Subgroups — #33241286

Order 18688.

598 conjugacy classes of subgroups.

The subgroup lattice has 340 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	#22620360	C₂	1	1	yes	maximal, characteristic
3	#22612164	C₂	1	4	yes	maximal, complemented
4	#22612158	C₂	1	1	yes	maximal, characteristic, complemented
5	#22612156	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612155	C₂	1	1	yes	maximal, characteristic, complemented
7	#22611637	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14556614	—	2	1	no	—
11	C₂×#14556614	C₂²	1	2	yes	—
12	#17989142	—	2	1	no	—
13	#17989138	—	2	1	no	—
14	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
15	#17988992	C₂²	1	1	yes	characteristic
16	#17988988	—	2	1	no	—
17	#17988988	C₂²	1	2	yes	—
18	#17988983	—	2	1	no	—
19	#17988972	—	2	1	no	—
20	#17988972	C₂²	1	2	yes	—
21	#17988964	C₂²	1	1	yes	characteristic
22	#17988961	—	2	1	no	—
23	#17988624	—	2	1	no	—
24	#17988624	C₂²	1	2	yes	—
25	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
26	#17988391	C₂²	1	4	yes	—
27	#17988389	C₂²	1	4	yes	—
28	#17988387	C₂²	1	4	yes	—
29	C₄×#11924003	C₂²	1	1	yes	characteristic
30	C₁₆×Dic₇₃	C₂²	1	2	yes	—
31	#500×C₇₃	C₂²	1	1	yes	characteristic
32	#494×C₇₃	C₂²	1	1	yes	characteristic
33	#358×C₇₃	C₂²	1	4	yes	—
34	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
35	C₂×#11924029	—	2	2	no	—
36	#14556614	—	2	8	no	—
37	#14556614	—	4	4	no	—
38	C₂×#11924015	—	2	2	no	—
39	C₂×#11924015	—	2	2	no	—
40	C₂×D₅₈₄	—	2	1	no	—
41	C₂×D₅₈₄	C₂³	1	2	yes	—
42	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
43	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
44	#14556591	—	2	1	no	—
45	#14556588	—	2	1	no	—
46	#14556582	—	2	2	no	—
47	#14556580	—	2	2	no	—
48	#14556571	—	4	1	no	—
49	#14556567	—	4	1	no	—
50	#14556566	—	4	1	no	—
51	#14556566	—	4	1	no	—
52	Q₈×Dic₇₃	—	4	1	no	—
53	#14556556	—	4	1	no	—
54	#14556553	—	4	1	no	—
55	#14556553	—	4	1	no	—
56	#14556549	—	2	2	no	—
57	C₄²×D₇₃	D₄	1	1	yes	characteristic
58	C₂×#11924003	C₂³	1	2	yes	—
59	C₁₆×D₇₃	—	2	4	no	—
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×Q₁₆×C₇₃	—	2	1	no	—
68	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₇₃	—	2	1	no	—
70	C₄⋊Q₈×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	#11924029	—	4	4	no	—
76	#11924029	—	4	4	no	—
77	C₂×Dic₁₄₆	—	4	1	no	—
78	C₂×D₂₉₂	—	2	2	no	—
79	C₂×D₂₉₂	—	4	2	no	—
80	C₂×D₂₉₂	—	4	1	no	—
81	C₂×D₂₉₂	—	4	1	no	—
82	C₂×C₄×D₇₃	—	4	2	no	—
83	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
84	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
85	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
86	#11924015	—	4	4	no	—
87	#11924015	—	4	4	no	—
88	D₅₈₄	—	2	4	no	—
89	D₅₈₄	—	4	2	no	—
90	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
91	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
92	C₈×D₇₃	SD₁₆	1	4	yes	—
93	C₈×D₇₃	#41	1	4	yes	—
94	C₄⋊Dic₇₃	—	2	2	no	—
95	C₄⋊Dic₇₃	—	4	1	no	—
96	C₄⋊Dic₇₃	—	4	2	no	—
97	#11924007	—	4	2	no	—
98	#11924005	—	4	2	no	—
99	#11924005	—	4	2	no	—
100	#11924005	—	4	2	no	—
101	C₄×Dic₇₃	—	4	2	no	—
102	C₄×Dic₇₃	D₈	1	1	yes	characteristic
103	C₄×Dic₇₃	D₈	1	1	yes	characteristic
104	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
105	#11924003	—	2	2	no	—
106	C₂×Q₈×C₇₃	—	2	2	no	—
107	C₂×Q₈×C₇₃	—	4	1	no	—
