Subgroups — #36712128

Order 22784.

472 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033242	C₂	1	1	yes	maximal, characteristic
2	#26033145	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022485	C₂	1	2	yes	maximal
4	#26022414	C₂	1	2	yes	maximal, complemented
5	#26022386	C₂	1	1	yes	maximal, characteristic
6	#26022317	C₂	1	2	yes	maximal
7	#26022299	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022108	C₂	1	2	yes	maximal, complemented
10	C₄×#15667957	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2241	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152323	—	2	1	no	—
13	#19152322	—	2	1	no	—
14	C₂×#15667958	C₂²	1	2	yes	—
15	C₂×#15667957	C₂²	1	1	yes	characteristic
16	#19151999	C₂²	1	2	yes	—
17	#19151983	C₂²	1	2	yes	—
18	#19151904	—	2	2	no	—
19	#19151880	—	2	1	no	—
20	#19151822	—	2	2	no	—
21	#19151800	—	2	1	no	—
22	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
23	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
24	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
25	#19151692	C₂²	1	2	yes	—
26	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
27	#19151426	C₂²	1	1	yes	characteristic
28	#19151423	C₂²	1	2	yes	—
29	C₄⋊#13083549	C₂²	1	1	yes	characteristic
30	#19151416	C₂²	1	2	yes	—
31	#19151403	C₂²	1	1	yes	characteristic
32	#19151402	C₂²	1	2	yes	—
33	#19151400	C₂²	1	1	yes	characteristic
34	#19151398	C₂²	1	2	yes	—
35	C₁₆×Dic₈₉	C₂²	1	2	yes	—
36	#500×C₈₉	C₂²	1	1	yes	characteristic
37	#494×C₈₉	C₂²	1	1	yes	characteristic
38	#358×C₈₉	C₂²	1	2	yes	—
39	#358×C₈₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
42	C₂×#13083561	—	2	2	no	—
43	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
44	#15668010	—	4	1	no	—
45	#15668007	—	4	2	no	—
46	#15668003	—	2	2	no	—
47	#15668002	—	4	2	no	—
48	#15668001	—	2	2	no	—
49	#15667992	—	4	1	no	—
50	Q₈×Dic₈₉	—	2	2	no	—
51	Q₈×Dic₈₉	—	4	1	no	—
52	C₄×Dic₁₇₈	—	2	1	no	—
53	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
54	#15667977	—	2	2	no	—
55	C₄×D₃₅₆	—	2	1	no	—
56	#15667970	—	4	1	no	—
57	#15667958	—	2	4	no	—
58	#15667957	—	2	2	no	—
59	C₂×#13083549	C₂³	1	2	yes	—
60	#15667946	C₂³	1	1	yes	characteristic
61	#15667946	C₂³	1	1	yes	characteristic
62	#15667943	—	2	2	no	—
63	C₄⋊#10623494	D₄	1	1	yes	characteristic
64	#15667941	D₄	1	2	yes	—
65	#15667937	—	2	1	no	—
66	#15667936	—	2	2	no	—
67	#15667935	—	2	2	no	—
68	#15667934	—	2	1	no	—
69	C₈×Dic₈₉	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₈₉	—	2	1	no	—
71	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
72	C₄⋊Q₈×C₈₉	—	2	1	no	—
73	C₄⋊Q₈×C₈₉	—	2	1	no	—
74	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
75	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	#107×C₈₉	—	2	1	no	—
77	#107×C₈₉	C₂³	1	2	yes	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₇₈	—	2	1	no	—
80	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
81	C₂×D₃₅₆	—	2	1	no	—
82	C₂×C₄×D₈₉	—	4	1	no	—
83	Dic₃₅₆	—	2	1	no	—
84	Dic₃₅₆	#41	1	2	yes	—
85	#13083561	—	8	2	no	—
86	D₇₁₂	—	2	1	no	—
87	D₇₁₂	—	2	1	no	—
88	C₂×#10623494	C₂×D₄	1	2	yes	—
89	C₄⋊Dic₈₉	—	2	1	no	—
90	C₄⋊Dic₈₉	—	2	1	no	—
91	C₄⋊Dic₈₉	—	4	2	no	—
92	C₄⋊Dic₈₉	—	4	2	no	—
93	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
94	#13083553	—	4	1	no	—
95	#13083553	—	4	2	no	—
96	#13083553	—	4	2	no	—
97	#13083553	D₈	1	2	yes	—
98	#13083551	—	4	1	no	—
