Subgroups — C₇₃×#23507 #32932911

Order 18688.

1446 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#419×C₇₃	C₂	1	2	yes	maximal
3	C₂×#419×C₇₃	C₂	1	2	yes	maximal
4	C₇₃×#2296	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#2043	C₂	1	8	yes	maximal, complemented
6	C₇₃×#1728	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₇₃	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Q₈×C₇₃	—	2	8	no	—
10	C₂×C₄⋊Q₈×C₇₃	—	2	2	no	—
11	C₂×C₄⋊Q₈×C₇₃	C₂²	1	4	yes	—
12	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
13	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₇₃	—	2	2	no	—
15	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
16	#419×C₇₃	—	2	8	no	—
17	#419×C₇₃	—	2	8	no	—
18	C₂×#103×C₇₃	C₂²	1	8	yes	—
19	C₂×#103×C₇₃	C₂²	1	8	yes	—
20	#389×C₇₃	C₂²	1	4	yes	—
21	#385×C₇₃	—	2	2	no	—
22	C₄×C₄⋊C₄×C₇₃	—	2	2	no	—
23	#336×C₇₃	C₂²	1	4	yes	—
24	C₂²×Q₈×C₇₃	—	2	4	no	—
25	C₂²×Q₈×C₇₃	—	2	2	no	—
26	C₂²×Q₈×C₇₃	—	2	4	no	—
27	C₂²×Q₈×C₇₃	—	2	4	no	—
28	C₂²×Q₈×C₇₃	—	2	4	no	—
29	C₂²×Q₈×C₇₃	—	2	4	no	—
30	C₂²×Q₈×C₇₃	D₄	1	8	yes	—
31	C₂²×Q₈×C₇₃	C₂³	1	4	yes	—
32	C₂³×C₄×C₇₃	—	2	2	no	—
33	C₂³×C₄×C₇₃	D₄	1	2	yes	—
34	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₇₃	—	2	4	no	—
36	C₂×M₁₆×C₇₃	—	2	4	no	—
37	C₂×M₁₆×C₇₃	—	2	4	no	—
38	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
39	C₄⋊Q₈×C₇₃	—	2	16	no	—
40	C₄⋊Q₈×C₇₃	—	2	16	no	—
41	#117×C₇₃	—	2	2	no	—
42	#117×C₇₃	—	2	4	no	—
43	#117×C₇₃	D₄	1	4	yes	—
44	C₂×C₄⋊C₄×C₇₃	—	2	16	no	—
45	C₂×C₄⋊C₄×C₇₃	—	2	8	no	—
46	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
47	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
50	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
51	C₂×C₄²×C₇₃	—	2	1	no	—
52	C₂×C₄²×C₇₃	—	2	4	no	—
53	C₂×C₄²×C₇₃	—	2	1	no	—
54	C₂×C₄²×C₇₃	D₄	1	2	yes	—
55	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
56	#103×C₇₃	—	2	16	no	—
57	#103×C₇₃	—	2	16	no	—
58	#95×C₇₃	—	2	4	no	—
59	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
60	C₂×Q₈×C₇₃	—	2	8	no	—
61	C₂×Q₈×C₇₃	—	2	4	no	—
62	C₂×Q₈×C₇₃	—	2	16	no	—
63	C₂×Q₈×C₇₃	—	2	16	no	—
64	C₂×Q₈×C₇₃	—	2	4	no	—
65	C₂×Q₈×C₇₃	—	2	4	no	—
66	C₂×Q₈×C₇₃	—	2	4	no	—
67	C₂×Q₈×C₇₃	—	2	4	no	—
68	C₂×Q₈×C₇₃	—	4	4	no	—
69	C₂×Q₈×C₇₃	—	4	8	no	—
70	C₂×Q₈×C₇₃	—	4	8	no	—
71	C₂×Q₈×C₇₃	—	4	4	no	—
72	C₂×Q₈×C₇₃	C₂×D₄	1	8	yes	—
73	C₂²×C₄×C₇₃	—	2	4	no	—
