Subgroups — #33077938

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622945	C₂	1	1	yes	maximal, characteristic, complemented
2	#22622728	C₂	1	2	yes	maximal, complemented
3	#22621681	C₂	1	1	yes	maximal, characteristic, complemented
4	#22609496	C₂	1	2	yes	maximal, complemented
5	#22609298	C₂	1	1	yes	maximal, characteristic, complemented
6	#22609283	C₂	1	1	yes	maximal, characteristic
7	#22609114	C₂	1	2	yes	maximal, complemented
8	#22609031	C₂	1	2	yes	maximal, complemented
9	#22608829	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11924017	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#1709	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₃	—	2	1	no	—
13	#17989323	C₂²	1	2	yes	—
14	#17989252	C₂²	1	2	yes	—
15	#17988970	—	2	1	no	—
16	#17988945	—	2	2	no	—
17	#17988875	—	2	2	no	—
18	#17988869	C₂²	1	2	yes	—
19	#17988825	—	2	1	no	—
20	#17988801	—	2	2	no	—
21	#17988800	—	2	2	no	—
22	#17988794	C₂²	1	2	yes	—
23	#17988784	C₂²	1	2	yes	—
24	#17988779	C₂²	1	2	yes	—
25	#17988741	—	2	2	no	—
26	#17988717	C₂²	1	2	yes	—
27	C₈×Dic₁₄₆	C₂²	1	1	yes	characteristic
28	#17988696	—	2	2	no	—
29	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
30	C₄×#11924017	C₂²	1	1	yes	characteristic
31	D₈×Dic₇₃	—	2	1	no	—
32	C₈×D₂₉₂	C₂²	1	1	yes	characteristic
33	#17988340	C₂²	1	1	yes	characteristic
34	#17988338	C₂²	1	1	yes	characteristic
35	#17988333	C₂²	1	2	yes	—
36	#17988331	C₂²	1	2	yes	—
37	#17988319	C₂²	1	1	yes	characteristic
38	#17988319	C₂²	1	2	yes	—
39	#17988316	C₂²	1	1	yes	characteristic
40	#494×C₇₃	C₂²	1	1	yes	characteristic
41	#493×C₇₃	C₂²	1	1	yes	characteristic
42	#486×C₇₃	C₂²	1	2	yes	—
43	#486×C₇₃	C₂²	1	2	yes	—
44	C₈²×C₇₃	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₇₃	—	2	2	no	—
46	C₂×Dic₂₉₂	D₄	1	1	yes	characteristic
47	C₂×#11924018	—	2	2	no	—
48	Q₁₆×D₇₃	—	4	4	no	—
49	C₂×#11924017	D₄	1	1	yes	characteristic
50	C₂×#11924016	—	2	2	no	—
51	C₂×#11924013	—	2	2	no	—
52	C₂×C₈×D₇₃	—	2	1	no	—
53	#14556590	—	2	2	no	—
54	#14556578	—	2	2	no	—
55	#14556578	—	2	2	no	—
56	#14556572	—	4	2	no	—
57	#14556569	—	4	2	no	—
58	#14556567	—	2	2	no	—
59	C₄×Dic₁₄₆	C₂³	1	1	yes	characteristic
60	C₄×Dic₁₄₆	C₂³	1	2	yes	—
61	C₄×D₂₉₂	C₂³	1	1	yes	characteristic
62	D₄×Dic₇₃	—	2	2	no	—
63	D₄×Dic₇₃	—	2	2	no	—
64	#14556553	—	2	2	no	—
65	#14556525	D₄	1	1	yes	characteristic
66	#14556523	D₄	1	1	yes	characteristic
67	#14556522	—	2	2	no	—
68	C₄⋊#10596831	C₂³	1	2	yes	—
