Subgroups — #33056659

Order 18688.

560 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620514	C₂	1	1	yes	maximal, characteristic
2	#22619027	C₂	1	2	yes	maximal
3	#22608841	C₂	1	1	yes	maximal, characteristic, complemented
4	#22608826	C₂	1	2	yes	maximal, complemented
5	#22608750	C₂	1	1	yes	maximal, characteristic, complemented
6	#22608749	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608735	C₂	1	2	yes	maximal, complemented
8	#22608691	C₂	1	1	yes	maximal, characteristic
9	#22608402	C₂	1	2	yes	maximal
10	#22608216	C₂	1	1	yes	maximal, characteristic
11	C₇₃×#1593	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989141	C₂²	1	2	yes	—
13	#17988991	C₄	1	1	yes	characteristic
14	#17988965	C₄	1	1	yes	characteristic
15	#17988936	C₄	1	2	yes	—
16	#17988829	C₄	1	1	yes	characteristic
17	#17988827	C₄	1	2	yes	—
18	#17988825	C₄	1	1	yes	characteristic
19	#17988717	C₂²	1	2	yes	—
20	#17988715	—	2	2	no	—
21	#17988714	—	2	2	no	—
22	Q₈×#10596831	C₂²	1	2	yes	—
23	#17988711	C₂²	1	2	yes	—
24	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
25	C₄×#11924018	C₂²	1	2	yes	—
26	#17988649	—	2	2	no	—
27	#17988647	—	2	2	no	—
28	#17988643	C₂²	1	2	yes	—
29	#17988638	C₂²	1	1	yes	characteristic
30	#17988638	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
32	#17988339	C₂²	1	2	yes	—
33	#17988338	C₂²	1	1	yes	characteristic
34	#17988319	C₂²	1	2	yes	—
35	Q₈⋊#10596831	C₂²	1	2	yes	—
36	#17988317	C₂²	1	2	yes	—
37	#17988308	C₂²	1	1	yes	characteristic
38	C₈×#10596831	C₂²	1	1	yes	characteristic
39	C₈×#10596831	C₂²	1	1	yes	characteristic
40	#446×C₇₃	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
42	#334×C₇₃	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₇₃	C₂²	1	2	yes	—
44	#322×C₇₃	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
46	C₂×#11924018	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
49	#14556590	C₂×C₄	1	2	yes	—
50	#14556589	C₂×C₄	1	2	yes	—
51	#14556581	—	2	2	no	—
52	#14556571	C₂×C₄	1	2	yes	—
53	#14556569	—	2	2	no	—
54	#14556567	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₄₆	—	2	2	no	—
56	C₄×Dic₁₄₆	C₂³	1	2	yes	—
57	#14556548	—	2	2	no	—
58	C₄²×D₇₃	C₂³	1	1	yes	characteristic
59	#14556525	C₂×C₄	1	1	yes	characteristic
60	#14556524	C₂×C₄	1	2	yes	—
61	C₄⋊#10596831	—	2	2	no	—
62	C₄⋊#10596831	C₂³	1	2	yes	—
63	#14556520	—	2	2	no	—
64	#14556520	—	2	1	no	—
65	#14556520	—	2	2	no	—
66	#14556520	D₄	1	2	yes	—
67	#14556516	C₂×C₄	1	2	yes	—
68	#14556515	C₂×C₄	1	2	yes	—
69	#14556514	C₂×C₄	1	2	yes	—
70	#14556503	—	2	2	no	—
71	#14556503	—	2	1	no	—
72	#14556503	D₄	1	2	yes	—
73	#14556501	—	2	2	no	—
74	#14556501	—	2	2	no	—
75	C₄×#10596831	—	2	2	no	—
76	C₄×#10596831	C₂³	1	1	yes	characteristic
77	C₄×#10596831	C₂³	1	1	yes	characteristic
78	C₄×#10596831	C₂³	1	1	yes	characteristic
