Subgroups — C₇₃×#25827 #32935231

Order 18688.

910 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₃×#3502	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#3166	C₂	1	2	yes	maximal
3	C₇₃×#2565	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#2061	C₂	1	2	yes	maximal
5	C₇₃×#2060	C₂	1	2	yes	maximal
6	C₇₃×#1986	C₂	1	2	yes	maximal
7	C₇₃×#1857	C₂	1	2	yes	maximal
8	C₇₃×#1857	C₂	1	2	yes	maximal
9	C₄×#101×C₇₃	C₂	1	1	yes	maximal, characteristic
10	Q₈²×C₇₃	—	2	2	no	—
11	#557×C₇₃	—	2	2	no	—
12	#557×C₇₃	—	2	4	no	—
13	#544×C₇₃	—	2	2	no	—
14	#544×C₇₃	C₂²	1	2	yes	—
15	#533×C₇₃	—	2	2	no	—
16	#533×C₇₃	C₂²	1	2	yes	—
17	C₂×#125×C₇₃	C₂²	1	1	yes	characteristic
18	#519×C₇₃	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
20	#501×C₇₃	—	2	2	no	—
21	#501×C₇₃	—	2	2	no	—
22	#499×C₇₃	—	2	4	no	—
23	#428×C₇₃	C₂²	1	4	yes	—
24	#428×C₇₃	C₂²	1	4	yes	—
25	C₂×#101×C₇₃	C₂²	1	2	yes	—
26	C₄×M₁₆×C₇₃	C₂²	1	2	yes	—
27	#398×C₇₃	—	2	2	no	—
28	#391×C₇₃	C₂²	1	1	yes	characteristic
29	#391×C₇₃	C₂²	1	1	yes	characteristic
30	#387×C₇₃	—	2	2	no	—
31	#384×C₇₃	C₂²	1	1	yes	characteristic
32	#382×C₇₃	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
34	#343×C₇₃	C₂²	1	2	yes	—
35	#339×C₇₃	C₂²	1	4	yes	—
36	#339×C₇₃	C₂²	1	4	yes	—
37	C₂²×Q₈×C₇₃	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₇₃	C₂³	1	4	yes	—
39	C₄⋊Q₈×C₇₃	—	2	2	no	—
40	C₄⋊Q₈×C₇₃	—	2	2	no	—
41	C₄⋊Q₈×C₇₃	—	2	8	no	—
42	C₄⋊Q₈×C₇₃	—	2	4	no	—
43	C₄⋊Q₈×C₇₃	D₄	1	2	yes	—
44	C₄⋊Q₈×C₇₃	D₄	1	2	yes	—
45	#125×C₇₃	—	2	4	no	—
46	#125×C₇₃	—	2	8	no	—
47	#125×C₇₃	—	2	4	no	—
48	#125×C₇₃	—	2	2	no	—
49	#125×C₇₃	—	2	2	no	—
50	#125×C₇₃	D₄	1	4	yes	—
51	C₂²⋊Q₈×C₇₃	—	2	2	no	—
52	C₂²⋊Q₈×C₇₃	—	2	4	no	—
53	C₂²⋊Q₈×C₇₃	—	2	2	no	—
54	C₄×Q₈×C₇₃	—	2	4	no	—
55	C₄×Q₈×C₇₃	—	2	4	no	—
56	C₄×Q₈×C₇₃	—	2	4	no	—
57	C₄×Q₈×C₇₃	—	2	4	no	—
58	C₄×Q₈×C₇₃	—	2	2	no	—
59	C₄×Q₈×C₇₃	—	2	2	no	—
60	C₄×Q₈×C₇₃	—	2	2	no	—
61	C₄×Q₈×C₇₃	—	2	2	no	—
62	#117×C₇₃	—	2	4	no	—
63	#117×C₇₃	C₂³	1	4	yes	—
64	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
65	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
66	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
67	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
