Subgroups — C₇₃×#29338 #32938742

Order 18688.

770 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₃×#2857	C₂	1	1	yes	maximal, characteristic
2	C₇₃×#2296	C₂	1	1	yes	maximal, characteristic
3	C₇₃×#2100	C₂	1	2	yes	maximal, complemented
4	C₇₃×#2043	C₂	1	2	yes	maximal, complemented
5	C₇₃×#1935	C₂	1	4	yes	maximal, complemented
6	C₇₃×#1873	C₂	1	2	yes	maximal
7	C₇₃×#1873	C₂	1	2	yes	maximal
8	C₇₃×#1749	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊Q₈×C₇₃	C₂²	1	2	yes	—
11	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
12	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
14	C₂×#103×C₇₃	C₂²	1	4	yes	—
15	C₂×#103×C₇₃	C₂²	1	4	yes	—
16	C₂×C₂²⋊C₈×C₇₃	C₂²	1	2	yes	—
17	C₂×#97×C₇₃	C₂²	1	2	yes	—
18	#391×C₇₃	C₂²	1	2	yes	—
19	#389×C₇₃	—	2	2	no	—
20	#389×C₇₃	C₂²	1	2	yes	—
21	#389×C₇₃	C₂²	1	2	yes	—
22	#385×C₇₃	—	2	2	no	—
23	#384×C₇₃	C₂²	1	1	yes	characteristic
24	#380×C₇₃	—	2	2	no	—
25	C₄×C₄⋊C₄×C₇₃	—	2	2	no	—
26	#340×C₇₃	C₂²	1	4	yes	—
27	#340×C₇₃	C₂²	1	4	yes	—
28	#336×C₇₃	C₂²	1	2	yes	—
29	C₂²×Q₈×C₇₃	C₂³	1	2	yes	—
30	C₂³×C₄×C₇₃	D₄	1	2	yes	—
31	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
32	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
33	C₄⋊Q₈×C₇₃	—	4	2	no	—
34	C₂²⋊Q₈×C₇₃	—	2	8	no	—
35	#117×C₇₃	—	2	4	no	—
36	#117×C₇₃	—	4	2	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	—	2	2	no	—
41	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
42	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
43	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
44	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
45	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
46	C₂×C₂²⋊C₄×C₇₃	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
48	C₂×C₄²×C₇₃	—	2	1	no	—
49	C₂×C₄²×C₇₃	—	2	1	no	—
50	C₂×C₄²×C₇₃	—	4	2	no	—
51	C₂×C₄²×C₇₃	D₄	1	2	yes	—
52	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
54	#103×C₇₃	—	2	8	no	—
55	#103×C₇₃	—	4	4	no	—
56	C₂²⋊C₈×C₇₃	—	2	4	no	—
57	#97×C₇₃	—	4	2	no	—
58	#95×C₇₃	—	2	2	no	—
59	#95×C₇₃	—	2	2	no	—
60	#95×C₇₃	—	2	4	no	—
61	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
62	C₂×Q₈×C₇₃	—	2	4	no	—
63	C₂×Q₈×C₇₃	—	2	4	no	—
64	C₂×Q₈×C₇₃	—	2	4	no	—
65	C₂²×C₄×C₇₃	—	2	2	no	—
66	C₂²×C₄×C₇₃	—	2	2	no	—
67	C₂²×C₄×C₇₃	—	2	2	no	—
68	C₂²×C₄×C₇₃	—	2	4	no	—
69	C₂²×C₄×C₇₃	—	2	2	no	—
70	C₂²×C₄×C₇₃	—	2	2	no	—
71	C₂²×C₄×C₇₃	—	2	2	no	—
72	C₂²×C₄×C₇₃	—	4	2	no	—
73	C₂²×C₄×C₇₃	—	4	2	no	—
74	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₇₃	#41	1	4	yes	—
78	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₇₃	—	2	4	no	—
80	C₂×C₈×C₇₃	—	2	4	no	—
81	C₂×C₈×C₇₃	—	2	4	no	—
82	C₄⋊C₄×C₇₃	—	2	2	no	—
83	C₄⋊C₄×C₇₃	—	2	4	no	—
84	C₄⋊C₄×C₇₃	—	2	4	no	—
85	C₄⋊C₄×C₇₃	—	4	4	no	—
86	C₄⋊C₄×C₇₃	—	4	4	no	—
87	C₄⋊C₄×C₇₃	—	4	4	no	—
88	C₄⋊C₄×C₇₃	—	4	4	no	—
89	C₄⋊C₄×C₇₃	—	4	4	no	—
90	C₄⋊C₄×C₇₃	#41	1	4	yes	—
91	C₂²⋊C₄×C₇₃	—	2	4	no	—
