Subgroups — C₇₃×#7724 #32917128

Order 18688.

634 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₃×#2856	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#1817	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₃×#1797	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₃×#1270	C₂	1	4	yes	maximal, complemented
5	C₂×#127×C₇₃	C₄	1	1	yes	characteristic
6	C₂×#121×C₇₃	—	2	1	no	—
7	C₂×#121×C₇₃	—	2	1	no	—
8	C₂×#121×C₇₃	C₄	1	2	yes	—
9	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₇₃	—	2	1	no	—
11	C₂×C₂²⋊C₈×C₇₃	—	2	1	no	—
12	C₂×C₂²⋊C₈×C₇₃	—	2	1	no	—
13	#390×C₇₃	—	2	1	no	—
14	#390×C₇₃	—	2	1	no	—
15	#390×C₇₃	C₂²	1	2	yes	—
16	#384×C₇₃	C₄	1	1	yes	characteristic
17	#336×C₇₃	—	2	1	no	—
18	#336×C₇₃	C₂²	1	2	yes	—
19	#336×C₇₃	C₂²	1	2	yes	—
20	C₂²×D₄×C₇₃	—	2	2	no	—
21	C₂²×D₄×C₇₃	—	2	1	no	—
22	C₂²×D₄×C₇₃	—	2	1	no	—
23	C₂²×D₄×C₇₃	C₂×C₄	1	2	yes	—
24	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
26	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₃	—	2	2	no	—
28	C₂²×C₈×C₇₃	—	2	2	no	—
29	#127×C₇₃	—	4	2	no	—
30	#121×C₇₃	—	4	4	no	—
31	#121×C₇₃	—	4	2	no	—
32	#121×C₇₃	—	4	2	no	—
33	#121×C₇₃	—	4	4	no	—
34	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
35	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
36	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
37	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
39	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
40	C₂×C₄²×C₇₃	—	4	2	no	—
41	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₇₃	—	4	2	no	—
46	C₂²⋊C₈×C₇₃	—	4	2	no	—
47	C₂²⋊C₈×C₇₃	—	4	2	no	—
48	C₂⁴×C₇₃	—	2	2	no	—
49	C₂⁴×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×D₄×C₇₃	—	2	4	no	—
51	C₂×D₄×C₇₃	—	4	2	no	—
52	C₂×D₄×C₇₃	—	4	2	no	—
53	C₂×D₄×C₇₃	—	4	2	no	—
54	C₂×D₄×C₇₃	—	4	2	no	—
55	C₂×D₄×C₇₃	—	4	2	no	—
56	C₂×D₄×C₇₃	—	4	2	no	—
57	C₂×D₄×C₇₃	—	4	4	no	—
58	C₂×D₄×C₇₃	—	4	4	no	—
59	C₂×D₄×C₇₃	—	4	4	no	—
60	C₂×D₄×C₇₃	—	4	2	no	—
61	C₂×D₄×C₇₃	—	4	2	no	—
62	C₂²×C₄×C₇₃	—	2	2	no	—
63	C₂²×C₄×C₇₃	—	2	2	no	—
64	C₂²×C₄×C₇₃	—	2	2	no	—
65	C₂²×C₄×C₇₃	—	2	2	no	—
66	C₂²×C₄×C₇₃	—	2	2	no	—
67	C₂²×C₄×C₇₃	—	4	1	no	—
68	C₂²×C₄×C₇₃	—	4	1	no	—
69	C₂²×C₄×C₇₃	—	4	1	no	—
70	C₂²×C₄×C₇₃	—	4	1	no	—
71	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₃	D₈	1	2	yes	—
75	C₂²×C₄×C₇₃	SD₁₆	1	2	yes	—
76	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₇₃	—	4	2	no	—
80	C₂×C₈×C₇₃	—	4	2	no	—
81	C₂×C₈×C₇₃	—	4	2	no	—
82	C₂×C₈×C₇₃	—	4	2	no	—
83	C₂×C₈×C₇₃	—	4	2	no	—
84	C₂×C₈×C₇₃	—	4	2	no	—
