Subgroups — #33075041

Order 18688.

730 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22621082	C₂	1	2	yes	maximal, complemented
2	#22621063	C₂	1	2	yes	maximal
3	#22620676	C₂	1	1	yes	maximal, characteristic, complemented
4	#22609606	C₂	1	2	yes	maximal, complemented
5	#22609600	C₂	1	1	yes	maximal, characteristic, complemented
6	#22609582	C₂	1	2	yes	maximal, complemented
7	#22609562	C₂	1	1	yes	maximal, characteristic, complemented
8	#22609547	C₂	1	1	yes	maximal, characteristic, complemented
9	#22609530	C₂	1	1	yes	maximal, characteristic, complemented
10	#22608228	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#3152	C₂	1	1	yes	maximal, characteristic
12	#17989144	C₂²	1	2	yes	—
13	#17988990	C₂²	1	2	yes	—
14	#17988990	C₂²	1	2	yes	—
15	#17988990	C₂²	1	2	yes	—
16	#17988985	C₂²	1	1	yes	characteristic
17	#17988983	C₂²	1	2	yes	—
18	#17988967	C₂²	1	1	yes	characteristic
19	#17988939	C₂²	1	2	yes	—
20	#17988939	C₂²	1	2	yes	—
21	#17988938	C₂²	1	2	yes	—
22	#17988933	C₂²	1	2	yes	—
23	#17988933	C₂²	1	2	yes	—
24	#17988922	—	2	4	no	—
25	#17988908	—	2	4	no	—
26	#17988887	—	2	4	no	—
27	#17988879	—	2	4	no	—
28	#17988845	—	2	2	no	—
29	#17988695	—	2	2	no	—
30	#17988640	C₂²	1	2	yes	—
31	#17988309	C₂²	1	2	yes	—
32	#17988308	C₂²	1	1	yes	characteristic
33	C₈×#10596831	C₂²	1	1	yes	characteristic
34	C₂×#137×C₇₃	—	2	2	no	—
35	#532×C₇₃	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
37	#497×C₇₃	—	2	2	no	—
38	#494×C₇₃	C₂²	1	1	yes	characteristic
39	#494×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	#14556591	C₂³	1	2	yes	—
45	#14556591	C₂³	1	2	yes	—
46	#14556583	—	2	2	no	—
47	#14556570	—	2	2	no	—
48	#14556570	—	2	2	no	—
49	C₄×Dic₁₄₆	—	2	2	no	—
50	C₄×Dic₁₄₆	—	2	2	no	—
51	#14556550	—	2	2	no	—
52	#14556524	—	2	4	no	—
53	#14556524	—	2	4	no	—
54	#14556524	—	2	4	no	—
55	#14556523	—	2	2	no	—
56	#14556522	—	2	4	no	—
57	C₄⋊#10596831	D₄	1	4	yes	—
58	#14556516	—	2	4	no	—
59	#14556514	—	2	4	no	—
60	#14556514	—	2	4	no	—
61	#14556512	—	2	4	no	—
62	#14556512	—	2	4	no	—
63	C₂²⋊#10596831	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×Q₈×C₇₃	—	2	1	no	—
68	#137×C₇₃	—	4	8	no	—
69	C₂×Q₁₆×C₇₃	—	2	2	no	—
70	C₂×Q₁₆×C₇₃	—	2	2	no	—
71	C₂×SD₁₆×C₇₃	—	2	2	no	—
72	C₂×SD₁₆×C₇₃	—	2	2	no	—
73	C₂×M₁₆×C₇₃	D₄	1	2	yes	—
74	C₄⋊Q₈×C₇₃	—	2	2	no	—
75	C₄⋊Q₈×C₇₃	—	2	1	no	—
76	C₄⋊Q₈×C₇₃	C₂³	1	1	yes	characteristic
77	C₄⋊Q₈×C₇₃	C₂³	1	1	yes	characteristic
78	C₄⋊Q₈×C₇₃	C₂³	1	1	yes	characteristic
79	#127×C₇₃	C₂³	1	1	yes	characteristic
80	#124×C₇₃	—	2	1	no	—
81	#121×C₇₃	—	2	2	no	—
82	C₄×D₄×C₇₃	—	2	2	no	—
