Subgroups — C₇₃×#21910 #32931314

Order 18688.

950 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₃×#2280	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#1924	C₂	1	4	yes	maximal, complemented
3	C₇₃×#1923	C₂	1	4	yes	maximal, complemented
4	C₇₃×#1798	C₂	1	2	yes	maximal, complemented
5	C₇₃×#1779	C₂	1	2	yes	maximal
6	C₇₃×#1750	C₂	1	2	yes	maximal, complemented
7	C₂⁴×C₄×C₇₃	C₂²	1	1	yes	characteristic
8	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
9	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₇₃	C₂²	1	4	yes	complemented
11	C₂×C₂²⋊C₈×C₇₃	C₂²	1	4	yes	—
12	C₂×C₂²⋊C₈×C₇₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₈×C₇₃	C₂²	1	4	yes	—
14	C₂×C₂²⋊C₈×C₇₃	C₂²	1	2	yes	—
15	C₂×#97×C₇₃	C₂²	1	2	yes	complemented
16	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	complemented
17	#381×C₇₃	C₄	1	2	yes	—
18	#380×C₇₃	C₄	1	2	yes	—
19	#380×C₇₃	C₄	1	2	yes	—
20	#379×C₇₃	C₄	1	2	yes	—
21	C₄×C₂²⋊C₄×C₇₃	C₂²	1	4	yes	—
22	#336×C₇₃	C₂²	1	4	yes	complemented
23	#336×C₇₃	C₂²	1	2	yes	complemented
24	#336×C₇₃	C₂²	1	2	yes	complemented
25	C₂⁵×C₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₇₃	—	2	4	no	—
27	C₂³×C₄×C₇₃	—	4	2	no	—
28	C₂³×C₄×C₇₃	—	4	2	no	—
29	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₂³×C₄×C₇₃	C₂³	1	2	yes	—
32	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
34	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
35	#117×C₇₃	—	4	2	no	—
36	#117×C₇₃	—	4	2	no	—
37	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
46	C₄⋊C₈×C₇₃	—	4	4	no	—
47	C₂²⋊C₈×C₇₃	—	2	8	no	—
48	C₂²⋊C₈×C₇₃	—	4	4	no	—
49	C₂²⋊C₈×C₇₃	—	4	2	no	—
50	C₂²⋊C₈×C₇₃	—	4	2	no	—
51	#97×C₇₃	—	4	2	no	—
52	C₄×C₈×C₇₃	—	4	2	no	—
53	#95×C₇₃	C₂×C₄	1	4	yes	—
54	#95×C₇₃	C₂×C₄	1	4	yes	—
55	C₂⁴×C₇₃	—	2	2	no	—
56	C₂⁴×C₇₃	—	2	4	no	—
57	C₂⁴×C₇₃	—	4	1	no	—
58	C₂⁴×C₇₃	—	4	1	no	—
59	C₂⁴×C₇₃	—	4	2	no	—
60	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₂⁴×C₇₃	C₂²×C₄	1	2	yes	—
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	4	no	—
66	C₂²×C₄×C₇₃	—	2	4	no	—
67	C₂²×C₄×C₇₃	—	2	2	no	—
68	C₂²×C₄×C₇₃	—	2	2	no	—
69	C₂²×C₄×C₇₃	—	2	4	no	—
70	C₂²×C₄×C₇₃	—	4	2	no	—
71	C₂²×C₄×C₇₃	—	4	2	no	—
72	C₂²×C₄×C₇₃	—	4	1	no	—
73	C₂²×C₄×C₇₃	—	4	1	no	—
74	C₂²×C₄×C₇₃	—	4	16	no	—
75	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	#41	1	4	yes	—
81	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₇₃	—	2	4	no	—
83	C₂×C₈×C₇₃	—	2	4	no	—
84	C₂×C₈×C₇₃	—	2	4	no	—
85	C₂×C₈×C₇₃	—	2	4	no	—
86	C₂×C₈×C₇₃	—	2	4	no	—
