Subgroups — #33057769

Order 18688.

600 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620501	C₂	1	1	yes	maximal, characteristic
2	#22618929	C₂	1	2	yes	maximal, complemented
3	#22608827	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608815	C₂	1	1	yes	maximal, characteristic, complemented
5	#22608773	C₂	1	2	yes	maximal, complemented
6	#22608766	C₂	1	2	yes	maximal, complemented
7	#22608737	C₂	1	1	yes	maximal, characteristic, complemented
8	#22608725	C₂	1	1	yes	maximal, characteristic, complemented
9	#22608337	C₂	1	2	yes	maximal, complemented
10	C₈×#11924012	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989121	C₂²	1	2	yes	—
13	#17988989	C₄	1	1	yes	characteristic
14	#17988971	C₄	1	1	yes	characteristic
15	#17988942	C₄	1	2	yes	—
16	#17988828	C₄	1	1	yes	characteristic
17	#17988823	C₄	1	1	yes	characteristic
18	#17988816	C₄	1	2	yes	—
19	#17988714	C₂²	1	2	yes	—
20	#17988711	C₂²	1	2	yes	—
21	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
22	C₄×#11924017	C₂²	1	2	yes	—
23	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
24	#17988664	C₂²	1	2	yes	—
25	#17988662	—	2	2	no	—
26	#17988661	—	2	2	no	—
27	#17988652	C₂²	1	2	yes	—
28	#17988646	—	2	2	no	—
29	#17988640	C₂²	1	1	yes	characteristic
30	#17988541	—	2	2	no	—
31	C₄×#11924012	C₂²	1	1	yes	characteristic
32	#17988317	C₂²	1	2	yes	—
33	#17988317	C₂²	1	2	yes	—
34	#17988314	C₂²	1	2	yes	—
35	#17988314	C₂²	1	2	yes	—
36	#17988309	C₂²	1	1	yes	characteristic
37	#17988309	C₂²	1	1	yes	characteristic
38	C₈×#10596831	C₂²	1	1	yes	characteristic
39	C₈×#10596831	C₂²	1	1	yes	characteristic
40	#431×C₇₃	C₂²	1	2	yes	—
41	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
42	#335×C₇₃	C₂²	1	1	yes	characteristic
43	#335×C₇₃	C₂²	1	1	yes	characteristic
44	#334×C₇₃	C₂²	1	2	yes	—
45	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
46	C₂×#11924017	C₂×C₄	1	2	yes	—
47	C₂×D₅₈₄	C₂×C₄	1	1	yes	characteristic
48	C₂×#11924012	D₄	1	1	yes	characteristic
49	C₂×#11924012	D₄	1	1	yes	characteristic
50	C₂×#11924011	—	2	2	no	—
51	C₂×C₈×D₇₃	—	2	2	no	—
52	#14556585	—	2	2	no	—
53	#14556583	C₂×C₄	1	2	yes	—
54	#14556579	C₂×C₄	1	2	yes	—
55	#14556574	—	2	2	no	—
56	#14556570	C₂×C₄	1	2	yes	—
57	#14556565	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₄₆	C₂³	1	2	yes	—
59	C₄×D₂₉₂	C₂³	1	2	yes	—
60	C₄×C₂²⋊D₇₃	—	2	2	no	—
61	#14556548	—	2	2	no	—
62	#14556525	C₂×C₄	1	1	yes	characteristic
63	#14556525	C₂×C₄	1	1	yes	characteristic
64	#14556523	C₂×C₄	1	2	yes	—
65	C₄⋊#10596831	—	2	1	no	—
66	#14556520	—	2	2	no	—
67	#14556520	—	2	2	no	—
68	#14556516	C₂×C₄	1	2	yes	—
69	#14556516	C₂×C₄	1	2	yes	—
70	#14556513	C₂×C₄	1	2	yes	—
71	#14556513	C₂×C₄	1	2	yes	—
72	C₂²⋊#10596831	—	2	1	no	—
73	#14556503	—	2	2	no	—
74	#14556503	—	2	2	no	—
75	#14556502	D₄	1	1	yes	characteristic
76	#14556502	D₄	1	1	yes	characteristic
77	#14556501	—	2	2	no	—
78	C₄×#10596831	C₂³	1	1	yes	characteristic
79	C₄×#10596831	C₂³	1	1	yes	characteristic
80	C₄×#10596831	C₂³	1	1	yes	characteristic
81	C₄×#10596831	C₂³	1	1	yes	characteristic
82	C₈×Dic₇₃	—	2	2	no	—
83	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
