Subgroups — #34018979

Order 18688.

592 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626877	C₂	1	2	yes	maximal, complemented
2	#22625573	C₂	1	2	yes	maximal, complemented
3	#22625570	C₂	1	1	yes	maximal, characteristic, complemented
4	#22625563	C₂	1	1	yes	maximal, characteristic, complemented
5	#22625523	C₂	1	2	yes	maximal, complemented
6	#22625519	C₂	1	1	yes	maximal, characteristic, complemented
7	#22625512	C₂	1	1	yes	maximal, characteristic, complemented
8	#22625489	C₂	1	1	yes	maximal, characteristic, complemented
9	#22625289	C₂	1	2	yes	maximal, complemented
10	#22625135	C₂	1	1	yes	maximal, characteristic, complemented
11	#22620503	C₂	1	1	yes	maximal, characteristic
12	#17989637	C₂²	1	2	yes	—
13	#17989636	C₂²	1	2	yes	—
14	#17989621	—	2	2	no	—
15	#17989621	C₂²	1	2	yes	—
16	#17989617	—	2	2	no	—
17	#17989615	C₂²	1	2	yes	—
18	#17989612	—	2	2	no	—
19	#17989605	C₂²	1	1	yes	characteristic
20	#17989605	C₂²	1	1	yes	characteristic
21	#17989594	—	2	2	no	—
22	#17989591	C₂²	1	2	yes	—
23	#17989450	C₂²	1	1	yes	characteristic
24	#17989450	C₂²	1	1	yes	characteristic
25	#17989450	C₂²	1	2	yes	—
26	#17989437	C₂²	1	2	yes	—
27	#17989437	C₂²	1	2	yes	—
28	#17989435	C₂²	1	2	yes	—
29	#17989435	C₂²	1	2	yes	—
30	#17989428	C₂²	1	1	yes	characteristic
31	#17989428	C₂²	1	1	yes	characteristic
32	C₈×#10596833	C₂²	1	1	yes	characteristic
33	C₈×#10596833	C₂²	1	1	yes	characteristic
34	#17989122	C₂²	1	2	yes	—
35	#17988971	C₄	1	1	yes	characteristic
36	#17988944	C₄	1	2	yes	—
37	#17988829	C₄	1	1	yes	characteristic
38	#17988824	C₄	1	1	yes	characteristic
39	#17988817	C₄	1	2	yes	—
40	#17988681	C₂²	1	1	yes	characteristic
41	#17988674	C₂²	1	2	yes	—
42	#17988669	C₂²	1	1	yes	characteristic
43	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
44	#499×C₇₃	C₄	1	1	yes	characteristic
45	C₂×#11924038	—	2	2	no	—
46	C₂×#11924038	—	2	2	no	—
47	C₂×#11924036	—	2	2	no	—
48	#14556658	—	2	2	no	—
49	#14556658	—	2	1	no	—
50	#14556658	D₄	1	1	yes	characteristic
51	#14556658	D₄	1	1	yes	characteristic
52	C₄⋊#10596833	—	2	2	no	—
53	C₄⋊#10596833	—	2	2	no	—
54	C₄⋊#10596833	C₂³	1	2	yes	—
55	C₄⋊#10596833	C₂³	1	2	yes	—
56	#14556648	—	2	2	no	—
57	#14556648	—	2	2	no	—
58	#14556648	—	2	1	no	—
59	#14556648	D₄	1	1	yes	characteristic
60	#14556648	D₄	1	1	yes	characteristic
61	#14556646	—	2	2	no	—
62	C₄×#10596833	—	2	2	no	—
63	C₄×#10596833	C₂³	1	1	yes	characteristic
64	C₄×#10596833	C₂³	1	1	yes	characteristic
