Subgroups — #33073989

Order 18688.

754 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620852	C₂	1	2	yes	maximal, complemented
2	#22620508	C₂	1	1	yes	maximal, characteristic
3	#500×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#22620321	C₂	1	2	yes	maximal, complemented
5	#22609582	C₂	1	2	yes	maximal, complemented
6	#22609581	C₂	1	2	yes	maximal, complemented
7	#22609541	C₂	1	1	yes	maximal, characteristic, complemented
8	#22609540	C₂	1	1	yes	maximal, characteristic, complemented
9	#22609520	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11924011	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#1706	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989135	C₂²	1	2	yes	—
13	C₄⋊Q₈×D₇₃	C₂²	1	2	yes	—
14	#17988992	C₂²	1	1	yes	characteristic
15	#17988992	C₂²	1	1	yes	characteristic
16	#17988991	—	2	2	no	—
17	#17988990	C₂²	1	2	yes	—
18	#17988959	C₂²	1	2	yes	—
19	#17988945	—	2	4	no	—
20	#17988939	C₂²	1	2	yes	—
21	#17988938	C₂²	1	2	yes	—
22	#17988938	C₂²	1	2	yes	—
23	#17988937	C₂²	1	2	yes	—
24	#17988933	C₂²	1	2	yes	—
25	#17988933	C₂²	1	2	yes	—
26	#17988820	—	2	2	no	—
27	#17988790	—	2	4	no	—
28	#17988724	—	2	4	no	—
29	D₈×Dic₇₃	—	2	2	no	—
30	#17988638	C₂²	1	1	yes	characteristic
31	#107×D₇₃	—	2	2	no	—
32	#17988585	—	2	4	no	—
33	C₄×#11924011	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
35	#17988309	C₂²	1	1	yes	characteristic
36	#17988308	C₂²	1	1	yes	characteristic
37	C₈×#10596831	C₂²	1	1	yes	characteristic
38	#500×C₇₃	C₂²	1	1	yes	characteristic
39	#493×C₇₃	C₂²	1	1	yes	characteristic
40	#492×C₇₃	C₂²	1	2	yes	—
41	#486×C₇₃	C₂²	1	2	yes	—
42	C₈²×C₇₃	C₂²	1	1	yes	characteristic
43	C₂×#11924027	—	2	2	no	—
44	C₂×Q₈×D₇₃	—	2	2	no	—
45	C₂×#11924016	—	2	4	no	—
46	C₂×#11924011	D₄	1	2	yes	—
47	C₂×C₈×D₇₃	D₄	1	2	yes	—
48	#14556591	C₂³	1	2	yes	—
49	#14556591	C₂³	1	2	yes	—
50	#14556590	—	2	2	no	—
51	#14556590	—	2	2	no	—
52	#14556589	—	2	2	no	—
53	#14556578	—	2	2	no	—
54	#14556562	—	2	2	no	—
55	#14556562	—	2	2	no	—
56	D₄×Dic₇₃	—	2	2	no	—
57	D₄×Dic₇₃	—	2	2	no	—
58	C₄⋊C₄×D₇₃	—	2	2	no	—
59	C₄⋊C₄×D₇₃	—	2	2	no	—
60	C₄²×D₇₃	C₂³	1	1	yes	characteristic
61	#14556525	—	2	2	no	—
62	#14556525	—	2	2	no	—
63	#14556524	—	2	4	no	—
64	#14556520	D₄	1	2	yes	—
65	#14556516	—	2	4	no	—
66	#14556516	—	2	4	no	—
67	#14556515	—	2	4	no	—
68	#14556514	—	2	4	no	—
69	#14556512	—	2	4	no	—
70	#14556512	—	2	4	no	—
71	#14556503	D₄	1	2	yes	—
72	#14556501	D₄	1	2	yes	—
73	C₄×#10596831	C₂³	1	1	yes	characteristic
74	C₄×#10596831	C₂³	1	1	yes	characteristic
75	C₄×#10596831	C₂³	1	1	yes	characteristic
76	C₈×Dic₇₃	D₄	1	2	yes	—
77	C₂×SD₁₆×C₇₃	—	2	4	no	—
78	C₂×D₈×C₇₃	—	2	2	no	—