87	C₂×C₈×C₇₃	—	2	4	no	—
88	C₄⋊C₄×C₇₃	—	4	2	no	—
89	C₄⋊C₄×C₇₃	—	4	1	no	—
90	C₄⋊C₄×C₇₃	—	4	1	no	—
91	C₂²⋊C₄×C₇₃	—	2	8	no	—
92	C₂²⋊C₄×C₇₃	—	4	4	no	—
93	C₂²⋊C₄×C₇₃	—	4	2	no	—
94	C₂²⋊C₄×C₇₃	—	4	2	no	—
95	C₄²×C₇₃	—	2	2	no	—
96	C₄²×C₇₃	—	2	2	no	—
97	C₄²×C₇₃	—	2	4	no	—
98	C₂³×C₇₃	—	2	2	no	—
99	C₂³×C₇₃	—	2	1	no	—
100	C₂³×C₇₃	—	2	1	no	—
101	C₂³×C₇₃	—	2	2	no	—
102	C₂³×C₇₃	—	2	2	no	—
103	C₂³×C₇₃	—	2	2	no	—
104	C₂³×C₇₃	—	2	2	no	—
105	C₂³×C₇₃	—	2	2	no	—
106	C₂³×C₇₃	—	2	2	no	—
107	C₂³×C₇₃	—	2	2	no	—
108	C₂³×C₇₃	—	4	8	no	—
109	C₂³×C₇₃	—	4	4	no	—
110	C₂³×C₇₃	—	4	4	no	—
111	C₂³×C₇₃	—	4	8	no	—
112	C₂³×C₇₃	—	4	4	no	—
113	C₂³×C₇₃	#117	1	2	yes	—
114	C₂³×C₇₃	#131	1	4	yes	—
115	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	—	2	1	no	—
117	C₂×C₄×C₇₃	—	2	1	no	—
118	C₂×C₄×C₇₃	—	2	8	no	—
119	C₂×C₄×C₇₃	—	2	2	no	—
120	C₂×C₄×C₇₃	—	2	4	no	—
121	C₂×C₄×C₇₃	—	2	2	no	—
122	C₂×C₄×C₇₃	—	2	2	no	—
123	C₂×C₄×C₇₃	—	2	2	no	—
124	C₂×C₄×C₇₃	—	2	2	no	—
125	C₂×C₄×C₇₃	—	2	4	no	—
126	C₂×C₄×C₇₃	—	2	4	no	—
127	C₂×C₄×C₇₃	—	2	4	no	—
128	C₂×C₄×C₇₃	—	2	4	no	—
129	C₂×C₄×C₇₃	—	4	8	no	—
130	C₂×C₄×C₇₃	—	4	4	no	—
131	C₂×C₄×C₇₃	—	4	4	no	—
132	C₂×C₄×C₇₃	—	4	4	no	—
133	C₂×C₄×C₇₃	#117	1	2	yes	—
134	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
135	C₂×C₄×C₇₃	H(5,𝔽₂)	1	2	yes	—
136	C₂×C₄×C₇₃	H(5,𝔽₂)	1	2	yes	—
137	C₈×C₇₃	—	4	4	no	—
138	C₈×C₇₃	—	4	4	no	—
139	C₂²×C₇₃	—	2	2	no	—
140	C₂²×C₇₃	—	2	4	no	—
141	C₂²×C₇₃	—	2	4	no	—
142	C₂²×C₇₃	—	2	4	no	—
143	C₂²×C₇₃	—	2	4	no	—
144	C₂²×C₇₃	—	4	16	no	—
145	C₂²×C₇₃	—	4	2	no	—
146	C₂²×C₇₃	—	4	2	no	—
147	C₂²×C₇₃	—	4	2	no	—
148	C₂²×C₇₃	—	4	2	no	—
149	C₂²×C₇₃	—	4	1	no	—
150	C₂²×C₇₃	—	4	1	no	—
151	C₂²×C₇₃	—	4	2	no	—
152	C₂²×C₇₃	C₂×#117	1	1	yes	characteristic
153	C₂²×C₇₃	#518	1	1	yes	characteristic
154	C₂²×C₇₃	#518	1	1	yes	characteristic
155	C₂²×C₇₃	C₂×#131	1	2	yes	—
156	C₂²×C₇₃	#568	1	2	yes	—
157	C₄×C₇₃	—	2	2	no	—
158	C₄×C₇₃	—	4	2	no	—
159	C₄×C₇₃	—	4	4	no	—
160	C₄×C₇₃	—	4	2	no	—
161	C₄×C₇₃	—	4	2	no	—
162	C₄×C₇₃	#538	1	4	yes	—
163	#21910	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂×C₇₃	—	4	2	no	—
165	C₂×C₇₃	—	4	4	no	—
166	C₂×C₇₃	#2312	1	1	yes	characteristic
167	C₂×C₇₃	#2924	1	2	yes	—
168	C₂×C₇₃	#2976	1	2	yes	—
169	C₂×C₇₃	#2977	1	1	yes	characteristic
170	C₂×C₇₃	#2978	1	1	yes	characteristic
171	#2280	C₂×C₇₃	1	1	yes	characteristic
172	#1924	C₂×C₇₃	1	4	yes	complemented
173	#1923	C₂×C₇₃	1	4	yes	complemented
174	#1798	C₂×C₇₃	1	2	yes	complemented
175	#1779	C₂×C₇₃	1	2	yes	—
176	#1750	C₂×C₇₃	1	2	yes	complemented
