Subgroups — C₇₃×#23732 #32933136

Order 18688.

1098 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#428×C₇₃	C₂	1	2	yes	maximal
2	C₂×#419×C₇₃	C₂	1	2	yes	maximal
3	C₂×#391×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#2285	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#2097	C₂	1	4	yes	maximal, complemented
6	C₇₃×#2096	C₂	1	2	yes	maximal, complemented
7	C₇₃×#2095	C₂	1	2	yes	maximal, complemented
8	C₇₃×#1728	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₇₃	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
11	C₂×#117×C₇₃	C₂²	1	2	yes	—
12	C₂²×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
14	#428×C₇₃	—	2	8	no	—
15	C₂×#107×C₇₃	C₂²	1	4	yes	—
16	C₂×#106×C₇₃	C₂²	1	4	yes	—
17	#419×C₇₃	—	2	8	no	—
18	C₂×#103×C₇₃	C₂²	1	4	yes	—
19	C₂×#103×C₇₃	C₂²	1	4	yes	—
20	#391×C₇₃	—	2	2	no	—
21	#391×C₇₃	C₂²	1	2	yes	—
22	#391×C₇₃	C₂²	1	2	yes	—
23	#384×C₇₃	C₂²	1	1	yes	characteristic
24	#384×C₇₃	C₂²	1	1	yes	characteristic
25	#382×C₇₃	C₂²	1	2	yes	—
26	#380×C₇₃	—	2	2	no	—
27	C₄×C₄⋊C₄×C₇₃	—	2	2	no	—
28	C₂×#95×C₇₃	—	2	2	no	—
29	#336×C₇₃	C₂²	1	2	yes	—
30	#336×C₇₃	C₂²	1	2	yes	—
31	C₂²×Q₈×C₇₃	—	2	1	no	—
32	C₂²×Q₈×C₇₃	—	2	2	no	—
33	C₂²×Q₈×C₇₃	—	2	4	no	—
34	C₂²×Q₈×C₇₃	—	2	4	no	—
35	C₂²×Q₈×C₇₃	Q₈	1	4	yes	—
36	C₂²×Q₈×C₇₃	C₂³	1	2	yes	—
37	C₂³×C₄×C₇₃	—	2	1	no	—
38	C₂³×C₄×C₇₃	—	2	1	no	—
39	C₂³×C₄×C₇₃	D₄	1	2	yes	—
40	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
41	C₂×M₁₆×C₇₃	—	2	4	no	—
42	C₂×M₁₆×C₇₃	—	2	4	no	—
43	C₂×M₁₆×C₇₃	D₄	1	8	yes	—
44	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
45	#117×C₇₃	—	2	4	no	—
46	#117×C₇₃	—	2	4	no	—
47	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
48	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
50	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
51	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
54	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
55	C₂×C₄²×C₇₃	—	2	4	no	—
56	C₂×C₄²×C₇₃	—	2	2	no	—
57	C₂×C₄²×C₇₃	D₄	1	2	yes	—
58	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
60	#107×C₇₃	—	2	8	no	—
61	#106×C₇₃	—	2	8	no	—
62	#103×C₇₃	—	2	8	no	—
63	#103×C₇₃	—	2	8	no	—
64	#95×C₇₃	—	2	4	no	—
65	#95×C₇₃	—	2	4	no	—
66	#95×C₇₃	—	2	4	no	—
67	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
68	C₂×Q₈×C₇₃	—	2	4	no	—
69	C₂×Q₈×C₇₃	—	2	2	no	—
70	C₂×Q₈×C₇₃	—	2	16	no	—
71	C₂×Q₈×C₇₃	—	2	4	no	—
72	C₂×Q₈×C₇₃	—	2	4	no	—
73	C₂×Q₈×C₇₃	—	4	2	no	—
74	C₂×Q₈×C₇₃	—	4	8	no	—
75	C₂×Q₈×C₇₃	—	4	2	no	—
76	C₂×Q₈×C₇₃	C₂×Q₈	1	4	yes	—
77	C₂²×C₄×C₇₃	—	2	4	no	—
78	C₂²×C₄×C₇₃	—	2	4	no	—
79	C₂²×C₄×C₇₃	—	2	2	no	—
80	C₂²×C₄×C₇₃	—	2	4	no	—
81	C₂²×C₄×C₇₃	—	2	2	no	—
82	C₂²×C₄×C₇₃	—	2	2	no	—
83	C₂²×C₄×C₇₃	—	2	2	no	—
84	C₂²×C₄×C₇₃	—	4	2	no	—
85	C₂²×C₄×C₇₃	—	4	2	no	—
