Subgroups — #33077856

Order 18688.

622 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	#22622848	C₂	1	2	yes	maximal
2	Q₁₆×D₂₉₂	C₂	1	1	yes	maximal, characteristic
3	#22609547	C₂	1	1	yes	maximal, characteristic, complemented
4	#22609447	C₂	1	2	yes	maximal, complemented
5	#22609301	C₂	1	1	yes	maximal, characteristic, complemented
6	#22609293	C₂	1	1	yes	maximal, characteristic
7	#22609178	C₂	1	2	yes	maximal
8	#22609173	C₂	1	2	yes	maximal, complemented
9	#22608828	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#11924015	C₂	1	1	yes	maximal, characteristic
11	C₇₃×#3395	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₃	—	2	1	no	—
13	Q₈×D₂₉₂	C₂²	1	2	yes	—
14	#17988997	—	2	2	no	—
15	#17988990	C₂²	1	2	yes	—
16	#17988990	C₂²	1	2	yes	—
17	#17988967	C₂²	1	1	yes	characteristic
18	#17988884	—	2	2	no	—
19	#17988882	—	2	2	no	—
20	#17988872	C₂²	1	2	yes	—
21	#17988864	C₂²	1	2	yes	—
22	#17988829	—	2	1	no	—
23	#17988797	C₂²	1	2	yes	—
24	#17988744	—	2	2	no	—
25	#17988718	C₂²	1	2	yes	—
26	#17988685	—	2	2	no	—
27	C₄×#11924015	C₂²	1	1	yes	characteristic
28	C₈×D₂₉₂	C₂²	1	1	yes	characteristic
29	#17988340	C₂²	1	1	yes	characteristic
30	#17988337	C₂²	1	1	yes	characteristic
31	#17988334	C₂²	1	2	yes	—
32	#17988333	C₂²	1	2	yes	—
33	Q₈⋊#10596831	C₂²	1	1	yes	characteristic
34	Q₈⋊#10596831	C₂²	1	2	yes	—
35	#17988314	C₂²	1	1	yes	characteristic
36	C₈×#10596831	C₂²	1	1	yes	characteristic
37	#557×C₇₃	C₂²	1	2	yes	—
38	#494×C₇₃	—	2	1	no	—
39	#494×C₇₃	C₂²	1	1	yes	characteristic
40	#462×C₇₃	—	2	2	no	—
41	#462×C₇₃	C₂²	1	2	yes	—
42	C₈×Q₈×C₇₃	C₂²	1	1	yes	characteristic
43	C₄×Q₁₆×C₇₃	—	2	1	no	—
44	C₄×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₇₃	—	2	2	no	—
46	C₂×Dic₂₉₂	—	2	1	no	—
47	C₂×Dic₂₉₂	—	2	1	no	—
48	C₂×#11924018	—	2	2	no	—
49	Q₁₆×D₇₃	—	4	4	no	—
50	C₂×#11924015	D₄	1	1	yes	characteristic
51	C₂×C₈×D₇₃	—	2	1	no	—
52	#14556591	—	2	2	no	—
53	#14556591	C₂³	1	2	yes	—
54	#14556571	—	2	2	no	—
55	#14556571	—	4	2	no	—
56	C₄×D₂₉₂	—	2	2	no	—
57	C₄×D₂₉₂	C₂³	1	1	yes	characteristic
58	#14556524	—	2	2	no	—
59	#14556524	—	2	2	no	—
60	#14556524	—	2	2	no	—
61	#14556524	—	4	2	no	—
62	#14556522	—	2	2	no	—
63	#14556522	D₄	1	1	yes	characteristic
64	C₄⋊#10596831	—	2	2	no	—
65	C₄⋊#10596831	C₂³	1	2	yes	—
66	C₄⋊#10596831	C₂³	1	1	yes	characteristic
67	#14556515	—	2	2	no	—
68	#14556515	D₄	1	1	yes	characteristic
69	#14556514	—	2	2	no	—
