Subgroups — C₇₃×#23463 #32932867

Order 18688.

1306 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₃×#3458	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#418×C₇₃	C₂	1	2	yes	maximal
3	C₇₃×#2309	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#2027	C₂	1	8	yes	maximal, complemented
5	C₇₃×#1742	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇₃×#1733	C₂	1	2	yes	maximal, complemented
7	C₂³×D₄×C₇₃	C₂²	1	1	yes	characteristic
8	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
9	#544×C₇₃	—	2	4	no	—
10	#544×C₇₃	C₂²	1	4	yes	complemented
11	#543×C₇₃	—	2	2	no	—
12	#533×C₇₃	—	2	4	no	—
13	#533×C₇₃	—	2	2	no	—
14	C₂×#117×C₇₃	C₂²	1	2	yes	—
15	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
17	#418×C₇₃	—	2	4	no	—
18	#418×C₇₃	C₂²	1	8	yes	—
19	C₂×#102×C₇₃	—	2	4	no	—
20	#390×C₇₃	—	2	2	no	—
21	#390×C₇₃	C₂²	1	4	yes	—
22	#379×C₇₃	—	2	2	no	—
23	#343×C₇₃	C₂²	1	4	yes	complemented
24	#339×C₇₃	C₂²	1	8	yes	complemented
25	C₂⁵×C₇₃	—	2	1	no	—
26	C₂²×D₄×C₇₃	—	2	2	no	—
27	C₂²×D₄×C₇₃	—	2	2	no	—
28	C₂²×D₄×C₇₃	—	2	1	no	—
29	C₂²×D₄×C₇₃	—	2	4	no	—
30	C₂²×D₄×C₇₃	—	2	2	no	—
31	C₂²×D₄×C₇₃	Q₈	1	4	yes	complemented
32	C₂²×D₄×C₇₃	C₂³	1	2	yes	—
33	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
34	C₂×M₁₆×C₇₃	—	2	2	no	—
35	C₂×M₁₆×C₇₃	—	2	4	no	—
36	C₂×M₁₆×C₇₃	—	2	4	no	—
37	C₂×M₁₆×C₇₃	C₂³	1	4	yes	—
38	C₂²×C₈×C₇₃	—	2	2	no	—
39	C₄⋊Q₈×C₇₃	—	2	4	no	—
40	C₄⋊Q₈×C₇₃	—	2	8	no	—
41	#125×C₇₃	—	2	4	no	—
42	#125×C₇₃	—	2	8	no	—
43	C₂²⋊Q₈×C₇₃	—	2	16	no	—
44	C₂²⋊Q₈×C₇₃	—	2	8	no	—
45	C₄×Q₈×C₇₃	—	2	8	no	—
46	#117×C₇₃	—	2	2	no	—
47	#117×C₇₃	—	2	2	no	—
48	#117×C₇₃	—	2	2	no	—
49	#117×C₇₃	—	2	2	no	—
50	#117×C₇₃	—	2	4	no	—
51	#117×C₇₃	D₄	1	4	yes	—
52	#117×C₇₃	C₂³	1	4	yes	—
53	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₇₃	D₄	1	4	yes	—
55	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
56	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
58	C₂×C₂²⋊C₄×C₇₃	—	2	8	no	—
59	C₂×C₂²⋊C₄×C₇₃	D₄	1	2	yes	—
60	C₂×C₄²×C₇₃	—	2	2	no	—
61	C₂×C₄²×C₇₃	D₄	1	2	yes	—
62	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
63	#102×C₇₃	—	2	16	no	—
64	C₂⁴×C₇₃	—	2	2	no	—
65	C₂⁴×C₇₃	—	2	4	no	—
66	C₂⁴×C₇₃	—	4	2	no	—
67	C₂⁴×C₇₃	—	4	2	no	—
68	C₂⁴×C₇₃	—	8	4	no	—
