Subgroups — C₇₃×#29127 #32938531

Order 18688.

842 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₃×#2864	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#127×C₇₃	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#2099	C₂	1	2	yes	maximal, complemented
4	C₇₃×#2041	C₂	1	2	yes	maximal
5	C₇₃×#1936	C₂	1	4	yes	maximal, complemented
6	C₇₃×#1804	C₂	1	2	yes	maximal, complemented
7	C₇₃×#1804	C₂	1	2	yes	maximal, complemented
8	C₇₃×#1767	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#127×C₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₇₃	—	2	2	no	—
11	C₂×C₄×D₄×C₇₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
13	C₂×#102×C₇₃	C₂²	1	4	yes	—
14	C₂×#102×C₇₃	C₂²	1	4	yes	—
15	C₂×#97×C₇₃	C₂²	1	2	yes	—
16	#390×C₇₃	—	2	2	no	—
17	#390×C₇₃	C₂²	1	2	yes	—
18	#389×C₇₃	C₂²	1	2	yes	—
19	#389×C₇₃	C₂²	1	2	yes	—
20	#387×C₇₃	—	2	2	no	—
21	#387×C₇₃	—	2	2	no	—
22	#384×C₇₃	C₂²	1	1	yes	characteristic
23	#382×C₇₃	C₂²	1	1	yes	characteristic
24	#382×C₇₃	C₂²	1	1	yes	characteristic
25	C₄³×C₇₃	C₂²	1	1	yes	characteristic
26	#340×C₇₃	C₂²	1	4	yes	—
27	#340×C₇₃	C₂²	1	4	yes	—
28	#336×C₇₃	C₂²	1	2	yes	—
29	C₂²×D₄×C₇₃	—	2	2	no	—
30	C₂²×D₄×C₇₃	C₂³	1	2	yes	—
31	C₂³×C₄×C₇₃	—	2	2	no	—
32	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
33	#127×C₇₃	—	2	4	no	—
34	C₄×D₄×C₇₃	—	2	4	no	—
35	C₄×D₄×C₇₃	—	2	4	no	—
36	C₄×D₄×C₇₃	—	4	8	no	—
37	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
40	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
41	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
42	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
43	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
44	C₂×C₂²⋊C₄×C₇₃	—	2	4	no	—
45	C₂×C₄²×C₇₃	D₄	1	2	yes	—
46	C₂×C₄²×C₇₃	D₄	1	2	yes	—
47	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
50	C₄⋊C₈×C₇₃	—	2	4	no	—
51	#102×C₇₃	—	2	8	no	—
52	#102×C₇₃	—	4	4	no	—
53	#97×C₇₃	—	4	2	no	—
54	#95×C₇₃	—	2	1	no	—
55	#95×C₇₃	—	2	1	no	—
56	#95×C₇₃	—	2	1	no	—
57	#95×C₇₃	—	2	1	no	—
58	#95×C₇₃	—	2	2	no	—
59	#95×C₇₃	—	2	2	no	—
60	C₂⁴×C₇₃	—	2	2	no	—
61	C₂×D₄×C₇₃	—	2	8	no	—
62	C₂×D₄×C₇₃	—	2	2	no	—
63	C₂×D₄×C₇₃	—	2	2	no	—
64	C₂×D₄×C₇₃	—	2	2	no	—
65	C₂×D₄×C₇₃	—	2	2	no	—
66	C₂×D₄×C₇₃	—	2	2	no	—
67	C₂×D₄×C₇₃	—	4	4	no	—
68	C₂×D₄×C₇₃	—	4	4	no	—
69	C₂×D₄×C₇₃	#41	1	4	yes	—
70	C₂²×C₄×C₇₃	—	2	2	no	—
71	C₂²×C₄×C₇₃	—	2	2	no	—
72	C₂²×C₄×C₇₃	—	2	2	no	—
73	C₂²×C₄×C₇₃	—	4	8	no	—
74	C₂²×C₄×C₇₃	—	4	4	no	—
75	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
80	C₂×C₈×C₇₃	—	2	4	no	—
81	C₂×C₈×C₇₃	—	2	4	no	—
82	C₂×C₈×C₇₃	—	2	4	no	—
83	C₄⋊C₄×C₇₃	—	2	4	no	—
84	C₄⋊C₄×C₇₃	—	2	2	no	—
85	C₄⋊C₄×C₇₃	—	2	2	no	—
86	C₄⋊C₄×C₇₃	—	2	2	no	—
87	C₄⋊C₄×C₇₃	—	2	2	no	—
88	C₄⋊C₄×C₇₃	—	4	4	no	—
89	C₄⋊C₄×C₇₃	—	4	4	no	—
90	C₄⋊C₄×C₇₃	—	4	4	no	—
91	C₂²⋊C₄×C₇₃	—	4	8	no	—
92	C₂²⋊C₄×C₇₃	—	4	4	no	—
93	C₄²×C₇₃	—	2	4	no	—
94	C₄²×C₇₃	—	2	2	no	—
95	C₄²×C₇₃	—	2	1	no	—
96	C₄²×C₇₃	—	2	1	no	—
97	C₄²×C₇₃	—	4	2	no	—
98	C₄²×C₇₃	#41	1	4	yes	—
99	C₂³×C₇₃	—	2	4	no	—
100	C₂³×C₇₃	—	4	2	no	—
