Subgroups — #33073962

Order 18688.

878 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620868	C₂	1	2	yes	maximal, complemented
2	#22620518	C₂	1	1	yes	maximal, characteristic, complemented
3	#493×D₇₃	C₂	1	1	yes	maximal, characteristic
4	#22620302	C₂	1	2	yes	maximal, complemented
5	#22609571	C₂	1	2	yes	maximal, complemented
6	#22609568	C₂	1	2	yes	maximal, complemented
7	#22609537	C₂	1	1	yes	maximal, characteristic, complemented
8	#22609536	C₂	1	1	yes	maximal, characteristic, complemented
9	#22609524	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	C₂×D₈×D₇₃	—	2	2	no	—
13	#17989142	C₂²	1	2	yes	—
14	#127×D₇₃	C₂²	1	2	yes	—
15	#17988967	C₂²	1	1	yes	characteristic, complemented
16	#17988964	C₂²	1	1	yes	characteristic
17	#17988963	C₂²	1	2	yes	—
18	#17988962	—	2	2	no	—
19	#17988961	C₂²	1	2	yes	complemented
20	#17988950	—	2	4	no	—
21	#17988939	C₂²	1	2	yes	complemented
22	#17988935	C₂²	1	2	yes	complemented
23	#17988934	C₂²	1	2	yes	complemented
24	#17988934	C₂²	1	2	yes	complemented
25	#17988933	C₂²	1	2	yes	complemented
26	#17988933	C₂²	1	2	yes	complemented
27	#17988829	—	2	2	no	—
28	#17988785	—	2	4	no	—
29	#17988736	—	2	4	no	—
30	#17988681	—	2	2	no	—
31	#17988638	C₂²	1	1	yes	characteristic
32	#17988590	—	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, complemented
36	#17988308	C₂²	1	1	yes	characteristic, complemented
37	C₈×#10596831	C₂²	1	1	yes	characteristic, complemented
38	#500×C₇₃	C₂²	1	1	yes	characteristic, complemented
39	#493×C₇₃	C₂²	1	1	yes	characteristic
40	#492×C₇₃	C₂²	1	2	yes	complemented
41	#486×C₇₃	C₂²	1	2	yes	complemented
42	C₈²×C₇₃	C₂²	1	1	yes	characteristic, complemented
43	C₂×D₄×D₇₃	—	2	2	no	—
44	C₂×D₄×D₇₃	—	2	2	no	—
45	C₂×D₄×D₇₃	—	2	2	no	—
46	C₂×Dic₂₉₂	—	2	2	no	—
47	C₂×#11924015	—	2	4	no	—
48	C₂×D₅₈₄	—	2	2	no	—
49	C₂×#11924013	—	2	4	no	—
50	D₈×D₇₃	—	4	8	no	—
51	C₂×#11924011	D₄	1	2	yes	—
52	C₂×C₈×D₇₃	D₄	1	2	yes	—
53	#14556591	C₂³	1	2	yes	—
54	#14556588	C₂³	1	2	yes	—
55	#14556587	—	2	2	no	—
56	#14556587	—	2	2	no	—
57	#14556587	—	2	2	no	—
58	#14556582	—	2	2	no	—
59	#14556571	—	2	2	no	—
60	#14556567	—	2	2	no	—
61	Q₈×Dic₇₃	—	2	2	no	—
62	#14556556	—	2	2	no	—
63	#14556549	—	2	2	no	—
64	C₄²×D₇₃	C₂³	1	1	yes	characteristic
65	#14556520	D₄	1	2	yes	—
66	#14556514	—	2	4	no	—
67	#14556513	—	2	4	no	—
68	#14556513	—	2	4	no	—
69	#14556512	—	2	4	no	—
70	#14556512	—	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×SD₁₆×C₇₃	—	2	4	no	—
79	C₂×D₈×C₇₃	—	2	2	no	—
80	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
81	#127×C₇₃	C₂³	1	2	yes	—
82	#107×C₇₃	—	2	2	no	—
83	#102×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	C₂³×D₇₃	—	4	2	no	—
88	C₂×Dic₁₄₆	—	2	2	no	—
89	C₂×D₂₉₂	—	2	2	no	—
90	C₂×D₂₉₂	—	2	2	no	—
91	C₂×D₂₉₂	—	2	2	no	—
92	C₂×C₂²⋊D₇₃	—	4	2	no	—
93	C₂×C₂²⋊D₇₃	—	4	2	no	—
