Subgroups — #32988763

Order 18688.

670 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988966	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17988935	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988934	C₂	1	1	yes	maximal, characteristic
4	#22610891	C₂	1	1	yes	maximal, characteristic, complemented
5	#22610736	C₂	1	1	yes	maximal, characteristic
6	#22607758	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11924018	—	2	2	no	—
9	C₂×#14556591	C₂²	1	1	yes	characteristic
10	C₂×#14556588	C₂²	1	1	yes	characteristic
11	#17988966	—	2	2	no	—
12	#17988966	—	2	2	no	—
13	#17988935	—	2	2	no	—
14	#17988935	—	2	2	no	—
15	#17988934	—	2	2	no	—
16	#17988934	—	2	2	no	—
17	C₂×#14556513	—	2	2	no	—
18	C₂×#14556511	—	2	2	no	—
19	C₂×#14556511	—	2	2	no	—
20	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
21	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
22	#17988349	—	2	2	no	—
23	C₂×C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
25	#384×C₇₃	C₂²	1	1	yes	characteristic
26	#340×C₇₃	—	2	2	no	—
27	C₂²×Dic₁₄₆	—	2	1	no	—
28	C₂²×D₂₉₂	—	2	1	no	—
29	C₂²×D₂₉₂	—	2	1	no	—
30	C₂²×D₂₉₂	—	2	1	no	—
31	C₂×#11924018	—	2	8	no	—
32	C₂×#11924018	—	4	4	no	—
33	C₂×#11924018	—	4	4	no	—
34	C₂²×#10596831	D₄	1	2	yes	—
35	C₂²×#10596831	D₄	1	2	yes	—
36	#14556591	—	2	1	no	—
37	#14556591	—	2	1	no	—
38	#14556591	—	2	1	no	—
39	#14556591	D₄	1	2	yes	—
40	#14556588	—	2	1	no	—
41	#14556588	—	2	1	no	—
42	#14556588	—	2	1	no	—
43	#14556588	D₄	1	2	yes	—
44	C₂×C₄⋊Dic₇₃	—	2	1	no	—
45	C₂×C₄⋊Dic₇₃	—	2	1	no	—
46	#14556513	—	2	8	no	—
47	#14556511	—	2	8	no	—
48	#14556511	—	4	4	no	—
49	C₄×#10596831	—	2	1	no	—
50	C₄×#10596831	—	2	1	no	—
51	C₄×#10596831	—	2	1	no	—
52	C₄×#10596831	—	2	1	no	—
53	C₂²×Q₈×C₇₃	—	2	1	no	—
54	C₂²×C₈×C₇₃	D₄	1	2	yes	—
55	C₄⋊Q₈×C₇₃	—	2	1	no	—
56	C₄⋊Q₈×C₇₃	—	2	1	no	—
57	C₄⋊Q₈×C₇₃	—	2	1	no	—
58	C₄⋊Q₈×C₇₃	D₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
60	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
62	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
63	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
64	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₇₃	—	2	1	no	—
66	C₄×C₈×C₇₃	—	2	1	no	—
67	C₂³×D₇₃	—	4	1	no	—
68	C₂×Dic₁₄₆	—	2	2	no	—
69	C₂×Dic₁₄₆	—	4	2	no	—
70	C₂×Dic₁₄₆	—	4	2	no	—
71	C₂×Dic₁₄₆	—	4	1	no	—
72	C₂×D₂₉₂	—	2	2	no	—
73	C₂×D₂₉₂	—	2	2	no	—
74	C₂×D₂₉₂	—	2	2	no	—
75	C₂×D₂₉₂	—	4	2	no	—
76	C₂×D₂₉₂	—	4	2	no	—
77	C₂×D₂₉₂	—	4	2	no	—
78	C₂×D₂₉₂	—	4	2	no	—
79	C₂×D₂₉₂	—	4	2	no	—
80	C₂×D₂₉₂	—	4	2	no	—
81	C₂×D₂₉₂	—	4	1	no	—
82	C₂×D₂₉₂	—	4	1	no	—
83	C₂×D₂₉₂	—	4	1	no	—
84	C₂²×Dic₇₃	—	4	1	no	—
85	#11924018	—	4	16	no	—
86	C₂×#10596831	—	2	2	no	—
87	C₂×#10596831	—	2	2	no	—
88	C₂×#10596831	—	2	2	no	—
89	C₂×#10596831	—	2	2	no	—
90	C₂×#10596831	—	2	2	no	—
91	C₂×#10596831	D₈	1	4	yes	—
92	C₄⋊Dic₇₃	—	2	2	no	—
93	C₄⋊Dic₇₃	—	2	2	no	—
94	C₄⋊Dic₇₃	—	4	1	no	—
