Subgroups — #33073920

Order 18688.

858 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22621053	C₂	1	2	yes	maximal, complemented
2	#22620511	C₂	1	1	yes	maximal, characteristic
3	#500×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#22620389	C₂	1	2	yes	maximal, complemented
5	#22609603	C₂	1	2	yes	maximal, complemented
6	#22609601	C₂	1	2	yes	maximal, complemented
7	#22609548	C₂	1	1	yes	maximal, characteristic, complemented
8	#22609524	C₂	1	1	yes	maximal, characteristic, complemented
9	#22609523	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11924011	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#1713	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556623	—	2	4	no	—
13	#17989138	C₂²	1	2	yes	—
14	C₄⋊Q₈×D₇₃	C₂²	1	2	yes	—
15	#17988992	C₂²	1	1	yes	characteristic
16	#17988991	—	2	2	no	—
17	#17988990	C₂²	1	2	yes	—
18	#17988990	C₂²	1	2	yes	—
19	#17988980	—	2	4	no	—
20	#17988966	C₂²	1	2	yes	—
21	#17988964	C₂²	1	1	yes	characteristic
22	#17988961	C₂²	1	2	yes	—
23	#17988960	C₂²	1	2	yes	—
24	#17988935	C₂²	1	2	yes	—
25	#17988935	C₂²	1	2	yes	—
26	#17988934	C₂²	1	2	yes	—
27	#17988902	—	2	4	no	—
28	#17988866	—	2	4	no	—
29	#17988824	—	2	2	no	—
30	#17988669	—	2	2	no	—
31	#17988638	C₂²	1	1	yes	characteristic
32	#107×D₇₃	—	2	2	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	#500×C₇₃	C₂²	1	1	yes	characteristic
40	#500×C₇₃	C₂²	1	2	yes	—
41	#494×C₇₃	C₂²	1	2	yes	—
42	C₈²×C₇₃	C₂²	1	1	yes	characteristic
43	C₂×#11924029	—	2	2	no	—
44	C₂×Q₈×D₇₃	—	2	2	no	—
45	#14556623	—	4	16	no	—
46	C₂×#11924018	—	2	4	no	—
47	C₂×#11924017	—	2	4	no	—
48	C₂×#11924015	—	2	4	no	—
49	C₂×D₅₈₄	—	2	2	no	—
50	C₂×#11924011	D₄	1	2	yes	—
51	C₂×C₈×D₇₃	D₄	1	2	yes	—
52	#14556591	C₂³	1	2	yes	—
53	#14556590	—	2	2	no	—
54	#14556590	—	2	2	no	—
55	#14556589	—	2	2	no	—
56	#14556588	C₂³	1	2	yes	—
57	#14556580	—	2	2	no	—
58	#14556566	—	2	2	no	—
59	#14556566	—	2	2	no	—
60	#14556553	—	2	2	no	—
61	#14556553	—	2	2	no	—
62	C₄⋊C₄×D₇₃	—	2	2	no	—
63	C₄⋊C₄×D₇₃	—	2	2	no	—
64	C₄²×D₇₃	C₂³	1	1	yes	characteristic
65	#14556525	—	2	2	no	—
66	#14556524	—	2	4	no	—
67	#14556524	—	2	4	no	—
68	#14556520	D₄	1	2	yes	—
69	#14556513	—	2	4	no	—
70	#14556511	—	2	4	no	—
71	#14556511	—	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	4	no	—
79	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
80	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
81	#107×C₇₃	—	2	4	no	—
82	#107×C₇₃	—	2	2	no	—
83	#107×C₇₃	—	2	2	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	#11924029	—	4	8	no	—
88	C₂×Dic₁₄₆	—	2	2	no	—