94	C₂×C₂²⋊D₇₃	—	4	2	no	—
95	D₄×D₇₃	—	4	8	no	—
96	D₄×D₇₃	—	4	8	no	—
97	D₄×D₇₃	—	8	4	no	—
98	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
101	Dic₂₉₂	—	4	4	no	—
102	#11924015	—	8	4	no	—
103	D₅₈₄	—	4	4	no	—
104	#11924013	—	4	8	no	—
105	#11924011	—	2	4	no	—
106	C₂×#10596831	C₂×D₄	1	2	yes	—
107	C₂×#10596831	C₂×D₄	1	2	yes	—
108	C₂×#10596831	C₂×D₄	1	2	yes	—
109	C₈×D₇₃	—	2	4	no	—
110	C₄⋊Dic₇₃	—	2	2	no	—
111	C₄⋊Dic₇₃	—	2	2	no	—
112	C₄⋊Dic₇₃	—	4	2	no	—
113	#11924007	—	4	2	no	—
114	#11924005	—	4	2	no	—
115	C₄×Dic₇₃	—	4	2	no	—
116	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
117	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
118	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
119	C₂×Q₈×C₇₃	—	2	2	no	—
120	C₂×D₄×C₇₃	—	2	2	no	—
121	C₂×D₄×C₇₃	—	2	2	no	—
122	C₂×D₄×C₇₃	—	2	2	no	—
123	SD₁₆×C₇₃	—	8	4	no	—
124	D₈×C₇₃	—	4	4	no	—
125	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
127	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
128	C₄⋊C₄×C₇₃	—	2	2	no	—
129	C₄⋊C₄×C₇₃	—	2	2	no	—
130	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
131	C₂³×C₇₃	—	4	2	no	—
132	C₂²×D₇₃	—	4	2	no	—
133	C₂²×D₇₃	—	8	4	no	—
134	C₂²×D₇₃	—	8	4	no	—
135	C₂²×D₇₃	—	8	4	no	—
136	C₂²×D₇₃	#127	1	1	yes	characteristic
137	Q₈×C₇₃	—	4	4	no	—
138	D₄×C₇₃	—	4	4	no	—
139	D₄×C₇₃	—	4	4	no	—
140	D₄×C₇₃	—	4	4	no	—
141	C₂×C₄×C₇₃	—	4	2	no	—
142	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
145	C₂²⋊D₇₃	—	8	4	no	—
146	C₂²⋊D₇₃	—	8	4	no	—
147	C₂²⋊D₇₃	—	8	4	no	—
148	C₂×Dic₇₃	—	4	2	no	—
149	C₂×Dic₇₃	#127	1	1	yes	characteristic
150	C₂×Dic₇₃	#127	1	1	yes	characteristic
151	C₂×Dic₇₃	#127	1	1	yes	characteristic
152	D₂₉₂	—	4	4	no	—
153	D₂₉₂	—	4	4	no	—
154	D₂₉₂	—	4	4	no	—
155	C₄×D₇₃	—	2	2	no	—
156	C₄×D₇₃	—	2	2	no	—
157	C₄×D₇₃	#135	1	4	yes	—
158	Dic₁₄₆	—	4	4	no	—
159	C₈×C₇₃	—	2	2	no	—
160	C₈×C₇₃	—	2	2	no	—
161	C₈×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
162	#10596831	—	2	2	no	—
163	#10596831	—	2	2	no	—
164	#10596831	—	2	2	no	—
165	C₂²×C₇₃	—	8	2	no	—
166	C₂²×C₇₃	—	8	2	no	—
167	C₂²×C₇₃	—	8	2	no	—
168	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
169	C₂×D₇₃	—	2	1	no	—
170	C₂×D₇₃	—	2	1	no	—
171	C₂×D₇₃	—	8	2	no	—
172	C₂×D₇₃	—	8	2	no	—
173	C₂×D₇₃	—	8	2	no	—
174	C₂×D₇₃	—	16	8	no	—
175	C₂×D₇₃	#495	1	2	yes	—
176	C₄×C₇₃	—	8	2	no	—
177	C₄×C₇₃	C₂×#135	1	2	yes	—
178	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
179	C₄×C₇₃	#576	1	2	yes	—
180	Dic₇₃	—	2	1	no	—
181	Dic₇₃	—	2	1	no	—
182	Dic₇₃	—	8	2	no	—
183	Dic₇₃	#495	1	2	yes	—
184	#19635	—	73	1	no	maximal, Sylow
185	C₂×C₇₃	—	16	2	no	—
186	C₂×C₇₃	C₂×#495	1	1	yes	characteristic
187	C₂×C₇₃	#3151	1	1	yes	characteristic
188	C₂×C₇₃	#3155	1	1	yes	characteristic
189	D₇₃	—	2	2	no	—
190	D₇₃	—	16	2	no	—
191	#3208	—	73	2	no	—
192	#3170	—	73	1	no	—
193	C₂×#493	—	73	1	no	—