69	C₄⋊#10596831	C₂³	1	1	yes	characteristic
70	C₄⋊#10596831	C₂³	1	1	yes	characteristic
71	#14556520	—	2	2	no	—
72	#14556520	—	2	1	no	—
73	#14556516	D₄	1	1	yes	characteristic
74	#14556516	D₄	1	2	yes	—
75	#14556515	—	2	2	no	—
76	#14556514	—	2	2	no	—
77	#14556513	D₄	1	1	yes	characteristic
78	#14556512	—	2	2	no	—
79	#14556512	—	2	2	no	—
80	#14556503	—	2	1	no	—
81	#14556501	—	2	2	no	—
82	C₈×Dic₇₃	—	2	1	no	—
83	C₂×Q₁₆×C₇₃	—	2	1	no	—
84	C₂×Q₁₆×C₇₃	—	2	1	no	—
85	C₂×D₈×C₇₃	—	2	1	no	—
86	C₂×D₈×C₇₃	—	2	1	no	—
87	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
88	#127×C₇₃	C₂³	1	2	yes	—
89	#102×C₇₃	—	2	2	no	—
90	#102×C₇₃	—	2	2	no	—
91	#102×C₇₃	—	4	2	no	—
92	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
95	C₂×Dic₁₄₆	—	4	2	no	—
96	C₂×Dic₁₄₆	C₂×D₄	1	1	yes	characteristic
97	C₂×Dic₁₄₆	C₂×D₄	1	2	yes	—
98	Q₈×D₇₃	—	4	4	no	—
99	Q₈×D₇₃	—	8	2	no	—
100	C₂×D₂₉₂	C₂×D₄	1	1	yes	characteristic
101	C₂²×Dic₇₃	—	4	2	no	—
102	C₂×C₄×D₇₃	—	2	1	no	—
103	C₂×C₄×D₇₃	—	4	2	no	—
104	Dic₂₉₂	—	2	2	no	—
105	#11924018	—	4	4	no	—
106	#11924017	—	2	2	no	—
107	#11924016	—	8	2	no	—
108	#11924013	—	4	4	no	—
109	C₂×#10596831	—	2	1	no	—
110	C₂×#10596831	—	2	2	no	—
111	C₂×#10596831	—	2	1	no	—
112	C₈×D₇₃	—	4	2	no	—
113	C₄⋊Dic₇₃	C₂×D₄	1	1	yes	characteristic
114	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
115	C₄⋊Dic₇₃	C₂×D₄	1	1	yes	characteristic
116	#11924007	—	2	2	no	—
117	#11924007	—	2	1	no	—
118	#11924007	—	4	2	no	—
119	C₂²⋊Dic₇₃	—	4	2	no	—
120	C₂²⋊Dic₇₃	—	4	2	no	—
121	C₂²⋊Dic₇₃	—	4	2	no	—
122	#11924005	—	2	1	no	—
123	#11924005	—	4	2	no	—
124	C₄×Dic₇₃	—	2	1	no	—
125	C₄×Dic₇₃	—	2	2	no	—
126	C₂×Q₈×C₇₃	—	2	2	no	—
127	C₂×D₄×C₇₃	—	2	2	no	—
128	C₂×D₄×C₇₃	—	2	2	no	—
129	C₂×D₄×C₇₃	—	2	2	no	—
130	Q₁₆×C₇₃	—	4	2	no	—
131	Q₁₆×C₇₃	—	8	1	no	—
132	D₈×C₇₃	—	4	2	no	—
133	D₈×C₇₃	—	8	1	no	—
134	C₂×C₈×C₇₃	—	2	1	no	—
135	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
138	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
139	C₄⋊C₄×C₇₃	—	2	2	no	—
140	C₄⋊C₄×C₇₃	—	2	2	no	—
141	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
142	C₂³×C₇₃	—	4	2	no	—
143	C₂²×D₇₃	—	2	1	no	—
144	Q₈×C₇₃	—	4	2	no	—
145	Q₈×C₇₃	—	4	2	no	—
146	D₄×C₇₃	—	4	2	no	—
147	D₄×C₇₃	—	4	2	no	—
148	D₄×C₇₃	—	4	2	no	—
149	D₄×C₇₃	—	4	2	no	—
150	D₄×C₇₃	—	8	2	no	—
151	C₂×C₄×C₇₃	—	4	2	no	—
152	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×Dic₇₃	—	2	2	no	—
156	C₂×Dic₇₃	—	2	1	no	—
157	C₂×Dic₇₃	—	2	1	no	—
