Subgroups — C₇₃×#33634 #32943038

Order 18688.

534 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₃×#3164	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₃×#2951	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#2229	C₂	1	2	yes	maximal, complemented
4	C₇₃×#2228	C₂	1	2	yes	maximal, complemented
5	C₇₃×#2226	C₂	1	2	yes	maximal, complemented
6	C₇₃×#2225	C₂	1	2	yes	maximal, complemented
7	C₇₃×#2159	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#367×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₃×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇₃×#2139	C₂	1	1	yes	maximal, characteristic
11	C₇₃×#2138	C₂	1	1	yes	maximal, characteristic
12	C₂×#135×C₇₃	C₂²	1	1	yes	characteristic
13	#544×C₇₃	—	2	1	no	—
14	#520×C₇₃	—	2	1	no	—
15	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	#501×C₇₃	—	2	2	no	—
18	#500×C₇₃	C₂²	1	2	yes	—
19	#499×C₇₃	C₂²	1	2	yes	—
20	#440×C₇₃	—	2	2	no	—
21	C₄×Q₁₆×C₇₃	C₂²	1	2	yes	—
22	C₄×D₈×C₇₃	C₂²	1	2	yes	—
23	#428×C₇₃	—	2	1	no	—
24	#427×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
26	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
27	#417×C₇₃	—	2	1	no	—
28	#405×C₇₃	C₂²	1	1	yes	characteristic
29	#367×C₇₃	C₂²	1	2	yes	—
30	#367×C₇₃	C₂²	1	2	yes	—
31	#367×C₇₃	C₂²	1	2	yes	—
32	#366×C₇₃	C₂²	1	2	yes	—
33	C₄⋊C₁₆×C₇₃	C₂²	1	2	yes	—
34	C₄⋊C₁₆×C₇₃	C₂²	1	2	yes	—
35	#358×C₇₃	C₂²	1	1	yes	characteristic
36	#358×C₇₃	C₂²	1	2	yes	—
37	#358×C₇₃	C₂²	1	1	yes	characteristic
38	#357×C₇₃	C₂²	1	1	yes	characteristic
39	#357×C₇₃	C₂²	1	2	yes	—
40	#357×C₇₃	C₂²	1	1	yes	characteristic
41	C₂×#41×C₇₃	—	2	1	no	—
42	#135×C₇₃	—	2	2	no	—
43	#135×C₇₃	Q₈	1	4	yes	complemented
44	C₂×Q₁₆×C₇₃	C₂³	1	1	yes	characteristic
45	C₂×SD₁₆×C₇₃	—	2	1	no	—
46	C₂×D₈×C₇₃	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₇₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₇₃	—	2	2	no	—
50	C₄⋊Q₈×C₇₃	—	4	1	no	—
51	#125×C₇₃	—	2	2	no	—
52	#125×C₇₃	—	4	1	no	—
53	C₂²⋊Q₈×C₇₃	—	4	2	no	—
54	C₄×Q₈×C₇₃	—	2	2	no	—
55	C₄×D₄×C₇₃	—	2	2	no	—
56	C₄×D₄×C₇₃	—	4	2	no	—
57	#117×C₇₃	—	2	1	no	—
58	#117×C₇₃	—	4	1	no	—
59	#117×C₇₃	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
62	C₂×C₄⋊C₄×C₇₃	—	4	1	no	—
63	M₃₂×C₇₃	—	2	2	no	—
64	C₂×C₁₆×C₇₃	—	2	2	no	—
65	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
66	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
68	#107×C₇₃	—	2	1	no	—
69	#107×C₇₃	—	2	1	no	—
70	#107×C₇₃	—	2	2	no	—
71	#107×C₇₃	C₂³	1	2	yes	—
72	#107×C₇₃	C₂³	1	2	yes	—
73	#107×C₇₃	C₂³	1	1	yes	characteristic
74	#107×C₇₃	C₂³	1	1	yes	characteristic
75	#106×C₇₃	—	2	1	no	—
76	#106×C₇₃	—	2	2	no	—
77	#103×C₇₃	—	2	2	no	—
78	#102×C₇₃	—	2	2	no	—
79	#97×C₇₃	D₄	1	2	yes	—
80	C₄×C₈×C₇₃	C₂³	1	2	yes	—
81	#41×C₇₃	—	2	4	no	—
82	#41×C₇₃	—	4	2	no	—
83	C₂×Q₈×C₇₃	—	2	1	no	—
