Subgroups — #33073788

Order 18688.

690 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620517	C₂	1	1	yes	maximal, characteristic
2	#22620513	C₂	1	2	yes	maximal
3	#22620441	C₂	1	2	yes	maximal
4	#22609611	C₂	1	2	yes	maximal, complemented
5	#22609610	C₂	1	2	yes	maximal, complemented
6	#22609607	C₂	1	2	yes	maximal, complemented
7	#22609606	C₂	1	2	yes	maximal, complemented
8	C₂×#17988309	C₂	1	1	yes	maximal, characteristic
9	C₇₃×#3170	C₂	1	1	yes	maximal, characteristic
10	#17989144	C₂²	1	2	yes	—
11	#17988992	C₂²	1	1	yes	characteristic
12	#17988990	C₂²	1	2	yes	—
13	#17988990	C₂²	1	2	yes	—
14	#17988990	C₂²	1	2	yes	—
15	#17988986	—	2	4	no	—
16	#17988983	C₂²	1	2	yes	—
17	#17988983	C₂²	1	2	yes	—
18	#17988967	C₂²	1	1	yes	characteristic
19	#17988966	C₂²	1	2	yes	—
20	#17988939	C₂²	1	2	yes	—
21	#17988938	C₂²	1	2	yes	—
22	#17988937	C₂²	1	2	yes	—
23	#17988828	—	2	2	no	—
24	#17988826	—	2	4	no	—
25	C₄×Dic₂₉₂	—	2	2	no	—
26	C₄×#11924018	—	2	4	no	—
27	#17988623	—	2	4	no	—
28	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
29	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
30	#17988309	C₂²	1	2	yes	—
31	#17988309	C₂²	1	2	yes	—
32	#533×C₇₃	C₂²	1	2	yes	—
33	#500×C₇₃	C₂²	1	1	yes	characteristic
34	#494×C₇₃	C₂²	1	1	yes	characteristic
35	#487×C₇₃	C₂²	1	2	yes	—
36	#467×C₇₃	—	2	2	no	—
37	C₄×Q₁₆×C₇₃	—	2	2	no	—
38	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
39	C₂×Dic₂₉₂	—	2	2	no	—
40	C₂×#11924018	—	2	4	no	—
41	C₂²×#10596831	D₄	1	2	yes	—
42	C₂²×#10596831	D₄	1	2	yes	—
43	#14556591	C₂³	1	2	yes	—
44	#14556591	C₂³	1	2	yes	—
45	#14556583	—	2	2	no	—
46	#14556570	—	2	2	no	—
47	#14556570	—	2	2	no	—
48	C₄×Dic₁₄₆	—	2	2	no	—
49	C₄×Dic₁₄₆	—	2	2	no	—
50	#14556550	—	2	2	no	—
51	#14556525	—	2	2	no	—
52	#14556524	—	2	4	no	—
53	#14556524	—	2	4	no	—
54	#14556524	—	2	4	no	—
55	#14556522	—	2	4	no	—
56	#14556522	—	2	4	no	—
57	#14556516	—	2	4	no	—
58	#14556515	—	2	4	no	—
59	#14556514	—	2	4	no	—
60	C₄×#10596831	D₄	1	2	yes	—
61	C₄×#10596831	D₄	1	2	yes	—
62	C₄×#10596831	C₂³	1	2	yes	—
63	C₄×#10596831	C₂³	1	2	yes	—
64	C₂×Q₁₆×C₇₃	—	2	2	no	—
65	C₂²×C₈×C₇₃	D₄	1	2	yes	—
66	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
67	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
68	#125×C₇₃	—	2	2	no	—
69	C₂²⋊Q₈×C₇₃	—	2	2	no	—
70	C₂²⋊Q₈×C₇₃	—	2	2	no	—
71	C₄×Q₈×C₇₃	—	2	2	no	—
72	C₄×Q₈×C₇₃	—	2	2	no	—
73	#117×C₇₃	—	2	2	no	—
74	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
75	#107×C₇₃	—	2	2	no	—
76	#103×C₇₃	—	2	4	no	—
77	C₄×C₈×C₇₃	D₄	1	2	yes	—
78	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
80	C₂×Dic₁₄₆	—	2	2	no	—
81	C₂×Dic₁₄₆	—	2	2	no	—
82	Dic₂₉₂	—	4	4	no	—
83	#11924018	—	4	8	no	—
84	C₂×#10596831	—	4	2	no	—
85	C₂×#10596831	Q₁₆	1	8	yes	complemented
86	C₂×#10596831	C₂×D₄	1	4	yes	—
87	C₂×#10596831	C₂×D₄	1	4	yes	—
88	C₄⋊Dic₇₃	—	2	2	no	—
89	C₄⋊Dic₇₃	—	2	2	no	—
90	C₄⋊Dic₇₃	—	2	2	no	—
91	C₄⋊Dic₇₃	—	2	2	no	—
92	#11924007	—	4	2	no	—
93	#11924007	—	4	2	no	—
94	C₂²⋊Dic₇₃	—	4	2	no	—
95	C₄×Dic₇₃	—	4	2	no	—
96	C₂×Q₈×C₇₃	—	2	2	no	—
97	C₂×Q₈×C₇₃	—	2	2	no	—
98	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
101	Q₁₆×C₇₃	—	4	4	no	—
