Subgroups — #33078029

Order 18688.

590 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22622944	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₂₉₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#22622743	C₂	1	2	yes	maximal, complemented
4	#22609523	C₂	1	1	yes	maximal, characteristic
5	#22609472	C₂	1	2	yes	maximal, complemented
6	#22609335	C₂	1	1	yes	maximal, characteristic, complemented
7	#22609291	C₂	1	1	yes	maximal, characteristic
8	#22609054	C₂	1	2	yes	maximal, complemented
9	#22609011	C₂	1	2	yes	maximal, complemented
10	#22608829	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989322	C₂²	1	2	yes	—
13	Q₈×Dic₁₄₆	C₂²	1	2	yes	—
14	#17988992	—	2	1	no	—
15	#17988992	C₂²	1	1	yes	characteristic
16	#17988971	—	2	1	no	—
17	#17988964	C₂²	1	1	yes	characteristic
18	#17988960	C₂²	1	2	yes	—
19	#17988948	—	2	2	no	—
20	#17988934	C₂²	1	2	yes	—
21	#17988883	—	2	2	no	—
22	#17988883	C₂²	1	2	yes	—
23	#17988882	C₂²	1	2	yes	—
24	#17988866	—	2	2	no	—
25	#17988811	—	2	2	no	—
26	#17988803	—	2	2	no	—
27	#17988788	C₂²	1	2	yes	—
28	#17988787	C₂²	1	2	yes	—
29	#17988717	C₂²	1	2	yes	—
30	#17988690	—	2	2	no	—
31	C₄×Dic₂₉₂	—	2	1	no	—
32	C₄×Dic₂₉₂	C₂²	1	1	yes	characteristic
33	C₄×D₅₈₄	—	2	1	no	—
34	#17988340	C₂²	1	1	yes	characteristic
35	#17988335	C₂²	1	2	yes	—
36	#17988330	C₂²	1	2	yes	—
37	#17988319	C₂²	1	2	yes	—
38	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
39	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
41	#335×C₇₃	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
44	C₈²×C₇₃	C₂²	1	1	yes	characteristic
45	C₂×Dic₂₉₂	D₄	1	1	yes	characteristic
46	C₂×#11924017	—	2	2	no	—
47	C₂×#11924017	—	4	2	no	—
48	C₂×#11924015	—	2	2	no	—
49	C₂×D₅₈₄	—	2	1	no	—
50	C₂×D₅₈₄	—	2	1	no	—
51	#14556591	—	2	2	no	—
52	#14556591	C₂³	1	2	yes	—
53	#14556590	—	2	2	no	—
54	#14556590	—	4	2	no	—
55	#14556588	C₂³	1	2	yes	—
56	#14556580	—	2	2	no	—
57	#14556579	—	2	2	no	—
58	#14556576	—	4	2	no	—
59	Q₈×Dic₇₃	—	2	2	no	—
60	C₄×Dic₁₄₆	—	2	2	no	—
61	C₄×Dic₁₄₆	C₂³	1	2	yes	—
62	C₄×D₂₉₂	—	2	2	no	—
63	#14556553	—	2	2	no	—
64	#14556525	—	2	2	no	—
65	#14556525	—	2	1	no	—
66	#14556525	—	2	1	no	—
67	#14556525	D₄	1	1	yes	characteristic
68	#14556524	—	2	2	no	—
69	C₄⋊#10596831	C₂³	1	2	yes	—
70	#14556520	—	2	2	no	—
71	#14556516	—	2	2	no	—
72	#14556516	D₄	1	2	yes	—
73	#14556515	—	2	2	no	—
74	#14556513	—	2	2	no	—
75	#14556513	—	2	2	no	—
76	#14556513	—	2	2	no	—
77	#14556511	—	2	2	no	—
78	#14556501	—	2	2	no	—
79	C₂×Q₁₆×C₇₃	D₄	1	1	yes	characteristic
80	C₄×Q₈×C₇₃	C₂³	1	1	yes	characteristic
81	C₄×Q₈×C₇₃	C₂³	1	1	yes	characteristic
82	#107×C₇₃	D₄	1	1	yes	characteristic
83	C₄⋊C₈×C₇₃	—	2	1	no	—
84	C₄⋊C₈×C₇₃	—	2	1	no	—
85	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
87	#103×C₇₃	D₄	1	1	yes	characteristic
88	#103×C₇₃	D₄	1	1	yes	characteristic
