Subgroups — #36712056

Order 22784.

470 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032684	C₂	1	1	yes	maximal, characteristic
2	#26032644	C₂	1	1	yes	maximal, characteristic
3	#26022532	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022462	C₂	1	2	yes	maximal, complemented
5	#26022448	C₂	1	2	yes	maximal, complemented
6	#26022395	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022359	C₂	1	2	yes	maximal, complemented
8	C₄×#15667958	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022083	C₂	1	1	yes	maximal, characteristic
10	#26022037	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2227	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152321	—	2	1	no	—
13	#19152309	—	2	1	no	—
14	C₂×#15667958	C₂²	1	1	yes	characteristic
15	C₂×#15667951	C₂²	1	2	yes	—
16	#19152003	C₂²	1	1	yes	characteristic
17	#19151938	—	2	2	no	—
18	#19151935	—	2	2	no	—
19	#19151890	—	2	1	no	—
20	#19151883	—	2	1	no	—
21	#19151840	C₂²	1	2	yes	—
22	#19151836	C₂²	1	2	yes	—
23	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
24	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
25	#19151635	C₂²	1	2	yes	—
26	#107×D₈₉	C₂²	1	2	yes	—
27	#19151426	C₂²	1	2	yes	—
28	#19151425	C₂²	1	2	yes	—
29	#19151425	C₂²	1	2	yes	—
30	#19151423	C₂²	1	1	yes	characteristic
31	#19151403	C₂²	1	2	yes	—
32	#19151402	C₂²	1	1	yes	characteristic
33	#19151401	C₂²	1	1	yes	characteristic
34	#19151377	C₂²	1	2	yes	—
35	C₄×#13083549	C₂²	1	1	yes	characteristic
36	#500×C₈₉	C₂²	1	1	yes	characteristic
37	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	#366×C₈₉	C₂²	1	2	yes	—
39	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	#358×C₈₉	C₂²	1	2	yes	—
41	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
42	C₂×C₈×D₈₉	C₂³	1	2	yes	—
43	#15668012	—	2	1	no	—
44	#15668007	—	4	2	no	—
45	#15668003	—	4	1	no	—
46	#15668000	—	4	1	no	—
47	#15667992	—	2	2	no	—
48	#15667988	—	2	2	no	—
49	#15667988	—	4	2	no	—
50	C₄×Dic₁₇₈	—	2	1	no	—
51	C₄×Dic₁₇₈	—	4	1	no	—
52	C₄×D₃₅₆	D₄	1	1	yes	characteristic
53	#15667974	—	4	1	no	—
54	#15667970	—	2	2	no	—
55	C₄⋊C₄×D₈₉	—	2	2	no	—
56	#15667958	—	2	2	no	—
57	#15667951	—	2	4	no	—
58	C₂×#13083549	C₂³	1	1	yes	characteristic
59	C₂×#13083549	C₂³	1	1	yes	characteristic
60	#15667946	—	2	1	no	—
61	#15667946	C₂³	1	1	yes	characteristic
62	#15667946	C₂³	1	2	yes	—
63	#15667945	—	2	2	no	—
64	#15667943	—	2	2	no	—
65	C₄⋊#10623494	D₄	1	1	yes	characteristic
66	#15667937	—	2	1	no	—
67	#15667936	—	2	2	no	—
68	#15667935	—	2	2	no	—
69	#15667924	D₄	1	2	yes	—
70	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
71	C₄⋊Q₈×C₈₉	—	2	1	no	—
72	C₄×Q₈×C₈₉	—	2	1	no	—
73	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
74	#107×C₈₉	—	2	1	no	—
75	#107×C₈₉	C₂³	1	2	yes	—
76	#107×C₈₉	C₂³	1	1	yes	characteristic
77	#103×C₈₉	—	2	1	no	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₇₈	—	2	1	no	—
80	C₂×Dic₁₇₈	—	4	1	no	—
81	C₂×D₃₅₆	D₈	1	1	yes	characteristic
82	C₂×C₄×D₈₉	—	4	2	no	—
83	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
84	Dic₃₅₆	—	2	1	no	—
85	Dic₃₅₆	—	2	1	no	—
86	C₂×#10623494	C₂×D₄	1	2	yes	—
87	C₈×D₈₉	—	2	2	no	—
88	C₈×D₈₉	—	2	2	no	—
89	C₄⋊Dic₈₉	—	2	2	no	—
90	C₄⋊Dic₈₉	—	2	1	no	—
91	C₄⋊Dic₈₉	—	4	1	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	#13083551	—	4	2	no	—
98	#13083551	—	4	2	no	—
99	#13083551	—	4	2	no	—
