Subgroups — #33242772

Order 18688.

646 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	#22623079	C₂	1	2	yes	maximal, complemented
2	#22621718	C₂	1	2	yes	maximal, complemented
3	#22612192	C₂	1	1	yes	maximal, characteristic, complemented
4	#22612061	C₂	1	2	yes	maximal, complemented
5	#22612001	C₂	1	2	yes	maximal, complemented
6	#22611911	C₂	1	2	yes	maximal, complemented
7	#22611890	C₂	1	2	yes	maximal, complemented
8	#22611813	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₃×#2170	C₂	1	1	yes	maximal, characteristic, complemented
10	#17989404	C₂²	1	2	yes	—
11	#17989394	—	2	2	no	—
12	#17989392	C₂²	1	2	yes	complemented
13	#17989030	—	2	2	no	—
14	#17989030	C₂²	1	2	yes	complemented
15	#17989025	—	2	2	no	—
16	#17989025	C₂²	1	2	yes	—
17	#17989019	—	2	2	no	—
18	#17989019	C₂²	1	1	yes	characteristic
19	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic, complemented
20	C₂×#14556536	—	2	2	no	—
21	C₂×#14556536	C₂²	1	2	yes	complemented
22	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic
23	#17988971	C₂²	1	2	yes	complemented
24	#17988862	C₂²	1	2	yes	complemented
25	#17988857	C₂²	1	2	yes	—
26	#17988851	—	2	2	no	—
27	#17988766	—	2	2	no	—
28	#17988761	—	2	2	no	—
29	#17988705	C₂²	1	2	yes	complemented
30	#17988417	C₂²	1	1	yes	characteristic, complemented
31	#17988404	C₂²	1	2	yes	complemented
32	#17988401	C₂²	1	2	yes	complemented
33	#17988395	C₂²	1	2	yes	complemented
34	#504×C₇₃	C₂²	1	2	yes	complemented
35	#443×C₇₃	C₂²	1	1	yes	characteristic
36	#364×C₇₃	C₂²	1	1	yes	characteristic, complemented
37	#350×C₇₃	C₂²	1	2	yes	complemented
38	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic, complemented
39	#14556638	—	2	2	no	—
40	#14556637	—	2	2	no	—
41	#14556624	—	2	2	no	—
42	#14556624	—	4	2	no	—
43	#14556623	—	4	2	no	—
44	#14556621	—	2	2	no	—
45	#14556619	—	2	2	no	—
46	#14556619	—	4	2	no	—
47	#14556616	—	2	2	no	—
48	#14556615	—	4	2	no	—
49	#14556613	—	2	2	no	—
50	#14556613	—	2	2	no	—
51	#14556613	C₂³	1	2	yes	—
52	#14556612	—	4	2	no	—
53	C₂×Dic₂₉₂	—	2	2	no	—
54	C₂×Dic₂₉₂	C₂³	1	2	yes	—
55	C₂×#11924017	—	2	2	no	—
56	SD₁₆×D₇₃	—	4	2	no	—
57	C₂×D₅₈₄	—	2	2	no	—
58	C₂×D₅₈₄	C₂³	1	2	yes	—
59	#14556597	—	2	2	no	—
60	#14556579	—	2	2	no	—
61	Dic₅₈₄	—	2	2	no	—
62	Dic₅₈₄	—	4	1	no	—
63	Dic₅₈₄	D₄	1	1	yes	characteristic
64	#14556536	—	2	2	no	—