84	C₂×Q₈×C₇₃	—	4	2	no	—
85	C₂×D₄×C₇₃	—	2	1	no	—
86	C₂×D₄×C₇₃	—	4	1	no	—
87	C₂²×C₄×C₇₃	—	4	1	no	—
88	C₂²×C₄×C₇₃	—	4	1	no	—
89	C₂²×C₄×C₇₃	—	4	1	no	—
90	C₂²×C₄×C₇₃	—	4	2	no	—
91	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
92	Q₁₆×C₇₃	—	2	1	no	—
93	Q₁₆×C₇₃	C₂×Q₈	1	2	yes	—
94	SD₁₆×C₇₃	—	2	4	no	—
95	D₈×C₇₃	—	2	1	no	—
96	D₈×C₇₃	C₂×Q₈	1	2	yes	—
97	M₁₆×C₇₃	—	2	2	no	—
98	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
101	C₂×C₈×C₇₃	#41	1	2	yes	—
102	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₇₃	—	2	2	no	—
104	C₄⋊C₄×C₇₃	—	2	1	no	—
105	C₄⋊C₄×C₇₃	—	2	2	no	—
106	C₄⋊C₄×C₇₃	—	2	1	no	—
107	C₄⋊C₄×C₇₃	—	4	1	no	—
108	C₄⋊C₄×C₇₃	—	4	1	no	—
109	C₄⋊C₄×C₇₃	—	4	2	no	—
110	C₄⋊C₄×C₇₃	—	4	2	no	—
111	C₄⋊C₄×C₇₃	—	4	2	no	—
112	C₄⋊C₄×C₇₃	—	4	2	no	—
113	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
114	C₄⋊C₄×C₇₃	D₈	1	1	yes	characteristic
115	C₂²⋊C₄×C₇₃	—	4	1	no	—
116	C₂²⋊C₄×C₇₃	—	4	2	no	—
117	C₂²⋊C₄×C₇₃	D₈	1	2	yes	—
118	C₄²×C₇₃	—	4	1	no	—
119	C₄²×C₇₃	—	4	2	no	—
120	C₄²×C₇₃	C₂×D₄	1	2	yes	—
121	C₁₆×C₇₃	—	2	1	no	—
122	C₁₆×C₇₃	—	2	1	no	—
123	C₁₆×C₇₃	—	2	2	no	—
124	C₂³×C₇₃	—	4	1	no	—
125	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
126	Q₈×C₇₃	—	2	2	no	—
127	Q₈×C₇₃	—	4	1	no	—
128	Q₈×C₇₃	—	16	2	no	—
129	D₄×C₇₃	—	2	2	no	—
130	D₄×C₇₃	—	4	4	no	—
131	D₄×C₇₃	—	4	1	no	—
132	C₂×C₄×C₇₃	—	4	1	no	—
133	C₂×C₄×C₇₃	—	4	4	no	—
134	C₂×C₄×C₇₃	—	4	1	no	—
135	C₂×C₄×C₇₃	—	4	2	no	—
136	C₂×C₄×C₇₃	—	4	1	no	—
137	C₂×C₄×C₇₃	—	4	2	no	—
138	C₂×C₄×C₇₃	—	16	2	no	—
139	C₂×C₄×C₇₃	—	16	1	no	—
140	C₂×C₄×C₇₃	—	16	1	no	—
141	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
142	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₇₃	#137	1	2	yes	—
146	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
147	C₈×C₇₃	—	2	2	no	—
148	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
149	C₈×C₇₃	C₂²×Q₈	1	1	yes	characteristic
150	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
151	C₂²×C₇₃	—	2	1	no	—
152	C₂²×C₇₃	—	2	1	no	—
153	C₂²×C₇₃	—	4	2	no	—
154	C₂²×C₇₃	—	8	1	no	—
155	C₂²×C₇₃	—	8	1	no	—
156	C₂²×C₇₃	#474	1	2	yes	—
157	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
158	C₄×C₇₃	—	2	2	no	—
159	C₄×C₇₃	—	2	2	no	—
160	C₄×C₇₃	—	4	2	no	—
161	C₄×C₇₃	—	8	1	no	—
162	C₄×C₇₃	—	8	2	no	—
163	C₄×C₇₃	—	8	2	no	—
164	C₄×C₇₃	—	8	1	no	—
165	C₄×C₇₃	#474	1	2	yes	—
166	C₄×C₇₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
167	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
168	#33634	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
169	C₂×C₇₃	—	2	2	no	—
170	C₂×C₇₃	—	8	2	no	—
171	C₂×C₇₃	C₂×#474	1	1	yes	characteristic
172	C₂×C₇₃	#3416	1	1	yes	characteristic
173	C₂×C₇₃	#3416	1	1	yes	characteristic
174	#3164	C₂×C₇₃	1	1	yes	characteristic, complemented
175	#2951	C₂×C₇₃	1	1	yes	characteristic