69	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
70	C₂×Q₈×C₇₃	—	2	8	no	—
71	C₂×D₄×C₇₃	—	2	8	no	—
72	C₂×D₄×C₇₃	—	2	8	no	—
73	C₂×D₄×C₇₃	—	2	4	no	—
74	C₂×D₄×C₇₃	—	2	4	no	—
75	C₂×D₄×C₇₃	—	2	4	no	—
76	C₂×D₄×C₇₃	—	2	2	no	—
77	C₂×D₄×C₇₃	—	4	1	no	—
78	C₂×D₄×C₇₃	—	4	1	no	—
79	C₂×D₄×C₇₃	—	4	4	no	—
80	C₂×D₄×C₇₃	—	4	4	no	—
81	C₂×D₄×C₇₃	—	4	2	no	—
82	C₂×D₄×C₇₃	C₂×Q₈	1	4	yes	—
83	C₂²×C₄×C₇₃	—	2	4	no	—
84	C₂²×C₄×C₇₃	—	2	2	no	—
85	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
90	C₂²×C₄×C₇₃	#41	1	4	yes	—
91	C₂²×C₄×C₇₃	#41	1	2	yes	—
92	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
93	M₁₆×C₇₃	—	2	4	no	—
94	M₁₆×C₇₃	—	2	4	no	—
95	M₁₆×C₇₃	—	4	4	no	—
96	C₂×C₈×C₇₃	—	2	4	no	—
97	C₂×C₈×C₇₃	—	4	4	no	—
98	C₄⋊C₄×C₇₃	—	2	16	no	—
99	C₄⋊C₄×C₇₃	—	2	8	no	—
100	C₄⋊C₄×C₇₃	—	2	4	no	—
101	C₄⋊C₄×C₇₃	—	2	4	no	—
102	C₄⋊C₄×C₇₃	—	4	2	no	—
103	C₄⋊C₄×C₇₃	C₂×D₄	1	8	yes	—
104	C₂²⋊C₄×C₇₃	—	2	4	no	—
105	C₂²⋊C₄×C₇₃	—	2	4	no	—
106	C₂²⋊C₄×C₇₃	—	2	4	no	—
107	C₂²⋊C₄×C₇₃	—	4	2	no	—
108	C₂²⋊C₄×C₇₃	—	8	8	no	—
109	C₄²×C₇₃	—	2	4	no	—
110	C₄²×C₇₃	—	2	2	no	—
111	C₄²×C₇₃	—	2	2	no	—
112	C₄²×C₇₃	—	2	4	no	—
113	C₄²×C₇₃	—	2	4	no	—
114	C₂³×C₇₃	—	2	4	no	—
115	C₂³×C₇₃	—	4	2	no	—
116	C₂³×C₇₃	—	4	8	no	—
117	C₂³×C₇₃	—	4	2	no	—
118	C₂³×C₇₃	—	4	2	no	—
119	C₂³×C₇₃	—	4	8	no	—
120	C₂³×C₇₃	—	8	8	no	—
121	C₂³×C₇₃	—	8	8	no	—
122	C₂³×C₇₃	—	8	8	no	—
123	C₂³×C₇₃	#120	1	2	yes	—
124	C₂³×C₇₃	C₂²⋊Q₈	1	2	yes	—
125	C₂³×C₇₃	#124	1	2	yes	—
126	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
127	Q₈×C₇₃	—	8	8	no	—
128	D₄×C₇₃	—	2	8	no	—
129	D₄×C₇₃	—	4	4	no	—
130	D₄×C₇₃	—	4	4	no	—
131	D₄×C₇₃	—	4	4	no	—
132	C₂×C₄×C₇₃	—	2	4	no	—
133	C₂×C₄×C₇₃	—	2	8	no	—
134	C₂×C₄×C₇₃	—	4	4	no	—
135	C₂×C₄×C₇₃	—	4	4	no	—
136	C₂×C₄×C₇₃	—	4	2	no	—
137	C₂×C₄×C₇₃	—	8	4	no	—
138	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
139	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
140	C₂×C₄×C₇₃	C₂²⋊Q₈	1	4	yes	—
141	C₂×C₄×C₇₃	C₂²⋊Q₈	1	4	yes	—
142	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₇₃	#124	1	4	yes	—
144	C₂×C₄×C₇₃	#124	1	4	yes	—
145	C₂×C₄×C₇₃	#124	1	2	yes	—
146	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
147	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