74	C₂²×C₄×C₇₃	—	2	8	no	—
75	C₂²×C₄×C₇₃	—	2	8	no	—
76	C₂²×C₄×C₇₃	—	2	2	no	—
77	C₂²×C₄×C₇₃	—	2	4	no	—
78	C₂²×C₄×C₇₃	—	2	2	no	—
79	C₂²×C₄×C₇₃	—	4	4	no	—
80	C₂²×C₄×C₇₃	—	4	4	no	—
81	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
82	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
84	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₇₃	#41	1	4	yes	—
86	C₂²×C₄×C₇₃	#41	1	2	yes	—
87	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
88	M₁₆×C₇₃	—	4	8	no	—
89	C₂×C₈×C₇₃	—	2	4	no	—
90	C₂×C₈×C₇₃	—	2	4	no	—
91	C₂×C₈×C₇₃	—	2	4	no	—
92	C₄⋊C₄×C₇₃	—	2	32	no	—
93	C₄⋊C₄×C₇₃	—	2	2	no	—
94	C₄⋊C₄×C₇₃	—	2	4	no	—
95	C₄⋊C₄×C₇₃	—	4	8	no	—
96	C₄⋊C₄×C₇₃	C₂×D₄	1	4	yes	—
97	C₂²⋊C₄×C₇₃	—	2	4	no	—
98	C₂²⋊C₄×C₇₃	—	4	2	no	—
99	C₄²×C₇₃	—	2	4	no	—
100	C₄²×C₇₃	—	2	4	no	—
101	C₄²×C₇₃	—	2	4	no	—
102	C₄²×C₇₃	—	4	2	no	—
103	C₂³×C₇₃	—	2	4	no	—
104	C₂³×C₇₃	#120	1	2	yes	—
105	C₂³×C₇₃	#121	1	2	yes	—
106	C₂³×C₇₃	#124	1	2	yes	—
107	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
108	Q₈×C₇₃	—	2	8	no	—
109	Q₈×C₇₃	—	2	8	no	—
110	Q₈×C₇₃	—	4	4	no	—
111	Q₈×C₇₃	—	4	4	no	—
112	Q₈×C₇₃	—	4	16	no	—
113	C₂×C₄×C₇₃	—	2	16	no	—
114	C₂×C₄×C₇₃	—	2	4	no	—
115	C₂×C₄×C₇₃	—	2	4	no	—
116	C₂×C₄×C₇₃	—	2	8	no	—
117	C₂×C₄×C₇₃	—	2	4	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	16	no	—
122	C₂×C₄×C₇₃	—	4	2	no	—
123	C₂×C₄×C₇₃	—	4	4	no	—
124	C₂×C₄×C₇₃	#120	1	2	yes	—
125	C₂×C₄×C₇₃	#121	1	8	yes	—
126	C₂×C₄×C₇₃	#121	1	2	yes	—
127	C₂×C₄×C₇₃	#123	1	8	yes	—
128	C₂×C₄×C₇₃	#124	1	2	yes	—
129	C₂×C₄×C₇₃	#137	1	8	yes	—
130	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
132	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
133	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
134	C₈×C₇₃	—	4	4	no	—
135	C₂²×C₇₃	—	2	4	no	—
136	C₂²×C₇₃	—	2	4	no	—
137	C₂²×C₇₃	—	2	4	no	—
138	C₂²×C₇₃	#394	1	4	yes	—
139	C₂²×C₇₃	C₂×#120	1	1	yes	characteristic
140	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
141	C₂²×C₇₃	C₂×#124	1	1	yes	characteristic
142	C₂²×C₇₃	C₂×#137	1	4	yes	—
143	C₄×C₇₃	—	2	8	no	—
144	C₄×C₇₃	—	4	8	no	—
145	C₄×C₇₃	—	4	4	no	—
146	C₄×C₇₃	—	8	2	no	—
147	C₄×C₇₃	#394	1	4	yes	—
148	C₄×C₇₃	C₂×#121	1	2	yes	—
149	C₄×C₇₃	C₂×#123	1	2	yes	—
150	#23507	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂×C₇₃	—	2	4	no	—
152	C₂×C₇₃	C₂×#394	1	1	yes	characteristic
