Subgroups — C₇₃×#23733 #32933137

Order 18688.

1130 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#428×C₇₃	C₂	1	2	yes	maximal, complemented
2	C₂×#417×C₇₃	C₂	1	2	yes	maximal
3	C₇₃×#2297	C₂	1	1	yes	maximal, characteristic
4	C₂×#382×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₃×#2097	C₂	1	4	yes	maximal, complemented
6	C₇₃×#2092	C₂	1	4	yes	maximal, complemented
7	C₇₃×#1728	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₇₃	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
10	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
12	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
13	#428×C₇₃	—	2	8	no	—
14	C₂×#107×C₇₃	C₂²	1	4	yes	—
15	C₂×#106×C₇₃	C₂²	1	4	yes	—
16	#417×C₇₃	—	2	8	no	—
17	C₂×#103×C₇₃	C₂²	1	4	yes	—
18	C₂×#102×C₇₃	C₂²	1	4	yes	—
19	#391×C₇₃	C₂²	1	2	yes	—
20	#390×C₇₃	C₂²	1	2	yes	—
21	#386×C₇₃	—	2	2	no	—
22	#382×C₇₃	—	2	2	no	—
23	#382×C₇₃	C₂²	1	4	yes	—
24	C₄×C₂²⋊C₄×C₇₃	—	2	2	no	—
25	C₂×#95×C₇₃	—	2	2	no	—
26	#336×C₇₃	C₂²	1	4	yes	—
27	C₂×#41×C₇₃	—	2	2	no	—
28	C₂×#41×C₇₃	—	2	4	no	—
29	C₂×#41×C₇₃	—	2	4	no	—
30	C₂×#41×C₇₃	Q₈	1	4	yes	—
31	C₂²×Q₈×C₇₃	C₂³	1	1	yes	characteristic
32	C₂²×D₄×C₇₃	—	2	1	no	—
33	C₂²×D₄×C₇₃	C₂³	1	1	yes	characteristic
34	C₂³×C₄×C₇₃	—	2	1	no	—
35	C₂³×C₄×C₇₃	—	2	2	no	—
36	C₂³×C₄×C₇₃	D₄	1	2	yes	—
37	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₇₃	—	2	4	no	—
39	C₂×M₁₆×C₇₃	—	2	4	no	—
40	C₂×M₁₆×C₇₃	D₄	1	8	yes	—
41	C₂²×C₈×C₇₃	C₂³	1	4	yes	—
42	#117×C₇₃	—	2	4	no	—
43	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
44	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
45	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₇₃	C₂³	1	4	yes	—
47	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
48	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
49	C₂×C₂²⋊C₄×C₇₃	—	2	4	no	—
50	C₂×C₄²×C₇₃	—	2	4	no	—
51	C₂×C₄²×C₇₃	—	2	1	no	—
52	C₂×C₄²×C₇₃	D₄	1	2	yes	—
53	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
54	#107×C₇₃	—	2	8	no	—
55	#106×C₇₃	—	2	8	no	—
56	#103×C₇₃	—	2	8	no	—
57	#102×C₇₃	—	2	8	no	—
58	#95×C₇₃	—	2	4	no	—
59	#95×C₇₃	—	2	8	no	—
60	C₂⁴×C₇₃	—	2	1	no	—
61	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
62	#41×C₇₃	—	2	16	no	—
63	#41×C₇₃	—	4	8	no	—
64	C₂×Q₈×C₇₃	—	2	1	no	—
65	C₂×Q₈×C₇₃	—	2	2	no	—
66	C₂×Q₈×C₇₃	—	2	2	no	—
67	C₂×Q₈×C₇₃	C₂×Q₈	1	2	yes	—
68	C₂×D₄×C₇₃	—	2	4	no	—
69	C₂×D₄×C₇₃	—	2	2	no	—
70	C₂×D₄×C₇₃	—	2	2	no	—
71	C₂×D₄×C₇₃	—	2	1	no	—
72	C₂×D₄×C₇₃	—	4	2	no	—
73	C₂×D₄×C₇₃	—	4	2	no	—
74	C₂×D₄×C₇₃	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₇₃	—	2	4	no	—
76	C₂²×C₄×C₇₃	—	2	1	no	—
77	C₂²×C₄×C₇₃	—	2	4	no	—
