Subgroups — #33073939

Order 18688.

750 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22621083	C₂	1	2	yes	maximal
2	#22620515	C₂	1	1	yes	maximal, characteristic
3	#22620474	C₂	1	2	yes	maximal
4	#494×D₇₃	C₂	1	1	yes	maximal, characteristic
5	#22609609	C₂	1	2	yes	maximal, complemented
6	#22609607	C₂	1	2	yes	maximal, complemented
7	#22609547	C₂	1	1	yes	maximal, characteristic, complemented
8	#22609528	C₂	1	1	yes	maximal, characteristic, complemented
9	#22609525	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11924011	C₂	1	1	yes	maximal, characteristic
11	C₇₃×#1713	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₃	—	2	2	no	—
13	#17989142	C₂²	1	2	yes	—
14	C₄⋊Q₈×D₇₃	C₂²	1	2	yes	—
15	#17988995	—	2	4	no	—
16	#17988992	C₂²	1	1	yes	characteristic
17	#17988990	C₂²	1	2	yes	—
18	#17988990	C₂²	1	2	yes	—
19	#17988985	C₂²	1	2	yes	—
20	#17988983	C₂²	1	2	yes	—
21	#17988967	C₂²	1	1	yes	characteristic
22	#17988966	C₂²	1	2	yes	—
23	#17988965	—	2	2	no	—
24	#17988939	C₂²	1	2	yes	—
25	#17988938	C₂²	1	2	yes	—
26	#17988937	C₂²	1	2	yes	—
27	#17988924	—	2	4	no	—
28	#17988881	—	2	4	no	—
29	#17988825	—	2	2	no	—
30	Q₁₆×Dic₇₃	—	2	2	no	—
31	#17988638	C₂²	1	1	yes	characteristic
32	#17988628	—	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	2	yes	—
39	#500×C₇₃	C₂²	1	2	yes	—
40	#494×C₇₃	C₂²	1	1	yes	characteristic
41	#494×C₇₃	C₂²	1	1	yes	characteristic
42	C₈²×C₇₃	C₂²	1	1	yes	characteristic
43	C₂×Q₈×D₇₃	—	2	2	no	—
44	C₂×Dic₂₉₂	—	2	2	no	—
45	C₂×#11924018	—	2	4	no	—
46	Q₁₆×D₇₃	—	4	8	no	—
47	C₂×#11924011	D₄	1	2	yes	—
48	C₂×C₈×D₇₃	D₄	1	2	yes	—
49	#14556591	C₂³	1	2	yes	—
50	#14556591	C₂³	1	2	yes	—
51	#14556590	—	2	2	no	—
52	#14556590	—	2	2	no	—
53	#14556589	—	2	2	no	—
54	#14556582	—	2	2	no	—
55	#14556571	—	2	2	no	—
56	#14556567	—	2	2	no	—
57	Q₈×Dic₇₃	—	2	2	no	—
58	#14556556	—	2	2	no	—
59	#14556549	—	2	2	no	—
60	C₄⋊C₄×D₇₃	—	2	2	no	—
61	C₄⋊C₄×D₇₃	—	2	2	no	—
62	C₄²×D₇₃	C₂³	1	1	yes	characteristic
63	#14556525	—	2	2	no	—
64	#14556524	—	2	4	no	—
65	#14556524	—	2	4	no	—
66	#14556523	—	2	4	no	—
67	#14556522	—	2	4	no	—
68	#14556520	D₄	1	2	yes	—
69	#14556516	—	2	4	no	—
70	#14556515	—	2	4	no	—
71	#14556514	—	2	4	no	—
72	#14556503	D₄	1	2	yes	—
73	#14556501	D₄	1	2	yes	—
74	C₄×#10596831	C₂³	1	1	yes	characteristic
75	C₄×#10596831	C₂³	1	1	yes	characteristic
76	C₄×#10596831	C₂³	1	1	yes	characteristic
77	C₈×Dic₇₃	D₄	1	2	yes	—
78	C₂×Q₁₆×C₇₃	—	2	2	no	—
79	C₂×Q₁₆×C₇₃	—	2	2	no	—
80	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
81	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
82	#107×C₇₃	—	2	4	no	—
83	#107×C₇₃	—	2	4	no	—