101	C₂³×C₇₃	—	4	2	no	—
102	C₂³×C₇₃	—	4	4	no	—
103	C₂³×C₇₃	—	4	4	no	—
104	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
105	D₄×C₇₃	—	2	4	no	—
106	D₄×C₇₃	—	2	4	no	—
107	D₄×C₇₃	—	4	16	no	—
108	D₄×C₇₃	—	4	2	no	—
109	D₄×C₇₃	—	4	2	no	—
110	C₂×C₄×C₇₃	—	2	2	no	—
111	C₂×C₄×C₇₃	—	2	2	no	—
112	C₂×C₄×C₇₃	—	2	2	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	2	no	—
117	C₂×C₄×C₇₃	—	4	2	no	—
118	C₂×C₄×C₇₃	—	4	2	no	—
119	C₂×C₄×C₇₃	—	4	2	no	—
120	C₂×C₄×C₇₃	—	4	4	no	—
121	C₂×C₄×C₇₃	—	4	2	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	—	8	8	no	—
126	C₂×C₄×C₇₃	#124	1	2	yes	—
127	C₂×C₄×C₇₃	#124	1	2	yes	—
128	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
129	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
130	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
131	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
133	C₈×C₇₃	—	4	4	no	—
134	C₂²×C₇₃	—	4	4	no	—
135	C₂²×C₇₃	—	4	8	no	—
136	C₂²×C₇₃	—	8	2	no	—
137	C₂²×C₇₃	—	8	2	no	—
138	C₂²×C₇₃	—	8	4	no	—
139	C₂²×C₇₃	#525	1	1	yes	characteristic
140	C₂²×C₇₃	C₂×#124	1	1	yes	characteristic
141	C₂²×C₇₃	#537	1	1	yes	characteristic
142	C₂²×C₇₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
143	C₂²×C₇₃	#575	1	2	yes	—
144	C₄×C₇₃	—	2	4	no	—
145	C₄×C₇₃	—	2	2	no	—
146	C₄×C₇₃	—	2	2	no	—
147	C₄×C₇₃	—	4	2	no	—
148	C₄×C₇₃	—	8	2	no	—
149	C₄×C₇₃	—	8	2	no	—
150	C₄×C₇₃	—	8	2	no	—
151	C₄×C₇₃	#552	1	2	yes	—
152	C₄×C₇₃	#553	1	2	yes	—
153	#29127	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂×C₇₃	—	8	4	no	—
155	C₂×C₇₃	#2562	1	1	yes	characteristic
156	C₂×C₇₃	#3114	1	2	yes	—
157	C₂×C₇₃	#3135	1	1	yes	characteristic
158	C₂×C₇₃	#3138	1	1	yes	characteristic
159	C₂×C₇₃	#3251	1	2	yes	—
160	#2864	C₂×C₇₃	1	1	yes	characteristic, complemented
161	C₄×#127	C₂×C₇₃	1	1	yes	characteristic
162	#2099	C₂×C₇₃	1	2	yes	complemented
163	#2041	C₂×C₇₃	1	2	yes	—
164	#1936	C₂×C₇₃	1	4	yes	complemented
165	#1804	C₂×C₇₃	1	2	yes	complemented
166	#1804	C₂×C₇₃	1	2	yes	complemented
167	#1767	C₂×C₇₃	1	1	yes	characteristic, complemented
168	C₇₃	#29127	1	1	yes	characteristic, direct product factor, Sylow
169	C₂×#127	C₂²×C₇₃	1	1	yes	characteristic
170	C₂×C₄×D₄	—	2	2	no	—
171	C₂×C₄×D₄	C₂²×C₇₃	1	2	yes	—
172	C₂×C₄⋊C₈	C₂²×C₇₃	1	2	yes	—
173	C₂×#102	C₂²×C₇₃	1	4	yes	—
174	C₂×#102	C₂²×C₇₃	1	4	yes	—
175	C₂×#97	C₂²×C₇₃	1	2	yes	—
176	#390	—	2	2	no	—
177	#390	C₂²×C₇₃	1	2	yes	—
178	#389	C₂²×C₇₃	1	2	yes	—
179	#389	C₂²×C₇₃	1	2	yes	—
180	#387	—	2	2	no	—
181	#387	—	2	2	no	—
182	#384	C₂²×C₇₃	1	1	yes	characteristic
183	#382	C₂²×C₇₃	1	1	yes	characteristic
184	#382	C₂²×C₇₃	1	1	yes	characteristic
185	C₄³	C₂²×C₇₃	1	1	yes	characteristic
186	#340	C₂²×C₇₃	1	4	yes	—
187	#340	C₂²×C₇₃	1	4	yes	—
188	#336	C₂²×C₇₃	1	2	yes	—
189	C₂²×D₄	—	2	2	no	—
190	C₂²×D₄	C₂³×C₇₃	1	2	yes	—
191	C₂³×C₄	—	2	2	no	—
192	C₂²×C₈	C₂³×C₇₃	1	4	yes	—
193	#127	—	2	4	no	—
194	C₄×D₄	—	2	4	no	—
195	C₄×D₄	—	2	4	no	—
196	C₄×D₄	—	4	8	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
