Subgroups — #34019621

Order 18688.

600 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626763	C₂	1	2	yes	maximal, complemented
2	#22625571	C₂	1	1	yes	maximal, characteristic, complemented
3	#22625564	C₂	1	1	yes	maximal, characteristic, complemented
4	#22625540	C₂	1	2	yes	maximal, complemented
5	#22625536	C₂	1	2	yes	maximal, complemented
6	Q₁₆×#10596833	C₂	1	1	yes	maximal, characteristic, complemented
7	D₈×#10596833	C₂	1	1	yes	maximal, characteristic, complemented
8	#22625481	C₂	1	1	yes	maximal, characteristic, complemented
9	#22625243	C₂	1	2	yes	maximal, complemented
10	C₈×#11924038	C₂	1	1	yes	maximal, characteristic, complemented
11	#22620498	C₂	1	1	yes	maximal, characteristic
12	#17989638	C₂²	1	2	yes	—
13	Q₈×#10596833	C₂²	1	2	yes	—
14	#17989622	C₂²	1	2	yes	—
15	#17989620	—	2	2	no	—
16	#17989616	—	2	2	no	—
17	D₄×#10596833	C₂²	1	2	yes	—
18	#17989611	—	2	2	no	—
19	#17989606	C₂²	1	2	yes	—
20	#17989606	C₂²	1	1	yes	characteristic
21	#17989551	—	2	2	no	—
22	C₄×#11924038	C₂²	1	1	yes	characteristic
23	#17989450	C₂²	1	1	yes	characteristic
24	#17989450	C₂²	1	1	yes	characteristic
25	#17989449	C₂²	1	2	yes	—
26	Q₈⋊#10596833	C₂²	1	2	yes	—
27	Q₈⋊#10596833	C₂²	1	2	yes	—
28	D₄⋊#10596833	C₂²	1	2	yes	—
29	D₄⋊#10596833	C₂²	1	2	yes	—
30	#17989429	C₂²	1	1	yes	characteristic
31	#17989429	C₂²	1	1	yes	characteristic
32	C₈×#10596833	C₂²	1	1	yes	characteristic
33	C₈×#10596833	C₂²	1	1	yes	characteristic
34	#17989116	C₂²	1	2	yes	—
35	#17988989	C₄	1	1	yes	characteristic
36	#17988941	C₄	1	2	yes	—
37	#17988825	C₄	1	1	yes	characteristic
38	#17988820	C₄	1	1	yes	characteristic
39	#17988813	C₄	1	2	yes	—
40	Q₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
41	SD₁₆×Dic₇₃	C₂²	1	2	yes	—
42	D₈×Dic₇₃	C₂²	1	1	yes	characteristic
43	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
44	#495×C₇₃	C₄	1	1	yes	characteristic
45	C₂×#11924038	D₄	1	1	yes	characteristic
46	C₂×#11924038	D₄	1	1	yes	characteristic
47	C₂×#11924037	—	2	2	no	—
48	C₂²×#10596833	—	2	2	no	—
49	#14556658	—	2	2	no	—
50	C₄⋊#10596833	—	2	2	no	—
51	C₄⋊#10596833	—	2	1	no	—
52	C₄⋊#10596833	C₂³	1	2	yes	—
53	C₄⋊#10596833	C₂³	1	2	yes	—
54	#14556648	—	2	2	no	—
55	#14556648	—	2	1	no	—
56	C₂²⋊#10596833	—	2	2	no	—
57	C₂²⋊#10596833	—	2	2	no	—
58	#14556646	—	2	2	no	—
59	#14556646	D₄	1	1	yes	characteristic
60	#14556646	D₄	1	1	yes	characteristic
61	C₄×#10596833	—	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×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
69	#14556584	—	2	2	no	—
70	#14556583	C₂×C₄	1	2	yes	—
71	#14556573	—	2	2	no	—
72	#14556567	C₂×C₄	1	2	yes	—
73	#14556562	C₂×C₄	1	2	yes	—
74	Q₈×Dic₇₃	C₂³	1	2	yes	—
75	D₄×Dic₇₃	C₂³	1	2	yes	—
76	#14556551	—	2	2	no	—
77	#14556547	—	2	2	no	—
78	C₄²×D₇₃	C₂³	1	1	yes	characteristic
79	#14556525	C₂×C₄	1	1	yes	characteristic
80	#14556525	C₂×C₄	1	1	yes	characteristic
81	#14556523	C₂×C₄	1	2	yes	—
82	#14556515	C₂×C₄	1	2	yes	—
83	#14556515	C₂×C₄	1	2	yes	—