86	C₂²×C₄×C₇₃	—	4	2	no	—
87	C₂²×C₄×C₇₃	—	4	4	no	—
88	C₂²×C₄×C₇₃	—	4	2	no	—
89	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
93	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
94	C₂²×C₄×C₇₃	#41	1	4	yes	—
95	C₂²×C₄×C₇₃	#41	1	2	yes	—
96	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
97	M₁₆×C₇₃	—	2	16	no	—
98	C₂×C₈×C₇₃	—	2	4	no	—
99	C₂×C₈×C₇₃	—	2	4	no	—
100	C₂×C₈×C₇₃	C₂×D₄	1	8	yes	—
101	C₄⋊C₄×C₇₃	—	2	2	no	—
102	C₄⋊C₄×C₇₃	—	2	2	no	—
103	C₄⋊C₄×C₇₃	—	2	2	no	—
104	C₄⋊C₄×C₇₃	—	2	4	no	—
105	C₄⋊C₄×C₇₃	—	2	2	no	—
106	C₄⋊C₄×C₇₃	—	4	4	no	—
107	C₄⋊C₄×C₇₃	—	4	2	no	—
108	C₄⋊C₄×C₇₃	—	4	4	no	—
109	C₂²⋊C₄×C₇₃	—	4	4	no	—
110	C₄²×C₇₃	—	2	4	no	—
111	C₄²×C₇₃	—	2	2	no	—
112	C₄²×C₇₃	—	2	2	no	—
113	C₄²×C₇₃	—	4	4	no	—
114	C₂³×C₇₃	—	2	2	no	—
115	C₂³×C₇₃	—	2	2	no	—
116	C₂³×C₇₃	C₂²⋊Q₈	1	2	yes	—
117	C₂³×C₇₃	#123	1	2	yes	—
118	C₂³×C₇₃	#127	1	2	yes	—
119	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
120	Q₈×C₇₃	—	2	8	no	—
121	Q₈×C₇₃	—	4	4	no	—
122	Q₈×C₇₃	—	4	8	no	—
123	C₂×C₄×C₇₃	—	2	2	no	—
124	C₂×C₄×C₇₃	—	2	2	no	—
125	C₂×C₄×C₇₃	—	2	4	no	—
126	C₂×C₄×C₇₃	—	2	2	no	—
127	C₂×C₄×C₇₃	—	2	2	no	—
128	C₂×C₄×C₇₃	—	2	2	no	—
129	C₂×C₄×C₇₃	—	4	8	no	—
130	C₂×C₄×C₇₃	—	4	2	no	—
131	C₂×C₄×C₇₃	—	4	2	no	—
132	C₂×C₄×C₇₃	—	4	8	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	4	no	—
137	C₂×C₄×C₇₃	—	4	4	no	—
138	C₂×C₄×C₇₃	—	4	4	no	—
139	C₂×C₄×C₇₃	—	8	4	no	—
140	C₂×C₄×C₇₃	C₂²⋊Q₈	1	8	yes	—
141	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
142	C₂×C₄×C₇₃	#123	1	8	yes	—
143	C₂×C₄×C₇₃	#123	1	2	yes	—
144	C₂×C₄×C₇₃	#127	1	2	yes	—
145	C₂×C₄×C₇₃	#137	1	4	yes	—
146	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
147	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
148	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
149	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
150	C₈×C₇₃	—	2	8	no	—
151	C₂²×C₇₃	—	2	4	no	—
152	C₂²×C₇₃	—	2	4	no	—
153	C₂²×C₇₃	—	2	2	no	—
154	C₂²×C₇₃	—	2	2	no	—
155	C₂²×C₇₃	#399	1	4	yes	—
156	C₂²×C₇₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
157	C₂²×C₇₃	C₂×#123	1	1	yes	characteristic
158	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
159	C₂²×C₇₃	C₂×#137	1	2	yes	—
160	C₂²×C₇₃	#575	1	2	yes	—
161	C₄×C₇₃	—	2	4	no	—
162	C₄×C₇₃	—	4	2	no	—
163	C₄×C₇₃	—	4	4	no	—
164	C₄×C₇₃	—	8	2	no	—
165	C₄×C₇₃	—	8	2	no	—
166	C₄×C₇₃	—	8	2	no	—
167	C₄×C₇₃	#399	1	4	yes	—
168	C₄×C₇₃	C₂×C₂²⋊Q₈	1	2	yes	—
169	C₄×C₇₃	C₂×#123	1	2	yes	—
170	#23732	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂×C₇₃	—	2	4	no	—
172	C₂×C₇₃	C₂×#399	1	1	yes	characteristic
173	C₂×C₇₃	#3078	1	2	yes	—
174	C₂×C₇₃	#3095	1	2	yes	—
175	C₂×C₇₃	#3152	1	1	yes	characteristic
176	C₂×C₇₃	#3153	1	1	yes	characteristic