89	C₂×Dic₁₄₆	—	4	2	no	—
90	Q₈×D₇₃	—	4	8	no	—
91	C₂×D₂₉₂	—	2	2	no	—
92	C₂×D₂₉₂	—	2	2	no	—
93	C₂×D₂₉₂	—	2	2	no	—
94	C₂×D₂₉₂	—	4	2	no	—
95	C₂×C₄×D₇₃	—	4	2	no	—
96	C₂×C₄×D₇₃	—	4	2	no	—
97	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
100	#11924018	—	4	8	no	—
101	#11924017	—	4	8	no	—
102	#11924015	—	4	8	no	—
103	D₅₈₄	—	4	4	no	—
104	#11924011	D₈	1	8	yes	complemented
105	C₂×#10596831	C₂×D₄	1	2	yes	—
106	C₂×#10596831	C₂×D₄	1	2	yes	—
107	C₂×#10596831	C₂×D₄	1	2	yes	—
108	C₈×D₇₃	—	2	4	no	—
109	C₄⋊Dic₇₃	—	2	2	no	—
110	C₄⋊Dic₇₃	—	2	2	no	—
111	#11924007	—	4	2	no	—
112	#11924007	—	4	2	no	—
113	#11924005	—	4	2	no	—
114	#11924005	—	4	2	no	—
115	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
116	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
117	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
118	C₂×Q₈×C₇₃	—	2	2	no	—
119	C₂×Q₈×C₇₃	—	2	2	no	—
120	Q₁₆×C₇₃	—	4	8	no	—
121	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
123	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
124	C₄⋊C₄×C₇₃	—	2	2	no	—
125	C₄⋊C₄×C₇₃	—	2	2	no	—
126	C₄⋊C₄×C₇₃	—	2	2	no	—
127	C₄⋊C₄×C₇₃	—	2	2	no	—
128	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
129	C₂²×D₇₃	—	4	2	no	—
130	C₂²×D₇₃	#127	1	1	yes	characteristic
131	Q₈×C₇₃	—	4	4	no	—
132	Q₈×C₇₃	—	4	4	no	—
133	C₂×C₄×C₇₃	—	4	2	no	—
134	C₂×C₄×C₇₃	—	4	2	no	—
135	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×Dic₇₃	—	4	2	no	—
139	C₂×Dic₇₃	#127	1	1	yes	characteristic
140	C₂×Dic₇₃	#127	1	1	yes	characteristic
141	C₂×Dic₇₃	#127	1	1	yes	characteristic
142	D₂₉₂	—	4	4	no	—
143	D₂₉₂	—	4	4	no	—
144	D₂₉₂	—	4	4	no	—
145	D₂₉₂	—	8	4	no	—
146	C₄×D₇₃	—	2	2	no	—
147	C₄×D₇₃	—	2	2	no	—
148	C₄×D₇₃	—	8	4	no	—
149	C₄×D₇₃	—	8	4	no	—
150	C₄×D₇₃	C₂×D₈	1	4	yes	—
151	Dic₁₄₆	—	4	4	no	—
152	Dic₁₄₆	—	8	4	no	—
153	C₈×C₇₃	—	2	2	no	—
154	C₈×C₇₃	C₂×D₈	1	4	yes	—
155	C₈×C₇₃	#137	1	4	yes	complemented
156	#10596831	—	2	2	no	—
157	#10596831	—	2	2	no	—
158	#10596831	C₂×D₈	1	4	yes	—
159	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
160	C₂×D₇₃	—	2	1	no	—
161	C₂×D₇₃	—	2	1	no	—
162	C₂×D₇₃	—	8	2	no	—
163	C₂×D₇₃	—	8	2	no	—
164	C₂×D₇₃	—	8	2	no	—
165	C₂×D₇₃	#493	1	2	yes	—
166	C₄×C₇₃	—	8	2	no	—
167	C₄×C₇₃	—	8	2	no	—
168	C₄×C₇₃	C₂²×D₈	1	2	yes	—
169	C₄×C₇₃	C₂×#137	1	2	yes	—
170	C₄×C₇₃	#578	1	2	yes	—
171	Dic₇₃	—	2	1	no	—
172	Dic₇₃	—	2	1	no	—
173	Dic₇₃	—	8	2	no	—
174	Dic₇₃	#493	1	2	yes	—
175	#19630	—	73	1	no	maximal, Sylow
176	C₂×C₇₃	C₂×#493	1	1	yes	characteristic
177	C₂×C₇₃	#3152	1	1	yes	characteristic
178	C₂×C₇₃	#3156	1	1	yes	characteristic