70	#14556511	—	2	2	no	—
71	#14556511	D₄	1	1	yes	characteristic
72	#14556503	—	2	1	no	—
73	#14556502	—	2	2	no	—
74	C₄×#10596831	C₂³	1	1	yes	characteristic
75	C₄×#10596831	C₂³	1	1	yes	characteristic
76	C₂×Q₁₆×C₇₃	—	2	2	no	—
77	C₂×Q₁₆×C₇₃	—	2	1	no	—
78	C₂×Q₁₆×C₇₃	—	2	1	no	—
79	C₂×Q₁₆×C₇₃	D₄	1	1	yes	characteristic
80	C₄⋊Q₈×C₇₃	—	2	2	no	—
81	C₄⋊Q₈×C₇₃	C₂³	1	2	yes	—
82	#125×C₇₃	—	2	2	no	—
83	#125×C₇₃	—	4	2	no	—
84	C₄×Q₈×C₇₃	—	2	2	no	—
85	C₄×Q₈×C₇₃	—	2	2	no	—
86	C₄×Q₈×C₇₃	C₂³	1	1	yes	characteristic
87	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
88	#107×C₇₃	D₄	1	1	yes	characteristic
89	C₄⋊C₈×C₇₃	—	2	1	no	—
90	C₄⋊C₈×C₇₃	C₂³	1	1	yes	characteristic
91	#103×C₇₃	—	2	2	no	—
92	#103×C₇₃	D₄	1	2	yes	—
93	C₄×C₈×C₇₃	—	2	1	no	—
94	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
95	C₂×Dic₁₄₆	—	2	2	no	—
96	C₂×Dic₁₄₆	—	4	2	no	—
97	Q₈×D₇₃	—	4	4	no	—
98	Q₈×D₇₃	—	8	2	no	—
99	C₂×D₂₉₂	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×D₇₃	—	2	1	no	—
101	C₂×C₄×D₇₃	—	4	2	no	—
102	Dic₂₉₂	—	4	2	no	—
103	Dic₂₉₂	—	8	1	no	—
104	#11924018	—	4	4	no	—
105	#11924015	Q₁₆	1	4	yes	complemented
106	C₂×#10596831	—	2	1	no	—
107	C₂×#10596831	—	2	2	no	—
108	C₂×#10596831	—	2	1	no	—
109	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
110	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
111	C₈×D₇₃	—	4	2	no	—
112	C₄⋊Dic₇₃	—	2	2	no	—
113	C₄⋊Dic₇₃	—	2	2	no	—
114	C₄⋊Dic₇₃	—	4	2	no	—
115	C₄⋊Dic₇₃	C₂×D₄	1	1	yes	characteristic
116	#11924005	—	2	1	no	—
117	#11924005	—	4	2	no	—
118	C₂×Q₈×C₇₃	—	2	2	no	—
119	C₂×Q₈×C₇₃	C₂×D₄	1	1	yes	characteristic
120	C₂×Q₈×C₇₃	C₂×D₄	1	2	yes	—
121	Q₁₆×C₇₃	—	2	2	no	—
122	Q₁₆×C₇₃	—	4	2	no	—
123	Q₁₆×C₇₃	—	8	1	no	—
124	Q₁₆×C₇₃	—	8	2	no	—
125	C₂×C₈×C₇₃	—	2	1	no	—
126	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
128	C₄⋊C₄×C₇₃	—	2	1	no	—
129	C₄⋊C₄×C₇₃	—	2	2	no	—
130	C₄⋊C₄×C₇₃	—	2	2	no	—
131	C₄⋊C₄×C₇₃	—	2	2	no	—
132	C₄⋊C₄×C₇₃	—	4	2	no	—
133	C₄⋊C₄×C₇₃	—	4	2	no	—
134	C₄⋊C₄×C₇₃	—	4	2	no	—
135	C₄⋊C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
136	C₄⋊C₄×C₇₃	C₂×D₄	1	2	yes	—
137	C₄²×C₇₃	—	2	1	no	—
138	C₄²×C₇₃	—	2	2	no	—
139	C₄²×C₇₃	—	4	2	no	—
140	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
141	C₂²×D₇₃	—	2	1	no	—
142	Q₈×C₇₃	—	2	1	no	—
143	Q₈×C₇₃	—	2	2	no	—
144	Q₈×C₇₃	—	2	2	no	—
145	Q₈×C₇₃	—	4	2	no	—
146	Q₈×C₇₃	—	4	2	no	—
147	Q₈×C₇₃	C₂×Q₁₆	1	2	yes	complemented