68	C₂×C₄⋊C₄×C₇₃	D₄	1	2	yes	—
69	C₂×C₄⋊C₄×C₇₃	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₇₃	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₇₃	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₇₃	D₄	1	2	yes	—
73	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
76	#107×C₇₃	—	2	4	no	—
77	#107×C₇₃	—	2	4	no	—
78	#106×C₇₃	—	2	4	no	—
79	#106×C₇₃	—	2	4	no	—
80	#101×C₇₃	Q₈	1	8	yes	complemented
81	#97×C₇₃	—	2	2	no	—
82	C₄×C₈×C₇₃	—	2	2	no	—
83	#95×C₇₃	—	2	1	no	—
84	#95×C₇₃	—	2	2	no	—
85	#95×C₇₃	—	2	2	no	—
86	#95×C₇₃	—	2	2	no	—
87	C₂×Q₈×C₇₃	—	2	4	no	—
88	C₂×Q₈×C₇₃	—	2	2	no	—
89	C₂×Q₈×C₇₃	—	2	2	no	—
90	C₂×Q₈×C₇₃	—	2	2	no	—
91	C₂×Q₈×C₇₃	—	2	2	no	—
92	C₂×Q₈×C₇₃	C₂×Q₈	1	4	yes	—
93	C₂²×C₄×C₇₃	—	2	1	no	—
94	C₂²×C₄×C₇₃	—	2	1	no	—
95	C₂²×C₄×C₇₃	—	2	1	no	—
96	C₂²×C₄×C₇₃	—	2	1	no	—
97	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
98	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
102	M₁₆×C₇₃	—	2	4	no	—
103	M₁₆×C₇₃	C₂×Q₈	1	8	yes	—
104	C₂×C₈×C₇₃	—	2	4	no	—
105	C₄⋊C₄×C₇₃	—	2	4	no	—
106	C₄⋊C₄×C₇₃	—	2	4	no	—
107	C₄⋊C₄×C₇₃	—	2	4	no	—
108	C₄⋊C₄×C₇₃	—	2	8	no	—
109	C₄⋊C₄×C₇₃	—	2	4	no	—
110	C₄⋊C₄×C₇₃	—	2	4	no	—
111	C₄⋊C₄×C₇₃	—	2	2	no	—
112	C₄⋊C₄×C₇₃	—	2	2	no	—
113	C₄⋊C₄×C₇₃	—	2	4	no	—
114	C₄⋊C₄×C₇₃	—	2	2	no	—
115	C₄⋊C₄×C₇₃	—	2	2	no	—
116	C₄⋊C₄×C₇₃	—	2	2	no	—
117	C₄⋊C₄×C₇₃	—	4	2	no	—
118	C₄⋊C₄×C₇₃	—	4	4	no	—
119	C₄⋊C₄×C₇₃	—	4	1	no	—
120	C₄⋊C₄×C₇₃	—	4	1	no	—
121	C₄⋊C₄×C₇₃	—	4	2	no	—
122	C₄⋊C₄×C₇₃	C₂×D₄	1	4	yes	—
123	C₄⋊C₄×C₇₃	C₂×D₄	1	4	yes	—
124	C₂²⋊C₄×C₇₃	—	2	4	no	—
125	C₄²×C₇₃	—	2	8	no	—
126	C₄²×C₇₃	—	2	2	no	—
127	C₄²×C₇₃	—	2	2	no	—
128	C₄²×C₇₃	—	2	4	no	—
129	C₄²×C₇₃	—	2	2	no	—
130	C₄²×C₇₃	—	2	2	no	—
131	C₄²×C₇₃	C₂×D₄	1	2	yes	—
132	C₄²×C₇₃	C₂×D₄	1	2	yes	—
133	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
134	Q₈×C₇₃	—	2	8	no	—
135	Q₈×C₇₃	—	4	4	no	—
136	Q₈×C₇₃	—	8	2	no	—
137	Q₈×C₇₃	—	8	4	no	—
138	C₂×C₄×C₇₃	—	2	2	no	—
139	C₂×C₄×C₇₃	—	2	2	no	—
140	C₂×C₄×C₇₃	—	2	2	no	—
141	C₂×C₄×C₇₃	—	2	4	no	—
142	C₂×C₄×C₇₃	—	2	4	no	—
143	C₂×C₄×C₇₃	—	2	2	no	—
144	C₂×C₄×C₇₃	—	4	1	no	—
145	C₂×C₄×C₇₃	—	4	1	no	—
146	C₂×C₄×C₇₃	—	4	1	no	—
147	C₂×C₄×C₇₃	—	4	1	no	—
