Subgroups — C₇₃×#33933 #32943337

Order 18688.

532 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×#3391	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₃×#3380	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₃×#2244	C₂	1	2	yes	maximal, complemented
4	C₇₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#2228	C₂	1	2	yes	maximal, complemented
6	C₇₃×#2226	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#2206	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₃×#2206	C₂	1	2	yes	maximal, complemented
9	C₇₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇₃×#2177	C₂	1	2	yes	maximal, complemented
11	C₄×D₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#554×C₇₃	—	2	1	no	—
13	#550×C₇₃	—	2	1	no	—
14	C₂×D₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×D₁₆×C₇₃	C₂²	1	2	yes	—
16	#500×C₇₃	C₂²	1	1	yes	characteristic
17	#499×C₇₃	C₂²	1	2	yes	—
18	#493×C₇₃	C₂²	1	1	yes	characteristic
19	#493×C₇₃	C₂²	1	2	yes	—
20	#478×C₇₃	—	2	2	no	—
21	#476×C₇₃	—	2	1	no	—
22	#459×C₇₃	—	2	1	no	—
23	#459×C₇₃	—	2	2	no	—
24	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
25	C₄×D₈×C₇₃	C₂²	1	1	yes	characteristic
26	C₄×D₈×C₇₃	C₂²	1	2	yes	—
27	C₄×D₈×C₇₃	C₂²	1	1	yes	characteristic
28	C₄×D₈×C₇₃	C₂²	1	2	yes	—
29	#367×C₇₃	C₂²	1	2	yes	—
30	#366×C₇₃	C₂²	1	1	yes	characteristic
31	C₄⋊C₁₆×C₇₃	C₂²	1	1	yes	characteristic
32	C₄⋊C₁₆×C₇₃	C₂²	1	2	yes	—
33	#357×C₇₃	C₂²	1	2	yes	—
34	#357×C₇₃	C₂²	1	2	yes	—
35	#357×C₇₃	C₂²	1	2	yes	—
36	#357×C₇₃	C₂²	1	2	yes	—
37	#357×C₇₃	C₂²	1	1	yes	characteristic
38	#357×C₇₃	C₂²	1	1	yes	characteristic
39	#346×C₇₃	C₂²	1	2	yes	—
40	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
41	C₂×#41×C₇₃	—	4	1	no	—
42	C₂×D₈×C₇₃	—	2	2	no	—
43	C₂×D₈×C₇₃	—	2	1	no	—
44	C₂×D₈×C₇₃	C₂³	1	1	yes	characteristic
45	C₂×D₈×C₇₃	C₂³	1	2	yes	—
46	C₂×D₈×C₇₃	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₇₃	—	2	1	no	—
48	#127×C₇₃	—	2	2	no	—
49	#127×C₇₃	—	2	1	no	—
50	#125×C₇₃	—	2	2	no	—
51	C₂²⋊Q₈×C₇₃	—	4	1	no	—
52	C₂²⋊Q₈×C₇₃	—	4	2	no	—
53	#121×C₇₃	—	4	1	no	—
54	#121×C₇₃	—	4	2	no	—
55	C₄×Q₈×C₇₃	D₄	1	1	yes	characteristic
56	C₄×D₄×C₇₃	—	2	2	no	—
57	C₄×D₄×C₇₃	—	2	1	no	—
58	C₄×D₄×C₇₃	—	2	1	no	—
59	C₄×D₄×C₇₃	—	2	2	no	—
60	C₂×C₄⋊C₄×C₇₃	—	4	1	no	—
61	D₁₆×C₇₃	—	2	2	no	—
62	D₁₆×C₇₃	—	2	4	no	—
