Subgroups — C₇₃×#25814 #32935218

Order 18688.

1034 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#499×C₇₃	C₂	1	2	yes	maximal, complemented
3	C₇₃×#2851	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#2058	C₂	1	2	yes	maximal, complemented
5	C₇₃×#2057	C₂	1	2	yes	maximal, complemented
6	C₇₃×#1991	C₂	1	2	yes	maximal, complemented
7	C₇₃×#1854	C₂	1	4	yes	maximal
8	C₄×C₂²⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
9	#533×C₇₃	—	2	8	no	—
10	C₂×C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
11	C₂×#125×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×#125×C₇₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊Q₈×C₇₃	—	2	2	no	—
14	C₂×C₄×Q₈×C₇₃	—	2	2	no	—
15	C₂×#117×C₇₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
17	#499×C₇₃	—	2	8	no	—
18	C₂×#107×C₇₃	C₂²	1	4	yes	—
19	C₂×#106×C₇₃	C₂²	1	4	yes	—
20	C₂×C₂²⋊C₈×C₇₃	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
22	#389×C₇₃	C₂²	1	2	yes	—
23	#387×C₇₃	—	2	2	no	—
24	#385×C₇₃	—	2	2	no	—
25	#382×C₇₃	C₂²	1	1	yes	characteristic
26	#380×C₇₃	C₂²	1	2	yes	—
27	#340×C₇₃	C₂²	1	4	yes	—
28	#340×C₇₃	C₂²	1	4	yes	—
29	#336×C₇₃	C₂²	1	1	yes	characteristic
30	#336×C₇₃	C₂²	1	1	yes	characteristic
31	C₂²×Q₈×C₇₃	—	2	2	no	—
32	C₂³×C₄×C₇₃	D₄	1	2	yes	—
33	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
36	C₄⋊Q₈×C₇₃	—	2	8	no	—
37	#125×C₇₃	—	2	4	no	—
38	#125×C₇₃	—	2	4	no	—
39	#125×C₇₃	D₄	1	8	yes	—
40	C₂²⋊Q₈×C₇₃	—	2	16	no	—
41	C₄×Q₈×C₇₃	—	2	16	no	—
42	#117×C₇₃	—	2	8	no	—
43	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
44	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
50	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
51	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
52	C₂×C₄²×C₇₃	—	2	2	no	—
53	C₂×C₄²×C₇₃	—	2	4	no	—
54	C₂×C₄²×C₇₃	D₄	1	2	yes	—
55	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
57	#107×C₇₃	—	2	8	no	—
58	#106×C₇₃	—	2	8	no	—
59	C₂²⋊C₈×C₇₃	Q₈	1	8	yes	complemented
60	C₄×C₈×C₇₃	—	2	4	no	—
61	#95×C₇₃	—	2	2	no	—
62	#95×C₇₃	—	2	2	no	—
63	#95×C₇₃	—	2	2	no	—
64	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
65	C₂×Q₈×C₇₃	—	2	8	no	—
66	C₂×Q₈×C₇₃	—	4	4	no	—
67	C₂×Q₈×C₇₃	—	4	4	no	—
68	C₂²×C₄×C₇₃	—	2	2	no	—
69	C₂²×C₄×C₇₃	—	2	1	no	—
70	C₂²×C₄×C₇₃	—	2	1	no	—
71	C₂²×C₄×C₇₃	—	2	4	no	—
72	C₂²×C₄×C₇₃	—	2	4	no	—
73	C₂²×C₄×C₇₃	—	2	4	no	—
74	C₂²×C₄×C₇₃	—	2	2	no	—
75	C₂²×C₄×C₇₃	—	2	2	no	—
