Subgroups — C₇₃×#33782 #32943186

Order 18688.

588 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₃×#3147	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#2251	C₂	1	2	yes	maximal, complemented
4	C₇₃×#2243	C₂	1	2	yes	maximal, complemented
5	C₇₃×#2209	C₂	1	2	yes	maximal, complemented
6	C₇₃×#2207	C₂	1	2	yes	maximal, complemented
7	C₇₃×#2189	C₂	1	4	yes	maximal, complemented
8	C₇₃×#2163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#135×C₇₃	—	2	1	no	—
10	C₂×#135×C₇₃	C₂²	1	2	yes	—
11	#533×C₇₃	—	2	1	no	—
12	#532×C₇₃	—	2	1	no	—
13	C₂×Q₃₂×C₇₃	C₂²	1	4	yes	—
14	C₂×SD₃₂×C₇₃	C₂²	1	4	yes	—
15	#500×C₇₃	C₂²	1	1	yes	characteristic
16	#495×C₇₃	—	2	1	no	—
17	#495×C₇₃	C₂²	1	2	yes	—
18	#494×C₇₃	C₂²	1	1	yes	characteristic
19	#494×C₇₃	C₂²	1	1	yes	characteristic
20	#493×C₇₃	C₂²	1	1	yes	characteristic
21	#492×C₇₃	—	2	1	no	—
22	#487×C₇₃	—	2	1	no	—
23	#467×C₇₃	—	2	1	no	—
24	#467×C₇₃	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₇₃	—	2	1	no	—
26	C₄×Q₁₆×C₇₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
28	C₄⋊C₁₆×C₇₃	C₂²	1	2	yes	—
29	#358×C₇₃	C₂²	1	4	yes	—
30	#357×C₇₃	C₂²	1	4	yes	—
31	C₂²⋊C₁₆×C₇₃	C₂²	1	2	yes	—
32	#346×C₇₃	C₂²	1	2	yes	—
33	C₂×#41×C₇₃	—	2	2	no	—
34	#135×C₇₃	—	4	4	no	—
35	#135×C₇₃	—	4	4	no	—
36	C₂×Q₁₆×C₇₃	—	2	1	no	—
37	C₂×Q₁₆×C₇₃	—	2	1	no	—
38	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
39	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
40	C₂×SD₁₆×C₇₃	—	2	2	no	—
41	C₂×SD₁₆×C₇₃	—	2	2	no	—
42	C₂×D₈×C₇₃	—	2	1	no	—
43	C₂×D₈×C₇₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈×C₇₃	—	2	1	no	—
47	C₄⋊Q₈×C₇₃	—	2	1	no	—
48	C₄⋊Q₈×C₇₃	—	2	1	no	—
49	#127×C₇₃	—	2	1	no	—
50	#125×C₇₃	—	4	1	no	—
51	#124×C₇₃	—	2	2	no	—
52	C₂²⋊Q₈×C₇₃	—	2	2	no	—
53	C₂²⋊Q₈×C₇₃	—	4	1	no	—
54	#121×C₇₃	—	4	1	no	—
55	#121×C₇₃	—	4	1	no	—
56	C₄×Q₈×C₇₃	—	2	2	no	—
57	C₄×Q₈×C₇₃	—	4	1	no	—
58	C₄×D₄×C₇₃	—	4	1	no	—
59	C₄×D₄×C₇₃	—	4	1	no	—
60	#117×C₇₃	—	4	1	no	—
61	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
62	Q₃₂×C₇₃	—	4	4	no	—
63	SD₃₂×C₇₃	—	4	4	no	—
64	C₂×C₁₆×C₇₃	C₂³	1	4	yes	—
65	#107×C₇₃	—	2	1	no	—
66	#107×C₇₃	C₂³	1	2	yes	—
67	#103×C₇₃	—	2	2	no	—
68	#103×C₇₃	—	2	2	no	—
69	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
70	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
71	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
73	#41×C₇₃	—	4	4	no	—
74	#41×C₇₃	—	4	4	no	—
75	C₂×Q₈×C₇₃	—	2	2	no	—
76	C₂×Q₈×C₇₃	—	2	2	no	—
77	C₂×Q₈×C₇₃	—	4	1	no	—