177	C₂×#428	C₂×C₇₃	1	2	yes	—
178	C₂×#419	C₂×C₇₃	1	2	yes	—
179	C₂×#391	C₂×C₇₃	1	1	yes	characteristic
180	#2285	C₂×C₇₃	1	1	yes	characteristic
181	#2097	C₂×C₇₃	1	4	yes	complemented
182	#2096	C₂×C₇₃	1	2	yes	complemented
183	#2095	C₂×C₇₃	1	2	yes	complemented
184	#1728	C₂×C₇₃	1	1	yes	characteristic
185	C₇₃	#23732	1	1	yes	characteristic, direct product factor, Sylow
186	C₂³×Q₈	C₂²×C₇₃	1	1	yes	characteristic
187	C₂²×M₁₆	C₂²×C₇₃	1	2	yes	—
188	C₂×#117	C₂²×C₇₃	1	2	yes	—
189	C₂²×C₄⋊C₄	C₂²×C₇₃	1	1	yes	characteristic
190	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
191	#428	—	2	8	no	—
192	C₂×#107	C₂²×C₇₃	1	4	yes	—
193	C₂×#106	C₂²×C₇₃	1	4	yes	—
194	#419	—	2	8	no	—
195	C₂×#103	C₂²×C₇₃	1	4	yes	—
196	C₂×#103	C₂²×C₇₃	1	4	yes	—
197	#391	—	2	2	no	—
198	#391	C₂²×C₇₃	1	2	yes	—
199	#391	C₂²×C₇₃	1	2	yes	—
200	#384	C₂²×C₇₃	1	1	yes	characteristic
201	#384	C₂²×C₇₃	1	1	yes	characteristic
202	#382	C₂²×C₇₃	1	2	yes	—
203	#380	—	2	2	no	—
204	C₄×C₄⋊C₄	—	2	2	no	—
205	C₂×#95	—	2	2	no	—
206	#336	C₂²×C₇₃	1	2	yes	—
207	#336	C₂²×C₇₃	1	2	yes	—
208	C₂²×Q₈	—	2	1	no	—
209	C₂²×Q₈	—	2	2	no	—
210	C₂²×Q₈	—	2	4	no	—
211	C₂²×Q₈	—	2	4	no	—
212	C₂²×Q₈	Q₈×C₇₃	1	4	yes	—
213	C₂²×Q₈	C₂³×C₇₃	1	2	yes	—
214	C₂³×C₄	—	2	1	no	—
215	C₂³×C₄	—	2	1	no	—
216	C₂³×C₄	D₄×C₇₃	1	2	yes	—
217	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
218	C₂×M₁₆	—	2	4	no	—
219	C₂×M₁₆	—	2	4	no	—
220	C₂×M₁₆	D₄×C₇₃	1	8	yes	—
221	C₂²×C₈	C₂³×C₇₃	1	4	yes	—
222	#117	—	2	4	no	—
223	#117	—	2	4	no	—
224	C₂×C₄⋊C₄	—	2	4	no	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
229	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
230	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
231	C₂×C₂²⋊C₄	—	2	2	no	—
232	C₂×C₄²	—	2	4	no	—
233	C₂×C₄²	—	2	2	no	—
234	C₂×C₄²	D₄×C₇₃	1	2	yes	—
235	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
236	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
237	#107	—	2	8	no	—
238	#106	—	2	8	no	—
239	#103	—	2	8	no	—
240	#103	—	2	8	no	—
241	#95	—	2	4	no	—
242	#95	—	2	4	no	—
243	#95	—	2	4	no	—
244	C₂⁴	C₂×D₄×C₇₃	1	1	yes	characteristic
245	C₂×Q₈	—	2	4	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	16	no	—
248	C₂×Q₈	—	2	4	no	—
249	C₂×Q₈	—	2	4	no	—
250	C₂×Q₈	—	4	8	no	—
251	C₂×Q₈	—	4	2	no	—
252	C₂×Q₈	—	4	2	no	—
253	C₂×Q₈	C₂×Q₈×C₇₃	1	4	yes	—
254	C₂²×C₄	—	2	4	no	—
255	C₂²×C₄	—	2	4	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	4	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	—	2	2	no	—
260	C₂²×C₄	—	2	2	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	4	2	no	—
263	C₂²×C₄	—	4	2	no	—
264	C₂²×C₄	—	4	4	no	—
265	C₂²×C₄	—	4	2	no	—
266	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
267	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
