Subgroups — #32979047

Order 18688.

526 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988652	C₂	1	1	yes	maximal, characteristic
2	#22610058	C₂	1	1	yes	maximal, characteristic, complemented
3	#22609932	C₂	1	1	yes	maximal, characteristic
4	#22609928	C₂	1	1	yes	maximal, characteristic
5	C₂×#17988337	C₂	1	1	yes	maximal, characteristic, complemented
6	#22607815	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₇₃	—	2	1	no	—
9	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
10	#17988652	—	2	2	no	—
11	#17988652	C₄	1	4	yes	complemented
12	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
13	C₂×#14556513	—	2	1	no	—
14	C₂×#14556511	—	2	1	no	—
15	C₂×#14556503	—	2	1	no	—
16	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
17	#17988457	C₂²	1	1	yes	characteristic
18	#17988350	—	2	1	no	—
19	#17988349	—	2	1	no	—
20	#17988349	—	2	1	no	—
21	#17988349	—	2	1	no	—
22	#17988337	—	2	2	no	—
23	C₂×C₄⋊C₈×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	C₂²×D₂₉₂	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₇₃	—	2	1	no	—
28	C₂²×#10596831	—	2	1	no	—
29	C₂²×#10596831	D₄	1	1	yes	characteristic
30	C₂²×#10596831	D₄	1	1	yes	characteristic
31	C₂×C₈×D₇₃	—	2	4	no	—
32	C₂×C₈×D₇₃	—	4	2	no	—
33	C₂×C₈×D₇₃	—	4	2	no	—
34	C₄×D₂₉₂	—	2	1	no	—
35	C₄×D₂₉₂	—	2	1	no	—
36	C₄×D₂₉₂	C₂×C₄	1	2	yes	—
37	C₄×D₂₉₂	Q₈	1	2	yes	—
38	C₂×C₄⋊Dic₇₃	—	2	1	no	—
39	C₂×C₄⋊Dic₇₃	—	2	1	no	—
40	C₂×C₄⋊Dic₇₃	—	2	1	no	—
41	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
42	C₂×#11924005	—	2	1	no	—
43	C₄⋊#10596831	—	2	1	no	—
44	C₄⋊#10596831	—	2	1	no	—
45	#14556513	—	4	2	no	—
46	#14556511	—	4	2	no	—
47	#14556503	—	2	4	no	—
48	C₄×#10596831	C₂×C₄	1	2	yes	—
49	C₄×#10596831	D₄	1	2	yes	—
50	C₂²×C₈×C₇₃	—	2	1	no	—
51	C₂²×C₈×C₇₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
53	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
55	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₇₃	—	2	1	no	—
57	C₄⋊C₈×C₇₃	—	2	1	no	—
58	C₄⋊C₈×C₇₃	—	2	1	no	—
59	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
60	C₂³×D₇₃	—	2	1	no	—
61	C₂×D₂₉₂	—	2	1	no	—
62	C₂×D₂₉₂	—	2	1	no	—
63	C₂×D₂₉₂	—	2	1	no	—
64	C₂×D₂₉₂	C₄⋊C₄	1	2	yes	—
65	C₂×D₂₉₂	SD₁₆	1	2	yes	—
66	C₂×D₂₉₂	SD₁₆	1	2	yes	—
67	C₂²×Dic₇₃	—	2	1	no	—
68	C₂²×Dic₇₃	—	4	1	no	—
69	C₂×C₄×D₇₃	—	2	2	no	—
70	C₂×C₄×D₇₃	—	2	2	no	—
71	C₂×C₄×D₇₃	—	4	1	no	—
72	C₂×C₄×D₇₃	—	4	1	no	—
73	C₂×C₄×D₇₃	—	4	1	no	—
74	C₂×C₄×D₇₃	—	4	1	no	—
75	C₂×#10596831	—	2	1	no	—
76	C₂×#10596831	—	2	1	no	—
77	C₂×#10596831	—	2	1	no	—
78	C₂×#10596831	—	2	1	no	—
79	C₂×#10596831	—	4	1	no	—
80	C₂×#10596831	—	4	1	no	—
81	C₂×#10596831	—	4	1	no	—
82	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
83	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
84	C₈×D₇₃	—	4	8	no	—
85	C₄⋊Dic₇₃	—	2	1	no	—
86	C₄⋊Dic₇₃	—	4	1	no	—
87	C₄⋊Dic₇₃	—	4	1	no	—
88	C₄⋊Dic₇₃	—	4	1	no	—
89	C₄⋊Dic₇₃	—	4	1	no	—
90	C₄⋊Dic₇₃	—	8	1	no	—
91	C₄⋊Dic₇₃	C₄⋊C₄	1	2	yes	—
92	#11924005	—	2	2	no	—
93	#11924005	—	2	2	no	—
94	C₂²×C₄×C₇₃	—	4	1	no	—