89	C₄×C₈×C₇₃	—	2	1	no	—
90	C₄×C₈×C₇₃	—	2	1	no	—
91	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₄₆	—	2	2	no	—
95	C₂×Dic₁₄₆	—	4	2	no	—
96	C₂×Dic₁₄₆	C₂×D₄	1	2	yes	—
97	C₂×D₂₉₂	—	2	2	no	—
98	C₂×D₂₉₂	—	2	2	no	—
99	C₂×D₂₉₂	—	2	2	no	—
100	C₂×C₄×D₇₃	—	4	2	no	—
101	Dic₂₉₂	—	2	2	no	—
102	#11924017	—	8	4	no	—
103	#11924017	—	8	2	no	—
104	#11924015	—	4	4	no	—
105	D₅₈₄	—	4	2	no	—
106	D₅₈₄	—	8	1	no	—
107	C₂×#10596831	—	2	2	no	—
108	C₄⋊Dic₇₃	—	2	2	no	—
109	C₄⋊Dic₇₃	—	2	2	no	—
110	C₄⋊Dic₇₃	—	4	2	no	—
111	C₄⋊Dic₇₃	C₂×D₄	1	2	yes	—
112	#11924007	—	2	2	no	—
113	#11924007	—	4	2	no	—
114	#11924005	—	4	2	no	—
115	#11924005	—	4	2	no	—
116	#11924005	—	4	2	no	—
117	C₄×Dic₇₃	—	2	2	no	—
118	C₄×Dic₇₃	—	4	2	no	—
119	C₂×Q₈×C₇₃	C₂×D₄	1	1	yes	characteristic
120	C₂×Q₈×C₇₃	C₂×D₄	1	1	yes	characteristic
121	Q₁₆×C₇₃	—	2	2	no	—
122	C₂×C₈×C₇₃	—	2	1	no	—
123	C₂×C₈×C₇₃	—	2	1	no	—
124	C₂×C₈×C₇₃	—	2	1	no	—
125	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
129	C₄⋊C₄×C₇₃	—	2	1	no	—
130	C₄⋊C₄×C₇₃	—	2	1	no	—
131	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
133	C₄²×C₇₃	—	2	1	no	—
134	C₄²×C₇₃	—	2	1	no	—
135	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
136	C₂²×D₇₃	—	4	2	no	—
137	Q₈×C₇₃	—	2	1	no	—
138	Q₈×C₇₃	—	2	1	no	—
139	Q₈×C₇₃	—	2	1	no	—
140	Q₈×C₇₃	#135	1	2	yes	—
141	C₂×C₄×C₇₃	—	2	1	no	—
142	C₂×C₄×C₇₃	—	2	1	no	—
143	C₂×C₄×C₇₃	—	2	1	no	—
144	C₂×C₄×C₇₃	—	2	1	no	—
145	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×Dic₇₃	—	2	2	no	—
149	C₂×Dic₇₃	—	2	2	no	—
150	C₂×Dic₇₃	—	4	2	no	—
151	D₂₉₂	—	4	2	no	—
152	D₂₉₂	—	4	2	no	—
153	D₂₉₂	—	4	2	no	—
154	D₂₉₂	—	4	2	no	—
155	D₂₉₂	—	8	2	no	—
156	C₄×D₇₃	—	16	2	no	—
157	Dic₁₄₆	—	2	2	no	—
158	Dic₁₄₆	—	2	2	no	—
159	Dic₁₄₆	—	8	2	no	—
160	Dic₁₄₆	—	16	2	no	—
161	C₈×C₇₃	—	2	2	no	—
162	C₈×C₇₃	—	2	1	no	—
163	C₈×C₇₃	—	2	1	no	—
164	C₈×C₇₃	—	2	1	no	—
165	C₈×C₇₃	—	4	1	no	—
166	C₈×C₇₃	—	4	1	no	—
167	C₈×C₇₃	#137	1	2	yes	—
168	#10596831	—	4	2	no	—
169	C₂²×C₇₃	D₄²	1	1	yes	characteristic
170	C₂×D₇₃	—	8	2	no	—
171	C₂×D₇₃	—	8	2	no	—
172	C₂×D₇₃	—	8	2	no	—
173	C₄×C₇₃	—	2	2	no	—
174	C₄×C₇₃	—	2	1	no	—
175	C₄×C₇₃	—	4	1	no	—
176	C₄×C₇₃	—	4	1	no	—
177	C₄×C₇₃	—	4	1	no	—
178	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
179	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
180	C₄×C₇₃	#577	1	1	yes	characteristic
181	C₄×C₇₃	#578	1	1	yes	characteristic
182	Dic₇₃	—	4	2	no	—
183	Dic₇₃	—	4	2	no	—
184	Dic₇₃	—	8	2	no	—
185	#19942	—	73	1	no	maximal, Sylow
186	C₂×C₇₃	#3278	1	1	yes	characteristic
187	C₂×C₇₃	#3285	1	1	yes	characteristic
188	C₂×C₇₃	#3292	1	1	yes	characteristic
189	D₇₃	—	16	2	no	—
190	#3396	—	73	1	no	—
191	Q₈×Q₁₆	—	73	1	no	—
192	#3375	—	73	2	no	—
193	#1706	—	73	1	no	—
194	#1698	—	73	2	no	—
195	#1676	—	73	1	no	—
196	#1668	—	73	1	no	—
197	#1630	—	73	2	no	—
198	#1623	—	73	2	no	—