92	C₂²⋊C₄×C₇₃	—	4	2	no	—
93	C₂²⋊C₄×C₇₃	—	4	4	no	—
94	C₄²×C₇₃	—	2	4	no	—
95	C₄²×C₇₃	—	2	2	no	—
96	C₄²×C₇₃	—	2	2	no	—
97	C₄²×C₇₃	—	4	2	no	—
98	C₄²×C₇₃	—	4	2	no	—
99	C₂³×C₇₃	—	2	1	no	—
100	C₂³×C₇₃	—	2	1	no	—
101	C₂³×C₇₃	—	2	2	no	—
102	C₂³×C₇₃	—	2	2	no	—
103	C₂³×C₇₃	#124	1	2	yes	—
104	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
105	Q₈×C₇₃	—	4	8	no	—
106	C₂×C₄×C₇₃	—	2	1	no	—
107	C₂×C₄×C₇₃	—	2	2	no	—
108	C₂×C₄×C₇₃	—	2	1	no	—
109	C₂×C₄×C₇₃	—	2	2	no	—
110	C₂×C₄×C₇₃	—	2	8	no	—
111	C₂×C₄×C₇₃	—	2	4	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	—	4	8	no	—
116	C₂×C₄×C₇₃	—	4	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	2	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	4	no	—
125	C₂×C₄×C₇₃	—	4	2	no	—
126	C₂×C₄×C₇₃	#124	1	2	yes	—
127	C₂×C₄×C₇₃	#137	1	4	yes	—
128	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
129	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
130	C₂×C₄×C₇₃	H(5,𝔽₂)	1	2	yes	—
131	C₈×C₇₃	—	4	4	no	—
132	C₂²×C₇₃	—	2	2	no	—
133	C₂²×C₇₃	—	2	2	no	—
134	C₂²×C₇₃	—	2	2	no	—
135	C₂²×C₇₃	—	4	2	no	—
136	C₂²×C₇₃	—	4	2	no	—
137	C₂²×C₇₃	#525	1	1	yes	characteristic
138	C₂²×C₇₃	C₂×#124	1	1	yes	characteristic
139	C₂²×C₇₃	#534	1	1	yes	characteristic
140	C₂²×C₇₃	C₂×#137	1	2	yes	—
141	C₂²×C₇₃	#575	1	2	yes	—
142	C₄×C₇₃	—	2	2	no	—
143	C₄×C₇₃	—	2	4	no	—
144	C₄×C₇₃	—	4	4	no	—
145	C₄×C₇₃	—	4	2	no	—
146	C₄×C₇₃	—	8	2	no	—
147	C₄×C₇₃	—	8	2	no	—
148	C₄×C₇₃	—	8	2	no	—
149	C₄×C₇₃	#553	1	4	yes	—
150	#29338	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂×C₇₃	—	4	2	no	—
152	C₂×C₇₃	#2591	1	1	yes	characteristic
153	C₂×C₇₃	#3116	1	2	yes	—
154	C₂×C₇₃	#3135	1	1	yes	characteristic
155	C₂×C₇₃	#3139	1	1	yes	characteristic
156	C₂×C₇₃	#3196	1	2	yes	—
157	#2857	C₂×C₇₃	1	1	yes	characteristic
158	#2296	C₂×C₇₃	1	1	yes	characteristic
159	#2100	C₂×C₇₃	1	2	yes	complemented
160	#2043	C₂×C₇₃	1	2	yes	complemented
161	#1935	C₂×C₇₃	1	4	yes	complemented
162	#1873	C₂×C₇₃	1	2	yes	—
163	#1873	C₂×C₇₃	1	2	yes	—
164	#1749	C₂×C₇₃	1	1	yes	characteristic
165	C₇₃	#29338	1	1	yes	characteristic, direct product factor, Sylow
166	C₂×C₄⋊Q₈	C₂²×C₇₃	1	1	yes	characteristic
167	C₂×C₂²⋊Q₈	C₂²×C₇₃	1	2	yes	—
168	C₂×#117	C₂²×C₇₃	1	1	yes	characteristic
169	C₂×#117	C₂²×C₇₃	1	1	yes	characteristic
170	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
171	C₂×#103	C₂²×C₇₃	1	4	yes	—
172	C₂×#103	C₂²×C₇₃	1	4	yes	—
173	C₂×C₂²⋊C₈	C₂²×C₇₃	1	2	yes	—
174	C₂×#97	C₂²×C₇₃	1	2	yes	—
175	#391	C₂²×C₇₃	1	2	yes	—
176	#389	—	2	2	no	—
177	#389	C₂²×C₇₃	1	2	yes	—
178	#389	C₂²×C₇₃	1	2	yes	—
179	#385	—	2	2	no	—
180	#384	C₂²×C₇₃	1	1	yes	characteristic
181	#380	—	2	2	no	—
182	C₄×C₄⋊C₄	—	2	2	no	—
183	#340	C₂²×C₇₃	1	4	yes	—
184	#340	C₂²×C₇₃	1	4	yes	—
185	#336	C₂²×C₇₃	1	2	yes	—
186	C₂²×Q₈	C₂³×C₇₃	1	2	yes	—
187	C₂³×C₄	D₄×C₇₃	1	2	yes	—
188	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₇₃	1	4	yes	—