65	C₄×#10596833	C₂³	1	1	yes	characteristic
66	C₄×#10596833	C₂³	1	1	yes	characteristic
67	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic
68	C₂×#11924017	C₂×C₄	1	2	yes	—
69	C₂×D₅₈₄	C₂×C₄	1	1	yes	characteristic
70	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
71	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
72	#14556586	—	2	2	no	—
73	#14556579	C₂×C₄	1	2	yes	—
74	#14556576	—	2	2	no	—
75	#14556571	C₂×C₄	1	2	yes	—
76	#14556566	C₂×C₄	1	2	yes	—
77	#14556556	C₂³	1	2	yes	—
78	#14556553	—	2	2	no	—
79	#14556553	C₂³	1	2	yes	—
80	#14556549	—	2	2	no	—
81	C₄²×D₇₃	C₂³	1	1	yes	characteristic
82	#14556514	C₂×C₄	1	2	yes	—
83	#14556514	C₂×C₄	1	2	yes	—
84	#14556511	C₂×C₄	1	2	yes	—
85	#14556511	C₂×C₄	1	2	yes	—
86	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
87	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
88	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
89	#125×C₇₃	C₂×C₄	1	2	yes	—
90	#107×C₇₃	C₂×C₄	1	1	yes	characteristic
91	#107×C₇₃	C₂×C₄	1	1	yes	characteristic
92	#106×C₇₃	C₂×C₄	1	2	yes	—
93	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
94	#11924038	—	8	2	no	—
95	#11924038	—	8	2	no	—
96	#11924036	—	8	2	no	—
97	C₂×#10596833	—	2	2	no	—
98	C₂×#10596833	—	2	2	no	—
99	C₂×#10596833	—	2	1	no	—
100	C₂×#10596833	—	2	1	no	—
101	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
102	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
103	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
104	C₂×#10596833	C₂×D₄	1	1	yes	characteristic
105	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
106	C₂×D₂₉₂	—	2	2	no	—
107	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
108	C₂×C₄×D₇₃	—	2	2	no	—
109	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
112	Dic₂₉₂	—	2	2	no	—
113	#11924017	—	2	4	no	—
114	D₅₈₄	—	2	2	no	—
115	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
116	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
117	C₈×D₇₃	—	2	2	no	—
118	C₈×D₇₃	—	2	2	no	—
119	C₄⋊Dic₇₃	—	2	2	no	—
120	#11924007	—	2	2	no	—
121	#11924005	—	2	2	no	—
122	#11924005	—	2	2	no	—
123	#11924005	—	2	2	no	—
124	#11924005	—	2	2	no	—
125	#11924005	—	2	2	no	—
126	C₄×Dic₇₃	—	2	2	no	—
127	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
128	C₄×Dic₇₃	#41	1	1	yes	characteristic
129	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
130	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
132	C₄⋊C₄×C₇₃	—	2	2	no	—
133	C₄⋊C₄×C₇₃	—	2	2	no	—