148	C₂×C₄×C₇₃	—	2	1	no	—
149	C₂×C₄×C₇₃	—	2	2	no	—
150	C₂×C₄×C₇₃	—	2	1	no	—
151	C₂×C₄×C₇₃	—	2	2	no	—
152	C₂×C₄×C₇₃	—	4	2	no	—
153	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂×Dic₇₃	—	2	1	no	—
157	C₂×Dic₇₃	—	4	2	no	—
158	D₂₉₂	—	2	1	no	—
159	D₂₉₂	C₂×Q₁₆	1	2	yes	—
160	C₄×D₇₃	—	4	1	no	—
161	C₄×D₇₃	—	4	1	no	—
162	C₄×D₇₃	—	8	4	no	—
163	Dic₁₄₆	—	4	2	no	—
164	Dic₁₄₆	—	4	2	no	—
165	Dic₁₄₆	—	8	4	no	—
166	C₈×C₇₃	—	2	1	no	—
167	C₈×C₇₃	—	4	1	no	—
168	C₈×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
169	#10596831	—	2	2	no	—
170	#10596831	—	2	1	no	—
171	#10596831	—	4	1	no	—
172	#10596831	—	4	2	no	—
173	#10596831	C₂×Q₁₆	1	2	yes	complemented
174	C₂²×C₇₃	D₄²	1	1	yes	characteristic
175	C₂×D₇₃	—	2	2	no	—
176	C₂×D₇₃	—	4	1	no	—
177	C₂×D₇₃	—	4	1	no	—
178	C₄×C₇₃	—	2	2	no	—
179	C₄×C₇₃	—	2	1	no	—
180	C₄×C₇₃	—	4	2	no	—
181	C₄×C₇₃	—	4	2	no	—
182	C₄×C₇₃	—	4	1	no	—
183	C₄×C₇₃	—	8	2	no	—
184	C₄×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
185	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₄×C₇₃	#576	1	1	yes	characteristic
187	C₄×C₇₃	#578	1	1	yes	characteristic
188	Dic₇₃	—	4	1	no	—
189	Dic₇₃	—	8	2	no	—
190	#19928	—	73	1	no	maximal, Sylow
191	C₂×C₇₃	#3274	1	1	yes	characteristic
192	C₂×C₇₃	D₄×Q₁₆	1	1	yes	characteristic
193	C₂×C₇₃	#3293	1	1	yes	characteristic
194	D₇₃	—	4	2	no	—
195	#3395	D₇₃	1	1	yes	characteristic, complemented
196	#3386	—	73	2	no	—
197	D₄×Q₁₆	—	73	1	no	—
198	#1713	—	73	1	no	—
199	#1694	—	73	2	no	—
200	#1670	—	73	1	no	—
201	#1669	—	73	1	no	—
202	#1649	—	73	2	no	—
203	#1648	—	73	2	no	—
204	#1591	—	73	1	no	—
205	C₈×SD₁₆	—	73	1	no	—
206	C₇₃	#19928	1	1	yes	characteristic, complemented, Sylow
207	C₂²×Q₁₆	—	146	1	no	—
208	#557	C₂×D₇₃	1	2	yes	—
209	D₄×Q₈	—	73	2	no	—
210	#501	—	146	2	no	—
211	#500	—	73	2	no	—
212	#500	—	73	2	no	—
213	#494	—	2	1	no	—
214	#494	—	73	1	no	—
215	#494	C₂×D₇₃	1	1	yes	characteristic
216	#478	—	146	2	no	—
217	#477	—	146	2	no	—
218	#475	—	73	2	no	—
219	#474	—	73	2	no	—
220	#467	—	146	1	no	—
221	#462	—	2	2	no	—
222	#462	—	73	2	no	—
223	#462	C₂×D₇₃	1	2	yes	—
224	#451	—	146	2	no	—
225	#446	—	73	2	no	—
226	C₈×Q₈	C₂×D₇₃	1	1	yes	characteristic
227	#440	—	146	2	no	—
228	C₄×Q₁₆	—	2	1	no	—
229	C₄×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
230	C₄×SD₁₆	—	73	1	no	—
231	C₈×D₄	—	73	1	no	—
232	#335	—	73	1	no	—
233	#334	—	73	1	no	—
234	#332	—	73	2	no	—