190	C₄⋊Q₈	—	4	2	no	—
191	C₂²⋊Q₈	—	2	8	no	—
192	#117	—	2	4	no	—
193	#117	—	4	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
200	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
202	C₂×C₂²⋊C₄	—	2	1	no	—
203	C₂×C₂²⋊C₄	—	2	1	no	—
204	C₂×C₂²⋊C₄	—	2	2	no	—
205	C₂×C₄²	—	2	1	no	—
206	C₂×C₄²	—	2	1	no	—
207	C₂×C₄²	—	4	2	no	—
208	C₂×C₄²	D₄×C₇₃	1	2	yes	—
209	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
210	C₂×C₄²	C₂³×C₇₃	1	1	yes	characteristic
211	#103	—	2	8	no	—
212	#103	—	4	4	no	—
213	C₂²⋊C₈	—	2	4	no	—
214	#97	—	4	2	no	—
215	#95	—	2	2	no	—
216	#95	—	2	2	no	—
217	#95	—	2	4	no	—
218	C₂⁴	C₂×D₄×C₇₃	1	1	yes	characteristic
219	C₂×Q₈	—	2	4	no	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
232	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
234	C₂²×C₄	#41×C₇₃	1	4	yes	—
235	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	4	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	4	4	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₄⋊C₄	—	4	4	no	—
245	C₄⋊C₄	—	4	4	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₄⋊C₄	#41×C₇₃	1	4	yes	—
248	C₂²⋊C₄	—	2	4	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₂²⋊C₄	—	4	4	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	4	2	no	—
255	C₄²	—	4	2	no	—
256	C₂³	—	2	1	no	—
257	C₂³	—	2	1	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	#124×C₇₃	1	2	yes	—
261	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
262	Q₈	—	4	8	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	8	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	#124×C₇₃	1	2	yes	—
284	C₂×C₄	#137×C₇₃	1	4	yes	—
285	C₂×C₄	C₂×#41×C₇₃	1	2	yes	—
286	C₂×C₄	C₂×#41×C₇₃	1	2	yes	—
287	C₂×C₄	H(5,𝔽₂)×C₇₃	1	2	yes	—
288	C₈	—	4	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	#525×C₇₃	1	1	yes	characteristic
295	C₂²	C₂×#124×C₇₃	1	1	yes	characteristic
296	C₂²	#534×C₇₃	1	1	yes	characteristic
297	C₂²	C₂×#137×C₇₃	1	2	yes	—
298	C₂²	#575×C₇₃	1	2	yes	—
299	C₄	—	2	2	no	—
300	C₄	—	2	4	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	2	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	2	no	—
306	C₄	#553×C₇₃	1	4	yes	—
307	C₂	—	4	2	no	—
308	C₂	C₇₃×#2591	1	1	yes	characteristic
309	C₂	C₇₃×#3116	1	2	yes	—
310	C₂	C₇₃×#3135	1	1	yes	characteristic
311	C₂	C₇₃×#3139	1	1	yes	characteristic
312	C₂	C₇₃×#3196	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
3. C₇₃×#2100	C₂	semidirect
4. C₇₃×#2043	C₂	semidirect
5. C₇₃×#1935	C₂	semidirect
150. #29338	C₇₃	direct (N × H)
159. #2100	C₂×C₇₃	semidirect
160. #2043	C₂×C₇₃	semidirect
161. #1935	C₂×C₇₃	semidirect
165. C₇₃	#29338	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×#2857	2
C₇₃×#2296	2
C₇₃×#2100	2
C₇₃×#2043	2
C₇₃×#1935	2
C₇₃×#1873	2
C₇₃×#1749	2
#29338	73

Subgroups — C73×#29338 #32938742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#29338 #32938742