176	#2229	C₂×C₇₃	1	2	yes	complemented
177	#2228	C₂×C₇₃	1	2	yes	complemented
178	#2226	C₂×C₇₃	1	2	yes	complemented
179	#2225	C₂×C₇₃	1	2	yes	complemented
180	#2159	C₂×C₇₃	1	1	yes	characteristic, complemented
181	C₂×#367	C₂×C₇₃	1	1	yes	characteristic, complemented
182	#2151	C₂×C₇₃	1	1	yes	characteristic, complemented
183	#2139	C₂×C₇₃	1	1	yes	characteristic
184	#2138	C₂×C₇₃	1	1	yes	characteristic
185	C₇₃	#33634	1	1	yes	characteristic, direct product factor, Sylow
186	C₂×#135	C₂²×C₇₃	1	1	yes	characteristic
187	#544	—	2	1	no	—
188	#520	—	2	1	no	—
189	C₂×M₃₂	C₂²×C₇₃	1	1	yes	characteristic
190	C₂²×C₁₆	C₂²×C₇₃	1	1	yes	characteristic
191	#501	—	2	2	no	—
192	#500	C₂²×C₇₃	1	2	yes	—
193	#499	C₂²×C₇₃	1	2	yes	—
194	#440	—	2	2	no	—
195	C₄×Q₁₆	C₂²×C₇₃	1	2	yes	—
196	C₄×D₈	C₂²×C₇₃	1	2	yes	—
197	#428	—	2	1	no	—
198	#427	C₂²×C₇₃	1	1	yes	characteristic
199	C₂×#107	C₂²×C₇₃	1	1	yes	characteristic
200	C₂×#107	C₂²×C₇₃	1	1	yes	characteristic
201	#417	—	2	1	no	—
202	#405	C₂²×C₇₃	1	1	yes	characteristic
203	#367	C₂²×C₇₃	1	2	yes	—
204	#367	C₂²×C₇₃	1	2	yes	—
205	#367	C₂²×C₇₃	1	2	yes	—
206	#366	C₂²×C₇₃	1	2	yes	—
207	C₄⋊C₁₆	C₂²×C₇₃	1	2	yes	—
208	C₄⋊C₁₆	C₂²×C₇₃	1	2	yes	—
209	#358	C₂²×C₇₃	1	1	yes	characteristic
210	#358	C₂²×C₇₃	1	1	yes	characteristic
211	#358	C₂²×C₇₃	1	2	yes	—
212	#357	C₂²×C₇₃	1	1	yes	characteristic
213	#357	C₂²×C₇₃	1	1	yes	characteristic
214	#357	C₂²×C₇₃	1	2	yes	—
215	C₂×#41	—	2	1	no	—
216	#135	—	2	2	no	—
217	#135	Q₈×C₇₃	1	4	yes	complemented
218	C₂×Q₁₆	C₂³×C₇₃	1	1	yes	characteristic
219	C₂×SD₁₆	—	2	1	no	—
220	C₂×D₈	C₂³×C₇₃	1	1	yes	characteristic
221	C₂×M₁₆	D₄×C₇₃	1	1	yes	characteristic
222	C₂²×C₈	C₂³×C₇₃	1	1	yes	characteristic
223	C₄⋊Q₈	—	2	2	no	—
224	C₄⋊Q₈	—	4	1	no	—
225	#125	—	2	2	no	—
226	#125	—	4	1	no	—
227	C₂²⋊Q₈	—	4	2	no	—
228	C₄×Q₈	—	2	2	no	—
229	C₄×D₄	—	2	2	no	—
230	C₄×D₄	—	4	2	no	—
231	#117	—	2	1	no	—
232	#117	—	4	1	no	—
233	#117	D₄×C₇₃	1	1	yes	characteristic
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	4	1	no	—
237	M₃₂	—	2	2	no	—
238	C₂×C₁₆	—	2	2	no	—
239	C₂×C₁₆	C₂³×C₇₃	1	2	yes	—
240	C₂×C₁₆	C₂³×C₇₃	1	1	yes	characteristic
241	C₂×C₁₆	C₂³×C₇₃	1	1	yes	characteristic
242	#107	—	2	1	no	—
243	#107	—	2	1	no	—
244	#107	—	2	2	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	#107	C₂³×C₇₃	1	2	yes	—
249	#106	—	2	1	no	—
250	#106	—	2	2	no	—
251	#103	—	2	2	no	—
252	#102	—	2	2	no	—
253	#97	D₄×C₇₃	1	2	yes	—
254	C₄×C₈	C₂³×C₇₃	1	2	yes	—
255	#41	—	2	4	no	—
256	#41	—	4	2	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	—	4	2	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂²×C₄	—	4	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	4	2	no	—
265	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
266	Q₁₆	—	2	1	no	—
267	Q₁₆	C₂×Q₈×C₇₃	1	2	yes	—
268	SD₁₆	—	2	4	no	—
269	D₈	—	2	1	no	—