153	C₂×C₇₃	#3004	1	2	yes	—
154	C₂×C₇₃	#3039	1	2	yes	—
155	C₂×C₇₃	#3139	1	2	yes	—
156	C₂²×C₄⋊Q₈	C₂×C₇₃	1	1	yes	characteristic
157	C₂×#419	C₂×C₇₃	1	2	yes	—
158	C₂×#419	C₂×C₇₃	1	2	yes	—
159	#2296	C₂×C₇₃	1	1	yes	characteristic
160	#2043	C₂×C₇₃	1	8	yes	complemented
161	#1728	C₂×C₇₃	1	1	yes	characteristic
162	C₇₃	#23507	1	1	yes	characteristic, direct product factor, Sylow
163	C₂³×Q₈	C₂²×C₇₃	1	2	yes	—
164	C₂²×M₁₆	C₂²×C₇₃	1	2	yes	—
165	C₂×C₄⋊Q₈	—	2	8	no	—
166	C₂×C₄⋊Q₈	—	2	2	no	—
167	C₂×C₄⋊Q₈	C₂²×C₇₃	1	4	yes	—
168	C₂×#117	C₂²×C₇₃	1	1	yes	characteristic
169	C₂×#117	C₂²×C₇₃	1	1	yes	characteristic
170	C₂²×C₄⋊C₄	—	2	2	no	—
171	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
172	#419	—	2	8	no	—
173	#419	—	2	8	no	—
174	C₂×#103	C₂²×C₇₃	1	8	yes	—
175	C₂×#103	C₂²×C₇₃	1	8	yes	—
176	#389	C₂²×C₇₃	1	4	yes	—
177	#385	—	2	2	no	—
178	C₄×C₄⋊C₄	—	2	2	no	—
179	#336	C₂²×C₇₃	1	4	yes	—
180	C₂²×Q₈	—	2	4	no	—
181	C₂²×Q₈	—	2	2	no	—
182	C₂²×Q₈	—	2	4	no	—
183	C₂²×Q₈	—	2	4	no	—
184	C₂²×Q₈	—	2	4	no	—
185	C₂²×Q₈	—	2	4	no	—
186	C₂²×Q₈	D₄×C₇₃	1	8	yes	—
187	C₂²×Q₈	C₂³×C₇₃	1	4	yes	—
188	C₂³×C₄	—	2	2	no	—
189	C₂³×C₄	D₄×C₇₃	1	2	yes	—
190	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
191	C₂×M₁₆	—	2	4	no	—
192	C₂×M₁₆	—	2	4	no	—
193	C₂×M₁₆	—	2	4	no	—
194	C₂²×C₈	C₂³×C₇₃	1	4	yes	—
195	C₄⋊Q₈	—	2	16	no	—
196	C₄⋊Q₈	—	2	16	no	—
197	#117	—	2	2	no	—
198	#117	—	2	4	no	—
199	#117	D₄×C₇₃	1	4	yes	—
200	C₂×C₄⋊C₄	—	2	16	no	—
201	C₂×C₄⋊C₄	—	2	8	no	—
202	C₂×C₄⋊C₄	—	2	4	no	—
203	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
204	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
205	C₂×C₂²⋊C₄	—	2	1	no	—
206	C₂×C₂²⋊C₄	—	2	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	2	4	no	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	D₄×C₇₃	1	2	yes	—
211	C₂×C₄²	C₂³×C₇₃	1	2	yes	—
212	#103	—	2	16	no	—
213	#103	—	2	16	no	—
214	#95	—	2	4	no	—
215	C₂⁴	C₂×D₄×C₇₃	1	1	yes	characteristic
216	C₂×Q₈	—	2	8	no	—
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	—	2	16	no	—
219	C₂×Q₈	—	2	16	no	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	2	4	no	—
223	C₂×Q₈	—	2	4	no	—
224	C₂×Q₈	—	4	8	no	—
225	C₂×Q₈	—	4	4	no	—
226	C₂×Q₈	—	4	8	no	—
227	C₂×Q₈	—	4	4	no	—
228	C₂×Q₈	C₂×D₄×C₇₃	1	8	yes	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	8	no	—
