Subgroups — #33090368

Order 18688.

770 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618702	C₂	1	1	yes	maximal, characteristic
2	#22618574	C₂	1	2	yes	maximal, complemented
3	#22610819	C₂	1	1	yes	maximal, characteristic
4	#22610817	C₂	1	1	yes	maximal, characteristic
5	#22610680	C₂	1	1	yes	maximal, characteristic, complemented
6	#22610636	C₂	1	2	yes	maximal, complemented
7	#22610634	C₂	1	2	yes	maximal, complemented
8	#108×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#22609735	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#1945	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556613	C₂²	1	1	yes	characteristic
13	#17989055	C₂²	1	2	yes	—
14	#17988681	C₄	1	2	yes	complemented
15	#17988674	C₄	1	4	yes	complemented
16	#17988669	C₄	1	2	yes	complemented
17	#17988647	—	2	4	no	—
18	#17988639	—	2	4	no	—
19	C₂×#14556527	C₂²	1	1	yes	characteristic
20	#17988614	C₂²	1	2	yes	—
21	C₂×#14556519	C₂²	1	2	yes	—
22	C₂×#14556517	C₂²	1	2	yes	—
23	#17988581	C₂²	1	2	yes	—
24	#17988555	C₂²	1	2	yes	—
25	#17988553	C₂²	1	2	yes	—
26	M₁₆×Dic₇₃	C₂²	1	2	yes	—
27	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
28	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
29	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
30	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
31	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
32	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
33	#17988351	C₂²	1	2	yes	—
34	#17988346	C₂²	1	2	yes	—
35	C₂×#108×C₇₃	C₂²	1	1	yes	characteristic
36	#427×C₇₃	C₂²	1	1	yes	characteristic
37	#424×C₇₃	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
39	#339×C₇₃	C₂²	1	2	yes	—
40	C₂×#11924028	C₂³	1	2	yes	—
41	#14556613	—	2	2	no	—
42	#14556613	—	2	2	no	—
43	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
44	C₂×#11924017	C₂×C₄	1	2	yes	—
45	C₂×D₅₈₄	C₂×C₄	1	1	yes	characteristic
46	C₂×#11924012	C₂³	1	2	yes	—
47	C₂²×#10596831	C₂³	1	1	yes	characteristic
48	C₂²×#10596831	C₂³	1	1	yes	characteristic
49	#14556556	C₂×C₄	1	4	yes	—
50	#14556553	C₂×C₄	1	4	yes	—
51	#14556551	—	2	4	no	—
52	#14556548	—	2	2	no	—
53	C₂×C₄×Dic₇₃	—	2	2	no	—
54	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
55	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
56	C₄²×D₇₃	—	2	2	no	—
57	#14556527	—	2	2	no	—
58	C₄⋊#10596831	—	2	4	no	—
59	#14556519	—	2	4	no	—
60	#14556517	—	2	4	no	—
61	#14556514	C₂×C₄	1	4	yes	—
62	#14556511	C₂×C₄	1	4	yes	—
63	#14556503	—	2	4	no	—
64	#14556503	—	2	4	no	—
65	#14556501	—	2	2	no	—
66	C₄×#10596831	—	2	2	no	—
67	C₈×Dic₇₃	—	2	2	no	—
68	C₈×Dic₇₃	—	2	2	no	—
69	C₈×Dic₇₃	—	2	2	no	—
70	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
71	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
72	C₈×Dic₇₃	D₄	1	2	yes	—
73	C₈×Dic₇₃	D₄	1	2	yes	—
74	C₂×M₁₆×C₇₃	C₂³	1	2	yes	—
75	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
77	#117×C₇₃	C₂³	1	2	yes	—
78	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
79	#108×C₇₃	—	2	2	no	—
80	#107×C₇₃	C₂×C₄	1	2	yes	—
81	#106×C₇₃	C₂×C₄	1	2	yes	—