78	C₂×D₄×C₇₃	—	2	2	no	—
79	C₂×D₄×C₇₃	—	4	2	no	—
80	C₂×D₄×C₇₃	—	4	1	no	—
81	C₂×D₄×C₇₃	—	4	1	no	—
82	C₂²×C₄×C₇₃	—	4	2	no	—
83	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
86	Q₁₆×C₇₃	—	2	2	no	—
87	Q₁₆×C₇₃	—	2	2	no	—
88	Q₁₆×C₇₃	—	2	4	no	—
89	Q₁₆×C₇₃	—	4	2	no	—
90	Q₁₆×C₇₃	—	4	2	no	—
91	SD₁₆×C₇₃	—	4	4	no	—
92	SD₁₆×C₇₃	—	4	4	no	—
93	D₈×C₇₃	—	2	4	no	—
94	D₈×C₇₃	—	4	2	no	—
95	C₂×C₈×C₇₃	—	2	2	no	—
96	C₂×C₈×C₇₃	—	2	2	no	—
97	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₇₃	—	2	2	no	—
102	C₄⋊C₄×C₇₃	—	4	1	no	—
103	C₄⋊C₄×C₇₃	—	4	1	no	—
104	C₄⋊C₄×C₇₃	—	4	1	no	—
105	C₄⋊C₄×C₇₃	—	4	1	no	—
106	C₄⋊C₄×C₇₃	—	4	2	no	—
107	C₄⋊C₄×C₇₃	—	4	2	no	—
108	C₄⋊C₄×C₇₃	—	4	1	no	—
109	C₂²⋊C₄×C₇₃	—	4	2	no	—
110	C₂²⋊C₄×C₇₃	—	4	2	no	—
111	C₂²⋊C₄×C₇₃	—	4	2	no	—
112	C₄²×C₇₃	—	4	2	no	—
113	C₄²×C₇₃	D₈	1	1	yes	characteristic
114	C₄²×C₇₃	D₈	1	1	yes	characteristic
115	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₇₃	—	2	4	no	—
118	C₂³×C₇₃	—	4	2	no	—
119	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₇₃	—	4	2	no	—
121	Q₈×C₇₃	—	4	2	no	—
122	Q₈×C₇₃	—	4	2	no	—
123	Q₈×C₇₃	—	4	2	no	—
124	Q₈×C₇₃	—	8	2	no	—
125	D₄×C₇₃	—	4	2	no	—
126	D₄×C₇₃	—	4	2	no	—
127	D₄×C₇₃	—	8	2	no	—
128	D₄×C₇₃	—	8	4	no	—
129	D₄×C₇₃	—	8	2	no	—
130	C₂×C₄×C₇₃	—	2	1	no	—
131	C₂×C₄×C₇₃	—	2	1	no	—
132	C₂×C₄×C₇₃	—	2	2	no	—
133	C₂×C₄×C₇₃	—	4	1	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	—	8	4	no	—
138	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
144	C₈×C₇₃	—	2	1	no	—
145	C₈×C₇₃	—	2	1	no	—
146	C₈×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
147	C₈×C₇₃	H(5,𝔽₂)	1	2	yes	—
148	C₂²×C₇₃	—	2	1	no	—
149	C₂²×C₇₃	—	2	1	no	—
150	C₂²×C₇₃	—	2	1	no	—
151	C₂²×C₇₃	—	8	2	no	—
152	C₂²×C₇₃	—	8	2	no	—
153	C₂²×C₇₃	—	8	2	no	—
154	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
155	C₄×C₇₃	—	2	1	no	—
156	C₄×C₇₃	—	2	1	no	—
157	C₄×C₇₃	—	2	1	no	—
158	C₄×C₇₃	—	2	1	no	—
159	C₄×C₇₃	—	8	2	no	—
160	C₄×C₇₃	—	8	2	no	—
161	C₄×C₇₃	—	8	2	no	—
162	C₄×C₇₃	#510	1	2	yes	—
163	C₄×C₇₃	#535	1	1	yes	characteristic
164	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	#33782	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂×C₇₃	—	4	1	no	—
167	C₂×C₇₃	—	16	2	no	—
168	C₂×C₇₃	#3205	1	1	yes	characteristic
169	C₂×C₇₃	C₂×#510	1	1	yes	characteristic
170	C₂×C₇₃	#3414	1	1	yes	characteristic
171	#3170	C₂×C₇₃	1	1	yes	characteristic, complemented
172	#3147	C₂×C₇₃	1	1	yes	characteristic
173	#2251	C₂×C₇₃	1	2	yes	complemented