63	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
64	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
65	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
66	#107×C₇₃	—	2	1	no	—
67	#107×C₇₃	C₂³	1	1	yes	characteristic
68	#107×C₇₃	C₂³	1	1	yes	characteristic
69	#107×C₇₃	C₂³	1	2	yes	—
70	#106×C₇₃	—	2	2	no	—
71	C₄⋊C₈×C₇₃	D₄	1	1	yes	characteristic
72	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
73	#102×C₇₃	—	2	2	no	—
74	#102×C₇₃	—	2	1	no	—
75	#102×C₇₃	—	2	2	no	—
76	#102×C₇₃	—	2	1	no	—
77	C₄×C₈×C₇₃	C₂³	1	2	yes	—
78	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
79	#41×C₇₃	—	8	4	no	—
80	C₂×Q₈×C₇₃	—	4	1	no	—
81	C₂×Q₈×C₇₃	D₈	1	1	yes	characteristic
82	C₂×D₄×C₇₃	—	2	1	no	—
83	C₂×D₄×C₇₃	—	2	2	no	—
84	C₂×D₄×C₇₃	—	2	1	no	—
85	C₂×D₄×C₇₃	—	4	2	no	—
86	C₂×D₄×C₇₃	—	4	1	no	—
87	C₂×D₄×C₇₃	—	4	1	no	—
88	C₂×D₄×C₇₃	—	4	2	no	—
89	C₂²×C₄×C₇₃	—	4	1	no	—
90	C₂²×C₄×C₇₃	—	4	2	no	—
91	C₂²×C₄×C₇₃	—	4	1	no	—
92	C₂²×C₄×C₇₃	—	4	2	no	—
93	D₈×C₇₃	—	2	1	no	—
94	D₈×C₇₃	—	2	2	no	—
95	D₈×C₇₃	—	2	1	no	—
96	D₈×C₇₃	—	2	1	no	—
97	D₈×C₇₃	—	2	2	no	—
98	D₈×C₇₃	—	8	2	no	—
99	D₈×C₇₃	—	8	1	no	—
100	D₈×C₇₃	#41	1	2	yes	complemented
101	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₇₃	—	2	1	no	—
105	C₄⋊C₄×C₇₃	—	2	1	no	—
106	C₄⋊C₄×C₇₃	—	2	2	no	—
107	C₄⋊C₄×C₇₃	—	4	2	no	—
108	C₄⋊C₄×C₇₃	—	4	1	no	—
109	C₄⋊C₄×C₇₃	—	4	2	no	—
110	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
111	C₄⋊C₄×C₇₃	D₈	1	2	yes	—
112	C₂²⋊C₄×C₇₃	—	4	1	no	—
113	C₂²⋊C₄×C₇₃	—	4	2	no	—
114	C₂²⋊C₄×C₇₃	—	4	1	no	—
115	C₂²⋊C₄×C₇₃	—	4	2	no	—
116	C₄²×C₇₃	C₂×D₄	1	2	yes	—
117	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₇₃	—	2	1	no	—
119	C₁₆×C₇₃	—	2	2	no	—
120	C₁₆×C₇₃	—	2	1	no	—
121	C₂³×C₇₃	—	4	1	no	—
122	C₂³×C₇₃	—	4	2	no	—
123	C₂³×C₇₃	—	4	1	no	—
124	Q₈×C₇₃	—	2	2	no	—
125	Q₈×C₇₃	—	16	1	no	—
126	D₄×C₇₃	—	2	2	no	—
127	D₄×C₇₃	—	4	1	no	—
128	D₄×C₇₃	—	4	2	no	—
129	D₄×C₇₃	—	4	1	no	—
130	D₄×C₇₃	—	4	1	no	—
131	D₄×C₇₃	—	4	2	no	—
132	D₄×C₇₃	—	8	4	no	—
133	D₄×C₇₃	—	8	2	no	—
134	D₄×C₇₃	—	16	2	no	—
135	D₄×C₇₃	—	16	1	no	—
136	C₂×C₄×C₇₃	—	4	1	no	—
137	C₂×C₄×C₇₃	—	4	1	no	—
138	C₂×C₄×C₇₃	—	4	2	no	—
139	C₂×C₄×C₇₃	—	8	4	no	—
140	C₂×C₄×C₇₃	—	8	2	no	—
141	C₂×C₄×C₇₃	—	8	2	no	—
142	C₂×C₄×C₇₃	—	16	2	no	—
143	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₇₃	—	2	2	no	—
149	C₈×C₇₃	—	2	1	no	—
