Subgroups — #33070023

Order 18688.

706 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620018	C₂	1	2	yes	maximal, complemented
2	C₂×#17988666	C₂	1	1	yes	maximal, characteristic, complemented
3	#22609461	C₂	1	4	yes	maximal, complemented
4	#22609451	C₂	1	2	yes	maximal, complemented
5	#22609407	C₂	1	2	yes	maximal, complemented
6	#22608694	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608626	C₂	1	2	yes	maximal, complemented
8	C₇₃×#3169	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11924011	—	2	1	no	—
10	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
11	#17988903	—	2	2	no	—
12	#17988874	C₂²	1	4	yes	—
13	#17988779	C₂²	1	4	yes	—
14	#17988720	—	2	2	no	—
15	#17988697	—	2	4	no	—
16	#17988666	—	2	1	no	—
17	#17988666	—	2	1	no	—
18	#17988666	C₂²	1	2	yes	—
19	#17988666	C₂²	1	2	yes	—
20	#17988644	C₂²	1	2	yes	—
21	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
22	C₂×#14556503	—	2	1	no	—
23	#17988340	C₂²	1	2	yes	complemented
24	#17988338	C₂²	1	2	yes	complemented
25	#17988336	C₂²	1	4	yes	complemented
26	#17988331	C₂²	1	4	yes	complemented
27	#532×C₇₃	C₂²	1	2	yes	—
28	#500×C₇₃	C₂²	1	1	yes	characteristic, complemented
29	#493×C₇₃	C₂²	1	1	yes	characteristic, complemented
30	#486×C₇₃	C₂²	1	2	yes	complemented
31	#466×C₇₃	—	2	1	no	—
32	#466×C₇₃	—	2	1	no	—
33	C₄×D₈×C₇₃	—	2	1	no	—
34	C₄×D₈×C₇₃	—	2	1	no	—
35	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
36	C₂²×D₂₉₂	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₇₃	—	2	1	no	—
38	C₂×#11924013	—	2	4	no	—
39	C₂×#11924011	—	2	2	no	—
40	C₂×#11924011	—	2	2	no	—
41	C₂×#11924011	—	4	4	no	—
42	C₂²×#10596831	—	2	1	no	—
43	C₄×D₂₉₂	—	2	1	no	—
44	C₄×D₂₉₂	—	2	1	no	—
45	C₄×D₂₉₂	D₄	1	2	yes	—
46	C₄×D₂₉₂	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
48	C₂×#11924005	—	2	1	no	—
49	#14556522	—	2	4	no	—
50	C₄⋊#10596831	—	2	1	no	—
51	C₄⋊#10596831	C₂³	1	2	yes	—
52	C₄⋊#10596831	C₂³	1	4	yes	—
53	#14556512	—	2	4	no	—
54	#14556511	—	2	4	no	—
55	C₂²⋊#10596831	—	2	2	no	—
56	#14556503	—	2	2	no	—
57	#14556503	—	2	2	no	—
58	#14556502	—	2	2	no	—
59	C₂×#41×C₇₃	—	2	2	no	—
60	C₂×D₈×C₇₃	—	2	1	no	—
61	C₂×D₈×C₇₃	—	2	1	no	—
62	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
64	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
65	#127×C₇₃	C₂³	1	2	yes	—
66	#124×C₇₃	—	2	2	no	—
67	#121×C₇₃	—	2	2	no	—
68	#121×C₇₃	—	2	2	no	—
69	C₄×D₄×C₇₃	—	2	2	no	—
70	C₄×D₄×C₇₃	—	2	2	no	—
71	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
72	#107×C₇₃	—	2	1	no	—
73	#107×C₇₃	—	2	1	no	—
74	#102×C₇₃	—	2	2	no	—
75	#102×C₇₃	—	2	2	no	—
76	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
78	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
80	C₂³×D₇₃	—	2	1	no	—
81	C₂×D₂₉₂	—	2	1	no	—
82	C₂×D₂₉₂	—	4	2	no	—
83	C₂×D₂₉₂	C₂×D₄	1	2	yes	—
84	C₂²×Dic₇₃	—	2	1	no	—
85	C₂×C₄×D₇₃	—	2	2	no	—
86	C₂×C₄×D₇₃	—	2	2	no	—