84	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
87	C₂×Dic₁₄₆	—	2	2	no	—
88	C₂×Dic₁₄₆	—	2	2	no	—
89	C₂×Dic₁₄₆	—	4	2	no	—
90	Q₈×D₇₃	—	4	8	no	—
91	C₂×C₄×D₇₃	—	4	2	no	—
92	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
95	Dic₂₉₂	—	4	4	no	—
96	#11924018	—	4	8	no	—
97	#11924011	Q₁₆	1	8	yes	complemented
98	C₂×#10596831	C₂×D₄	1	2	yes	—
99	C₂×#10596831	C₂×D₄	1	2	yes	—
100	C₂×#10596831	C₂×D₄	1	2	yes	—
101	C₈×D₇₃	—	2	4	no	—
102	C₄⋊Dic₇₃	—	2	2	no	—
103	C₄⋊Dic₇₃	—	2	2	no	—
104	C₄⋊Dic₇₃	—	2	2	no	—
105	C₄⋊Dic₇₃	—	2	2	no	—
106	C₄⋊Dic₇₃	—	4	2	no	—
107	#11924007	—	4	2	no	—
108	#11924007	—	4	2	no	—
109	#11924007	—	4	2	no	—
110	#11924005	—	4	2	no	—
111	C₄×Dic₇₃	—	4	2	no	—
112	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
113	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
114	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
115	C₂×Q₈×C₇₃	—	2	2	no	—
116	C₂×Q₈×C₇₃	—	2	2	no	—
117	Q₁₆×C₇₃	—	4	4	no	—
118	Q₁₆×C₇₃	—	4	4	no	—
119	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
120	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
121	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
122	C₄⋊C₄×C₇₃	—	2	2	no	—
123	C₄⋊C₄×C₇₃	—	2	2	no	—
124	C₄⋊C₄×C₇₃	—	2	2	no	—
125	C₄⋊C₄×C₇₃	—	2	2	no	—
126	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
127	C₂²×D₇₃	#127	1	1	yes	characteristic
128	Q₈×C₇₃	—	4	4	no	—
129	Q₈×C₇₃	—	4	4	no	—
130	C₂×C₄×C₇₃	—	4	2	no	—
131	C₂×C₄×C₇₃	—	4	2	no	—
132	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
134	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
135	C₂×Dic₇₃	—	4	2	no	—
136	C₂×Dic₇₃	—	4	2	no	—
137	C₂×Dic₇₃	#127	1	1	yes	characteristic
138	C₂×Dic₇₃	#127	1	1	yes	characteristic
139	C₂×Dic₇₃	#127	1	1	yes	characteristic
140	C₄×D₇₃	—	2	2	no	—
141	C₄×D₇₃	—	2	2	no	—
142	C₄×D₇₃	—	8	4	no	—
143	C₄×D₇₃	C₂×Q₁₆	1	4	yes	—
144	Dic₁₄₆	—	4	4	no	—
145	Dic₁₄₆	—	4	4	no	—
146	Dic₁₄₆	—	8	4	no	—
147	C₈×C₇₃	—	2	2	no	—
148	C₈×C₇₃	C₂×Q₁₆	1	4	yes	complemented
149	C₈×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
150	#10596831	—	2	2	no	—
151	#10596831	—	2	2	no	—
152	#10596831	C₂×Q₁₆	1	4	yes	complemented
153	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
154	C₂×D₇₃	—	2	1	no	—
155	C₂×D₇₃	—	2	1	no	—
156	C₂×D₇₃	#494	1	2	yes	—
157	C₄×C₇₃	—	8	2	no	—
158	C₄×C₇₃	—	8	2	no	—
159	C₄×C₇₃	C₂²×Q₁₆	1	2	yes	—
160	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
161	C₄×C₇₃	#578	1	2	yes	—
162	Dic₇₃	—	2	1	no	—
163	Dic₇₃	—	2	1	no	—
164	Dic₇₃	—	8	2	no	—
165	Dic₇₃	—	8	2	no	—
166	Dic₇₃	#494	1	2	yes	—
167	#19632	—	73	1	no	maximal, Sylow
168	C₂×C₇₃	C₂×#494	1	1	yes	characteristic
