidirect
25. #17988365	C₂²	semidirect
26. #17988364	C₂²	semidirect
29. #347×C₇₃	C₂²	semidirect
52. #14556531	C₂×C₄	semidirect
55. #14556530	C₂×C₄	semidirect
56. C₂×#11924003	C₂×C₄	semidirect
58. C₁₆×D₇₃	C₂×C₄	semidirect
68. M₃₂×C₇₃	C₂×C₄	semidirect
70. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
180. D₇₃	#2110	direct (N × H)
186. #2110	D₇₃	direct (N × H)
188. C₇₃	C₂×#2110	semidirect
210. #347	C₂×D₇₃	semidirect
240. M₃₂	C₄×D₇₃	semidirect
245. C₂×C₁₆	C₄×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₇₃	2
#405×D₇₃	2
#22611536	2
#22611535	2
#22611534	2
#347×D₇₃	2
C₇₃×#2110	2
C₂×#2110	73

Subgroups — #2110×D73 #33238639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2110×D₇₃ #33238639