148	C₂×C₄×C₇₃	—	4	2	no	—
149	C₂×C₄×C₇₃	—	4	1	no	—
150	C₂×C₄×C₇₃	—	4	2	no	—
151	C₂×C₄×C₇₃	—	4	1	no	—
152	C₂×C₄×C₇₃	—	4	2	no	—
153	C₂×C₄×C₇₃	—	4	1	no	—
154	C₂×C₄×C₇₃	—	4	1	no	—
155	C₂×C₄×C₇₃	—	4	1	no	—
156	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
157	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
158	C₂×C₄×C₇₃	#120	1	2	yes	—
159	C₂×C₄×C₇₃	C₄⋊Q₈	1	4	yes	—
160	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
161	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
162	C₂×C₄×C₇₃	2₋¹⁺⁴	1	2	yes	—
163	C₈×C₇₃	—	2	8	no	—
164	C₂²×C₇₃	—	2	1	no	—
165	C₂²×C₇₃	—	2	1	no	—
166	C₂²×C₇₃	C₂×#120	1	1	yes	characteristic
167	C₂²×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
168	C₂²×C₇₃	#536	1	1	yes	characteristic
169	C₄×C₇₃	—	2	4	no	—
170	C₄×C₇₃	—	2	2	no	—
171	C₄×C₇₃	—	2	1	no	—
172	C₄×C₇₃	—	2	1	no	—
173	C₄×C₇₃	—	4	2	no	—
174	C₄×C₇₃	—	4	2	no	—
175	C₄×C₇₃	—	4	4	no	—
176	C₄×C₇₃	—	4	4	no	—
177	C₄×C₇₃	—	8	1	no	—
178	C₄×C₇₃	—	8	1	no	—
179	C₄×C₇₃	—	8	1	no	—
180	C₄×C₇₃	D₄×Q₈	1	4	yes	—
181	#25827	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
182	C₂×C₇₃	—	2	1	no	—
183	C₂×C₇₃	—	2	1	no	—
184	C₂×C₇₃	#2434	1	1	yes	characteristic
185	C₂×C₇₃	#3018	1	1	yes	characteristic
186	C₂×C₇₃	#3020	1	1	yes	characteristic
187	#3502	C₂×C₇₃	1	1	yes	characteristic
188	#3166	C₂×C₇₃	1	2	yes	—
189	#2565	C₂×C₇₃	1	1	yes	characteristic
190	#2061	C₂×C₇₃	1	2	yes	—
191	#2060	C₂×C₇₃	1	2	yes	—
192	#1986	C₂×C₇₃	1	2	yes	—
193	#1857	C₂×C₇₃	1	2	yes	—
194	#1857	C₂×C₇₃	1	2	yes	—
195	C₄×#101	C₂×C₇₃	1	1	yes	characteristic
196	C₇₃	#25827	1	1	yes	characteristic, direct product factor, Sylow
197	Q₈²	—	2	2	no	—
198	#557	—	2	2	no	—
199	#557	—	2	4	no	—
200	#544	—	2	2	no	—
201	#544	C₂²×C₇₃	1	2	yes	—
202	#533	—	2	2	no	—
203	#533	C₂²×C₇₃	1	2	yes	—
204	C₂×#125	C₂²×C₇₃	1	1	yes	characteristic
205	#519	C₂²×C₇₃	1	2	yes	—
206	C₂×C₄×Q₈	C₂²×C₇₃	1	1	yes	characteristic
207	#501	—	2	2	no	—
208	#501	—	2	2	no	—
209	#499	—	2	4	no	—
210	#428	C₂²×C₇₃	1	4	yes	—
211	#428	C₂²×C₇₃	1	4	yes	—
212	C₂×#101	C₂²×C₇₃	1	2	yes	—
213	C₄×M₁₆	C₂²×C₇₃	1	2	yes	—
214	#398	—	2	2	no	—
215	#391	C₂²×C₇₃	1	1	yes	characteristic
216	#391	C₂²×C₇₃	1	1	yes	characteristic
