Subgroups — C₇₃×#21978 #32931382

Order 18688.

870 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#432×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#423×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₄×#117×C₇₃	C₂	1	1	yes	maximal, characteristic
4	C₇₃×#1940	C₂	1	4	yes	maximal, complemented
5	C₇₃×#1851	C₂	1	4	yes	maximal
6	C₇₃×#1842	C₂	1	2	yes	maximal, complemented
7	C₇₃×#1831	C₂	1	2	yes	maximal, complemented
8	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
9	C₂×#117×C₇₃	C₄	1	2	yes	—
10	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
11	#432×C₇₃	—	2	4	no	—
12	#423×C₇₃	—	2	4	no	—
13	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
16	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₇₃	C₂²	1	8	yes	—
18	C₂×C₂²⋊C₈×C₇₃	C₂²	1	2	yes	—
19	C₂×#97×C₇₃	C₂²	1	2	yes	—
20	C₄×C₄⋊C₄×C₇₃	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
22	C₄×C₂²⋊C₄×C₇₃	C₄	1	2	yes	—
23	C₄×C₂²⋊C₄×C₇₃	C₂²	1	2	yes	—
24	#376×C₇₃	C₄	1	1	yes	characteristic
25	#376×C₇₃	C₄	1	1	yes	characteristic
26	C₄³×C₇₃	C₂²	1	2	yes	—
27	#336×C₇₃	C₂²	1	8	yes	—
28	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₇₃	—	2	4	no	—
32	C₂×M₁₆×C₇₃	—	4	4	no	—
33	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
36	#117×C₇₃	—	2	8	no	—
37	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
39	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₇₃	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₇₃	D₄	1	4	yes	—
45	C₂×C₄²×C₇₃	Q₈	1	4	yes	—
46	C₂×C₄²×C₇₃	C₂³	1	4	yes	—
47	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
49	C₄⋊C₈×C₇₃	—	2	4	no	—
50	C₄⋊C₈×C₇₃	—	2	4	no	—
51	C₄⋊C₈×C₇₃	—	2	2	no	—
52	C₄⋊C₈×C₇₃	—	2	2	no	—
53	C₄⋊C₈×C₇₃	—	4	8	no	—
54	C₂²⋊C₈×C₇₃	—	2	4	no	—
55	#97×C₇₃	—	2	4	no	—
56	#95×C₇₃	C₂×C₄	1	2	yes	—
57	#95×C₇₃	C₂×C₄	1	2	yes	—
58	#95×C₇₃	C₂×C₄	1	2	yes	—
59	#95×C₇₃	C₂×C₄	1	2	yes	—
60	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₃	—	2	4	no	—
62	C₂²×C₄×C₇₃	—	2	2	no	—
63	C₂²×C₄×C₇₃	—	2	2	no	—
64	C₂²×C₄×C₇₃	—	4	2	no	—
65	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
66	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
77	M₁₆×C₇₃	—	4	8	no	—
78	C₂×C₈×C₇₃	—	2	2	no	—
79	C₂×C₈×C₇₃	—	2	2	no	—
80	C₂×C₈×C₇₃	—	2	4	no	—
81	C₂×C₈×C₇₃	—	2	4	no	—
82	C₂×C₈×C₇₃	—	2	8	no	—
83	C₂×C₈×C₇₃	—	2	4	no	—