83	C₂×C₄⋊C₄×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	C₂×#10596831	C₂×D₄	1	4	yes	—
92	C₂×#10596831	C₂×D₄	1	4	yes	—
93	C₄⋊Dic₇₃	—	2	2	no	—
94	C₄⋊Dic₇₃	—	2	2	no	—
95	C₄⋊Dic₇₃	—	2	2	no	—
96	C₄⋊Dic₇₃	—	2	2	no	—
97	#11924007	—	4	2	no	—
98	#11924007	—	4	2	no	—
99	C₂²⋊Dic₇₃	—	4	2	no	—
100	C₄×Dic₇₃	—	4	2	no	—
101	#41×C₇₃	—	4	4	no	—
102	C₂×Q₈×C₇₃	—	2	1	no	—
103	C₂×Q₈×C₇₃	—	2	1	no	—
104	C₂×Q₈×C₇₃	—	2	1	no	—
105	C₂×Q₈×C₇₃	—	4	4	no	—
106	C₂×Q₈×C₇₃	—	4	1	no	—
107	C₂×D₄×C₇₃	—	2	2	no	—
108	C₂×D₄×C₇₃	—	2	1	no	—
109	C₂×D₄×C₇₃	—	4	1	no	—
110	C₂²×C₄×C₇₃	—	4	1	no	—
111	C₂²×C₄×C₇₃	—	4	1	no	—
112	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
114	Q₁₆×C₇₃	—	4	4	no	—
115	Q₁₆×C₇₃	—	4	4	no	—
116	SD₁₆×C₇₃	—	4	4	no	—
117	SD₁₆×C₇₃	—	4	4	no	—
118	M₁₆×C₇₃	—	2	4	no	—
119	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
120	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
121	C₄⋊C₄×C₇₃	—	2	2	no	—
122	C₄⋊C₄×C₇₃	—	2	2	no	—
123	C₄⋊C₄×C₇₃	—	2	2	no	—
124	C₄⋊C₄×C₇₃	—	4	2	no	—
125	C₂²⋊C₄×C₇₃	—	4	2	no	—
126	C₄²×C₇₃	C₂×D₄	1	2	yes	—
127	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
129	C₂³×C₇₃	—	4	1	no	—
130	C₂³×C₇₃	#127	1	1	yes	characteristic
131	Q₈×C₇₃	—	4	2	no	—
132	Q₈×C₇₃	—	4	2	no	—
133	Q₈×C₇₃	—	4	2	no	—
134	Q₈×C₇₃	—	8	2	no	—
135	D₄×C₇₃	—	4	2	no	—
136	D₄×C₇₃	—	4	4	no	—
137	D₄×C₇₃	—	8	2	no	—
138	C₂×C₄×C₇₃	—	2	2	no	—
139	C₂×C₄×C₇₃	—	2	4	no	—
140	C₂×C₄×C₇₃	—	4	1	no	—
141	C₂×C₄×C₇₃	—	4	1	no	—
142	C₂×C₄×C₇₃	—	4	1	no	—
143	C₂×C₄×C₇₃	—	8	2	no	—
144	C₂×C₄×C₇₃	—	8	2	no	—
145	C₂×C₄×C₇₃	#127	1	2	yes	—
146	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
147	C₂×C₄×C₇₃	C₂²×D₄	1	2	yes	—
148	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×Dic₇₃	—	4	2	no	—
150	C₂×Dic₇₃	—	4	2	no	—
151	Dic₁₄₆	—	4	4	no	—
152	Dic₁₄₆	—	4	4	no	—
153	C₈×C₇₃	—	2	2	no	—
154	C₈×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
155	#10596831	—	2	4	no	—
156	#10596831	—	2	4	no	—
157	C₂²×C₇₃	—	2	1	no	—
158	C₂²×C₇₃	—	2	2	no	—
159	C₂²×C₇₃	—	8	1	no	—
160	C₂²×C₇₃	—	8	2	no	—
161	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
162	C₄×C₇₃	—	2	1	no	—
163	C₄×C₇₃	—	2	2	no	—
164	C₄×C₇₃	—	8	1	no	—
165	C₄×C₇₃	—	8	1	no	—
166	C₄×C₇₃	—	8	1	no	—
167	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
168	C₄×C₇₃	#578	1	4	yes	—
169	Dic₇₃	—	8	2	no	—
170	Dic₇₃	—	8	2	no	—
171	#19731	—	73	1	no	maximal, Sylow
172	C₂×C₇₃	—	4	1	no	—
173	C₂×C₇₃	—	16	1	no	—
174	C₂×C₇₃	#3151	1	1	yes	characteristic
175	C₂×C₇₃	#3156	1	2	yes	—
176	#3230	—	73	2	no	—
177	#3228	—	73	2	no	—
178	#3185	—	73	1	no	—
179	#3152	D₇₃	1	1	yes	characteristic