99	#13083551	—	4	2	no	—
100	#13083551	—	4	2	no	—
101	C₄×Dic₈₉	—	4	1	no	—
102	C₄×Dic₈₉	—	4	2	no	—
103	C₄×Dic₈₉	—	4	2	no	—
104	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
105	#13083549	—	2	2	no	—
106	C₂×Q₈×C₈₉	—	2	2	no	—
107	C₂×Q₈×C₈₉	—	4	1	no	—
108	Q₁₆×C₈₉	—	2	2	no	—
109	Q₁₆×C₈₉	—	2	2	no	—
110	Q₁₆×C₈₉	—	8	1	no	—
111	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₈₉	—	2	2	no	—
114	C₄⋊C₄×C₈₉	—	4	1	no	—
115	C₄⋊C₄×C₈₉	—	4	1	no	—
116	C₄⋊C₄×C₈₉	—	4	1	no	—
117	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₈₉	—	2	1	no	—
119	C₁₆×C₈₉	—	2	1	no	—
120	C₂²×D₈₉	—	4	1	no	—
121	Q₈×C₈₉	—	4	2	no	—
122	Q₈×C₈₉	—	4	2	no	—
123	Q₈×C₈₉	—	8	2	no	—
124	C₂×C₄×C₈₉	—	4	2	no	—
125	C₂×C₄×C₈₉	—	4	2	no	—
126	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₈₉	—	4	1	no	—
130	C₂×Dic₈₉	—	4	1	no	—
131	C₂×Dic₈₉	—	4	1	no	—
132	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
133	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
134	D₃₅₆	—	4	1	no	—
135	D₃₅₆	—	4	1	no	—
136	C₄×D₈₉	—	8	2	no	—
137	Dic₁₇₈	—	2	2	no	—
138	Dic₁₇₈	—	2	2	no	—
139	Dic₁₇₈	—	4	1	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
142	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
143	#10623494	—	2	2	no	—
144	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
145	C₂×D₈₉	—	8	1	no	—
146	C₂×D₈₉	—	8	1	no	—
147	C₂×D₈₉	—	8	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	8	2	no	—
150	C₄×C₈₉	—	8	2	no	—
151	C₄×C₈₉	#508	1	2	yes	—
152	C₄×C₈₉	#548	1	1	yes	characteristic
153	C₄×C₈₉	#577	1	1	yes	characteristic
154	Dic₈₉	—	2	2	no	—
155	Dic₈₉	—	2	2	no	—
156	Dic₈₉	—	4	2	no	—
157	Dic₈₉	—	8	1	no	—
158	Dic₈₉	—	8	1	no	—
159	#33931	—	89	1	no	maximal, Sylow
160	C₂×C₈₉	#3358	1	1	yes	characteristic
161	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
162	C₂×C₈₉	#3417	1	1	yes	characteristic
163	D₈₉	—	16	1	no	—
164	#3393	—	89	1	no	—
165	#3383	—	89	1	no	—
166	#2242	—	89	2	no	—
167	#2241	D₈₉	1	1	yes	characteristic, complemented
168	#2229	—	89	2	no	—
169	#2224	—	89	1	no	—
170	#2211	—	89	2	no	—
171	#2209	—	89	1	no	—
172	Q₈×C₁₆	—	89	1	no	—
173	C₄×Q₃₂	—	89	2	no	—
174	C₄×SD₃₂	—	89	1	no	—
175	C₈₉	#33931	1	1	yes	characteristic, complemented, Sylow
176	#557	—	178	1	no	—
177	#556	—	178	1	no	—
178	C₂×Q₃₂	—	89	2	no	—
179	C₂×SD₃₂	—	89	1	no	—
180	#500	C₂×D₈₉	1	1	yes	characteristic
181	#499	—	89	2	no	—
182	#495	—	89	2	no	—
183	#494	C₂×D₈₉	1	1	yes	characteristic
184	#479	—	178	2	no	—
185	#475	—	178	1	no	—
186	#464	—	178	2	no	—
187	#461	—	178	1	no	—
188	C₈×Q₈	—	89	1	no	—
189	C₄×Q₁₆	—	89	2	no	—
190	C₄×Q₁₆	—	89	2	no	—
191	C₄×Q₁₆	—	89	1	no	—
192	C₄×D₈	—	89	1	no	—
193	#367	—	89	1	no	—
194	#366	—	89	2	no	—
195	C₄⋊C₁₆	—	89	2	no	—
196	C₄⋊C₁₆	—	89	1	no	—
197	#358	—	89	2	no	—
198	#358	—	89	1	no	—
199	#358	C₂×D₈₉	1	2	yes	—
200	#358	C₂×D₈₉	1	2	yes	—
201	#357	—	89	1	no	—
202	#357	—	89	2	no	—
203	C₄×C₁₆	—	89	2	no	—
204	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
205	C₂×Q₁₆	—	2	1	no	—
206	C₂×Q₁₆	—	89	1	no	—
207	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
208	C₂×SD₁₆	—	178	2	no	—
209	C₂×D₈	—	89	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	356	1	no	—
213	#126	—	356	2	no	—
214	#125	—	356	2	no	—
215	#125	—	178	2	no	—
216	#124	—	178	2	no	—
217	C₂²⋊Q₈	—	356	1	no	—
218	C₄×Q₈	—	89	1	no	—
219	C₄×Q₈	—	356	1	no	—
220	C₄×Q₈	—	178	2	no	—
221	C₄×Q₈	—	178	2	no	—