84	C₄⋊C₄×C₇₃	—	2	8	no	—
85	C₂²⋊C₄×C₇₃	—	2	8	no	—
86	C₄²×C₇₃	—	2	4	no	—
87	C₄²×C₇₃	—	2	4	no	—
88	C₄²×C₇₃	—	2	4	no	—
89	C₄²×C₇₃	—	4	4	no	—
90	C₄²×C₇₃	C₄⋊C₄	1	4	yes	—
91	C₄²×C₇₃	C₄⋊C₄	1	4	yes	—
92	C₄²×C₇₃	M₁₆	1	8	yes	complemented
93	C₄²×C₇₃	C₂×D₄	1	2	yes	—
94	C₄²×C₇₃	C₂×D₄	1	2	yes	—
95	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
96	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
97	C₂³×C₇₃	—	2	2	no	—
98	C₂³×C₇₃	—	2	2	no	—
99	C₂³×C₇₃	—	2	2	no	—
100	C₂³×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₇₃	—	2	4	no	—
103	C₂×C₄×C₇₃	—	2	2	no	—
104	C₂×C₄×C₇₃	—	2	2	no	—
105	C₂×C₄×C₇₃	—	2	2	no	—
106	C₂×C₄×C₇₃	—	2	4	no	—
107	C₂×C₄×C₇₃	—	2	2	no	—
108	C₂×C₄×C₇₃	—	2	2	no	—
109	C₂×C₄×C₇₃	—	2	2	no	—
110	C₂×C₄×C₇₃	—	2	4	no	—
111	C₂×C₄×C₇₃	—	2	8	no	—
112	C₂×C₄×C₇₃	—	2	4	no	—
113	C₂×C₄×C₇₃	—	2	4	no	—
114	C₂×C₄×C₇₃	—	2	4	no	—
115	C₂×C₄×C₇₃	—	4	4	no	—
116	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
118	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
119	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₇₃	C₂×M₁₆	1	8	yes	—
122	C₂×C₄×C₇₃	C₂×M₁₆	1	4	yes	—
123	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₇₃	H(5,𝔽₂)	1	2	yes	—
126	C₂×C₄×C₇₃	2₋¹⁺⁴	1	2	yes	—
127	C₈×C₇₃	—	4	2	no	—
128	C₈×C₇₃	—	4	4	no	—
129	C₈×C₇₃	—	4	2	no	—
130	C₂²×C₇₃	—	2	2	no	—
131	C₂²×C₇₃	—	2	2	no	—
132	C₂²×C₇₃	—	2	2	no	—
133	C₂²×C₇₃	—	4	4	no	—
134	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₂²×C₇₃	#520	1	2	yes	—
136	C₂²×C₇₃	C₂²×M₁₆	1	2	yes	—
137	C₂²×C₇₃	#568	1	2	yes	—
138	C₄×C₇₃	—	2	2	no	—
139	C₄×C₇₃	—	2	2	no	—
140	C₄×C₇₃	—	2	2	no	—
141	C₄×C₇₃	—	2	2	no	—
142	C₄×C₇₃	—	2	8	no	—
143	C₄×C₇₃	—	4	4	no	—
144	C₄×C₇₃	#423	1	4	yes	—
145	C₄×C₇₃	#535	1	1	yes	characteristic
146	C₄×C₇₃	#536	1	1	yes	characteristic
147	C₄×C₇₃	#544	1	2	yes	—
148	#21978	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂×C₇₃	—	4	2	no	—
150	C₂×C₇₃	#2317	1	1	yes	characteristic
151	C₂×C₇₃	C₂×#423	1	1	yes	characteristic
152	C₂×C₇₃	#2905	1	1	yes	characteristic
153	C₂×C₇₃	#2989	1	4	yes	—
154	C₂×#432	C₂×C₇₃	1	1	yes	characteristic
155	C₂×#423	C₂×C₇₃	1	1	yes	characteristic
156	C₄×#117	C₂×C₇₃	1	1	yes	characteristic
157	#1940	C₂×C₇₃	1	4	yes	complemented
158	#1851	C₂×C₇₃	1	4	yes	—
159	#1842	C₂×C₇₃	1	2	yes	complemented
160	#1831	C₂×C₇₃	1	2	yes	complemented
