Subgroups — C₇₃×#14276 #32923680

Order 18688.

1406 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#389×C₇₃	C₂	1	2	yes	maximal
2	C₂×#384×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#382×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#376×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1441	C₂	1	8	yes	maximal
6	C₇₃×C₄⋊#95	C₂	1	2	yes	maximal
7	C₂²×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
8	C₂²×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₇₃	C₂²	1	4	yes	—
10	C₂²×C₄⋊C₄×C₇₃	C₂²	1	4	yes	—
11	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
14	#389×C₇₃	—	2	8	no	—
15	#384×C₇₃	C₄	1	8	yes	complemented
16	#382×C₇₃	—	2	4	no	—
17	#376×C₇₃	—	2	4	no	—
18	C₂×#95×C₇₃	C₂²	1	4	yes	—
19	C₂×#95×C₇₃	C₂²	1	4	yes	—
20	C₂×#95×C₇₃	C₂²	1	4	yes	—
21	C₂×#95×C₇₃	C₂²	1	8	yes	—
22	C₂³×C₄×C₇₃	C₂³	1	4	yes	—
23	C₂³×C₄×C₇₃	C₂³	1	4	yes	—
24	C₂³×C₄×C₇₃	C₂³	1	2	yes	—
25	C₂³×C₄×C₇₃	C₂³	1	2	yes	—
26	C₂³×C₄×C₇₃	C₂³	1	2	yes	—
27	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
29	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
30	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
31	C₂×C₄⋊C₄×C₇₃	—	2	8	no	—
32	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
33	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
34	C₂×C₄⋊C₄×C₇₃	—	4	8	no	—
35	C₂×C₄⋊C₄×C₇₃	—	4	8	no	—
36	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	8	yes	—
37	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	16	yes	—
38	C₂×C₄²×C₇₃	—	2	2	no	—
39	C₂×C₄²×C₇₃	—	2	2	no	—
40	C₂×C₄²×C₇₃	—	2	2	no	—
41	C₂×C₄²×C₇₃	—	2	2	no	—
42	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₇₃	D₄	1	4	yes	—
44	C₂×C₄²×C₇₃	D₄	1	4	yes	—
45	C₂×C₄²×C₇₃	Q₈	1	4	yes	—
46	C₂×C₄²×C₇₃	Q₈	1	4	yes	—
47	#95×C₇₃	—	2	8	no	—
48	#95×C₇₃	—	2	8	no	—
49	#95×C₇₃	—	2	8	no	—
50	#95×C₇₃	—	4	8	no	—
51	C₂⁴×C₇₃	C₂⁴	1	1	yes	characteristic
52	C₂²×C₄×C₇₃	—	2	4	no	—
53	C₂²×C₄×C₇₃	—	2	4	no	—
54	C₂²×C₄×C₇₃	—	2	2	no	—
55	C₂²×C₄×C₇₃	—	2	2	no	—
56	C₂²×C₄×C₇₃	—	2	4	no	—
57	C₂²×C₄×C₇₃	—	2	2	no	—
58	C₂²×C₄×C₇₃	—	2	2	no	—
59	C₂²×C₄×C₇₃	—	2	4	no	—
60	C₂²×C₄×C₇₃	—	2	4	no	—
61	C₂²×C₄×C₇₃	—	2	8	no	—
62	C₂²×C₄×C₇₃	—	2	8	no	—
63	C₂²×C₄×C₇₃	—	2	4	no	—
64	C₂²×C₄×C₇₃	—	2	4	no	—
65	C₂²×C₄×C₇₃	—	2	4	no	—
66	C₂²×C₄×C₇₃	—	2	4	no	—
67	C₂²×C₄×C₇₃	—	2	4	no	—
68	C₂²×C₄×C₇₃	—	2	4	no	—
69	C₂²×C₄×C₇₃	—	2	8	no	—
70	C₂²×C₄×C₇₃	—	2	8	no	—
71	C₂²×C₄×C₇₃	C₂²×C₄	1	8	yes	—
72	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
75	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
76	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