65	#14556536	—	2	2	no	—
66	#14556536	—	2	2	no	—
67	#14556536	—	2	2	no	—
68	#14556536	D₄	1	2	yes	—
69	D₁₁₆₈	—	2	2	no	—
70	D₁₁₆₈	—	4	1	no	—
71	D₁₁₆₈	D₄	1	1	yes	characteristic
72	#14556527	C₂³	1	2	yes	—
73	#14556519	—	2	2	no	—
74	#14556509	—	2	2	no	—
75	#14556507	—	2	2	no	—
76	#135×C₇₃	D₄	1	1	yes	characteristic
77	#131×C₇₃	C₂³	1	2	yes	—
78	M₃₂×C₇₃	—	2	2	no	—
79	M₃₂×C₇₃	C₂³	1	2	yes	—
80	C₂×C₁₆×C₇₃	—	2	2	no	—
81	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
83	#108×C₇₃	D₄	1	1	yes	characteristic
84	#104×C₇₃	D₄	1	2	yes	—
85	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
86	#11924029	—	4	2	no	—
87	#11924028	—	2	2	no	—
88	#11924028	—	4	2	no	—
89	#11924028	—	4	2	no	—
90	#11924027	—	4	2	no	—
91	#11924027	—	8	2	no	—
92	C₂×Dic₁₄₆	—	2	2	no	—
93	C₂×Dic₁₄₆	—	4	2	no	—
94	Q₈×D₇₃	—	4	2	no	—
95	C₂×D₂₉₂	—	2	2	no	—
96	C₂×D₂₉₂	—	4	2	no	—
97	D₄×D₇₃	—	4	2	no	—
98	D₄×D₇₃	—	8	2	no	—
99	Dic₂₉₂	—	2	2	no	—
100	Dic₂₉₂	—	2	2	no	—
101	Dic₂₉₂	—	4	2	no	—
102	Dic₂₉₂	C₂×D₄	1	2	yes	—
103	#11924018	—	4	2	no	—
104	#11924017	—	2	2	no	—
105	#11924017	—	4	2	no	—
106	#11924017	—	4	2	no	—
107	#11924017	—	4	2	no	—
108	#11924017	—	4	2	no	—
109	#11924016	—	4	2	no	—
110	#11924015	—	4	2	no	—
111	D₅₈₄	—	2	2	no	—
112	D₅₈₄	—	2	2	no	—
113	D₅₈₄	—	4	2	no	—
114	D₅₈₄	C₂×D₄	1	2	yes	—
115	#11924013	—	4	2	no	—
116	#11924012	—	2	2	no	—
117	#11924011	—	4	2	no	—
118	C₈×D₇₃	—	4	2	no	—
119	#11924005	—	8	2	no	—
120	#41×C₇₃	C₂×D₄	1	2	yes	—
121	Q₁₆×C₇₃	D₈	1	1	yes	characteristic
122	SD₁₆×C₇₃	D₈	1	2	yes	—
123	D₈×C₇₃	D₈	1	1	yes	characteristic
124	M₁₆×C₇₃	—	2	2	no	—
125	M₁₆×C₇₃	C₂×D₄	1	2	yes	—
126	C₂×C₈×C₇₃	—	2	2	no	—
127	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
129	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
130	C₁₆×C₇₃	—	2	2	no	—
131	C₁₆×C₇₃	—	2	1	no	—
132	C₁₆×C₇₃	C₂×D₄	1	1	yes	characteristic
133	C₁₆×C₇₃	C₂×D₄	1	1	yes	characteristic
134	C₂²×D₇₃	—	4	2	no	—
135	C₂²×D₇₃	—	8	2	no	—
136	Q₈×C₇₃	C₂×D₈	1	2	yes	—
137	D₄×C₇₃	—	2	2	no	—
138	D₄×C₇₃	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₇₃	—	2	2	no	—
140	C₂×C₄×C₇₃	—	2	1	no	—
141	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
142	C₂²⋊D₇₃	—	4	2	no	—
143	C₂²⋊D₇₃	—	8	2	no	—
144	C₂²⋊D₇₃	—	8	2	no	—
145	C₂×Dic₇₃	—	4	2	no	—
146	C₂×Dic₇₃	—	8	2	no	—
147	D₂₉₂	—	2	2	no	—
148	D₂₉₂	—	4	2	no	—
149	D₂₉₂	—	4	2	no	—
150	D₂₉₂	—	8	2	no	—