87	C₂×C₄×D₇₃	—	4	2	no	—
88	C₂×C₄×D₇₃	—	4	2	no	—
89	#11924013	—	4	8	no	—
90	#11924011	—	4	4	no	—
91	#11924011	—	4	4	no	—
92	C₂×#10596831	—	2	4	no	—
93	C₂×#10596831	—	2	2	no	—
94	C₂×#10596831	—	4	2	no	—
95	C₄⋊Dic₇₃	—	2	1	no	—
96	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
97	#11924005	—	2	2	no	—
98	#11924005	—	2	2	no	—
99	#41×C₇₃	—	8	4	no	—
100	C₂×Q₈×C₇₃	—	2	2	no	—
101	C₂×D₄×C₇₃	—	2	2	no	—
102	C₂×D₄×C₇₃	—	2	2	no	—
103	C₂×D₄×C₇₃	—	2	2	no	—
104	C₂×D₄×C₇₃	—	4	2	no	—
105	C₂²×C₄×C₇₃	—	4	2	no	—
106	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
109	D₈×C₇₃	—	4	2	no	—
110	D₈×C₇₃	—	4	2	no	—
111	C₂×C₈×C₇₃	—	2	2	no	—
112	C₂×C₈×C₇₃	—	4	1	no	—
113	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
117	C₄⋊C₄×C₇₃	—	2	2	no	—
118	C₄⋊C₄×C₇₃	—	2	2	no	—
119	C₂²⋊C₄×C₇₃	—	4	2	no	—
120	C₂²⋊C₄×C₇₃	—	4	2	no	—
121	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
122	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₇₃	#41	1	1	yes	characteristic
124	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
125	C₂³×C₇₃	—	4	2	no	—
126	C₂³×C₇₃	#121	1	1	yes	characteristic
127	C₂²×D₇₃	—	2	2	no	—
128	C₂²×D₇₃	—	4	2	no	—
129	C₂²×D₇₃	—	4	2	no	—
130	C₂²×D₇₃	—	4	2	no	—
131	Q₈×C₇₃	—	8	2	no	—
132	D₄×C₇₃	—	4	2	no	—
133	D₄×C₇₃	—	4	2	no	—
134	D₄×C₇₃	—	4	4	no	—
135	D₄×C₇₃	—	8	2	no	—
136	D₄×C₇₃	—	8	4	no	—
137	C₂×C₄×C₇₃	—	2	2	no	—
138	C₂×C₄×C₇₃	—	2	1	no	—
139	C₂×C₄×C₇₃	—	2	1	no	—
140	C₂×C₄×C₇₃	—	2	2	no	—
141	C₂×C₄×C₇₃	—	4	2	no	—
142	C₂×C₄×C₇₃	—	8	4	no	—
143	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
146	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
149	C₂×Dic₇₃	—	2	2	no	—
150	C₂×Dic₇₃	—	4	2	no	—
151	D₂₉₂	—	2	4	no	—
152	D₂₉₂	—	4	2	no	—
153	C₄×D₇₃	—	4	4	no	—
154	C₄×D₇₃	—	4	2	no	—
155	C₄×D₇₃	—	4	2	no	—
156	C₈×C₇₃	—	2	1	no	—
157	C₈×C₇₃	—	2	1	no	—
158	C₈×C₇₃	—	2	1	no	—
159	C₈×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
160	#10596831	—	4	2	no	—
161	#10596831	—	4	4	no	—
162	C₂²×C₇₃	—	2	1	no	—
163	C₂²×C₇₃	—	2	1	no	—
164	C₂²×C₇₃	—	8	2	no	—
165	C₂²×C₇₃	—	8	2	no	—
166	C₂²×C₇₃	—	8	2	no	—
167	C₂²×C₇₃	#472	1	2	yes	—
168	C₂²×C₇₃	C₂×#121	1	1	yes	characteristic
169	C₂×D₇₃	—	4	2	no	—
170	C₂×D₇₃	—	4	2	no	—
171	C₂×D₇₃	—	4	2	no	—
172	C₂×D₇₃	—	8	4	no	—
173	C₄×C₇₃	—	2	1	no	—
174	C₄×C₇₃	—	2	1	no	—
175	C₄×C₇₃	—	2	1	no	—
176	C₄×C₇₃	—	2	1	no	—
177	C₄×C₇₃	—	8	2	no	—
178	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
179	C₄×C₇₃	#575	1	2	yes	—
180	C₄×C₇₃	#575	1	1	yes	characteristic
181	Dic₇₃	—	4	2	no	—
182	#19283	—	73	1	no	maximal, Sylow
183	C₂×C₇₃	—	2	2	no	—
184	C₂×C₇₃	—	16	2	no	—
185	C₂×C₇₃	#3036	1	1	yes	characteristic
186	C₂×C₇₃	#3054	1	1	yes	characteristic
187	C₂×C₇₃	C₂×#472	1	1	yes	characteristic
188	D₇₃	—	8	2	no	—
189	#3169	D₇₃	1	1	yes	characteristic, complemented
190	#3109	—	73	2	no	—
191	C₂×#436	—	73	1	no	—
192	#1697	—	73	4	no	—
193	#1695	—	73	2	no	—
194	#1687	—	73	2	no	—
