Subgroups — C₇₃×#19747 #32929151

Order 18688.

908 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₃×#3184	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₃×#3182	C₂	1	2	yes	maximal, complemented
3	C₇₃×#3151	C₂	1	2	yes	maximal, complemented
4	C₇₃×#3138	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#1721	C₂	1	2	yes	maximal, complemented
6	C₇₃×#1717	C₂	1	2	yes	maximal, complemented
7	C₇₃×#1714	C₂	1	2	yes	maximal, complemented
8	C₇₃×#1706	C₂	1	2	yes	maximal, complemented
9	C₇₃×#1455	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×AGL(1,ℤ/8)×C₇₃	—	2	4	no	—
11	#533×C₇₃	C₂²	1	1	yes	characteristic
12	#532×C₇₃	C₂²	1	2	yes	—
13	C₂×#127×C₇₃	C₂²	1	1	yes	characteristic
14	#500×C₇₃	C₂²	1	2	yes	complemented
15	#498×C₇₃	C₂²	1	1	yes	characteristic, complemented
16	#496×C₇₃	—	2	4	no	—
17	#493×C₇₃	C₂²	1	2	yes	—
18	#493×C₇₃	C₂²	1	2	yes	—
19	#493×C₇₃	C₂²	1	1	yes	characteristic, complemented
20	#492×C₇₃	C₂²	1	2	yes	—
21	#492×C₇₃	C₂²	1	2	yes	—
22	#492×C₇₃	C₂²	1	2	yes	complemented
23	#490×C₇₃	—	2	2	no	—
24	#487×C₇₃	C₂²	1	2	yes	complemented
25	#486×C₇₃	C₂²	1	2	yes	—
26	#486×C₇₃	C₂²	1	2	yes	—
27	#486×C₇₃	C₂²	1	2	yes	complemented
28	#486×C₇₃	C₂²	1	2	yes	complemented
29	#469×C₇₃	—	2	2	no	—
30	#468×C₇₃	—	2	4	no	—
31	#442×C₇₃	—	2	2	no	—
32	#440×C₇₃	—	2	4	no	—
33	#418×C₇₃	—	2	2	no	—
34	C₄×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
35	C₄×M₁₆×C₇₃	C₂²	1	2	yes	—
36	#322×C₇₃	C₂²	1	2	yes	complemented
37	#322×C₇₃	C₂²	1	1	yes	characteristic, complemented
38	C₈²×C₇₃	C₂²	1	1	yes	characteristic, complemented
39	C₂×#41×C₇₃	—	2	2	no	—
40	C₂²×D₄×C₇₃	—	2	1	no	—
41	C₂²×D₄×C₇₃	—	2	2	no	—
42	AGL(1,ℤ/8)×C₇₃	—	4	16	no	—
43	C₂×SD₁₆×C₇₃	—	2	4	no	—
44	C₂×SD₁₆×C₇₃	—	2	4	no	—
45	C₂×SD₁₆×C₇₃	—	2	4	no	—
46	C₂×D₈×C₇₃	—	2	2	no	—
47	C₂×D₈×C₇₃	—	2	4	no	—
48	C₂×D₈×C₇₃	—	2	4	no	—
49	C₂×M₁₆×C₇₃	D₄	1	4	yes	—
50	C₂×M₁₆×C₇₃	D₄	1	2	yes	—
51	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
52	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
53	#127×C₇₃	—	2	1	no	—
54	#127×C₇₃	—	2	2	no	—
55	#127×C₇₃	C₂³	1	2	yes	—
56	#127×C₇₃	C₂³	1	2	yes	—
57	#125×C₇₃	—	2	1	no	—
58	#124×C₇₃	—	2	2	no	—
59	C₂²⋊Q₈×C₇₃	—	2	2	no	—
60	#121×C₇₃	—	2	2	no	—
61	#121×C₇₃	—	2	2	no	—
62	C₄×Q₈×C₇₃	—	2	2	no	—
