Subgroups — #33185826

Order 18688.

810 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618329	C₂	1	1	yes	maximal, characteristic
2	#22611072	C₂	1	4	yes	maximal
3	#22610734	C₂	1	2	yes	maximal
4	#22610058	C₂	1	2	yes	maximal
5	#22609936	C₂	1	4	yes	maximal
6	C₄×C₄⋊#10596831	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556591	—	2	1	no	—
9	C₂×#14556591	—	2	1	no	—
10	C₂×#14556591	C₂²	1	2	yes	—
11	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
12	C₂×#14556515	C₂²	1	8	yes	—
13	C₂×C₄×#10596831	C₂²	1	2	yes	—
14	#17988457	—	2	1	no	—
15	#17988457	—	2	1	no	—
16	#17988457	C₂²	1	2	yes	—
17	#17988457	C₂²	1	2	yes	—
18	#17988349	C₂²	1	8	yes	—
19	#17988342	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₇₃	C₂²	1	2	yes	—
22	#391×C₇₃	C₂²	1	2	yes	—
23	#389×C₇₃	—	2	1	no	—
24	#389×C₇₃	—	2	1	no	—
25	#384×C₇₃	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₇₃	—	2	1	no	—
27	C₄×C₄⋊C₄×C₇₃	—	2	1	no	—
28	C₄³×C₇₃	C₂²	1	1	yes	characteristic
29	C₂²×Dic₁₄₆	—	2	2	no	—
30	C₂²×#10596831	C₂³	1	4	yes	—
31	#14556591	—	4	4	no	—
32	#14556591	—	4	4	no	—
33	#14556591	—	4	4	no	—
34	C₂×C₄⋊Dic₇₃	—	2	2	no	—
35	C₂×C₄⋊Dic₇₃	—	2	2	no	—
36	C₂×C₄⋊Dic₇₃	—	2	2	no	—
37	C₂×C₄⋊Dic₇₃	—	2	2	no	—
38	C₂×C₄⋊Dic₇₃	—	2	2	no	—
39	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
40	C₄⋊#10596831	—	2	4	no	—
41	#14556515	—	4	8	no	—
42	C₄×#10596831	—	2	4	no	—
43	C₂²×Q₈×C₇₃	C₂³	1	2	yes	—
44	C₄⋊Q₈×C₇₃	—	2	2	no	—
45	C₄⋊Q₈×C₇₃	Q₈	1	4	yes	complemented
46	C₄×Q₈×C₇₃	—	2	8	no	—
47	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
48	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
49	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
50	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
51	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₇₃	—	2	2	no	—
54	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
61	#95×C₇₃	—	2	2	no	—
62	#95×C₇₃	—	2	2	no	—
63	C₂×Dic₁₄₆	—	4	4	no	—
64	C₂×Dic₁₄₆	—	4	4	no	—
65	C₂×Dic₁₄₆	—	4	4	no	—
66	C₂²×Dic₇₃	—	2	4	no	—
67	C₂×#10596831	—	2	4	no	—
68	C₂×#10596831	—	2	4	no	—
69	C₂×#10596831	—	2	4	no	—
70	C₄⋊Dic₇₃	—	2	8	no	—
71	C₄⋊Dic₇₃	—	4	4	no	—
72	C₄⋊Dic₇₃	—	4	4	no	—
73	C₄⋊Dic₇₃	—	4	4	no	—
74	C₄⋊Dic₇₃	—	4	4	no	—
75	C₄⋊Dic₇₃	—	4	4	no	—
76	C₂×Q₈×C₇₃	—	2	2	no	—
77	C₂×Q₈×C₇₃	—	2	4	no	—
78	C₂×Q₈×C₇₃	—	2	4	no	—
79	C₂×Q₈×C₇₃	C₂×Q₈	1	4	yes	—
80	C₂²×C₄×C₇₃	—	2	2	no	—
81	C₂²×C₄×C₇₃	—	2	2	no	—
82	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
89	C₄⋊C₄×C₇₃	—	2	4	no	—
90	C₄⋊C₄×C₇₃	—	2	4	no	—
91	C₄⋊C₄×C₇₃	—	2	4	no	—
92	C₄⋊C₄×C₇₃	—	4	4	no	—
93	C₄⋊C₄×C₇₃	—	4	2	no	—
94	C₄⋊C₄×C₇₃	—	4	2	no	—
95	C₄²×C₇₃	—	2	2	no	—
96	C₄²×C₇₃	—	2	2	no	—
97	C₄²×C₇₃	—	2	2	no	—
98	C₄²×C₇₃	—	4	4	no	—
99	C₄²×C₇₃	SD₁₆	1	4	yes	—
100	C₄²×C₇₃	Q₁₆	1	4	yes	—
101	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
102	C₄²×C₇₃	#41	1	4	yes	—
103	C₄²×C₇₃	#41	1	2	yes	—
104	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
105	Q₈×C₇₃	—	2	8	no	—
106	Q₈×C₇₃	—	4	4	no	—
107	C₂×C₄×C₇₃	—	2	1	no	—
108	C₂×C₄×C₇₃	—	2	1	no	—
109	C₂×C₄×C₇₃	—	2	1	no	—
110	C₂×C₄×C₇₃	—	2	1	no	—
111	C₂×C₄×C₇₃	—	2	1	no	—
112	C₂×C₄×C₇₃	—	2	1	no	—
113	C₂×C₄×C₇₃	—	2	4	no	—
114	C₂×C₄×C₇₃	—	4	2	no	—
115	C₂×C₄×C₇₃	—	4	2	no	—
116	C₂×C₄×C₇₃	—	4	2	no	—
117	C₂×C₄×C₇₃	—	4	2	no	—
118	C₂×C₄×C₇₃	—	4	2	no	—
119	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₇₃	#124	1	2	yes	—
