Subgroups — #33241333

Order 18688.

682 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620442	C₂	1	1	yes	maximal, characteristic, complemented
2	#493×D₇₃	C₂	1	1	yes	maximal, characteristic
3	#22612171	C₂	1	4	yes	maximal, complemented
4	#22612156	C₂	1	1	yes	maximal, characteristic, complemented
5	#22612155	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612154	C₂	1	1	yes	maximal, characteristic, complemented
7	#22611645	C₂	1	4	yes	maximal, complemented
8	C₄×#14556530	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₈×D₇₃	—	2	1	no	—
11	C₂×D₈×D₇₃	C₂²	1	2	yes	—
12	#17989142	—	2	1	no	—
13	#127×D₇₃	—	2	1	no	—
14	C₂×#14556530	C₂²	1	2	yes	—
15	#17988992	C₂²	1	1	yes	characteristic, complemented
16	#17988988	—	2	1	no	—
17	#17988988	C₂²	1	2	yes	complemented
18	#17988983	—	2	1	no	—
19	#17988964	C₂²	1	1	yes	characteristic
20	#17988963	—	2	1	no	—
21	#17988962	—	2	1	no	—
22	#17988962	C₂²	1	2	yes	—
23	#17988624	—	2	1	no	—
24	#17988624	C₂²	1	2	yes	—
25	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
26	#17988389	C₂²	1	4	yes	complemented
27	#17988388	C₂²	1	4	yes	complemented
28	#17988387	C₂²	1	4	yes	complemented
29	#17988362	C₂²	1	2	yes	complemented
30	C₄×#11924003	C₂²	1	1	yes	characteristic, complemented
31	#500×C₇₃	C₂²	1	1	yes	characteristic, complemented
32	#493×C₇₃	C₂²	1	1	yes	characteristic
33	#357×C₇₃	C₂²	1	4	yes	complemented
34	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic, complemented
35	C₂×D₄×D₇₃	—	2	2	no	—
36	C₂×D₄×D₇₃	—	4	1	no	—
37	C₂×D₄×D₇₃	—	4	1	no	—
38	C₂×D₅₈₄	—	2	1	no	—
39	C₂×D₅₈₄	C₂³	1	2	yes	—
40	C₂×#11924013	—	2	2	no	—
41	C₂×#11924013	—	2	2	no	—
42	D₈×D₇₃	—	2	8	no	—
43	D₈×D₇₃	—	4	4	no	—
44	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
45	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
46	#14556591	—	2	1	no	—
47	#14556588	—	2	1	no	—
48	#14556587	—	2	2	no	—
49	#14556587	—	4	1	no	—
50	#14556587	—	4	1	no	—
51	#14556582	—	2	2	no	—
52	#14556571	—	4	1	no	—
53	#14556567	—	4	1	no	—
54	Q₈×Dic₇₃	—	4	1	no	—
55	#14556556	—	4	1	no	—
56	#14556549	—	2	2	no	—
57	C₄²×D₇₃	D₄	1	1	yes	characteristic
58	#14556530	—	2	4	no	—
59	C₂×#11924003	C₂³	1	2	yes	—
60	#14556525	—	2	1	no	—
61	#14556525	C₂³	1	2	yes	—
62	#14556522	—	2	2	no	—
63	#14556522	—	2	2	no	—
64	C₄×#10596831	D₄	1	1	yes	characteristic
65	C₈×Dic₇₃	D₄	1	1	yes	characteristic
66	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
67	C₂×D₈×C₇₃	—	2	1	no	—
68	C₂×D₈×C₇₃	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₇₃	—	2	1	no	—
70	#127×C₇₃	—	2	1	no	—
71	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
72	#107×C₇₃	—	2	1	no	—
73	#107×C₇₃	C₂³	1	2	yes	—
74	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
75	C₂³×D₇₃	—	4	2	no	—
76	C₂×Dic₁₄₆	—	4	1	no	—
77	C₂×D₂₉₂	—	2	2	no	—
78	C₂×D₂₉₂	—	4	1	no	—
79	C₂×D₂₉₂	—	4	1	no	—
80	C₂×C₂²⋊D₇₃	—	4	2	no	—