63	C₄×D₄×C₇₃	—	2	2	no	—
64	C₄×D₄×C₇₃	—	2	2	no	—
65	#117×C₇₃	—	2	1	no	—
66	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
67	#107×C₇₃	—	2	4	no	—
68	#106×C₇₃	—	2	2	no	—
69	#103×C₇₃	—	2	4	no	—
70	#102×C₇₃	—	2	4	no	—
71	#102×C₇₃	—	2	4	no	—
72	#102×C₇₃	—	2	4	no	—
73	#102×C₇₃	—	2	4	no	—
74	#97×C₇₃	D₄	1	4	yes	—
75	#97×C₇₃	D₄	1	2	yes	—
76	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₇₃	C₂³	1	2	yes	—
79	C₄×C₈×C₇₃	C₂³	1	2	yes	—
80	C₂⁴×C₇₃	—	4	1	no	—
81	#41×C₇₃	—	4	8	no	—
82	C₂×Q₈×C₇₃	—	2	2	no	—
83	C₂×Q₈×C₇₃	—	2	2	no	—
84	C₂×D₄×C₇₃	—	2	2	no	—
85	C₂×D₄×C₇₃	—	2	2	no	—
86	C₂×D₄×C₇₃	—	2	2	no	—
87	C₂×D₄×C₇₃	—	2	2	no	—
88	C₂×D₄×C₇₃	—	2	2	no	—
89	C₂×D₄×C₇₃	—	2	2	no	—
90	C₂×D₄×C₇₃	—	4	8	no	—
91	C₂×D₄×C₇₃	—	4	4	no	—
92	C₂×D₄×C₇₃	—	4	2	no	—
93	C₂×D₄×C₇₃	—	4	1	no	—
94	C₂×D₄×C₇₃	—	4	2	no	—
95	C₂²×C₄×C₇₃	—	4	2	no	—
96	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
98	SD₁₆×C₇₃	—	4	8	no	—
99	SD₁₆×C₇₃	—	4	8	no	—
100	SD₁₆×C₇₃	—	4	8	no	—
101	D₈×C₇₃	—	4	4	no	—
102	D₈×C₇₃	—	4	8	no	—
103	D₈×C₇₃	—	4	8	no	—
104	M₁₆×C₇₃	—	2	8	no	—
105	M₁₆×C₇₃	—	4	2	no	—
106	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
109	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
110	C₄⋊C₄×C₇₃	—	2	2	no	—
111	C₄⋊C₄×C₇₃	—	2	2	no	—
112	C₄⋊C₄×C₇₃	—	2	2	no	—
113	C₄⋊C₄×C₇₃	—	2	2	no	—
114	C₄⋊C₄×C₇₃	—	4	2	no	—
115	C₂²⋊C₄×C₇₃	—	4	1	no	—
116	C₂²⋊C₄×C₇₃	—	4	2	no	—
117	C₂²⋊C₄×C₇₃	—	4	2	no	—
118	C₄²×C₇₃	—	4	1	no	—
119	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₇₃	C₂×D₄	1	2	yes	—
121	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
122	C₂³×C₇₃	—	4	2	no	—
123	C₂³×C₇₃	—	4	2	no	—
124	C₂³×C₇₃	—	8	4	no	—
125	C₂³×C₇₃	—	8	2	no	—
126	C₂³×C₇₃	#127	1	1	yes	characteristic
127	Q₈×C₇₃	—	4	4	no	—
128	Q₈×C₇₃	—	4	4	no	—
129	D₄×C₇₃	—	4	4	no	—
130	D₄×C₇₃	—	4	4	no	—
131	D₄×C₇₃	—	4	4	no	—
132	D₄×C₇₃	—	4	4	no	—
133	D₄×C₇₃	—	4	4	no	—
134	D₄×C₇₃	—	4	4	no	—
135	D₄×C₇₃	—	8	4	no	—
136	D₄×C₇₃	—	8	2	no	—
137	D₄×C₇₃	—	8	4	no	—
138	C₂×C₄×C₇₃	—	2	4	no	—
139	C₂×C₄×C₇₃	—	2	2	no	—
140	C₂×C₄×C₇₃	—	4	2	no	—
141	C₂×C₄×C₇₃	—	4	2	no	—
142	C₂×C₄×C₇₃	—	8	4	no	—
143	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
144	C₂×C₄×C₇₃	#127	1	2	yes	—
145	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