95	C₄⋊Dic₇₃	—	4	1	no	—
96	C₂×Q₈×C₇₃	—	2	2	no	—
97	C₂×Q₈×C₇₃	—	4	1	no	—
98	C₂×Q₈×C₇₃	—	4	2	no	—
99	C₂×Q₈×C₇₃	—	4	2	no	—
100	C₂²×C₄×C₇₃	—	4	1	no	—
101	C₂²×C₄×C₇₃	—	4	1	no	—
102	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₇₃	—	2	2	no	—
106	C₂×C₈×C₇₃	—	2	2	no	—
107	C₂×C₈×C₇₃	—	2	2	no	—
108	C₄⋊C₄×C₇₃	—	2	2	no	—
109	C₄⋊C₄×C₇₃	—	2	2	no	—
110	C₄⋊C₄×C₇₃	—	4	1	no	—
111	C₄⋊C₄×C₇₃	—	4	1	no	—
112	C₄⋊C₄×C₇₃	—	4	2	no	—
113	C₄⋊C₄×C₇₃	—	4	2	no	—
114	C₄⋊C₄×C₇₃	—	4	2	no	—
115	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₇₃	#41	1	1	yes	characteristic
119	C₂³×C₇₃	#127	1	1	yes	characteristic
120	C₂²×D₇₃	—	4	2	no	—
121	C₂²×D₇₃	—	8	1	no	—
122	C₂²×D₇₃	—	8	1	no	—
123	C₂²×D₇₃	—	8	1	no	—
124	C₂²×D₇₃	—	8	1	no	—
125	C₂²×D₇₃	—	8	1	no	—
126	C₂²×D₇₃	—	8	1	no	—
127	Q₈×C₇₃	—	4	4	no	—
128	Q₈×C₇₃	—	8	2	no	—
129	C₂×C₄×C₇₃	—	4	2	no	—
130	C₂×C₄×C₇₃	—	8	1	no	—
131	C₂×C₄×C₇₃	—	8	1	no	—
132	C₂×C₄×C₇₃	—	8	1	no	—
133	C₂×C₄×C₇₃	—	8	1	no	—
134	C₂×C₄×C₇₃	—	8	1	no	—
135	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
136	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
137	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
138	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
139	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
140	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
141	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
142	C₂×C₄×C₇₃	#135	1	2	yes	—
143	C₂×C₄×C₇₃	#135	1	2	yes	—
144	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
145	C₂×C₄×C₇₃	#137	1	2	yes	—
146	C₂×C₄×C₇₃	#137	1	2	yes	—
147	C₂×Dic₇₃	—	4	2	no	—
148	C₂×Dic₇₃	—	8	1	no	—
149	C₂×Dic₇₃	—	8	1	no	—
150	D₂₉₂	—	4	4	no	—
151	D₂₉₂	—	4	4	no	—
152	D₂₉₂	—	4	4	no	—
153	D₂₉₂	—	8	2	no	—
154	D₂₉₂	—	8	2	no	—
155	D₂₉₂	—	8	2	no	—
156	Dic₁₄₆	—	4	4	no	—
157	Dic₁₄₆	—	8	2	no	—
158	C₈×C₇₃	—	4	2	no	—
159	#10596831	—	2	4	no	—
160	#10596831	—	4	2	no	—
161	C₂²×C₇₃	#392	1	1	yes	characteristic
162	C₂²×C₇₃	#493	1	1	yes	characteristic
163	C₂²×C₇₃	#495	1	1	yes	characteristic
164	C₂²×C₇₃	#495	1	1	yes	characteristic
165	C₂²×C₇₃	#496	1	1	yes	characteristic
166	C₂²×C₇₃	#497	1	1	yes	characteristic
167	C₂²×C₇₃	#497	1	1	yes	characteristic
168	C₂×D₇₃	—	8	2	no	—
169	C₂×D₇₃	—	8	2	no	—
170	C₂×D₇₃	—	8	2	no	—
171	C₂×D₇₃	—	16	1	no	—
172	C₂×D₇₃	—	16	1	no	—
173	C₂×D₇₃	—	16	1	no	—
174	C₂×D₇₃	—	16	1	no	—
175	C₄×C₇₃	—	8	2	no	—
176	C₄×C₇₃	—	16	1	no	—
177	C₄×C₇₃	#447	1	2	yes	—
178	C₄×C₇₃	#452	1	2	yes	—
179	C₄×C₇₃	#452	1	2	yes	—
180	C₄×C₇₃	#459	1	2	yes	—
181	C₄×C₇₃	#464	1	2	yes	—
182	C₄×C₇₃	#464	1	2	yes	—
183	Dic₇₃	—	8	2	no	—
184	#10306	—	73	1	no	maximal, Sylow
185	C₂×C₇₃	#1709	1	1	yes	characteristic
186	C₂×C₇₃	#1720	1	1	yes	characteristic
187	C₂×C₇₃	#1720	1	1	yes	characteristic
188	C₂×C₇₃	#1722	1	1	yes	characteristic
189	C₂×C₇₃	#2011	1	1	yes	characteristic
190	C₂×C₇₃	#2015	1	1	yes	characteristic
191	C₂×C₇₃	#2015	1	1	yes	characteristic
192	D₇₃	—	16	2	no	—