79	C₈×Dic₇₃	—	2	2	no	—
80	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
82	C₂×Q₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
84	#107×C₇₃	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₈×C₇₃	—	2	2	no	—
86	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
87	#103×C₇₃	C₂×C₄	1	2	yes	—
88	#97×C₇₃	—	2	2	no	—
89	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
92	C₂×Dic₁₄₆	—	2	2	no	—
93	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
97	Dic₂₉₂	—	2	2	no	—
98	#11924018	—	2	4	no	—
99	C₂×#10596831	—	2	2	no	—
100	C₂×#10596831	—	2	1	no	—
101	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
102	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
103	C₂×#10596831	C₂×D₄	1	2	yes	—
104	C₈×D₇₃	—	2	2	no	—
105	C₈×D₇₃	—	2	2	no	—
106	C₄⋊Dic₇₃	—	2	2	no	—
107	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
108	#11924007	—	2	2	no	—
109	#11924007	—	2	2	no	—
110	#11924007	—	2	2	no	—
111	#11924007	—	2	2	no	—
112	#11924007	—	2	2	no	—
113	#11924005	—	2	2	no	—
114	#11924005	—	2	2	no	—
115	C₄×Dic₇₃	—	2	2	no	—
116	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
117	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
118	C₄×Dic₇₃	#41	1	1	yes	characteristic
119	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
120	Q₁₆×C₇₃	—	2	2	no	—
121	C₂×C₈×C₇₃	—	2	2	no	—
122	C₂×C₈×C₇₃	—	2	1	no	—
123	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
126	C₄⋊C₄×C₇₃	—	2	2	no	—
127	C₄⋊C₄×C₇₃	C₂²×C₄	1	2	yes	—
128	C₄²×C₇₃	—	2	2	no	—
129	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
130	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
131	Q₈×C₇₃	—	2	2	no	—
132	Q₈×C₇₃	—	2	2	no	—
133	C₂×C₄×C₇₃	—	2	2	no	—
134	C₂×C₄×C₇₃	—	2	2	no	—
135	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₇₃	—	2	2	no	—
139	C₂×Dic₇₃	—	2	2	no	—
140	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
141	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
142	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
143	C₄×D₇₃	—	2	1	no	—
144	C₄×D₇₃	—	2	1	no	—
145	C₄×D₇₃	—	2	2	no	—
146	C₄×D₇₃	—	4	1	no	—
147	Dic₁₄₆	—	2	2	no	—
148	Dic₁₄₆	—	2	2	no	—
149	Dic₁₄₆	—	8	2	no	—
150	C₈×C₇₃	—	2	1	no	—
151	C₈×C₇₃	—	2	2	no	—
152	C₈×C₇₃	—	4	1	no	—
153	C₈×C₇₃	—	4	2	no	—
154	C₈×C₇₃	#131	1	2	yes	—
155	#10596831	—	2	1	no	—
156	#10596831	—	2	2	no	—
157	#10596831	—	4	1	no	—
158	#10596831	—	4	2	no	—
159	#10596831	#131	1	2	yes	—
160	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
161	C₂×D₇₃	—	2	1	no	—
162	C₂×D₇₃	—	2	1	no	—
163	C₂×D₇₃	—	2	1	no	—
164	C₄×C₇₃	—	2	1	no	—
165	C₄×C₇₃	—	4	2	no	—
166	C₄×C₇₃	—	4	2	no	—
167	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
168	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
169	C₄×C₇₃	#578	1	2	yes	—
170	Dic₇₃	—	2	1	no	—
171	Dic₇₃	—	2	1	no	—