148	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
149	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
150	C₈×C₇₃	—	4	4	no	—
151	C₂²×C₇₃	—	2	2	no	—
152	C₂²×C₇₃	—	2	4	no	—
153	C₂²×C₇₃	—	4	2	no	—
154	C₂²×C₇₃	—	4	2	no	—
155	C₂²×C₇₃	—	4	8	no	—
156	C₂²×C₇₃	—	4	4	no	—
157	C₂²×C₇₃	—	8	4	no	—
158	C₂²×C₇₃	—	8	16	no	—
159	C₂²×C₇₃	—	8	4	no	—
160	C₂²×C₇₃	#393	1	4	yes	—
161	C₂²×C₇₃	C₂×#120	1	1	yes	characteristic
162	C₂²×C₇₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
163	C₂²×C₇₃	C₂×#124	1	1	yes	characteristic
164	C₄×C₇₃	—	4	4	no	—
165	C₄×C₇₃	—	4	8	no	—
166	C₄×C₇₃	—	8	2	no	—
167	C₄×C₇₃	#393	1	2	yes	—
168	C₄×C₇₃	#393	1	2	yes	—
169	C₄×C₇₃	C₂×C₂²⋊Q₈	1	2	yes	—
170	C₄×C₇₃	C₂×#124	1	2	yes	—
171	#23463	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂×C₇₃	—	2	1	no	—
173	C₂×C₇₃	—	2	1	no	—
174	C₂×C₇₃	—	2	4	no	—
175	C₂×C₇₃	—	8	4	no	—
176	C₂×C₇₃	C₂×#393	1	1	yes	characteristic
177	C₂×C₇₃	#3021	1	2	yes	—
178	#3458	C₂×C₇₃	1	1	yes	characteristic, complemented
179	C₂×#418	C₂×C₇₃	1	2	yes	—
180	#2309	C₂×C₇₃	1	1	yes	characteristic
181	#2027	C₂×C₇₃	1	8	yes	complemented
182	#1742	C₂×C₇₃	1	1	yes	characteristic, complemented
183	#1733	C₂×C₇₃	1	2	yes	complemented
184	C₇₃	#23463	1	1	yes	characteristic, direct product factor, Sylow
185	C₂³×D₄	C₂²×C₇₃	1	1	yes	characteristic
186	C₂²×M₁₆	C₂²×C₇₃	1	2	yes	—
187	#544	—	2	4	no	—
188	#544	C₂²×C₇₃	1	4	yes	complemented
189	#543	—	2	2	no	—
190	#533	—	2	4	no	—
191	#533	—	2	2	no	—
192	C₂×#117	C₂²×C₇₃	1	2	yes	—
193	C₂×#117	C₂²×C₇₃	1	1	yes	characteristic
194	C₂×#117	C₂²×C₇₃	1	1	yes	characteristic
195	#418	—	2	4	no	—
196	#418	C₂²×C₇₃	1	8	yes	—
197	C₂×#102	—	2	4	no	—
198	#390	—	2	2	no	—
199	#390	C₂²×C₇₃	1	4	yes	—
200	#379	—	2	2	no	—
201	#343	C₂²×C₇₃	1	4	yes	complemented
202	#339	C₂²×C₇₃	1	8	yes	complemented
203	C₂⁵	—	2	1	no	—
204	C₂²×D₄	—	2	2	no	—
205	C₂²×D₄	—	2	2	no	—
206	C₂²×D₄	—	2	1	no	—
207	C₂²×D₄	—	2	4	no	—
208	C₂²×D₄	—	2	2	no	—
209	C₂²×D₄	Q₈×C₇₃	1	4	yes	complemented
210	C₂²×D₄	C₂³×C₇₃	1	2	yes	—
211	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
212	C₂×M₁₆	—	2	2	no	—
213	C₂×M₁₆	—	2	4	no	—
214	C₂×M₁₆	—	2	4	no	—
215	C₂×M₁₆	C₂³×C₇₃	1	4	yes	—
216	C₂²×C₈	—	2	2	no	—
217	C₄⋊Q₈	—	2	4	no	—
218	C₄⋊Q₈	—	2	8	no	—
219	#125	—	2	4	no	—