82	C₄⋊C₈×C₇₃	—	2	4	no	—
83	C₄×C₈×C₇₃	—	2	2	no	—
84	#11924028	—	2	2	no	—
85	#11924028	—	2	2	no	—
86	#11924028	—	2	2	no	—
87	#11924028	—	2	2	no	—
88	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
89	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
90	C₂×C₂²⋊D₇₃	—	2	2	no	—
91	C₂²×Dic₇₃	—	2	1	no	—
92	C₂²×Dic₇₃	#41	1	2	yes	—
93	C₂×C₄×D₇₃	—	2	2	no	—
94	C₂×C₄×D₇₃	—	2	4	no	—
95	Dic₂₉₂	—	2	2	no	—
96	#11924017	—	2	4	no	—
97	D₅₈₄	—	2	2	no	—
98	#11924012	—	2	2	no	—
99	#11924012	—	2	2	no	—
100	C₂×#10596831	—	2	1	no	—
101	C₂×#10596831	—	2	1	no	—
102	C₂×#10596831	—	2	1	no	—
103	C₂×#10596831	—	2	1	no	—
104	C₂×#10596831	—	2	2	no	—
105	C₂×#10596831	C₂²×C₄	1	2	yes	—
106	C₂×#10596831	C₂×D₄	1	2	yes	—
107	#11924007	—	2	4	no	—
108	#11924005	—	2	4	no	—
109	C₄×Dic₇₃	—	2	8	no	—
110	C₄×Dic₇₃	—	2	1	no	—
111	C₄×Dic₇₃	—	2	4	no	—
112	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
113	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
114	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
115	C₄×Dic₇₃	#41	1	2	yes	—
116	C₂²×C₄×C₇₃	—	2	1	no	—
117	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
118	M₁₆×C₇₃	—	2	2	no	—
119	M₁₆×C₇₃	—	2	2	no	—
120	C₂×C₈×C₇₃	—	2	1	no	—
121	C₂×C₈×C₇₃	—	2	1	no	—
122	C₂×C₈×C₇₃	—	2	2	no	—
123	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₇₃	#41	1	2	yes	—
127	C₂×C₈×C₇₃	#41	1	2	yes	—
128	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
129	C₂²⋊C₄×C₇₃	—	2	2	no	—
130	C₄²×C₇₃	—	2	2	no	—
131	C₄²×C₇₃	—	2	1	no	—
132	C₄²×C₇₃	#41	1	2	yes	—
133	C₂³×C₇₃	C₂×#41	1	1	yes	characteristic
134	C₂²×D₇₃	—	2	2	no	—
135	C₂×C₄×C₇₃	—	2	4	no	—
136	C₂×C₄×C₇₃	—	2	2	no	—
137	C₂×C₄×C₇₃	—	2	4	no	—
138	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
142	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
144	C₂²⋊D₇₃	—	4	2	no	—
145	C₂²⋊D₇₃	—	4	2	no	—
146	C₂×Dic₇₃	—	2	4	no	—
147	C₂×Dic₇₃	—	2	2	no	—
148	C₂×Dic₇₃	—	2	4	no	—
149	C₂×Dic₇₃	—	2	2	no	—
150	C₂×Dic₇₃	C₄×D₄	1	2	yes	—
151	C₂×Dic₇₃	C₄×D₄	1	2	yes	—
152	D₂₉₂	—	2	2	no	—
153	D₂₉₂	—	2	2	no	—
154	C₄×D₇₃	—	4	2	no	—
155	C₄×D₇₃	—	4	2	no	—
156	C₄×D₇₃	—	4	8	no	—
157	Dic₁₄₆	—	2	2	no	—
158	Dic₁₄₆	—	2	2	no	—
159	C₈×C₇₃	—	2	2	no	—
160	C₈×C₇₃	—	4	2	no	—
161	C₈×C₇₃	#117	1	2	yes	—
162	C₈×C₇₃	#117	1	2	yes	—
163	#10596831	—	2	2	no	—
164	#10596831	—	2	2	no	—
165	#10596831	—	4	2	no	—
166	C₂²×C₇₃	—	2	2	no	—
167	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂²×C₇₃	#529	1	1	yes	characteristic
169	C₂²×C₇₃	#529	1	1	yes	characteristic
170	C₂×D₇₃	—	4	4	no	—
171	C₂×D₇₃	—	4	2	no	—
172	C₄×C₇₃	—	2	4	no	—
173	C₄×C₇₃	—	4	4	no	—
174	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
175	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
176	C₄×C₇₃	#532	1	1	yes	characteristic
177	C₄×C₇₃	#532	1	1	yes	characteristic
178	Dic₇₃	—	2	4	no	—
179	Dic₇₃	—	2	2	no	—
180	Dic₇₃	—	4	2	no	—
181	Dic₇₃	#443	1	4	yes	—
182	#21292	—	73	1	no	maximal, Sylow
183	C₂×C₇₃	—	2	1	no	—
184	C₂×C₇₃	—	2	1	no	—
185	C₂×C₇₃	#2293	1	1	yes	characteristic
186	C₂×C₇₃	C₂×#443	1	2	yes	—
187	D₇₃	—	8	2	no	—
