Subgroups — C₇₃×#25823 #32935227

Order 18688.

1002 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₃×#3501	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#3165	C₂	1	2	yes	maximal, complemented
3	C₇₃×#2563	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#2061	C₂	1	4	yes	maximal, complemented
5	C₇₃×#1986	C₂	1	2	yes	maximal, complemented
6	C₇₃×#1856	C₂	1	4	yes	maximal
7	C₄×#100×C₇₃	C₂	1	1	yes	maximal, characteristic
8	#554×C₇₃	—	2	2	no	—
9	#554×C₇₃	—	2	4	no	—
10	D₄×Q₈×C₇₃	—	2	2	no	—
11	#544×C₇₃	C₂²	1	2	yes	—
12	#543×C₇₃	—	2	2	no	—
13	#533×C₇₃	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₂²⋊Q₈×C₇₃	—	2	2	no	—
16	#518×C₇₃	C₂²	1	2	yes	—
17	C₂×C₄×D₄×C₇₃	C₂²	1	1	yes	characteristic
18	#501×C₇₃	—	2	4	no	—
19	#500×C₇₃	—	2	2	no	—
20	#498×C₇₃	—	2	2	no	—
21	#428×C₇₃	C₂²	1	8	yes	—
22	C₂×#100×C₇₃	C₂²	1	2	yes	—
23	C₄×M₁₆×C₇₃	C₂²	1	2	yes	—
24	#393×C₇₃	—	2	2	no	—
25	#390×C₇₃	C₂²	1	2	yes	—
26	#382×C₇₃	C₂²	1	2	yes	—
27	#380×C₇₃	—	2	2	no	—
28	C₄×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
29	#343×C₇₃	C₂²	1	2	yes	—
30	#339×C₇₃	C₂²	1	8	yes	—
31	C₂²×Q₈×C₇₃	—	2	2	no	—
32	C₂²×Q₈×C₇₃	—	2	1	no	—
33	C₂²×D₄×C₇₃	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₇₃	D₄	1	2	yes	—
35	C₂×M₁₆×C₇₃	C₂³	1	4	yes	—
36	C₄⋊Q₈×C₇₃	—	2	4	no	—
37	C₄⋊Q₈×C₇₃	—	2	2	no	—
38	C₄⋊Q₈×C₇₃	—	2	2	no	—
39	C₄⋊Q₈×C₇₃	D₄	1	2	yes	—
40	C₄⋊Q₈×C₇₃	D₄	1	2	yes	—
41	#125×C₇₃	—	2	4	no	—
42	#125×C₇₃	D₄	1	4	yes	—
43	C₂²⋊Q₈×C₇₃	—	2	4	no	—
44	C₂²⋊Q₈×C₇₃	—	2	4	no	—
45	C₂²⋊Q₈×C₇₃	—	2	8	no	—
46	C₂²⋊Q₈×C₇₃	—	2	8	no	—
47	C₂²⋊Q₈×C₇₃	—	2	2	no	—
48	C₂²⋊Q₈×C₇₃	—	2	4	no	—
49	C₂²⋊Q₈×C₇₃	—	2	2	no	—
50	C₄×Q₈×C₇₃	—	2	2	no	—
51	C₄×Q₈×C₇₃	—	2	2	no	—
52	C₄×D₄×C₇₃	—	2	4	no	—
53	C₄×D₄×C₇₃	—	2	4	no	—
54	C₄×D₄×C₇₃	—	2	2	no	—
55	C₄×D₄×C₇₃	—	2	2	no	—
56	#117×C₇₃	C₂³	1	4	yes	—
57	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
58	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
59	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
60	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₇₃	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊C₄×C₇₃	—	2	4	no	—
64	C₂×C₂²⋊C₄×C₇₃	—	2	4	no	—
65	C₂×C₂²⋊C₄×C₇₃	D₄	1	2	yes	—
66	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
67	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
68	#107×C₇₃	—	2	8	no	—
69	#106×C₇₃	—	2	8	no	—
70	#100×C₇₃	Q₈	1	8	yes	complemented
71	#97×C₇₃	—	2	2	no	—
72	C₄×C₈×C₇₃	—	2	2	no	—
73	#95×C₇₃	—	2	1	no	—
74	#95×C₇₃	—	2	4	no	—
75	C₂⁴×C₇₃	C₂×D₄	1	2	yes	—
76	C₂×Q₈×C₇₃	—	2	4	no	—
77	C₂×Q₈×C₇₃	—	2	2	no	—
78	C₂×Q₈×C₇₃	—	2	2	no	—
79	C₂×Q₈×C₇₃	—	4	2	no	—
80	C₂×Q₈×C₇₃	—	8	2	no	—