195	#1568	—	73	1	no	—
196	#1551	—	73	2	no	—
197	C₇₃	#19283	1	1	yes	characteristic, complemented, Sylow
198	C₂²×M₁₆	—	146	1	no	—
199	#532	C₂×D₇₃	1	2	yes	—
200	C₂×C₄×D₄	—	73	1	no	—
201	#500	C₂×D₇₃	1	1	yes	characteristic, complemented
202	#493	C₂×D₇₃	1	1	yes	characteristic, complemented
203	#486	C₂×D₇₃	1	2	yes	complemented
204	#481	—	146	2	no	—
205	#476	—	73	4	no	—
206	#466	—	2	1	no	—
207	#466	—	2	1	no	—
208	#459	—	73	4	no	—
209	#447	—	146	2	no	—
210	#442	—	146	4	no	—
211	C₄×D₈	—	2	1	no	—
212	C₄×D₈	—	2	1	no	—
213	#436	—	73	2	no	—
214	#436	—	73	2	no	—
215	#436	—	146	1	no	—
216	#436	—	146	1	no	—
217	#432	—	73	2	no	—
218	C₂×C₄⋊C₈	—	73	1	no	—
219	C₂×C₂²⋊C₈	—	146	1	no	—
220	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
221	#335	—	73	2	no	—
222	#334	—	73	2	no	—
223	#333	—	73	4	no	—
224	#331	—	73	4	no	—
225	C₂×#41	—	2	2	no	—
226	C₂²×D₄	—	73	1	no	—
227	C₂³×C₄	—	146	1	no	—
228	C₂×D₈	—	2	1	no	—
229	C₂×D₈	—	2	1	no	—
230	C₂×D₈	—	146	4	no	—
231	C₂×M₁₆	—	292	4	no	—
232	C₂×M₁₆	—	146	2	no	—
233	C₂×M₁₆	—	146	2	no	—
234	C₂²×C₈	—	146	1	no	—
235	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
236	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
237	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
238	#127	C₂²×D₇₃	1	2	yes	—
239	#124	—	2	2	no	—
240	#121	—	2	2	no	—
241	#121	—	2	2	no	—
242	C₄×D₄	—	2	2	no	—
243	C₄×D₄	—	2	2	no	—
244	C₄×D₄	—	73	2	no	—
245	C₄×D₄	—	73	2	no	—
246	C₄×D₄	—	146	1	no	—
247	C₄×D₄	—	146	1	no	—
248	C₂×C₄⋊C₄	—	73	1	no	—
249	C₂×C₂²⋊C₄	—	146	1	no	—
250	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
251	#107	—	2	1	no	—
252	#107	—	2	1	no	—
253	#106	—	146	4	no	—
254	C₄⋊C₈	—	73	2	no	—
255	C₄⋊C₈	—	73	4	no	—
256	C₄⋊C₈	—	146	1	no	—
257	#102	—	2	2	no	—
258	#102	—	2	2	no	—
259	#102	—	146	4	no	—
260	#102	—	146	4	no	—
261	C₂²⋊C₈	—	146	2	no	—
262	C₂²⋊C₈	—	146	2	no	—
263	C₂²⋊C₈	—	146	2	no	—
264	#97	—	146	2	no	—
265	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
266	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
268	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
269	C₂⁴	—	146	1	no	—
270	#41	—	8	4	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	4	2	no	—
276	C₂×D₄	—	292	2	no	—
277	C₂×D₄	—	73	2	no	—
278	C₂×D₄	—	146	1	no	—
279	C₂²×C₄	—	4	2	no	—
280	C₂²×C₄	—	292	2	no	—
281	C₂²×C₄	—	292	2	no	—
282	C₂²×C₄	—	146	1	no	—
283	C₂²×C₄	—	146	2	no	—
284	C₂²×C₄	—	146	2	no	—
285	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
287	C₂²×C₄	#11924027	1	1	yes	characteristic
288	D₈	—	4	2	no	—
289	D₈	—	4	2	no	—
290	D₈	—	292	8	no	—
291	M₁₆	—	292	4	no	—
292	M₁₆	—	292	4	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	4	1	no	—
295	C₂×C₈	—	292	2	no	—
296	C₂×C₈	—	146	4	no	—
297	C₂×C₈	—	146	2	no	—
298	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
299	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