76	C₂²×C₄×C₇₃	—	4	2	no	—
77	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
81	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₇₃	—	2	2	no	—
83	C₂×C₈×C₇₃	—	2	2	no	—
84	C₂×C₈×C₇₃	—	2	2	no	—
85	C₂×C₈×C₇₃	—	2	2	no	—
86	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
87	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
88	C₄⋊C₄×C₇₃	—	2	8	no	—
89	C₄⋊C₄×C₇₃	—	2	16	no	—
90	C₄⋊C₄×C₇₃	—	2	4	no	—
91	C₄⋊C₄×C₇₃	—	2	4	no	—
92	C₄⋊C₄×C₇₃	—	2	2	no	—
93	C₄⋊C₄×C₇₃	—	2	2	no	—
94	C₄⋊C₄×C₇₃	—	4	4	no	—
95	C₄⋊C₄×C₇₃	—	4	4	no	—
96	C₄⋊C₄×C₇₃	C₂×D₄	1	8	yes	—
97	C₂²⋊C₄×C₇₃	—	2	8	no	—
98	C₂²⋊C₄×C₇₃	—	4	4	no	—
99	C₄²×C₇₃	—	2	8	no	—
100	C₄²×C₇₃	—	2	4	no	—
101	C₄²×C₇₃	—	2	2	no	—
102	C₄²×C₇₃	C₂×D₄	1	4	yes	—
103	C₂³×C₇₃	—	2	1	no	—
104	C₂³×C₇₃	—	2	1	no	—
105	C₂³×C₇₃	—	2	2	no	—
106	C₂³×C₇₃	—	2	2	no	—
107	C₂³×C₇₃	C₄⋊Q₈	1	2	yes	—
108	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₇₃	—	4	16	no	—
110	C₂×C₄×C₇₃	—	2	1	no	—
111	C₂×C₄×C₇₃	—	2	2	no	—
112	C₂×C₄×C₇₃	—	2	1	no	—
113	C₂×C₄×C₇₃	—	2	2	no	—
114	C₂×C₄×C₇₃	—	2	2	no	—
115	C₂×C₄×C₇₃	—	2	2	no	—
116	C₂×C₄×C₇₃	—	2	4	no	—
117	C₂×C₄×C₇₃	—	2	4	no	—
118	C₂×C₄×C₇₃	—	2	8	no	—
119	C₂×C₄×C₇₃	—	4	8	no	—
120	C₂×C₄×C₇₃	—	4	4	no	—
121	C₂×C₄×C₇₃	—	4	4	no	—
122	C₂×C₄×C₇₃	—	4	2	no	—
123	C₂×C₄×C₇₃	—	4	2	no	—
124	C₂×C₄×C₇₃	—	4	2	no	—
125	C₂×C₄×C₇₃	—	4	2	no	—
126	C₂×C₄×C₇₃	—	4	2	no	—
127	C₂×C₄×C₇₃	—	4	2	no	—
128	C₂×C₄×C₇₃	#120	1	4	yes	—
129	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
130	C₂×C₄×C₇₃	#135	1	4	yes	—
131	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
132	C₂×C₄×C₇₃	C₂²×Q₈	1	2	yes	—
133	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
134	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
135	C₈×C₇₃	—	2	4	no	—
136	C₈×C₇₃	—	2	4	no	—
137	C₂²×C₇₃	—	2	2	no	—
138	C₂²×C₇₃	—	2	2	no	—
139	C₂²×C₇₃	—	2	2	no	—
140	C₂²×C₇₃	—	4	2	no	—
141	C₂²×C₇₃	—	4	2	no	—
142	C₂²×C₇₃	C₂×#120	1	1	yes	characteristic
143	C₂²×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
144	C₂²×C₇₃	#536	1	1	yes	characteristic
145	C₂²×C₇₃	C₂×#135	1	2	yes	—
146	C₂²×C₇₃	#575	1	2	yes	—
147	C₄×C₇₃	—	2	2	no	—
148	C₄×C₇₃	—	2	4	no	—
149	C₄×C₇₃	—	2	4	no	—
150	C₄×C₇₃	—	4	8	no	—
151	C₄×C₇₃	—	8	2	no	—
152	C₄×C₇₃	—	8	2	no	—
153	C₄×C₇₃	D₄×Q₈	1	2	yes	—
154	C₄×C₇₃	D₄×Q₈	1	2	yes	—
155	#25814	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂×C₇₃	—	4	2	no	—
157	C₂×C₇₃	#2434	1	1	yes	characteristic