85	C₄⋊C₄×C₇₃	—	4	2	no	—
86	C₄⋊C₄×C₇₃	—	4	2	no	—
87	C₄⋊C₄×C₇₃	—	4	2	no	—
88	C₂²⋊C₄×C₇₃	—	4	4	no	—
89	C₂²⋊C₄×C₇₃	—	4	2	no	—
90	C₂²⋊C₄×C₇₃	—	4	2	no	—
91	C₂²⋊C₄×C₇₃	—	4	4	no	—
92	C₄²×C₇₃	—	2	2	no	—
93	C₄²×C₇₃	—	2	2	no	—
94	C₄²×C₇₃	—	4	1	no	—
95	C₄²×C₇₃	—	4	1	no	—
96	C₂³×C₇₃	—	2	1	no	—
97	C₂³×C₇₃	—	2	1	no	—
98	C₂³×C₇₃	—	2	1	no	—
99	C₂³×C₇₃	—	2	1	no	—
100	C₂³×C₇₃	—	2	1	no	—
101	C₂³×C₇₃	—	2	1	no	—
102	C₂³×C₇₃	—	4	2	no	—
103	C₂³×C₇₃	—	4	2	no	—
104	C₂³×C₇₃	—	4	2	no	—
105	C₂³×C₇₃	—	4	2	no	—
106	C₂³×C₇₃	—	4	2	no	—
107	C₂³×C₇₃	—	4	2	no	—
108	C₂³×C₇₃	—	4	2	no	—
109	C₂³×C₇₃	#102	1	2	yes	—
110	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	D₄×C₇₃	—	8	4	no	—
112	D₄×C₇₃	—	8	4	no	—
113	D₄×C₇₃	—	8	4	no	—
114	D₄×C₇₃	—	8	8	no	—
115	C₂×C₄×C₇₃	—	2	1	no	—
116	C₂×C₄×C₇₃	—	2	1	no	—
117	C₂×C₄×C₇₃	—	2	1	no	—
118	C₂×C₄×C₇₃	—	2	1	no	—
119	C₂×C₄×C₇₃	—	2	1	no	—
120	C₂×C₄×C₇₃	—	2	1	no	—
121	C₂×C₄×C₇₃	—	2	1	no	—
122	C₂×C₄×C₇₃	—	2	1	no	—
123	C₂×C₄×C₇₃	—	2	1	no	—
124	C₂×C₄×C₇₃	—	2	1	no	—
125	C₂×C₄×C₇₃	—	2	1	no	—
126	C₂×C₄×C₇₃	—	2	1	no	—
127	C₂×C₄×C₇₃	—	4	2	no	—
128	C₂×C₄×C₇₃	—	4	2	no	—
129	C₂×C₄×C₇₃	—	4	4	no	—
130	C₂×C₄×C₇₃	—	4	4	no	—
131	C₂×C₄×C₇₃	—	4	2	no	—
132	C₂×C₄×C₇₃	—	4	2	no	—
133	C₂×C₄×C₇₃	—	4	2	no	—
134	C₂×C₄×C₇₃	#99	1	2	yes	—
135	C₂×C₄×C₇₃	#100	1	2	yes	—
136	C₂×C₄×C₇₃	#102	1	2	yes	—
137	C₂×C₄×C₇₃	#131	1	2	yes	—
138	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
140	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
141	C₈×C₇₃	—	8	2	no	—
142	C₈×C₇₃	—	8	2	no	—
143	C₂²×C₇₃	—	2	2	no	—
144	C₂²×C₇₃	—	2	2	no	—
145	C₂²×C₇₃	—	2	2	no	—
146	C₂²×C₇₃	—	4	1	no	—
147	C₂²×C₇₃	—	4	1	no	—
148	C₂²×C₇₃	—	4	1	no	—
149	C₂²×C₇₃	—	4	1	no	—
150	C₂²×C₇₃	—	8	2	no	—
151	C₂²×C₇₃	—	8	2	no	—
152	C₂²×C₇₃	—	8	2	no	—
153	C₂²×C₇₃	—	8	2	no	—
154	C₂²×C₇₃	—	8	2	no	—
155	C₂²×C₇₃	—	8	2	no	—
156	C₂²×C₇₃	—	8	2	no	—
157	C₂²×C₇₃	#408	1	1	yes	characteristic
158	C₂²×C₇₃	C₂×#99	1	1	yes	characteristic
159	C₂²×C₇₃	C₂×#100	1	1	yes	characteristic
160	C₂²×C₇₃	C₂×#102	1	1	yes	characteristic
161	C₂²×C₇₃	#418	1	1	yes	characteristic
162	C₂²×C₇₃	#421	1	1	yes	characteristic
163	C₂²×C₇₃	#421	1	1	yes	characteristic
164	C₄×C₇₃	—	2	2	no	—
165	C₄×C₇₃	—	2	2	no	—
166	C₄×C₇₃	—	2	2	no	—
167	C₄×C₇₃	—	2	2	no	—
168	C₄×C₇₃	—	2	1	no	—
169	C₄×C₇₃	—	2	2	no	—
170	C₄×C₇₃	—	2	1	no	—
171	C₄×C₇₃	—	8	2	no	—
172	#7724	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂×C₇₃	—	4	2	no	—
174	C₂×C₇₃	—	16	2	no	—
175	C₂×C₇₃	#1471	1	1	yes	characteristic
176	C₂×C₇₃	#1480	1	1	yes	characteristic
