Subgroups — C₇₃×#21842 #32931246

Order 18688.

1010 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#432×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#2283	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#1993	C₂	1	2	yes	maximal, complemented
5	C₇₃×#1993	C₂	1	2	yes	maximal, complemented
6	C₇₃×#1989	C₂	1	4	yes	maximal, complemented
7	C₇₃×#1825	C₂	1	4	yes	maximal
8	#533×C₇₃	—	2	4	no	—
9	#533×C₇₃	C₄	1	8	yes	complemented
10	C₂×C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×#125×C₇₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊Q₈×C₇₃	—	2	1	no	—
14	C₂×C₂²⋊Q₈×C₇₃	—	2	1	no	—
15	C₂×C₄×Q₈×C₇₃	—	2	1	no	—
16	C₂×C₄×Q₈×C₇₃	—	2	1	no	—
17	C₂×#117×C₇₃	C₂²	1	2	yes	—
18	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
19	#432×C₇₃	—	2	4	no	—
20	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₈×C₇₃	C₂²	1	2	yes	—
22	C₂×#97×C₇₃	C₂²	1	2	yes	—
23	#383×C₇₃	—	2	1	no	—
24	#382×C₇₃	C₂²	1	2	yes	—
25	#381×C₇₃	—	2	1	no	—
26	#380×C₇₃	C₂²	1	2	yes	—
27	C₄×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
28	C₄×C₂²⋊C₄×C₇₃	—	2	1	no	—
29	#376×C₇₃	—	2	1	no	—
30	#340×C₇₃	C₂²	1	8	yes	—
31	#340×C₇₃	C₂²	1	8	yes	—
32	C₂²×Q₈×C₇₃	—	2	1	no	—
33	C₂²×Q₈×C₇₃	—	2	1	no	—
34	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
35	C₂³×C₄×C₇₃	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
38	C₄⋊Q₈×C₇₃	—	2	2	no	—
39	C₄⋊Q₈×C₇₃	—	2	2	no	—
40	C₄⋊Q₈×C₇₃	C₂×C₄	1	4	yes	—
41	C₄⋊Q₈×C₇₃	C₂×C₄	1	4	yes	—
42	#125×C₇₃	—	2	4	no	—
43	#125×C₇₃	C₂×C₄	1	8	yes	—
44	C₂²⋊Q₈×C₇₃	—	2	8	no	—
45	C₂²⋊Q₈×C₇₃	—	2	8	no	—
46	C₄×Q₈×C₇₃	—	2	8	no	—
47	C₄×Q₈×C₇₃	—	2	8	no	—
48	#117×C₇₃	—	2	4	no	—
49	#117×C₇₃	C₂×C₄	1	8	yes	—
50	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
55	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
56	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
57	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
58	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
59	C₂×C₄²×C₇₃	—	2	2	no	—
60	C₂×C₄²×C₇₃	—	2	2	no	—
61	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
62	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
64	C₂×C₄²×C₇₃	D₄	1	4	yes	—
65	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₇₃	—	2	4	no	—
68	C₂²⋊C₈×C₇₃	—	2	4	no	—
69	#97×C₇₃	—	2	4	no	—
70	#95×C₇₃	—	2	2	no	—
71	#95×C₇₃	—	2	2	no	—
72	#95×C₇₃	—	2	2	no	—
73	#95×C₇₃	—	2	2	no	—
74	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
75	C₂×Q₈×C₇₃	—	2	4	no	—
76	C₂×Q₈×C₇₃	—	2	4	no	—
77	C₂×Q₈×C₇₃	—	4	4	no	—
78	C₂×Q₈×C₇₃	—	4	4	no	—
79	C₂²×C₄×C₇₃	—	2	4	no	—
80	C₂²×C₄×C₇₃	—	2	2	no	—
81	C₂²×C₄×C₇₃	—	2	2	no	—
82	C₂²×C₄×C₇₃	—	2	4	no	—
83	C₂²×C₄×C₇₃	—	4	2	no	—