199	#1591	—	73	1	no	—
200	C₈×Q₁₆	D₇₃	1	1	yes	characteristic, complemented
201	C₇₃	#19942	1	1	yes	characteristic, complemented, Sylow
202	#559	—	73	2	no	—
203	Q₈²	—	73	2	no	—
204	#500	—	73	1	no	—
205	#500	—	146	1	no	—
206	#495	—	146	1	no	—
207	#493	—	73	1	no	—
208	#492	—	73	2	no	—
209	#489	—	146	2	no	—
210	#486	—	73	2	no	—
211	#477	—	73	2	no	—
212	#477	—	73	2	no	—
213	#477	—	146	2	no	—
214	#474	—	146	2	no	—
215	#464	—	146	2	no	—
216	#463	—	146	2	no	—
217	#460	—	73	2	no	—
218	#460	—	73	2	no	—
219	#446	—	73	2	no	—
220	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
221	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
222	#440	—	146	2	no	—
223	C₄×Q₁₆	—	73	1	no	—
224	C₄×Q₁₆	—	146	1	no	—
225	C₄×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
226	C₄×D₈	—	146	1	no	—
227	#335	—	73	1	no	—
228	#335	C₂×D₇₃	1	1	yes	characteristic
229	#333	—	73	2	no	—
230	#331	—	73	2	no	—
231	Q₈⋊C₈	—	73	2	no	—
232	Q₈⋊C₈	C₂×D₇₃	1	1	yes	characteristic
233	Q₈⋊C₈	C₂×D₇₃	1	1	yes	characteristic
234	C₈²	C₂×D₇₃	1	1	yes	characteristic
235	C₂×Q₁₆	—	73	1	no	—
236	C₂×Q₁₆	D₂₉₂	1	1	yes	characteristic
237	C₂×SD₁₆	—	292	2	no	—
238	C₂×SD₁₆	—	146	2	no	—
239	C₂×SD₁₆	—	146	2	no	—
240	C₂×D₈	—	146	1	no	—
241	C₂×D₈	—	146	1	no	—
242	C₄⋊Q₈	—	292	2	no	—
243	C₄⋊Q₈	—	73	2	no	—
244	C₄⋊Q₈	—	146	2	no	—
245	C₄⋊Q₈	—	146	2	no	—
246	#127	—	73	2	no	—
247	#124	—	146	2	no	—
248	#124	—	146	2	no	—
249	#123	—	292	2	no	—
250	C₄×Q₈	—	73	2	no	—
251	C₄×Q₈	—	146	2	no	—
252	C₄×Q₈	—	146	2	no	—
253	C₄×Q₈	C₂²×D₇₃	1	1	yes	characteristic
254	C₄×Q₈	C₂²×D₇₃	1	1	yes	characteristic
255	C₄×D₄	—	146	2	no	—
256	C₄×D₄	—	146	2	no	—
257	#107	—	73	1	no	—
258	#107	—	146	2	no	—
259	#107	—	146	1	no	—
260	#107	—	146	1	no	—
261	#107	—	146	2	no	—
262	#107	D₂₉₂	1	1	yes	characteristic
263	C₄⋊C₈	—	2	1	no	—
264	C₄⋊C₈	—	2	1	no	—
265	C₄⋊C₈	—	73	2	no	—
266	C₄⋊C₈	—	146	2	no	—
267	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
268	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
269	#103	—	73	2	no	—
270	#103	—	146	2	no	—
271	#103	—	146	2	no	—
272	#103	D₂₉₂	1	1	yes	characteristic
273	#103	D₂₉₂	1	1	yes	characteristic
274	#102	—	146	2	no	—
275	#102	—	146	2	no	—
276	#102	—	146	2	no	—
277	#102	—	146	2	no	—
278	#97	—	146	2	no	—
279	C₄×C₈	—	2	1	no	—
280	C₄×C₈	—	2	1	no	—
281	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
282	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
284	C₂×Q₈	—	292	2	no	—
285	C₂×Q₈	—	73	2	no	—
286	C₂×Q₈	—	146	2	no	—
287	C₂×Q₈	C₂×D₂₉₂	1	1	yes	characteristic
288	C₂×Q₈	C₂×D₂₉₂	1	1	yes	characteristic
289	C₂×D₄	—	146	2	no	—
290	C₂×D₄	—	146	2	no	—
291	C₂×D₄	—	146	2	no	—
292	C₂²×C₄	—	292	2	no	—
293	Q₁₆	—	2	2	no	—
294	Q₁₆	—	146	2	no	—
295	SD₁₆	—	292	4	no	—
296	SD₁₆	—	584	4	no	—
297	SD₁₆	—	584	2	no	—
298	D₈	—	292	2	no	—
299	D₈	—	584	1	no	—