169	C₂×C₇₃	#3151	1	1	yes	characteristic
170	C₂×C₇₃	#3156	1	1	yes	characteristic
171	D₇₃	—	2	2	no	—
172	#3230	—	73	2	no	—
173	#3170	—	73	1	no	—
174	#3165	—	73	2	no	—
175	C₂×#494	—	73	1	no	—
176	#1723	—	73	2	no	—
177	#1723	—	73	2	no	—
178	#1713	—	73	1	no	—
179	#1713	D₇₃	1	1	yes	characteristic, complemented
180	#1707	—	73	1	no	—
181	#1707	—	73	1	no	—
182	C₈×M₁₆	—	73	1	no	—
183	C₇₃	#19632	1	1	yes	characteristic, complemented, Sylow
184	C₂²×Q₁₆	—	146	2	no	—
185	#533	—	73	2	no	—
186	C₂×C₄⋊Q₈	—	73	2	no	—
187	#501	—	146	4	no	—
188	#500	—	73	2	no	—
189	#500	—	73	2	no	—
190	#500	—	73	1	no	—
191	#500	C₂×D₇₃	1	2	yes	—
192	#500	C₂×D₇₃	1	2	yes	—
193	#498	—	73	2	no	—
194	#498	—	73	2	no	—
195	#494	—	73	2	no	—
196	#494	—	73	1	no	—
197	#494	—	146	2	no	—
198	#494	C₂×D₇₃	1	1	yes	characteristic
199	#494	C₂×D₇₃	1	1	yes	characteristic
200	#487	—	73	2	no	—
201	#487	—	73	2	no	—
202	#487	—	73	2	no	—
203	#484	—	146	4	no	—
204	#477	—	146	4	no	—
205	#467	—	146	2	no	—
206	C₄×Q₁₆	—	146	2	no	—
207	#431	—	73	1	no	—
208	#428	—	146	4	no	—
209	C₄×M₁₆	—	73	1	no	—
210	C₂×C₄×C₈	—	73	1	no	—
211	#322	—	73	1	no	—
212	#322	—	73	1	no	—
213	C₈²	—	73	1	no	—
214	C₈²	C₂×D₇₃	1	1	yes	characteristic
215	C₂²×Q₈	—	146	2	no	—
216	C₂×Q₁₆	—	2	2	no	—
217	C₂×Q₁₆	—	2	2	no	—
218	C₂×Q₁₆	—	292	8	no	—
219	C₂×Q₁₆	—	146	4	no	—
220	C₂×Q₁₆	—	146	2	no	—
221	C₂×M₁₆	—	73	2	no	—
222	C₂²×C₈	—	73	2	no	—
223	C₄⋊Q₈	—	73	2	no	—
224	C₄⋊Q₈	—	73	2	no	—
225	C₄⋊Q₈	—	146	2	no	—
226	C₄⋊Q₈	—	146	2	no	—
227	C₄⋊Q₈	—	146	2	no	—
228	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
229	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
230	#125	—	146	2	no	—
231	C₂²⋊Q₈	—	146	2	no	—
232	C₂²⋊Q₈	—	146	2	no	—
233	C₄×Q₈	—	146	2	no	—
234	C₄×Q₈	—	146	2	no	—
235	#117	—	146	2	no	—
236	C₂×C₄⋊C₄	—	146	2	no	—
237	C₂×C₄⋊C₄	—	146	2	no	—
238	C₂×C₄²	—	73	1	no	—
239	#107	—	2	4	no	—
240	#107	—	2	4	no	—
241	#107	—	146	4	no	—
242	#107	—	146	4	no	—
243	#107	—	146	2	no	—
244	#106	—	146	4	no	—
245	#106	—	146	4	no	—
246	C₄⋊C₈	—	73	2	no	—
247	#103	—	146	4	no	—
248	#103	—	146	4	no	—
249	#103	—	146	4	no	—
250	C₂²⋊C₈	—	73	2	no	—
251	#97	—	73	2	no	—
252	C₄×C₈	—	73	2	no	—
253	C₄×C₈	—	73	1	no	—
254	C₄×C₈	—	73	1	no	—
255	C₄×C₈	—	73	1	no	—
256	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
257	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
258	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	292	2	no	—
262	C₂×Q₈	—	292	8	no	—
263	C₂×Q₈	—	146	2	no	—
264	C₂×Q₈	—	146	2	no	—
265	C₂²×C₄	—	292	2	no	—
266	C₂²×C₄	—	73	1	no	—
267	C₂²×C₄	—	73	1	no	—
268	C₂²×C₄	—	73	1	no	—
269	Q₁₆	—	4	4	no	—
270	Q₁₆	—	4	4	no	—
271	Q₁₆	—	292	8	no	—