177	C₇₃	#21910	1	1	yes	characteristic, direct product factor, Sylow
178	C₂⁴×C₄	C₂²×C₇₃	1	1	yes	characteristic
179	C₂×#117	C₂²×C₇₃	1	1	yes	characteristic
180	C₂×#117	C₂²×C₇₃	1	1	yes	characteristic
181	C₂×C₄⋊C₈	C₂²×C₇₃	1	4	yes	complemented
182	C₂×C₂²⋊C₈	C₂²×C₇₃	1	4	yes	—
183	C₂×C₂²⋊C₈	C₂²×C₇₃	1	2	yes	—
184	C₂×C₂²⋊C₈	C₂²×C₇₃	1	4	yes	—
185	C₂×C₂²⋊C₈	C₂²×C₇₃	1	2	yes	—
186	C₂×#97	C₂²×C₇₃	1	2	yes	complemented
187	C₂×C₄×C₈	C₂²×C₇₃	1	2	yes	complemented
188	#381	C₄×C₇₃	1	2	yes	—
189	#380	C₄×C₇₃	1	2	yes	—
190	#380	C₄×C₇₃	1	2	yes	—
191	#379	C₄×C₇₃	1	2	yes	—
192	C₄×C₂²⋊C₄	C₂²×C₇₃	1	4	yes	—
193	#336	C₂²×C₇₃	1	2	yes	complemented
194	#336	C₂²×C₇₃	1	2	yes	complemented
195	#336	C₂²×C₇₃	1	4	yes	complemented
196	C₂⁵	C₂×C₄×C₇₃	1	1	yes	characteristic
197	C₂³×C₄	—	2	4	no	—
198	C₂³×C₄	—	4	2	no	—
199	C₂³×C₄	—	4	2	no	—
200	C₂³×C₄	C₂×C₄×C₇₃	1	2	yes	—
201	C₂³×C₄	C₂×C₄×C₇₃	1	1	yes	characteristic
202	C₂³×C₄	C₂³×C₇₃	1	2	yes	—
203	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₇₃	1	4	yes	—
205	C₂²×C₈	C₂³×C₇₃	1	4	yes	—
206	#117	—	4	2	no	—
207	#117	—	4	2	no	—
208	C₂×C₄⋊C₄	C₂×C₄×C₇₃	1	2	yes	—
209	C₂×C₄⋊C₄	C₂×C₄×C₇₃	1	1	yes	characteristic
210	C₂×C₄⋊C₄	C₂×C₄×C₇₃	1	1	yes	characteristic
211	C₂×C₂²⋊C₄	C₂×C₄×C₇₃	1	4	yes	—
212	C₂×C₂²⋊C₄	C₂×C₄×C₇₃	1	2	yes	—
213	C₂×C₂²⋊C₄	C₂×C₄×C₇₃	1	2	yes	—
214	C₂×C₂²⋊C₄	C₂×C₄×C₇₃	1	4	yes	—
215	C₂×C₄²	C₂³×C₇₃	1	2	yes	—
216	C₂×C₄²	C₂³×C₇₃	1	2	yes	—
217	C₄⋊C₈	—	4	4	no	—
218	C₂²⋊C₈	—	2	8	no	—
219	C₂²⋊C₈	—	4	4	no	—
220	C₂²⋊C₈	—	4	2	no	—
221	C₂²⋊C₈	—	4	2	no	—
222	#97	—	4	2	no	—
223	C₄×C₈	—	4	2	no	—
224	#95	C₂×C₄×C₇₃	1	4	yes	—
225	#95	C₂×C₄×C₇₃	1	4	yes	—
226	C₂⁴	—	2	2	no	—
227	C₂⁴	—	2	4	no	—
228	C₂⁴	—	4	1	no	—
229	C₂⁴	—	4	1	no	—
230	C₂⁴	—	4	2	no	—
231	C₂⁴	C₂²×C₄×C₇₃	1	1	yes	characteristic
232	C₂⁴	C₂²×C₄×C₇₃	1	2	yes	—
233	C₂²×C₄	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	4	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	4	2	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	4	1	no	—
245	C₂²×C₄	—	4	16	no	—
246	C₂²×C₄	C₂²×C₄×C₇₃	1	2	yes	—
247	C₂²×C₄	C₂²×C₄×C₇₃	1	2	yes	—
248	C₂²×C₄	C₂²×C₄×C₇₃	1	4	yes	—
249	C₂²×C₄	C₂²×C₄×C₇₃	1	2	yes	—
250	C₂²×C₄	C₂²×C₄×C₇₃	1	1	yes	characteristic
251	C₂²×C₄	#41×C₇₃	1	4	yes	—
252	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	2	4	no	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	4	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	2	8	no	—
263	C₂²⋊C₄	—	4	4	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	4	no	—