81	C₂×Q₈×C₇₃	—	8	4	no	—
82	C₂×D₄×C₇₃	—	2	2	no	—
83	C₂×D₄×C₇₃	—	2	2	no	—
84	C₂×D₄×C₇₃	C₂×Q₈	1	4	yes	—
85	C₂²×C₄×C₇₃	—	2	4	no	—
86	C₂²×C₄×C₇₃	—	2	4	no	—
87	C₂²×C₄×C₇₃	—	2	2	no	—
88	C₂²×C₄×C₇₃	—	2	2	no	—
89	C₂²×C₄×C₇₃	—	2	2	no	—
90	C₂²×C₄×C₇₃	—	2	2	no	—
91	C₂²×C₄×C₇₃	—	4	4	no	—
92	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
96	M₁₆×C₇₃	—	2	4	no	—
97	M₁₆×C₇₃	C₂×Q₈	1	8	yes	complemented
98	C₂×C₈×C₇₃	—	2	4	no	—
99	C₄⋊C₄×C₇₃	—	2	4	no	—
100	C₄⋊C₄×C₇₃	—	2	8	no	—
101	C₄⋊C₄×C₇₃	—	2	4	no	—
102	C₄⋊C₄×C₇₃	—	2	2	no	—
103	C₄⋊C₄×C₇₃	—	2	4	no	—
104	C₄⋊C₄×C₇₃	—	2	2	no	—
105	C₄⋊C₄×C₇₃	—	2	2	no	—
106	C₄⋊C₄×C₇₃	—	2	4	no	—
107	C₄⋊C₄×C₇₃	—	4	2	no	—
108	C₄⋊C₄×C₇₃	—	4	1	no	—
109	C₄⋊C₄×C₇₃	—	4	1	no	—
110	C₄⋊C₄×C₇₃	C₂×D₄	1	4	yes	—
111	C₄⋊C₄×C₇₃	C₂×D₄	1	4	yes	—
112	C₂²⋊C₄×C₇₃	—	2	8	no	—
113	C₂²⋊C₄×C₇₃	—	2	4	no	—
114	C₂²⋊C₄×C₇₃	—	2	4	no	—
115	C₂²⋊C₄×C₇₃	—	8	4	no	—
116	C₄²×C₇₃	—	2	4	no	—
117	C₄²×C₇₃	—	2	4	no	—
118	C₄²×C₇₃	C₂×D₄	1	2	yes	—
119	C₄²×C₇₃	C₂×D₄	1	2	yes	—
120	C₂³×C₇₃	—	2	2	no	—
121	C₂³×C₇₃	—	2	2	no	—
122	C₂³×C₇₃	—	4	2	no	—
123	C₂³×C₇₃	—	4	2	no	—
124	C₂³×C₇₃	C₄⋊Q₈	1	4	yes	—
125	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
126	Q₈×C₇₃	—	4	4	no	—
127	Q₈×C₇₃	—	8	2	no	—
128	Q₈×C₇₃	—	8	4	no	—
129	Q₈×C₇₃	—	8	4	no	—
130	D₄×C₇₃	—	2	8	no	—
131	C₂×C₄×C₇₃	—	2	2	no	—
132	C₂×C₄×C₇₃	—	2	4	no	—
133	C₂×C₄×C₇₃	—	2	4	no	—
134	C₂×C₄×C₇₃	—	2	2	no	—
135	C₂×C₄×C₇₃	—	4	1	no	—
136	C₂×C₄×C₇₃	—	4	4	no	—
137	C₂×C₄×C₇₃	—	4	2	no	—
138	C₂×C₄×C₇₃	—	4	2	no	—
139	C₂×C₄×C₇₃	—	4	4	no	—
140	C₂×C₄×C₇₃	—	4	2	no	—
141	C₂×C₄×C₇₃	—	4	2	no	—
142	C₂×C₄×C₇₃	—	4	2	no	—
143	C₂×C₄×C₇₃	—	8	2	no	—
144	C₂×C₄×C₇₃	—	8	4	no	—
145	C₂×C₄×C₇₃	—	8	4	no	—
146	C₂×C₄×C₇₃	#120	1	2	yes	—
147	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
148	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
149	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
150	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
151	C₂×C₄×C₇₃	2₋¹⁺⁴	1	2	yes	—
152	C₈×C₇₃	—	2	8	no	—
153	C₂²×C₇₃	—	2	1	no	—
154	C₂²×C₇₃	—	2	1	no	—
155	C₂²×C₇₃	—	2	4	no	—
156	C₂²×C₇₃	—	4	4	no	—
157	C₂²×C₇₃	—	4	2	no	—
158	C₂²×C₇₃	—	4	2	no	—
159	C₂²×C₇₃	—	8	2	no	—
160	C₂²×C₇₃	C₂×#120	1	1	yes	characteristic
161	C₂²×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
162	C₂²×C₇₃	#536	1	1	yes	characteristic
163	C₄×C₇₃	—	2	1	no	—
164	C₄×C₇₃	—	2	2	no	—
165	C₄×C₇₃	—	4	2	no	—
166	C₄×C₇₃	—	4	4	no	—
167	C₄×C₇₃	—	4	4	no	—
168	C₄×C₇₃	—	8	2	no	—
169	C₄×C₇₃	—	8	2	no	—
170	C₄×C₇₃	#455	1	2	yes	—
171	C₄×C₇₃	D₄×Q₈	1	4	yes	—
172	#25823	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂×C₇₃	—	2	1	no	—
174	C₂×C₇₃	—	2	1	no	—