300	C₂×C₈	—	2	1	no	—
301	C₂×C₈	—	2	1	no	—
302	C₂×C₈	—	2	1	no	—
303	C₂×C₈	—	146	2	no	—
304	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
305	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
306	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
307	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	292	2	no	—
311	C₄⋊C₄	—	292	2	no	—
312	C₄⋊C₄	—	73	2	no	—
313	C₄⋊C₄	—	146	2	no	—
314	C₄⋊C₄	—	146	2	no	—
315	C₄⋊C₄	—	146	2	no	—
316	C₄⋊C₄	C₂×D₂₉₂	1	1	yes	characteristic
317	C₄⋊C₄	C₂×D₂₉₂	1	1	yes	characteristic
318	C₂²⋊C₄	—	292	2	no	—
319	C₂²⋊C₄	—	292	2	no	—
320	C₂²⋊C₄	—	292	2	no	—
321	C₄²	—	2	1	no	—
322	C₄²	—	2	1	no	—
323	C₄²	—	292	2	no	—
324	C₄²	—	146	2	no	—
325	C₄²	C₂³×D₇₃	1	1	yes	characteristic
326	C₂³	—	292	2	no	—
327	Q₈	—	2	1	no	—
328	Q₈	—	2	1	no	—
329	Q₈	—	2	1	no	—
330	Q₈	—	584	2	no	—
331	Q₈	—	1168	2	no	—
332	Q₈	—	146	2	no	—
333	Q₈	—	146	2	no	—
334	Q₈	#14556613	1	2	yes	—
335	D₄	—	292	2	no	—
336	D₄	—	292	2	no	—
337	D₄	—	292	2	no	—
338	D₄	—	292	2	no	—
339	D₄	—	584	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	292	2	no	—
345	C₂×C₄	—	1168	2	no	—
346	C₂×C₄	—	146	2	no	—
347	C₂×C₄	—	146	2	no	—
348	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
349	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
350	C₂×C₄	#14556637	1	1	yes	characteristic
351	C₈	—	2	2	no	—
352	C₈	—	2	1	no	—
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	4	1	no	—
356	C₈	—	4	1	no	—
357	C₈	—	292	2	no	—
358	C₈	#14556623	1	2	yes	—
359	C₂²	—	584	2	no	—
360	C₂²	—	584	2	no	—
361	C₂²	—	584	2	no	—
362	C₂²	D₄×D₂₉₂	1	1	yes	characteristic
363	C₄	—	2	1	no	—
364	C₄	—	2	2	no	—
365	C₄	—	4	1	no	—
366	C₄	—	4	1	no	—
367	C₄	—	4	1	no	—
368	C₄	—	292	2	no	—
369	C₄	—	292	2	no	—
370	C₄	—	584	2	no	—
371	C₄	C₂×#14556613	1	1	yes	characteristic
372	C₄	C₂×#14556623	1	1	yes	characteristic
373	C₄	#17989394	1	1	yes	characteristic
374	C₄	#17989404	1	1	yes	characteristic
375	C₂	—	1168	2	no	—
376	C₂	#22621598	1	1	yes	characteristic
377	C₂	#22621674	1	1	yes	characteristic
378	C₂	#22621731	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₉₂
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₈×Q₁₆×C₇₃	C₂
socle	C₂²×C₇₃	D₄²
prime radical	#17988964	C₂²
hypercenter	C₈×Q₁₆	D₇₃
hypocenter	C₇₃	#19942
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₂₉₂	C₂	semidirect
3. #22622743	C₂	semidirect
5. #22609472	C₂	semidirect
6. #22609335	C₂	semidirect
8. #22609054	C₂	semidirect
9. #22609011	C₂	semidirect
10. #22608829	C₂	semidirect
11. C₈×Q₁₆×C₇₃	C₂	semidirect
200. C₈×Q₁₆	D₇₃	semidirect
201. C₇₃	#19942	semidirect

Maximal subgroups

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

subgroup	index
#22622944	2
Q₈×Dic₂₉₂	2
#22622743	2
#22609523	2
#22609472	2
#22609335	2
#22609291	2
#22609054	2
#22609011	2
#22608829	2
C₈×Q₁₆×C₇₃	2
#19942	73