84	#14556512	C₂×C₄	1	2	yes	—
85	#14556512	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	C₂×Q₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
90	C₂×SD₁₆×C₇₃	C₂×C₄	1	2	yes	—
91	C₂×D₈×C₇₃	C₂×C₄	1	1	yes	characteristic
92	#124×C₇₃	C₂×C₄	1	2	yes	—
93	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
94	#11924038	—	2	2	no	—
95	#11924038	—	2	2	no	—
96	#11924037	—	4	4	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	—	4	4	no	—
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×#10596833	C₂×D₄	1	1	yes	characteristic
106	C₂×C₂²⋊D₇₃	—	2	2	no	—
107	C₂²×Dic₇₃	—	2	2	no	—
108	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×D₇₃	#41	1	1	yes	characteristic
111	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
112	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
113	C₈×D₇₃	—	2	2	no	—
114	C₈×D₇₃	—	4	1	no	—
115	C₄⋊Dic₇₃	—	2	2	no	—
116	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
117	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
118	#11924007	—	2	2	no	—
119	#11924007	—	2	2	no	—
120	#11924007	—	2	2	no	—
121	C₂²⋊Dic₇₃	—	2	2	no	—
122	C₂²⋊Dic₇₃	—	2	2	no	—
123	#11924005	—	2	2	no	—
124	C₄×Dic₇₃	—	2	2	no	—
125	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
126	C₄×Dic₇₃	#41	1	1	yes	characteristic
127	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
128	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
129	C₂×D₄×C₇₃	C₂²×C₄	1	2	yes	—
130	Q₁₆×C₇₃	—	2	2	no	—
131	SD₁₆×C₇₃	—	4	2	no	—
132	D₈×C₇₃	—	2	2	no	—
133	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
135	C₂²⋊C₄×C₇₃	—	2	2	no	—
136	C₂²⋊C₄×C₇₃	—	2	2	no	—
137	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
138	C₂³×C₇₃	—	2	2	no	—
139	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
140	Q₈×C₇₃	—	2	2	no	—
141	Q₈×C₇₃	—	2	2	no	—
142	D₄×C₇₃	—	2	2	no	—
143	D₄×C₇₃	—	2	2	no	—
144	C₂×C₄×C₇₃	—	2	2	no	—
145	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
148	C₂²⋊D₇₃	—	8	2	no	—
149	C₂×Dic₇₃	—	2	2	no	—
150	C₂×Dic₇₃	—	2	2	no	—
151	C₂×Dic₇₃	—	4	2	no	—
152	C₂×Dic₇₃	—	4	2	no	—
153	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
154	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×Dic₇₃	C₂×#41	1	1	yes	characteristic
156	C₄×D₇₃	—	2	1	no	—
157	C₄×D₇₃	—	2	1	no	—
158	C₄×D₇₃	—	2	1	no	—
159	C₄×D₇₃	—	2	1	no	—
160	C₄×D₇₃	—	4	1	no	—
161	#10596833	—	2	1	no	—
162	#10596833	—	2	1	no	—
163	#10596833	—	4	1	no	—
164	#10596833	—	4	1	no	—
165	#10596833	—	4	2	no	—
166	#10596833	—	4	2	no	—
167	#10596833	#135	1	2	yes	—
168	#10596833	#135	1	2	yes	—
169	C₈×C₇₃	—	2	1	no	—
170	C₈×C₇₃	#131	1	2	yes	—
171	#10596831	—	2	1	no	—
172	#10596831	—	2	1	no	—
173	C₂²×C₇₃	—	4	2	no	—
174	C₂²×C₇₃	—	4	2	no	—
175	C₂²×C₇₃	—	4	2	no	—
176	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂×D₇₃	—	2	1	no	—
178	C₂×D₇₃	—	2	1	no	—
179	C₂×D₇₃	—	2	1	no	—
180	C₄×C₇₃	—	2	1	no	—
181	C₄×C₇₃	—	2	1	no	—
182	C₄×C₇₃	—	4	2	no	—
183	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