134	C₄⋊C₄×C₇₃	C₂²×C₄	1	2	yes	—
135	C₄⋊C₄×C₇₃	C₂²×C₄	1	2	yes	—
136	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
137	C₂²×D₇₃	—	2	2	no	—
138	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₇₃	—	2	2	no	—
140	C₂×C₄×C₇₃	—	2	2	no	—
141	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×Dic₇₃	—	2	2	no	—
145	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
146	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×Dic₇₃	C₂×#41	1	1	yes	characteristic
148	D₂₉₂	—	2	2	no	—
149	D₂₉₂	—	2	2	no	—
150	D₂₉₂	—	8	2	no	—
151	C₄×D₇₃	—	2	1	no	—
152	C₄×D₇₃	—	2	1	no	—
153	C₄×D₇₃	—	2	1	no	—
154	C₄×D₇₃	—	2	1	no	—
155	C₄×D₇₃	—	4	1	no	—
156	C₄×D₇₃	—	4	2	no	—
157	C₄×D₇₃	—	4	2	no	—
158	Dic₁₄₆	—	2	2	no	—
159	Dic₁₄₆	—	2	2	no	—
160	#10596833	—	2	1	no	—
161	#10596833	—	2	1	no	—
162	#10596833	—	2	1	no	—
163	#10596833	—	2	1	no	—
164	#10596833	—	4	1	no	—
165	#10596833	—	4	1	no	—
166	#10596833	—	4	2	no	—
167	#10596833	—	4	2	no	—
168	C₈×C₇₃	#131	1	2	yes	—
169	C₈×C₇₃	#131	1	2	yes	—
170	#10596831	—	2	1	no	—
171	#10596831	—	2	1	no	—
172	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂×D₇₃	—	2	1	no	—
174	C₂×D₇₃	—	2	1	no	—
175	C₂×D₇₃	—	2	1	no	—
176	C₂×D₇₃	—	4	2	no	—
177	C₂×D₇₃	—	4	2	no	—
178	C₂×D₇₃	—	4	2	no	—
179	C₄×C₇₃	—	2	1	no	—
180	C₄×C₇₃	—	2	1	no	—
181	C₄×C₇₃	—	4	2	no	—
182	C₄×C₇₃	—	4	2	no	—
183	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
184	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
185	Dic₇₃	—	2	1	no	—
186	Dic₇₃	—	2	1	no	—
187	Dic₇₃	—	4	2	no	—
188	Dic₇₃	#576	1	1	yes	characteristic
189	Dic₇₃	#578	1	1	yes	characteristic
190	#17973	—	73	1	no	maximal, Sylow
191	C₂×C₇₃	#2931	1	1	yes	characteristic
192	C₂×C₇₃	#2951	1	1	yes	characteristic
193	C₂×C₇₃	#2957	1	1	yes	characteristic
194	D₇₃	—	4	1	no	—
195	D₇₃	—	8	2	no	—
196	#3168	—	73	1	no	—
197	#2993	—	73	2	no	—
198	#1592	—	73	2	no	—
199	#1591	—	73	1	no	—
200	#1589	—	73	1	no	—
201	SD₁₆⋊C₈	—	73	2	no	—
202	C₈×Q₁₆	—	73	1	no	—
203	C₈×D₈	—	73	1	no	—
204	#1567	—	73	1	no	—
205	#1495	—	73	2	no	—
206	#1450	—	73	1	no	—
207	C₇₃	#17973	1	1	yes	characteristic, complemented, Sylow
208	#529	—	73	2	no	—
209	#499	#64408	1	1	yes	characteristic
210	#495	—	73	1	no	—
211	#488	—	73	2	no	—
212	#467	—	73	1	no	—
213	#466	—	73	1	no	—
214	#465	—	73	2	no	—
215	#446	—	73	2	no	—
216	C₈×Q₈	—	73	2	no	—
217	C₄×Q₁₆	—	73	1	no	—
218	C₄×SD₁₆	—	73	2	no	—
219	C₄×D₈	—	73	1	no	—
220	#436	—	73	2	no	—
221	#436	—	146	2	no	—
222	#435	—	146	2	no	—
223	C₈×D₄	—	73	2	no	—
