Subgroups — #34018894

Order 18688.

630 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626745	C₂	1	2	yes	maximal, complemented
2	#22625571	C₂	1	1	yes	maximal, characteristic, complemented
3	#22625567	C₂	1	2	yes	maximal, complemented
4	#22625564	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₁₆×#10596833	C₂	1	1	yes	maximal, characteristic, complemented
6	SD₁₆×#10596833	C₂	1	2	yes	maximal, complemented
7	D₈×#10596833	C₂	1	1	yes	maximal, characteristic, complemented
8	#22625492	C₂	1	1	yes	maximal, characteristic, complemented
9	#22625283	C₂	1	2	yes	maximal, complemented
10	C₈×#11924036	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	#17989619	—	2	2	no	—
16	#17989614	—	2	2	no	—
17	D₄×#10596833	C₂²	1	2	yes	—
18	#17989611	—	2	2	no	—
19	#17989609	C₂²	1	2	yes	—
20	#17989551	—	2	2	no	—
21	C₄×#11924036	C₂²	1	1	yes	characteristic
22	C₄×#11924036	C₂²	1	1	yes	characteristic
23	#17989450	C₂²	1	1	yes	characteristic
24	#17989450	C₂²	1	1	yes	characteristic
25	#17989448	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	C₈×#10596833	C₂²	1	1	yes	characteristic
31	C₈×#10596833	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×#11924037	—	2	2	no	—
46	C₂×#11924036	—	2	1	no	—
47	C₂×#11924036	D₄	1	1	yes	characteristic
48	C₂×#11924036	D₄	1	1	yes	characteristic
49	C₂²×#10596833	—	2	2	no	—
50	#14556658	—	2	2	no	—
51	C₄⋊#10596833	—	2	2	no	—
52	C₄⋊#10596833	C₂³	1	2	yes	—
53	C₄⋊#10596833	C₂³	1	2	yes	—
54	#14556648	—	2	2	no	—
55	C₂²⋊#10596833	—	2	2	no	—
56	C₂²⋊#10596833	—	2	2	no	—
57	#14556646	—	2	2	no	—
58	#14556646	—	2	2	no	—
59	C₄×#10596833	—	2	2	no	—
60	C₄×#10596833	—	2	1	no	—
61	C₄×#10596833	D₄	1	1	yes	characteristic
62	C₄×#10596833	D₄	1	1	yes	characteristic
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	#11924037	—	4	4	no	—
95	#11924036	—	2	4	no	—
96	#11924036	—	2	2	no	—
97	#11924036	—	2	2	no	—
98	C₂×#10596833	—	2	1	no	—
99	C₂×#10596833	—	2	1	no	—
100	C₂×#10596833	—	2	2	no	—
101	C₂×#10596833	—	2	2	no	—
102	C₂×#10596833	—	4	4	no	—
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×#10596833	C₂×D₄	1	1	yes	characteristic
107	C₂×C₂²⋊D₇₃	—	2	2	no	—
108	C₂²×Dic₇₃	—	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	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
113	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
114	C₈×D₇₃	—	2	2	no	—
115	C₈×D₇₃	—	2	2	no	—
116	C₄⋊Dic₇₃	—	2	2	no	—
117	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
118	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
119	#11924007	—	2	2	no	—