202	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
203	C₂×C₂²⋊C₄	—	2	4	no	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₄²	D₄×C₇₃	1	2	yes	—
206	C₂×C₄²	D₄×C₇₃	1	2	yes	—
207	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
210	C₄⋊C₈	—	2	4	no	—
211	#102	—	2	8	no	—
212	#102	—	4	4	no	—
213	#97	—	4	2	no	—
214	#95	—	2	1	no	—
215	#95	—	2	1	no	—
216	#95	—	2	1	no	—
217	#95	—	2	1	no	—
218	#95	—	2	2	no	—
219	#95	—	2	2	no	—
220	C₂⁴	—	2	2	no	—
221	C₂×D₄	—	2	8	no	—
222	C₂×D₄	—	2	2	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	4	4	no	—
228	C₂×D₄	—	4	4	no	—
229	C₂×D₄	#41×C₇₃	1	4	yes	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	4	8	no	—
234	C₂²×C₄	—	4	4	no	—
235	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
239	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	—	4	4	no	—
251	C₂²⋊C₄	—	4	8	no	—
252	C₂²⋊C₄	—	4	4	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	4	2	no	—
258	C₄²	#41×C₇₃	1	4	yes	—
259	C₂³	—	2	4	no	—
260	C₂³	—	4	2	no	—
261	C₂³	—	4	2	no	—
262	C₂³	—	4	4	no	—
263	C₂³	—	4	4	no	—
264	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
265	D₄	—	2	4	no	—
266	D₄	—	2	4	no	—
267	D₄	—	4	16	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	8	8	no	—
286	C₂×C₄	#124×C₇₃	1	2	yes	—
287	C₂×C₄	#124×C₇₃	1	2	yes	—
288	C₂×C₄	AGL(1,ℤ/8)×C₇₃	1	4	yes	—
289	C₂×C₄	C₂×#41×C₇₃	1	2	yes	—
290	C₂×C₄	C₂×#41×C₇₃	1	2	yes	—
291	C₂×C₄	H(5,𝔽₂)×C₇₃	1	1	yes	characteristic
292	C₂×C₄	2₋¹⁺⁴×C₇₃	1	1	yes	characteristic
293	C₈	—	4	4	no	—
294	C₂²	—	4	4	no	—
295	C₂²	—	4	8	no	—
296	C₂²	—	8	4	no	—
297	C₂²	—	8	2	no	—
298	C₂²	—	8	2	no	—
299	C₂²	#525×C₇₃	1	1	yes	characteristic
300	C₂²	C₂×#124×C₇₃	1	1	yes	characteristic
301	C₂²	#537×C₇₃	1	1	yes	characteristic
302	C₂²	C₂×AGL(1,ℤ/8)×C₇₃	1	2	yes	—
303	C₂²	#575×C₇₃	1	2	yes	—
304	C₄	—	2	4	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	8	2	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	#552×C₇₃	1	2	yes	—
312	C₄	#553×C₇₃	1	2	yes	—
313	C₂	—	8	4	no	—
314	C₂	C₇₃×#2562	1	1	yes	characteristic
315	C₂	C₇₃×#3114	1	2	yes	—
316	C₂	C₇₃×#3135	1	1	yes	characteristic
317	C₂	C₇₃×#3138	1	1	yes	characteristic
318	C₂	C₇₃×#3251	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₃
prime radical	C₂×#127×C₇₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₃ ◁ C₄³×C₇₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇₃×#2864	C₂	semidirect
3. C₇₃×#2099	C₂	semidirect
5. C₇₃×#1936	C₂	semidirect
6. C₇₃×#1804	C₂	semidirect
7. C₇₃×#1804	C₂	semidirect
8. C₇₃×#1767	C₂	semidirect
153. #29127	C₇₃	direct (N × H)
160. #2864	C₂×C₇₃	semidirect
162. #2099	C₂×C₇₃	semidirect
164. #1936	C₂×C₇₃	semidirect
165. #1804	C₂×C₇₃	semidirect
166. #1804	C₂×C₇₃	semidirect
167. #1767	C₂×C₇₃	semidirect
168. C₇₃	#29127	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×#2864	2
C₄×#127×C₇₃	2
C₇₃×#2099	2
C₇₃×#2041	2
C₇₃×#1936	2
C₇₃×#1804	2
C₇₃×#1767	2
#29127	73

Subgroups — C73×#29127 #32938531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#29127 #32938531