161	C₇₃	#21978	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²×M₁₆	C₂²×C₇₃	1	2	yes	—
163	C₂×#117	C₄×C₇₃	1	2	yes	—
164	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
165	#432	—	2	4	no	—
166	#423	—	2	4	no	—
167	C₂×C₄⋊C₈	C₂²×C₇₃	1	1	yes	characteristic
168	C₂×C₄⋊C₈	C₂²×C₇₃	1	2	yes	—
169	C₂×C₄⋊C₈	C₂²×C₇₃	1	1	yes	characteristic
170	C₂×C₄⋊C₈	C₂²×C₇₃	1	2	yes	—
171	C₂×C₄⋊C₈	C₂²×C₇₃	1	8	yes	—
172	C₂×C₂²⋊C₈	C₂²×C₇₃	1	2	yes	—
173	C₂×#97	C₂²×C₇₃	1	2	yes	—
174	C₄×C₄⋊C₄	C₄×C₇₃	1	2	yes	—
175	C₄×C₄⋊C₄	C₂²×C₇₃	1	2	yes	—
176	C₄×C₂²⋊C₄	C₄×C₇₃	1	2	yes	—
177	C₄×C₂²⋊C₄	C₂²×C₇₃	1	2	yes	—
178	#376	C₄×C₇₃	1	1	yes	characteristic
179	#376	C₄×C₇₃	1	1	yes	characteristic
180	C₄³	C₂²×C₇₃	1	2	yes	—
181	#336	C₂²×C₇₃	1	8	yes	—
182	C₂³×C₄	C₂×C₄×C₇₃	1	1	yes	characteristic
183	C₂³×C₄	C₂×C₄×C₇₃	1	1	yes	characteristic
184	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
185	C₂×M₁₆	—	2	4	no	—
186	C₂×M₁₆	—	4	4	no	—
187	C₂²×C₈	C₂³×C₇₃	1	4	yes	—
188	C₂²×C₈	C₂³×C₇₃	1	2	yes	—
189	C₂²×C₈	C₂³×C₇₃	1	2	yes	—
190	#117	—	2	8	no	—
191	C₂×C₄⋊C₄	C₂×C₄×C₇₃	1	4	yes	—
192	C₂×C₂²⋊C₄	C₂×C₄×C₇₃	1	4	yes	—
193	C₂×C₄²	C₂×C₄×C₇₃	1	2	yes	—
194	C₂×C₄²	C₂×C₄×C₇₃	1	4	yes	—
195	C₂×C₄²	C₂×C₄×C₇₃	1	1	yes	characteristic
196	C₂×C₄²	C₂×C₄×C₇₃	1	1	yes	characteristic
197	C₂×C₄²	C₂×C₄×C₇₃	1	2	yes	—
198	C₂×C₄²	D₄×C₇₃	1	4	yes	—
199	C₂×C₄²	Q₈×C₇₃	1	4	yes	—
200	C₂×C₄²	C₂³×C₇₃	1	4	yes	—
201	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
203	C₄⋊C₈	—	2	4	no	—
204	C₄⋊C₈	—	2	4	no	—
205	C₄⋊C₈	—	2	2	no	—
206	C₄⋊C₈	—	2	2	no	—
207	C₄⋊C₈	—	4	8	no	—
208	C₂²⋊C₈	—	2	4	no	—
209	#97	—	2	4	no	—
210	#95	C₂×C₄×C₇₃	1	2	yes	—
211	#95	C₂×C₄×C₇₃	1	2	yes	—
212	#95	C₂×C₄×C₇₃	1	2	yes	—
213	#95	C₂×C₄×C₇₃	1	2	yes	—
214	C₂⁴	C₂²×C₄×C₇₃	1	1	yes	characteristic
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	C₄⋊C₄×C₇₃	1	4	yes	—
220	C₂²×C₄	C₄⋊C₄×C₇₃	1	4	yes	—
221	C₂²×C₄	C₂²×C₄×C₇₃	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₇₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²×C₄×C₇₃	1	1	yes	characteristic
224	C₂²×C₄	C₂²×C₄×C₇₃	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₇₃	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₇₃	1	4	yes	—
227	C₂²×C₄	C₂²×C₄×C₇₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
229	C₂²×C₄	C₂×Q₈×C₇₃	1	2	yes	—
230	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