84	C₂²×C₄×C₇₃	—	4	2	no	—
85	C₂²×C₄×C₇₃	C₂²⋊C₄	1	4	yes	—
86	C₂²×C₄×C₇₃	C₂²⋊C₄	1	4	yes	—
87	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
95	C₂×C₈×C₇₃	—	2	4	no	—
96	C₂×C₈×C₇₃	—	2	8	no	—
97	C₄⋊C₄×C₇₃	—	2	8	no	—
98	C₄⋊C₄×C₇₃	—	2	8	no	—
99	C₄⋊C₄×C₇₃	—	2	4	no	—
100	C₄⋊C₄×C₇₃	—	2	4	no	—
101	C₄⋊C₄×C₇₃	—	2	2	no	—
102	C₄⋊C₄×C₇₃	—	2	2	no	—
103	C₄⋊C₄×C₇₃	—	2	8	no	—
104	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
105	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
106	C₂²⋊C₄×C₇₃	—	2	8	no	—
107	C₂²⋊C₄×C₇₃	—	4	4	no	—
108	C₄²×C₇₃	—	2	8	no	—
109	C₄²×C₇₃	—	4	4	no	—
110	C₄²×C₇₃	C₂²⋊C₄	1	4	yes	—
111	C₄²×C₇₃	C₂²⋊C₄	1	4	yes	—
112	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
113	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
114	C₄²×C₇₃	C₂×D₄	1	2	yes	—
115	C₄²×C₇₃	C₂×D₄	1	2	yes	—
116	C₂³×C₇₃	—	2	2	no	—
117	C₂³×C₇₃	—	2	2	no	—
118	C₂³×C₇₃	—	2	2	no	—
119	C₂³×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
120	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
121	Q₈×C₇₃	—	4	8	no	—
122	Q₈×C₇₃	—	4	8	no	—
123	C₂×C₄×C₇₃	—	2	2	no	—
124	C₂×C₄×C₇₃	—	2	2	no	—
125	C₂×C₄×C₇₃	—	2	2	no	—
126	C₂×C₄×C₇₃	—	2	2	no	—
127	C₂×C₄×C₇₃	—	2	2	no	—
128	C₂×C₄×C₇₃	—	2	4	no	—
129	C₂×C₄×C₇₃	—	2	4	no	—
130	C₂×C₄×C₇₃	—	2	4	no	—
131	C₂×C₄×C₇₃	—	2	4	no	—
132	C₂×C₄×C₇₃	—	4	4	no	—
133	C₂×C₄×C₇₃	—	4	4	no	—
134	C₂×C₄×C₇₃	—	4	4	no	—
135	C₂×C₄×C₇₃	—	4	4	no	—
136	C₂×C₄×C₇₃	—	4	4	no	—
137	C₂×C₄×C₇₃	—	4	4	no	—
138	C₂×C₄×C₇₃	—	4	4	no	—
139	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
140	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
141	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
142	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
143	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
144	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇₃	2₋¹⁺⁴	1	4	yes	—
147	C₈×C₇₃	—	4	4	no	—
148	C₂²×C₇₃	—	2	2	no	—
149	C₂²×C₇₃	—	2	2	no	—
150	C₂²×C₇₃	—	2	2	no	—
151	C₂²×C₇₃	—	4	4	no	—
152	C₂²×C₇₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
153	C₂²×C₇₃	#536	1	2	yes	—
154	C₂²×C₇₃	#575	1	2	yes	—
155	C₂²×C₇₃	#575	1	2	yes	—
156	C₄×C₇₃	—	2	2	no	—
157	C₄×C₇₃	—	2	2	no	—
158	C₄×C₇₃	—	2	2	no	—
159	C₄×C₇₃	—	2	2	no	—
160	C₄×C₇₃	—	2	2	no	—
161	C₄×C₇₃	—	4	4	no	—
162	C₄×C₇₃	—	4	4	no	—
163	C₄×C₇₃	—	8	4	no	—
164	C₄×C₇₃	#519	1	2	yes	—
165	C₄×C₇₃	#536	1	2	yes	—
166	#21842	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
167	C₂×C₇₃	—	4	2	no	—
168	C₂×C₇₃	#2310	1	1	yes	characteristic
169	C₂×C₇₃	#2896	1	1	yes	characteristic
170	C₂×C₇₃	#2896	1	1	yes	characteristic
171	C₂×C₇₃	#3240	1	4	yes	—
172	C₂×#533	C₂×C₇₃	1	1	yes	characteristic
173	C₂×#432	C₂×C₇₃	1	1	yes	characteristic