188	C₄×#135	—	73	1	no	—
189	#2922	—	73	2	no	—
190	#1974	—	73	1	no	—
191	#1974	—	73	1	no	—
192	#1945	—	73	1	no	—
193	#1945	D₇₃	1	1	yes	characteristic, complemented
194	#1937	—	73	2	no	—
195	#1937	—	73	2	no	—
196	C₄×#108	—	73	1	no	—
197	C₄×#104	—	73	2	no	—
198	C₄²×C₈	—	73	1	no	—
199	C₇₃	#21292	1	1	yes	characteristic, complemented, Sylow
200	C₂×#135	—	73	1	no	—
201	C₄×#41	—	73	2	no	—
202	C₄×Q₁₆	—	73	2	no	—
203	C₄×SD₁₆	—	73	4	no	—
204	C₄×D₈	—	73	2	no	—
205	#433	—	146	4	no	—
206	#431	—	146	4	no	—
207	C₂×#108	—	73	1	no	—
208	C₂×#108	C₂×D₇₃	1	1	yes	characteristic
209	#427	C₂×D₇₃	1	1	yes	characteristic
210	#424	—	73	2	no	—
211	#424	C₂×D₇₃	1	2	yes	—
212	C₂×#104	—	73	2	no	—
213	C₂×#104	—	73	2	no	—
214	#416	—	73	2	no	—
215	#408	—	73	2	no	—
216	#408	—	73	2	no	—
217	C₄×M₁₆	—	73	2	no	—
218	C₂×C₄×C₈	—	73	1	no	—
219	C₂×C₄×C₈	—	73	1	no	—
220	C₂×C₄×C₈	—	73	1	no	—
221	C₂×C₄×C₈	—	73	1	no	—
222	C₂×C₄×C₈	—	73	1	no	—
223	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
224	C₄³	—	73	1	no	—
225	#341	—	73	2	no	—
226	#339	—	73	2	no	—
227	#339	C₂×D₇₃	1	2	yes	—
228	C₂×#41	—	73	2	no	—
229	#135	—	146	2	no	—
230	#135	—	146	2	no	—
231	C₂×Q₁₆	—	73	1	no	—
232	C₂×SD₁₆	—	73	2	no	—
233	C₂×D₈	—	73	1	no	—
234	C₂×M₁₆	—	73	2	no	—
235	C₂×M₁₆	C₂²×D₇₃	1	2	yes	—
236	C₂²×C₈	—	73	1	no	—
237	C₂²×C₈	—	73	1	no	—
238	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
239	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
240	C₄×Q₈	—	73	4	no	—
241	C₄×D₄	—	73	4	no	—
242	C₄×D₄	—	146	4	no	—
243	#117	—	146	2	no	—
244	#117	C₂²×D₇₃	1	2	yes	—
245	C₂×C₄²	—	73	1	no	—
246	C₂×C₄²	—	73	1	no	—
247	C₂×C₄²	—	146	2	no	—
248	C₂×C₄²	—	146	2	no	—
249	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
250	#108	—	2	2	no	—
251	#108	—	146	2	no	—
252	#107	C₄×D₇₃	1	2	yes	—
253	#106	C₄×D₇₃	1	2	yes	—
254	C₄⋊C₈	—	2	4	no	—
255	C₄⋊C₈	—	146	4	no	—
256	#104	—	146	4	no	—
257	#104	—	146	4	no	—
258	#103	—	73	4	no	—
259	#102	—	73	4	no	—
260	C₂²⋊C₈	—	146	4	no	—
261	C₂²⋊C₈	—	146	4	no	—
262	#97	—	146	2	no	—
263	C₄×C₈	—	2	2	no	—
264	C₄×C₈	—	73	2	no	—
265	C₄×C₈	—	73	2	no	—
266	C₄×C₈	—	73	2	no	—
267	C₄×C₈	—	73	2	no	—
268	C₄×C₈	—	146	2	no	—
269	C₄×C₈	—	146	2	no	—
270	C₄×C₈	—	146	2	no	—
271	C₄×C₈	—	146	2	no	—
272	#41	—	146	2	no	—
273	#41	—	146	2	no	—
274	#41	—	146	2	no	—
275	#41	—	146	2	no	—
276	C₂×Q₈	—	73	2	no	—
277	C₂×D₄	—	73	2	no	—
278	C₂×D₄	—	146	2	no	—
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	73	2	no	—
281	C₂²×C₄	—	146	4	no	—
282	C₂²×C₄	—	146	2	no	—
283	C₂²×C₄	—	146	1	no	—
284	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
285	Q₁₆	—	146	2	no	—
286	SD₁₆	—	146	4	no	—
287	D₈	—	146	2	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	2	2	no	—
290	M₁₆	—	146	2	no	—
291	M₁₆	—	146	2	no	—
292	C₂×C₈	—	2	1	no	—
293	C₂×C₈	—	2	1	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	73	2	no	—
296	C₂×C₈	—	73	2	no	—
297	C₂×C₈	—	146	2	no	—
298	C₂×C₈	—	146	1	no	—
299	C₂×C₈	—	146	1	no	—
300	C₂×C₈	—	146	1	no	—
301	C₂×C₈	—	146	1	no	—
302	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