231	M₁₆	—	4	8	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	8	no	—
237	C₂×C₈	—	2	4	no	—
238	C₄⋊C₄	—	2	8	no	—
239	C₂²⋊C₄	—	2	8	no	—
240	C₄²	—	2	4	no	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	4	4	no	—
244	C₄²	C₄⋊C₄×C₇₃	1	4	yes	—
245	C₄²	C₄⋊C₄×C₇₃	1	4	yes	—
246	C₄²	M₁₆×C₇₃	1	8	yes	complemented
247	C₄²	C₂×D₄×C₇₃	1	2	yes	—
248	C₄²	C₂×D₄×C₇₃	1	2	yes	—
249	C₄²	C₂×Q₈×C₇₃	1	2	yes	—
250	C₄²	C₂×Q₈×C₇₃	1	2	yes	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	C₂×C₄⋊C₄×C₇₃	1	2	yes	—
255	C₂³	C₂³×C₄×C₇₃	1	1	yes	characteristic
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	2	yes	—
271	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	2	yes	—
272	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	2	yes	—
273	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	2	yes	—
274	C₂×C₄	C₂×C₄⋊C₄×C₇₃	1	2	yes	—
275	C₂×C₄	C₂×M₁₆×C₇₃	1	8	yes	—
276	C₂×C₄	C₂×M₁₆×C₇₃	1	4	yes	—
277	C₂×C₄	C₂²×D₄×C₇₃	1	1	yes	characteristic
278	C₂×C₄	C₂²×Q₈×C₇₃	1	1	yes	characteristic
279	C₂×C₄	H(5,𝔽₂)×C₇₃	1	2	yes	—
280	C₂×C₄	2₋¹⁺⁴×C₇₃	1	2	yes	—
281	C₈	—	4	4	no	—
282	C₈	—	4	2	no	—
283	C₈	—	4	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	—	4	4	no	—
288	C₂²	C₂²×C₄⋊C₄×C₇₃	1	1	yes	characteristic
289	C₂²	#520×C₇₃	1	2	yes	—
290	C₂²	C₂²×M₁₆×C₇₃	1	2	yes	—
291	C₂²	#568×C₇₃	1	2	yes	—
292	C₄	—	2	8	no	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	4	no	—
298	C₄	#423×C₇₃	1	4	yes	—
299	C₄	#535×C₇₃	1	1	yes	characteristic
300	C₄	#536×C₇₃	1	1	yes	characteristic
301	C₄	#544×C₇₃	1	2	yes	—
302	C₂	—	4	2	no	—
303	C₂	C₇₃×#2317	1	1	yes	characteristic
304	C₂	C₂×#423×C₇₃	1	1	yes	characteristic
305	C₂	C₇₃×#2905	1	1	yes	characteristic
306	C₂	C₇₃×#2989	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₇₃×#1940	C₂	semidirect
6. C₇₃×#1842	C₂	semidirect
7. C₇₃×#1831	C₂	semidirect
92. C₄²×C₇₃	M₁₆	semidirect
148. #21978	C₇₃	direct (N × H)
157. #1940	C₂×C₇₃	semidirect
159. #1842	C₂×C₇₃	semidirect
160. #1831	C₂×C₇₃	semidirect
161. C₇₃	#21978	direct (N × H)
246. C₄²	M₁₆×C₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#432×C₇₃	2
C₂×#423×C₇₃	2
C₄×#117×C₇₃	2
C₇₃×#1940	2
C₇₃×#1851	2
C₇₃×#1842	2
C₇₃×#1831	2
#21978	73

Subgroups — C73×#21978 #32931382

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#21978 #32931382