Subgroups — C₇₃×#29136 #32938540

Order 18688.

754 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×#2647	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₃×#2328	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#2101	C₂	1	2	yes	maximal, complemented
4	C₇₃×#2041	C₂	1	2	yes	maximal
5	C₇₃×#1936	C₂	1	2	yes	maximal, complemented
6	C₇₃×#1934	C₂	1	2	yes	maximal, complemented
7	C₇₃×#1804	C₂	1	2	yes	maximal, complemented
8	C₇₃×#1802	C₂	1	2	yes	maximal, complemented
9	C₇₃×#1768	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#127×C₇₃	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₄×C₇₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
13	C₂×#102×C₇₃	C₂²	1	2	yes	—
14	C₂×#102×C₇₃	C₂²	1	2	yes	—
15	C₂×#102×C₇₃	C₂²	1	2	yes	—
16	C₂×#102×C₇₃	C₂²	1	2	yes	—
17	C₂×#97×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
19	#400×C₇₃	—	2	2	no	—
20	#398×C₇₃	—	2	2	no	—
21	#390×C₇₃	C₂²	1	2	yes	—
22	#389×C₇₃	C₂²	1	2	yes	—
23	#389×C₇₃	C₂²	1	2	yes	—
24	#386×C₇₃	—	2	2	no	—
25	#386×C₇₃	—	2	2	no	—
26	#384×C₇₃	C₂²	1	1	yes	characteristic
27	#382×C₇₃	C₂²	1	1	yes	characteristic
28	#382×C₇₃	C₂²	1	1	yes	characteristic
29	#376×C₇₃	C₂²	1	1	yes	characteristic
30	#340×C₇₃	C₂²	1	2	yes	—
31	#340×C₇₃	C₂²	1	2	yes	—
32	#340×C₇₃	C₂²	1	2	yes	—
33	#340×C₇₃	C₂²	1	2	yes	—
34	#336×C₇₃	C₂²	1	2	yes	—
35	C₂²×D₄×C₇₃	—	2	2	no	—
36	C₂²×D₄×C₇₃	C₂³	1	2	yes	—
37	C₂³×C₄×C₇₃	—	2	2	no	—
38	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
40	#127×C₇₃	—	2	4	no	—
41	C₄×D₄×C₇₃	—	2	4	no	—
42	C₄×D₄×C₇₃	—	2	4	no	—
43	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
52	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₇₃	—	2	4	no	—
56	#102×C₇₃	—	2	4	no	—
57	#102×C₇₃	—	2	4	no	—
58	#102×C₇₃	—	4	2	no	—
59	#102×C₇₃	—	4	2	no	—
60	#97×C₇₃	—	4	1	no	—
61	C₄×C₈×C₇₃	—	4	1	no	—
62	#95×C₇₃	—	2	2	no	—
63	#95×C₇₃	—	2	2	no	—
64	#95×C₇₃	—	2	2	no	—
65	#95×C₇₃	—	2	2	no	—
66	#95×C₇₃	D₄	1	2	yes	—
67	#95×C₇₃	D₄	1	2	yes	—
68	C₂⁴×C₇₃	—	2	2	no	—
69	C₂×D₄×C₇₃	—	2	8	no	—
70	C₂×D₄×C₇₃	—	2	2	no	—
71	C₂×D₄×C₇₃	—	2	2	no	—
72	C₂×D₄×C₇₃	—	2	2	no	—
73	C₂×D₄×C₇₃	—	2	2	no	—
74	C₂×D₄×C₇₃	—	2	2	no	—
75	C₂×D₄×C₇₃	—	8	2	no	—
76	C₂×D₄×C₇₃	—	8	2	no	—
77	C₂×D₄×C₇₃	#41	1	4	yes	—
78	C₂²×C₄×C₇₃	—	2	2	no	—
79	C₂²×C₄×C₇₃	—	2	2	no	—
80	C₂²×C₄×C₇₃	—	2	2	no	—
81	C₂²×C₄×C₇₃	—	4	4	no	—
82	C₂²×C₄×C₇₃	—	8	2	no	—
83	C₂²×C₄×C₇₃	—	8	2	no	—
84	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₇₃	—	2	2	no	—