79	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
80	#127×C₇₃	C₂³	1	2	yes	—
81	#107×C₇₃	—	2	2	no	—
82	#102×C₇₃	—	2	4	no	—
83	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
86	#11924027	—	4	8	no	—
87	C₂×Dic₁₄₆	—	2	2	no	—
88	C₂×Dic₁₄₆	—	2	2	no	—
89	C₂×Dic₁₄₆	—	4	2	no	—
90	Q₈×D₇₃	—	8	4	no	—
91	C₂×C₂²⋊D₇₃	—	4	2	no	—
92	C₂²×Dic₇₃	—	4	2	no	—
93	C₂×C₄×D₇₃	—	4	2	no	—
94	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
97	#11924016	—	8	4	no	—
98	#11924011	—	2	4	no	—
99	C₂×#10596831	C₂×D₄	1	2	yes	—
100	C₂×#10596831	C₂×D₄	1	2	yes	—
101	C₂×#10596831	C₂×D₄	1	2	yes	—
102	C₈×D₇₃	—	2	4	no	—
103	C₄⋊Dic₇₃	—	2	2	no	—
104	C₄⋊Dic₇₃	—	2	2	no	—
105	C₄⋊Dic₇₃	—	2	2	no	—
106	C₄⋊Dic₇₃	—	2	2	no	—
107	#11924007	—	4	2	no	—
108	#11924007	—	4	2	no	—
109	C₂²⋊Dic₇₃	—	4	2	no	—
110	C₂²⋊Dic₇₃	—	4	2	no	—
111	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
112	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
113	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
114	C₂×Q₈×C₇₃	—	2	2	no	—
115	C₂×D₄×C₇₃	—	2	2	no	—
116	C₂×D₄×C₇₃	—	2	2	no	—
117	C₂×D₄×C₇₃	—	2	2	no	—
118	SD₁₆×C₇₃	—	8	4	no	—
119	D₈×C₇₃	—	4	4	no	—
120	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
123	C₄⋊C₄×C₇₃	—	2	2	no	—
124	C₄⋊C₄×C₇₃	—	2	2	no	—
125	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
126	C₂³×C₇₃	—	4	2	no	—
127	C₂²×D₇₃	#127	1	1	yes	characteristic
128	Q₈×C₇₃	—	4	4	no	—
129	D₄×C₇₃	—	4	4	no	—
130	D₄×C₇₃	—	4	4	no	—
131	D₄×C₇₃	—	4	4	no	—
132	C₂×C₄×C₇₃	—	4	2	no	—
133	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
136	C₂²⋊D₇₃	—	8	4	no	—
137	C₂×Dic₇₃	—	4	2	no	—
138	C₂×Dic₇₃	—	4	2	no	—
139	C₂×Dic₇₃	—	8	4	no	—
140	C₂×Dic₇₃	#127	1	1	yes	characteristic
141	C₂×Dic₇₃	#127	1	1	yes	characteristic
142	C₂×Dic₇₃	#127	1	1	yes	characteristic
143	C₄×D₇₃	—	2	2	no	—
144	C₄×D₇₃	—	2	2	no	—
145	C₄×D₇₃	—	8	4	no	—
146	C₄×D₇₃	#135	1	4	yes	—
147	Dic₁₄₆	—	4	4	no	—
148	Dic₁₄₆	—	4	4	no	—
149	Dic₁₄₆	—	8	4	no	—
150	C₈×C₇₃	—	2	2	no	—
151	C₈×C₇₃	—	2	2	no	—
152	C₈×C₇₃	#137	1	4	yes	—
153	#10596831	—	2	2	no	—
154	#10596831	—	2	2	no	—
155	#10596831	—	2	2	no	—
156	C₂²×C₇₃	—	8	2	no	—
157	C₂²×C₇₃	—	8	2	no	—
158	C₂²×C₇₃	—	8	2	no	—
159	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
160	C₂×D₇₃	—	2	1	no	—
161	C₂×D₇₃	—	2	1	no	—
162	C₂×D₇₃	#495	1	2	yes	—
163	C₄×C₇₃	—	8	2	no	—
164	C₄×C₇₃	C₂×#135	1	2	yes	—
165	C₄×C₇₃	C₂×#137	1	2	yes	—
166	C₄×C₇₃	#576	1	2	yes	—
167	Dic₇₃	—	2	1	no	—
168	Dic₇₃	—	2	1	no	—
169	Dic₇₃	—	8	2	no	—
170	Dic₇₃	—	8	2	no	—
171	Dic₇₃	#495	1	2	yes	—
172	#19638	—	73	1	no	maximal, Sylow
173	C₂×C₇₃	—	16	2	no	—
174	C₂×C₇₃	C₂×#495	1	1	yes	characteristic
175	C₂×C₇₃	#3152	1	1	yes	characteristic
176	C₂×C₇₃	#3155	1	1	yes	characteristic