235	#332	—	73	2	no	—
236	Q₈⋊C₈	—	73	1	no	—
237	Q₈⋊C₈	—	73	2	no	—
238	D₄⋊C₈	—	73	1	no	—
239	C₈²	—	73	1	no	—
240	C₂²×Q₈	—	146	2	no	—
241	C₂×Q₁₆	—	2	2	no	—
242	C₂×Q₁₆	—	2	1	no	—
243	C₂×Q₁₆	—	2	1	no	—
244	C₂×Q₁₆	—	292	4	no	—
245	C₂×Q₁₆	—	146	2	no	—
246	C₂×Q₁₆	—	146	1	no	—
247	C₂×Q₁₆	—	146	1	no	—
248	C₂×Q₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
249	C₂×SD₁₆	—	73	1	no	—
250	C₂²×C₈	—	146	1	no	—
251	C₄⋊Q₈	—	2	2	no	—
252	C₄⋊Q₈	—	73	2	no	—
253	C₄⋊Q₈	—	146	2	no	—
254	C₄⋊Q₈	C₂²×D₇₃	1	2	yes	—
255	#125	—	2	2	no	—
256	#125	—	4	2	no	—
257	C₂²⋊Q₈	—	292	2	no	—
258	C₂²⋊Q₈	—	146	2	no	—
259	C₄×Q₈	—	2	2	no	—
260	C₄×Q₈	—	2	2	no	—
261	C₄×Q₈	C₂²×D₇₃	1	1	yes	characteristic
262	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
263	C₄×D₄	—	73	1	no	—
264	C₄×D₄	—	146	2	no	—
265	#107	—	292	2	no	—
266	#107	—	146	2	no	—
267	#107	—	146	2	no	—
268	#107	—	146	2	no	—
269	#107	C₂²⋊D₇₃	1	1	yes	characteristic
270	#106	—	73	1	no	—
271	#106	—	146	2	no	—
272	C₄⋊C₈	—	2	1	no	—
273	C₄⋊C₈	—	73	1	no	—
274	C₄⋊C₈	—	73	2	no	—
275	C₄⋊C₈	—	146	2	no	—
276	C₄⋊C₈	C₂²×D₇₃	1	1	yes	characteristic
277	#103	—	2	2	no	—
278	#103	—	73	1	no	—
279	#103	—	146	2	no	—
280	#103	—	146	2	no	—
281	#103	C₂²⋊D₇₃	1	2	yes	—
282	#102	—	73	1	no	—
283	#102	—	146	2	no	—
284	C₂²⋊C₈	—	146	1	no	—
285	#97	—	146	2	no	—
286	C₄×C₈	—	2	1	no	—
287	C₄×C₈	—	73	1	no	—
288	C₄×C₈	—	73	1	no	—
289	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	—	292	2	no	—
292	C₂×Q₈	—	292	4	no	—
293	C₂×Q₈	—	584	2	no	—
294	C₂×Q₈	—	146	2	no	—
295	C₂×Q₈	D₄×D₇₃	1	1	yes	characteristic
296	C₂×Q₈	C₂×C₂²⋊D₇₃	1	2	yes	—
297	C₂×D₄	—	73	1	no	—
298	C₂²×C₄	—	292	2	no	—
299	C₂²×C₄	—	146	1	no	—
300	Q₁₆	—	2	2	no	—
301	Q₁₆	—	4	2	no	—
302	Q₁₆	—	8	1	no	—
303	Q₁₆	—	8	2	no	—
304	Q₁₆	—	292	4	no	—
305	Q₁₆	—	292	2	no	—
306	Q₁₆	—	584	1	no	—
307	SD₁₆	—	73	4	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	292	2	no	—
310	C₂×C₈	—	73	1	no	—
311	C₂×C₈	—	73	1	no	—
312	C₂×C₈	—	146	2	no	—
313	C₂×C₈	—	146	1	no	—
314	C₂×C₈	—	146	1	no	—
315	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
316	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	4	2	no	—
322	C₄⋊C₄	—	4	2	no	—
323	C₄⋊C₄	—	4	2	no	—
324	C₄⋊C₄	—	292	2	no	—
325	C₄⋊C₄	—	73	1	no	—
326	C₄⋊C₄	—	146	2	no	—
327	C₄⋊C₄	—	146	2	no	—
328	C₄⋊C₄	D₄×D₇₃	1	1	yes	characteristic
329	C₄⋊C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
330	C₂²⋊C₄	—	292	2	no	—