151	C₄×D₇₃	—	4	2	no	—
152	C₄×D₇₃	—	8	2	no	—
153	C₄×D₇₃	—	8	2	no	—
154	Dic₁₄₆	—	2	2	no	—
155	Dic₁₄₆	—	4	2	no	—
156	Dic₁₄₆	—	4	2	no	—
157	Dic₁₄₆	—	8	2	no	—
158	C₈×C₇₃	—	2	2	no	—
159	C₈×C₇₃	C₂×D₈	1	1	yes	characteristic
160	C₈×C₇₃	C₂×D₈	1	1	yes	characteristic
161	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
162	C₈×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
163	#10596831	—	4	2	no	—
164	C₂²×C₇₃	—	2	2	no	—
165	C₂²×C₇₃	#576	1	1	yes	characteristic
166	C₂×D₇₃	—	4	2	no	—
167	C₂×D₇₃	—	8	2	no	—
168	C₂×D₇₃	—	16	2	no	—
169	C₂×D₇₃	—	32	2	no	—
170	C₄×C₇₃	—	2	2	no	—
171	C₄×C₇₃	—	4	1	no	—
172	C₄×C₇₃	D₄²	1	1	yes	characteristic
173	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
174	Dic₇₃	—	4	2	no	—
175	Dic₇₃	—	8	2	no	—
176	#33857	—	73	1	no	maximal, Sylow
177	C₂×C₇₃	—	2	1	no	—
178	C₂×C₇₃	—	4	2	no	—
179	C₂×C₇₃	D₄×D₈	1	1	yes	characteristic
180	D₇₃	—	8	2	no	—
181	D₇₃	—	16	2	no	—
182	#3416	—	73	2	no	—
183	#3290	—	73	2	no	—
184	#2249	—	73	1	no	—
185	#2223	—	73	2	no	—
186	#2212	—	73	2	no	—
187	#2195	—	73	2	no	—
188	#2192	—	73	2	no	—
189	#2178	—	73	1	no	—
190	#2170	D₇₃	1	1	yes	characteristic, complemented
191	C₇₃	#33857	1	1	yes	characteristic, complemented, Sylow
192	#578	—	73	2	no	—
193	#577	—	146	2	no	—
194	#576	—	73	2	no	—
195	#510	—	73	2	no	—
196	#510	—	146	2	no	—
197	#509	—	73	2	no	—
198	#509	—	146	2	no	—
199	#508	—	73	1	no	—
200	#508	—	146	2	no	—
201	C₂×Q₃₂	—	73	1	no	—
202	C₂×SD₃₂	—	73	2	no	—
203	C₂×SD₃₂	—	146	2	no	—
204	C₂×D₁₆	—	73	1	no	—
205	#504	C₂×D₇₃	1	2	yes	complemented
206	#495	—	73	2	no	—
207	#473	—	73	2	no	—
208	#472	—	73	2	no	—
209	#471	—	146	2	no	—
210	#455	—	146	2	no	—
211	#454	—	146	2	no	—
212	#443	—	73	2	no	—
213	#443	C₂×D₇₃	1	1	yes	characteristic
214	#368	—	73	1	no	—
215	#364	C₂×D₇₃	1	1	yes	characteristic, complemented
216	#362	—	73	2	no	—
217	#361	—	73	2	no	—
218	#359	—	73	2	no	—
219	#350	C₂×D₇₃	1	2	yes	complemented
220	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic, complemented
221	2₋¹⁺⁴	—	146	2	no	—
222	H(5,𝔽₂)	—	146	2	no	—
223	#137	—	292	2	no	—
224	#137	—	292	2	no	—
225	#137	—	146	2	no	—
226	#137	—	146	2	no	—
227	AGL(1,ℤ/8)	—	292	2	no	—
228	AGL(1,ℤ/8)	—	292	2	no	—
229	AGL(1,ℤ/8)	—	146	2	no	—
230	AGL(1,ℤ/8)	—	146	2	no	—
231	#135	—	292	2	no	—
232	#135	—	73	2	no	—
233	#135	—	146	2	no	—
234	#135	—	146	2	no	—