217	#387	—	2	2	no	—
218	#384	C₂²×C₇₃	1	1	yes	characteristic
219	#382	C₂²×C₇₃	1	1	yes	characteristic
220	C₄×C₄⋊C₄	C₂²×C₇₃	1	1	yes	characteristic
221	#343	C₂²×C₇₃	1	2	yes	—
222	#339	C₂²×C₇₃	1	4	yes	—
223	#339	C₂²×C₇₃	1	4	yes	—
224	C₂²×Q₈	C₂³×C₇₃	1	1	yes	characteristic
225	C₂×M₁₆	C₂³×C₇₃	1	4	yes	—
226	C₄⋊Q₈	—	2	2	no	—
227	C₄⋊Q₈	—	2	2	no	—
228	C₄⋊Q₈	—	2	8	no	—
229	C₄⋊Q₈	—	2	4	no	—
230	C₄⋊Q₈	D₄×C₇₃	1	2	yes	—
231	C₄⋊Q₈	D₄×C₇₃	1	2	yes	—
232	#125	—	2	4	no	—
233	#125	—	2	8	no	—
234	#125	—	2	4	no	—
235	#125	—	2	2	no	—
236	#125	—	2	2	no	—
237	#125	D₄×C₇₃	1	4	yes	—
238	C₂²⋊Q₈	—	2	2	no	—
239	C₂²⋊Q₈	—	2	4	no	—
240	C₂²⋊Q₈	—	2	2	no	—
241	C₄×Q₈	—	2	4	no	—
242	C₄×Q₈	—	2	4	no	—
243	C₄×Q₈	—	2	4	no	—
244	C₄×Q₈	—	2	4	no	—
245	C₄×Q₈	—	2	2	no	—
246	C₄×Q₈	—	2	2	no	—
247	C₄×Q₈	—	2	2	no	—
248	C₄×Q₈	—	2	2	no	—
249	#117	—	2	4	no	—
250	#117	C₂³×C₇₃	1	4	yes	—
251	C₂×C₄⋊C₄	—	2	1	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	2	2	no	—
255	C₂×C₄⋊C₄	D₄×C₇₃	1	2	yes	—
256	C₂×C₄⋊C₄	C₂³×C₇₃	1	1	yes	characteristic
257	C₂×C₄⋊C₄	C₂³×C₇₃	1	1	yes	characteristic
258	C₂×C₄⋊C₄	C₂³×C₇₃	1	1	yes	characteristic
259	C₂×C₄²	D₄×C₇₃	1	2	yes	—
260	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
261	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
262	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
263	#107	—	2	4	no	—
264	#107	—	2	4	no	—
265	#106	—	2	4	no	—
266	#106	—	2	4	no	—
267	#101	Q₈×C₇₃	1	8	yes	complemented
268	#97	—	2	2	no	—
269	C₄×C₈	—	2	2	no	—
270	#95	—	2	1	no	—
271	#95	—	2	2	no	—
272	#95	—	2	2	no	—
273	#95	—	2	2	no	—
274	C₂×Q₈	—	2	4	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	—	2	2	no	—
277	C₂×Q₈	—	2	2	no	—
278	C₂×Q₈	—	2	2	no	—
279	C₂×Q₈	C₂×Q₈×C₇₃	1	4	yes	—
280	C₂²×C₄	—	2	1	no	—
281	C₂²×C₄	—	2	1	no	—
282	C₂²×C₄	—	2	1	no	—
283	C₂²×C₄	—	2	1	no	—
284	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
285	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
286	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
287	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
288	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
289	M₁₆	—	2	4	no	—
290	M₁₆	C₂×Q₈×C₇₃	1	8	yes	—