95	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
98	C₂×C₈×C₇₃	—	2	2	no	—
99	C₂×C₈×C₇₃	—	4	1	no	—
100	C₂×C₈×C₇₃	—	4	1	no	—
101	C₂×C₈×C₇₃	—	4	1	no	—
102	C₂×C₈×C₇₃	—	4	1	no	—
103	C₂×C₈×C₇₃	—	4	1	no	—
104	C₄⋊C₄×C₇₃	—	4	1	no	—
105	C₄⋊C₄×C₇₃	—	4	1	no	—
106	C₄⋊C₄×C₇₃	—	4	1	no	—
107	C₄⋊C₄×C₇₃	—	4	1	no	—
108	C₄⋊C₄×C₇₃	—	8	1	no	—
109	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
112	C₄²×C₇₃	#41	1	1	yes	characteristic
113	C₂³×C₇₃	#121	1	1	yes	characteristic
114	C₂²×D₇₃	—	2	2	no	—
115	C₂²×D₇₃	—	2	2	no	—
116	C₂²×D₇₃	—	2	2	no	—
117	C₂²×D₇₃	—	4	1	no	—
118	C₂²×D₇₃	—	4	1	no	—
119	C₂²×D₇₃	—	4	1	no	—
120	C₂²×D₇₃	—	4	1	no	—
121	C₂×C₄×C₇₃	—	2	1	no	—
122	C₂×C₄×C₇₃	—	2	1	no	—
123	C₂×C₄×C₇₃	—	8	1	no	—
124	C₂×C₄×C₇₃	—	8	1	no	—
125	C₂×C₄×C₇₃	—	8	1	no	—
126	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
129	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
130	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
135	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
136	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
137	C₂×C₄×C₇₃	#135	1	1	yes	characteristic
138	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₂×Dic₇₃	—	2	2	no	—
141	C₂×Dic₇₃	—	4	1	no	—
142	C₂×Dic₇₃	—	4	1	no	—
143	C₂×Dic₇₃	—	8	1	no	—
144	C₂×Dic₇₃	—	8	1	no	—
145	C₂×Dic₇₃	—	8	1	no	—
146	D₂₉₂	—	2	2	no	—
147	D₂₉₂	—	2	2	no	—
148	D₂₉₂	—	2	2	no	—
149	D₂₉₂	#106	1	4	yes	—
150	C₄×D₇₃	—	4	2	no	—
151	C₄×D₇₃	—	4	2	no	—
152	C₄×D₇₃	—	4	2	no	—
153	C₄×D₇₃	—	4	2	no	—
154	C₈×C₇₃	—	4	2	no	—
155	C₈×C₇₃	—	8	1	no	—
156	#10596831	—	2	2	no	—
157	#10596831	—	2	2	no	—
158	#10596831	—	8	1	no	—
159	C₂²×C₇₃	#380	1	1	yes	characteristic
160	C₂²×C₇₃	#463	1	1	yes	characteristic
161	C₂²×C₇₃	#463	1	1	yes	characteristic
162	C₂²×C₇₃	#465	1	1	yes	characteristic
163	C₂²×C₇₃	#465	1	1	yes	characteristic
164	C₂²×C₇₃	#472	1	1	yes	characteristic
165	C₂²×C₇₃	#473	1	1	yes	characteristic
166	C₂×D₇₃	—	2	4	no	—
167	C₂×D₇₃	—	4	2	no	—
168	C₂×D₇₃	—	4	2	no	—
169	C₂×D₇₃	—	4	2	no	—
170	C₂×D₇₃	—	4	2	no	—
171	C₂×D₇₃	—	4	2	no	—
172	C₂×D₇₃	—	4	2	no	—
173	C₄×C₇₃	—	2	1	no	—
174	C₄×C₇₃	—	2	1	no	—
175	C₄×C₇₃	—	16	1	no	—
176	C₄×C₇₃	#416	1	1	yes	characteristic
177	C₄×C₇₃	#418	1	1	yes	characteristic
178	C₄×C₇₃	C₂×#106	1	1	yes	characteristic
179	C₄×C₇₃	#427	1	1	yes	characteristic
180	C₄×C₇₃	#465	1	1	yes	characteristic
181	C₄×C₇₃	#465	1	1	yes	characteristic
182	C₄×C₇₃	#478	1	1	yes	characteristic
183	C₄×C₇₃	#478	1	1	yes	characteristic
184	Dic₇₃	—	4	2	no	—
185	Dic₇₃	—	16	1	no	—
186	#8646	—	73	1	no	maximal, Sylow
187	C₂×C₇₃	#1677	1	1	yes	characteristic
188	C₂×C₇₃	#1677	1	1	yes	characteristic
189	C₂×C₇₃	#1679	1	1	yes	characteristic
190	C₂×C₇₃	#1679	1	1	yes	characteristic
191	C₂×C₇₃	#1804	1	1	yes	characteristic
192	C₂×C₇₃	#1807	1	1	yes	characteristic
193	C₂×C₇₃	#1816	1	1	yes	characteristic
194	D₇₃	—	4	4	no	—
195	C₂×C₈×D₄	—	73	1	no	—
196	#1834	—	73	1	no	—
197	#1802	—	73	1	no	—
198	#1802	—	73	1	no	—
199	C₂×#334	—	73	1	no	—
200	#1296	—	73	1	no	—