84	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
87	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
88	C₂²⋊C₈×C₇₃	—	2	2	no	—
89	C₄×C₈×C₇₃	—	2	2	no	—
90	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
91	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
92	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
95	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
96	C₂×C₂²⋊D₇₃	—	2	2	no	—
97	C₂×C₄×D₇₃	—	2	2	no	—
98	Dic₂₉₂	—	2	2	no	—
99	#11924017	—	4	2	no	—
100	D₅₈₄	—	2	2	no	—
101	#11924012	—	2	2	no	—
102	#11924012	—	2	2	no	—
103	#11924011	—	4	4	no	—
104	C₂×#10596831	—	2	1	no	—
105	C₂×#10596831	—	2	1	no	—
106	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
107	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
108	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
109	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
110	C₈×D₇₃	—	4	4	no	—
111	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
112	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
113	#11924007	—	2	2	no	—
114	#11924007	—	2	2	no	—
115	#11924007	—	2	2	no	—
116	#11924007	—	2	2	no	—
117	C₂²⋊Dic₇₃	—	2	2	no	—
118	#11924005	—	2	2	no	—
119	#11924005	—	2	2	no	—
120	#11924005	—	2	2	no	—
121	#11924005	—	2	2	no	—
122	C₄×Dic₇₃	—	2	2	no	—
123	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
124	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
126	C₂×C₈×C₇₃	—	2	2	no	—
127	C₂×C₈×C₇₃	—	2	2	no	—
128	C₂×C₈×C₇₃	—	2	2	no	—
129	C₂×C₈×C₇₃	—	4	1	no	—
130	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
134	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
135	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₇₃	#41	1	1	yes	characteristic
137	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
138	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
139	C₂²×D₇₃	—	2	2	no	—
140	C₂×C₄×C₇₃	—	2	1	no	—
141	C₂×C₄×C₇₃	—	2	1	no	—
142	C₂×C₄×C₇₃	—	2	1	no	—
143	C₂×C₄×C₇₃	—	2	1	no	—
144	C₂×C₄×C₇₃	—	4	1	no	—
145	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
151	C₂²⋊D₇₃	—	8	2	no	—
152	C₂×Dic₇₃	—	2	2	no	—
153	C₂×Dic₇₃	—	2	2	no	—
154	C₂×Dic₇₃	—	2	2	no	—
155	D₂₉₂	—	2	2	no	—
156	D₂₉₂	—	2	2	no	—
157	C₄×D₇₃	—	4	2	no	—
158	C₄×D₇₃	—	4	2	no	—
159	Dic₁₄₆	—	2	2	no	—
160	Dic₁₄₆	—	2	2	no	—
161	C₈×C₇₃	—	2	1	no	—
162	C₈×C₇₃	—	2	1	no	—
163	C₈×C₇₃	—	2	1	no	—
164	C₈×C₇₃	—	4	2	no	—
165	C₈×C₇₃	—	4	2	no	—
166	C₈×C₇₃	#131	1	2	yes	—
167	#10596831	—	2	1	no	—
168	#10596831	—	2	1	no	—
169	#10596831	—	4	1	no	—
170	#10596831	—	4	1	no	—
171	#10596831	#135	1	2	yes	—
172	#10596831	#135	1	2	yes	—
173	C₂²×C₇₃	—	2	1	no	—
174	C₂²×C₇₃	—	2	1	no	—
175	C₂²×C₇₃	—	2	1	no	—
176	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂×D₇₃	—	4	2	no	—
178	C₂×D₇₃	—	4	2	no	—
179	C₂×D₇₃	—	4	2	no	—
180	C₄×C₇₃	—	2	1	no	—
181	C₄×C₇₃	—	2	1	no	—
182	C₄×C₇₃	—	2	1	no	—
183	C₄×C₇₃	—	2	1	no	—
184	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
185	C₄×C₇₃	#568	1	1	yes	characteristic
186	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
187	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
188	Dic₇₃	—	4	2	no	—