79	C₂²×C₄×C₇₃	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₇₃	#41	1	2	yes	—
83	C₂²×C₄×C₇₃	#41	1	2	yes	—
84	C₄⋊C₄×C₇₃	—	2	16	no	—
85	C₄⋊C₄×C₇₃	—	4	16	no	—
86	C₄⋊C₄×C₇₃	—	4	16	no	—
87	C₄⋊C₄×C₇₃	—	4	8	no	—
88	C₄²×C₇₃	—	2	8	no	—
89	C₄²×C₇₃	—	2	8	no	—
90	C₄²×C₇₃	C₄⋊C₄	1	16	yes	complemented
91	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
92	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
93	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
94	C₂³×C₇₃	C₂²×Q₈	1	1	yes	characteristic
95	C₂³×C₇₃	C₂²×Q₈	1	1	yes	characteristic
96	C₂³×C₇₃	C₂×#41	1	1	yes	characteristic
97	C₂³×C₇₃	C₂×#41	1	1	yes	characteristic
98	C₂³×C₇₃	H(5,𝔽₂)	1	4	yes	—
99	C₂³×C₇₃	2₋¹⁺⁴	1	4	yes	—
100	C₂×C₄×C₇₃	—	2	4	no	—
101	C₂×C₄×C₇₃	—	2	4	no	—
102	C₂×C₄×C₇₃	—	2	8	no	—
103	C₂×C₄×C₇₃	—	2	8	no	—
104	C₂×C₄×C₇₃	—	2	8	no	—
105	C₂×C₄×C₇₃	—	2	16	no	—
106	C₂×C₄×C₇₃	—	2	4	no	—
107	C₂×C₄×C₇₃	—	2	4	no	—
108	C₂×C₄×C₇₃	—	2	4	no	—
109	C₂×C₄×C₇₃	—	2	4	no	—
110	C₂×C₄×C₇₃	—	2	8	no	—
111	C₂×C₄×C₇₃	—	4	4	no	—
112	C₂×C₄×C₇₃	—	4	4	no	—
113	C₂×C₄×C₇₃	—	4	4	no	—
114	C₂×C₄×C₇₃	—	4	4	no	—
115	C₂×C₄×C₇₃	—	4	8	no	—
116	C₂×C₄×C₇₃	—	4	2	no	—
117	C₂×C₄×C₇₃	—	4	2	no	—
118	C₂×C₄×C₇₃	—	4	4	no	—
119	C₂×C₄×C₇₃	—	4	4	no	—
120	C₂×C₄×C₇₃	—	4	4	no	—
121	C₂×C₄×C₇₃	—	4	4	no	—
122	C₂×C₄×C₇₃	—	4	8	no	—
123	C₂×C₄×C₇₃	—	4	8	no	—
124	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	8	yes	—
125	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
126	C₂×C₄×C₇₃	C₄×D₄	1	4	yes	—
127	C₂×C₄×C₇₃	C₄×Q₈	1	4	yes	—
128	C₂×C₄×C₇₃	#121	1	4	yes	—
129	C₂×C₄×C₇₃	C₂²⋊Q₈	1	4	yes	—
130	C₂×C₄×C₇₃	#125	1	4	yes	—
131	C₂×C₄×C₇₃	C₄⋊Q₈	1	4	yes	—
132	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂²×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
135	C₂²×C₇₃	#518	1	2	yes	—
136	C₂²×C₇₃	#519	1	2	yes	—
137	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
138	C₂²×C₇₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
139	C₂²×C₇₃	C₂×#125	1	1	yes	characteristic
140	C₂²×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
141	C₂²×C₇₃	#535	1	2	yes	—
142	C₂²×C₇₃	#536	1	2	yes	—
143	C₂²×C₇₃	#537	1	4	yes	—
144	C₂²×C₇₃	#540	1	2	yes	—
145	C₂²×C₇₃	#541	1	2	yes	—
146	C₂²×C₇₃	#542	1	4	yes	—
147	C₂²×C₇₃	#544	1	4	yes	—
148	C₂²×C₇₃	#544	1	4	yes	—
149	C₄×C₇₃	—	2	8	no	—
150	C₄×C₇₃	—	4	8	no	—
151	C₄×C₇₃	—	4	4	no	—
152	C₄×C₇₃	—	4	4	no	—
153	C₄×C₇₃	—	8	4	no	—
154	C₄×C₇₃	#389	1	8	yes	—
155	#14276	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂×C₇₃	C₂×#389	1	1	yes	characteristic
157	C₂×C₇₃	#2316	1	2	yes	—
158	C₂×C₇₃	#2327	1	1	yes	characteristic