177	D₇₃	—	2	2	no	—
178	#3206	—	73	2	no	—
179	#3169	—	73	1	no	—
180	C₂×#500	—	73	1	no	—
181	#3141	—	73	2	no	—
182	#1719	—	73	2	no	—
183	#1719	—	73	2	no	—
184	#1710	—	73	1	no	—
185	#1710	—	73	1	no	—
186	#1706	—	73	1	no	—
187	#1706	D₇₃	1	1	yes	characteristic, complemented
188	C₈×M₁₆	—	73	1	no	—
189	C₇₃	#19638	1	1	yes	characteristic, complemented, Sylow
190	#532	—	73	2	no	—
191	C₂×C₄⋊Q₈	—	73	2	no	—
192	#500	—	73	2	no	—
193	#500	—	73	1	no	—
194	#500	—	73	1	no	—
195	#500	—	146	2	no	—
196	#500	C₂×D₇₃	1	1	yes	characteristic
197	#493	C₂×D₇₃	1	1	yes	characteristic
198	#492	—	73	2	no	—
199	#492	C₂×D₇₃	1	2	yes	—
200	#489	—	146	4	no	—
201	#487	—	73	2	no	—
202	#487	—	73	2	no	—
203	#487	—	73	2	no	—
204	#487	—	73	2	no	—
205	#486	—	73	2	no	—
206	#486	—	73	2	no	—
207	#486	C₂×D₇₃	1	2	yes	—
208	#466	—	146	2	no	—
209	#461	—	146	4	no	—
210	#448	—	146	4	no	—
211	C₄×D₈	—	146	2	no	—
212	#431	—	73	1	no	—
213	C₂×#107	—	146	2	no	—
214	#417	—	146	4	no	—
215	C₄×M₁₆	—	73	1	no	—
216	C₂×C₄×C₈	—	73	1	no	—
217	#322	—	73	1	no	—
218	#322	—	73	1	no	—
219	C₈²	—	73	1	no	—
220	C₈²	C₂×D₇₃	1	1	yes	characteristic
221	C₂×#41	—	146	2	no	—
222	C₂²×Q₈	—	146	2	no	—
223	C₂×SD₁₆	—	2	4	no	—
224	C₂×SD₁₆	—	146	4	no	—
225	C₂×D₈	—	2	2	no	—
226	C₂×M₁₆	—	73	2	no	—
227	C₂²×C₈	—	73	2	no	—
228	C₄⋊Q₈	—	73	2	no	—
229	C₄⋊Q₈	—	73	2	no	—
230	C₄⋊Q₈	—	146	2	no	—
231	C₄⋊Q₈	—	146	2	no	—
232	C₄⋊Q₈	—	146	2	no	—
233	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
234	#127	C₂²×D₇₃	1	2	yes	—
235	#124	—	146	2	no	—
236	#121	—	146	2	no	—
237	#121	—	146	2	no	—
238	C₄×D₄	—	146	2	no	—
239	C₄×D₄	—	146	2	no	—
240	C₂×C₄⋊C₄	—	146	2	no	—
241	C₂×C₄⋊C₄	—	146	2	no	—
242	C₂×C₄²	—	73	1	no	—
243	#107	—	2	2	no	—
244	#107	—	146	4	no	—
245	#107	—	146	2	no	—
246	#107	—	146	2	no	—
247	C₄⋊C₈	—	73	2	no	—
248	#103	—	146	4	no	—
249	#103	—	146	4	no	—
250	#103	—	146	4	no	—
251	#103	—	146	4	no	—
252	#102	—	2	4	no	—
253	#102	—	146	4	no	—
254	#102	—	146	4	no	—
255	C₂²⋊C₈	—	73	2	no	—
256	#97	—	73	2	no	—
257	C₄×C₈	—	73	2	no	—
258	C₄×C₈	—	73	1	no	—
259	C₄×C₈	—	73	1	no	—
260	C₄×C₈	—	73	1	no	—
261	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
262	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
264	#41	—	292	8	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	292	2	no	—
267	C₂×Q₈	—	584	4	no	—
268	C₂×Q₈	—	146	2	no	—
269	C₂×Q₈	—	146	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	292	2	no	—
274	C₂²×C₄	—	292	2	no	—
275	C₂²×C₄	—	292	2	no	—
276	C₂²×C₄	—	73	1	no	—
277	C₂²×C₄	—	73	1	no	—
278	C₂²×C₄	—	73	1	no	—
279	SD₁₆	—	8	4	no	—
280	SD₁₆	—	584	4	no	—
281	D₈	—	4	4	no	—
282	M₁₆	—	146	4	no	—
283	C₂×C₈	—	73	2	no	—
284	C₂×C₈	—	73	2	no	—
