Subgroups — #33075265

Order 18688.

822 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	#22620676	C₂	1	1	yes	maximal, characteristic, complemented
2	#22620645	C₂	1	2	yes	maximal, complemented
3	#22620305	C₂	1	2	yes	maximal, complemented
4	#22609600	C₂	1	1	yes	maximal, characteristic, complemented
5	#22609582	C₂	1	2	yes	maximal, complemented
6	#22609569	C₂	1	2	yes	maximal, complemented
7	#22609557	C₂	1	1	yes	maximal, characteristic, complemented
8	#22609536	C₂	1	1	yes	maximal, characteristic, complemented
9	#22609530	C₂	1	1	yes	maximal, characteristic, complemented
10	#22608234	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#3151	C₂	1	1	yes	maximal, characteristic
12	#17989144	C₂²	1	2	yes	—
13	#17988990	C₂²	1	2	yes	complemented
14	#17988985	C₂²	1	1	yes	characteristic, complemented
15	#17988967	C₂²	1	1	yes	characteristic, complemented
16	#17988959	C₂²	1	2	yes	complemented
17	#17988951	—	2	4	no	—
18	#17988939	C₂²	1	2	yes	complemented
19	#17988938	C₂²	1	2	yes	complemented
20	#17988933	C₂²	1	2	yes	—
21	#17988933	C₂²	1	2	yes	—
22	#17988933	C₂²	1	2	yes	—
23	#17988933	C₂²	1	2	yes	—
24	#17988933	C₂²	1	2	yes	complemented
25	#17988845	—	2	2	no	—
26	#17988831	—	2	4	no	—
27	#17988695	—	2	2	no	—
28	#17988687	—	2	4	no	—
29	#17988592	—	2	4	no	—
30	C₄×#11924012	C₂²	1	2	yes	—
31	#17988309	C₂²	1	2	yes	complemented
32	#17988308	C₂²	1	1	yes	characteristic, complemented
33	C₈×#10596831	C₂²	1	1	yes	characteristic, complemented
34	C₂×AGL(1,ℤ/8)×C₇₃	—	2	2	no	—
35	#532×C₇₃	C₂²	1	1	yes	characteristic
36	C₂×#127×C₇₃	C₂²	1	1	yes	characteristic
37	#496×C₇₃	—	2	2	no	—
38	#493×C₇₃	C₂²	1	1	yes	characteristic
39	#493×C₇₃	C₂²	1	1	yes	characteristic
40	#492×C₇₃	C₂²	1	1	yes	characteristic
41	#492×C₇₃	C₂²	1	1	yes	characteristic
42	C₄×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
43	C₂×Dic₂₉₂	—	2	2	no	—
44	C₂×#11924016	—	2	4	no	—
45	C₂×#11924012	D₄	1	4	yes	—
46	#14556591	C₂³	1	2	yes	—
47	#14556591	C₂³	1	2	yes	—
48	#14556583	—	2	2	no	—
49	#14556570	—	2	2	no	—
50	#14556570	—	2	2	no	—
51	C₄×Dic₁₄₆	—	2	2	no	—
52	C₄×Dic₁₄₆	—	2	2	no	—
53	#14556550	—	2	2	no	—
54	#14556524	—	2	4	no	—
55	#14556523	—	2	2	no	—
56	#14556516	—	2	4	no	—
57	#14556514	—	2	4	no	—
58	#14556512	—	2	4	no	—
59	#14556512	—	2	4	no	—
60	#14556512	—	2	4	no	—
61	#14556512	—	2	4	no	—
62	#14556512	—	2	4	no	—