235	#135	D₂₉₂	1	1	yes	characteristic
236	C₂×Q₁₆	—	73	2	no	—
237	C₂×Q₁₆	—	146	2	no	—
238	C₂×SD₁₆	—	292	2	no	—
239	C₂×SD₁₆	—	146	2	no	—
240	C₂×D₈	—	73	2	no	—
241	C₂×D₈	—	146	2	no	—
242	#131	—	146	2	no	—
243	#131	C₂²×D₇₃	1	2	yes	—
244	#124	—	146	2	no	—
245	Q₃₂	—	292	1	no	—
246	Q₃₂	—	73	1	no	—
247	Q₃₂	—	146	2	no	—
248	SD₃₂	—	73	2	no	—
249	SD₃₂	—	146	2	no	—
250	SD₃₂	—	146	2	no	—
251	SD₃₂	—	146	2	no	—
252	SD₃₂	—	146	2	no	—
253	D₁₆	—	292	1	no	—
254	D₁₆	—	73	1	no	—
255	D₁₆	—	146	2	no	—
256	M₃₂	—	2	2	no	—
257	M₃₂	C₂²×D₇₃	1	2	yes	—
258	C₂×C₁₆	—	2	2	no	—
259	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
260	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
261	#108	—	73	2	no	—
262	#108	D₂₉₂	1	1	yes	characteristic
263	#104	—	146	2	no	—
264	#104	D₂₉₂	1	2	yes	—
265	#101	—	146	2	no	—
266	#100	—	146	2	no	—
267	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
268	#41	—	292	2	no	—
269	#41	—	292	2	no	—
270	#41	—	292	2	no	—
271	#41	—	292	2	no	—
272	#41	—	584	2	no	—
273	#41	—	146	2	no	—
274	#41	C₂×D₂₉₂	1	2	yes	—
275	C₂×Q₈	—	292	2	no	—
276	C₂×Q₈	—	292	2	no	—
277	C₂×Q₈	—	146	2	no	—
278	C₂×D₄	—	292	2	no	—
279	C₂×D₄	—	292	2	no	—
280	C₂×D₄	—	584	2	no	—
281	C₂×D₄	—	146	2	no	—
282	Q₁₆	—	292	2	no	—
283	Q₁₆	—	292	2	no	—
284	Q₁₆	—	73	2	no	—
285	Q₁₆	—	146	2	no	—
286	Q₁₆	—	146	2	no	—
287	Q₁₆	D₅₈₄	1	1	yes	characteristic
288	SD₁₆	—	292	2	no	—
289	SD₁₆	—	292	2	no	—
290	SD₁₆	—	292	2	no	—
291	SD₁₆	—	292	2	no	—
292	SD₁₆	—	292	2	no	—
293	SD₁₆	—	292	2	no	—
294	SD₁₆	—	146	2	no	—
295	SD₁₆	D₅₈₄	1	2	yes	—
296	D₈	—	292	2	no	—
297	D₈	—	292	2	no	—
298	D₈	—	73	2	no	—
299	D₈	—	146	2	no	—
300	D₈	—	146	2	no	—
301	D₈	D₅₈₄	1	1	yes	characteristic
302	M₁₆	—	2	2	no	—
303	M₁₆	—	292	2	no	—
304	M₁₆	—	146	2	no	—
305	M₁₆	C₂×D₂₉₂	1	2	yes	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	292	2	no	—
308	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
309	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
310	C₂²⋊C₄	—	584	2	no	—
311	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
312	C₁₆	—	2	2	no	—
313	C₁₆	—	2	1	no	—
314	C₁₆	D₄×D₇₃	1	1	yes	characteristic
315	C₁₆	D₄×D₇₃	1	1	yes	characteristic
316	C₂³	—	292	2	no	—
317	C₂³	—	584	2	no	—
318	Q₈	—	292	2	no	—
319	Q₈	—	292	2	no	—
320	Q₈	—	584	2	no	—
321	Q₈	—	146	2	no	—
322	Q₈	C₂×D₅₈₄	1	2	yes	—
323	D₄	—	2	2	no	—