159	C₂×C₇₃	#2328	1	1	yes	characteristic
160	C₂×C₇₃	#2331	1	2	yes	—
161	C₂×C₇₃	#2620	1	2	yes	—
162	C₂×C₇₃	#2629	1	2	yes	—
163	C₂×C₇₃	#2647	1	2	yes	—
164	C₂×C₇₃	#2660	1	2	yes	—
165	C₂×#389	C₂×C₇₃	1	2	yes	—
166	C₂×#384	C₂×C₇₃	1	1	yes	characteristic
167	C₂×#382	C₂×C₇₃	1	1	yes	characteristic
168	C₂×#376	C₂×C₇₃	1	1	yes	characteristic
169	#1441	C₂×C₇₃	1	8	yes	—
170	C₄⋊#95	C₂×C₇₃	1	2	yes	—
171	C₇₃	#14276	1	1	yes	characteristic, direct product factor, Sylow
172	C₂²×C₄⋊C₄	C₂²×C₇₃	1	2	yes	—
173	C₂²×C₄⋊C₄	C₂²×C₇₃	1	2	yes	—
174	C₂²×C₄⋊C₄	C₂²×C₇₃	1	4	yes	—
175	C₂²×C₄⋊C₄	C₂²×C₇₃	1	4	yes	—
176	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
177	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
178	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
179	#389	—	2	8	no	—
180	#384	C₄×C₇₃	1	8	yes	complemented
181	#382	—	2	4	no	—
182	#376	—	2	4	no	—
183	C₂×#95	C₂²×C₇₃	1	4	yes	—
184	C₂×#95	C₂²×C₇₃	1	4	yes	—
185	C₂×#95	C₂²×C₇₃	1	4	yes	—
186	C₂×#95	C₂²×C₇₃	1	8	yes	—
187	C₂³×C₄	C₂³×C₇₃	1	4	yes	—
188	C₂³×C₄	C₂³×C₇₃	1	4	yes	—
189	C₂³×C₄	C₂³×C₇₃	1	2	yes	—
190	C₂³×C₄	C₂³×C₇₃	1	2	yes	—
191	C₂³×C₄	C₂³×C₇₃	1	2	yes	—
192	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
193	C₂×C₄⋊C₄	—	2	4	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₄⋊C₄	—	2	8	no	—
197	C₂×C₄⋊C₄	—	2	4	no	—
198	C₂×C₄⋊C₄	—	2	4	no	—
199	C₂×C₄⋊C₄	—	4	8	no	—
200	C₂×C₄⋊C₄	—	4	8	no	—
201	C₂×C₄⋊C₄	C₂×C₄×C₇₃	1	8	yes	—
202	C₂×C₄⋊C₄	C₂×C₄×C₇₃	1	16	yes	—
203	C₂×C₄²	—	2	2	no	—
204	C₂×C₄²	—	2	2	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	C₂×C₄×C₇₃	1	4	yes	—
208	C₂×C₄²	D₄×C₇₃	1	4	yes	—
209	C₂×C₄²	D₄×C₇₃	1	4	yes	—
210	C₂×C₄²	Q₈×C₇₃	1	4	yes	—
211	C₂×C₄²	Q₈×C₇₃	1	4	yes	—
212	#95	—	2	8	no	—
213	#95	—	2	8	no	—
214	#95	—	2	8	no	—
215	#95	—	4	8	no	—
216	C₂⁴	C₂⁴×C₇₃	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	8	no	—
227	C₂²×C₄	—	2	8	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	2	4	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	8	no	—
235	C₂²×C₄	—	2	8	no	—
236	C₂²×C₄	C₂²×C₄×C₇₃	1	8	yes	—
237	C₂²×C₄	C₂²×C₄×C₇₃	1	4	yes	—
238	C₂²×C₄	C₂²×C₄×C₇₃	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
240	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
241	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
242	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
243	C₂²×C₄	C₂×Q₈×C₇₃	1	4	yes	—
244	C₂²×C₄	C₂×Q₈×C₇₃	1	4	yes	—
245	C₂²×C₄	C₂×Q₈×C₇₃	1	2	yes	—
246	C₂²×C₄	C₂×Q₈×C₇₃	1	2	yes	—
247	C₂²×C₄	#41×C₇₃	1	2	yes	—
248	C₂²×C₄	#41×C₇₃	1	2	yes	—
249	C₄⋊C₄	—	2	16	no	—
250	C₄⋊C₄	—	4	16	no	—