146	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
147	C₈×C₇₃	—	2	4	no	—
148	C₈×C₇₃	—	2	2	no	—
149	C₈×C₇₃	—	2	2	no	—
150	C₈×C₇₃	AGL(1,ℤ/8)	1	8	yes	complemented
151	C₂²×C₇₃	—	2	2	no	—
152	C₂²×C₇₃	—	2	1	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	—	8	2	no	—
158	C₂²×C₇₃	—	8	2	no	—
159	C₂²×C₇₃	—	16	4	no	—
160	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
161	C₄×C₇₃	—	2	2	no	—
162	C₄×C₇₃	—	2	1	no	—
163	C₄×C₇₃	—	8	2	no	—
164	C₄×C₇₃	—	8	2	no	—
165	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
166	C₄×C₇₃	#575	1	2	yes	—
167	#19747	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂×C₇₃	—	4	1	no	—
169	C₂×C₇₃	—	16	2	no	—
170	C₂×C₇₃	—	16	2	no	—
171	C₂×C₇₃	#3151	1	2	yes	—
172	C₂×C₇₃	#3153	1	1	yes	characteristic
173	#3184	C₂×C₇₃	1	1	yes	characteristic, complemented
174	#3182	C₂×C₇₃	1	2	yes	complemented
175	#3151	C₂×C₇₃	1	2	yes	complemented
176	#3138	C₂×C₇₃	1	1	yes	characteristic, complemented
177	#1721	C₂×C₇₃	1	2	yes	complemented
178	#1717	C₂×C₇₃	1	2	yes	complemented
179	#1714	C₂×C₇₃	1	2	yes	complemented
180	#1706	C₂×C₇₃	1	2	yes	complemented
181	#1455	C₂×C₇₃	1	1	yes	characteristic, complemented
182	C₇₃	#19747	1	1	yes	characteristic, direct product factor, Sylow
183	C₂×AGL(1,ℤ/8)	—	2	4	no	—
184	#533	C₂²×C₇₃	1	1	yes	characteristic
185	#532	C₂²×C₇₃	1	2	yes	—
186	C₂×#127	C₂²×C₇₃	1	1	yes	characteristic
187	#500	C₂²×C₇₃	1	2	yes	complemented
188	#498	C₂²×C₇₃	1	1	yes	characteristic, complemented
189	#496	—	2	4	no	—
190	#493	C₂²×C₇₃	1	2	yes	—
191	#493	C₂²×C₇₃	1	2	yes	—
192	#493	C₂²×C₇₃	1	1	yes	characteristic, complemented
193	#492	C₂²×C₇₃	1	2	yes	—
194	#492	C₂²×C₇₃	1	2	yes	—
195	#492	C₂²×C₇₃	1	2	yes	complemented
196	#490	—	2	2	no	—
197	#487	C₂²×C₇₃	1	2	yes	complemented
198	#486	C₂²×C₇₃	1	2	yes	—
199	#486	C₂²×C₇₃	1	2	yes	—
200	#486	C₂²×C₇₃	1	2	yes	complemented
201	#486	C₂²×C₇₃	1	2	yes	complemented
202	#469	—	2	2	no	—
203	#468	—	2	4	no	—
204	#442	—	2	2	no	—
205	#440	—	2	4	no	—
206	#418	—	2	2	no	—
207	C₄×M₁₆	C₂²×C₇₃	1	1	yes	characteristic
208	C₄×M₁₆	C₂²×C₇₃	1	2	yes	—
209	#322	C₂²×C₇₃	1	2	yes	complemented
210	#322	C₂²×C₇₃	1	1	yes	characteristic, complemented
211	C₈²	C₂²×C₇₃	1	1	yes	characteristic, complemented
212	C₂×#41	—	2	2	no	—