63	#14556502	D₄	1	4	yes	—
64	C₄×#10596831	C₂³	1	2	yes	—
65	C₄×#10596831	C₂³	1	2	yes	—
66	C₂×#41×C₇₃	—	2	1	no	—
67	C₂²×D₄×C₇₃	—	2	1	no	—
68	C₂²×D₄×C₇₃	—	2	2	no	—
69	AGL(1,ℤ/8)×C₇₃	—	4	8	no	—
70	C₂×SD₁₆×C₇₃	—	2	2	no	—
71	C₂×SD₁₆×C₇₃	—	2	2	no	—
72	C₂×D₈×C₇₃	—	2	2	no	—
73	C₂×D₈×C₇₃	—	2	2	no	—
74	C₂×M₁₆×C₇₃	D₄	1	2	yes	—
75	C₄⋊Q₈×C₇₃	C₂³	1	1	yes	characteristic
76	#127×C₇₃	—	2	2	no	—
77	#127×C₇₃	—	2	1	no	—
78	#127×C₇₃	C₂³	1	1	yes	characteristic
79	#127×C₇₃	C₂³	1	1	yes	characteristic
80	#127×C₇₃	C₂³	1	1	yes	characteristic
81	#124×C₇₃	—	2	1	no	—
82	#121×C₇₃	—	2	2	no	—
83	C₄×D₄×C₇₃	—	2	2	no	—
84	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
85	#97×C₇₃	D₄	1	2	yes	—
86	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
88	C₂×Dic₁₄₆	—	2	2	no	—
89	C₂×Dic₁₄₆	—	2	2	no	—
90	Dic₂₉₂	—	4	4	no	—
91	#11924016	—	4	8	no	—
92	#11924012	—	4	4	no	—
93	C₂×#10596831	C₂×D₄	1	4	yes	—
94	C₂×#10596831	C₂×D₄	1	4	yes	—
95	C₄⋊Dic₇₃	—	2	2	no	—
96	C₄⋊Dic₇₃	—	2	2	no	—
97	C₄⋊Dic₇₃	—	2	2	no	—
98	C₄⋊Dic₇₃	—	2	2	no	—
99	#11924007	—	4	2	no	—
100	#11924007	—	4	2	no	—
101	C₂²⋊Dic₇₃	—	4	2	no	—
102	C₄×Dic₇₃	—	4	2	no	—
103	C₂⁴×C₇₃	—	4	1	no	—
104	#41×C₇₃	—	4	4	no	—
105	C₂×Q₈×C₇₃	—	2	1	no	—
106	C₂×D₄×C₇₃	—	2	2	no	—
107	C₂×D₄×C₇₃	—	2	1	no	—
108	C₂×D₄×C₇₃	—	2	1	no	—
109	C₂×D₄×C₇₃	—	2	1	no	—
110	C₂×D₄×C₇₃	—	2	2	no	—
111	C₂×D₄×C₇₃	—	2	2	no	—
112	C₂×D₄×C₇₃	—	4	4	no	—
113	C₂×D₄×C₇₃	—	4	8	no	—
114	C₂×D₄×C₇₃	—	4	1	no	—
115	C₂×D₄×C₇₃	—	4	2	no	—
116	C₂×D₄×C₇₃	—	4	1	no	—
117	C₂²×C₄×C₇₃	—	4	1	no	—
118	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
120	SD₁₆×C₇₃	—	4	4	no	—
121	SD₁₆×C₇₃	—	4	4	no	—
122	D₈×C₇₃	—	4	4	no	—
123	D₈×C₇₃	—	4	4	no	—
124	M₁₆×C₇₃	—	2	4	no	—
125	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
127	C₄⋊C₄×C₇₃	—	2	2	no	—
128	C₂²⋊C₄×C₇₃	—	4	2	no	—
129	C₄²×C₇₃	C₂×D₄	1	2	yes	—
130	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
132	C₂³×C₇₃	—	4	1	no	—
133	C₂³×C₇₃	—	4	1	no	—
134	C₂³×C₇₃	—	4	1	no	—
135	C₂³×C₇₃	—	8	2	no	—
136	C₂³×C₇₃	—	8	4	no	—
137	C₂³×C₇₃	#127	1	1	yes	characteristic
138	Q₈×C₇₃	—	4	2	no	—
139	D₄×C₇₃	—	4	2	no	—
140	D₄×C₇₃	—	4	2	no	—
141	D₄×C₇₃	—	4	2	no	—
142	D₄×C₇₃	—	4	4	no	—
143	D₄×C₇₃	—	4	4	no	—
144	D₄×C₇₃	—	4	4	no	—
145	D₄×C₇₃	—	8	2	no	—
146	D₄×C₇₃	—	8	4	no	—
147	D₄×C₇₃	—	8	2	no	—
148	C₂×C₄×C₇₃	—	2	2	no	—
149	C₂×C₄×C₇₃	—	2	4	no	—
150	C₂×C₄×C₇₃	—	4	1	no	—
151	C₂×C₄×C₇₃	—	8	2	no	—