300	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
301	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	73	2	no	—
305	C₄⋊C₄	—	146	1	no	—
306	C₂²⋊C₄	—	4	2	no	—
307	C₂²⋊C₄	—	4	2	no	—
308	C₂²⋊C₄	—	146	2	no	—
309	C₂²⋊C₄	—	146	2	no	—
310	C₄²	D₄×D₇₃	1	1	yes	characteristic
311	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
312	C₄²	#11924027	1	1	yes	characteristic
313	C₄²	C₂³×D₇₃	1	1	yes	characteristic
314	C₂³	—	4	2	no	—
315	C₂³	—	292	2	no	—
316	C₂³	—	292	2	no	—
317	C₂³	—	292	2	no	—
318	C₂³	—	146	2	no	—
319	C₂³	#14556562	1	1	yes	characteristic
320	Q₈	—	8	2	no	—
321	D₄	—	4	4	no	—
322	D₄	—	4	2	no	—
323	D₄	—	4	2	no	—
324	D₄	—	8	2	no	—
325	D₄	—	8	4	no	—
326	D₄	—	292	2	no	—
327	D₄	—	146	4	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	8	4	no	—
334	C₂×C₄	—	292	4	no	—
335	C₂×C₄	—	292	2	no	—
336	C₂×C₄	—	292	2	no	—
337	C₂×C₄	—	292	2	no	—
338	C₂×C₄	—	146	2	no	—
339	C₂×C₄	#14556562	1	1	yes	characteristic
340	C₂×C₄	#14556562	1	1	yes	characteristic
341	C₂×C₄	#14556562	1	1	yes	characteristic
342	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
343	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
344	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	2	1	no	—
348	C₈	—	292	4	no	—
349	C₈	—	292	2	no	—
350	C₈	#14556616	1	2	yes	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	8	2	no	—
354	C₂²	—	8	2	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	292	2	no	—
357	C₂²	—	292	2	no	—
358	C₂²	—	292	2	no	—
359	C₂²	—	584	4	no	—
360	C₂²	#17988854	1	2	yes	—
361	C₂²	C₂×#14556562	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	8	2	no	—
367	C₄	—	292	2	no	—
368	C₄	C₂×#14556616	1	1	yes	characteristic
369	C₄	#17989384	1	2	yes	—
370	C₄	#17989385	1	1	yes	characteristic
371	C₂	—	2	2	no	—
372	C₂	—	16	2	no	—
373	C₂	—	584	2	no	—
374	C₂	#22619391	1	1	yes	characteristic
375	C₂	#22619536	1	1	yes	characteristic
376	C₂	C₂×#17988854	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14556562
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#3169	C₂
socle	C₂²×C₇₃	C₂×#121
hypercenter	#3169	D₇₃
hypocenter	C₇₃	#19283
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #22620018	C₂	semidirect
2. C₂×#17988666	C₂	semidirect
3. #22609461	C₂	semidirect
4. #22609451	C₂	semidirect
5. #22609407	C₂	semidirect
6. #22608694	C₂	semidirect
7. #22608626	C₂	semidirect
8. C₇₃×#3169	C₂	semidirect
23. #17988340	C₂²	semidirect
24. #17988338	C₂²	semidirect
25. #17988336	C₂²	semidirect
26. #17988331	C₂²	semidirect
28. #500×C₇₃	C₂²	semidirect
29. #493×C₇₃	C₂²	semidirect
30. #486×C₇₃	C₂²	semidirect
189. #3169	D₇₃	semidirect
197. C₇₃	#19283	semidirect
201. #500	C₂×D₇₃	semidirect
202. #493	C₂×D₇₃	semidirect
203. #486	C₂×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22620018	2
C₂×#17988666	2
#22609461	2
#22609451	2
#22609407	2
#22608694	2
#22608626	2
C₇₃×#3169	2
#19283	73