201	#1296	D₇₃	1	1	yes	characteristic, complemented
202	C₇₃	#8646	1	1	yes	characteristic, complemented, Sylow
203	C₂³×C₈	—	146	1	no	—
204	C₂×C₄×D₄	—	73	1	no	—
205	C₈×D₄	—	73	4	no	—
206	C₈×D₄	—	146	2	no	—
207	C₂×C₄⋊C₈	—	73	1	no	—
208	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
209	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
210	C₂×#102	—	146	1	no	—
211	C₂×#102	—	146	1	no	—
212	C₂×C₂²⋊C₈	—	146	1	no	—
213	C₂×C₄×C₈	—	73	1	no	—
214	#384	—	73	1	no	—
215	#384	C₂×D₇₃	1	1	yes	characteristic
216	#340	—	146	1	no	—
217	#340	—	146	1	no	—
218	#340	—	146	1	no	—
219	#340	—	146	1	no	—
220	#334	—	146	2	no	—
221	C₂²×D₄	—	73	1	no	—
222	C₂³×C₄	—	146	1	no	—
223	C₂²×C₈	—	2	1	no	—
224	C₂²×C₈	—	2	1	no	—
225	C₂²×C₈	—	292	2	no	—
226	C₂²×C₈	—	292	2	no	—
227	C₂²×C₈	—	73	1	no	—
228	C₂²×C₈	—	73	1	no	—
229	C₂²×C₈	—	146	4	no	—
230	C₂²×C₈	—	146	1	no	—
231	C₄×D₄	—	73	2	no	—
232	C₄×D₄	—	73	2	no	—
233	C₄×D₄	—	146	1	no	—
234	C₄×D₄	—	146	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	—	73	1	no	—
239	C₂×C₄⋊C₄	—	146	1	no	—
240	C₂×C₄⋊C₄	—	146	1	no	—
241	C₂×C₄⋊C₄	—	146	1	no	—
242	C₂×C₂²⋊C₄	—	146	1	no	—
243	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	—	2	1	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	146	1	no	—
248	C₄⋊C₈	—	146	1	no	—
249	C₄⋊C₈	C₄×D₇₃	1	2	yes	—
250	#102	—	292	2	no	—
251	#102	—	292	2	no	—
252	C₂²⋊C₈	—	146	4	no	—
253	C₄×C₈	—	73	2	no	—
254	C₄×C₈	—	73	2	no	—
255	C₂⁴	—	146	1	no	—
256	C₂×D₄	—	73	2	no	—
257	C₂×D₄	—	73	2	no	—
258	C₂×D₄	—	73	2	no	—
259	C₂×D₄	—	146	1	no	—
260	C₂×D₄	—	146	1	no	—
261	C₂×D₄	—	146	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	292	1	no	—
264	C₂²×C₄	—	292	1	no	—
265	C₂²×C₄	—	292	1	no	—
266	C₂²×C₄	—	292	1	no	—
267	C₂²×C₄	—	292	1	no	—
268	C₂²×C₄	—	146	1	no	—
269	C₂²×C₄	—	146	2	no	—
270	C₂²×C₄	—	146	2	no	—
271	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
272	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
273	C₂²×C₄	#11924028	1	1	yes	characteristic
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	4	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	292	8	no	—
281	C₂×C₈	—	292	1	no	—
282	C₂×C₈	—	292	1	no	—
283	C₂×C₈	—	292	1	no	—
284	C₂×C₈	—	73	2	no	—
285	C₂×C₈	—	73	2	no	—
286	C₂×C₈	—	146	1	no	—
287	C₂×C₈	—	146	1	no	—
288	C₂×C₈	—	146	1	no	—
289	C₂×C₈	—	146	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	8	1	no	—
295	C₄⋊C₄	—	292	1	no	—
296	C₄⋊C₄	—	292	1	no	—
297	C₄⋊C₄	—	292	1	no	—
298	C₄⋊C₄	—	292	1	no	—
299	C₄⋊C₄	—	584	1	no	—
300	C₄⋊C₄	—	73	2	no	—
301	C₄⋊C₄	—	146	1	no	—
302	C₂²⋊C₄	—	146	2	no	—
303	C₂²⋊C₄	—	146	2	no	—
304	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
305	C₄²	D₄×D₇₃	1	1	yes	characteristic
306	C₄²	Q₈×D₇₃	1	1	yes	characteristic
307	C₄²	#11924028	1	1	yes	characteristic
308	C₂³	—	292	1	no	—
309	C₂³	—	292	1	no	—
310	C₂³	—	292	1	no	—
311	C₂³	—	292	1	no	—
312	C₂³	—	146	2	no	—
313	C₂³	—	146	2	no	—
314	C₂³	—	146	2	no	—
315	C₂³	#14556564	1	1	yes	characteristic
316	D₄	—	73	4	no	—
317	D₄	—	146	2	no	—