150	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
151	C₈×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
152	C₂²×C₇₃	—	4	2	no	—
153	C₂²×C₇₃	—	8	1	no	—
154	C₂²×C₇₃	—	8	1	no	—
155	C₂²×C₇₃	—	8	2	no	—
156	C₂²×C₇₃	—	8	1	no	—
157	C₂²×C₇₃	—	8	2	no	—
158	C₂²×C₇₃	—	8	1	no	—
159	C₂²×C₇₃	—	8	1	no	—
160	C₂²×C₇₃	—	8	2	no	—
161	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
162	C₄×C₇₃	—	2	2	no	—
163	C₄×C₇₃	—	2	2	no	—
164	C₄×C₇₃	—	2	1	no	—
165	C₄×C₇₃	—	8	1	no	—
166	C₄×C₇₃	—	8	1	no	—
167	C₄×C₇₃	—	8	2	no	—
168	C₄×C₇₃	#509	1	2	yes	—
169	C₄×C₇₃	#548	1	1	yes	characteristic
170	C₄×C₇₃	#577	1	1	yes	characteristic
171	#33933	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂×C₇₃	—	8	2	no	—
173	C₂×C₇₃	—	16	2	no	—
174	C₂×C₇₃	—	16	1	no	—
175	C₂×C₇₃	#3358	1	1	yes	characteristic
176	C₂×C₇₃	C₂×#509	1	1	yes	characteristic
177	C₂×C₇₃	#3415	1	1	yes	characteristic
178	#3391	C₂×C₇₃	1	1	yes	characteristic, complemented
179	#3380	C₂×C₇₃	1	1	yes	characteristic, complemented
180	#2244	C₂×C₇₃	1	2	yes	complemented
181	#2240	C₂×C₇₃	1	1	yes	characteristic, complemented
182	#2228	C₂×C₇₃	1	2	yes	complemented
183	#2226	C₂×C₇₃	1	1	yes	characteristic, complemented
184	#2206	C₂×C₇₃	1	2	yes	complemented
185	#2206	C₂×C₇₃	1	1	yes	characteristic, complemented
186	#2183	C₂×C₇₃	1	1	yes	characteristic, complemented
187	#2177	C₂×C₇₃	1	2	yes	complemented
188	C₄×D₁₆	C₂×C₇₃	1	1	yes	characteristic, complemented
189	C₇₃	#33933	1	1	yes	characteristic, direct product factor, Sylow
190	#554	—	2	1	no	—
191	#550	—	2	1	no	—
192	C₂×D₁₆	C₂²×C₇₃	1	1	yes	characteristic
193	C₂×D₁₆	C₂²×C₇₃	1	2	yes	—
194	#500	C₂²×C₇₃	1	1	yes	characteristic
195	#499	C₂²×C₇₃	1	2	yes	—
196	#493	C₂²×C₇₃	1	1	yes	characteristic
197	#493	C₂²×C₇₃	1	2	yes	—
198	#478	—	2	2	no	—
199	#476	—	2	1	no	—
200	#459	—	2	1	no	—
201	#459	—	2	2	no	—
202	C₈×Q₈	C₂²×C₇₃	1	1	yes	characteristic
203	C₄×D₈	C₂²×C₇₃	1	1	yes	characteristic
204	C₄×D₈	C₂²×C₇₃	1	2	yes	—
205	C₄×D₈	C₂²×C₇₃	1	1	yes	characteristic
206	C₄×D₈	C₂²×C₇₃	1	2	yes	—
207	#367	C₂²×C₇₃	1	2	yes	—
208	#366	C₂²×C₇₃	1	1	yes	characteristic
209	C₄⋊C₁₆	C₂²×C₇₃	1	1	yes	characteristic
210	C₄⋊C₁₆	C₂²×C₇₃	1	2	yes	—
211	#357	C₂²×C₇₃	1	1	yes	characteristic
212	#357	C₂²×C₇₃	1	1	yes	characteristic
213	#357	C₂²×C₇₃	1	2	yes	—
214	#357	C₂²×C₇₃	1	2	yes	—