172	Dic₇₃	—	2	1	no	—
173	Dic₇₃	—	4	2	no	—
174	Dic₇₃	—	4	2	no	—
175	#17972	—	73	1	no	maximal, Sylow
176	C₂×C₇₃	#2931	1	1	yes	characteristic
177	C₂×C₇₃	#2950	1	1	yes	characteristic
178	C₂×C₇₃	#2957	1	1	yes	characteristic
179	D₇₃	—	4	1	no	—
180	#3170	—	73	1	no	—
181	#2985	—	73	2	no	—
182	#1593	—	73	1	no	—
183	#1593	D₇₃	1	1	yes	characteristic, complemented
184	#1591	—	73	2	no	—
185	#1579	—	73	1	no	—
186	#1579	—	73	1	no	—
187	C₈×Q₁₆	—	73	2	no	—
188	#1568	—	73	1	no	—
189	#1497	—	73	2	no	—
190	#1450	—	73	1	no	—
191	C₇₃	#17972	1	1	yes	characteristic, complemented, Sylow
192	#533	—	73	2	no	—
193	#500	—	73	1	no	—
194	#494	—	73	1	no	—
195	#487	—	73	2	no	—
196	#467	—	73	1	no	—
197	#467	—	73	1	no	—
198	#467	—	73	2	no	—
199	#446	—	73	2	no	—
200	#446	—	146	2	no	—
201	#446	—	146	2	no	—
202	#446	C₂×D₇₃	1	2	yes	—
203	C₈×Q₈	—	73	2	no	—
204	C₈×Q₈	—	73	2	no	—
205	C₄×Q₁₆	—	73	2	no	—
206	C₄×Q₁₆	—	73	1	no	—
207	C₄×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
208	#433	—	146	2	no	—
209	#433	—	146	2	no	—
210	#432	—	73	2	no	—
211	#431	—	73	1	no	—
212	#431	—	73	1	no	—
213	C₂×C₄×C₈	—	73	1	no	—
214	#335	—	73	2	no	—
215	#334	—	73	1	no	—
216	#334	C₂×D₇₃	1	1	yes	characteristic
217	Q₈⋊C₈	—	73	2	no	—
218	Q₈⋊C₈	—	73	2	no	—
219	Q₈⋊C₈	—	73	2	no	—
220	Q₈⋊C₈	C₂×D₇₃	1	2	yes	—
221	#322	—	73	1	no	—
222	#322	C₂×D₇₃	1	1	yes	characteristic
223	C₈²	—	73	1	no	—
224	C₈²	—	73	1	no	—
225	C₂×Q₁₆	—	73	2	no	—
226	C₂×Q₁₆	—	73	1	no	—
227	C₂×Q₁₆	C₄×D₇₃	1	1	yes	characteristic
228	C₂²×C₈	—	73	1	no	—
229	C₂²×C₈	—	73	1	no	—
230	C₄⋊Q₈	—	73	2	no	—
231	C₄⋊Q₈	—	73	2	no	—
232	#125	—	146	2	no	—
233	C₂²⋊Q₈	—	73	2	no	—
234	C₂²⋊Q₈	—	73	2	no	—
235	C₂²⋊Q₈	—	146	2	no	—
236	C₄×Q₈	—	73	2	no	—
237	C₄×Q₈	—	146	2	no	—
238	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
239	#117	—	146	2	no	—
240	C₂×C₄²	—	73	1	no	—
241	#107	—	73	1	no	—
242	#107	—	73	2	no	—
243	#107	C₄×D₇₃	1	1	yes	characteristic
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	73	2	no	—
246	C₄⋊C₈	—	73	2	no	—
247	C₄⋊C₈	—	146	2	no	—
248	C₄⋊C₈	—	146	1	no	—
249	C₄⋊C₈	—	146	2	no	—
250	C₄⋊C₈	—	146	2	no	—
251	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
252	#103	—	73	2	no	—
253	#103	—	73	2	no	—
254	#103	—	73	2	no	—
255	#103	C₄×D₇₃	1	2	yes	—
256	C₂²⋊C₈	—	73	2	no	—
257	C₂²⋊C₈	—	146	1	no	—
258	C₂²⋊C₈	—	146	2	no	—
259	#97	—	2	2	no	—
260	#97	—	146	2	no	—
261	#97	—	146	2	no	—
262	C₄×C₈	—	73	1	no	—
263	C₄×C₈	—	73	1	no	—
264	C₄×C₈	—	73	1	no	—
265	C₄×C₈	—	73	1	no	—
266	C₄×C₈	—	73	1	no	—
267	C₄×C₈	—	146	2	no	—
268	C₄×C₈	—	146	2	no	—
269	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
271	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
272	C₂×Q₈	—	73	2	no	—
273	C₂×Q₈	—	146	2	no	—
274	C₂×Q₈	C₂×C₄×D₇₃	1	2	yes	—
275	C₂²×C₄	—	73	1	no	—
276	C₂²×C₄	—	73	1	no	—