158	C₂×C₇₃	#3002	1	2	yes	—
159	C₂×C₇₃	#3160	1	1	yes	characteristic
160	C₂×C₇₃	#3166	1	1	yes	characteristic
161	C₂×C₇₃	#3233	1	2	yes	—
162	C₂×#533	C₂×C₇₃	1	1	yes	characteristic
163	C₂×#499	C₂×C₇₃	1	2	yes	complemented
164	#2851	C₂×C₇₃	1	1	yes	characteristic
165	#2058	C₂×C₇₃	1	2	yes	complemented
166	#2057	C₂×C₇₃	1	2	yes	complemented
167	#1991	C₂×C₇₃	1	2	yes	complemented
168	#1854	C₂×C₇₃	1	4	yes	—
169	C₄×C₂²⋊C₈	C₂×C₇₃	1	1	yes	characteristic
170	C₇₃	#25814	1	1	yes	characteristic, direct product factor, Sylow
171	#533	—	2	8	no	—
172	C₂×C₄⋊Q₈	C₂²×C₇₃	1	2	yes	—
173	C₂×#125	C₂²×C₇₃	1	1	yes	characteristic
174	C₂×#125	C₂²×C₇₃	1	2	yes	—
175	C₂×C₂²⋊Q₈	—	2	2	no	—
176	C₂×C₄×Q₈	—	2	2	no	—
177	C₂×#117	C₂²×C₇₃	1	2	yes	—
178	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
179	#499	—	2	8	no	—
180	C₂×#107	C₂²×C₇₃	1	4	yes	—
181	C₂×#106	C₂²×C₇₃	1	4	yes	—
182	C₂×C₂²⋊C₈	C₂²×C₇₃	1	2	yes	—
183	C₂×C₄×C₈	C₂²×C₇₃	1	2	yes	—
184	#389	C₂²×C₇₃	1	2	yes	—
185	#387	—	2	2	no	—
186	#385	—	2	2	no	—
187	#382	C₂²×C₇₃	1	1	yes	characteristic
188	#380	C₂²×C₇₃	1	2	yes	—
189	#340	C₂²×C₇₃	1	4	yes	—
190	#340	C₂²×C₇₃	1	4	yes	—
191	#336	C₂²×C₇₃	1	1	yes	characteristic
192	#336	C₂²×C₇₃	1	1	yes	characteristic
193	C₂²×Q₈	—	2	2	no	—
194	C₂³×C₄	D₄×C₇₃	1	2	yes	—
195	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₇₃	1	2	yes	—
197	C₂²×C₈	C₂³×C₇₃	1	2	yes	—
198	C₄⋊Q₈	—	2	8	no	—
199	#125	—	2	4	no	—
200	#125	—	2	4	no	—
201	#125	D₄×C₇₃	1	8	yes	—
202	C₂²⋊Q₈	—	2	16	no	—
203	C₄×Q₈	—	2	16	no	—
204	#117	—	2	8	no	—
205	C₂×C₄⋊C₄	—	2	4	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
210	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
211	C₂×C₄⋊C₄	C₂³×C₇₃	1	4	yes	—
212	C₂×C₂²⋊C₄	—	2	2	no	—
213	C₂×C₂²⋊C₄	—	2	2	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	2	4	no	—
216	C₂×C₄²	D₄×C₇₃	1	2	yes	—
217	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
218	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
219	#107	—	2	8	no	—
220	#106	—	2	8	no	—
221	C₂²⋊C₈	Q₈×C₇₃	1	8	yes	complemented
222	C₄×C₈	—	2	4	no	—
223	#95	—	2	2	no	—
224	#95	—	2	2	no	—
225	#95	—	2	2	no	—
226	C₂⁴	C₂×D₄×C₇₃	1	1	yes	characteristic
227	C₂×Q₈	—	2	8	no	—
228	C₂×Q₈	—	4	4	no	—
229	C₂×Q₈	—	4	4	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	4	2	no	—
239	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
240	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