100	#13083551	D₈	1	2	yes	—
101	C₄×Dic₈₉	—	4	1	no	—
102	C₄×Dic₈₉	—	4	2	no	—
103	#13083549	—	2	1	no	—
104	#13083549	#41	1	2	yes	—
105	C₂×Q₈×C₈₉	—	2	1	no	—
106	C₂×Q₈×C₈₉	—	4	1	no	—
107	Q₁₆×C₈₉	—	2	1	no	—
108	Q₁₆×C₈₉	—	2	1	no	—
109	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	—	2	1	no	—
112	C₄⋊C₄×C₈₉	—	2	2	no	—
113	C₄⋊C₄×C₈₉	—	4	1	no	—
114	C₄⋊C₄×C₈₉	—	4	1	no	—
115	C₄²×C₈₉	—	4	1	no	—
116	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₈₉	—	2	2	no	—
118	C₂²×D₈₉	C₂×D₈	1	2	yes	—
119	Q₈×C₈₉	—	4	1	no	—
120	Q₈×C₈₉	—	4	1	no	—
121	Q₈×C₈₉	—	16	1	no	—
122	C₂×C₄×C₈₉	—	4	1	no	—
123	C₂×C₄×C₈₉	—	4	1	no	—
124	C₂×C₄×C₈₉	—	4	2	no	—
125	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₈₉	—	4	2	no	—
129	C₂×Dic₈₉	—	4	1	no	—
130	C₂×Dic₈₉	—	4	1	no	—
131	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
132	D₃₅₆	—	2	2	no	—
133	C₄×D₈₉	—	2	2	no	—
134	C₄×D₈₉	—	2	2	no	—
135	C₄×D₈₉	—	16	2	no	—
136	Dic₁₇₈	—	4	1	no	—
137	Dic₁₇₈	—	4	1	no	—
138	Dic₁₇₈	—	8	2	no	—
139	C₈×C₈₉	—	2	1	no	—
140	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
141	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
142	#10623494	—	2	2	no	—
143	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
144	C₂×D₈₉	—	2	2	no	—
145	C₂×D₈₉	—	2	2	no	—
146	C₂×D₈₉	—	2	2	no	—
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	—	8	1	no	—
149	C₄×C₈₉	—	8	2	no	—
150	C₄×C₈₉	—	8	1	no	—
151	C₄×C₈₉	#510	1	2	yes	—
152	C₄×C₈₉	#547	1	1	yes	characteristic
153	C₄×C₈₉	#578	1	1	yes	characteristic
154	Dic₈₉	—	2	2	no	—
155	Dic₈₉	—	8	1	no	—
156	Dic₈₉	—	8	2	no	—
157	Dic₈₉	—	8	1	no	—
158	#33923	—	89	1	no	maximal, Sylow
159	C₂×C₈₉	#3334	1	1	yes	characteristic
160	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
161	C₂×C₈₉	#3416	1	1	yes	characteristic
162	D₈₉	—	4	2	no	—
163	#3342	—	89	1	no	—
164	#3338	—	89	1	no	—
165	#2254	—	89	1	no	—
166	#2237	—	89	2	no	—
167	#2235	—	89	2	no	—
168	#2227	D₈₉	1	1	yes	characteristic, complemented
169	#2225	—	89	1	no	—
170	#2219	—	89	2	no	—
171	C₄×Q₃₂	—	89	1	no	—
172	#2169	—	89	1	no	—
173	#2159	—	89	2	no	—
174	C₈₉	#33923	1	1	yes	characteristic, complemented, Sylow
175	#556	—	178	1	no	—
176	#554	—	178	1	no	—
177	C₂×Q₃₂	—	89	1	no	—
178	C₂×M₃₂	—	89	2	no	—
179	#500	—	89	1	no	—
180	#500	C₂×D₈₉	1	1	yes	characteristic
181	#485	—	178	2	no	—
182	#484	—	178	2	no	—
183	#477	—	178	1	no	—
184	#475	—	178	1	no	—
185	#467	—	89	2	no	—
186	#467	—	89	2	no	—
187	C₄×Q₁₆	—	89	1	no	—
188	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
189	C₈×D₄	—	89	1	no	—
190	#427	—	89	2	no	—
191	C₂×#107	—	89	2	no	—
192	#367	—	89	2	no	—
193	#367	—	89	2	no	—
194	#367	—	89	2	no	—
195	#366	—	89	1	no	—
196	#366	C₂×D₈₉	1	2	yes	—
197	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
198	#358	—	89	1	no	—
199	#358	—	89	2	no	—
200	#358	—	89	1	no	—
201	#358	C₂×D₈₉	1	2	yes	—
202	C₂²⋊C₁₆	—	89	2	no	—
203	C₄×C₁₆	—	89	1	no	—
204	C₂×Q₁₆	—	89	1	no	—
205	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
206	C₂²×C₈	—	89	2	no	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	178	1	no	—
209	#126	—	356	2	no	—
210	#125	—	356	1	no	—
211	#124	—	356	1	no	—
212	C₂²⋊Q₈	—	356	2	no	—
213	C₂²⋊Q₈	—	178	2	no	—
214	C₂²⋊Q₈	—	178	2	no	—
215	C₄×Q₈	—	2	1	no	—
216	C₄×Q₈	—	356	1	no	—
217	C₄×Q₈	—	178	1	no	—
218	C₄×D₄	—	89	1	no	—
219	C₄×D₄	—	356	1	no	—
220	#117	—	178	2	no	—
221	C₂×C₄⋊C₄	—	178	2	no	—