179	D₇₃	—	2	2	no	—
180	D₇₃	—	16	2	no	—
181	#3226	—	73	2	no	—
182	#3169	—	73	1	no	—
183	C₂×#500	—	73	1	no	—
184	#3152	—	73	2	no	—
185	#1722	—	73	2	no	—
186	#1722	—	73	2	no	—
187	#1713	—	73	1	no	—
188	#1713	D₇₃	1	1	yes	characteristic, complemented
189	#1706	—	73	1	no	—
190	#1706	—	73	1	no	—
191	C₈×M₁₆	—	73	1	no	—
192	C₇₃	#19630	1	1	yes	characteristic, complemented, Sylow
193	C₂×#137	—	146	4	no	—
194	#532	—	73	2	no	—
195	C₂×C₄⋊Q₈	—	73	2	no	—
196	#500	—	73	2	no	—
197	#500	—	73	2	no	—
198	#500	—	73	1	no	—
199	#500	—	146	2	no	—
200	#500	C₂×D₇₃	1	1	yes	characteristic
201	#500	C₂×D₇₃	1	1	yes	characteristic
202	#500	C₂×D₇₃	1	2	yes	—
203	#497	—	146	4	no	—
204	#494	—	73	2	no	—
205	#494	C₂×D₇₃	1	2	yes	—
206	#493	—	73	1	no	—
207	#492	—	73	2	no	—
208	#492	—	73	2	no	—
209	#486	—	73	2	no	—
210	#486	—	73	2	no	—
211	#486	—	73	2	no	—
212	#480	—	146	4	no	—
213	#474	—	146	4	no	—
214	#466	—	146	2	no	—
215	C₄×D₈	—	146	2	no	—
216	#431	—	73	1	no	—
217	C₂×#107	—	146	2	no	—
218	C₄×M₁₆	—	73	1	no	—
219	C₂×C₄×C₈	—	73	1	no	—
220	#322	—	73	1	no	—
221	#322	—	73	1	no	—
222	C₈²	—	73	1	no	—
223	C₈²	C₂×D₇₃	1	1	yes	characteristic
224	C₂×#41	—	146	2	no	—
225	C₂²×Q₈	—	146	2	no	—
226	#137	—	292	16	no	—
227	C₂×Q₁₆	—	2	4	no	—
228	C₂×Q₁₆	—	146	4	no	—
229	C₂×SD₁₆	—	146	4	no	—
230	C₂×SD₁₆	—	146	4	no	—
231	C₂×D₈	—	146	2	no	—
232	C₂×M₁₆	—	73	2	no	—
233	C₂²×C₈	—	73	2	no	—
234	C₄⋊Q₈	—	73	2	no	—
235	C₄⋊Q₈	—	146	2	no	—
236	C₄⋊Q₈	—	146	2	no	—
237	C₄⋊Q₈	—	146	2	no	—
238	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
239	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
240	#127	—	73	2	no	—
241	#124	—	146	2	no	—
242	#121	—	146	2	no	—
243	#121	—	146	2	no	—
244	C₄×D₄	—	146	2	no	—
245	C₄×D₄	—	146	2	no	—
246	C₂×C₄⋊C₄	—	146	2	no	—
247	C₂×C₄⋊C₄	—	146	2	no	—
248	C₂×C₄²	—	73	1	no	—
249	#107	—	2	2	no	—
250	#107	—	2	2	no	—
251	#107	—	2	4	no	—
252	#107	—	146	4	no	—
253	#107	—	146	4	no	—
254	#107	—	146	2	no	—
255	C₄⋊C₈	—	73	2	no	—
256	#102	—	146	4	no	—
257	#102	—	146	4	no	—
258	#102	—	146	4	no	—
259	C₂²⋊C₈	—	73	2	no	—
260	#97	—	73	2	no	—
261	C₄×C₈	—	73	2	no	—
262	C₄×C₈	—	73	1	no	—
263	C₄×C₈	—	73	1	no	—
264	C₄×C₈	—	73	1	no	—
265	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
268	#41	—	292	8	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	292	2	no	—
272	C₂×Q₈	—	292	8	no	—
273	C₂×Q₈	—	146	2	no	—
274	C₂×D₄	—	292	2	no	—
275	C₂×D₄	—	146	2	no	—
276	C₂×D₄	—	146	2	no	—
277	C₂×D₄	—	146	2	no	—
278	C₂²×C₄	—	292	2	no	—
279	C₂²×C₄	—	292	2	no	—
280	C₂²×C₄	—	73	1	no	—
281	C₂²×C₄	—	73	1	no	—