158	C₂×Dic₇₃	—	2	2	no	—
159	C₂×Dic₇₃	—	2	1	no	—
160	C₂×Dic₇₃	—	16	2	no	—
161	D₂₉₂	—	2	1	no	—
162	D₂₉₂	#135	1	2	yes	—
163	C₄×D₇₃	—	4	1	no	—
164	C₄×D₇₃	—	4	1	no	—
165	C₄×D₇₃	—	8	4	no	—
166	Dic₁₄₆	—	2	1	no	—
167	Dic₁₄₆	—	2	2	no	—
168	Dic₁₄₆	—	2	2	no	—
169	Dic₁₄₆	—	2	1	no	—
170	Dic₁₄₆	—	8	4	no	—
171	C₈×C₇₃	—	2	1	no	—
172	C₈×C₇₃	—	2	1	no	—
173	C₈×C₇₃	—	2	1	no	—
174	C₈×C₇₃	—	4	1	no	—
175	C₈×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
176	#10596831	—	4	2	no	—
177	#10596831	—	4	1	no	—
178	#10596831	—	4	1	no	—
179	C₂²×C₇₃	—	8	2	no	—
180	C₂²×C₇₃	—	8	2	no	—
181	C₂²×C₇₃	—	8	2	no	—
182	C₂²×C₇₃	D₄²	1	1	yes	characteristic
183	C₂×D₇₃	—	2	2	no	—
184	C₂×D₇₃	—	4	1	no	—
185	C₂×D₇₃	—	4	1	no	—
186	C₄×C₇₃	—	2	1	no	—
187	C₄×C₇₃	—	8	2	no	—
188	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
189	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
190	C₄×C₇₃	#576	1	1	yes	characteristic
191	C₄×C₇₃	#578	1	1	yes	characteristic
192	Dic₇₃	—	4	2	no	—
193	Dic₇₃	—	4	1	no	—
194	Dic₇₃	—	4	1	no	—
195	Dic₇₃	—	4	2	no	—
196	Dic₇₃	—	4	1	no	—
197	#19935	—	73	1	no	maximal, Sylow
198	C₂×C₇₃	—	16	2	no	—
199	C₂×C₇₃	#3274	1	1	yes	characteristic
200	C₂×C₇₃	#3285	1	1	yes	characteristic
201	C₂×C₇₃	#3290	1	1	yes	characteristic
202	D₇₃	—	4	2	no	—
203	#3396	—	73	1	no	—
204	#3373	—	73	2	no	—
205	D₄×Q₁₆	—	73	1	no	—
206	#1709	D₇₃	1	1	yes	characteristic, complemented
207	#1702	—	73	2	no	—
208	#1670	—	73	1	no	—
209	#1667	—	73	1	no	—
210	#1640	—	73	2	no	—
211	#1627	—	73	2	no	—
212	#1591	—	73	1	no	—
213	C₈×SD₁₆	—	73	1	no	—
214	C₇₃	#19935	1	1	yes	characteristic, complemented, Sylow
215	C₂²×Q₁₆	—	146	1	no	—
216	#559	—	73	2	no	—
217	D₄×Q₈	—	73	2	no	—
218	#495	—	146	1	no	—
219	#494	C₂×D₇₃	1	1	yes	characteristic
220	#493	C₂×D₇₃	1	1	yes	characteristic
221	#489	—	146	2	no	—
222	#486	C₂×D₇₃	1	2	yes	—
223	#486	C₂×D₇₃	1	2	yes	—
224	#476	—	146	2	no	—
225	#475	—	73	2	no	—
226	#467	—	146	1	no	—
227	#463	—	146	2	no	—
228	#463	—	146	2	no	—
229	#462	—	73	2	no	—
230	#460	—	73	2	no	—
231	#459	—	73	2	no	—
232	#451	—	146	2	no	—
233	#446	—	73	2	no	—
234	C₈×Q₈	—	73	1	no	—
235	#442	—	146	2	no	—
236	C₄×Q₁₆	—	73	1	no	—
237	C₄×SD₁₆	—	73	1	no	—
238	C₄×D₈	—	146	1	no	—
239	C₈×D₄	—	73	1	no	—
240	#335	—	73	1	no	—
241	#334	—	73	1	no	—
242	#332	—	73	2	no	—
243	#331	—	73	2	no	—
244	Q₈⋊C₈	—	73	1	no	—
245	Q₈⋊C₈	—	73	2	no	—
246	D₄⋊C₈	—	73	1	no	—
247	C₈²	C₂×D₇₃	1	1	yes	characteristic
248	C₂²×Q₈	—	146	2	no	—