122	C₂×C₄×C₇₃	#124	1	2	yes	—
123	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
124	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
125	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
126	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
127	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
128	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
129	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
130	C₂×C₄×C₇₃	#137	1	2	yes	—
131	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
132	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
133	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₇₃	H(5,𝔽₂)	1	2	yes	—
135	C₂×Dic₇₃	—	4	4	no	—
136	C₂×Dic₇₃	—	4	4	no	—
137	C₂×Dic₇₃	—	4	4	no	—
138	Dic₁₄₆	—	8	8	no	—
139	#10596831	—	4	4	no	—
140	C₂²×C₇₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
141	C₂²×C₇₃	C₂×#124	1	1	yes	characteristic
142	C₂²×C₇₃	#535	1	1	yes	characteristic
143	C₂²×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
144	C₂²×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
145	C₂²×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂²×C₇₃	C₂×#137	1	1	yes	characteristic
147	C₄×C₇₃	—	2	2	no	—
148	C₄×C₇₃	—	2	2	no	—
149	C₄×C₇₃	—	2	2	no	—
150	C₄×C₇₃	—	4	4	no	—
151	C₄×C₇₃	—	8	2	no	—
152	C₄×C₇₃	#475	1	4	yes	—
153	C₄×C₇₃	#477	1	4	yes	—
154	C₄×C₇₃	#487	1	4	yes	—
155	C₄×C₇₃	#554	1	2	yes	—
156	C₄×C₇₃	#555	1	2	yes	—
157	Dic₇₃	—	8	4	no	—
158	#29141	—	73	1	no	maximal, Sylow
159	C₂×C₇₃	#2563	1	1	yes	characteristic
160	C₂×C₇₃	C₂×#475	1	1	yes	characteristic
161	C₂×C₇₃	C₂×#477	1	1	yes	characteristic
162	C₂×C₇₃	C₂×#487	1	1	yes	characteristic
163	C₂×C₇₃	#3141	1	1	yes	characteristic
164	C₂×C₇₃	#3208	1	1	yes	characteristic
165	C₂×C₇₃	#3209	1	1	yes	characteristic
166	#2868	—	73	1	no	—
167	C₄×C₄⋊Q₈	D₇₃	1	1	yes	characteristic
168	#2024	—	73	4	no	—
169	#1956	—	73	2	no	—
170	#1834	—	73	2	no	—
171	#1803	—	73	4	no	—
172	C₄×C₄⋊C₈	—	73	1	no	—
173	C₇₃	#29141	1	1	yes	characteristic, complemented, Sylow
174	C₂×C₄⋊Q₈	—	73	2	no	—
175	C₂×C₄⋊Q₈	—	146	1	no	—
176	C₂×C₄⋊Q₈	—	146	1	no	—
177	C₂×C₄⋊Q₈	C₂×D₇₃	1	1	yes	characteristic
178	C₂×C₄×Q₈	C₂×D₇₃	1	2	yes	—
179	C₂×C₄⋊C₈	—	73	2	no	—
180	C₂×#103	—	73	8	no	—
181	C₂×C₄×C₈	—	73	2	no	—
182	#391	C₂×D₇₃	1	2	yes	—
183	#389	—	2	1	no	—
184	#389	—	2	1	no	—
185	#384	—	73	2	no	—
186	#384	—	73	2	no	—
187	#384	—	146	1	no	—
188	#384	—	146	1	no	—
189	#384	C₂×D₇₃	1	1	yes	characteristic
190	C₄×C₄⋊C₄	—	2	1	no	—
191	C₄×C₄⋊C₄	—	2	1	no	—
192	C₄³	C₂×D₇₃	1	1	yes	characteristic
193	#340	—	73	8	no	—
194	#336	—	73	2	no	—
195	C₂²×Q₈	—	146	2	no	—
196	C₂²×Q₈	C₂²×D₇₃	1	2	yes	—
197	C₂²×C₈	—	73	4	no	—
198	C₄⋊Q₈	—	2	2	no	—
199	C₄⋊Q₈	—	292	4	no	—
200	C₄⋊Q₈	—	292	4	no	—
201	C₄⋊Q₈	—	292	4	no	—
202	C₄⋊Q₈	Dic₁₄₆	1	4	yes	complemented
203	C₄×Q₈	—	2	8	no	—
204	C₂×C₄⋊C₄	—	2	1	no	—
205	C₂×C₄⋊C₄	—	2	1	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	73	4	no	—
210	C₂×C₄⋊C₄	—	146	2	no	—
211	C₂×C₄⋊C₄	—	146	2	no	—
212	C₂×C₄⋊C₄	—	146	2	no	—
213	C₂×C₄⋊C₄	—	146	2	no	—
214	C₂×C₄⋊C₄	—	146	2	no	—
215	C₂×C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
218	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
219	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
220	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