324	D₄	—	292	2	no	—
325	D₄	—	292	2	no	—
326	D₄	—	292	2	no	—
327	D₄	—	584	2	no	—
328	D₄	—	584	2	no	—
329	D₄	—	584	2	no	—
330	D₄	—	146	2	no	—
331	D₄	C₂×D₅₈₄	1	2	yes	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	292	2	no	—
335	C₂×C₄	—	292	2	no	—
336	C₂×C₄	—	584	2	no	—
337	C₂×C₄	—	584	2	no	—
338	C₂×C₄	—	584	2	no	—
339	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	292	2	no	—
342	C₈	C₂×D₅₈₄	1	1	yes	characteristic
343	C₈	C₂×D₅₈₄	1	1	yes	characteristic
344	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
345	C₈	#14556637	1	1	yes	characteristic
346	C₂²	—	2	2	no	—
347	C₂²	—	2336	2	no	—
348	C₂²	—	292	2	no	—
349	C₂²	—	584	2	no	—
350	C₂²	—	1168	2	no	—
351	C₂²	#17989392	1	1	yes	characteristic
352	C₄	—	2	2	no	—
353	C₄	—	4	1	no	—
354	C₄	—	292	2	no	—
355	C₄	—	584	2	no	—
356	C₄	D₄×D₂₉₂	1	1	yes	characteristic
357	C₄	C₂²×D₅₈₄	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	4	2	no	—
360	C₂	—	584	2	no	—
361	C₂	—	1168	2	no	—
362	C₂	D₄×D₅₈₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₅₈₄
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2170	C₂
socle	C₂×C₇₃	D₄×D₈
hypercenter	#2170	D₇₃
hypocenter	C₇₃	#33857
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443 ◁ #2170

Complements

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

N	H ≅ G/N	type
1. #22623079	C₂	semidirect
2. #22621718	C₂	semidirect
3. #22612192	C₂	semidirect
4. #22612061	C₂	semidirect
5. #22612001	C₂	semidirect
6. #22611911	C₂	semidirect
7. #22611890	C₂	semidirect
8. #22611813	C₂	semidirect
9. C₇₃×#2170	C₂	semidirect
12. #17989392	C₂²	semidirect
14. #17989030	C₂²	semidirect
19. C₂×Dic₅₈₄	C₂²	semidirect
21. C₂×#14556536	C₂²	semidirect
23. #17988971	C₂²	semidirect
24. #17988862	C₂²	semidirect
29. #17988705	C₂²	semidirect
30. #17988417	C₂²	semidirect
31. #17988404	C₂²	semidirect
32. #17988401	C₂²	semidirect
33. #17988395	C₂²	semidirect
34. #504×C₇₃	C₂²	semidirect
36. #364×C₇₃	C₂²	semidirect
37. #350×C₇₃	C₂²	semidirect
38. C₄×C₁₆×C₇₃	C₂²	semidirect
190. #2170	D₇₃	semidirect
191. C₇₃	#33857	semidirect
205. #504	C₂×D₇₃	semidirect
215. #364	C₂×D₇₃	semidirect
219. #350	C₂×D₇₃	semidirect
220. 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
#22623079	2
#22621718	2
#22612192	2
#22612061	2
#22612001	2
#22611911	2
#22611890	2
#22611813	2
C₇₃×#2170	2
#33857	73