249	C₂×Q₁₆	—	2	1	no	—
250	C₂×Q₁₆	—	2	1	no	—
251	C₂×Q₁₆	—	292	4	no	—
252	C₂×Q₁₆	—	73	1	no	—
253	C₂×Q₁₆	—	146	2	no	—
254	C₂×SD₁₆	—	73	1	no	—
255	C₂×SD₁₆	—	146	2	no	—
256	C₂×D₈	—	2	1	no	—
257	C₂×D₈	—	2	1	no	—
258	C₂×D₈	—	146	2	no	—
259	C₂²×C₈	—	146	1	no	—
260	C₄⋊Q₈	—	146	2	no	—
261	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
262	#127	C₂²×D₇₃	1	2	yes	—
263	#124	—	146	2	no	—
264	#124	—	146	2	no	—
265	#123	—	292	2	no	—
266	C₂²⋊Q₈	—	292	2	no	—
267	C₂²⋊Q₈	—	146	2	no	—
268	C₄×Q₈	—	73	2	no	—
269	C₄×Q₈	—	73	1	no	—
270	C₄×D₄	—	73	1	no	—
271	C₄×D₄	—	146	2	no	—
272	C₄×D₄	—	146	2	no	—
273	C₄×D₄	—	146	2	no	—
274	#107	—	73	1	no	—
275	#106	—	73	1	no	—
276	#106	—	146	2	no	—
277	C₄⋊C₈	—	73	1	no	—
278	C₄⋊C₈	—	73	1	no	—
279	C₄⋊C₈	—	73	2	no	—
280	C₄⋊C₈	—	146	1	no	—
281	C₄⋊C₈	—	146	2	no	—
282	#103	—	73	1	no	—
283	#103	—	73	2	no	—
284	#103	—	146	2	no	—
285	#103	—	146	2	no	—
286	#102	—	2	2	no	—
287	#102	—	2	2	no	—
288	#102	—	4	2	no	—
289	#102	—	73	1	no	—
290	#102	—	146	2	no	—
291	#102	—	146	2	no	—
292	C₂²⋊C₈	—	146	1	no	—
293	#97	—	146	2	no	—
294	C₄×C₈	—	146	1	no	—
295	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
296	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
297	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	292	2	no	—
300	C₂×Q₈	—	292	4	no	—
301	C₂×Q₈	—	584	2	no	—
302	C₂×Q₈	—	73	1	no	—
303	C₂×Q₈	—	73	2	no	—
304	C₂×D₄	—	2	2	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	73	1	no	—
308	C₂²×C₄	—	292	2	no	—
309	C₂²×C₄	—	292	2	no	—
310	C₂²×C₄	—	146	1	no	—
311	Q₁₆	—	4	2	no	—
312	Q₁₆	—	8	1	no	—
313	Q₁₆	—	292	4	no	—
314	Q₁₆	—	146	2	no	—
315	SD₁₆	—	584	2	no	—
316	SD₁₆	—	146	2	no	—
317	D₈	—	4	2	no	—
318	D₈	—	8	1	no	—
319	D₈	—	292	4	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	292	2	no	—
322	C₂×C₈	—	146	1	no	—
323	C₂×C₈	—	146	2	no	—
324	C₂×C₈	—	146	1	no	—
325	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
326	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
327	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
328	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	292	2	no	—
332	C₄⋊C₄	—	73	1	no	—
333	C₄⋊C₄	—	73	1	no	—
334	C₄⋊C₄	—	73	2	no	—
335	C₄⋊C₄	—	146	2	no	—
336	C₄⋊C₄	—	146	1	no	—
337	C₂²⋊C₄	—	292	2	no	—
338	C₂²⋊C₄	—	292	2	no	—
339	C₂²⋊C₄	—	292	2	no	—
340	C₂²⋊C₄	—	292	2	no	—
341	C₂²⋊C₄	—	146	1	no	—
342	C₄²	—	146	1	no	—
343	C₄²	—	146	2	no	—
344	C₄²	C₂³×D₇₃	1	1	yes	characteristic
345	C₂³	—	4	2	no	—