152	C₂×C₄×C₇₃	#127	1	2	yes	—
153	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
154	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
155	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×Dic₇₃	—	4	2	no	—
157	C₂×Dic₇₃	—	4	2	no	—
158	Dic₁₄₆	—	4	4	no	—
159	Dic₁₄₆	—	4	4	no	—
160	C₈×C₇₃	—	2	2	no	—
161	C₈×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
162	#10596831	—	2	4	no	—
163	#10596831	—	2	4	no	—
164	C₂²×C₇₃	—	2	1	no	—
165	C₂²×C₇₃	—	2	2	no	—
166	C₂²×C₇₃	—	8	1	no	—
167	C₂²×C₇₃	—	8	1	no	—
168	C₂²×C₇₃	—	8	1	no	—
169	C₂²×C₇₃	—	8	2	no	—
170	C₂²×C₇₃	—	8	2	no	—
171	C₂²×C₇₃	—	8	2	no	—
172	C₂²×C₇₃	—	16	4	no	—
173	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
174	C₄×C₇₃	—	2	1	no	—
175	C₄×C₇₃	—	2	2	no	—
176	C₄×C₇₃	—	8	1	no	—
177	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
178	C₄×C₇₃	#575	1	4	yes	—
179	Dic₇₃	—	8	2	no	—
180	Dic₇₃	—	8	2	no	—
181	#19749	—	73	1	no	maximal, Sylow
182	C₂×C₇₃	—	4	1	no	—
183	C₂×C₇₃	—	16	1	no	—
184	C₂×C₇₃	—	16	1	no	—
185	C₂×C₇₃	—	16	1	no	—
186	C₂×C₇₃	#3151	1	1	yes	characteristic
187	C₂×C₇₃	#3153	1	2	yes	—
188	#3185	—	73	1	no	—
189	#3182	—	73	2	no	—
190	#3151	D₇₃	1	1	yes	characteristic
191	#3138	—	73	2	no	—
192	#1722	—	73	1	no	—
193	#1719	—	73	2	no	—
194	#1717	—	73	2	no	—
195	#1715	—	73	1	no	—
196	#1709	—	73	1	no	—
197	#1708	—	73	1	no	—
198	#1455	—	73	1	no	—
199	C₇₃	#19749	1	1	yes	characteristic, complemented, Sylow
200	C₂×AGL(1,ℤ/8)	—	2	2	no	—
201	#533	—	73	2	no	—
202	#532	C₂×D₇₃	1	1	yes	characteristic
203	C₂×#127	C₂×D₇₃	1	1	yes	characteristic
204	#500	—	73	2	no	—
205	#498	—	73	1	no	—
206	#496	—	2	2	no	—
207	#494	—	73	1	no	—
208	#493	C₂×D₇₃	1	1	yes	characteristic
209	#493	C₂×D₇₃	1	1	yes	characteristic
210	#492	—	73	2	no	—
211	#492	C₂×D₇₃	1	1	yes	characteristic
212	#492	C₂×D₇₃	1	1	yes	characteristic
213	#490	—	146	4	no	—
214	#487	—	73	2	no	—
215	#487	—	73	2	no	—
216	#486	—	73	2	no	—
217	#486	—	73	2	no	—
218	#486	—	73	2	no	—
219	#486	—	73	2	no	—
220	#486	—	73	2	no	—
221	#470	—	146	2	no	—
222	#468	—	146	4	no	—
223	#441	—	146	2	no	—
224	#440	—	146	4	no	—
225	#418	—	146	4	no	—
226	C₄×M₁₆	—	73	2	no	—
227	C₄×M₁₆	C₂×D₇₃	1	1	yes	characteristic
228	#322	—	73	1	no	—
229	#322	—	73	2	no	—
230	C₈²	—	73	1	no	—
231	C₂×#41	—	2	1	no	—
232	C₂²×D₄	—	2	2	no	—
233	C₂²×D₄	—	2	1	no	—
234	AGL(1,ℤ/8)	—	4	8	no	—
235	C₂×Q₁₆	—	146	2	no	—