108	Q₁₆×C₇₃	—	2	4	no	—
109	Q₁₆×C₇₃	—	4	2	no	—
110	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₇₃	—	2	2	no	—
113	C₄⋊C₄×C₇₃	—	4	1	no	—
114	C₄⋊C₄×C₇₃	—	4	1	no	—
115	C₄⋊C₄×C₇₃	—	4	1	no	—
116	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₇₃	—	2	2	no	—
118	C₂²×D₇₃	—	4	2	no	—
119	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₇₃	—	4	2	no	—
121	Q₈×C₇₃	—	4	2	no	—
122	Q₈×C₇₃	—	8	2	no	—
123	C₂×C₄×C₇₃	—	4	2	no	—
124	C₂×C₄×C₇₃	—	4	2	no	—
125	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₇₃	—	4	2	no	—
129	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
132	D₂₉₂	—	4	2	no	—
133	D₂₉₂	—	4	2	no	—
134	D₂₉₂	—	8	4	no	—
135	D₂₉₂	—	8	2	no	—
136	D₂₉₂	—	8	2	no	—
137	C₄×D₇₃	—	2	2	no	—
138	C₄×D₇₃	—	2	2	no	—
139	C₄×D₇₃	—	8	4	no	—
140	C₄×D₇₃	#124	1	2	yes	—
141	C₄×D₇₃	C₂×SD₁₆	1	2	yes	—
142	Dic₁₄₆	—	8	2	no	—
143	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
144	C₈×C₇₃	C₂×#41	1	2	yes	—
145	#10596831	#124	1	2	yes	—
146	#10596831	C₂×SD₁₆	1	2	yes	—
147	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₇₃	—	2	1	no	—
149	C₂×D₇₃	—	2	1	no	—
150	C₂×D₇₃	—	8	2	no	—
151	C₂×D₇₃	—	8	2	no	—
152	C₂×D₇₃	—	8	2	no	—
153	C₂×D₇₃	#486	1	2	yes	—
154	C₄×C₇₃	—	8	2	no	—
155	C₄×C₇₃	—	8	2	no	—
156	C₄×C₇₃	#508	1	2	yes	—
157	C₄×C₇₃	#508	1	2	yes	—
158	C₄×C₇₃	C₂×#124	1	1	yes	characteristic
159	C₄×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
160	Dic₇₃	—	2	1	no	—
161	Dic₇₃	—	2	1	no	—
162	Dic₇₃	—	8	2	no	—
163	Dic₇₃	#486	1	2	yes	—
164	#33697	—	73	1	no	maximal, Sylow
165	C₂×C₇₃	C₂×#486	1	1	yes	characteristic
166	C₂×C₇₃	C₂×#508	1	1	yes	characteristic
167	C₂×C₇₃	C₂×#508	1	1	yes	characteristic
168	D₇₃	—	2	2	no	—
169	D₇₃	—	16	2	no	—
170	#3160	—	73	1	no	—
171	#3147	—	73	1	no	—
172	#2242	—	73	4	no	—
173	#2241	—	73	1	no	—
174	#2241	D₇₃	1	1	yes	characteristic, complemented
175	#2240	—	73	1	no	—
176	#2240	—	73	1	no	—
177	#2138	—	73	4	no	—
178	C₂×C₄×C₁₆	—	73	1	no	—
179	C₇₃	#33697	1	1	yes	characteristic, complemented, Sylow
180	C₂×#135	—	73	2	no	—
181	C₂×#135	—	146	1	no	—
182	#533	—	146	1	no	—
183	#532	—	146	1	no	—
184	C₂²×C₁₆	—	73	2	no	—
185	#500	—	73	1	no	—
186	#500	C₂×D₇₃	1	1	yes	characteristic
187	#499	—	73	2	no	—
188	#499	—	146	1	no	—
189	#498	—	146	1	no	—
190	#495	—	73	2	no	—
191	#495	—	146	1	no	—
192	#494	C₂×D₇₃	1	1	yes	characteristic
193	#493	—	73	1	no	—
194	#492	—	146	1	no	—
195	#427	—	73	2	no	—
196	#427	—	146	1	no	—
197	C₂×C₄×C₈	—	73	1	no	—
198	#358	—	73	4	no	—
199	#358	C₂×D₇₃	1	4	yes	—
200	#357	—	73	4	no	—
201	#357	—	73	4	no	—
202	C₄×C₁₆	—	73	2	no	—
203	C₄×C₁₆	—	73	1	no	—
204	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
205	C₂×#41	—	146	2	no	—
206	#135	—	292	4	no	—
207	#135	—	146	8	no	—
208	C₂×Q₁₆	—	2	1	no	—
209	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
210	C₂×SD₁₆	—	146	2	no	—
211	C₂×SD₁₆	—	146	2	no	—
212	C₂×D₈	—	73	2	no	—
213	C₂×D₈	—	146	1	no	—
214	C₂²×C₈	—	73	1	no	—
215	C₂²×C₈	—	73	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	146	1	no	—
219	#127	—	146	1	no	—
220	#125	—	146	2	no	—
221	#124	—	146	2	no	—
222	C₂²⋊Q₈	—	292	1	no	—
223	C₂²⋊Q₈	—	292	1	no	—
224	#121	—	292	1	no	—
225	#121	—	292	1	no	—
226	C₄×Q₈	—	292	1	no	—
227	C₄×Q₈	—	292	1	no	—
228	C₄×D₄	—	292	1	no	—
229	C₄×D₄	—	292	1	no	—
230	#117	—	146	2	no	—