194	#3138	—	73	2	no	—
195	#1717	—	73	2	no	—
196	#1717	—	73	2	no	—
197	#1709	—	73	1	no	—
198	#1709	—	73	1	no	—
199	#1706	—	73	1	no	—
200	#1706	D₇₃	1	1	yes	characteristic, complemented
201	C₈×M₁₆	—	73	1	no	—
202	C₇₃	#19635	1	1	yes	characteristic, complemented, Sylow
203	C₂²×D₈	—	146	2	no	—
204	#533	—	73	2	no	—
205	C₂×#127	—	73	2	no	—
206	#500	C₂×D₇₃	1	1	yes	characteristic, complemented
207	#494	—	73	1	no	—
208	#493	—	73	2	no	—
209	#493	—	73	1	no	—
210	#493	—	146	2	no	—
211	#493	C₂×D₇₃	1	1	yes	characteristic
212	#492	—	73	2	no	—
213	#492	C₂×D₇₃	1	2	yes	complemented
214	#490	—	146	4	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	#486	C₂×D₇₃	1	2	yes	complemented
222	#467	—	146	2	no	—
223	#460	—	146	4	no	—
224	#450	—	146	4	no	—
225	C₄×Q₁₆	—	146	2	no	—
226	#431	—	73	1	no	—
227	#418	—	146	4	no	—
228	C₄×M₁₆	—	73	1	no	—
229	C₂×C₄×C₈	—	73	1	no	—
230	#322	—	73	1	no	—
231	#322	—	73	1	no	—
232	C₈²	—	73	1	no	—
233	C₈²	C₂×D₇₃	1	1	yes	characteristic, complemented
234	C₂²×D₄	—	146	2	no	—
235	C₂²×D₄	—	146	2	no	—
236	C₂²×D₄	—	146	2	no	—
237	C₂×Q₁₆	—	146	2	no	—
238	C₂×SD₁₆	—	2	4	no	—
239	C₂×SD₁₆	—	146	4	no	—
240	C₂×D₈	—	2	2	no	—
241	C₂×D₈	—	292	8	no	—
242	C₂×D₈	—	146	4	no	—
243	C₂×D₈	—	146	2	no	—
244	C₂×M₁₆	—	73	2	no	—
245	C₂²×C₈	—	73	2	no	—
246	C₄⋊Q₈	—	73	2	no	—
247	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
248	#127	—	73	2	no	—
249	#127	—	146	2	no	—
250	#127	—	146	2	no	—
251	#127	—	146	2	no	—
252	#127	C₂²×D₇₃	1	2	yes	—
253	#125	—	146	2	no	—
254	C₂²⋊Q₈	—	146	2	no	—
255	C₂²⋊Q₈	—	146	2	no	—
256	C₄×Q₈	—	146	2	no	—
257	C₄×Q₈	—	146	2	no	—
258	#117	—	146	2	no	—
259	C₂×C₄²	—	73	1	no	—
260	#107	—	2	2	no	—
261	C₄⋊C₈	—	73	2	no	—
262	#103	—	146	4	no	—
263	#102	—	2	4	no	—
264	#102	—	146	4	no	—
265	#102	—	146	4	no	—
266	#102	—	146	4	no	—
267	#102	—	146	4	no	—
268	#102	—	146	4	no	—
269	C₂²⋊C₈	—	73	2	no	—
270	#97	—	73	2	no	—
271	C₄×C₈	—	73	2	no	—
272	C₄×C₈	—	73	1	no	—
273	C₄×C₈	—	73	1	no	—
274	C₄×C₈	—	73	1	no	—
275	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
278	C₂⁴	—	292	2	no	—
279	C₂×Q₈	—	2	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	—	292	8	no	—
285	C₂×D₄	—	292	2	no	—
286	C₂×D₄	—	292	2	no	—
287	C₂×D₄	—	292	2	no	—
288	C₂×D₄	—	292	8	no	—
289	C₂×D₄	—	584	4	no	—
290	C₂×D₄	—	146	2	no	—
291	C₂×D₄	—	146	2	no	—
292	C₂×D₄	—	146	2	no	—
293	C₂²×C₄	—	73	1	no	—
294	C₂²×C₄	—	73	1	no	—
295	C₂²×C₄	—	73	1	no	—
296	Q₁₆	—	292	4	no	—
297	SD₁₆	—	8	4	no	—
298	SD₁₆	—	584	4	no	—
299	D₈	—	4	4	no	—
300	D₈	—	292	8	no	—
301	D₈	—	292	4	no	—
302	M₁₆	—	146	4	no	—
303	C₂×C₈	—	73	2	no	—
304	C₂×C₈	—	73	2	no	—
305	C₂×C₈	—	73	2	no	—
306	C₂×C₈	—	146	4	no	—
307	C₂×C₈	D₄×D₇₃	1	2	yes	—
308	C₂×C₈	D₄×D₇₃	1	2	yes	—
309	C₂×C₈	D₄×D₇₃	1	2	yes	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	292	2	no	—
313	C₄⋊C₄	—	292	2	no	—