220	#125	—	2	8	no	—
221	C₂²⋊Q₈	—	2	16	no	—
222	C₂²⋊Q₈	—	2	8	no	—
223	C₄×Q₈	—	2	8	no	—
224	#117	—	2	2	no	—
225	#117	—	2	2	no	—
226	#117	—	2	2	no	—
227	#117	—	2	2	no	—
228	#117	—	2	4	no	—
229	#117	D₄×C₇₃	1	4	yes	—
230	#117	C₂³×C₇₃	1	4	yes	—
231	C₂×C₄⋊C₄	—	2	1	no	—
232	C₂×C₄⋊C₄	D₄×C₇₃	1	4	yes	—
233	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
234	C₂×C₂²⋊C₄	—	2	2	no	—
235	C₂×C₂²⋊C₄	—	2	1	no	—
236	C₂×C₂²⋊C₄	—	2	8	no	—
237	C₂×C₂²⋊C₄	D₄×C₇₃	1	2	yes	—
238	C₂×C₄²	—	2	2	no	—
239	C₂×C₄²	D₄×C₇₃	1	2	yes	—
240	C₂×C₄²	C₂³×C₇₃	1	2	yes	—
241	#102	—	2	16	no	—
242	C₂⁴	—	2	2	no	—
243	C₂⁴	—	2	4	no	—
244	C₂⁴	—	4	2	no	—
245	C₂⁴	—	4	2	no	—
246	C₂⁴	—	8	4	no	—
247	C₂⁴	C₂×D₄×C₇₃	1	1	yes	characteristic
248	C₂×Q₈	—	2	8	no	—
249	C₂×D₄	—	2	8	no	—
250	C₂×D₄	—	2	8	no	—
251	C₂×D₄	—	2	4	no	—
252	C₂×D₄	—	2	4	no	—
253	C₂×D₄	—	2	4	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	1	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	4	4	no	—
259	C₂×D₄	—	4	4	no	—
260	C₂×D₄	C₂×Q₈×C₇₃	1	4	yes	—
261	C₂²×C₄	—	2	4	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
264	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
265	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
266	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
267	C₂²×C₄	C₂×Q₈×C₇₃	1	2	yes	—
268	C₂²×C₄	#41×C₇₃	1	4	yes	—
269	C₂²×C₄	#41×C₇₃	1	2	yes	—
270	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
271	M₁₆	—	2	4	no	—
272	M₁₆	—	2	4	no	—
273	M₁₆	—	4	4	no	—
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	4	4	no	—
276	C₄⋊C₄	—	2	16	no	—
277	C₄⋊C₄	—	2	8	no	—
278	C₄⋊C₄	—	2	4	no	—
279	C₄⋊C₄	—	2	4	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	C₂×D₄×C₇₃	1	8	yes	—
282	C₂²⋊C₄	—	2	4	no	—
283	C₂²⋊C₄	—	2	4	no	—
284	C₂²⋊C₄	—	2	4	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	8	8	no	—
287	C₄²	—	2	4	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	4	no	—
291	C₄²	—	2	4	no	—
292	C₂³	—	2	4	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	8	no	—
295	C₂³	—	4	8	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	8	8	no	—
299	C₂³	—	8	8	no	—
300	C₂³	—	8	8	no	—
301	C₂³	#120×C₇₃	1	2	yes	—
302	C₂³	C₂²⋊Q₈×C₇₃	1	2	yes	—
303	C₂³	#124×C₇₃	1	2	yes	—