189	Dic₇₃	—	4	2	no	—
190	Dic₇₃	—	4	2	no	—
191	#18069	—	73	1	no	maximal, Sylow
192	C₂×C₇₃	—	4	1	no	—
193	C₂×C₇₃	#2935	1	1	yes	characteristic
194	C₂×C₇₃	C₄×#135	1	1	yes	characteristic
195	C₂×C₇₃	#2957	1	1	yes	characteristic
196	D₇₃	—	8	2	no	—
197	#3168	—	73	1	no	—
198	#2969	—	73	2	no	—
199	#1591	—	73	1	no	—
200	#1589	—	73	1	no	—
201	#1583	—	73	2	no	—
202	#1582	—	73	2	no	—
203	C₈×Q₁₆	—	73	1	no	—
204	C₈×D₈	—	73	1	no	—
205	#1564	D₇₃	1	1	yes	characteristic, complemented
206	#1482	—	73	2	no	—
207	C₈×M₁₆	—	73	1	no	—
208	C₇₃	#18069	1	1	yes	characteristic, complemented, Sylow
209	#529	—	73	2	no	—
210	#499	—	73	1	no	—
211	#495	—	73	1	no	—
212	#488	—	73	2	no	—
213	#467	—	73	1	no	—
214	#466	—	73	1	no	—
215	#465	—	73	2	no	—
216	#446	—	73	2	no	—
217	C₈×Q₈	—	73	2	no	—
218	C₄×Q₁₆	—	73	1	no	—
219	C₄×SD₁₆	—	73	2	no	—
220	C₄×D₈	—	73	1	no	—
221	#436	—	73	2	no	—
222	#436	—	146	2	no	—
223	#435	—	146	2	no	—
224	C₈×D₄	—	73	2	no	—
225	#433	—	146	2	no	—
226	#431	—	73	1	no	—
227	#431	C₂×D₇₃	1	2	yes	—
228	#405	—	146	2	no	—
229	C₄×M₁₆	—	73	1	no	—
230	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
231	#335	C₂×D₇₃	1	1	yes	characteristic
232	#335	C₂×D₇₃	1	1	yes	characteristic
233	#334	C₂×D₇₃	1	2	yes	—
234	Q₈⋊C₈	—	73	2	no	—
235	Q₈⋊C₈	—	73	2	no	—
236	D₄⋊C₈	—	73	2	no	—
237	D₄⋊C₈	—	73	2	no	—
238	#322	—	73	1	no	—
239	#322	—	73	1	no	—
240	C₈²	—	73	1	no	—
241	C₈²	—	73	1	no	—
242	C₂×Q₁₆	—	73	1	no	—
243	C₂×SD₁₆	—	73	2	no	—
244	C₂×D₈	—	73	1	no	—
245	C₂×M₁₆	—	73	1	no	—
246	C₂×M₁₆	—	73	1	no	—
247	C₂×M₁₆	—	146	2	no	—
248	C₂²×C₈	—	146	2	no	—
249	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
250	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
251	#126	—	146	2	no	—
252	#125	—	73	2	no	—
253	#124	—	73	2	no	—
254	#123	—	146	2	no	—
255	C₂²⋊Q₈	—	73	2	no	—
256	#121	—	73	2	no	—
257	C₄×Q₈	—	73	2	no	—
258	C₄×D₄	—	73	2	no	—
259	C₄×D₄	—	146	2	no	—
260	#117	—	146	2	no	—
261	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
262	#107	—	73	1	no	—
263	#107	—	73	1	no	—
264	#106	—	73	2	no	—
265	C₄⋊C₈	—	146	2	no	—
266	C₄⋊C₈	—	146	2	no	—
267	C₄⋊C₈	—	146	1	no	—
268	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
269	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
270	#103	—	73	2	no	—
271	#103	—	73	2	no	—
272	#102	—	73	2	no	—
273	#102	—	73	2	no	—
274	C₂²⋊C₈	—	2	2	no	—
275	C₂²⋊C₈	—	146	2	no	—
276	C₂²⋊C₈	—	146	2	no	—
277	C₂²⋊C₈	—	146	1	no	—
278	#97	—	73	1	no	—
279	#97	—	73	1	no	—
280	#97	—	146	2	no	—
281	C₄×C₈	—	2	2	no	—
282	C₄×C₈	—	73	1	no	—
283	C₄×C₈	—	73	1	no	—
284	C₄×C₈	—	73	1	no	—
285	C₄×C₈	—	73	1	no	—
286	C₄×C₈	—	146	2	no	—
287	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
288	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
291	C₂×Q₈	—	73	2	no	—
292	C₂×D₄	—	73	2	no	—
293	C₂×D₄	—	146	2	no	—
294	C₂²×C₄	—	146	2	no	—
295	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
296	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
297	C₂²×C₄	#11924029	1	1	yes	characteristic
298	Q₁₆	—	146	2	no	—
299	SD₁₆	—	292	2	no	—
300	D₈	—	146	2	no	—
301	M₁₆	—	292	4	no	—
302	M₁₆	—	146	2	no	—