81	C₂×C₂²⋊D₇₃	—	4	2	no	—
82	C₂×C₂²⋊D₇₃	—	4	2	no	—
83	D₄×D₇₃	—	4	4	no	—
84	D₄×D₇₃	—	4	4	no	—
85	D₄×D₇₃	—	8	4	no	—
86	D₄×D₇₃	—	8	4	no	—
87	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
88	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
89	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
90	D₅₈₄	—	2	4	no	—
91	D₅₈₄	—	4	2	no	—
92	#11924013	—	4	4	no	—
93	#11924013	—	4	4	no	—
94	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
95	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
96	C₈×D₇₃	SD₁₆	1	4	yes	—
97	C₈×D₇₃	#41	1	4	yes	—
98	C₄⋊Dic₇₃	—	2	2	no	—
99	C₄⋊Dic₇₃	—	4	1	no	—
100	C₄⋊Dic₇₃	—	4	2	no	—
101	#11924007	—	4	2	no	—
102	#11924005	—	4	2	no	—
103	C₄×Dic₇₃	—	4	2	no	—
104	C₄×Dic₇₃	D₈	1	1	yes	characteristic
105	C₄×Dic₇₃	D₈	1	1	yes	characteristic
106	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
107	#11924003	—	2	2	no	—
108	C₂×Q₈×C₇₃	—	4	1	no	—
109	C₂×D₄×C₇₃	—	2	2	no	—
110	C₂×D₄×C₇₃	—	4	1	no	—
111	C₂×D₄×C₇₃	—	4	1	no	—
112	D₈×C₇₃	—	2	4	no	—
113	D₈×C₇₃	—	4	2	no	—
114	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄×C₇₃	—	2	2	no	—
117	C₄⋊C₄×C₇₃	—	4	1	no	—
118	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
119	C₁₆×C₇₃	—	2	2	no	—
120	C₂³×C₇₃	—	4	2	no	—
121	C₂²×D₇₃	—	4	2	no	—
122	C₂²×D₇₃	—	8	4	no	—
123	C₂²×D₇₃	—	8	4	no	—
124	C₂²×D₇₃	—	8	4	no	—
125	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
126	Q₈×C₇₃	—	8	2	no	—
127	D₄×C₇₃	—	4	2	no	—
128	D₄×C₇₃	—	4	2	no	—
129	D₄×C₇₃	—	8	2	no	—
130	D₄×C₇₃	—	8	2	no	—
131	C₂×C₄×C₇₃	—	4	2	no	—
132	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
135	C₂²⋊D₇₃	—	8	4	no	—
136	C₂²⋊D₇₃	—	8	4	no	—
137	C₂²⋊D₇₃	—	8	4	no	—
138	C₂×Dic₇₃	—	4	2	no	—
139	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
140	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
141	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
142	D₂₉₂	—	4	2	no	—
143	D₂₉₂	—	4	2	no	—
144	D₂₉₂	—	8	2	no	—
145	D₂₉₂	—	8	2	no	—
146	C₄×D₇₃	—	2	2	no	—
147	C₄×D₇₃	—	2	2	no	—
148	C₄×D₇₃	#124	1	2	yes	—
149	C₄×D₇₃	C₂×SD₁₆	1	2	yes	—
150	Dic₁₄₆	—	8	2	no	—
151	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
152	C₈×C₇₃	C₂×#41	1	2	yes	—
153	#10596831	#124	1	2	yes	—
154	#10596831	C₂×SD₁₆	1	2	yes	—
155	C₂²×C₇₃	—	8	2	no	—
156	C₂²×C₇₃	—	8	2	no	—
157	C₂²×C₇₃	—	8	2	no	—
158	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
159	C₂×D₇₃	—	2	1	no	—
160	C₂×D₇₃	—	2	1	no	—
161	C₂×D₇₃	—	8	2	no	—
162	C₂×D₇₃	—	8	2	no	—
163	C₂×D₇₃	—	8	2	no	—
164	C₂×D₇₃	—	16	8	no	—
165	C₂×D₇₃	#486	1	2	yes	—
166	C₄×C₇₃	—	8	2	no	—
167	C₄×C₇₃	#509	1	2	yes	—
168	C₄×C₇₃	#509	1	2	yes	—
169	C₄×C₇₃	C₂×#124	1	1	yes	characteristic
170	C₄×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
171	Dic₇₃	—	2	1	no	—
172	Dic₇₃	—	2	1	no	—
173	Dic₇₃	—	8	2	no	—
174	Dic₇₃	#486	1	2	yes	—
175	#33704	—	73	1	no	maximal, Sylow
176	C₂×C₇₃	—	16	2	no	—