222	Q₃₂	—	178	2	no	—
223	M₃₂	—	178	4	no	—
224	C₂×C₁₆	—	89	1	no	—
225	C₂×C₁₆	—	89	1	no	—
226	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
227	#107	—	2	1	no	—
228	#107	—	89	1	no	—
229	#107	—	89	2	no	—
230	#107	—	178	2	no	—
231	#107	—	178	1	no	—
232	#107	C₂²×D₈₉	1	1	yes	characteristic
233	#107	C₂²×D₈₉	1	2	yes	—
234	#106	—	178	2	no	—
235	C₄⋊C₈	—	89	1	no	—
236	#103	—	2	1	no	—
237	#103	—	178	2	no	—
238	#103	—	178	2	no	—
239	#103	—	178	1	no	—
240	C₂²⋊C₈	—	89	2	no	—
241	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	356	1	no	—
245	C₂×Q₈	—	178	1	no	—
246	C₂×D₄	—	89	1	no	—
247	C₂²×C₄	—	89	2	no	—
248	C₂²×C₄	—	356	2	no	—
249	Q₁₆	—	2	1	no	—
250	Q₁₆	—	2	1	no	—
251	Q₁₆	—	178	1	no	—
252	Q₁₆	—	178	1	no	—
253	C₂×C₈	—	89	2	no	—
254	C₂×C₈	—	178	2	no	—
255	C₂×C₈	—	178	2	no	—
256	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
257	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	89	1	no	—
263	C₄⋊C₄	—	356	2	no	—
264	C₄⋊C₄	—	356	2	no	—
265	C₄⋊C₄	—	356	2	no	—
266	C₄⋊C₄	—	356	1	no	—
267	C₄⋊C₄	—	356	1	no	—
268	C₄⋊C₄	—	178	2	no	—
269	C₄⋊C₄	—	178	1	no	—
270	C₂²⋊C₄	—	89	2	no	—
271	C₂²⋊C₄	—	356	2	no	—
272	C₂²⋊C₄	—	356	2	no	—
273	C₂²⋊C₄	—	356	2	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	356	2	no	—
276	C₄²	—	356	1	no	—
277	C₄²	D₄×D₈₉	1	1	yes	characteristic
278	C₁₆	—	2	2	no	—
279	C₁₆	—	89	2	no	—
280	C₁₆	—	178	1	no	—
281	C₂³	—	89	2	no	—
282	Q₈	—	4	1	no	—
283	Q₈	—	4	1	no	—
284	Q₈	—	16	1	no	—
285	Q₈	—	356	1	no	—
286	Q₈	—	356	1	no	—
287	Q₈	—	712	2	no	—
288	D₄	—	178	2	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	89	2	no	—
293	C₂×C₄	—	356	1	no	—
294	C₂×C₄	—	356	2	no	—
295	C₂×C₄	—	356	1	no	—
296	C₂×C₄	—	1424	2	no	—
297	C₂×C₄	—	178	2	no	—
298	C₂×C₄	—	178	2	no	—
299	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
300	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
301	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	178	2	no	—
304	C₈	#41×D₈₉	1	1	yes	characteristic
305	C₈	#15668060	1	1	yes	characteristic
306	C₂²	—	178	2	no	—
307	C₂²	—	178	2	no	—
308	C₂²	—	178	2	no	—
309	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	8	1	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	1	no	—
314	C₄	—	178	2	no	—
315	C₄	—	712	1	no	—
316	C₄	—	712	2	no	—
317	C₄	—	712	1	no	—
318	C₄	#19152040	1	2	yes	—
319	C₄	#19152250	1	1	yes	characteristic
320	C₄	#19152414	1	1	yes	characteristic
321	C₂	—	356	2	no	—
322	C₂	#26032598	1	1	yes	characteristic
323	C₂	C₂×#19152040	1	1	yes	characteristic
324	C₂	#26033398	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×#2227	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₃₅₆	D₈
hypercenter	#2227	D₈₉
hypocenter	C₈₉	#33923
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2227

Complements

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

N	H ≅ G/N	type
3. #26022532	C₂	semidirect
4. #26022462	C₂	semidirect
5. #26022448	C₂	semidirect
6. #26022395	C₂	semidirect
7. #26022359	C₂	semidirect
8. C₄×#15667958	C₂	semidirect
10. #26022037	C₂	semidirect
11. C₈₉×#2227	C₂	semidirect
168. #2227	D₈₉	semidirect
174. C₈₉	#33923	semidirect

Maximal subgroups

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

subgroup	index
#26032684	2
#26032644	2
#26022532	2
#26022462	2
#26022448	2
#26022395	2
#26022359	2
C₄×#15667958	2
#26022083	2
#26022037	2
C₈₉×#2227	2
#33923	89