120	#11924007	—	2	2	no	—
121	#11924007	—	2	2	no	—
122	C₂²⋊Dic₇₃	—	2	2	no	—
123	C₂²⋊Dic₇₃	—	2	2	no	—
124	#11924005	—	2	2	no	—
125	C₄×Dic₇₃	—	2	2	no	—
126	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
127	C₄×Dic₇₃	#41	1	1	yes	characteristic
128	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
129	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
130	C₂×D₄×C₇₃	C₂²×C₄	1	2	yes	—
131	Q₁₆×C₇₃	—	2	2	no	—
132	SD₁₆×C₇₃	—	2	4	no	—
133	D₈×C₇₃	—	2	2	no	—
134	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
135	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
136	C₂²⋊C₄×C₇₃	—	2	2	no	—
137	C₂²⋊C₄×C₇₃	—	2	2	no	—
138	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
139	C₂³×C₇₃	—	2	2	no	—
140	C₂²×D₇₃	C₄×D₄	1	1	yes	characteristic
141	Q₈×C₇₃	—	2	2	no	—
142	Q₈×C₇₃	—	2	2	no	—
143	D₄×C₇₃	—	2	2	no	—
144	D₄×C₇₃	—	2	2	no	—
145	C₂×C₄×C₇₃	—	2	2	no	—
146	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
149	C₂²⋊D₇₃	—	4	4	no	—
150	C₂×Dic₇₃	—	2	2	no	—
151	C₂×Dic₇₃	—	2	2	no	—
152	C₂×Dic₇₃	—	4	2	no	—
153	C₂×Dic₇₃	—	4	2	no	—
154	C₂×Dic₇₃	C₄×D₄	1	1	yes	characteristic
155	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×Dic₇₃	C₂×#41	1	1	yes	characteristic
157	C₄×D₇₃	—	2	2	no	—
158	C₄×D₇₃	—	2	1	no	—
159	C₄×D₇₃	—	2	1	no	—
160	C₄×D₇₃	—	2	1	no	—
161	C₄×D₇₃	—	2	1	no	—
162	#10596833	—	2	2	no	—
163	#10596833	—	2	2	no	—
164	#10596833	—	2	1	no	—
165	#10596833	—	2	1	no	—
166	#10596833	—	4	2	no	—
167	#10596833	—	4	2	no	—
168	#10596833	#135	1	2	yes	—
169	#10596833	#135	1	2	yes	—
170	C₈×C₇₃	#131	1	2	yes	—
171	C₈×C₇₃	#131	1	2	yes	—
172	#10596831	—	2	1	no	—
173	#10596831	—	2	1	no	—
174	C₂²×C₇₃	—	4	2	no	—
175	C₂²×C₇₃	—	4	2	no	—
176	C₂²×C₇₃	—	4	2	no	—
177	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂×D₇₃	—	2	1	no	—
179	C₂×D₇₃	—	2	1	no	—
180	C₂×D₇₃	#443	1	2	yes	—
181	C₄×C₇₃	—	2	1	no	—
182	C₄×C₇₃	—	2	1	no	—
183	C₄×C₇₃	—	4	2	no	—
184	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
185	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
186	Dic₇₃	—	2	1	no	—
187	Dic₇₃	—	2	1	no	—
188	Dic₇₃	—	4	2	no	—
189	Dic₇₃	—	4	2	no	—
190	Dic₇₃	C₂×#135	1	1	yes	characteristic
191	Dic₇₃	C₂×#135	1	1	yes	characteristic
192	#17959	—	73	1	no	maximal, Sylow
193	C₂×C₇₃	—	8	2	no	—
194	C₂×C₇₃	#2931	1	1	yes	characteristic
195	C₂×C₇₃	C₄×#135	1	1	yes	characteristic
196	C₂×C₇₃	C₂×#443	1	1	yes	characteristic
197	D₇₃	—	2	2	no	—
198	#3168	—	73	1	no	—
199	#2966	—	73	2	no	—
200	#1591	—	73	1	no	—
201	#1590	—	73	2	no	—
202	#1589	—	73	1	no	—
203	C₈×Q₁₆	—	73	1	no	—
204	C₈×SD₁₆	—	73	2	no	—