231	C₂²×C₄	—	2	8	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	4	4	no	—
236	C₂²×C₄	—	4	4	no	—
237	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
238	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
240	C₂²×C₄	C₂×D₄×C₇₃	1	2	yes	—
241	C₂²×C₄	#41×C₇₃	1	4	yes	—
242	C₂²×C₄	#41×C₇₃	1	2	yes	—
243	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
244	M₁₆	—	4	8	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₄⋊C₄	—	2	32	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	8	no	—
252	C₄⋊C₄	C₂×D₄×C₇₃	1	4	yes	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	2	4	no	—
257	C₄²	—	2	4	no	—
258	C₄²	—	4	2	no	—
259	C₂³	—	2	4	no	—
260	C₂³	#120×C₇₃	1	2	yes	—
261	C₂³	#121×C₇₃	1	2	yes	—
262	C₂³	#124×C₇₃	1	2	yes	—
263	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
264	Q₈	—	2	8	no	—
265	Q₈	—	2	8	no	—
266	Q₈	—	4	16	no	—
267	Q₈	—	4	4	no	—
268	Q₈	—	4	4	no	—
269	C₂×C₄	—	2	16	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	4	16	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	#120×C₇₃	1	2	yes	—
281	C₂×C₄	#121×C₇₃	1	2	yes	—
282	C₂×C₄	#121×C₇₃	1	8	yes	—
283	C₂×C₄	#123×C₇₃	1	8	yes	—
284	C₂×C₄	#124×C₇₃	1	2	yes	—
285	C₂×C₄	#137×C₇₃	1	8	yes	—
286	C₂×C₄	C₂²×D₄×C₇₃	1	1	yes	characteristic
287	C₂×C₄	C₂²×D₄×C₇₃	1	2	yes	—
288	C₂×C₄	C₂×#41×C₇₃	1	2	yes	—
289	C₂×C₄	C₂×#41×C₇₃	1	1	yes	characteristic
290	C₈	—	4	4	no	—
291	C₂²	—	2	4	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	2	4	no	—
294	C₂²	#394×C₇₃	1	4	yes	—
295	C₂²	C₂×#120×C₇₃	1	1	yes	characteristic
296	C₂²	C₂×#121×C₇₃	1	1	yes	characteristic
297	C₂²	C₂×#124×C₇₃	1	1	yes	characteristic
298	C₂²	C₂×#137×C₇₃	1	4	yes	—
299	C₄	—	2	8	no	—
300	C₄	—	4	4	no	—
301	C₄	—	4	8	no	—
302	C₄	—	8	2	no	—
303	C₄	#394×C₇₃	1	4	yes	—
304	C₄	C₂×#121×C₇₃	1	2	yes	—
305	C₄	C₂×#123×C₇₃	1	2	yes	—
306	C₂	—	2	4	no	—
307	C₂	C₂×#394×C₇₃	1	1	yes	characteristic
308	C₂	C₇₃×#3004	1	2	yes	—
309	C₂	C₇₃×#3039	1	2	yes	—
310	C₂	C₇₃×#3139	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
5. C₇₃×#2043	C₂	semidirect
150. #23507	C₇₃	direct (N × H)
160. #2043	C₂×C₇₃	semidirect
162. C₇₃	#23507	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₄⋊Q₈×C₇₃	2
C₂×#419×C₇₃	2
C₇₃×#2296	2
C₇₃×#2043	2
C₇₃×#1728	2
#23507	73

Subgroups — C73×#23507 #32932911

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#23507 #32932911