282	C₂²×C₄	—	73	1	no	—
283	Q₁₆	—	4	8	no	—
284	Q₁₆	—	292	8	no	—
285	SD₁₆	—	292	8	no	—
286	SD₁₆	—	292	8	no	—
287	D₈	—	292	4	no	—
288	M₁₆	—	73	8	no	—
289	C₂×C₈	—	73	2	no	—
290	C₂×C₈	—	73	2	no	—
291	C₂×C₈	—	73	2	no	—
292	C₂×C₈	—	146	4	no	—
293	C₂×C₈	D₄×D₇₃	1	2	yes	—
294	C₂×C₈	D₄×D₇₃	1	2	yes	—
295	C₂×C₈	D₄×D₇₃	1	2	yes	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	292	2	no	—
301	C₄⋊C₄	—	292	2	no	—
302	C₄⋊C₄	—	146	2	no	—
303	C₄⋊C₄	—	146	2	no	—
304	C₂²⋊C₄	—	292	2	no	—
305	C₂²⋊C₄	—	292	2	no	—
306	C₄²	—	73	1	no	—
307	C₄²	—	73	1	no	—
308	C₄²	—	73	1	no	—
309	C₄²	C₂³×D₇₃	1	1	yes	characteristic
310	C₂³	—	292	2	no	—
311	C₂³	—	73	1	no	—
312	Q₈	—	4	4	no	—
313	Q₈	—	4	4	no	—
314	Q₈	—	292	4	no	—
315	Q₈	—	584	4	no	—
316	D₄	—	292	4	no	—
317	D₄	—	292	4	no	—
318	D₄	—	292	4	no	—
319	D₄	—	584	4	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	292	2	no	—
323	C₂×C₄	—	584	4	no	—
324	C₂×C₄	—	584	4	no	—
325	C₂×C₄	—	73	1	no	—
326	C₂×C₄	—	73	4	no	—
327	C₂×C₄	—	73	1	no	—
328	C₂×C₄	—	73	1	no	—
329	C₂×C₄	—	146	2	no	—
330	C₂×C₄	—	146	2	no	—
331	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	73	4	no	—
336	C₈	—	146	2	no	—
337	C₈	—	146	2	no	—
338	C₈	D₈×D₇₃	1	4	yes	—
339	C₈	#14556623	1	4	yes	complemented
340	C₂²	—	584	2	no	—
341	C₂²	—	584	2	no	—
342	C₂²	—	584	2	no	—
343	C₂²	—	73	2	no	—
344	C₂²	—	146	1	no	—
345	C₂²	—	146	1	no	—
346	C₂²	#127×D₇₃	1	1	yes	characteristic
347	C₄	—	8	2	no	—
348	C₄	—	8	2	no	—
349	C₄	—	584	2	no	—
350	C₄	—	73	2	no	—
351	C₄	—	146	1	no	—
352	C₄	—	146	1	no	—
353	C₄	C₂×D₈×D₇₃	1	2	yes	—
354	C₄	C₂×#14556623	1	2	yes	—
355	C₄	#17989400	1	2	yes	—
356	C₂	—	1168	2	no	—
357	C₂	—	146	2	no	—
358	C₂	#493×D₇₃	1	1	yes	characteristic
359	C₂	#22620389	1	1	yes	characteristic
360	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	#22620511	C₂
hypercenter	#1713	D₇₃
hypocenter	C₇₃	#19630
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
1. #22621053	C₂	semidirect
3. #500×D₇₃	C₂	semidirect
4. #22620389	C₂	semidirect
5. #22609603	C₂	semidirect
6. #22609601	C₂	semidirect
7. #22609548	C₂	semidirect
8. #22609524	C₂	semidirect
9. #22609523	C₂	semidirect
10. C₈×#11924011	C₂	semidirect
11. C₇₃×#1713	C₂	semidirect
104. #11924011	D₈	semidirect
155. C₈×C₇₃	#137	semidirect
188. #1713	D₇₃	semidirect
192. C₇₃	#19630	semidirect
339. C₈	#14556623	semidirect

Maximal subgroups

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

subgroup	index
#22621053	2
#22620511	2
#500×D₇₃	2
#22620389	2
#22609603	2
#22609601	2
#22609548	2
#22609524	2
#22609523	2
C₈×#11924011	2
C₇₃×#1713	2
#19630	73