303	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
304	C₂×C₈	D₄×D₇₃	1	2	yes	—
305	C₂×C₈	#11924027	1	2	yes	—
306	C₂×C₈	#11924029	1	2	yes	—
307	C₄⋊C₄	—	146	4	no	—
308	C₄⋊C₄	C₂×C₄×D₇₃	1	4	yes	—
309	C₂²⋊C₄	—	2	2	no	—
310	C₂²⋊C₄	—	146	4	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	2	1	no	—
313	C₄²	—	73	2	no	—
314	C₄²	—	73	1	no	—
315	C₄²	—	73	1	no	—
316	C₄²	—	73	2	no	—
317	C₄²	—	146	4	no	—
318	C₄²	—	146	8	no	—
319	C₄²	—	146	1	no	—
320	C₄²	#11924029	1	2	yes	—
321	C₂³	—	146	2	no	—
322	C₂³	#41×D₇₃	1	1	yes	characteristic
323	Q₈	—	146	2	no	—
324	Q₈	—	146	2	no	—
325	D₄	—	292	2	no	—
326	D₄	—	292	2	no	—
327	D₄	—	146	2	no	—
328	D₄	—	146	2	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	2	4	no	—
332	C₂×C₄	—	292	8	no	—
333	C₂×C₄	—	292	2	no	—
334	C₂×C₄	—	292	2	no	—
335	C₂×C₄	—	73	2	no	—
336	C₂×C₄	—	73	2	no	—
337	C₂×C₄	—	146	2	no	—
338	C₂×C₄	—	146	4	no	—
339	C₂×C₄	—	146	4	no	—
340	C₂×C₄	—	146	2	no	—
341	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
342	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
343	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
344	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
345	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
346	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
347	C₈	—	2	2	no	—
348	C₈	—	4	2	no	—
349	C₈	—	292	2	no	—
350	C₈	—	146	2	no	—
351	C₈	—	146	2	no	—
352	C₈	#14556549	1	2	yes	—
353	C₈	#14556549	1	2	yes	—
354	C₂²	—	2	2	no	—
355	C₂²	—	292	2	no	—
356	C₂²	—	292	4	no	—
357	C₂²	C₄×D₄×D₇₃	1	1	yes	characteristic
358	C₂²	#17989122	1	1	yes	characteristic
359	C₂²	#17989123	1	1	yes	characteristic
360	C₄	—	2	4	no	—
361	C₄	—	4	4	no	—
362	C₄	—	292	2	no	—
363	C₄	—	73	4	no	—
364	C₄	—	146	4	no	—
365	C₄	—	146	2	no	—
366	C₄	C₂×#14556549	1	1	yes	characteristic
367	C₄	C₄×#11924029	1	1	yes	characteristic
368	C₄	#17989132	1	1	yes	characteristic
369	C₄	#17989138	1	1	yes	characteristic
370	C₂	—	2	1	no	—
371	C₂	—	2	1	no	—
372	C₂	—	584	2	no	—
373	C₂	#22612389	1	1	yes	characteristic
374	C₂	#443×D₇₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₇₃×#1945	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂×#14556613	C₂²
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	#1945	D₇₃
hypocenter	C₇₃	#21292
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #1945

Complements

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

N	H ≅ G/N	type
2. #22618574	C₂	semidirect
5. #22610680	C₂	semidirect
6. #22610636	C₂	semidirect
7. #22610634	C₂	semidirect
8. #108×Dic₇₃	C₂	semidirect
9. #22609735	C₂	semidirect
10. C₄×C₈×Dic₇₃	C₂	semidirect
11. C₇₃×#1945	C₂	semidirect
14. #17988681	C₄	semidirect
15. #17988674	C₄	semidirect
16. #17988669	C₄	semidirect
193. #1945	D₇₃	semidirect
199. C₇₃	#21292	semidirect

Maximal subgroups

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

subgroup	index
#22618702	2
#22618574	2
#22610819	2
#22610817	2
#22610680	2
#22610636	2
#22610634	2
#108×Dic₇₃	2
#22609735	2
C₄×C₈×Dic₇₃	2
C₇₃×#1945	2
#21292	73