269	C₂³	—	2	2	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	2	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	2	no	—
279	C₂³	—	4	4	no	—
280	C₂³	—	4	8	no	—
281	C₂³	—	4	4	no	—
282	C₂³	—	4	8	no	—
283	C₂³	—	4	4	no	—
284	C₂³	#117×C₇₃	1	2	yes	—
285	C₂³	#131×C₇₃	1	4	yes	—
286	C₂³	C₂³×C₄×C₇₃	1	1	yes	characteristic
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	8	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	4	no	—
300	C₂×C₄	—	4	8	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	—	4	4	no	—
304	C₂×C₄	#117×C₇₃	1	2	yes	—
305	C₂×C₄	C₂×#41×C₇₃	1	2	yes	—
306	C₂×C₄	H(5,𝔽₂)×C₇₃	1	2	yes	—
307	C₂×C₄	H(5,𝔽₂)×C₇₃	1	2	yes	—
308	C₈	—	4	4	no	—
309	C₈	—	4	4	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	2	4	no	—
315	C₂²	—	4	16	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	2	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	4	2	no	—
323	C₂²	C₂×#117×C₇₃	1	1	yes	characteristic
324	C₂²	#518×C₇₃	1	1	yes	characteristic
325	C₂²	#518×C₇₃	1	1	yes	characteristic
326	C₂²	C₂×#131×C₇₃	1	2	yes	—
327	C₂²	#568×C₇₃	1	2	yes	—
328	C₄	—	2	2	no	—
329	C₄	—	4	4	no	—
330	C₄	—	4	2	no	—
331	C₄	—	4	2	no	—
332	C₄	—	4	2	no	—
333	C₄	#538×C₇₃	1	4	yes	—
334	C₂	—	4	4	no	—
335	C₂	—	4	2	no	—
336	C₂	C₇₃×#2312	1	1	yes	characteristic
337	C₂	C₇₃×#2924	1	2	yes	—
338	C₂	C₇₃×#2976	1	2	yes	—
339	C₂	C₇₃×#2977	1	1	yes	characteristic
340	C₂	C₇₃×#2978	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₇₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₃
socle	C₂³×C₇₃	C₂³×C₄
prime radical	C₂⁵×C₇₃	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₃×#1924	C₂	semidirect
3. C₇₃×#1923	C₂	semidirect
4. C₇₃×#1798	C₂	semidirect
6. C₇₃×#1750	C₂	semidirect
10. C₂×C₄⋊C₈×C₇₃	C₂²	semidirect
15. C₂×#97×C₇₃	C₂²	semidirect
16. C₂×C₄×C₈×C₇₃	C₂²	semidirect
22. #336×C₇₃	C₂²	semidirect
23. #336×C₇₃	C₂²	semidirect
24. #336×C₇₃	C₂²	semidirect
163. #21910	C₇₃	direct (N × H)
172. #1924	C₂×C₇₃	semidirect
173. #1923	C₂×C₇₃	semidirect
174. #1798	C₂×C₇₃	semidirect
176. #1750	C₂×C₇₃	semidirect
177. C₇₃	#21910	direct (N × H)
181. C₂×C₄⋊C₈	C₂²×C₇₃	semidirect
186. C₂×#97	C₂²×C₇₃	semidirect
187. C₂×C₄×C₈	C₂²×C₇₃	semidirect
193. #336	C₂²×C₇₃	semidirect
194. #336	C₂²×C₇₃	semidirect
195. #336	C₂²×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₇₃×#2280	2
C₇₃×#1924	2
C₇₃×#1923	2
C₇₃×#1798	2
C₇₃×#1779	2
C₇₃×#1750	2
#21910	73

Subgroups — C73×#21910 #32931314

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#21910 #32931314