304	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
305	Q₈	—	8	8	no	—
306	D₄	—	2	8	no	—
307	D₄	—	4	4	no	—
308	D₄	—	4	4	no	—
309	D₄	—	4	4	no	—
310	C₂×C₄	—	2	4	no	—
311	C₂×C₄	—	2	8	no	—
312	C₂×C₄	—	4	4	no	—
313	C₂×C₄	—	4	4	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	8	4	no	—
316	C₂×C₄	#120×C₇₃	1	1	yes	characteristic
317	C₂×C₄	#120×C₇₃	1	1	yes	characteristic
318	C₂×C₄	C₂²⋊Q₈×C₇₃	1	2	yes	—
319	C₂×C₄	C₂²⋊Q₈×C₇₃	1	4	yes	—
320	C₂×C₄	C₂²⋊Q₈×C₇₃	1	4	yes	—
321	C₂×C₄	#124×C₇₃	1	2	yes	—
322	C₂×C₄	#124×C₇₃	1	4	yes	—
323	C₂×C₄	#124×C₇₃	1	4	yes	—
324	C₂×C₄	C₂²×D₄×C₇₃	1	2	yes	—
325	C₂×C₄	C₂²×Q₈×C₇₃	1	1	yes	characteristic
326	C₂×C₄	C₂×#41×C₇₃	1	2	yes	—
327	C₂×C₄	C₂×#41×C₇₃	1	1	yes	characteristic
328	C₈	—	4	4	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	4	no	—
331	C₂²	—	4	2	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	8	no	—
334	C₂²	—	4	4	no	—
335	C₂²	—	8	4	no	—
336	C₂²	—	8	16	no	—
337	C₂²	—	8	4	no	—
338	C₂²	#393×C₇₃	1	4	yes	—
339	C₂²	C₂×#120×C₇₃	1	1	yes	characteristic
340	C₂²	C₂×C₂²⋊Q₈×C₇₃	1	1	yes	characteristic
341	C₂²	C₂×#124×C₇₃	1	1	yes	characteristic
342	C₄	—	4	4	no	—
343	C₄	—	4	8	no	—
344	C₄	—	8	2	no	—
345	C₄	#393×C₇₃	1	2	yes	—
346	C₄	#393×C₇₃	1	2	yes	—
347	C₄	C₂×C₂²⋊Q₈×C₇₃	1	2	yes	—
348	C₄	C₂×#124×C₇₃	1	2	yes	—
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	2	4	no	—
352	C₂	—	8	4	no	—
353	C₂	C₂×#393×C₇₃	1	1	yes	characteristic
354	C₂	C₇₃×#3021	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₇₃×#3458	C₂	semidirect
4. C₇₃×#2027	C₂	semidirect
5. C₇₃×#1742	C₂	semidirect
6. C₇₃×#1733	C₂	semidirect
10. #544×C₇₃	C₂²	semidirect
23. #343×C₇₃	C₂²	semidirect
24. #339×C₇₃	C₂²	semidirect
31. C₂²×D₄×C₇₃	Q₈	semidirect
171. #23463	C₇₃	direct (N × H)
178. #3458	C₂×C₇₃	semidirect
181. #2027	C₂×C₇₃	semidirect
182. #1742	C₂×C₇₃	semidirect
183. #1733	C₂×C₇₃	semidirect
184. C₇₃	#23463	direct (N × H)
188. #544	C₂²×C₇₃	semidirect
201. #343	C₂²×C₇₃	semidirect
202. #339	C₂²×C₇₃	semidirect
209. C₂²×D₄	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×#3458	2
C₂×#418×C₇₃	2
C₇₃×#2309	2
C₇₃×#2027	2
C₇₃×#1742	2
C₇₃×#1733	2
#23463	73

Subgroups — C73×#23463 #32932867

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#23463 #32932867