224	#433	—	146	2	no	—
225	#431	—	73	1	no	—
226	#431	—	73	1	no	—
227	#424	—	146	2	no	—
228	#423	—	73	2	no	—
229	C₂×C₄×C₈	—	73	1	no	—
230	#335	—	73	1	no	—
231	#335	—	73	1	no	—
232	#335	—	73	2	no	—
233	Q₈⋊C₈	—	73	2	no	—
234	Q₈⋊C₈	—	73	2	no	—
235	D₄⋊C₈	—	73	2	no	—
236	D₄⋊C₈	—	73	2	no	—
237	#322	—	73	1	no	—
238	#322	—	73	1	no	—
239	C₈²	—	73	1	no	—
240	C₈²	—	73	1	no	—
241	C₂×Q₁₆	—	73	1	no	—
242	C₂×SD₁₆	—	73	2	no	—
243	C₂×D₈	—	73	1	no	—
244	C₂×M₁₆	—	146	2	no	—
245	C₂×M₁₆	—	146	2	no	—
246	C₂²×C₈	—	73	1	no	—
247	C₂²×C₈	—	73	1	no	—
248	C₂²×C₈	—	146	2	no	—
249	#126	—	146	2	no	—
250	#125	C₂×#64408	1	2	yes	—
251	#124	—	73	2	no	—
252	#123	—	146	2	no	—
253	C₂²⋊Q₈	—	73	2	no	—
254	#121	—	73	2	no	—
255	C₄×Q₈	—	73	2	no	—
256	C₄×D₄	—	73	2	no	—
257	C₄×D₄	—	146	2	no	—
258	#117	—	146	2	no	—
259	C₂×C₄²	—	73	1	no	—
260	#107	C₂×#64408	1	1	yes	characteristic
261	#107	C₂×#64408	1	1	yes	characteristic
262	#106	C₂×#64408	1	2	yes	—
263	C₄⋊C₈	—	73	1	no	—
264	C₄⋊C₈	—	73	1	no	—
265	C₄⋊C₈	—	73	2	no	—
266	C₄⋊C₈	—	73	2	no	—
267	C₄⋊C₈	—	146	2	no	—
268	C₄⋊C₈	—	146	1	no	—
269	C₄⋊C₈	—	146	2	no	—
270	C₄⋊C₈	—	146	2	no	—
271	#103	—	73	2	no	—
272	#103	—	73	2	no	—
273	#102	—	73	2	no	—
274	#102	—	73	2	no	—
275	C₂²⋊C₈	—	73	1	no	—
276	C₂²⋊C₈	—	73	1	no	—
277	C₂²⋊C₈	—	146	2	no	—
278	C₂²⋊C₈	—	146	2	no	—
279	C₂²⋊C₈	—	146	1	no	—
280	#97	—	146	2	no	—
281	C₄×C₈	—	73	1	no	—
282	C₄×C₈	—	73	1	no	—
283	C₄×C₈	—	73	1	no	—
284	C₄×C₈	—	73	1	no	—
285	C₄×C₈	—	73	1	no	—
286	C₄×C₈	—	73	1	no	—
287	C₄×C₈	—	73	1	no	—
288	C₄×C₈	—	146	2	no	—
289	C₄×C₈	C₂×#64408	1	1	yes	characteristic
290	C₂×Q₈	—	73	2	no	—
291	C₂×D₄	—	73	2	no	—
292	C₂×D₄	—	146	2	no	—
293	C₂²×C₄	—	73	1	no	—
294	C₂²×C₄	—	73	1	no	—
295	C₂²×C₄	—	73	1	no	—
296	C₂²×C₄	—	146	2	no	—
297	Q₁₆	—	146	2	no	—
298	SD₁₆	—	146	4	no	—
299	D₈	—	146	2	no	—
300	M₁₆	—	584	2	no	—
301	M₁₆	—	584	2	no	—
302	C₂×C₈	—	584	2	no	—
303	C₂×C₈	—	73	1	no	—
304	C₂×C₈	—	73	1	no	—
305	C₂×C₈	—	73	1	no	—
306	C₂×C₈	—	73	1	no	—
307	C₂×C₈	—	73	1	no	—
308	C₂×C₈	—	73	1	no	—
309	C₂×C₈	—	146	1	no	—
310	C₂×C₈	—	146	1	no	—
311	C₂×C₈	—	146	2	no	—
312	C₂×C₈	—	146	2	no	—
313	C₂×C₈	—	146	2	no	—
314	C₂×C₈	—	146	2	no	—
315	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
316	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	146	2	no	—
320	C₄⋊C₄	—	146	2	no	—
321	C₄⋊C₄	C₂²×#64408	1	2	yes	—
322	C₄⋊C₄	C₂²×#64408	1	2	yes	—
323	C₂²⋊C₄	—	146	2	no	—
324	C₂²⋊C₄	—	146	2	no	—
325	C₂²⋊C₄	—	146	2	no	—