314	C₄⋊C₄	—	146	2	no	—
315	C₄⋊C₄	—	146	2	no	—
316	C₂²⋊C₄	—	292	2	no	—
317	C₄²	—	292	2	no	—
318	C₄²	—	73	1	no	—
319	C₄²	—	73	1	no	—
320	C₄²	—	73	1	no	—
321	C₄²	C₂³×D₇₃	1	1	yes	characteristic
322	C₂³	—	4	2	no	—
323	C₂³	—	292	2	no	—
324	C₂³	—	584	4	no	—
325	C₂³	—	584	4	no	—
326	C₂³	—	584	4	no	—
327	C₂³	—	73	1	no	—
328	Q₈	—	4	4	no	—
329	Q₈	—	292	4	no	—
330	D₄	—	4	4	no	—
331	D₄	—	4	4	no	—
332	D₄	—	4	4	no	—
333	D₄	—	292	4	no	—
334	D₄	—	292	4	no	—
335	D₄	—	292	4	no	—
336	D₄	—	584	4	no	—
337	D₄	—	584	4	no	—
338	D₄	—	584	4	no	—
339	C₂×C₄	—	4	2	no	—
340	C₂×C₄	—	292	2	no	—
341	C₂×C₄	—	73	1	no	—
342	C₂×C₄	—	73	4	no	—
343	C₂×C₄	—	73	1	no	—
344	C₂×C₄	—	73	1	no	—
345	C₂×C₄	—	146	2	no	—
346	C₂×C₄	—	146	2	no	—
347	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
349	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
350	C₈	—	2	2	no	—
351	C₈	—	2	2	no	—
352	C₈	—	146	2	no	—
353	C₈	—	146	2	no	—
354	C₈	—	146	2	no	—
355	C₈	#14556618	1	4	yes	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	2	no	—
358	C₂²	—	8	2	no	—
359	C₂²	—	584	2	no	—
360	C₂²	—	584	2	no	—
361	C₂²	—	584	2	no	—
362	C₂²	—	73	2	no	—
363	C₂²	—	1168	8	no	—
364	C₂²	—	146	1	no	—
365	C₂²	—	146	1	no	—
366	C₂²	#127×D₇₃	1	1	yes	characteristic
367	C₄	—	8	2	no	—
368	C₄	—	584	2	no	—
369	C₄	—	73	2	no	—
370	C₄	—	146	1	no	—
371	C₄	—	146	1	no	—
372	C₄	#135×D₇₃	1	2	yes	—
373	C₄	C₂×#14556618	1	2	yes	—
374	C₄	#17989388	1	2	yes	—
375	C₂	—	16	2	no	—
376	C₂	—	1168	2	no	—
377	C₂	—	146	2	no	—
378	C₂	#495×D₇₃	1	1	yes	characteristic
379	C₂	#22620378	1	1	yes	characteristic
380	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	#493×D₇₃	C₂
hypercenter	#1706	D₇₃
hypocenter	C₇₃	#19635
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. #22620868	C₂	semidirect
2. #22620518	C₂	semidirect
4. #22620302	C₂	semidirect
5. #22609571	C₂	semidirect
6. #22609568	C₂	semidirect
7. #22609537	C₂	semidirect
8. #22609536	C₂	semidirect
9. #22609524	C₂	semidirect
10. C₈×#11924011	C₂	semidirect
11. C₇₃×#1706	C₂	semidirect
15. #17988967	C₂²	semidirect
19. #17988961	C₂²	semidirect
21. #17988939	C₂²	semidirect
22. #17988935	C₂²	semidirect
23. #17988934	C₂²	semidirect
24. #17988934	C₂²	semidirect
25. #17988933	C₂²	semidirect
26. #17988933	C₂²	semidirect
35. #17988309	C₂²	semidirect
36. #17988308	C₂²	semidirect
37. C₈×#10596831	C₂²	semidirect
38. #500×C₇₃	C₂²	semidirect
40. #492×C₇₃	C₂²	semidirect
41. #486×C₇₃	C₂²	semidirect
42. C₈²×C₇₃	C₂²	semidirect
200. #1706	D₇₃	semidirect
202. C₇₃	#19635	semidirect
206. #500	C₂×D₇₃	semidirect
213. #492	C₂×D₇₃	semidirect
221. #486	C₂×D₇₃	semidirect
233. C₈²	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22620868	2
#22620518	2
#493×D₇₃	2
#22620302	2
#22609571	2
#22609568	2
#22609537	2
#22609536	2
#22609524	2
C₈×#11924011	2
C₇₃×#1706	2
#19635	73