205	C₈×D₈	—	73	1	no	—
206	#1568	—	73	1	no	—
207	#1494	—	73	2	no	—
208	C₂×C₈²	—	73	1	no	—
209	C₇₃	#17959	1	1	yes	characteristic, complemented, Sylow
210	#529	—	73	2	no	—
211	#499	—	73	1	no	—
212	#495	#64408	1	1	yes	characteristic
213	#488	—	73	2	no	—
214	#467	—	73	1	no	—
215	#466	—	73	1	no	—
216	#465	—	73	2	no	—
217	#446	—	73	2	no	—
218	C₈×Q₈	—	73	2	no	—
219	C₄×Q₁₆	—	73	1	no	—
220	C₄×SD₁₆	—	73	2	no	—
221	C₄×D₈	—	73	1	no	—
222	#436	—	73	2	no	—
223	#435	—	146	2	no	—
224	C₈×D₄	—	73	2	no	—
225	C₈×D₄	—	146	2	no	—
226	#433	—	146	2	no	—
227	#432	—	73	2	no	—
228	#405	—	146	2	no	—
229	C₂×C₄×C₈	—	73	1	no	—
230	C₂×C₄×C₈	—	73	1	no	—
231	C₂×C₄×C₈	—	73	1	no	—
232	#335	—	73	1	no	—
233	#335	—	73	1	no	—
234	#334	—	73	2	no	—
235	Q₈⋊C₈	—	73	2	no	—
236	Q₈⋊C₈	—	73	2	no	—
237	D₄⋊C₈	—	73	2	no	—
238	D₄⋊C₈	—	73	2	no	—
239	C₈²	—	73	1	no	—
240	C₈²	—	73	1	no	—
241	C₈²	—	73	1	no	—
242	C₈²	—	73	1	no	—
243	C₂×Q₁₆	C₂×#64408	1	1	yes	characteristic
244	C₂×SD₁₆	C₂×#64408	1	2	yes	—
245	C₂×D₈	C₂×#64408	1	1	yes	characteristic
246	C₂×M₁₆	—	146	2	no	—
247	C₂²×C₈	—	73	1	no	—
248	C₂²×C₈	—	73	1	no	—
249	C₂²×C₈	—	73	1	no	—
250	C₂²×C₈	—	73	1	no	—
251	C₂²×C₈	—	146	2	no	—
252	C₂²×C₈	—	146	1	no	—
253	#126	—	146	2	no	—
254	#125	—	73	2	no	—
255	#124	C₂×#64408	1	2	yes	—
256	#123	—	146	2	no	—
257	C₂²⋊Q₈	—	73	2	no	—
258	#121	—	73	2	no	—
259	C₄×Q₈	—	73	2	no	—
260	C₄×D₄	—	73	2	no	—
261	C₄×D₄	—	146	2	no	—
262	#117	—	146	2	no	—
263	C₂×C₄²	—	73	1	no	—
264	#107	—	73	1	no	—
265	#107	—	73	1	no	—
266	#106	—	73	2	no	—
267	C₄⋊C₈	—	73	2	no	—
268	C₄⋊C₈	—	73	2	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	—	146	2	no	—
276	C₂²⋊C₈	—	146	2	no	—
277	C₂²⋊C₈	—	146	2	no	—
278	#97	—	146	2	no	—
279	#97	—	146	2	no	—
280	C₄×C₈	—	73	1	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	—	73	1	no	—
289	C₄×C₈	—	146	1	no	—
290	C₄×C₈	—	146	2	no	—
291	C₄×C₈	C₂×#64408	1	1	yes	characteristic
292	C₂×Q₈	C₂²×#64408	1	2	yes	—
293	C₂×D₄	—	146	2	no	—
294	C₂×D₄	C₂²×#64408	1	2	yes	—
295	C₂²×C₄	—	73	1	no	—
296	C₂²×C₄	—	73	1	no	—
297	C₂²×C₄	—	73	1	no	—
298	C₂²×C₄	—	146	2	no	—
299	Q₁₆	—	2	2	no	—
300	SD₁₆	—	2	4	no	—
301	D₈	—	2	2	no	—
302	M₁₆	—	292	4	no	—
303	C₂×C₈	—	292	4	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	—	73	1	no	—
310	C₂×C₈	—	146	2	no	—
311	C₂×C₈	—	146	2	no	—
312	C₂×C₈	—	146	2	no	—
313	C₂×C₈	—	146	2	no	—
314	C₂×C₈	—	146	1	no	—
315	C₂×C₈	—	146	1	no	—
316	C₂×C₈	—	146	4	no	—
317	C₂×C₈	—	146	2	no	—
318	C₂×C₈	—	146	2	no	—
319	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