213	C₂²×D₄	—	2	2	no	—
214	C₂²×D₄	—	2	1	no	—
215	AGL(1,ℤ/8)	—	4	16	no	—
216	C₂×SD₁₆	—	2	4	no	—
217	C₂×SD₁₆	—	2	4	no	—
218	C₂×SD₁₆	—	2	4	no	—
219	C₂×D₈	—	2	4	no	—
220	C₂×D₈	—	2	4	no	—
221	C₂×D₈	—	2	2	no	—
222	C₂×M₁₆	D₄×C₇₃	1	4	yes	—
223	C₂×M₁₆	D₄×C₇₃	1	2	yes	—
224	C₄⋊Q₈	C₂³×C₇₃	1	2	yes	—
225	C₄⋊Q₈	C₂³×C₇₃	1	2	yes	—
226	#127	—	2	1	no	—
227	#127	—	2	2	no	—
228	#127	C₂³×C₇₃	1	2	yes	—
229	#127	C₂³×C₇₃	1	2	yes	—
230	#125	—	2	1	no	—
231	#124	—	2	2	no	—
232	C₂²⋊Q₈	—	2	2	no	—
233	#121	—	2	2	no	—
234	#121	—	2	2	no	—
235	C₄×Q₈	—	2	2	no	—
236	C₄×D₄	—	2	2	no	—
237	C₄×D₄	—	2	2	no	—
238	#117	—	2	1	no	—
239	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
240	#107	—	2	4	no	—
241	#106	—	2	2	no	—
242	#103	—	2	4	no	—
243	#102	—	2	4	no	—
244	#102	—	2	4	no	—
245	#102	—	2	4	no	—
246	#102	—	2	4	no	—
247	#97	D₄×C₇₃	1	4	yes	—
248	#97	D₄×C₇₃	1	2	yes	—
249	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
250	C₄×C₈	C₂³×C₇₃	1	2	yes	—
251	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
252	C₄×C₈	C₂³×C₇₃	1	2	yes	—
253	C₂⁴	—	4	1	no	—
254	#41	—	4	8	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	4	8	no	—
264	C₂×D₄	—	4	4	no	—
265	C₂×D₄	—	4	2	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂²×C₄	—	4	2	no	—
269	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
270	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
271	SD₁₆	—	4	8	no	—
272	SD₁₆	—	4	8	no	—
273	SD₁₆	—	4	8	no	—
274	D₈	—	4	4	no	—
275	D₈	—	4	8	no	—
276	D₈	—	4	8	no	—
277	M₁₆	—	2	8	no	—
278	M₁₆	—	4	2	no	—
279	C₂×C₈	C₂×D₄×C₇₃	1	2	yes	—
280	C₂×C₈	C₂×D₄×C₇₃	1	2	yes	—
281	C₂×C₈	C₂×D₄×C₇₃	1	4	yes	—
282	C₂×C₈	C₂×D₄×C₇₃	1	4	yes	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₄²	—	4	1	no	—
292	C₄²	C₂×D₄×C₇₃	1	1	yes	characteristic
293	C₄²	C₂×D₄×C₇₃	1	2	yes	—
294	C₄²	C₂⁴×C₇₃	1	1	yes	characteristic
295	C₂³	—	4	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	8	4	no	—
298	C₂³	—	8	2	no	—
299	C₂³	#127×C₇₃	1	1	yes	characteristic
300	Q₈	—	4	4	no	—
301	Q₈	—	4	4	no	—
302	D₄	—	4	4	no	—
303	D₄	—	4	4	no	—
304	D₄	—	4	4	no	—
305	D₄	—	4	4	no	—
306	D₄	—	4	4	no	—
307	D₄	—	4	4	no	—
308	D₄	—	8	4	no	—
309	D₄	—	8	2	no	—
310	D₄	—	8	4	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	8	4	no	—