78	C₂²×C₄×C₇₃	—	2	4	no	—
79	C₂²×C₄×C₇₃	—	2	2	no	—
80	C₂²×C₄×C₇₃	—	4	4	no	—
81	C₂²×C₄×C₇₃	—	4	2	no	—
82	C₂²×C₄×C₇₃	—	4	2	no	—
83	C₂²×C₄×C₇₃	—	4	8	no	—
84	C₂²×C₄×C₇₃	—	4	4	no	—
85	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	C₂×D₄	1	4	yes	—
88	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
89	C₂²×C₄×C₇₃	#41	1	4	yes	—
90	C₂²×C₄×C₇₃	#41	1	2	yes	—
91	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
92	M₁₆×C₇₃	—	2	16	no	—
93	C₂×C₈×C₇₃	—	2	4	no	—
94	C₂×C₈×C₇₃	—	2	4	no	—
95	C₂×C₈×C₇₃	C₂×D₄	1	8	yes	—
96	C₄⋊C₄×C₇₃	—	2	4	no	—
97	C₄⋊C₄×C₇₃	—	2	2	no	—
98	C₄⋊C₄×C₇₃	—	2	4	no	—
99	C₄⋊C₄×C₇₃	—	2	2	no	—
100	C₄⋊C₄×C₇₃	—	4	4	no	—
101	C₂²⋊C₄×C₇₃	—	4	8	no	—
102	C₂²⋊C₄×C₇₃	—	4	2	no	—
103	C₄²×C₇₃	—	2	4	no	—
104	C₄²×C₇₃	—	2	4	no	—
105	C₄²×C₇₃	—	4	2	no	—
106	C₂³×C₇₃	—	2	4	no	—
107	C₂³×C₇₃	—	2	2	no	—
108	C₂³×C₇₃	—	4	1	no	—
109	C₂³×C₇₃	—	4	1	no	—
110	C₂³×C₇₃	—	4	4	no	—
111	C₂³×C₇₃	C₂²⋊Q₈	1	2	yes	—
112	C₂³×C₇₃	#123	1	2	yes	—
113	C₂³×C₇₃	#127	1	2	yes	—
114	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
115	Q₈×C₇₃	—	2	4	no	—
116	Q₈×C₇₃	—	4	2	no	—
117	D₄×C₇₃	—	2	4	no	—
118	D₄×C₇₃	—	4	8	no	—
119	D₄×C₇₃	—	4	2	no	—
120	C₂×C₄×C₇₃	—	2	2	no	—
121	C₂×C₄×C₇₃	—	2	4	no	—
122	C₂×C₄×C₇₃	—	2	4	no	—
123	C₂×C₄×C₇₃	—	2	4	no	—
124	C₂×C₄×C₇₃	—	4	8	no	—
125	C₂×C₄×C₇₃	—	4	1	no	—
126	C₂×C₄×C₇₃	—	4	1	no	—
127	C₂×C₄×C₇₃	—	4	8	no	—
128	C₂×C₄×C₇₃	—	4	2	no	—
129	C₂×C₄×C₇₃	—	4	4	no	—
130	C₂×C₄×C₇₃	—	4	4	no	—
131	C₂×C₄×C₇₃	—	4	8	no	—
132	C₂×C₄×C₇₃	—	8	8	no	—
133	C₂×C₄×C₇₃	C₂²⋊Q₈	1	8	yes	—
134	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₇₃	#123	1	8	yes	—
136	C₂×C₄×C₇₃	#123	1	2	yes	—
137	C₂×C₄×C₇₃	#127	1	2	yes	—
138	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
139	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
141	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
142	C₈×C₇₃	—	2	8	no	—
143	C₂²×C₇₃	—	2	4	no	—
144	C₂²×C₇₃	—	2	4	no	—
145	C₂²×C₇₃	—	2	4	no	—
146	C₂²×C₇₃	—	4	4	no	—
147	C₂²×C₇₃	—	4	2	no	—
148	C₂²×C₇₃	—	8	2	no	—
149	C₂²×C₇₃	—	8	2	no	—
150	C₂²×C₇₃	#399	1	4	yes	—
151	C₂²×C₇₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
152	C₂²×C₇₃	C₂×#123	1	1	yes	characteristic
153	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
154	C₂²×C₇₃	#575	1	4	yes	—
155	C₄×C₇₃	—	4	4	no	—
156	C₄×C₇₃	—	4	2	no	—
157	C₄×C₇₃	—	8	2	no	—
158	C₄×C₇₃	—	8	4	no	—
159	C₄×C₇₃	#399	1	4	yes	—
160	C₄×C₇₃	C₂×C₂²⋊Q₈	1	2	yes	—
161	C₄×C₇₃	C₂×#123	1	2	yes	—
162	#23733	C₇₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂×C₇₃	—	2	4	no	—
164	C₂×C₇₃	—	8	2	no	—
165	C₂×C₇₃	C₂×#399	1	1	yes	characteristic
166	C₂×C₇₃	#3080	1	2	yes	—