184	C₄×C₇₃	#568	1	1	yes	characteristic
185	Dic₇₃	—	2	1	no	—
186	Dic₇₃	—	2	1	no	—
187	Dic₇₃	—	4	2	no	—
188	Dic₇₃	—	4	2	no	—
189	Dic₇₃	C₂×#135	1	1	yes	characteristic
190	Dic₇₃	C₂×#135	1	1	yes	characteristic
191	#18069	—	73	1	no	maximal, Sylow
192	C₂×C₇₃	—	8	2	no	—
193	C₂×C₇₃	#2935	1	1	yes	characteristic
194	C₂×C₇₃	C₄×#135	1	1	yes	characteristic
195	C₂×C₇₃	#2957	1	1	yes	characteristic
196	D₇₃	—	4	1	no	—
197	#3168	—	73	1	no	—
198	#2969	—	73	2	no	—
199	#1591	—	73	1	no	—
200	#1589	—	73	1	no	—
201	#1583	—	73	2	no	—
202	#1582	—	73	2	no	—
203	C₈×Q₁₆	—	73	1	no	—
204	C₈×D₈	—	73	1	no	—
205	#1564	—	73	1	no	—
206	#1482	—	73	2	no	—
207	C₈×M₁₆	—	73	1	no	—
208	C₇₃	#18069	1	1	yes	characteristic, complemented, Sylow
209	#529	—	73	2	no	—
210	#499	—	73	1	no	—
211	#495	#64408	1	1	yes	characteristic
212	#488	—	73	2	no	—
213	#467	—	73	1	no	—
214	#466	—	73	1	no	—
215	#465	—	73	2	no	—
216	#446	—	73	2	no	—
217	C₈×Q₈	—	73	2	no	—
218	C₄×Q₁₆	—	73	1	no	—
219	C₄×SD₁₆	—	73	2	no	—
220	C₄×D₈	—	73	1	no	—
221	#436	—	73	2	no	—
222	#436	—	146	2	no	—
223	#435	—	146	2	no	—
224	C₈×D₄	—	73	2	no	—
225	#433	—	146	2	no	—
226	#431	—	73	1	no	—
227	#431	—	73	2	no	—
228	#405	—	146	2	no	—
229	C₄×M₁₆	—	73	1	no	—
230	C₂×C₄×C₈	—	73	1	no	—
231	#335	—	73	1	no	—
232	#335	—	73	1	no	—
233	#334	—	73	2	no	—
234	Q₈⋊C₈	—	73	2	no	—
235	Q₈⋊C₈	—	73	2	no	—
236	D₄⋊C₈	—	73	2	no	—
237	D₄⋊C₈	—	73	2	no	—
238	#322	—	73	1	no	—
239	#322	—	73	1	no	—
240	C₈²	—	73	1	no	—
241	C₈²	—	73	1	no	—
242	C₂×Q₁₆	C₂×#64408	1	1	yes	characteristic
243	C₂×SD₁₆	C₂×#64408	1	2	yes	—
244	C₂×D₈	C₂×#64408	1	1	yes	characteristic
245	C₂×M₁₆	—	73	1	no	—
246	C₂×M₁₆	—	73	1	no	—
247	C₂×M₁₆	—	146	2	no	—
248	C₂²×C₈	—	73	1	no	—
249	C₂²×C₈	—	73	1	no	—
250	C₂²×C₈	—	146	2	no	—
251	#126	—	146	2	no	—
252	#125	—	73	2	no	—
253	#124	C₂×#64408	1	2	yes	—
254	#123	—	146	2	no	—
255	C₂²⋊Q₈	—	73	2	no	—
256	#121	—	73	2	no	—
257	C₄×Q₈	—	73	2	no	—
258	C₄×D₄	—	73	2	no	—
259	C₄×D₄	—	146	2	no	—
260	#117	—	146	2	no	—
261	C₂×C₄²	—	73	1	no	—
262	#107	—	73	1	no	—
263	#107	—	73	1	no	—
264	#106	—	73	2	no	—
265	C₄⋊C₈	—	73	2	no	—
266	C₄⋊C₈	—	73	2	no	—
267	C₄⋊C₈	—	146	1	no	—
268	C₄⋊C₈	—	146	2	no	—
269	C₄⋊C₈	—	146	2	no	—
270	#103	—	73	2	no	—
271	#103	—	73	2	no	—
272	#102	—	73	2	no	—
273	#102	—	73	2	no	—
274	C₂²⋊C₈	—	146	2	no	—
275	C₂²⋊C₈	—	146	2	no	—
276	C₂²⋊C₈	—	146	2	no	—
277	C₂²⋊C₈	—	146	1	no	—
278	#97	—	73	1	no	—
279	#97	—	73	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	—	146	2	no	—
290	C₄×C₈	C₂×#64408	1	1	yes	characteristic
291	C₂×Q₈	C₂²×#64408	1	2	yes	—
292	C₂×D₄	—	146	2	no	—
293	C₂×D₄	C₂²×#64408	1	2	yes	—
294	C₂²×C₄	—	73	1	no	—
295	C₂²×C₄	—	73	1	no	—
296	C₂²×C₄	—	73	1	no	—
297	C₂²×C₄	—	146	2	no	—
298	Q₁₆	—	2	2	no	—
299	SD₁₆	—	4	2	no	—
300	D₈	—	2	2	no	—
301	M₁₆	—	292	4	no	—
302	M₁₆	—	146	2	no	—
303	M₁₆	—	146	2	no	—
304	C₂×C₈	—	292	1	no	—
305	C₂×C₈	—	292	4	no	—
306	C₂×C₈	—	73	1	no	—
307	C₂×C₈	—	73	1	no	—
308	C₂×C₈	—	73	1	no	—
309	C₂×C₈	—	73	1	no	—