303	M₁₆	—	146	2	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	4	1	no	—
308	C₂×C₈	—	292	4	no	—
309	C₂×C₈	—	73	1	no	—
310	C₂×C₈	—	73	1	no	—
311	C₂×C₈	—	73	1	no	—
312	C₂×C₈	—	73	1	no	—
313	C₂×C₈	—	146	1	no	—
314	C₂×C₈	—	146	1	no	—
315	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
316	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
317	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
318	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
319	C₄⋊C₄	—	73	2	no	—
320	C₄⋊C₄	—	73	2	no	—
321	C₄⋊C₄	—	146	2	no	—
322	C₄⋊C₄	—	146	2	no	—
323	C₄⋊C₄	—	146	2	no	—
324	C₄⋊C₄	—	146	2	no	—
325	C₂²⋊C₄	—	146	2	no	—
326	C₂²⋊C₄	—	146	2	no	—
327	C₂²⋊C₄	—	146	2	no	—
328	C₂²⋊C₄	—	146	2	no	—
329	C₂²⋊C₄	—	146	2	no	—
330	C₄²	—	146	2	no	—
331	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
332	C₄²	D₄×D₇₃	1	1	yes	characteristic
333	C₄²	#11924029	1	1	yes	characteristic
334	C₄²	C₂³×D₇₃	1	1	yes	characteristic
335	C₂³	—	146	2	no	—
336	C₂³	#14556553	1	1	yes	characteristic
337	Q₈	—	146	2	no	—
338	Q₈	—	146	2	no	—
339	D₄	—	584	2	no	—
340	D₄	—	146	2	no	—
341	D₄	—	146	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	292	2	no	—
348	C₂×C₄	—	292	2	no	—
349	C₂×C₄	—	146	2	no	—
350	C₂×C₄	—	146	2	no	—
351	C₂×C₄	—	146	2	no	—
352	C₂×C₄	#14556553	1	1	yes	characteristic
353	C₂×C₄	#14556553	1	1	yes	characteristic
354	C₂×C₄	#14556553	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
356	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
357	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	4	2	no	—
362	C₈	—	4	2	no	—
363	C₈	—	292	1	no	—
364	C₈	—	292	1	no	—
365	C₈	—	73	2	no	—
366	C₈	—	73	2	no	—
367	C₈	—	146	1	no	—
368	C₈	—	146	1	no	—
369	C₈	#14556597	1	2	yes	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	292	2	no	—
374	C₂²	—	292	2	no	—
375	C₂²	—	292	2	no	—
376	C₂²	C₂×#14556553	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	292	2	no	—
382	C₄	—	292	2	no	—
383	C₄	—	292	2	no	—
384	C₄	C₂×#14556597	1	1	yes	characteristic
385	C₄	#17989353	1	1	yes	characteristic
386	C₄	#135×D₇₃	1	1	yes	characteristic
387	C₄	#135×D₇₃	1	1	yes	characteristic
388	C₂	—	4	1	no	—
389	C₂	—	584	2	no	—
390	C₂	#22618674	1	1	yes	characteristic
391	C₂	#22618702	1	1	yes	characteristic
392	C₂	#22618847	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1564	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#17988823	C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1564	D₇₃
hypocenter	C₇₃	#18069
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
2. #22618929	C₂	semidirect
3. #22608827	C₂	semidirect
4. #22608815	C₂	semidirect
5. #22608773	C₂	semidirect
6. #22608766	C₂	semidirect
7. #22608737	C₂	semidirect
8. #22608725	C₂	semidirect
9. #22608337	C₂	semidirect
10. C₈×#11924012	C₂	semidirect
11. C₇₃×#1564	C₂	semidirect
205. #1564	D₇₃	semidirect
208. C₇₃	#18069	semidirect

Maximal subgroups

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

subgroup	index
#22620501	2
#22618929	2
#22608827	2
#22608815	2
#22608773	2
#22608766	2
#22608737	2
#22608725	2
#22608337	2
C₈×#11924012	2
C₇₃×#1564	2
#18069	73