174	#2243	C₂×C₇₃	1	2	yes	complemented
175	#2209	C₂×C₇₃	1	2	yes	complemented
176	#2207	C₂×C₇₃	1	2	yes	complemented
177	#2189	C₂×C₇₃	1	4	yes	complemented
178	#2163	C₂×C₇₃	1	1	yes	characteristic, complemented
179	C₇₃	#33782	1	1	yes	characteristic, direct product factor, Sylow
180	C₂×#135	—	2	1	no	—
181	C₂×#135	C₂²×C₇₃	1	2	yes	—
182	#533	—	2	1	no	—
183	#532	—	2	1	no	—
184	C₂×Q₃₂	C₂²×C₇₃	1	4	yes	—
185	C₂×SD₃₂	C₂²×C₇₃	1	4	yes	—
186	#500	C₂²×C₇₃	1	1	yes	characteristic
187	#495	—	2	1	no	—
188	#495	C₂²×C₇₃	1	2	yes	—
189	#494	C₂²×C₇₃	1	1	yes	characteristic
190	#494	C₂²×C₇₃	1	1	yes	characteristic
191	#493	C₂²×C₇₃	1	1	yes	characteristic
192	#492	—	2	1	no	—
193	#487	—	2	1	no	—
194	#467	—	2	1	no	—
195	#467	C₂²×C₇₃	1	2	yes	—
196	C₄×Q₁₆	—	2	1	no	—
197	C₄×Q₁₆	C₂²×C₇₃	1	2	yes	—
198	C₂×C₄×C₈	C₂²×C₇₃	1	1	yes	characteristic
199	C₄⋊C₁₆	C₂²×C₇₃	1	2	yes	—
200	#358	C₂²×C₇₃	1	4	yes	—
201	#357	C₂²×C₇₃	1	4	yes	—
202	C₂²⋊C₁₆	C₂²×C₇₃	1	2	yes	—
203	#346	C₂²×C₇₃	1	2	yes	—
204	C₂×#41	—	2	2	no	—
205	#135	—	4	4	no	—
206	#135	—	4	4	no	—
207	C₂×Q₁₆	—	2	1	no	—
208	C₂×Q₁₆	—	2	1	no	—
209	C₂×Q₁₆	C₂³×C₇₃	1	2	yes	—
210	C₂×Q₁₆	C₂³×C₇₃	1	2	yes	—
211	C₂×SD₁₆	—	2	2	no	—
212	C₂×SD₁₆	—	2	2	no	—
213	C₂×D₈	—	2	1	no	—
214	C₂×D₈	C₂³×C₇₃	1	2	yes	—
215	C₂²×C₈	D₄×C₇₃	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₇₃	1	1	yes	characteristic
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	2	1	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	#127	—	2	1	no	—
221	#125	—	4	1	no	—
222	#124	—	2	2	no	—
223	C₂²⋊Q₈	—	2	2	no	—
224	C₂²⋊Q₈	—	4	1	no	—
225	#121	—	4	1	no	—
226	#121	—	4	1	no	—
227	C₄×Q₈	—	2	2	no	—
228	C₄×Q₈	—	4	1	no	—
229	C₄×D₄	—	4	1	no	—
230	C₄×D₄	—	4	1	no	—
231	#117	—	4	1	no	—
232	C₂×C₄²	D₄×C₇₃	1	1	yes	characteristic
233	Q₃₂	—	4	4	no	—
234	SD₃₂	—	4	4	no	—
235	C₂×C₁₆	C₂³×C₇₃	1	4	yes	—
236	#107	—	2	1	no	—
237	#107	C₂³×C₇₃	1	2	yes	—
238	#103	—	2	2	no	—
239	#103	—	2	2	no	—
240	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
241	C₄×C₈	D₄×C₇₃	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₇₃	1	1	yes	characteristic
244	#41	—	4	4	no	—
245	#41	—	4	4	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	D₈×C₇₃	1	1	yes	characteristic
255	C₂²×C₄	D₈×C₇₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
257	Q₁₆	—	2	2	no	—
258	Q₁₆	—	2	2	no	—
259	Q₁₆	—	2	4	no	—
260	Q₁₆	—	4	2	no	—
261	Q₁₆	—	4	2	no	—
262	SD₁₆	—	4	4	no	—
263	SD₁₆	—	4	4	no	—
264	D₈	—	2	4	no	—