180	#1723	—	73	2	no	—
181	#1722	—	73	1	no	—
182	#1719	—	73	2	no	—
183	#1716	—	73	1	no	—
184	#1713	—	73	1	no	—
185	#1708	—	73	1	no	—
186	#1453	—	73	1	no	—
187	C₇₃	#19731	1	1	yes	characteristic, complemented, Sylow
188	C₂×#137	—	2	2	no	—
189	#533	—	73	2	no	—
190	#532	C₂×D₇₃	1	1	yes	characteristic
191	C₂×C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
192	#500	—	73	2	no	—
193	#500	—	73	2	no	—
194	#500	—	73	2	no	—
195	#498	—	73	2	no	—
196	#498	—	73	1	no	—
197	#497	—	2	2	no	—
198	#494	—	73	1	no	—
199	#494	C₂×D₇₃	1	1	yes	characteristic
200	#494	C₂×D₇₃	1	1	yes	characteristic
201	#492	C₂×D₇₃	1	1	yes	characteristic
202	#492	C₂×D₇₃	1	1	yes	characteristic
203	#487	—	73	2	no	—
204	#487	—	73	2	no	—
205	#487	—	73	2	no	—
206	#486	—	73	2	no	—
207	#486	—	73	2	no	—
208	#484	—	146	4	no	—
209	#482	—	146	4	no	—
210	#478	—	146	4	no	—
211	#477	—	146	4	no	—
212	#470	—	146	2	no	—
213	#441	—	146	2	no	—
214	#431	—	73	2	no	—
215	C₄×M₁₆	C₂×D₇₃	1	1	yes	characteristic
216	#322	—	73	1	no	—
217	#322	—	73	2	no	—
218	C₈²	—	73	1	no	—
219	C₂×#41	—	2	1	no	—
220	C₂²×Q₈	—	2	1	no	—
221	#137	—	4	8	no	—
222	C₂×Q₁₆	—	2	2	no	—
223	C₂×Q₁₆	—	2	2	no	—
224	C₂×Q₁₆	—	146	2	no	—
225	C₂×SD₁₆	—	2	2	no	—
226	C₂×SD₁₆	—	2	2	no	—
227	C₂×M₁₆	C₂²⋊D₇₃	1	2	yes	—
228	C₄⋊Q₈	—	2	2	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	73	2	no	—
231	C₄⋊Q₈	—	73	2	no	—
232	C₄⋊Q₈	C₂²×D₇₃	1	1	yes	characteristic
233	C₄⋊Q₈	C₂²×D₇₃	1	1	yes	characteristic
234	C₄⋊Q₈	C₂²×D₇₃	1	1	yes	characteristic
235	#127	C₂²×D₇₃	1	1	yes	characteristic
236	#125	—	146	2	no	—
237	#124	—	2	1	no	—
238	C₂²⋊Q₈	—	146	2	no	—
239	C₂²⋊Q₈	—	146	2	no	—
240	#121	—	2	2	no	—
241	C₄×Q₈	—	146	2	no	—
242	C₄×Q₈	—	146	2	no	—
243	C₄×D₄	—	2	2	no	—
244	#117	—	146	2	no	—
245	C₂×C₄⋊C₄	—	2	2	no	—
246	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
247	#107	—	146	4	no	—
248	#107	—	146	4	no	—
249	#107	—	146	4	no	—
250	#106	—	146	4	no	—
251	#106	—	146	2	no	—
252	C₄⋊C₈	—	73	4	no	—
253	#103	—	146	4	no	—
254	#103	—	146	4	no	—
255	#103	—	146	4	no	—
256	#102	—	146	4	no	—
257	#102	—	146	4	no	—
258	C₂²⋊C₈	—	73	4	no	—
259	#97	C₂²⋊D₇₃	1	2	yes	—
260	C₄×C₈	—	73	2	no	—
261	C₄×C₈	—	73	2	no	—
262	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
264	#41	—	4	4	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	2	1	no	—
267	C₂×Q₈	—	2	1	no	—
268	C₂×Q₈	—	4	4	no	—
269	C₂×Q₈	—	4	1	no	—
270	C₂×Q₈	—	146	2	no	—
271	C₂×Q₈	—	146	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	1	no	—
274	C₂×D₄	—	4	1	no	—
275	C₂²×C₄	—	4	1	no	—
276	C₂²×C₄	—	4	1	no	—
277	C₂²×C₄	D₄×D₇₃	1	2	yes	—