318	D₄	—	146	2	no	—
319	D₄	—	146	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	8	1	no	—
323	C₂×C₄	—	8	1	no	—
324	C₂×C₄	—	8	1	no	—
325	C₂×C₄	—	292	1	no	—
326	C₂×C₄	—	292	1	no	—
327	C₂×C₄	—	292	2	no	—
328	C₂×C₄	—	292	2	no	—
329	C₂×C₄	—	292	2	no	—
330	C₂×C₄	—	292	2	no	—
331	C₂×C₄	—	584	1	no	—
332	C₂×C₄	—	584	1	no	—
333	C₂×C₄	—	584	1	no	—
334	C₂×C₄	—	146	2	no	—
335	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
336	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
337	C₂×C₄	#14556548	1	1	yes	characteristic
338	C₂×C₄	#14556564	1	1	yes	characteristic
339	C₂×C₄	#14556569	1	1	yes	characteristic
340	C₂×C₄	#14556569	1	1	yes	characteristic
341	C₂×C₄	SD₁₆×D₇₃	1	1	yes	characteristic
342	C₂×C₄	SD₁₆×D₇₃	1	1	yes	characteristic
343	C₂×C₄	#14556610	1	1	yes	characteristic
344	C₂×C₄	#14556612	1	1	yes	characteristic
345	C₂×C₄	#14556614	1	1	yes	characteristic
346	C₂×C₄	#14556614	1	1	yes	characteristic
347	C₂×C₄	#14556615	1	1	yes	characteristic
348	C₂×C₄	#14556618	1	1	yes	characteristic
349	C₈	—	4	2	no	—
350	C₈	—	8	1	no	—
351	C₈	—	584	1	no	—
352	C₈	—	146	2	no	—
353	C₈	—	146	2	no	—
354	C₂²	—	292	2	no	—
355	C₂²	—	292	2	no	—
356	C₂²	—	292	2	no	—
357	C₂²	—	292	2	no	—
358	C₂²	—	292	2	no	—
359	C₂²	—	292	2	no	—
360	C₂²	—	146	4	no	—
361	C₂²	#17988446	1	1	yes	characteristic
362	C₂²	#17988801	1	1	yes	characteristic
363	C₂²	#17988805	1	1	yes	characteristic
364	C₂²	#17988815	1	1	yes	characteristic
365	C₂²	#17988819	1	1	yes	characteristic
366	C₂²	#17988856	1	1	yes	characteristic
367	C₂²	#17988861	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	16	1	no	—
371	C₄	—	292	2	no	—
372	C₄	—	1168	1	no	—
373	C₄	#17988582	1	1	yes	characteristic
374	C₄	#17988590	1	1	yes	characteristic
375	C₄	#106×D₇₃	1	1	yes	characteristic
376	C₄	#17988624	1	1	yes	characteristic
377	C₄	#17988815	1	1	yes	characteristic
378	C₄	#17988819	1	1	yes	characteristic
379	C₄	#17988886	1	1	yes	characteristic
380	C₄	#17988890	1	1	yes	characteristic
381	C₂	—	292	4	no	—
382	C₂	#22609339	1	1	yes	characteristic
383	C₂	#22609343	1	1	yes	characteristic
384	C₂	#22609353	1	1	yes	characteristic
385	C₂	#22609357	1	1	yes	characteristic
386	C₂	#22609940	1	1	yes	characteristic
387	C₂	#22609955	1	1	yes	characteristic
388	C₂	#22609994	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14556564
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1296	C₂
socle	C₂³×C₇₃	#121
prime radical	C₂²×D₂₉₂	D₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1296	D₇₃
hypocenter	C₇₃	#8646
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

Complements

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

N	H ≅ G/N	type
2. #22610058	C₂	semidirect
5. C₂×#17988337	C₂	semidirect
6. #22607815	C₂	semidirect
7. C₇₃×#1296	C₂	semidirect
11. #17988652	C₄	semidirect
201. #1296	D₇₃	semidirect
202. C₇₃	#8646	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988652	2
#22610058	2
#22609932	2
#22609928	2
C₂×#17988337	2
#22607815	2
C₇₃×#1296	2
#8646	73