346	C₂³	—	146	1	no	—
347	Q₈	—	4	2	no	—
348	Q₈	—	4	2	no	—
349	Q₈	—	584	4	no	—
350	Q₈	—	146	2	no	—
351	Q₈	—	146	1	no	—
352	Q₈	—	146	1	no	—
353	Q₈	—	146	2	no	—
354	D₄	—	4	2	no	—
355	D₄	—	4	2	no	—
356	D₄	—	4	2	no	—
357	D₄	—	4	2	no	—
358	D₄	—	8	2	no	—
359	D₄	—	73	2	no	—
360	D₄	—	146	1	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	292	1	no	—
363	C₂×C₄	—	292	1	no	—
364	C₂×C₄	—	584	4	no	—
365	C₂×C₄	—	1168	2	no	—
366	C₂×C₄	—	146	1	no	—
367	C₂×C₄	—	146	2	no	—
368	C₂×C₄	—	146	1	no	—
369	C₂×C₄	—	146	2	no	—
370	C₂×C₄	—	146	1	no	—
371	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
372	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
373	C₂×C₄	#14556636	1	1	yes	characteristic
374	C₈	—	2	1	no	—
375	C₈	—	2	1	no	—
376	C₈	—	2	1	no	—
377	C₈	—	4	1	no	—
378	C₈	—	292	1	no	—
379	C₈	—	292	2	no	—
380	C₈	—	292	1	no	—
381	C₈	#14556615	1	2	yes	—
382	C₂²	—	8	2	no	—
383	C₂²	—	8	2	no	—
384	C₂²	—	8	2	no	—
385	C₂²	—	292	1	no	—
386	C₂²	—	292	1	no	—
387	C₂²	—	146	2	no	—
388	C₂²	#17989222	1	1	yes	characteristic
389	C₄	—	2	1	no	—
390	C₄	—	8	2	no	—
391	C₄	—	292	1	no	—
392	C₄	—	292	2	no	—
393	C₄	—	292	1	no	—
394	C₄	—	292	2	no	—
395	C₄	—	292	1	no	—
396	C₄	#135×D₇₃	1	1	yes	characteristic
397	C₄	C₂×#14556615	1	1	yes	characteristic
398	C₄	#17989388	1	1	yes	characteristic
399	C₄	#17989401	1	1	yes	characteristic
400	C₂	—	16	2	no	—
401	C₂	—	292	2	no	—
402	C₂	#22621550	1	1	yes	characteristic
403	C₂	#22621671	1	1	yes	characteristic
404	C₂	#22621716	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17989222
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1709	C₂
socle	C₂²×C₇₃	D₄²
prime radical	#22609283	C₂
hypercenter	#1709	D₇₃
hypocenter	C₇₃	#19935
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1709

Complements

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

N	H ≅ G/N	type
1. #22622945	C₂	semidirect
2. #22622728	C₂	semidirect
3. #22621681	C₂	semidirect
4. #22609496	C₂	semidirect
5. #22609298	C₂	semidirect
7. #22609114	C₂	semidirect
8. #22609031	C₂	semidirect
9. #22608829	C₂	semidirect
10. C₈×#11924017	C₂	semidirect
11. C₇₃×#1709	C₂	semidirect
206. #1709	D₇₃	semidirect
214. C₇₃	#19935	semidirect

Maximal subgroups

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

subgroup	index
#22622945	2
#22622728	2
#22621681	2
#22609496	2
#22609298	2
#22609283	2
#22609114	2
#22609031	2
#22608829	2
C₈×#11924017	2
C₇₃×#1709	2
#19935	73