102	C₂×C₈×C₇₃	—	4	2	no	—
103	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
105	C₄⋊C₄×C₇₃	—	2	2	no	—
106	C₄⋊C₄×C₇₃	—	2	2	no	—
107	C₄⋊C₄×C₇₃	—	2	2	no	—
108	C₄⋊C₄×C₇₃	—	2	2	no	—
109	C₄⋊C₄×C₇₃	—	4	2	no	—
110	C₄⋊C₄×C₇₃	—	4	2	no	—
111	C₂²⋊C₄×C₇₃	—	4	2	no	—
112	C₄²×C₇₃	—	4	2	no	—
113	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
114	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
117	C₂³×C₇₃	#127	1	1	yes	characteristic
118	Q₈×C₇₃	—	4	4	no	—
119	Q₈×C₇₃	—	4	4	no	—
120	C₂×C₄×C₇₃	—	2	2	no	—
121	C₂×C₄×C₇₃	—	2	2	no	—
122	C₂×C₄×C₇₃	—	4	2	no	—
123	C₂×C₄×C₇₃	—	4	2	no	—
124	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
125	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
126	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂×Q₁₆	1	4	yes	—
128	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×Dic₇₃	—	4	2	no	—
132	C₂×Dic₇₃	—	4	2	no	—
133	Dic₁₄₆	—	4	4	no	—
134	Dic₁₄₆	—	4	4	no	—
135	C₈×C₇₃	—	2	2	no	—
136	C₈×C₇₃	—	2	2	no	—
137	#10596831	—	2	4	no	—
138	#10596831	C₂×Q₁₆	1	8	yes	—
139	C₂²×C₇₃	—	2	1	no	—
140	C₂²×C₇₃	—	2	1	no	—
141	C₂²×C₇₃	#494	1	2	yes	—
142	C₂²×C₇₃	C₂×#127	1	1	yes	characteristic
143	C₄×C₇₃	—	2	1	no	—
144	C₄×C₇₃	—	2	1	no	—
145	C₄×C₇₃	—	8	2	no	—
146	C₄×C₇₃	—	8	2	no	—
147	C₄×C₇₃	#494	1	2	yes	—
148	C₄×C₇₃	C₂²×Q₁₆	1	2	yes	—
149	C₄×C₇₃	#575	1	2	yes	—
150	C₄×C₇₃	#575	1	2	yes	—
151	Dic₇₃	—	8	2	no	—
152	Dic₇₃	—	8	2	no	—
153	#19618	—	73	1	no	maximal, Sylow
154	C₂×C₇₃	—	2	2	no	—
155	C₂×C₇₃	C₂×#494	1	1	yes	characteristic
156	C₂×C₇₃	#3150	1	1	yes	characteristic
157	C₂×C₇₃	#3150	1	1	yes	characteristic
158	#3170	—	73	1	no	—
159	#3170	—	73	2	no	—
160	#3170	D₇₃	1	1	yes	characteristic
161	#3160	—	73	2	no	—
162	#1723	—	73	2	no	—
163	#1723	—	73	2	no	—
164	#1723	—	73	2	no	—
165	#1723	—	73	2	no	—
166	C₂×#322	—	73	1	no	—
167	C₇₃	#19618	1	1	yes	characteristic, complemented, Sylow
168	#533	—	73	2	no	—
169	#533	C₂×D₇₃	1	2	yes	—
170	#500	—	73	2	no	—
171	#500	—	73	2	no	—
172	#500	—	73	2	no	—
173	#500	—	73	1	no	—
174	#500	C₂×D₇₃	1	1	yes	characteristic
175	#499	—	146	4	no	—
176	#498	—	73	2	no	—
177	#498	—	73	2	no	—
178	#494	—	73	2	no	—
179	#494	—	73	1	no	—
180	#494	C₂×D₇₃	1	1	yes	characteristic
181	#487	—	73	2	no	—
182	#487	—	73	2	no	—
183	#487	—	73	2	no	—
184	#487	C₂×D₇₃	1	2	yes	—
185	#467	—	2	2	no	—
186	#467	—	146	4	no	—
187	#467	—	146	2	no	—
188	C₄×Q₁₆	—	2	2	no	—
189	C₄×Q₁₆	—	146	4	no	—
190	C₄×Q₁₆	—	146	2	no	—
191	#427	—	146	4	no	—
192	C₂×C₄×C₈	—	73	1	no	—
193	C₂×C₄×C₈	—	73	1	no	—
194	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
195	#322	—	73	2	no	—
196	#322	—	73	2	no	—
197	C₂×Q₁₆	—	2	2	no	—
198	C₂×Q₁₆	—	146	4	no	—
199	C₂×Q₁₆	—	146	2	no	—
200	C₂²×C₈	—	73	2	no	—
201	C₂²×C₈	—	73	2	no	—
202	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
203	C₄⋊Q₈	—	73	2	no	—
204	C₄⋊Q₈	—	73	2	no	—
205	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
206	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
207	#125	—	2	2	no	—
208	#125	—	146	2	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₂²⋊Q₈	—	2	2	no	—
211	C₂²⋊Q₈	—	146	2	no	—
212	C₂²⋊Q₈	—	146	2	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	—	146	2	no	—