236	C₂×SD₁₆	—	2	2	no	—
237	C₂×SD₁₆	—	2	2	no	—
238	C₂×SD₁₆	—	146	4	no	—
239	C₂×D₈	—	2	2	no	—
240	C₂×D₈	—	2	2	no	—
241	C₂×M₁₆	—	73	4	no	—
242	C₂×M₁₆	C₂²⋊D₇₃	1	2	yes	—
243	C₄⋊Q₈	—	73	2	no	—
244	C₄⋊Q₈	—	73	2	no	—
245	C₄⋊Q₈	C₂²×D₇₃	1	1	yes	characteristic
246	#127	—	2	2	no	—
247	#127	—	2	1	no	—
248	#127	C₂²×D₇₃	1	1	yes	characteristic
249	#127	C₂²×D₇₃	1	1	yes	characteristic
250	#127	C₂²×D₇₃	1	1	yes	characteristic
251	#125	—	146	2	no	—
252	#124	—	2	1	no	—
253	C₂²⋊Q₈	—	146	2	no	—
254	C₂²⋊Q₈	—	146	2	no	—
255	#121	—	2	2	no	—
256	C₄×Q₈	—	146	2	no	—
257	C₄×Q₈	—	146	2	no	—
258	C₄×D₄	—	2	2	no	—
259	#117	—	146	2	no	—
260	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
261	#107	—	146	4	no	—
262	#106	—	146	2	no	—
263	#103	—	146	4	no	—
264	#103	—	146	4	no	—
265	#102	—	146	4	no	—
266	#102	—	146	4	no	—
267	#102	—	146	4	no	—
268	#102	—	146	4	no	—
269	#102	—	146	4	no	—
270	#97	—	73	4	no	—
271	#97	C₂²⋊D₇₃	1	2	yes	—
272	C₄×C₈	—	73	2	no	—
273	C₄×C₈	—	73	2	no	—
274	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
276	C₂⁴	—	4	1	no	—
277	#41	—	4	4	no	—
278	C₂×Q₈	—	2	1	no	—
279	C₂×Q₈	—	146	2	no	—
280	C₂×Q₈	—	146	2	no	—
281	C₂×D₄	—	2	2	no	—
282	C₂×D₄	—	2	2	no	—
283	C₂×D₄	—	2	2	no	—
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	2	1	no	—
286	C₂×D₄	—	2	1	no	—
287	C₂×D₄	—	4	4	no	—
288	C₂×D₄	—	4	8	no	—
289	C₂×D₄	—	4	1	no	—
290	C₂×D₄	—	4	2	no	—
291	C₂×D₄	—	4	1	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	D₄×D₇₃	1	2	yes	—
294	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
295	Q₁₆	—	292	4	no	—
296	SD₁₆	—	4	4	no	—
297	SD₁₆	—	4	4	no	—
298	SD₁₆	—	292	8	no	—
299	D₈	—	4	4	no	—
300	D₈	—	4	4	no	—
301	M₁₆	—	2	4	no	—
302	M₁₆	—	292	4	no	—
303	C₂×C₈	—	73	4	no	—
304	C₂×C₈	—	73	4	no	—
305	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
306	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
307	C₄⋊C₄	—	2	2	no	—
308	C₄⋊C₄	—	292	2	no	—
309	C₄⋊C₄	—	292	2	no	—
310	C₄⋊C₄	—	146	2	no	—
311	C₄⋊C₄	—	146	2	no	—
312	C₄⋊C₄	—	146	2	no	—
313	C₄⋊C₄	—	146	2	no	—
314	C₂²⋊C₄	—	4	2	no	—
315	C₂²⋊C₄	—	292	2	no	—
316	C₄²	—	292	2	no	—
317	C₄²	D₄×D₇₃	1	2	yes	—
318	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
319	C₄²	C₂³×D₇₃	1	1	yes	characteristic
320	C₂³	—	4	1	no	—
321	C₂³	—	4	1	no	—
322	C₂³	—	4	1	no	—
323	C₂³	—	8	2	no	—
324	C₂³	—	8	4	no	—
325	C₂³	#14556587	1	1	yes	characteristic