167	C₂×C₇₃	#3093	1	2	yes	—
168	C₂×C₇₃	#3153	1	2	yes	—
169	C₂×#428	C₂×C₇₃	1	2	yes	complemented
170	C₂×#417	C₂×C₇₃	1	2	yes	—
171	#2297	C₂×C₇₃	1	1	yes	characteristic
172	C₂×#382	C₂×C₇₃	1	1	yes	characteristic, complemented
173	#2097	C₂×C₇₃	1	4	yes	complemented
174	#2092	C₂×C₇₃	1	4	yes	complemented
175	#1728	C₂×C₇₃	1	1	yes	characteristic, complemented
176	C₇₃	#23733	1	1	yes	characteristic, direct product factor, Sylow
177	C₂²×#41	C₂²×C₇₃	1	1	yes	characteristic
178	C₂²×M₁₆	C₂²×C₇₃	1	2	yes	—
179	C₂×#117	C₂²×C₇₃	1	1	yes	characteristic
180	C₂²×C₄⋊C₄	C₂²×C₇₃	1	2	yes	—
181	C₂²×C₄²	C₂²×C₇₃	1	1	yes	characteristic
182	#428	—	2	8	no	—
183	C₂×#107	C₂²×C₇₃	1	4	yes	—
184	C₂×#106	C₂²×C₇₃	1	4	yes	—
185	#417	—	2	8	no	—
186	C₂×#103	C₂²×C₇₃	1	4	yes	—
187	C₂×#102	C₂²×C₇₃	1	4	yes	—
188	#391	C₂²×C₇₃	1	2	yes	—
189	#390	C₂²×C₇₃	1	2	yes	—
190	#386	—	2	2	no	—
191	#382	—	2	2	no	—
192	#382	C₂²×C₇₃	1	4	yes	—
193	C₄×C₂²⋊C₄	—	2	2	no	—
194	C₂×#95	—	2	2	no	—
195	#336	C₂²×C₇₃	1	4	yes	—
196	C₂×#41	—	2	2	no	—
197	C₂×#41	—	2	4	no	—
198	C₂×#41	—	2	4	no	—
199	C₂×#41	Q₈×C₇₃	1	4	yes	—
200	C₂²×Q₈	C₂³×C₇₃	1	1	yes	characteristic
201	C₂²×D₄	—	2	1	no	—
202	C₂²×D₄	C₂³×C₇₃	1	1	yes	characteristic
203	C₂³×C₄	—	2	1	no	—
204	C₂³×C₄	—	2	2	no	—
205	C₂³×C₄	D₄×C₇₃	1	2	yes	—
206	C₂³×C₄	C₂³×C₇₃	1	1	yes	characteristic
207	C₂×M₁₆	—	2	4	no	—
208	C₂×M₁₆	—	2	4	no	—
209	C₂×M₁₆	D₄×C₇₃	1	8	yes	—
210	C₂²×C₈	C₂³×C₇₃	1	4	yes	—
211	#117	—	2	4	no	—
212	C₂×C₄⋊C₄	—	2	4	no	—
213	C₂×C₄⋊C₄	—	2	4	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	C₂³×C₇₃	1	4	yes	—
216	C₂×C₄⋊C₄	C₂³×C₇₃	1	2	yes	—
217	C₂×C₂²⋊C₄	—	2	1	no	—
218	C₂×C₂²⋊C₄	—	2	4	no	—
219	C₂×C₄²	—	2	4	no	—
220	C₂×C₄²	—	2	1	no	—
221	C₂×C₄²	D₄×C₇₃	1	2	yes	—
222	C₂×C₄²	C₂³×C₇₃	1	2	yes	—
223	#107	—	2	8	no	—
224	#106	—	2	8	no	—
225	#103	—	2	8	no	—
226	#102	—	2	8	no	—
227	#95	—	2	8	no	—
228	#95	—	2	4	no	—
229	C₂⁴	—	2	1	no	—
230	C₂⁴	C₂×D₄×C₇₃	1	1	yes	characteristic
231	#41	—	2	16	no	—
232	#41	—	4	8	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	2	2	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	C₂×Q₈×C₇₃	1	2	yes	—
237	C₂×D₄	—	2	4	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	C₂×Q₈×C₇₃	1	2	yes	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	1	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	4	4	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	—	4	2	no	—
252	C₂²×C₄	—	4	8	no	—
253	C₂²×C₄	—	4	4	no	—
254	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
255	C₂²×C₄	C₂×D₄×C₇₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₇₃	1	4	yes	—
257	C₂²×C₄	C₂×Q₈×C₇₃	1	2	yes	—
258	C₂²×C₄	#41×C₇₃	1	4	yes	—