320	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
321	C₄⋊C₄	—	73	2	no	—
322	C₄⋊C₄	—	73	2	no	—
323	C₄⋊C₄	—	146	2	no	—
324	C₄⋊C₄	—	146	2	no	—
325	C₄⋊C₄	—	146	2	no	—
326	C₄⋊C₄	—	146	2	no	—
327	C₂²⋊C₄	—	2	2	no	—
328	C₂²⋊C₄	—	2	2	no	—
329	C₂²⋊C₄	—	146	2	no	—
330	C₂²⋊C₄	—	146	2	no	—
331	C₂²⋊C₄	—	146	2	no	—
332	C₄²	—	73	1	no	—
333	C₄²	—	73	1	no	—
334	C₄²	—	73	1	no	—
335	C₄²	—	146	2	no	—
336	C₄²	C₂²×#64408	1	1	yes	characteristic
337	C₂³	—	2	2	no	—
338	C₂³	—	73	1	no	—
339	Q₈	—	2	2	no	—
340	Q₈	—	2	2	no	—
341	D₄	—	2	2	no	—
342	D₄	—	2	2	no	—
343	D₄	—	292	4	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	292	2	no	—
346	C₂×C₄	—	292	2	no	—
347	C₂×C₄	—	73	1	no	—
348	C₂×C₄	—	73	1	no	—
349	C₂×C₄	—	73	1	no	—
350	C₂×C₄	—	146	1	no	—
351	C₂×C₄	—	146	1	no	—
352	C₂×C₄	—	146	2	no	—
353	C₂×C₄	—	146	2	no	—
354	C₂×C₄	—	146	2	no	—
355	C₂×C₄	—	146	1	no	—
356	C₂×C₄	—	146	1	no	—
357	C₂×C₄	D₄×#64408	1	1	yes	characteristic
358	C₂×C₄	D₄×#64408	1	1	yes	characteristic
359	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
360	C₈	—	292	2	no	—
361	C₈	—	292	2	no	—
362	C₈	—	73	2	no	—
363	C₈	—	73	2	no	—
364	C₈	—	146	1	no	—
365	C₈	—	146	1	no	—
366	C₈	—	146	2	no	—
367	C₈	—	146	2	no	—
368	C₈	—	146	1	no	—
369	C₈	—	146	1	no	—
370	C₈	#14556678	1	2	yes	—
371	C₈	#14556678	1	2	yes	—
372	C₂²	—	4	2	no	—
373	C₂²	—	4	2	no	—
374	C₂²	—	4	2	no	—
375	C₂²	—	73	2	no	—
376	C₂²	—	146	1	no	—
377	C₂²	—	146	1	no	—
378	C₂²	C₂×D₄×#64408	1	1	yes	characteristic
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	4	2	no	—
382	C₄	—	292	2	no	—
383	C₄	—	292	2	no	—
384	C₄	—	73	1	no	—
385	C₄	—	73	1	no	—
386	C₄	—	146	1	no	—
387	C₄	—	146	1	no	—
388	C₄	C₂×#14556678	1	1	yes	characteristic
389	C₄	C₂×#14556678	1	1	yes	characteristic
390	C₂	—	8	2	no	—
391	C₂	—	146	2	no	—
392	C₂	#22626653	1	1	yes	characteristic
393	C₂	#135×#64408	1	1	yes	characteristic
394	C₂	D₇₃⋊#443	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	#17959
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. #22626745	C₂	semidirect
2. #22625571	C₂	semidirect
3. #22625567	C₂	semidirect
4. #22625564	C₂	semidirect
5. Q₁₆×#10596833	C₂	semidirect
6. SD₁₆×#10596833	C₂	semidirect
7. D₈×#10596833	C₂	semidirect
8. #22625492	C₂	semidirect
9. #22625283	C₂	semidirect
10. C₈×#11924036	C₂	semidirect
209. C₇₃	#17959	semidirect

Maximal subgroups

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

subgroup	index
#22626745	2
#22625571	2
#22625567	2
#22625564	2
Q₁₆×#10596833	2
SD₁₆×#10596833	2
D₈×#10596833	2
#22625492	2
#22625283	2
C₈×#11924036	2
#22620498	2
#17959	73