174	#2283	C₂×C₇₃	1	1	yes	characteristic
175	#1993	C₂×C₇₃	1	2	yes	complemented
176	#1993	C₂×C₇₃	1	2	yes	complemented
177	#1989	C₂×C₇₃	1	4	yes	complemented
178	#1825	C₂×C₇₃	1	4	yes	—
179	C₇₃	#21842	1	1	yes	characteristic, direct product factor, Sylow
180	#533	—	2	4	no	—
181	#533	C₄×C₇₃	1	8	yes	complemented
182	C₂×C₄⋊Q₈	C₂²×C₇₃	1	1	yes	characteristic
183	C₂×C₄⋊Q₈	C₂²×C₇₃	1	1	yes	characteristic
184	C₂×#125	C₂²×C₇₃	1	2	yes	—
185	C₂×C₂²⋊Q₈	—	2	1	no	—
186	C₂×C₂²⋊Q₈	—	2	1	no	—
187	C₂×C₄×Q₈	—	2	1	no	—
188	C₂×C₄×Q₈	—	2	1	no	—
189	C₂×#117	C₂²×C₇₃	1	2	yes	—
190	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
191	#432	—	2	4	no	—
192	C₂×C₄⋊C₈	C₂²×C₇₃	1	2	yes	—
193	C₂×C₂²⋊C₈	C₂²×C₇₃	1	2	yes	—
194	C₂×#97	C₂²×C₇₃	1	2	yes	—
195	#383	—	2	1	no	—
196	#382	C₂²×C₇₃	1	2	yes	—
197	#381	—	2	1	no	—
198	#380	C₂²×C₇₃	1	2	yes	—
199	C₄×C₄⋊C₄	C₂²×C₇₃	1	2	yes	—
200	C₄×C₂²⋊C₄	—	2	1	no	—
201	#376	—	2	1	no	—
202	#340	C₂²×C₇₃	1	8	yes	—
203	#340	C₂²×C₇₃	1	8	yes	—
204	C₂²×Q₈	—	2	1	no	—
205	C₂²×Q₈	—	2	1	no	—
206	C₂³×C₄	D₄×C₇₃	1	1	yes	characteristic
207	C₂³×C₄	D₄×C₇₃	1	1	yes	characteristic
208	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
209	C₂²×C₈	C₂³×C₇₃	1	4	yes	—
210	C₄⋊Q₈	—	2	2	no	—
211	C₄⋊Q₈	—	2	2	no	—
212	C₄⋊Q₈	C₂×C₄×C₇₃	1	4	yes	—
213	C₄⋊Q₈	C₂×C₄×C₇₃	1	4	yes	—
214	#125	—	2	4	no	—
215	#125	C₂×C₄×C₇₃	1	8	yes	—
216	C₂²⋊Q₈	—	2	8	no	—
217	C₂²⋊Q₈	—	2	8	no	—
218	C₄×Q₈	—	2	8	no	—
219	C₄×Q₈	—	2	8	no	—
220	#117	—	2	4	no	—
221	#117	C₂×C₄×C₇₃	1	8	yes	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	—	2	1	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	C₂³×C₇₃	1	4	yes	—
227	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
228	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₄²	—	2	2	no	—
232	C₂×C₄²	—	2	2	no	—
233	C₂×C₄²	C₂×C₄×C₇₃	1	4	yes	—
234	C₂×C₄²	D₄×C₇₃	1	1	yes	characteristic
235	C₂×C₄²	D₄×C₇₃	1	1	yes	characteristic
236	C₂×C₄²	D₄×C₇₃	1	4	yes	—
237	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
238	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
239	C₄⋊C₈	—	2	4	no	—
240	C₂²⋊C₈	—	2	4	no	—
241	#97	—	2	4	no	—
242	#95	—	2	2	no	—
243	#95	—	2	2	no	—
244	#95	—	2	2	no	—
245	#95	—	2	2	no	—
246	C₂⁴	C₂×D₄×C₇₃	1	1	yes	characteristic
247	C₂×Q₈	—	2	4	no	—
248	C₂×Q₈	—	2	4	no	—
249	C₂×Q₈	—	4	4	no	—
250	C₂×Q₈	—	4	4	no	—
251	C₂²×C₄	—	2	4	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	C₂²⋊C₄×C₇₃	1	4	yes	—
258	C₂²×C₄	C₂²⋊C₄×C₇₃	1	4	yes	—
259	C₂²×C₄	C₂²×C₄×C₇₃	1	2	yes	—
260	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
262	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
264	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