177	C₂×C₇₃	#1496	1	1	yes	characteristic
178	C₂×C₇₃	#1510	1	1	yes	characteristic
179	C₂×C₇₃	#1519	1	1	yes	characteristic
180	C₂×C₇₃	#1536	1	1	yes	characteristic
181	C₂×C₇₃	#1545	1	1	yes	characteristic
182	#2856	C₂×C₇₃	1	1	yes	characteristic
183	#1817	C₂×C₇₃	1	1	yes	characteristic, complemented
184	#1797	C₂×C₇₃	1	1	yes	characteristic, complemented
185	#1270	C₂×C₇₃	1	4	yes	complemented
186	C₇₃	#7724	1	1	yes	characteristic, direct product factor, Sylow
187	C₂×#127	C₄×C₇₃	1	1	yes	characteristic
188	C₂×#121	—	2	1	no	—
189	C₂×#121	—	2	1	no	—
190	C₂×#121	C₄×C₇₃	1	2	yes	—
191	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
192	C₂×C₄⋊C₈	—	2	1	no	—
193	C₂×C₂²⋊C₈	—	2	1	no	—
194	C₂×C₂²⋊C₈	—	2	1	no	—
195	#390	—	2	1	no	—
196	#390	—	2	1	no	—
197	#390	C₂²×C₇₃	1	2	yes	—
198	#384	C₄×C₇₃	1	1	yes	characteristic
199	#336	—	2	1	no	—
200	#336	C₂²×C₇₃	1	2	yes	—
201	#336	C₂²×C₇₃	1	2	yes	—
202	C₂²×D₄	—	2	2	no	—
203	C₂²×D₄	—	2	1	no	—
204	C₂²×D₄	—	2	1	no	—
205	C₂²×D₄	C₂×C₄×C₇₃	1	2	yes	—
206	C₂³×C₄	C₂×C₄×C₇₃	1	1	yes	characteristic
207	C₂³×C₄	D₄×C₇₃	1	1	yes	characteristic
208	C₂³×C₄	D₄×C₇₃	1	1	yes	characteristic
209	C₂²×C₈	—	2	2	no	—
210	C₂²×C₈	—	2	2	no	—
211	#127	—	4	2	no	—
212	#121	—	4	4	no	—
213	#121	—	4	2	no	—
214	#121	—	4	2	no	—
215	#121	—	4	4	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	C₂×C₄×C₇₃	1	2	yes	—
219	C₂×C₂²⋊C₄	—	2	2	no	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₂²⋊C₄	—	2	2	no	—
222	C₂×C₄²	—	4	2	no	—
223	C₂×C₄²	C₂×C₄×C₇₃	1	1	yes	characteristic
224	C₂×C₄²	D₄×C₇₃	1	1	yes	characteristic
225	C₂×C₄²	D₄×C₇₃	1	1	yes	characteristic
226	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
227	C₄⋊C₈	—	4	2	no	—
228	C₂²⋊C₈	—	4	2	no	—
229	C₂²⋊C₈	—	4	2	no	—
230	C₂⁴	—	2	2	no	—
231	C₂⁴	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	4	2	no	—
234	C₂×D₄	—	4	2	no	—
235	C₂×D₄	—	4	2	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	—	4	4	no	—
240	C₂×D₄	—	4	4	no	—
241	C₂×D₄	—	4	4	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂²×C₄	—	2	2	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	2	2	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
254	C₂²×C₄	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
255	C₂²×C₄	C₂²⋊C₄×C₇₃	1	1	yes	characteristic
256	C₂²×C₄	D₈×C₇₃	1	2	yes	—
257	C₂²×C₄	SD₁₆×C₇₃	1	2	yes	—
258	C₂²×C₄	C₂²×C₄×C₇₃	1	1	yes	characteristic
259	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
261	C₂×C₈	—	4	2	no	—
262	C₂×C₈	—	4	2	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	4	2	no	—