316	C₂×C₄	#127×C₇₃	1	1	yes	characteristic
317	C₂×C₄	#127×C₇₃	1	2	yes	—
318	C₂×C₄	C₂²×D₄×C₇₃	1	2	yes	—
319	C₂×C₄	C₂²×D₄×C₇₃	1	1	yes	characteristic
320	C₈	—	2	4	no	—
321	C₈	—	2	2	no	—
322	C₈	—	2	2	no	—
323	C₈	AGL(1,ℤ/8)×C₇₃	1	8	yes	complemented
324	C₂²	—	2	2	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	16	4	no	—
333	C₂²	C₂×#127×C₇₃	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	2	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	C₂×AGL(1,ℤ/8)×C₇₃	1	4	yes	—
339	C₄	#575×C₇₃	1	2	yes	—
340	C₂	—	4	1	no	—
341	C₂	—	16	2	no	—
342	C₂	—	16	2	no	—
343	C₂	C₇₃×#3151	1	2	yes	—
344	C₂	C₇₃×#3153	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇₃
LCS	G ▷ C₄² ▷ C₂² ▷ 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
1. C₇₃×#3184	C₂	semidirect
2. C₇₃×#3182	C₂	semidirect
3. C₇₃×#3151	C₂	semidirect
4. C₇₃×#3138	C₂	semidirect
5. C₇₃×#1721	C₂	semidirect
6. C₇₃×#1717	C₂	semidirect
7. C₇₃×#1714	C₂	semidirect
8. C₇₃×#1706	C₂	semidirect
9. C₇₃×#1455	C₂	semidirect
14. #500×C₇₃	C₂²	semidirect
15. #498×C₇₃	C₂²	semidirect
19. #493×C₇₃	C₂²	semidirect
22. #492×C₇₃	C₂²	semidirect
24. #487×C₇₃	C₂²	semidirect
27. #486×C₇₃	C₂²	semidirect
28. #486×C₇₃	C₂²	semidirect
36. #322×C₇₃	C₂²	semidirect
37. #322×C₇₃	C₂²	semidirect
38. C₈²×C₇₃	C₂²	semidirect
150. C₈×C₇₃	AGL(1,ℤ/8)	semidirect
167. #19747	C₇₃	direct (N × H)
173. #3184	C₂×C₇₃	semidirect
174. #3182	C₂×C₇₃	semidirect
175. #3151	C₂×C₇₃	semidirect
176. #3138	C₂×C₇₃	semidirect
177. #1721	C₂×C₇₃	semidirect
178. #1717	C₂×C₇₃	semidirect
179. #1714	C₂×C₇₃	semidirect
180. #1706	C₂×C₇₃	semidirect
181. #1455	C₂×C₇₃	semidirect
182. C₇₃	#19747	direct (N × H)
187. #500	C₂²×C₇₃	semidirect
188. #498	C₂²×C₇₃	semidirect
192. #493	C₂²×C₇₃	semidirect
195. #492	C₂²×C₇₃	semidirect
197. #487	C₂²×C₇₃	semidirect
200. #486	C₂²×C₇₃	semidirect
201. #486	C₂²×C₇₃	semidirect
209. #322	C₂²×C₇₃	semidirect
210. #322	C₂²×C₇₃	semidirect
211. C₈²	C₂²×C₇₃	semidirect
323. C₈	AGL(1,ℤ/8)×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₇₃×#3184	2
C₇₃×#3182	2
C₇₃×#3151	2
C₇₃×#3138	2
C₇₃×#1721	2
C₇₃×#1717	2
C₇₃×#1714	2
C₇₃×#1706	2
C₇₃×#1455	2
#19747	73

Subgroups — C73×#19747 #32929151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#19747 #32929151