310	C₂×C₈	—	73	1	no	—
311	C₂×C₈	—	73	1	no	—
312	C₂×C₈	—	146	1	no	—
313	C₂×C₈	—	146	1	no	—
314	C₂×C₈	—	146	2	no	—
315	C₂×C₈	—	146	2	no	—
316	C₂×C₈	—	146	2	no	—
317	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
318	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
319	C₄⋊C₄	—	73	2	no	—
320	C₄⋊C₄	—	73	2	no	—
321	C₄⋊C₄	—	146	2	no	—
322	C₄⋊C₄	—	146	2	no	—
323	C₄⋊C₄	—	146	2	no	—
324	C₄⋊C₄	—	146	2	no	—
325	C₂²⋊C₄	—	2	2	no	—
326	C₂²⋊C₄	—	2	2	no	—
327	C₂²⋊C₄	—	146	2	no	—
328	C₂²⋊C₄	—	146	2	no	—
329	C₂²⋊C₄	—	146	2	no	—
330	C₄²	—	73	1	no	—
331	C₄²	—	73	1	no	—
332	C₄²	—	73	1	no	—
333	C₄²	—	146	2	no	—
334	C₄²	C₂²×#64408	1	1	yes	characteristic
335	C₂³	—	2	2	no	—
336	C₂³	—	73	1	no	—
337	Q₈	—	2	2	no	—
338	Q₈	—	2	2	no	—
339	D₄	—	2	2	no	—
340	D₄	—	2	2	no	—
341	D₄	—	584	2	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	292	1	no	—
344	C₂×C₄	—	292	2	no	—
345	C₂×C₄	—	292	2	no	—
346	C₂×C₄	—	73	1	no	—
347	C₂×C₄	—	73	1	no	—
348	C₂×C₄	—	73	1	no	—
349	C₂×C₄	—	146	1	no	—
350	C₂×C₄	—	146	1	no	—
351	C₂×C₄	—	146	2	no	—
352	C₂×C₄	—	146	2	no	—
353	C₂×C₄	—	146	1	no	—
354	C₂×C₄	—	146	1	no	—
355	C₂×C₄	D₄×#64408	1	1	yes	characteristic
356	C₂×C₄	D₄×#64408	1	1	yes	characteristic
357	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	292	2	no	—
360	C₈	—	292	2	no	—
361	C₈	—	292	1	no	—
362	C₈	—	292	1	no	—
363	C₈	—	73	2	no	—
364	C₈	—	73	2	no	—
365	C₈	—	146	1	no	—
366	C₈	—	146	1	no	—
367	C₈	—	146	1	no	—
368	C₈	—	146	1	no	—
369	C₈	#14556678	1	2	yes	—
370	C₂²	—	4	2	no	—
371	C₂²	—	4	2	no	—
372	C₂²	—	4	2	no	—
373	C₂²	—	146	1	no	—
374	C₂²	—	146	1	no	—
375	C₂²	—	146	1	no	—
376	C₂²	C₂×D₄×#64408	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	4	2	no	—
380	C₄	—	292	2	no	—
381	C₄	—	292	2	no	—
382	C₄	—	73	1	no	—
383	C₄	—	73	1	no	—
384	C₄	—	146	1	no	—
385	C₄	—	146	1	no	—
386	C₄	C₂×#14556678	1	1	yes	characteristic
387	C₄	#17989665	1	1	yes	characteristic
388	C₂	—	8	2	no	—
389	C₂	—	292	1	no	—
390	C₂	#22626674	1	1	yes	characteristic
391	C₂	#135×#64408	1	1	yes	characteristic
392	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	#495×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#17988820	C₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	#495	#64408
hypocenter	C₇₃	#18069
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #495

Complements

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

N	H ≅ G/N	type
1. #22626763	C₂	semidirect
2. #22625571	C₂	semidirect
3. #22625564	C₂	semidirect
4. #22625540	C₂	semidirect
5. #22625536	C₂	semidirect
6. Q₁₆×#10596833	C₂	semidirect
7. D₈×#10596833	C₂	semidirect
8. #22625481	C₂	semidirect
9. #22625243	C₂	semidirect
10. C₈×#11924038	C₂	semidirect
208. C₇₃	#18069	semidirect

Maximal subgroups

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

subgroup	index
#22626763	2
#22625571	2
#22625564	2
#22625540	2
#22625536	2
Q₁₆×#10596833	2
D₈×#10596833	2
#22625481	2
#22625243	2
C₈×#11924038	2
#22620498	2
#18069	73