242	C₂²×C₄	C₂×Q₈×C₇₃	1	4	yes	—
243	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	C₂×Q₈×C₇₃	1	4	yes	—
249	C₂×C₈	C₂×Q₈×C₇₃	1	4	yes	—
250	C₄⋊C₄	—	2	8	no	—
251	C₄⋊C₄	—	2	16	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	C₂×D₄×C₇₃	1	8	yes	—
259	C₂²⋊C₄	—	2	8	no	—
260	C₂²⋊C₄	—	4	4	no	—
261	C₄²	—	2	8	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	2	2	no	—
264	C₄²	C₂×D₄×C₇₃	1	4	yes	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	C₄⋊Q₈×C₇₃	1	2	yes	—
270	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
271	Q₈	—	4	16	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	4	8	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	#120×C₇₃	1	4	yes	—
291	C₂×C₄	C₄⋊Q₈×C₇₃	1	2	yes	—
292	C₂×C₄	#135×C₇₃	1	4	yes	—
293	C₂×C₄	C₂²×D₄×C₇₃	1	2	yes	—
294	C₂×C₄	C₂²×Q₈×C₇₃	1	2	yes	—
295	C₂×C₄	2₋¹⁺⁴×C₇₃	1	1	yes	characteristic
296	C₂×C₄	2₋¹⁺⁴×C₇₃	1	1	yes	characteristic
297	C₈	—	2	4	no	—
298	C₈	—	2	4	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	4	2	no	—
303	C₂²	—	4	2	no	—
304	C₂²	C₂×#120×C₇₃	1	1	yes	characteristic
305	C₂²	C₂×C₄⋊Q₈×C₇₃	1	1	yes	characteristic
306	C₂²	#536×C₇₃	1	1	yes	characteristic
307	C₂²	C₂×#135×C₇₃	1	2	yes	—
308	C₂²	#575×C₇₃	1	2	yes	—
309	C₄	—	2	4	no	—
310	C₄	—	2	4	no	—
311	C₄	—	2	2	no	—
312	C₄	—	4	8	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	D₄×Q₈×C₇₃	1	2	yes	—
316	C₄	D₄×Q₈×C₇₃	1	2	yes	—
317	C₂	—	4	2	no	—
318	C₂	C₇₃×#2434	1	1	yes	characteristic
319	C₂	C₇₃×#3002	1	2	yes	—
320	C₂	C₇₃×#3160	1	1	yes	characteristic
321	C₂	C₇₃×#3166	1	1	yes	characteristic
322	C₂	C₇₃×#3233	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₃
prime radical	C₂⁴×C₇₃	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ C₂²×C₄²×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#499×C₇₃	C₂	semidirect
4. C₇₃×#2058	C₂	semidirect
5. C₇₃×#2057	C₂	semidirect
6. C₇₃×#1991	C₂	semidirect
59. C₂²⋊C₈×C₇₃	Q₈	semidirect
155. #25814	C₇₃	direct (N × H)
163. C₂×#499	C₂×C₇₃	semidirect
165. #2058	C₂×C₇₃	semidirect
166. #2057	C₂×C₇₃	semidirect
167. #1991	C₂×C₇₃	semidirect
170. C₇₃	#25814	direct (N × H)
221. C₂²⋊C₈	Q₈×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#533×C₇₃	2
C₂×#499×C₇₃	2
C₇₃×#2851	2
C₇₃×#2058	2
C₇₃×#2057	2
C₇₃×#1991	2
C₇₃×#1854	2
C₄×C₂²⋊C₈×C₇₃	2
#25814	73

Subgroups — C73×#25814 #32935218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#25814 #32935218