222	C₄×Q₈	—	178	1	no	—
223	C₄×D₄	—	178	1	no	—
224	#117	—	356	1	no	—
225	Q₃₂	—	178	4	no	—
226	SD₃₂	—	178	2	no	—
227	C₂×C₁₆	—	89	2	no	—
228	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
229	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
230	#107	—	2	1	no	—
231	#107	—	89	1	no	—
232	#107	—	89	1	no	—
233	#107	C₂²×D₈₉	1	2	yes	—
234	#106	—	178	2	no	—
235	C₄⋊C₈	—	89	2	no	—
236	C₄⋊C₈	—	89	1	no	—
237	#103	—	178	2	no	—
238	#103	—	178	2	no	—
239	#103	—	178	1	no	—
240	#102	—	178	1	no	—
241	C₄×C₈	—	89	2	no	—
242	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×Q₈	—	89	1	no	—
246	C₂×Q₈	—	178	1	no	—
247	C₂×D₄	—	178	1	no	—
248	C₂²×C₄	—	356	1	no	—
249	Q₁₆	—	2	2	no	—
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	8	1	no	—
252	Q₁₆	—	89	2	no	—
253	Q₁₆	—	178	1	no	—
254	SD₁₆	—	712	2	no	—
255	D₈	—	178	1	no	—
256	D₈	—	178	1	no	—
257	C₂×C₈	—	89	2	no	—
258	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
259	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	89	1	no	—
265	C₄⋊C₄	—	89	2	no	—
266	C₄⋊C₄	—	356	2	no	—
267	C₄⋊C₄	—	356	2	no	—
268	C₄⋊C₄	—	356	2	no	—
269	C₄⋊C₄	—	356	2	no	—
270	C₄⋊C₄	—	356	1	no	—
271	C₄⋊C₄	—	178	1	no	—
272	C₄⋊C₄	—	178	1	no	—
273	C₂²⋊C₄	—	356	2	no	—
274	C₂²⋊C₄	—	356	2	no	—
275	C₂²⋊C₄	—	356	1	no	—
276	C₄²	—	89	2	no	—
277	C₄²	—	356	2	no	—
278	C₄²	—	356	2	no	—
279	C₄²	—	356	1	no	—
280	C₄²	D₄×D₈₉	1	1	yes	characteristic
281	C₁₆	—	2	1	no	—
282	C₁₆	—	2	1	no	—
283	C₁₆	—	178	2	no	—
284	C₂³	—	356	1	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	4	2	no	—
287	Q₈	—	8	2	no	—
288	Q₈	—	356	1	no	—
289	Q₈	—	178	2	no	—
290	Q₈	—	178	2	no	—
291	D₄	—	356	1	no	—
292	D₄	—	356	1	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	89	2	no	—
296	C₂×C₄	—	89	2	no	—
297	C₂×C₄	—	356	1	no	—
298	C₂×C₄	—	356	1	no	—
299	C₂×C₄	—	356	1	no	—
300	C₂×C₄	—	712	2	no	—
301	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
302	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
303	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	178	2	no	—
306	C₈	#41×D₈₉	1	1	yes	characteristic
307	C₈	#15668059	1	1	yes	characteristic
308	C₂²	—	712	1	no	—
309	C₂²	—	712	1	no	—
310	C₂²	—	712	1	no	—
311	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	—	356	2	no	—
316	C₄	—	178	2	no	—
317	C₄	—	178	2	no	—
318	C₄	—	712	1	no	—
319	C₄	—	712	1	no	—
320	C₄	#19152029	1	2	yes	—
321	C₄	#19152256	1	1	yes	characteristic
322	C₄	#19152404	1	1	yes	characteristic
323	C₂	—	1424	1	no	—
324	C₂	#26032884	1	1	yes	characteristic
325	C₂	C₂×#19152029	1	1	yes	characteristic
326	C₂	#26033401	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2241	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₇₁₂	C₂³
hypercenter	#2241	D₈₉
hypocenter	C₈₉	#33931
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2241

Complements

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

N	H ≅ G/N	type
2. #26033145	C₂	semidirect
4. #26022414	C₂	semidirect
7. #26022299	C₂	semidirect
8. C₁₆×Dic₁₇₈	C₂	semidirect
9. #26022108	C₂	semidirect
11. C₈₉×#2241	C₂	semidirect
167. #2241	D₈₉	semidirect
175. C₈₉	#33931	semidirect

Maximal subgroups

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

subgroup	index
#26033242	2
#26033145	2
#26022485	2
#26022414	2
#26022386	2
#26022317	2
#26022299	2
C₁₆×Dic₁₇₈	2
#26022108	2
C₄×#15667957	2
C₈₉×#2241	2
#33931	89