90	C₂×C₈×C₇₃	—	2	2	no	—
91	C₂×C₈×C₇₃	—	2	2	no	—
92	C₂×C₈×C₇₃	—	2	2	no	—
93	C₂×C₈×C₇₃	—	2	2	no	—
94	C₂×C₈×C₇₃	—	2	2	no	—
95	C₄⋊C₄×C₇₃	—	2	2	no	—
96	C₄⋊C₄×C₇₃	—	2	2	no	—
97	C₄⋊C₄×C₇₃	—	2	2	no	—
98	C₄⋊C₄×C₇₃	—	2	2	no	—
99	C₄⋊C₄×C₇₃	—	2	2	no	—
100	C₄⋊C₄×C₇₃	—	2	2	no	—
101	C₄⋊C₄×C₇₃	—	8	2	no	—
102	C₄⋊C₄×C₇₃	—	8	2	no	—
103	C₄⋊C₄×C₇₃	—	8	2	no	—
104	C₂²⋊C₄×C₇₃	—	4	4	no	—
105	C₂²⋊C₄×C₇₃	—	8	2	no	—
106	C₂²⋊C₄×C₇₃	—	8	2	no	—
107	C₄²×C₇₃	—	2	2	no	—
108	C₄²×C₇₃	—	2	1	no	—
109	C₄²×C₇₃	—	2	1	no	—
110	C₄²×C₇₃	#41	1	4	yes	—
111	C₂³×C₇₃	—	2	4	no	—
112	C₂³×C₇₃	—	4	2	no	—
113	C₂³×C₇₃	—	4	2	no	—
114	C₂³×C₇₃	—	4	2	no	—
115	C₂³×C₇₃	—	4	2	no	—
116	C₂³×C₇₃	—	4	2	no	—
117	C₂³×C₇₃	—	4	2	no	—
118	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
119	D₄×C₇₃	—	2	4	no	—
120	D₄×C₇₃	—	2	4	no	—
121	D₄×C₇₃	—	4	2	no	—
122	D₄×C₇₃	—	4	2	no	—
123	D₄×C₇₃	—	8	8	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	1	no	—
128	C₂×C₄×C₇₃	—	2	1	no	—
129	C₂×C₄×C₇₃	—	2	2	no	—
130	C₂×C₄×C₇₃	—	2	2	no	—
131	C₂×C₄×C₇₃	—	2	2	no	—
132	C₂×C₄×C₇₃	—	2	2	no	—
133	C₂×C₄×C₇₃	—	4	2	no	—
134	C₂×C₄×C₇₃	—	4	2	no	—
135	C₂×C₄×C₇₃	—	4	2	no	—
136	C₂×C₄×C₇₃	—	4	2	no	—
137	C₂×C₄×C₇₃	—	4	2	no	—
138	C₂×C₄×C₇₃	—	4	2	no	—
139	C₂×C₄×C₇₃	—	4	2	no	—
140	C₂×C₄×C₇₃	—	4	2	no	—
141	C₂×C₄×C₇₃	—	4	2	no	—
142	C₂×C₄×C₇₃	—	8	8	no	—
143	C₂×C₄×C₇₃	#124	1	4	yes	—
144	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
145	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
146	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
148	C₈×C₇₃	—	4	2	no	—
149	C₈×C₇₃	—	4	2	no	—
150	C₂²×C₇₃	—	4	4	no	—
151	C₂²×C₇₃	—	4	4	no	—
152	C₂²×C₇₃	—	4	4	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	#525	1	1	yes	characteristic
158	C₂²×C₇₃	C₂×#124	1	1	yes	characteristic
159	C₂²×C₇₃	#537	1	1	yes	characteristic
160	C₂²×C₇₃	#576	1	2	yes	—
161	C₂²×C₇₃	#577	1	2	yes	—
162	C₄×C₇₃	—	2	4	no	—
163	C₄×C₇₃	—	4	2	no	—
164	C₄×C₇₃	—	4	1	no	—
165	C₄×C₇₃	—	4	1	no	—
166	C₄×C₇₃	—	8	2	no	—
167	C₄×C₇₃	—	8	2	no	—
168	C₄×C₇₃	—	8	2	no	—
169	C₄×C₇₃	#552	1	2	yes	—
170	C₄×C₇₃	#553	1	2	yes	—
171	#29136	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
172	C₂×C₇₃	—	8	4	no	—
173	C₂×C₇₃	#2562	1	1	yes	characteristic
174	C₂×C₇₃	#3118	1	2	yes	—
175	C₂×C₇₃	#3136	1	2	yes	—
176	C₂×C₇₃	#3259	1	1	yes	characteristic
177	C₂×C₇₃	#3260	1	1	yes	characteristic
178	#2647	C₂×C₇₃	1	1	yes	characteristic, complemented