221	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
222	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
223	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
224	C₄⋊C₈	—	146	4	no	—
225	#103	—	292	8	no	—
226	C₄×C₈	—	146	4	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	4	no	—
231	C₂×Q₈	—	2	4	no	—
232	C₂×Q₈	—	292	4	no	—
233	C₂×Q₈	—	292	4	no	—
234	C₂×Q₈	—	292	4	no	—
235	C₂×Q₈	C₂×Dic₁₄₆	1	4	yes	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	146	4	no	—
239	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
245	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
246	C₂×C₈	—	146	4	no	—
247	C₂×C₈	—	146	4	no	—
248	C₂×C₈	—	146	4	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	292	4	no	—
256	C₄⋊C₄	—	292	4	no	—
257	C₄⋊C₄	—	292	4	no	—
258	C₄⋊C₄	—	292	4	no	—
259	C₄⋊C₄	—	292	4	no	—
260	C₄⋊C₄	—	146	8	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	4	4	no	—
265	C₄²	#11924015	1	4	yes	—
266	C₄²	#11924018	1	4	yes	—
267	C₄²	C₂×Dic₁₄₆	1	2	yes	—
268	C₄²	#11924027	1	4	yes	—
269	C₄²	#11924028	1	2	yes	—
270	C₂³	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
271	Q₈	—	2	8	no	—
272	Q₈	—	4	4	no	—
273	Q₈	—	584	8	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	292	4	no	—
287	C₂×C₄	—	292	4	no	—
288	C₂×C₄	—	292	4	no	—
289	C₂×C₄	#14556570	1	2	yes	—
290	C₂×C₄	#14556570	1	2	yes	—
291	C₂×C₄	#14556578	1	2	yes	—
292	C₂×C₄	#14556578	1	2	yes	—
293	C₂×C₄	C₂×#11924015	1	2	yes	—
294	C₂×C₄	C₂×#11924015	1	2	yes	—
295	C₂×C₄	C₂×#11924015	1	2	yes	—
296	C₂×C₄	C₂×#11924018	1	2	yes	—
297	C₂×C₄	C₂×#11924018	1	2	yes	—
298	C₂×C₄	C₂×#11924018	1	2	yes	—
299	C₂×C₄	#14556616	1	2	yes	—
300	C₂×C₄	#14556621	1	2	yes	—
301	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
302	C₂×C₄	C₂×#11924027	1	2	yes	—
303	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
304	C₂×C₄	#14556637	1	2	yes	—
305	C₈	—	292	4	no	—
306	C₂²	C₂×#14556570	1	1	yes	characteristic
307	C₂²	C₂×#14556578	1	1	yes	characteristic
308	C₂²	#17989151	1	1	yes	characteristic
309	C₂²	C₂²×#11924015	1	1	yes	characteristic
310	C₂²	C₂²×#11924018	1	1	yes	characteristic
311	C₂²	C₂×#14556616	1	1	yes	characteristic
312	C₂²	C₂×#14556621	1	1	yes	characteristic
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	4	no	—
317	C₄	—	8	2	no	—
318	C₄	—	584	4	no	—
319	C₄	#17988872	1	4	yes	—
320	C₄	#17988882	1	4	yes	—
321	C₄	#17988937	1	4	yes	—
322	C₄	#17989294	1	2	yes	—
323	C₄	#17989302	1	2	yes	—
324	C₂	#22615104	1	1	yes	characteristic
325	C₂	C₂×#17988872	1	1	yes	characteristic
326	C₂	C₂×#17988882	1	1	yes	characteristic
327	C₂	C₂×#17988937	1	1	yes	characteristic
328	C₂	#22620324	1	1	yes	characteristic
329	C₂	#22620869	1	1	yes	characteristic
330	C₂	#22620876	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₇₃
commutator, cosocle	C₂²×C₄×C₇₃	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₄×C₄⋊Q₈×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂²×D₄
hypercenter	C₄×C₄⋊Q₈	D₇₃
hypocenter	C₇₃	#29141
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
45. C₄⋊Q₈×C₇₃	Q₈	semidirect
173. C₇₃	#29141	semidirect
202. C₄⋊Q₈	Dic₁₄₆	semidirect

Maximal subgroups

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

subgroup	index
#22618329	2
#22611072	2
#22610734	2
#22610058	2
#22609936	2
C₄×C₄⋊#10596831	2
C₄×C₄⋊Q₈×C₇₃	2
#29141	73