215	#357	C₂²×C₇₃	1	2	yes	—
216	#357	C₂²×C₇₃	1	2	yes	—
217	#346	C₂²×C₇₃	1	2	yes	—
218	C₄×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
219	C₂×#41	—	4	1	no	—
220	C₂×D₈	—	2	2	no	—
221	C₂×D₈	—	2	1	no	—
222	C₂×D₈	C₂³×C₇₃	1	1	yes	characteristic
223	C₂×D₈	C₂³×C₇₃	1	2	yes	—
224	C₂×D₈	C₂³×C₇₃	1	1	yes	characteristic
225	C₄⋊Q₈	—	2	1	no	—
226	#127	—	2	2	no	—
227	#127	—	2	1	no	—
228	#125	—	2	2	no	—
229	C₂²⋊Q₈	—	4	1	no	—
230	C₂²⋊Q₈	—	4	2	no	—
231	#121	—	4	1	no	—
232	#121	—	4	2	no	—
233	C₄×Q₈	D₄×C₇₃	1	1	yes	characteristic
234	C₄×D₄	—	2	2	no	—
235	C₄×D₄	—	2	1	no	—
236	C₄×D₄	—	2	1	no	—
237	C₄×D₄	—	2	2	no	—
238	C₂×C₄⋊C₄	—	4	1	no	—
239	D₁₆	—	2	2	no	—
240	D₁₆	—	2	4	no	—
241	C₂×C₁₆	C₂³×C₇₃	1	1	yes	characteristic
242	C₂×C₁₆	C₂³×C₇₃	1	1	yes	characteristic
243	C₂×C₁₆	C₂³×C₇₃	1	2	yes	—
244	#107	—	2	1	no	—
245	#107	C₂³×C₇₃	1	1	yes	characteristic
246	#107	C₂³×C₇₃	1	2	yes	—
247	#107	C₂³×C₇₃	1	1	yes	characteristic
248	#106	—	2	2	no	—
249	C₄⋊C₈	D₄×C₇₃	1	1	yes	characteristic
250	C₄⋊C₈	D₄×C₇₃	1	2	yes	—
251	#102	—	2	2	no	—
252	#102	—	2	1	no	—
253	#102	—	2	2	no	—
254	#102	—	2	1	no	—
255	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
256	C₄×C₈	C₂³×C₇₃	1	2	yes	—
257	#41	—	8	4	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	D₈×C₇₃	1	1	yes	characteristic
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	4	2	no	—
269	C₂²×C₄	—	4	1	no	—
270	C₂²×C₄	—	4	2	no	—
271	D₈	—	2	1	no	—
272	D₈	—	2	2	no	—
273	D₈	—	2	1	no	—
274	D₈	—	2	1	no	—
275	D₈	—	2	2	no	—
276	D₈	—	8	2	no	—
277	D₈	—	8	1	no	—
278	D₈	#41×C₇₃	1	2	yes	complemented
279	C₂×C₈	C₂×D₄×C₇₃	1	2	yes	—
280	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
281	C₂×C₈	C₂⁴×C₇₃	1	1	yes	characteristic
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	D₈×C₇₃	1	1	yes	characteristic
289	C₄⋊C₄	D₈×C₇₃	1	2	yes	—
290	C₂²⋊C₄	—	4	1	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	4	1	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₄²	C₂×D₄×C₇₃	1	2	yes	—
295	C₄²	C₂×D₄×C₇₃	1	1	yes	characteristic
296	C₁₆	—	2	1	no	—
297	C₁₆	—	2	2	no	—
298	C₁₆	—	2	1	no	—
299	C₂³	—	4	1	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	1	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	16	1	no	—
304	D₄	—	2	2	no	—
305	D₄	—	4	1	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	1	no	—