285	C₂×C₈	—	73	2	no	—
286	C₂×C₈	—	146	4	no	—
287	C₂×C₈	D₄×D₇₃	1	2	yes	—
288	C₂×C₈	D₄×D₇₃	1	2	yes	—
289	C₂×C₈	D₄×D₇₃	1	2	yes	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	292	2	no	—
293	C₄⋊C₄	—	292	2	no	—
294	C₄⋊C₄	—	146	2	no	—
295	C₄⋊C₄	—	146	2	no	—
296	C₄⋊C₄	—	146	2	no	—
297	C₄⋊C₄	—	146	2	no	—
298	C₂²⋊C₄	—	292	2	no	—
299	C₂²⋊C₄	—	292	2	no	—
300	C₄²	—	73	1	no	—
301	C₄²	—	73	1	no	—
302	C₄²	—	73	1	no	—
303	C₄²	C₂³×D₇₃	1	1	yes	characteristic
304	C₂³	—	4	2	no	—
305	C₂³	—	73	1	no	—
306	Q₈	—	4	4	no	—
307	Q₈	—	292	4	no	—
308	Q₈	—	292	4	no	—
309	Q₈	—	584	4	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	4	no	—
313	D₄	—	584	4	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	292	2	no	—
316	C₂×C₄	—	292	2	no	—
317	C₂×C₄	—	584	4	no	—
318	C₂×C₄	—	584	4	no	—
319	C₂×C₄	—	73	1	no	—
320	C₂×C₄	—	73	4	no	—
321	C₂×C₄	—	73	1	no	—
322	C₂×C₄	—	73	1	no	—
323	C₂×C₄	—	146	2	no	—
324	C₂×C₄	—	146	2	no	—
325	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
326	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
327	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
328	C₈	—	2	2	no	—
329	C₈	—	2	2	no	—
330	C₈	—	146	2	no	—
331	C₈	—	146	2	no	—
332	C₈	—	146	2	no	—
333	C₈	#14556620	1	4	yes	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	73	2	no	—
338	C₂²	—	146	1	no	—
339	C₂²	—	146	1	no	—
340	C₂²	#127×D₇₃	1	1	yes	characteristic
341	C₄	—	8	2	no	—
342	C₄	—	584	2	no	—
343	C₄	—	584	2	no	—
344	C₄	—	73	2	no	—
345	C₄	—	146	1	no	—
346	C₄	—	146	1	no	—
347	C₄	#135×D₇₃	1	2	yes	—
348	C₄	C₂×#14556620	1	2	yes	—
349	C₄	#17989388	1	2	yes	—
350	C₂	—	16	2	no	—
351	C₂	—	146	2	no	—
352	C₂	#495×D₇₃	1	1	yes	characteristic
353	C₂	#22620385	1	1	yes	characteristic
354	C₂	#22620411	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₇₃
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1706	C₂
socle	C₂²×C₇₃	C₂×#127
prime radical	#22620508	C₂
hypercenter	#1706	D₇₃
hypocenter	C₇₃	#19638
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1706

Complements

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

N	H ≅ G/N	type
1. #22620852	C₂	semidirect
3. #500×D₇₃	C₂	semidirect
4. #22620321	C₂	semidirect
5. #22609582	C₂	semidirect
6. #22609581	C₂	semidirect
7. #22609541	C₂	semidirect
8. #22609540	C₂	semidirect
9. #22609520	C₂	semidirect
10. C₈×#11924011	C₂	semidirect
11. C₇₃×#1706	C₂	semidirect
187. #1706	D₇₃	semidirect
189. C₇₃	#19638	semidirect

Maximal subgroups

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

subgroup	index
#22620852	2
#22620508	2
#500×D₇₃	2
#22620321	2
#22609582	2
#22609581	2
#22609541	2
#22609540	2
#22609520	2
C₈×#11924011	2
C₇₃×#1706	2
#19638	73