193	C₂×#494	—	73	1	no	—
194	C₂×#486	—	73	1	no	—
195	C₂×#486	—	73	1	no	—
196	#1990	—	73	1	no	—
197	#1990	D₇₃	1	1	yes	characteristic, complemented
198	#1957	—	73	1	no	—
199	#1273	—	73	1	no	—
200	C₇₃	#10306	1	1	yes	characteristic, complemented, Sylow
201	C₂²×Q₁₆	—	146	2	no	—
202	C₂×C₄⋊Q₈	—	73	1	no	—
203	C₂×C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
204	C₂×#127	—	73	1	no	—
205	#494	—	146	2	no	—
206	#494	—	146	2	no	—
207	#486	—	146	2	no	—
208	#486	—	146	2	no	—
209	#486	—	146	2	no	—
210	#486	—	146	2	no	—
211	C₂×#102	—	146	2	no	—
212	C₂×#102	—	146	2	no	—
213	C₂×#102	—	146	2	no	—
214	C₂×C₄×C₈	—	73	1	no	—
215	C₂×C₄×C₈	—	73	1	no	—
216	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
217	#384	C₂×D₇₃	1	1	yes	characteristic
218	#340	—	2	2	no	—
219	#340	—	146	2	no	—
220	C₂²×Q₈	—	2	1	no	—
221	C₂²×Q₈	—	146	1	no	—
222	C₂²×D₄	—	146	1	no	—
223	C₂²×D₄	—	146	1	no	—
224	C₂²×D₄	—	146	1	no	—
225	C₂×Q₁₆	—	292	4	no	—
226	C₂×Q₁₆	—	292	4	no	—
227	C₂×Q₁₆	—	146	8	no	—
228	C₂²×C₈	—	73	2	no	—
229	C₂²×C₈	—	73	2	no	—
230	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	73	2	no	—
235	C₄⋊Q₈	—	146	1	no	—
236	C₄⋊Q₈	—	146	1	no	—
237	C₄⋊Q₈	—	146	1	no	—
238	C₄⋊Q₈	D₂₉₂	1	2	yes	—
239	#127	—	73	2	no	—
240	#127	—	146	1	no	—
241	#127	—	146	1	no	—
242	#127	—	146	1	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄⋊C₄	—	146	1	no	—
249	C₂×C₄⋊C₄	—	146	1	no	—
250	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
251	#102	—	292	4	no	—
252	#102	—	146	8	no	—
253	#102	—	146	8	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	—	2	1	no	—
256	C₄×C₈	—	146	1	no	—
257	C₄×C₈	—	146	1	no	—
258	C₄×C₈	—	146	1	no	—
259	C₄×C₈	—	146	1	no	—
260	C₂⁴	—	292	1	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	4	1	no	—
263	C₂×Q₈	—	4	2	no	—
264	C₂×Q₈	—	4	2	no	—
265	C₂×Q₈	—	292	1	no	—
266	C₂×Q₈	—	292	2	no	—
267	C₂×Q₈	—	292	2	no	—
268	C₂×Q₈	—	146	2	no	—
269	C₂×D₄	—	292	1	no	—
270	C₂×D₄	—	292	1	no	—
271	C₂×D₄	—	292	1	no	—
272	C₂×D₄	—	292	2	no	—
273	C₂×D₄	—	292	2	no	—
274	C₂×D₄	—	292	2	no	—
275	C₂×D₄	—	292	2	no	—
276	C₂×D₄	—	292	2	no	—
277	C₂×D₄	—	292	2	no	—
278	C₂×D₄	—	146	2	no	—
279	C₂×D₄	—	146	2	no	—
280	C₂×D₄	—	146	2	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	—	4	1	no	—
283	C₂²×C₄	—	292	1	no	—
284	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
285	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
287	Q₁₆	—	292	16	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	73	4	no	—
292	C₂×C₈	—	146	2	no	—
293	C₂×C₈	—	146	2	no	—
294	C₂×C₈	—	146	2	no	—
295	C₂×C₈	—	146	2	no	—
296	C₂×C₈	—	146	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	2	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	4	2	no	—
304	C₄⋊C₄	—	292	1	no	—
305	C₄⋊C₄	—	292	1	no	—
306	C₄⋊C₄	—	146	2	no	—
307	C₄⋊C₄	—	146	2	no	—
308	C₄²	D₄×D₇₃	1	1	yes	characteristic
309	C₄²	D₄×D₇₃	1	1	yes	characteristic