231	C₂×C₄²	—	73	1	no	—
232	C₂×C₁₆	—	73	2	no	—
233	C₂×C₁₆	—	146	4	no	—
234	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
235	#107	—	2	1	no	—
236	#107	—	73	2	no	—
237	#107	—	146	1	no	—
238	#107	C₂²×D₇₃	1	2	yes	—
239	#106	—	146	2	no	—
240	#106	—	146	2	no	—
241	C₄×C₈	—	73	1	no	—
242	C₄×C₈	—	73	1	no	—
243	C₄×C₈	—	73	1	no	—
244	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
245	#41	—	292	4	no	—
246	#41	—	292	4	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	292	1	no	—
250	C₂×D₄	—	292	2	no	—
251	C₂×D₄	—	292	1	no	—
252	C₂×D₄	—	292	1	no	—
253	C₂×D₄	—	146	2	no	—
254	C₂²×C₄	—	292	2	no	—
255	C₂²×C₄	—	73	1	no	—
256	C₂²×C₄	—	73	1	no	—
257	C₂²×C₄	—	73	1	no	—
258	Q₁₆	—	2	4	no	—
259	Q₁₆	—	4	2	no	—
260	SD₁₆	—	292	4	no	—
261	SD₁₆	—	292	4	no	—
262	D₈	—	292	2	no	—
263	D₈	—	146	4	no	—
264	C₂×C₈	—	73	1	no	—
265	C₂×C₈	—	73	1	no	—
266	C₂×C₈	—	73	4	no	—
267	C₂×C₈	—	73	4	no	—
268	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
269	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	292	2	no	—
275	C₄⋊C₄	—	292	2	no	—
276	C₄⋊C₄	—	292	1	no	—
277	C₄⋊C₄	—	146	2	no	—
278	C₂²⋊C₄	—	292	2	no	—
279	C₂²⋊C₄	—	292	2	no	—
280	C₂²⋊C₄	—	292	2	no	—
281	C₄²	—	292	2	no	—
282	C₄²	—	73	1	no	—
283	C₄²	—	73	1	no	—
284	C₄²	—	73	1	no	—
285	C₄²	D₄×D₇₃	1	1	yes	characteristic
286	C₁₆	—	2	2	no	—
287	C₁₆	—	146	2	no	—
288	C₂³	—	292	2	no	—
289	C₂³	—	73	1	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	8	2	no	—
293	Q₈	—	584	2	no	—
294	D₄	—	292	2	no	—
295	D₄	—	292	2	no	—
296	D₄	—	584	4	no	—
297	D₄	—	584	2	no	—
298	D₄	—	584	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	292	2	no	—
302	C₂×C₄	—	584	4	no	—
303	C₂×C₄	—	73	1	no	—
304	C₂×C₄	—	73	2	no	—
305	C₂×C₄	—	73	2	no	—
306	C₂×C₄	—	73	1	no	—
307	C₂×C₄	—	73	1	no	—
308	C₂×C₄	—	146	2	no	—
309	C₂×C₄	—	146	2	no	—
310	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
311	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
313	C₈	—	73	2	no	—
314	C₈	—	73	2	no	—
315	C₈	SD₁₆×D₇₃	1	2	yes	—
316	C₈	#41×D₇₃	1	2	yes	—
317	C₂²	—	584	2	no	—
318	C₂²	—	584	2	no	—
319	C₂²	—	584	2	no	—
320	C₂²	—	73	2	no	—
321	C₂²	—	146	1	no	—
322	C₂²	—	146	1	no	—
323	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	584	2	no	—
327	C₄	—	73	2	no	—
328	C₄	—	146	1	no	—
329	C₄	—	146	1	no	—
330	C₄	#17989018	1	2	yes	—
331	C₄	#17989020	1	2	yes	—
332	C₄	#124×D₇₃	1	1	yes	characteristic
333	C₄	C₂×SD₁₆×D₇₃	1	1	yes	characteristic
334	C₂	—	1168	2	no	—
335	C₂	—	146	2	no	—
336	C₂	#486×D₇₃	1	1	yes	characteristic
337	C₂	C₂×#17989018	1	1	yes	characteristic
338	C₂	C₂×#17989020	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×#2241	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	#22620360	C₂
hypercenter	#2241	D₇₃
hypocenter	C₇₃	#33697
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
1. #22620442	C₂	semidirect
3. #22612164	C₂	semidirect
4. #22612158	C₂	semidirect
5. #22612156	C₂	semidirect
6. #22612155	C₂	semidirect
7. #22611637	C₂	semidirect
8. C₄×C₁₆×D₇₃	C₂	semidirect
9. C₇₃×#2241	C₂	semidirect
174. #2241	D₇₃	semidirect
179. C₇₃	#33697	semidirect

Maximal subgroups

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

subgroup	index
#22620442	2
#22620360	2
#22612164	2
#22612158	2
#22612156	2
#22612155	2
#22611637	2
C₄×C₁₆×D₇₃	2
C₇₃×#2241	2
#33697	73