308	D₄	—	4	1	no	—
309	D₄	—	4	2	no	—
310	D₄	—	8	4	no	—
311	D₄	—	8	2	no	—
312	D₄	—	16	2	no	—
313	D₄	—	16	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	8	4	no	—
318	C₂×C₄	—	8	2	no	—
319	C₂×C₄	—	8	2	no	—
320	C₂×C₄	—	16	2	no	—
321	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
322	C₂×C₄	C₂×D₈×C₇₃	1	2	yes	—
323	C₂×C₄	C₂×D₈×C₇₃	1	2	yes	—
324	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
325	C₂×C₄	C₂²×D₄×C₇₃	1	1	yes	characteristic
326	C₈	—	2	2	no	—
327	C₈	—	2	1	no	—
328	C₈	C₂×#41×C₇₃	1	1	yes	characteristic
329	C₈	2₋¹⁺⁴×C₇₃	1	1	yes	characteristic
330	C₂²	—	4	2	no	—
331	C₂²	—	8	1	no	—
332	C₂²	—	8	1	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	1	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	8	2	no	—
339	C₂²	C₂²×D₈×C₇₃	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	8	1	no	—
344	C₄	—	8	1	no	—
345	C₄	—	8	2	no	—
346	C₄	#509×C₇₃	1	2	yes	—
347	C₄	#548×C₇₃	1	1	yes	characteristic
348	C₄	#577×C₇₃	1	1	yes	characteristic
349	C₂	—	8	2	no	—
350	C₂	—	16	2	no	—
351	C₂	—	16	1	no	—
352	C₂	C₇₃×#3358	1	1	yes	characteristic
353	C₂	C₂×#509×C₇₃	1	1	yes	characteristic
354	C₂	C₇₃×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₄×Q₈×C₇₃ ◁ C₈×Q₈×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇₃×#3391	C₂	semidirect
2. C₇₃×#3380	C₂	semidirect
3. C₇₃×#2244	C₂	semidirect
4. C₇₃×#2240	C₂	semidirect
5. C₇₃×#2228	C₂	semidirect
6. C₇₃×#2226	C₂	semidirect
7. C₇₃×#2206	C₂	semidirect
8. C₇₃×#2206	C₂	semidirect
9. C₇₃×#2183	C₂	semidirect
10. C₇₃×#2177	C₂	semidirect
11. C₄×D₁₆×C₇₃	C₂	semidirect
100. D₈×C₇₃	#41	semidirect
171. #33933	C₇₃	direct (N × H)
178. #3391	C₂×C₇₃	semidirect
179. #3380	C₂×C₇₃	semidirect
180. #2244	C₂×C₇₃	semidirect
181. #2240	C₂×C₇₃	semidirect
182. #2228	C₂×C₇₃	semidirect
183. #2226	C₂×C₇₃	semidirect
184. #2206	C₂×C₇₃	semidirect
185. #2206	C₂×C₇₃	semidirect
186. #2183	C₂×C₇₃	semidirect
187. #2177	C₂×C₇₃	semidirect
188. C₄×D₁₆	C₂×C₇₃	semidirect
189. C₇₃	#33933	direct (N × H)
278. D₈	#41×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₇₃×#3391	2
C₇₃×#3380	2
C₇₃×#2244	2
C₇₃×#2240	2
C₇₃×#2228	2
C₇₃×#2226	2
C₇₃×#2206	2
C₇₃×#2183	2
C₇₃×#2177	2
C₄×D₁₆×C₇₃	2
#33933	73

Subgroups — C73×#33933 #32943337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#33933 #32943337