326	Q₈	—	4	2	no	—
327	Q₈	—	292	4	no	—
328	Q₈	—	292	4	no	—
329	D₄	—	4	4	no	—
330	D₄	—	4	4	no	—
331	D₄	—	4	4	no	—
332	D₄	—	4	2	no	—
333	D₄	—	4	2	no	—
334	D₄	—	4	2	no	—
335	D₄	—	8	2	no	—
336	D₄	—	8	4	no	—
337	D₄	—	8	2	no	—
338	C₂×C₄	—	2	4	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	8	2	no	—
342	C₂×C₄	—	292	2	no	—
343	C₂×C₄	—	292	2	no	—
344	C₂×C₄	#14556587	1	2	yes	—
345	C₂×C₄	#14556587	1	1	yes	characteristic
346	C₂×C₄	C₂×D₄×D₇₃	1	2	yes	—
347	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
348	C₈	—	2	2	no	—
349	C₈	—	146	4	no	—
350	C₈	—	146	4	no	—
351	C₈	#14556617	1	4	yes	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	8	1	no	—
355	C₂²	—	8	1	no	—
356	C₂²	—	8	1	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	8	2	no	—
359	C₂²	—	8	2	no	—
360	C₂²	—	16	4	no	—
361	C₂²	C₂×#14556587	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	2	no	—
364	C₄	—	8	1	no	—
365	C₄	—	584	2	no	—
366	C₄	—	584	2	no	—
367	C₄	C₂×#14556617	1	2	yes	—
368	C₄	#17989383	1	4	yes	—
369	C₂	—	4	1	no	—
370	C₂	—	16	1	no	—
371	C₂	—	16	1	no	—
372	C₂	—	16	1	no	—
373	C₂	#22620381	1	1	yes	characteristic
374	C₂	#22620393	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14556587
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting, prime radical	C₇₃×#3151	C₂
socle	C₂²×C₇₃	C₂×#127
hypercenter	#3151	D₇₃
hypocenter	C₇₃	#19749
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3151

Complements

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

N	H ≅ G/N	type
1. #22620676	C₂	semidirect
2. #22620645	C₂	semidirect
3. #22620305	C₂	semidirect
4. #22609600	C₂	semidirect
5. #22609582	C₂	semidirect
6. #22609569	C₂	semidirect
7. #22609557	C₂	semidirect
8. #22609536	C₂	semidirect
9. #22609530	C₂	semidirect
10. #22608234	C₂	semidirect
13. #17988990	C₂²	semidirect
14. #17988985	C₂²	semidirect
15. #17988967	C₂²	semidirect
16. #17988959	C₂²	semidirect
18. #17988939	C₂²	semidirect
19. #17988938	C₂²	semidirect
24. #17988933	C₂²	semidirect
31. #17988309	C₂²	semidirect
32. #17988308	C₂²	semidirect
33. C₈×#10596831	C₂²	semidirect
199. C₇₃	#19749	semidirect

Maximal subgroups

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

subgroup	index
#22620676	2
#22620645	2
#22620305	2
#22609600	2
#22609582	2
#22609569	2
#22609557	2
#22609536	2
#22609530	2
#22608234	2
C₇₃×#3151	2
#19749	73