177	C₂×C₇₃	C₂×#486	1	1	yes	characteristic
178	C₂×C₇₃	C₂×#509	1	1	yes	characteristic
179	C₂×C₇₃	C₂×#509	1	1	yes	characteristic
180	D₇₃	—	2	2	no	—
181	D₇₃	—	16	2	no	—
182	#3160	—	73	1	no	—
183	C₂×#493	—	73	1	no	—
184	#2244	—	73	4	no	—
185	#2240	—	73	1	no	—
186	#2240	—	73	1	no	—
187	#2240	—	73	1	no	—
188	#2240	D₇₃	1	1	yes	characteristic, complemented
189	#2140	—	73	4	no	—
190	C₄×M₃₂	—	73	1	no	—
191	C₇₃	#33704	1	1	yes	characteristic, complemented, Sylow
192	C₂²×D₈	—	73	2	no	—
193	C₂²×D₈	—	146	1	no	—
194	#533	—	146	1	no	—
195	C₂×#127	—	146	1	no	—
196	C₂×M₃₂	—	73	2	no	—
197	#500	—	73	1	no	—
198	#500	C₂×D₇₃	1	1	yes	characteristic, complemented
199	#499	—	73	2	no	—
200	#499	—	146	1	no	—
201	#498	—	146	1	no	—
202	#493	—	73	2	no	—
203	#493	—	73	1	no	—
204	#493	—	146	1	no	—
205	#493	—	146	1	no	—
206	#493	C₂×D₇₃	1	1	yes	characteristic
207	#427	—	73	2	no	—
208	#427	—	146	1	no	—
209	C₂×C₄×C₈	—	73	1	no	—
210	#357	—	73	4	no	—
211	#357	—	73	4	no	—
212	#357	—	73	4	no	—
213	#357	C₂×D₇₃	1	4	yes	complemented
214	#346	—	73	2	no	—
215	C₄×C₁₆	—	73	1	no	—
216	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic, complemented
217	C₂²×D₄	—	292	1	no	—
218	C₂²×D₄	—	292	1	no	—
219	C₂²×D₄	—	146	2	no	—
220	C₂×D₈	—	2	1	no	—
221	C₂×D₈	—	292	4	no	—
222	C₂×D₈	—	73	2	no	—
223	C₂×D₈	—	146	8	no	—
224	C₂×D₈	—	146	2	no	—
225	C₂×D₈	—	146	2	no	—
226	C₂×D₈	—	146	1	no	—
227	C₂×D₈	C₂²×D₇₃	1	2	yes	—
228	C₂²×C₈	—	73	1	no	—
229	C₂²×C₈	—	73	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	146	1	no	—
232	#127	—	2	1	no	—
233	#127	—	292	1	no	—
234	#127	—	292	1	no	—
235	#127	—	146	2	no	—
236	#127	—	146	1	no	—
237	#125	—	146	2	no	—
238	C₂²⋊Q₈	—	292	1	no	—
239	C₂²⋊Q₈	—	292	1	no	—
240	C₄×Q₈	—	292	1	no	—
241	C₄×Q₈	—	292	1	no	—
242	#117	—	146	2	no	—
243	C₂×C₄²	—	73	1	no	—
244	M₃₂	—	146	4	no	—
245	C₂×C₁₆	—	73	2	no	—
246	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
247	#107	—	2	1	no	—
248	#107	—	73	2	no	—
249	#107	—	146	1	no	—
250	#107	C₂²×D₇₃	1	2	yes	—
251	#106	—	146	2	no	—
252	#106	—	146	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	C₂⁴	—	292	2	no	—
258	C₂×Q₈	—	4	1	no	—
259	C₂×Q₈	—	292	1	no	—
260	C₂×D₄	—	2	2	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂×D₄	—	4	1	no	—
263	C₂×D₄	—	292	4	no	—
264	C₂×D₄	—	292	4	no	—
265	C₂×D₄	—	292	2	no	—
266	C₂×D₄	—	292	2	no	—
267	C₂×D₄	—	292	2	no	—
268	C₂×D₄	—	292	1	no	—
269	C₂×D₄	—	292	1	no	—
270	C₂×D₄	—	584	4	no	—
271	C₂×D₄	—	584	4	no	—
272	C₂×D₄	—	146	2	no	—
273	C₂²×C₄	—	73	1	no	—
274	C₂²×C₄	—	73	1	no	—
275	C₂²×C₄	—	73	1	no	—
276	D₈	—	2	4	no	—
277	D₈	—	4	2	no	—
278	D₈	—	292	4	no	—
279	D₈	—	292	4	no	—
280	D₈	—	292	2	no	—
281	D₈	—	146	4	no	—
282	C₂×C₈	—	73	1	no	—
283	C₂×C₈	—	73	1	no	—
284	C₂×C₈	—	73	4	no	—
285	C₂×C₈	—	73	4	no	—
286	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
287	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	292	2	no	—