265	C₂×C₈	—	4	2	no	—
266	C₂×C₈	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	4	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	4	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	4	1	no	—
277	C₄²	—	4	1	no	—
278	C₂³	—	2	1	no	—
279	C₂³	—	2	1	no	—
280	C₂³	—	2	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	2	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	4	2	no	—
285	C₂³	—	4	2	no	—
286	C₂³	—	4	2	no	—
287	C₂³	—	4	2	no	—
288	C₂³	—	4	2	no	—
289	C₂³	—	4	2	no	—
290	C₂³	—	4	2	no	—
291	C₂³	#102×C₇₃	1	2	yes	—
292	C₂³	C₂×C₂²⋊C₄×C₇₃	1	1	yes	characteristic
293	D₄	—	8	4	no	—
294	D₄	—	8	4	no	—
295	D₄	—	8	4	no	—
296	D₄	—	8	8	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	#99×C₇₃	1	2	yes	—
317	C₂×C₄	#100×C₇₃	1	2	yes	—
318	C₂×C₄	#102×C₇₃	1	2	yes	—
319	C₂×C₄	#131×C₇₃	1	2	yes	—
320	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
321	C₂×C₄	C₂×SD₁₆×C₇₃	1	1	yes	characteristic
322	C₂×C₄	AGL(1,ℤ/8)×C₇₃	1	2	yes	—
323	C₈	—	8	2	no	—
324	C₈	—	8	2	no	—
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	#408×C₇₃	1	1	yes	characteristic
340	C₂²	C₂×#99×C₇₃	1	1	yes	characteristic
341	C₂²	C₂×#100×C₇₃	1	1	yes	characteristic
342	C₂²	C₂×#102×C₇₃	1	1	yes	characteristic
343	C₂²	#418×C₇₃	1	1	yes	characteristic
344	C₂²	#421×C₇₃	1	1	yes	characteristic
345	C₂²	#421×C₇₃	1	1	yes	characteristic
346	C₄	—	2	2	no	—
347	C₄	—	2	2	no	—
348	C₄	—	2	2	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	2	no	—
352	C₄	—	2	1	no	—
353	C₄	—	8	2	no	—
354	C₂	—	4	2	no	—
355	C₂	—	16	2	no	—
356	C₂	C₇₃×#1471	1	1	yes	characteristic
357	C₂	C₇₃×#1480	1	1	yes	characteristic
358	C₂	C₇₃×#1496	1	1	yes	characteristic
359	C₂	C₇₃×#1510	1	1	yes	characteristic
360	C₂	C₇₃×#1519	1	1	yes	characteristic
361	C₂	C₇₃×#1536	1	1	yes	characteristic
362	C₂	C₇₃×#1545	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₇₃
Frattini, cosocle	C₂×C₄²	C₂³×C₇₃
prime radical	C₇₃×#2856	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ C₂²×C₄²×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₃×#1817	C₂	semidirect
3. C₇₃×#1797	C₂	semidirect
4. C₇₃×#1270	C₂	semidirect
172. #7724	C₇₃	direct (N × H)
183. #1817	C₂×C₇₃	semidirect
184. #1797	C₂×C₇₃	semidirect
185. #1270	C₂×C₇₃	semidirect
186. C₇₃	#7724	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇₃×#2856	2
C₇₃×#1817	2
C₇₃×#1797	2
C₇₃×#1270	2
#7724	73

Subgroups — C73×#7724 #32917128

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#7724 #32917128