259	C₂²×C₄	#41×C₇₃	1	2	yes	—
260	C₂²×C₄	C₂⁴×C₇₃	1	1	yes	characteristic
261	M₁₆	—	2	16	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	4	no	—
264	C₂×C₈	C₂×D₄×C₇₃	1	8	yes	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	4	4	no	—
270	C₂²⋊C₄	—	4	8	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₄²	—	2	4	no	—
273	C₄²	—	2	4	no	—
274	C₄²	—	4	2	no	—
275	C₂³	—	2	4	no	—
276	C₂³	—	2	2	no	—
277	C₂³	—	4	1	no	—
278	C₂³	—	4	1	no	—
279	C₂³	—	4	4	no	—
280	C₂³	C₂²⋊Q₈×C₇₃	1	2	yes	—
281	C₂³	#123×C₇₃	1	2	yes	—
282	C₂³	#127×C₇₃	1	2	yes	—
283	C₂³	C₂²×D₄×C₇₃	1	1	yes	characteristic
284	Q₈	—	2	4	no	—
285	Q₈	—	4	2	no	—
286	D₄	—	2	4	no	—
287	D₄	—	4	8	no	—
288	D₄	—	4	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	4	8	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	8	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	8	no	—
301	C₂×C₄	—	8	8	no	—
302	C₂×C₄	C₂²⋊Q₈×C₇₃	1	2	yes	—
303	C₂×C₄	C₂²⋊Q₈×C₇₃	1	8	yes	—
304	C₂×C₄	#123×C₇₃	1	2	yes	—
305	C₂×C₄	#123×C₇₃	1	8	yes	—
306	C₂×C₄	#127×C₇₃	1	2	yes	—
307	C₂×C₄	C₂²×D₄×C₇₃	1	2	yes	—
308	C₂×C₄	C₂²×Q₈×C₇₃	1	1	yes	characteristic
309	C₂×C₄	C₂×#41×C₇₃	1	2	yes	—
310	C₂×C₄	C₂×#41×C₇₃	1	1	yes	characteristic
311	C₈	—	2	8	no	—
312	C₂²	—	2	4	no	—
313	C₂²	—	2	4	no	—
314	C₂²	—	2	4	no	—
315	C₂²	—	4	4	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	—	8	2	no	—
319	C₂²	#399×C₇₃	1	4	yes	—
320	C₂²	C₂×C₂²⋊Q₈×C₇₃	1	1	yes	characteristic
321	C₂²	C₂×#123×C₇₃	1	1	yes	characteristic
322	C₂²	C₂×#127×C₇₃	1	1	yes	characteristic
323	C₂²	#575×C₇₃	1	4	yes	—
324	C₄	—	4	4	no	—
325	C₄	—	4	2	no	—
326	C₄	—	8	4	no	—
327	C₄	—	8	2	no	—
328	C₄	#399×C₇₃	1	4	yes	—
329	C₄	C₂×C₂²⋊Q₈×C₇₃	1	2	yes	—
330	C₄	C₂×#123×C₇₃	1	2	yes	—
331	C₂	—	2	4	no	—
332	C₂	—	8	2	no	—
333	C₂	C₂×#399×C₇₃	1	1	yes	characteristic
334	C₂	C₇₃×#3080	1	2	yes	—
335	C₂	C₇₃×#3093	1	2	yes	—
336	C₂	C₇₃×#3153	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₃
prime radical	C₂²×#41×C₇₃	C₂²
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
1. C₂×#428×C₇₃	C₂	semidirect
4. C₂×#382×C₇₃	C₂	semidirect
5. C₇₃×#2097	C₂	semidirect
6. C₇₃×#2092	C₂	semidirect
7. C₇₃×#1728	C₂	semidirect
162. #23733	C₇₃	direct (N × H)
169. C₂×#428	C₂×C₇₃	semidirect
172. C₂×#382	C₂×C₇₃	semidirect
173. #2097	C₂×C₇₃	semidirect
174. #2092	C₂×C₇₃	semidirect
175. #1728	C₂×C₇₃	semidirect
176. C₇₃	#23733	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#428×C₇₃	2
C₂×#417×C₇₃	2
C₇₃×#2297	2
C₂×#382×C₇₃	2
C₇₃×#2097	2
C₇₃×#2092	2
C₇₃×#1728	2
#23733	73

Subgroups — C73×#23733 #32933137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₃×#23733 #32933137