268	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
269	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
270	C₂²×C₄	C₂×Q₈×C₇₃	1	2	yes	—
271	C₂²×C₄	#41×C₇₃	1	4	yes	—
272	C₂²×C₄	#41×C₇₃	1	2	yes	—
273	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
274	M₁₆	—	2	16	no	—
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	C₂×D₄×C₇₃	1	8	yes	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	4	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	4	4	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	4	no	—
286	C₂²⋊C₄	—	4	4	no	—
287	C₄²	—	2	4	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	4	4	no	—
291	C₂³	—	2	2	no	—
292	C₂³	—	2	2	no	—
293	C₂³	C₂²⋊Q₈×C₇₃	1	2	yes	—
294	C₂³	#123×C₇₃	1	2	yes	—
295	C₂³	#127×C₇₃	1	2	yes	—
296	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
297	Q₈	—	2	8	no	—
298	Q₈	—	4	8	no	—
299	Q₈	—	4	4	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	8	no	—
307	C₂×C₄	—	4	8	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	4	no	—
314	C₂×C₄	—	4	4	no	—
315	C₂×C₄	—	4	4	no	—
316	C₂×C₄	—	8	4	no	—
317	C₂×C₄	C₂²⋊Q₈×C₇₃	1	2	yes	—
318	C₂×C₄	C₂²⋊Q₈×C₇₃	1	8	yes	—
319	C₂×C₄	#123×C₇₃	1	2	yes	—
320	C₂×C₄	#123×C₇₃	1	8	yes	—
321	C₂×C₄	#127×C₇₃	1	2	yes	—
322	C₂×C₄	#137×C₇₃	1	4	yes	—
323	C₂×C₄	C₂²×D₄×C₇₃	1	2	yes	—
324	C₂×C₄	C₂²×Q₈×C₇₃	1	1	yes	characteristic
325	C₂×C₄	C₂×#41×C₇₃	1	2	yes	—
326	C₂×C₄	C₂×#41×C₇₃	1	1	yes	characteristic
327	C₈	—	2	8	no	—
328	C₂²	—	2	4	no	—
329	C₂²	—	2	4	no	—
330	C₂²	—	2	2	no	—
331	C₂²	—	2	2	no	—
332	C₂²	#399×C₇₃	1	4	yes	—
333	C₂²	C₂×C₂²⋊Q₈×C₇₃	1	1	yes	characteristic
334	C₂²	C₂×#123×C₇₃	1	1	yes	characteristic
335	C₂²	C₂×#127×C₇₃	1	1	yes	characteristic
336	C₂²	C₂×#137×C₇₃	1	2	yes	—
337	C₂²	#575×C₇₃	1	2	yes	—
338	C₄	—	2	4	no	—
339	C₄	—	4	2	no	—
340	C₄	—	4	4	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	—	8	2	no	—
344	C₄	#399×C₇₃	1	4	yes	—
345	C₄	C₂×C₂²⋊Q₈×C₇₃	1	2	yes	—
346	C₄	C₂×#123×C₇₃	1	2	yes	—
347	C₂	—	2	4	no	—
348	C₂	C₂×#399×C₇₃	1	1	yes	characteristic
349	C₂	C₇₃×#3078	1	2	yes	—
350	C₂	C₇₃×#3095	1	2	yes	—
351	C₂	C₇₃×#3152	1	1	yes	characteristic
352	C₂	C₇₃×#3153	1	1	yes	characteristic

	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
5. C₇₃×#2097	C₂	semidirect
6. C₇₃×#2096	C₂	semidirect
7. C₇₃×#2095	C₂	semidirect
170. #23732	C₇₃	direct (N × H)
181. #2097	C₂×C₇₃	semidirect
182. #2096	C₂×C₇₃	semidirect
183. #2095	C₂×C₇₃	semidirect
185. C₇₃	#23732	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#428×C₇₃	2
C₂×#419×C₇₃	2
C₂×#391×C₇₃	2
C₇₃×#2285	2
C₇₃×#2097	2
C₇₃×#2096	2
C₇₃×#2095	2
C₇₃×#1728	2
#23732	73

Subgroups — C73×#23732 #32933136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#23732 #32933136