278	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
279	Q₁₆	—	4	4	no	—
280	Q₁₆	—	4	4	no	—
281	Q₁₆	—	292	4	no	—
282	SD₁₆	—	4	4	no	—
283	SD₁₆	—	4	4	no	—
284	M₁₆	—	2	4	no	—
285	C₂×C₈	—	73	4	no	—
286	C₂×C₈	—	73	4	no	—
287	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
288	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	292	2	no	—
294	C₄⋊C₄	—	292	2	no	—
295	C₄⋊C₄	—	146	2	no	—
296	C₄⋊C₄	—	146	2	no	—
297	C₄⋊C₄	—	146	2	no	—
298	C₄⋊C₄	—	146	2	no	—
299	C₂²⋊C₄	—	4	2	no	—
300	C₂²⋊C₄	—	292	2	no	—
301	C₄²	—	292	2	no	—
302	C₄²	D₄×D₇₃	1	2	yes	—
303	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
304	C₄²	C₂³×D₇₃	1	1	yes	characteristic
305	C₂³	—	4	1	no	—
306	C₂³	#14556587	1	1	yes	characteristic
307	Q₈	—	4	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	8	2	no	—
311	Q₈	—	292	4	no	—
312	Q₈	—	292	4	no	—
313	D₄	—	4	4	no	—
314	D₄	—	4	2	no	—
315	D₄	—	8	2	no	—
316	C₂×C₄	—	2	4	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	8	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	292	2	no	—
324	C₂×C₄	—	292	2	no	—
325	C₂×C₄	#14556587	1	2	yes	—
326	C₂×C₄	#14556587	1	1	yes	characteristic
327	C₂×C₄	C₂×D₄×D₇₃	1	2	yes	—
328	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	146	4	no	—
331	C₈	—	146	4	no	—
332	C₈	#14556617	1	4	yes	—
333	C₂²	—	2	2	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	1	no	—
337	C₂²	C₂×#14556587	1	1	yes	characteristic
338	C₄	—	2	2	no	—
339	C₄	—	2	1	no	—
340	C₄	—	8	1	no	—
341	C₄	—	8	1	no	—
342	C₄	—	8	1	no	—
343	C₄	—	584	2	no	—
344	C₄	—	584	2	no	—
345	C₄	C₂×#14556617	1	2	yes	—
346	C₄	#17989403	1	4	yes	—
347	C₂	—	4	1	no	—
348	C₂	—	16	1	no	—
349	C₂	#22620381	1	1	yes	characteristic
350	C₂	#22620424	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#14556587
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#3152	C₂
socle	C₂²×C₇₃	C₂×#127
prime radical	#532×C₇₃	C₂²
hypercenter	#3152	D₇₃
hypocenter	C₇₃	#19731
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3152

Complements

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

N	H ≅ G/N	type
1. #22621082	C₂	semidirect
3. #22620676	C₂	semidirect
4. #22609606	C₂	semidirect
5. #22609600	C₂	semidirect
6. #22609582	C₂	semidirect
7. #22609562	C₂	semidirect
8. #22609547	C₂	semidirect
9. #22609530	C₂	semidirect
10. #22608228	C₂	semidirect
187. C₇₃	#19731	semidirect

Maximal subgroups

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

subgroup	index
#22621082	2
#22621063	2
#22620676	2
#22609606	2
#22609600	2
#22609582	2
#22609562	2
#22609547	2
#22609530	2
#22608228	2
C₇₃×#3152	2
#19731	73