277	C₂²×C₄	—	73	1	no	—
278	Q₁₆	—	2	2	no	—
279	Q₁₆	—	146	4	no	—
280	Q₁₆	—	146	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	73	1	no	—
284	C₂×C₈	—	73	1	no	—
285	C₂×C₈	—	73	2	no	—
286	C₂×C₈	—	146	2	no	—
287	C₂×C₈	—	146	2	no	—
288	C₂×C₈	—	146	1	no	—
289	C₂×C₈	—	146	2	no	—
290	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
292	C₂×C₈	D₄×D₇₃	1	2	yes	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	73	2	no	—
295	C₄⋊C₄	—	146	2	no	—
296	C₄⋊C₄	—	146	2	no	—
297	C₄⋊C₄	—	146	2	no	—
298	C₄⋊C₄	—	146	2	no	—
299	C₄⋊C₄	—	146	2	no	—
300	C₄⋊C₄	—	146	2	no	—
301	C₄⋊C₄	C₂×C₄×D₇₃	1	2	yes	—
302	C₂²⋊C₄	—	146	2	no	—
303	C₂²⋊C₄	—	146	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	73	1	no	—
306	C₄²	—	73	1	no	—
307	C₄²	—	73	1	no	—
308	C₄²	—	146	2	no	—
309	C₄²	C₂³×D₇₃	1	1	yes	characteristic
310	C₂³	—	73	1	no	—
311	Q₈	—	2	2	no	—
312	Q₈	—	2	2	no	—
313	Q₈	—	584	2	no	—
314	Q₈	—	146	2	no	—
315	Q₈	—	146	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	292	1	no	—
319	C₂×C₄	—	73	1	no	—
320	C₂×C₄	—	73	1	no	—
321	C₂×C₄	—	73	1	no	—
322	C₂×C₄	—	146	2	no	—
323	C₂×C₄	—	146	2	no	—
324	C₂×C₄	—	146	2	no	—
325	C₂×C₄	—	146	1	no	—
326	C₂×C₄	—	146	1	no	—
327	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
329	C₂×C₄	#41×D₇₃	1	1	yes	characteristic
330	C₈	—	2	2	no	—
331	C₈	—	2	1	no	—
332	C₈	—	4	1	no	—
333	C₈	—	4	2	no	—
334	C₈	—	292	1	no	—
335	C₈	—	292	2	no	—
336	C₈	—	73	2	no	—
337	C₈	—	146	2	no	—
338	C₈	—	146	1	no	—
339	C₈	#14556599	1	2	yes	—
340	C₂²	—	146	1	no	—
341	C₂²	—	146	1	no	—
342	C₂²	—	146	1	no	—
343	C₂²	C₄×D₄×D₇₃	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	2	no	—
347	C₄	—	292	2	no	—
348	C₄	—	292	2	no	—
349	C₄	—	146	1	no	—
350	C₄	—	146	1	no	—
351	C₄	—	146	1	no	—
352	C₄	#131×D₇₃	1	1	yes	characteristic
353	C₄	C₂×#14556599	1	1	yes	characteristic
354	C₄	#17989403	1	2	yes	—
355	C₂	—	292	1	no	—
356	C₂	#22618640	1	1	yes	characteristic
357	C₂	#22618796	1	1	yes	characteristic
358	C₂	#22618845	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1593	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂²×D₇₃	C₄×D₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1593	D₇₃
hypocenter	C₇₃	#17972
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1593

Complements

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

N	H ≅ G/N	type
3. #22608841	C₂	semidirect
4. #22608826	C₂	semidirect
5. #22608750	C₂	semidirect
6. #22608749	C₂	semidirect
7. #22608735	C₂	semidirect
11. C₇₃×#1593	C₂	semidirect
183. #1593	D₇₃	semidirect
191. C₇₃	#17972	semidirect

Maximal subgroups

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

subgroup	index
#22620514	2
#22619027	2
#22608841	2
#22608826	2
#22608750	2
#22608749	2
#22608735	2
#22608691	2
#22608402	2
#22608216	2
C₇₃×#1593	2
#17972	73