326	C₂²⋊C₄	—	146	2	no	—
327	C₂²⋊C₄	—	146	2	no	—
328	C₄²	—	73	1	no	—
329	C₄²	—	73	1	no	—
330	C₄²	—	73	1	no	—
331	C₄²	—	146	2	no	—
332	C₄²	C₂²×#64408	1	1	yes	characteristic
333	C₂³	—	73	1	no	—
334	C₂³	—	146	2	no	—
335	Q₈	—	146	2	no	—
336	Q₈	—	146	2	no	—
337	D₄	—	584	2	no	—
338	D₄	—	146	2	no	—
339	D₄	—	146	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	292	1	no	—
343	C₂×C₄	—	292	2	no	—
344	C₂×C₄	—	292	2	no	—
345	C₂×C₄	—	73	1	no	—
346	C₂×C₄	—	73	1	no	—
347	C₂×C₄	—	73	1	no	—
348	C₂×C₄	—	146	1	no	—
349	C₂×C₄	—	146	1	no	—
350	C₂×C₄	—	146	2	no	—
351	C₂×C₄	—	146	1	no	—
352	C₂×C₄	—	146	1	no	—
353	C₂×C₄	D₄×#64408	1	1	yes	characteristic
354	C₂×C₄	D₄×#64408	1	1	yes	characteristic
355	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
356	C₈	—	292	2	no	—
357	C₈	—	292	2	no	—
358	C₈	—	292	1	no	—
359	C₈	—	292	1	no	—
360	C₈	—	146	1	no	—
361	C₈	—	146	1	no	—
362	C₈	—	146	1	no	—
363	C₈	—	146	1	no	—
364	C₈	—	146	1	no	—
365	C₈	—	146	1	no	—
366	C₈	#14556679	1	2	yes	—
367	C₈	#14556679	1	2	yes	—
368	C₂²	—	292	2	no	—
369	C₂²	—	292	2	no	—
370	C₂²	—	292	2	no	—
371	C₂²	—	146	1	no	—
372	C₂²	—	146	1	no	—
373	C₂²	—	146	1	no	—
374	C₂²	C₂×D₄×#64408	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	2	no	—
379	C₄	—	292	2	no	—
380	C₄	—	73	1	no	—
381	C₄	—	73	1	no	—
382	C₄	—	146	1	no	—
383	C₄	—	146	1	no	—
384	C₄	C₂×#14556679	1	1	yes	characteristic
385	C₄	C₂×#14556679	1	1	yes	characteristic
386	C₂	—	292	1	no	—
387	C₂	—	584	2	no	—
388	C₂	#22626656	1	1	yes	characteristic
389	C₂	#22626693	1	1	yes	characteristic
390	C₂	#22626708	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	#499×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#17988824	C₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	#499	#64408
hypocenter	C₇₃	#17973
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

Complements

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

N	H ≅ G/N	type
1. #22626877	C₂	semidirect
2. #22625573	C₂	semidirect
3. #22625570	C₂	semidirect
4. #22625563	C₂	semidirect
5. #22625523	C₂	semidirect
6. #22625519	C₂	semidirect
7. #22625512	C₂	semidirect
8. #22625489	C₂	semidirect
9. #22625289	C₂	semidirect
10. #22625135	C₂	semidirect
207. C₇₃	#17973	semidirect

Maximal subgroups

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

subgroup	index
#22626877	2
#22625573	2
#22625570	2
#22625563	2
#22625523	2
#22625519	2
#22625512	2
#22625489	2
#22625289	2
#22625135	2
#22620503	2
#17973	73