310	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
311	C₄²	#11924028	1	1	yes	characteristic
312	C₂³	—	292	2	no	—
313	C₂³	—	584	1	no	—
314	C₂³	—	584	1	no	—
315	C₂³	—	584	1	no	—
316	C₂³	—	584	1	no	—
317	C₂³	—	584	1	no	—
318	C₂³	—	584	1	no	—
319	C₂³	#14556587	1	1	yes	characteristic
320	Q₈	—	4	4	no	—
321	Q₈	—	8	2	no	—
322	Q₈	—	292	4	no	—
323	Q₈	—	584	2	no	—
324	D₄	—	292	4	no	—
325	D₄	—	292	4	no	—
326	D₄	—	292	4	no	—
327	D₄	—	584	2	no	—
328	D₄	—	584	2	no	—
329	D₄	—	584	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	8	1	no	—
332	C₂×C₄	—	8	1	no	—
333	C₂×C₄	—	8	1	no	—
334	C₂×C₄	—	8	1	no	—
335	C₂×C₄	—	8	1	no	—
336	C₂×C₄	—	292	2	no	—
337	C₂×C₄	—	584	1	no	—
338	C₂×C₄	—	584	1	no	—
339	C₂×C₄	#14556559	1	1	yes	characteristic
340	C₂×C₄	#14556561	1	1	yes	characteristic
341	C₂×C₄	#14556561	1	1	yes	characteristic
342	C₂×C₄	#14556564	1	1	yes	characteristic
343	C₂×C₄	#14556564	1	1	yes	characteristic
344	C₂×C₄	#14556565	1	1	yes	characteristic
345	C₂×C₄	D₈×D₇₃	1	2	yes	—
346	C₂×C₄	#14556611	1	2	yes	—
347	C₂×C₄	#14556612	1	2	yes	—
348	C₂×C₄	#14556618	1	2	yes	—
349	C₂×C₄	#14556622	1	2	yes	—
350	C₂×C₄	#14556623	1	2	yes	—
351	C₈	—	4	2	no	—
352	C₈	—	292	2	no	—
353	C₈	—	146	4	no	—
354	C₂²	—	584	2	no	—
355	C₂²	—	584	2	no	—
356	C₂²	—	584	2	no	—
357	C₂²	—	1168	1	no	—
358	C₂²	—	1168	1	no	—
359	C₂²	—	1168	1	no	—
360	C₂²	—	1168	1	no	—
361	C₂²	#17988496	1	1	yes	characteristic
362	C₂²	#17988962	1	1	yes	characteristic
363	C₂²	#17988968	1	1	yes	characteristic
364	C₂²	#17988969	1	1	yes	characteristic
365	C₂²	#17988973	1	1	yes	characteristic
366	C₂²	#17988979	1	1	yes	characteristic
367	C₂²	#17988980	1	1	yes	characteristic
368	C₄	—	8	2	no	—
369	C₄	—	16	1	no	—
370	C₄	—	584	2	no	—
371	C₄	#17988722	1	2	yes	—
372	C₄	#17988747	1	2	yes	—
373	C₄	#17988752	1	2	yes	—
374	C₄	#17988781	1	2	yes	—
375	C₄	#17988809	1	2	yes	—
376	C₄	#17988812	1	2	yes	—
377	C₂	—	1168	2	no	—
378	C₂	#22609535	1	1	yes	characteristic
379	C₂	#22609588	1	1	yes	characteristic
380	C₂	#22609589	1	1	yes	characteristic
381	C₂	#22609601	1	1	yes	characteristic
382	C₂	#22611003	1	1	yes	characteristic
383	C₂	#22611024	1	1	yes	characteristic
384	C₂	#22611026	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556587
commutator, cosocle	C₂×C₄²×C₇₃	C₂³
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1990	C₂
socle	C₂³×C₇₃	#127
prime radical	C₂×#14556588	C₂²
hypercenter	#1990	D₇₃
hypocenter	C₇₃	#10306
LCS	G ▷ C₂×C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1990

Complements

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

N	H ≅ G/N	type
1. C₂×#17988966	C₂	semidirect
4. #22610891	C₂	semidirect
6. #22607758	C₂	semidirect
7. C₇₃×#1990	C₂	semidirect
197. #1990	D₇₃	semidirect
200. C₇₃	#10306	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988966	2
C₂×#17988935	2
C₂×#17988934	2
#22610891	2
#22610736	2
#22607758	2
C₇₃×#1990	2
#10306	73