272	Q₁₆	—	292	4	no	—
273	M₁₆	—	73	8	no	—
274	C₂×C₈	—	73	2	no	—
275	C₂×C₈	—	73	2	no	—
276	C₂×C₈	—	73	2	no	—
277	C₂×C₈	—	146	4	no	—
278	C₂×C₈	D₄×D₇₃	1	2	yes	—
279	C₂×C₈	D₄×D₇₃	1	2	yes	—
280	C₂×C₈	D₄×D₇₃	1	2	yes	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	292	2	no	—
286	C₄⋊C₄	—	292	2	no	—
287	C₄⋊C₄	—	292	2	no	—
288	C₄⋊C₄	—	292	2	no	—
289	C₄⋊C₄	—	146	2	no	—
290	C₄⋊C₄	—	146	2	no	—
291	C₄⋊C₄	—	146	2	no	—
292	C₄⋊C₄	—	146	2	no	—
293	C₂²⋊C₄	—	292	2	no	—
294	C₄²	—	292	2	no	—
295	C₄²	—	73	1	no	—
296	C₄²	—	73	1	no	—
297	C₄²	—	73	1	no	—
298	C₄²	C₂³×D₇₃	1	1	yes	characteristic
299	C₂³	—	73	1	no	—
300	Q₈	—	4	4	no	—
301	Q₈	—	4	4	no	—
302	Q₈	—	292	4	no	—
303	Q₈	—	292	4	no	—
304	Q₈	—	584	4	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	292	2	no	—
308	C₂×C₄	—	292	2	no	—
309	C₂×C₄	—	584	4	no	—
310	C₂×C₄	—	73	1	no	—
311	C₂×C₄	—	73	4	no	—
312	C₂×C₄	—	73	1	no	—
313	C₂×C₄	—	73	1	no	—
314	C₂×C₄	—	146	2	no	—
315	C₂×C₄	—	146	2	no	—
316	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
317	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
318	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	73	4	no	—
321	C₈	—	146	2	no	—
322	C₈	—	146	2	no	—
323	C₈	Q₁₆×D₇₃	1	4	yes	complemented
324	C₈	#14556615	1	4	yes	—
325	C₂²	—	73	2	no	—
326	C₂²	—	146	1	no	—
327	C₂²	—	146	1	no	—
328	C₂²	#127×D₇₃	1	1	yes	characteristic
329	C₄	—	8	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	584	2	no	—
332	C₄	—	584	2	no	—
333	C₄	—	73	2	no	—
334	C₄	—	146	1	no	—
335	C₄	—	146	1	no	—
336	C₄	C₂×Q₁₆×D₇₃	1	2	yes	—
337	C₄	C₂×#14556615	1	2	yes	—
338	C₄	#17989400	1	2	yes	—
339	C₂	—	146	2	no	—
340	C₂	#494×D₇₃	1	1	yes	characteristic
341	C₂	#22620382	1	1	yes	characteristic
342	C₂	#22620419	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₇₃
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1713	C₂
socle	C₂²×C₇₃	C₂×#127
prime radical	C₂²×D₇₃	#127
hypercenter	#1713	D₇₃
hypocenter	C₇₃	#19632
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1713

Complements

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

N	H ≅ G/N	type
5. #22609609	C₂	semidirect
6. #22609607	C₂	semidirect
7. #22609547	C₂	semidirect
8. #22609528	C₂	semidirect
9. #22609525	C₂	semidirect
11. C₇₃×#1713	C₂	semidirect
97. #11924011	Q₁₆	semidirect
148. C₈×C₇₃	C₂×Q₁₆	semidirect
152. #10596831	C₂×Q₁₆	semidirect
179. #1713	D₇₃	semidirect
183. C₇₃	#19632	semidirect
323. C₈	Q₁₆×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22621083	2
#22620515	2
#22620474	2
#494×D₇₃	2
#22609609	2
#22609607	2
#22609547	2
#22609528	2
#22609525	2
C₈×#11924011	2
C₇₃×#1713	2
#19632	73