266	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
267	C₂×C₈	—	2	4	no	—
268	C₂×C₈	—	2	8	no	—
269	C₄⋊C₄	—	2	8	no	—
270	C₄⋊C₄	—	2	8	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	2	4	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	8	no	—
276	C₄⋊C₄	C₂²×C₄×C₇₃	1	4	yes	—
277	C₄⋊C₄	C₂²×C₄×C₇₃	1	4	yes	—
278	C₂²⋊C₄	—	2	8	no	—
279	C₂²⋊C₄	—	4	4	no	—
280	C₄²	—	2	8	no	—
281	C₄²	—	4	4	no	—
282	C₄²	C₂²⋊C₄×C₇₃	1	4	yes	—
283	C₄²	C₂²⋊C₄×C₇₃	1	4	yes	—
284	C₄²	C₂²×C₄×C₇₃	1	2	yes	—
285	C₄²	C₂²×C₄×C₇₃	1	2	yes	—
286	C₄²	C₂×D₄×C₇₃	1	2	yes	—
287	C₄²	C₂×D₄×C₇₃	1	2	yes	—
288	C₂³	—	2	2	no	—
289	C₂³	—	2	2	no	—
290	C₂³	—	2	2	no	—
291	C₂³	C₂×C₂²⋊C₄×C₇₃	1	2	yes	—
292	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
293	Q₈	—	4	8	no	—
294	Q₈	—	4	8	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	4	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	4	no	—
304	C₂×C₄	—	4	4	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	4	no	—
309	C₂×C₄	—	4	4	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	2	yes	—
312	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	2	yes	—
313	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	2	yes	—
314	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	2	yes	—
315	C₂×C₄	C₂×C₂²⋊C₄×C₇₃	1	2	yes	—
316	C₂×C₄	C₂³×C₄×C₇₃	1	1	yes	characteristic
317	C₂×C₄	C₂²×D₄×C₇₃	1	1	yes	characteristic
318	C₂×C₄	2₋¹⁺⁴×C₇₃	1	4	yes	—
319	C₈	—	4	4	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	4	4	no	—
324	C₂²	C₂²×C₂²⋊C₄×C₇₃	1	1	yes	characteristic
325	C₂²	#536×C₇₃	1	2	yes	—
326	C₂²	#575×C₇₃	1	2	yes	—
327	C₂²	#575×C₇₃	1	2	yes	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	4	4	no	—
334	C₄	—	4	4	no	—
335	C₄	—	8	4	no	—
336	C₄	#519×C₇₃	1	2	yes	—
337	C₄	#536×C₇₃	1	2	yes	—
338	C₂	—	4	2	no	—
339	C₂	C₇₃×#2310	1	1	yes	characteristic
340	C₂	C₇₃×#2896	1	1	yes	characteristic
341	C₂	C₇₃×#2896	1	1	yes	characteristic
342	C₂	C₇₃×#3240	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₇₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₃
prime radical	C₂⁴×C₇₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
4. C₇₃×#1993	C₂	semidirect
5. C₇₃×#1993	C₂	semidirect
6. C₇₃×#1989	C₂	semidirect
9. #533×C₇₃	C₄	semidirect
166. #21842	C₇₃	direct (N × H)
175. #1993	C₂×C₇₃	semidirect
176. #1993	C₂×C₇₃	semidirect
177. #1989	C₂×C₇₃	semidirect
179. C₇₃	#21842	direct (N × H)
181. #533	C₄×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#533×C₇₃	2
C₂×#432×C₇₃	2
C₇₃×#2283	2
C₇₃×#1993	2
C₇₃×#1989	2
C₇₃×#1825	2
#21842	73

Subgroups — C73×#21842 #32931246

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#21842 #32931246