175	C₂×C₇₃	—	4	4	no	—
176	C₂×C₇₃	#2434	1	1	yes	characteristic
177	C₂×C₇₃	C₂×#455	1	1	yes	characteristic
178	C₂×C₇₃	#3021	1	1	yes	characteristic
179	#3501	C₂×C₇₃	1	1	yes	characteristic
180	#3165	C₂×C₇₃	1	2	yes	complemented
181	#2563	C₂×C₇₃	1	1	yes	characteristic
182	#2061	C₂×C₇₃	1	4	yes	complemented
183	#1986	C₂×C₇₃	1	2	yes	complemented
184	#1856	C₂×C₇₃	1	4	yes	—
185	C₄×#100	C₂×C₇₃	1	1	yes	characteristic
186	C₇₃	#25823	1	1	yes	characteristic, direct product factor, Sylow
187	#554	—	2	2	no	—
188	#554	—	2	4	no	—
189	D₄×Q₈	—	2	2	no	—
190	#544	C₂²×C₇₃	1	2	yes	—
191	#543	—	2	2	no	—
192	#533	C₂²×C₇₃	1	2	yes	—
193	C₂×C₄⋊Q₈	C₂²×C₇₃	1	1	yes	characteristic
194	C₂×C₂²⋊Q₈	—	2	2	no	—
195	#518	C₂²×C₇₃	1	2	yes	—
196	C₂×C₄×D₄	C₂²×C₇₃	1	1	yes	characteristic
197	#501	—	2	4	no	—
198	#500	—	2	2	no	—
199	#498	—	2	2	no	—
200	#428	C₂²×C₇₃	1	8	yes	—
201	C₂×#100	C₂²×C₇₃	1	2	yes	—
202	C₄×M₁₆	C₂²×C₇₃	1	2	yes	—
203	#393	—	2	2	no	—
204	#390	C₂²×C₇₃	1	2	yes	—
205	#382	C₂²×C₇₃	1	2	yes	—
206	#380	—	2	2	no	—
207	C₄×C₄⋊C₄	C₂²×C₇₃	1	1	yes	characteristic
208	#343	C₂²×C₇₃	1	2	yes	—
209	#339	C₂²×C₇₃	1	8	yes	—
210	C₂²×Q₈	—	2	2	no	—
211	C₂²×Q₈	—	2	1	no	—
212	C₂²×D₄	C₂³×C₇₃	1	1	yes	characteristic
213	C₂³×C₄	D₄×C₇₃	1	2	yes	—
214	C₂×M₁₆	C₂³×C₇₃	1	4	yes	—
215	C₄⋊Q₈	—	2	4	no	—
216	C₄⋊Q₈	—	2	2	no	—
217	C₄⋊Q₈	—	2	2	no	—
218	C₄⋊Q₈	D₄×C₇₃	1	2	yes	—
219	C₄⋊Q₈	D₄×C₇₃	1	2	yes	—
220	#125	—	2	4	no	—
221	#125	D₄×C₇₃	1	4	yes	—
222	C₂²⋊Q₈	—	2	4	no	—
223	C₂²⋊Q₈	—	2	4	no	—
224	C₂²⋊Q₈	—	2	8	no	—
225	C₂²⋊Q₈	—	2	8	no	—
226	C₂²⋊Q₈	—	2	2	no	—
227	C₂²⋊Q₈	—	2	4	no	—
228	C₂²⋊Q₈	—	2	2	no	—
229	C₄×Q₈	—	2	2	no	—
230	C₄×Q₈	—	2	2	no	—
231	C₄×D₄	—	2	4	no	—
232	C₄×D₄	—	2	4	no	—
233	C₄×D₄	—	2	2	no	—
234	C₄×D₄	—	2	2	no	—
235	#117	C₂³×C₇₃	1	4	yes	—
236	C₂×C₄⋊C₄	—	2	2	no	—
237	C₂×C₄⋊C₄	—	2	4	no	—
238	C₂×C₄⋊C₄	—	2	1	no	—
239	C₂×C₄⋊C₄	—	2	1	no	—
240	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
241	C₂×C₄⋊C₄	C₂³×C₇₃	1	1	yes	characteristic
242	C₂×C₂²⋊C₄	—	2	4	no	—
243	C₂×C₂²⋊C₄	—	2	4	no	—
244	C₂×C₂²⋊C₄	D₄×C₇₃	1	2	yes	—
245	C₂×C₄²	C₂³×C₇₃	1	2	yes	—
246	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
247	#107	—	2	8	no	—
248	#106	—	2	8	no	—
249	#100	Q₈×C₇₃	1	8	yes	complemented
250	#97	—	2	2	no	—
251	C₄×C₈	—	2	2	no	—
252	#95	—	2	1	no	—
253	#95	—	2	4	no	—
254	C₂⁴	C₂×D₄×C₇₃	1	2	yes	—
255	C₂×Q₈	—	2	4	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	4	2	no	—
259	C₂×Q₈	—	8	2	no	—
260	C₂×Q₈	—	8	4	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	C₂×Q₈×C₇₃	1	4	yes	—
264	C₂²×C₄	—	2	4	no	—
265	C₂²×C₄	—	2	4	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	2	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	4	4	no	—