216	C₄×Q₈	—	146	2	no	—
217	#117	—	2	2	no	—
218	#117	—	146	2	no	—
219	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
220	#107	—	2	2	no	—
221	#107	—	146	4	no	—
222	#107	—	146	4	no	—
223	#107	—	146	4	no	—
224	#107	—	146	2	no	—
225	#106	—	146	4	no	—
226	#106	—	146	4	no	—
227	#103	—	2	4	no	—
228	#103	—	146	4	no	—
229	#103	—	146	4	no	—
230	#103	—	146	4	no	—
231	C₄×C₈	—	73	2	no	—
232	C₄×C₈	—	73	2	no	—
233	C₄×C₈	—	73	2	no	—
234	C₄×C₈	—	73	2	no	—
235	C₄×C₈	C₂²⋊D₇₃	1	2	yes	—
236	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×Q₈	—	146	2	no	—
241	C₂×Q₈	—	146	2	no	—
242	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
243	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
245	Q₁₆	—	4	4	no	—
246	Q₁₆	—	292	4	no	—
247	Q₁₆	—	292	8	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	292	2	no	—
250	C₂×C₈	—	73	8	no	—
251	C₂×C₈	—	73	4	no	—
252	C₂×C₈	—	73	4	no	—
253	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
254	C₂×C₈	C₂×C₂²⋊D₇₃	1	2	yes	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	292	2	no	—
262	C₄⋊C₄	—	292	2	no	—
263	C₄⋊C₄	—	146	2	no	—
264	C₄⋊C₄	—	146	2	no	—
265	C₄⋊C₄	—	146	2	no	—
266	C₄⋊C₄	—	146	2	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	292	2	no	—
269	C₄²	—	4	2	no	—
270	C₄²	—	292	2	no	—
271	C₄²	D₄×D₇₃	1	1	yes	characteristic
272	C₄²	D₄×D₇₃	1	1	yes	characteristic
273	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
274	C₄²	C₂³×D₇₃	1	1	yes	characteristic
275	C₂³	#14556587	1	1	yes	characteristic
276	Q₈	—	4	4	no	—
277	Q₈	—	4	4	no	—
278	Q₈	—	292	4	no	—
279	Q₈	—	292	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	292	2	no	—
285	C₂×C₄	—	292	2	no	—
286	C₂×C₄	#14556587	1	1	yes	characteristic
287	C₂×C₄	#14556587	1	1	yes	characteristic
288	C₂×C₄	#14556587	1	1	yes	characteristic
289	C₂×C₄	Q₁₆×D₇₃	1	4	yes	—
290	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
291	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
293	C₈	—	2	2	no	—
294	C₈	—	2	2	no	—
295	C₈	—	73	8	no	—
296	C₈	—	146	4	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	#17988965	1	2	yes	—
300	C₂²	C₂×#14556587	1	1	yes	characteristic
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	584	2	no	—
306	C₄	—	584	2	no	—
307	C₄	#17988965	1	2	yes	—
308	C₄	C₂×Q₁₆×D₇₃	1	2	yes	—
309	C₄	#17989384	1	2	yes	—
310	C₄	#17989385	1	2	yes	—
311	C₂	—	2	2	no	—
312	C₂	C₂×#17988965	1	1	yes	characteristic
313	C₂	#22620374	1	1	yes	characteristic
314	C₂	#22620376	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556587
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#3170	C₂
socle	C₂²×C₇₃	C₂×#127
prime radical	C₂³×C₇₃	#127
hypercenter	#3170	D₇₃
hypocenter	C₇₃	#19618
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
4. #22609611	C₂	semidirect
5. #22609610	C₂	semidirect
6. #22609607	C₂	semidirect
7. #22609606	C₂	semidirect
85. C₂×#10596831	Q₁₆	semidirect
167. C₇₃	#19618	semidirect

Maximal subgroups

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

subgroup	index
#22620517	2
#22620513	2
#22620441	2
#22609611	2
#22609610	2
#22609607	2
#22609606	2
C₂×#17988309	2
C₇₃×#3170	2
#19618	73