265	D₈	—	4	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
269	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
270	C₂×C₈	C₂×D₄×C₇₃	1	1	yes	characteristic
271	C₂×C₈	C₂⁴×C₇₃	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	—	4	2	no	—
283	C₄²	—	4	2	no	—
284	C₄²	D₈×C₇₃	1	1	yes	characteristic
285	C₄²	D₈×C₇₃	1	1	yes	characteristic
286	C₄²	C₂×D₄×C₇₃	1	1	yes	characteristic
287	C₄²	C₂×D₄×C₇₃	1	1	yes	characteristic
288	C₁₆	—	2	4	no	—
289	C₂³	—	4	2	no	—
290	C₂³	C₂×D₈×C₇₃	1	1	yes	characteristic
291	Q₈	—	4	2	no	—
292	Q₈	—	4	2	no	—
293	Q₈	—	4	2	no	—
294	Q₈	—	4	2	no	—
295	Q₈	—	8	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	4	2	no	—
298	D₄	—	8	2	no	—
299	D₄	—	8	4	no	—
300	D₄	—	8	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
310	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
311	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
312	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
313	C₂×C₄	C₂×D₈×C₇₃	1	1	yes	characteristic
314	C₂×C₄	C₂²×D₄×C₇₃	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	AGL(1,ℤ/8)×C₇₃	1	2	yes	—
318	C₈	H(5,𝔽₂)×C₇₃	1	2	yes	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	C₂²×D₈×C₇₃	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	8	2	no	—
331	C₄	—	8	2	no	—
332	C₄	—	8	2	no	—
333	C₄	#510×C₇₃	1	2	yes	—
334	C₄	#535×C₇₃	1	1	yes	characteristic
335	C₄	C₂×AGL(1,ℤ/8)×C₇₃	1	1	yes	characteristic
336	C₂	—	4	1	no	—
337	C₂	—	16	2	no	—
338	C₂	C₇₃×#3205	1	1	yes	characteristic
339	C₂	C₂×#510×C₇₃	1	1	yes	characteristic
340	C₂	C₇₃×#3414	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇₃
prime radical	C₇₃×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₃ ◁ C₂×C₄²×C₇₃ ◁ C₂×C₄×C₈×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇₃×#3170	C₂	semidirect
3. C₇₃×#2251	C₂	semidirect
4. C₇₃×#2243	C₂	semidirect
5. C₇₃×#2209	C₂	semidirect
6. C₇₃×#2207	C₂	semidirect
7. C₇₃×#2189	C₂	semidirect
8. C₇₃×#2163	C₂	semidirect
165. #33782	C₇₃	direct (N × H)
171. #3170	C₂×C₇₃	semidirect
173. #2251	C₂×C₇₃	semidirect
174. #2243	C₂×C₇₃	semidirect
175. #2209	C₂×C₇₃	semidirect
176. #2207	C₂×C₇₃	semidirect
177. #2189	C₂×C₇₃	semidirect
178. #2163	C₂×C₇₃	semidirect
179. C₇₃	#33782	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×#3170	2
C₇₃×#3147	2
C₇₃×#2251	2
C₇₃×#2243	2
C₇₃×#2209	2
C₇₃×#2207	2
C₇₃×#2189	2
C₇₃×#2163	2
#33782	73

Subgroups — C73×#33782 #32943186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#33782 #32943186