271	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
273	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
274	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
275	M₁₆	—	2	4	no	—
276	M₁₆	C₂×Q₈×C₇₃	1	8	yes	complemented
277	C₂×C₈	—	2	4	no	—
278	C₄⋊C₄	—	2	4	no	—
279	C₄⋊C₄	—	2	8	no	—
280	C₄⋊C₄	—	2	4	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	4	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	C₂×D₄×C₇₃	1	4	yes	—
290	C₄⋊C₄	C₂×D₄×C₇₃	1	4	yes	—
291	C₂²⋊C₄	—	2	8	no	—
292	C₂²⋊C₄	—	2	4	no	—
293	C₂²⋊C₄	—	2	4	no	—
294	C₂²⋊C₄	—	8	4	no	—
295	C₄²	—	2	4	no	—
296	C₄²	—	2	4	no	—
297	C₄²	C₂×D₄×C₇₃	1	2	yes	—
298	C₄²	C₂×D₄×C₇₃	1	2	yes	—
299	C₂³	—	2	2	no	—
300	C₂³	—	2	2	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	C₄⋊Q₈×C₇₃	1	4	yes	—
304	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
305	Q₈	—	4	4	no	—
306	Q₈	—	8	2	no	—
307	Q₈	—	8	4	no	—
308	Q₈	—	8	4	no	—
309	D₄	—	2	8	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	8	4	no	—
324	C₂×C₄	—	8	4	no	—
325	C₂×C₄	#120×C₇₃	1	2	yes	—
326	C₂×C₄	#120×C₇₃	1	1	yes	characteristic
327	C₂×C₄	#120×C₇₃	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₄×C₇₃	1	2	yes	—
329	C₂×C₄	C₂²×Q₈×C₇₃	1	2	yes	—
330	C₂×C₄	2₋¹⁺⁴×C₇₃	1	2	yes	—
331	C₈	—	2	8	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	4	no	—
335	C₂²	—	4	4	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	C₂×#120×C₇₃	1	1	yes	characteristic
340	C₂²	C₂×C₄⋊Q₈×C₇₃	1	1	yes	characteristic
341	C₂²	#536×C₇₃	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	4	no	—
346	C₄	—	4	4	no	—
347	C₄	—	8	2	no	—
348	C₄	—	8	2	no	—
349	C₄	#455×C₇₃	1	2	yes	—
350	C₄	D₄×Q₈×C₇₃	1	4	yes	—
351	C₂	—	2	1	no	—
352	C₂	—	2	1	no	—
353	C₂	—	4	4	no	—
354	C₂	C₇₃×#2434	1	1	yes	characteristic
355	C₂	C₂×#455×C₇₃	1	1	yes	characteristic
356	C₂	C₇₃×#3021	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₃
prime radical	C₂²×D₄×C₇₃	C₂³
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
2. C₇₃×#3165	C₂	semidirect
4. C₇₃×#2061	C₂	semidirect
5. C₇₃×#1986	C₂	semidirect
70. #100×C₇₃	Q₈	semidirect
97. M₁₆×C₇₃	C₂×Q₈	semidirect
172. #25823	C₇₃	direct (N × H)
180. #3165	C₂×C₇₃	semidirect
182. #2061	C₂×C₇₃	semidirect
183. #1986	C₂×C₇₃	semidirect
186. C₇₃	#25823	direct (N × H)
249. #100	Q₈×C₇₃	semidirect
276. M₁₆	C₂×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×#3501	2
C₇₃×#3165	2
C₇₃×#2563	2
C₇₃×#2061	2
C₇₃×#1986	2
C₇₃×#1856	2
C₄×#100×C₇₃	2
#25823	73

Subgroups — C73×#25823 #32935227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#25823 #32935227