291	C₄⋊C₄	—	292	2	no	—
292	C₄⋊C₄	—	292	1	no	—
293	C₄⋊C₄	—	146	2	no	—
294	C₂²⋊C₄	—	292	2	no	—
295	C₄²	—	292	2	no	—
296	C₄²	—	73	1	no	—
297	C₄²	—	73	1	no	—
298	C₄²	—	73	1	no	—
299	C₄²	D₄×D₇₃	1	1	yes	characteristic
300	C₁₆	—	2	2	no	—
301	C₁₆	—	146	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	292	2	no	—
304	C₂³	—	584	4	no	—
305	C₂³	—	584	4	no	—
306	C₂³	—	584	4	no	—
307	C₂³	—	73	1	no	—
308	Q₈	—	8	2	no	—
309	Q₈	—	584	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	D₄	—	8	2	no	—
313	D₄	—	8	2	no	—
314	D₄	—	292	2	no	—
315	D₄	—	292	2	no	—
316	D₄	—	584	4	no	—
317	D₄	—	584	4	no	—
318	D₄	—	584	4	no	—
319	D₄	—	584	2	no	—
320	D₄	—	584	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	292	2	no	—
323	C₂×C₄	—	73	1	no	—
324	C₂×C₄	—	73	2	no	—
325	C₂×C₄	—	73	2	no	—
326	C₂×C₄	—	73	1	no	—
327	C₂×C₄	—	73	1	no	—
328	C₂×C₄	—	146	2	no	—
329	C₂×C₄	—	146	2	no	—
330	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
331	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
333	C₈	—	73	2	no	—
334	C₈	—	73	2	no	—
335	C₈	SD₁₆×D₇₃	1	2	yes	—
336	C₈	#41×D₇₃	1	2	yes	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	584	2	no	—
341	C₂²	—	584	2	no	—
342	C₂²	—	584	2	no	—
343	C₂²	—	73	2	no	—
344	C₂²	—	1168	8	no	—
345	C₂²	—	146	1	no	—
346	C₂²	—	146	1	no	—
347	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
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	#17989021	1	2	yes	—
354	C₄	#17989024	1	2	yes	—
355	C₄	#124×D₇₃	1	1	yes	characteristic
356	C₄	C₂×SD₁₆×D₇₃	1	1	yes	characteristic
357	C₂	—	16	2	no	—
358	C₂	—	1168	2	no	—
359	C₂	—	146	2	no	—
360	C₂	#486×D₇₃	1	1	yes	characteristic
361	C₂	C₂×#17989021	1	1	yes	characteristic
362	C₂	C₂×#17989024	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2240	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	#493×D₇₃	C₂
hypercenter	#2240	D₇₃
hypocenter	C₇₃	#33704
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2240

Complements

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

N	H ≅ G/N	type
1. #22620442	C₂	semidirect
3. #22612171	C₂	semidirect
4. #22612156	C₂	semidirect
5. #22612155	C₂	semidirect
6. #22612154	C₂	semidirect
7. #22611645	C₂	semidirect
8. C₄×#14556530	C₂	semidirect
9. C₇₃×#2240	C₂	semidirect
15. #17988992	C₂²	semidirect
17. #17988988	C₂²	semidirect
26. #17988389	C₂²	semidirect
27. #17988388	C₂²	semidirect
28. #17988387	C₂²	semidirect
29. #17988362	C₂²	semidirect
30. C₄×#11924003	C₂²	semidirect
31. #500×C₇₃	C₂²	semidirect
33. #357×C₇₃	C₂²	semidirect
34. C₄×C₁₆×C₇₃	C₂²	semidirect
188. #2240	D₇₃	semidirect
191. C₇₃	#33704	semidirect
198. #500	C₂×D₇₃	semidirect
213. #357	C₂×D₇₃	semidirect
216. C₄×C₁₆	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22620442	2
#493×D₇₃	2
#22612171	2
#22612156	2
#22612155	2
#22612154	2
#22611645	2
C₄×#14556530	2
C₇₃×#2240	2
#33704	73