179	#2328	C₂×C₇₃	1	1	yes	characteristic
180	#2101	C₂×C₇₃	1	2	yes	complemented
181	#2041	C₂×C₇₃	1	2	yes	—
182	#1936	C₂×C₇₃	1	2	yes	complemented
183	#1934	C₂×C₇₃	1	2	yes	complemented
184	#1804	C₂×C₇₃	1	2	yes	complemented
185	#1802	C₂×C₇₃	1	2	yes	complemented
186	#1768	C₂×C₇₃	1	1	yes	characteristic, complemented
187	C₇₃	#29136	1	1	yes	characteristic, direct product factor, Sylow
188	C₂×#127	C₂²×C₇₃	1	1	yes	characteristic
189	C₂×C₄×D₄	C₂²×C₇₃	1	2	yes	—
190	C₂×C₄⋊C₈	C₂²×C₇₃	1	2	yes	—
191	C₂×#102	C₂²×C₇₃	1	2	yes	—
192	C₂×#102	C₂²×C₇₃	1	2	yes	—
193	C₂×#102	C₂²×C₇₃	1	2	yes	—
194	C₂×#102	C₂²×C₇₃	1	2	yes	—
195	C₂×#97	C₂²×C₇₃	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂²×C₇₃	1	1	yes	characteristic
197	#400	—	2	2	no	—
198	#398	—	2	2	no	—
199	#390	C₂²×C₇₃	1	2	yes	—
200	#389	C₂²×C₇₃	1	2	yes	—
201	#389	C₂²×C₇₃	1	2	yes	—
202	#386	—	2	2	no	—
203	#386	—	2	2	no	—
204	#384	C₂²×C₇₃	1	1	yes	characteristic
205	#382	C₂²×C₇₃	1	1	yes	characteristic
206	#382	C₂²×C₇₃	1	1	yes	characteristic
207	#376	C₂²×C₇₃	1	1	yes	characteristic
208	#340	C₂²×C₇₃	1	2	yes	—
209	#340	C₂²×C₇₃	1	2	yes	—
210	#340	C₂²×C₇₃	1	2	yes	—
211	#340	C₂²×C₇₃	1	2	yes	—
212	#336	C₂²×C₇₃	1	2	yes	—
213	C₂²×D₄	—	2	2	no	—
214	C₂²×D₄	C₂³×C₇₃	1	2	yes	—
215	C₂³×C₄	—	2	2	no	—
216	C₂²×C₈	C₂³×C₇₃	1	2	yes	—
217	C₂²×C₈	C₂³×C₇₃	1	2	yes	—
218	#127	—	2	4	no	—
219	C₄×D₄	—	2	4	no	—
220	C₄×D₄	—	2	4	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
225	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
226	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	—	2	2	no	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
231	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
232	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
233	C₄⋊C₈	—	2	4	no	—
234	#102	—	2	4	no	—
235	#102	—	2	4	no	—
236	#102	—	4	2	no	—
237	#102	—	4	2	no	—
238	#97	—	4	1	no	—
239	C₄×C₈	—	4	1	no	—
240	#95	—	2	2	no	—
241	#95	—	2	2	no	—
242	#95	—	2	2	no	—
243	#95	—	2	2	no	—
244	#95	D₄×C₇₃	1	2	yes	—
245	#95	D₄×C₇₃	1	2	yes	—
246	C₂⁴	—	2	2	no	—
247	C₂×D₄	—	2	8	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	8	2	no	—
254	C₂×D₄	—	8	2	no	—
255	C₂×D₄	#41×C₇₃	1	4	yes	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	4	4	no	—
260	C₂²×C₄	—	8	2	no	—
261	C₂²×C₄	—	8	2	no	—
262	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
263	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