270	D₈	C₂×Q₈×C₇₃	1	2	yes	—
271	M₁₆	—	2	2	no	—
272	C₂×C₈	C₂×D₄×C₇₃	1	2	yes	—
273	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
274	C₂×C₈	C₂×Q₈×C₇₃	1	2	yes	—
275	C₂×C₈	#41×C₇₃	1	2	yes	—
276	C₂×C₈	C₂⁴×C₇₃	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	D₈×C₇₃	1	1	yes	characteristic
288	C₄⋊C₄	D₈×C₇₃	1	1	yes	characteristic
289	C₂²⋊C₄	—	4	1	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	D₈×C₇₃	1	2	yes	—
292	C₄²	—	4	1	no	—
293	C₄²	—	4	2	no	—
294	C₄²	C₂×D₄×C₇₃	1	2	yes	—
295	C₁₆	—	2	1	no	—
296	C₁₆	—	2	1	no	—
297	C₁₆	—	2	2	no	—
298	C₂³	—	4	1	no	—
299	C₂³	C₂×D₈×C₇₃	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	Q₈	—	4	1	no	—
302	Q₈	—	16	2	no	—
303	D₄	—	2	2	no	—
304	D₄	—	4	4	no	—
305	D₄	—	4	1	no	—
306	C₂×C₄	—	4	1	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	16	2	no	—
313	C₂×C₄	—	16	1	no	—
314	C₂×C₄	—	16	1	no	—
315	C₂×C₄	C₂²⋊Q₈×C₇₃	1	2	yes	—
316	C₂×C₄	C₂×D₈×C₇₃	1	2	yes	—
317	C₂×C₄	C₂×D₈×C₇₃	1	2	yes	—
318	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
319	C₂×C₄	#137×C₇₃	1	2	yes	—
320	C₂×C₄	C₂²×D₄×C₇₃	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	C₂²⋊Q₈×C₇₃	1	2	yes	—
323	C₈	C₂²×Q₈×C₇₃	1	1	yes	characteristic
324	C₈	C₂×#41×C₇₃	1	1	yes	characteristic
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	#474×C₇₃	1	2	yes	—
331	C₂²	C₂²×D₈×C₇₃	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	4	2	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	1	no	—
339	C₄	#474×C₇₃	1	2	yes	—
340	C₄	C₂×C₂²⋊Q₈×C₇₃	1	1	yes	characteristic
341	C₄	C₂×#137×C₇₃	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	8	2	no	—
344	C₂	C₂×#474×C₇₃	1	1	yes	characteristic
345	C₂	C₇₃×#3416	1	1	yes	characteristic
346	C₂	C₇₃×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇₃
prime radical	C₂×#135×C₇₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ #117×C₇₃ ◁ #405×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇₃×#3164	C₂	semidirect
3. C₇₃×#2229	C₂	semidirect
4. C₇₃×#2228	C₂	semidirect
5. C₇₃×#2226	C₂	semidirect
6. C₇₃×#2225	C₂	semidirect
7. C₇₃×#2159	C₂	semidirect
8. C₂×#367×C₇₃	C₂	semidirect
9. C₇₃×#2151	C₂	semidirect
43. #135×C₇₃	Q₈	semidirect
168. #33634	C₇₃	direct (N × H)
174. #3164	C₂×C₇₃	semidirect
176. #2229	C₂×C₇₃	semidirect
177. #2228	C₂×C₇₃	semidirect
178. #2226	C₂×C₇₃	semidirect
179. #2225	C₂×C₇₃	semidirect
180. #2159	C₂×C₇₃	semidirect
181. C₂×#367	C₂×C₇₃	semidirect
182. #2151	C₂×C₇₃	semidirect
185. C₇₃	#33634	direct (N × H)
217. #135	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×#3164	2
C₇₃×#2951	2
C₇₃×#2229	2
C₇₃×#2228	2
C₇₃×#2226	2
C₇₃×#2225	2
C₇₃×#2159	2
C₂×#367×C₇₃	2
C₇₃×#2151	2
C₇₃×#2139	2
C₇₃×#2138	2
#33634	73

Subgroups — C73×#33634 #32943038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#33634 #32943038