331	C₂²⋊C₄	—	146	1	no	—
332	C₄²	—	2	2	no	—
333	C₄²	—	2	1	no	—
334	C₄²	—	4	2	no	—
335	C₄²	C₂³×D₇₃	1	1	yes	characteristic
336	C₂³	—	146	1	no	—
337	Q₈	—	2	1	no	—
338	Q₈	—	2	2	no	—
339	Q₈	—	2	2	no	—
340	Q₈	—	4	2	no	—
341	Q₈	—	4	2	no	—
342	Q₈	—	292	2	no	—
343	Q₈	—	292	2	no	—
344	Q₈	—	584	4	no	—
345	Q₈	Q₁₆×D₇₃	1	2	yes	complemented
346	D₄	—	73	2	no	—
347	D₄	—	146	1	no	—
348	C₂×C₄	—	2	1	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	4	2	no	—
353	C₂×C₄	—	292	1	no	—
354	C₂×C₄	—	292	1	no	—
355	C₂×C₄	—	292	2	no	—
356	C₂×C₄	—	584	4	no	—
357	C₂×C₄	—	146	1	no	—
358	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
359	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
360	C₂×C₄	#14556636	1	1	yes	characteristic
361	C₈	—	2	1	no	—
362	C₈	—	4	1	no	—
363	C₈	—	292	2	no	—
364	C₈	—	292	1	no	—
365	C₈	—	73	2	no	—
366	C₈	—	146	2	no	—
367	C₈	—	146	1	no	—
368	C₈	#14556616	1	2	yes	—
369	C₂²	—	292	1	no	—
370	C₂²	—	292	1	no	—
371	C₂²	—	146	2	no	—
372	C₂²	D₄×C₂²⋊D₇₃	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	2	2	no	—
375	C₄	—	4	2	no	—
376	C₄	—	4	1	no	—
377	C₄	—	4	2	no	—
378	C₄	—	8	2	no	—
379	C₄	—	292	1	no	—
380	C₄	—	584	2	no	—
381	C₄	C₂×Q₁₆×D₇₃	1	1	yes	characteristic
382	C₄	C₂×#14556616	1	1	yes	characteristic
383	C₄	#17989390	1	1	yes	characteristic
384	C₄	#17989400	1	1	yes	characteristic
385	C₂	—	292	2	no	—
386	C₂	#22621551	1	1	yes	characteristic
387	C₂	Q₁₆×C₂²⋊D₇₃	1	1	yes	characteristic
388	C₂	#22621733	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₇₃
commutator, cosocle	C₄²×C₇₃	C₂⁴
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#3395	C₂
socle	C₂²×C₇₃	D₄²
prime radical	C₂×D₂₉₂	C₂×D₄
hypercenter	#3395	D₇₃
hypocenter	C₇₃	#19928
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #3395

Complements

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

N	H ≅ G/N	type
3. #22609547	C₂	semidirect
4. #22609447	C₂	semidirect
5. #22609301	C₂	semidirect
8. #22609173	C₂	semidirect
9. #22608828	C₂	semidirect
11. C₇₃×#3395	C₂	semidirect
105. #11924015	Q₁₆	semidirect
147. Q₈×C₇₃	C₂×Q₁₆	semidirect
173. #10596831	C₂×Q₁₆	semidirect
195. #3395	D₇₃	semidirect
206. C₇₃	#19928	semidirect
345. Q₈	Q₁₆×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22622848	2
Q₁₆×D₂₉₂	2
#22609547	2
#22609447	2
#22609301	2
#22609293	2
#22609178	2
#22609173	2
#22608828	2
C₈×#11924015	2
C₇₃×#3395	2
#19928	73