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	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	8	2	no	—
280	C₄⋊C₄	—	8	2	no	—
281	C₄⋊C₄	—	8	2	no	—
282	C₂²⋊C₄	—	4	4	no	—
283	C₂²⋊C₄	—	8	2	no	—
284	C₂²⋊C₄	—	8	2	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	2	1	no	—
288	C₄²	#41×C₇₃	1	4	yes	—
289	C₂³	—	2	4	no	—
290	C₂³	—	4	2	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
297	D₄	—	2	4	no	—
298	D₄	—	2	4	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	8	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	8	8	no	—
321	C₂×C₄	#124×C₇₃	1	4	yes	—
322	C₂×C₄	C₂×#41×C₇₃	1	2	yes	—
323	C₂×C₄	C₂×#41×C₇₃	1	2	yes	—
324	C₂×C₄	H(5,𝔽₂)×C₇₃	1	1	yes	characteristic
325	C₂×C₄	2₋¹⁺⁴×C₇₃	1	1	yes	characteristic
326	C₈	—	4	2	no	—
327	C₈	—	4	2	no	—
328	C₂²	—	4	4	no	—
329	C₂²	—	4	4	no	—
330	C₂²	—	4	4	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	#525×C₇₃	1	1	yes	characteristic
336	C₂²	C₂×#124×C₇₃	1	1	yes	characteristic
337	C₂²	#537×C₇₃	1	1	yes	characteristic
338	C₂²	#576×C₇₃	1	2	yes	—
339	C₂²	#577×C₇₃	1	2	yes	—
340	C₄	—	2	4	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	2	no	—
344	C₄	—	8	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	2	no	—
347	C₄	#552×C₇₃	1	2	yes	—
348	C₄	#553×C₇₃	1	2	yes	—
349	C₂	—	8	4	no	—
350	C₂	C₇₃×#2562	1	1	yes	characteristic
351	C₂	C₇₃×#3118	1	2	yes	—
352	C₂	C₇₃×#3136	1	2	yes	—
353	C₂	C₇₃×#3259	1	1	yes	characteristic
354	C₂	C₇₃×#3260	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₃
prime radical	C₂×#127×C₇₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ #376×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇₃×#2647	C₂	semidirect
3. C₇₃×#2101	C₂	semidirect
5. C₇₃×#1936	C₂	semidirect
6. C₇₃×#1934	C₂	semidirect
7. C₇₃×#1804	C₂	semidirect
8. C₇₃×#1802	C₂	semidirect
9. C₇₃×#1768	C₂	semidirect
171. #29136	C₇₃	direct (N × H)
178. #2647	C₂×C₇₃	semidirect
180. #2101	C₂×C₇₃	semidirect
182. #1936	C₂×C₇₃	semidirect
183. #1934	C₂×C₇₃	semidirect
184. #1804	C₂×C₇₃	semidirect
185. #1802	C₂×C₇₃	semidirect
186. #1768	C₂×C₇₃	semidirect
187. C₇₃	#29136	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇₃×#2647	2
C₇₃×#2328	2
C₇₃×#2101	2
C₇₃×#2041	2
C₇₃×#1936	2
C₇₃×#1934	2
C₇₃×#1804	2
C₇₃×#1802	2
C₇₃×#1768	2
#29136	73

Subgroups — C73×#29136 #32938540

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#29136 #32938540