251	C₄⋊C₄	—	4	16	no	—
252	C₄⋊C₄	—	4	8	no	—
253	C₄²	—	2	8	no	—
254	C₄²	—	2	8	no	—
255	C₄²	C₄⋊C₄×C₇₃	1	16	yes	complemented
256	C₂³	C₂³×C₄×C₇₃	1	1	yes	characteristic
257	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
258	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
259	C₂³	C₂²×Q₈×C₇₃	1	1	yes	characteristic
260	C₂³	C₂²×Q₈×C₇₃	1	1	yes	characteristic
261	C₂³	C₂×#41×C₇₃	1	1	yes	characteristic
262	C₂³	C₂×#41×C₇₃	1	1	yes	characteristic
263	C₂³	H(5,𝔽₂)×C₇₃	1	4	yes	—
264	C₂³	2₋¹⁺⁴×C₇₃	1	4	yes	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	2	16	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	4	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	8	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	4	no	—
287	C₂×C₄	—	4	8	no	—
288	C₂×C₄	—	4	8	no	—
289	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	8	yes	—
290	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	4	yes	—
291	C₂×C₄	C₄×D₄×C₇₃	1	4	yes	—
292	C₂×C₄	C₄×Q₈×C₇₃	1	4	yes	—
293	C₂×C₄	#121×C₇₃	1	4	yes	—
294	C₂×C₄	C₂²⋊Q₈×C₇₃	1	4	yes	—
295	C₂×C₄	#125×C₇₃	1	4	yes	—
296	C₂×C₄	C₄⋊Q₈×C₇₃	1	4	yes	—
297	C₂²	C₂²×C₄⋊C₄×C₇₃	1	1	yes	characteristic
298	C₂²	C₂×C₄×D₄×C₇₃	1	1	yes	characteristic
299	C₂²	C₂×C₄×Q₈×C₇₃	1	1	yes	characteristic
300	C₂²	#518×C₇₃	1	2	yes	—
301	C₂²	#519×C₇₃	1	2	yes	—
302	C₂²	C₂×#121×C₇₃	1	1	yes	characteristic
303	C₂²	C₂×C₂²⋊Q₈×C₇₃	1	1	yes	characteristic
304	C₂²	C₂×#125×C₇₃	1	1	yes	characteristic
305	C₂²	C₂×C₄⋊Q₈×C₇₃	1	1	yes	characteristic
306	C₂²	#535×C₇₃	1	2	yes	—
307	C₂²	#536×C₇₃	1	2	yes	—
308	C₂²	#537×C₇₃	1	4	yes	—
309	C₂²	#540×C₇₃	1	2	yes	—
310	C₂²	#541×C₇₃	1	2	yes	—
311	C₂²	#542×C₇₃	1	4	yes	—
312	C₂²	#544×C₇₃	1	4	yes	—
313	C₂²	#544×C₇₃	1	4	yes	—
314	C₄	—	2	8	no	—
315	C₄	—	4	8	no	—
316	C₄	—	4	4	no	—
317	C₄	—	4	4	no	—
318	C₄	—	8	4	no	—
319	C₄	#389×C₇₃	1	8	yes	—
320	C₂	C₂×#389×C₇₃	1	1	yes	characteristic
321	C₂	C₇₃×#2316	1	2	yes	—
322	C₂	C₇₃×#2327	1	1	yes	characteristic
323	C₂	C₇₃×#2328	1	1	yes	characteristic
324	C₂	C₇₃×#2331	1	2	yes	—
325	C₂	C₇₃×#2620	1	2	yes	—
326	C₂	C₇₃×#2629	1	2	yes	—
327	C₂	C₇₃×#2647	1	2	yes	—
328	C₂	C₇₃×#2660	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
15. #384×C₇₃	C₄	semidirect
90. C₄²×C₇₃	C₄⋊C₄	semidirect
155. #14276	C₇₃	direct (N × H)
171. C₇₃	#14276	direct (N × H)
180. #384	C₄×C₇₃	semidirect
255. C₄²	C₄⋊C₄×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#389×C₇₃	2
C₂×#384×C₇₃	2
C₂×#382×C₇₃	2
C₂×#376×C₇₃	2
C₇₃×#1441	2
C₇₃×C₄⋊#95	2
#14276	73

Subgroups — C73×#14276 #32923680

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#14276 #32923680