291	C₂×C₈	—	2	4	no	—
292	C₄⋊C₄	—	2	4	no	—
293	C₄⋊C₄	—	2	4	no	—
294	C₄⋊C₄	—	2	4	no	—
295	C₄⋊C₄	—	2	8	no	—
296	C₄⋊C₄	—	2	4	no	—
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	4	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	4	2	no	—
305	C₄⋊C₄	—	4	4	no	—
306	C₄⋊C₄	—	4	1	no	—
307	C₄⋊C₄	—	4	1	no	—
308	C₄⋊C₄	—	4	2	no	—
309	C₄⋊C₄	C₂×D₄×C₇₃	1	4	yes	—
310	C₄⋊C₄	C₂×D₄×C₇₃	1	4	yes	—
311	C₂²⋊C₄	—	2	4	no	—
312	C₄²	—	2	8	no	—
313	C₄²	—	2	2	no	—
314	C₄²	—	2	2	no	—
315	C₄²	—	2	4	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	2	2	no	—
318	C₄²	C₂×D₄×C₇₃	1	2	yes	—
319	C₄²	C₂×D₄×C₇₃	1	2	yes	—
320	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
321	Q₈	—	2	8	no	—
322	Q₈	—	4	4	no	—
323	Q₈	—	8	2	no	—
324	Q₈	—	8	4	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	4	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	#120×C₇₃	1	1	yes	characteristic
344	C₂×C₄	#120×C₇₃	1	1	yes	characteristic
345	C₂×C₄	#120×C₇₃	1	2	yes	—
346	C₂×C₄	C₄⋊Q₈×C₇₃	1	4	yes	—
347	C₂×C₄	C₂²×D₄×C₇₃	1	2	yes	—
348	C₂×C₄	C₂²×Q₈×C₇₃	1	2	yes	—
349	C₂×C₄	2₋¹⁺⁴×C₇₃	1	2	yes	—
350	C₈	—	2	8	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	C₂×#120×C₇₃	1	1	yes	characteristic
354	C₂²	C₂×C₄⋊Q₈×C₇₃	1	1	yes	characteristic
355	C₂²	#536×C₇₃	1	1	yes	characteristic
356	C₄	—	2	2	no	—
357	C₄	—	2	1	no	—
358	C₄	—	2	1	no	—
359	C₄	—	2	4	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	2	no	—
362	C₄	—	4	4	no	—
363	C₄	—	4	4	no	—
364	C₄	—	8	1	no	—
365	C₄	—	8	1	no	—
366	C₄	—	8	1	no	—
367	C₄	D₄×Q₈×C₇₃	1	4	yes	—
368	C₂	—	2	1	no	—
369	C₂	—	2	1	no	—
370	C₂	C₇₃×#2434	1	1	yes	characteristic
371	C₂	C₇₃×#3018	1	1	yes	characteristic
372	C₂	C₇₃×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₃
prime radical	C₂³×C₇₃	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
80. #101×C₇₃	Q₈	semidirect
181. #25827	C₇₃	direct (N × H)
196. C₇₃	#25827	direct (N × H)
267. #101	Q₈×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₇₃×#3502	2
C₇₃×#3166	2
C₇₃×#2565	2
C₇₃×#2061	2
C₇₃×#2060	2
C₇₃×#1986	2
C₇₃×#1857	2
C₄×#101×C₇₃	2
#25827	73

Subgroups — C73×#25827 #32935231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#25827 #32935231