Subgroups — #36542359

Order 22784.

1050 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26031167	C₂	1	2	yes	maximal, complemented
2	#26030705	C₂	1	2	yes	maximal, complemented
3	C₂×#19151975	C₂	1	1	yes	maximal, characteristic, complemented
4	#26019920	C₂	1	4	yes	maximal, complemented
5	#26019866	C₂	1	2	yes	maximal, complemented
6	#26019845	C₂	1	2	yes	maximal, complemented
7	C₈×#13083558	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#3169	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668037	—	2	4	no	—
10	C₂²×D₇₁₂	—	2	2	no	—
11	C₂×#15668009	C₂²	1	2	yes	—
12	#19151988	—	2	4	no	—
13	#19151975	—	2	2	no	—
14	#19151975	C₂²	1	2	yes	—
15	#19151975	C₂²	1	2	yes	—
16	#19151974	C₂²	1	4	yes	complemented
17	#19151974	C₂²	1	4	yes	complemented
18	#19151972	C₂²	1	4	yes	complemented
19	#19151946	C₂²	1	4	yes	complemented
20	#19151793	—	2	4	no	—
21	#19151731	—	2	4	no	—
22	#19151651	C₂²	1	1	yes	characteristic
23	C₄×#13083558	C₂²	1	1	yes	characteristic
24	#19151320	C₂²	1	2	yes	complemented
25	C₈×#10623494	C₂²	1	2	yes	complemented
26	#532×C₈₉	C₂²	1	2	yes	—
27	#500×C₈₉	C₂²	1	1	yes	characteristic, complemented
28	#493×C₈₉	C₂²	1	1	yes	characteristic, complemented
29	#486×C₈₉	C₂²	1	2	yes	complemented
30	#466×C₈₉	—	2	2	no	—
31	C₄×D₈×C₈₉	—	2	2	no	—
32	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
33	C₂²×D₃₅₆	—	2	2	no	—
34	C₂²×D₃₅₆	—	2	2	no	—
35	C₂²×D₃₅₆	—	2	2	no	—
36	#15668037	—	4	16	no	—
37	C₂×#13083561	—	2	8	no	—
38	C₂×D₇₁₂	—	2	4	no	—
39	C₂×D₇₁₂	—	2	4	no	—
40	C₂×D₇₁₂	—	4	8	no	—
41	C₂×#13083559	—	2	8	no	—
42	C₂×#13083559	—	2	8	no	—
43	C₂×#13083558	D₄	1	2	yes	—
44	#15668009	—	2	2	no	—
45	#15668009	—	2	2	no	—
46	#15668009	—	2	2	no	—
47	#15668009	C₂³	1	4	yes	—
48	C₄⋊#10623494	D₄	1	2	yes	—
49	#15667932	—	2	8	no	—
50	C₂²⋊#10623494	D₄	1	2	yes	—
51	#15667923	D₄	1	2	yes	—
52	C₄×#10623494	C₂³	1	2	yes	—
53	C₄×#10623494	C₂³	1	2	yes	—
54	C₂×#41×C₈₉	—	2	2	no	—
55	C₂×D₈×C₈₉	—	2	2	no	—
56	C₂²×C₈×C₈₉	D₄	1	2	yes	—
57	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
58	#127×C₈₉	C₂³	1	2	yes	—
59	#124×C₈₉	—	2	2	no	—
60	#121×C₈₉	—	2	2	no	—
61	#121×C₈₉	—	2	2	no	—
62	C₄×D₄×C₈₉	—	2	2	no	—
63	C₄×D₄×C₈₉	—	2	2	no	—
64	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
65	#107×C₈₉	—	2	2	no	—
66	#102×C₈₉	—	2	4	no	—
67	C₄×C₈×C₈₉	D₄	1	2	yes	—
68	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₂³×D₈₉	—	4	2	no	—
71	C₂×D₃₅₆	—	2	4	no	—
72	C₂×D₃₅₆	—	2	4	no	—
73	C₂×D₃₅₆	—	2	4	no	—
74	C₂×D₃₅₆	—	4	8	no	—
75	C₂×D₃₅₆	—	4	8	no	—
76	C₂×D₃₅₆	—	4	2	no	—
77	C₂×D₃₅₆	—	4	2	no	—
78	C₂×D₃₅₆	—	4	2	no	—
79	C₂×D₃₅₆	—	8	4	no	—
80	#13083561	—	4	16	no	—
81	D₇₁₂	—	4	8	no	—
82	D₇₁₂	—	4	8	no	—
83	#13083559	—	4	16	no	—
84	#13083559	—	8	8	no	—
85	#13083558	D₈	1	8	yes	complemented
86	C₂×#10623494	C₂×D₄	1	4	yes	—
87	C₂×#10623494	C₂×D₄	1	4	yes	—
88	#41×C₈₉	—	4	8	no	—
89	C₂×Q₈×C₈₉	—	2	2	no	—
90	C₂×D₄×C₈₉	—	2	2	no	—
91	C₂×D₄×C₈₉	—	2	2	no	—
92	C₂×D₄×C₈₉	—	2	2	no	—
93	C₂×D₄×C₈₉	—	4	2	no	—
94	C₂²×C₄×C₈₉	—	4	2	no	—
95	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
98	D₈×C₈₉	—	4	4	no	—
99	C₂×C₈×C₈₉	—	2	4	no	—
100	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
102	C₄⋊C₄×C₈₉	—	2	2	no	—
103	C₄⋊C₄×C₈₉	—	2	2	no	—
104	C₂²⋊C₄×C₈₉	—	4	2	no	—
105	C₂²⋊C₄×C₈₉	—	4	2	no	—
106	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
110	C₂³×C₈₉	—	4	2	no	—
111	C₂³×C₈₉	#127	1	1	yes	characteristic
112	C₂²×D₈₉	—	4	4	no	—
113	C₂²×D₈₉	—	8	4	no	—
114	C₂²×D₈₉	—	8	4	no	—
115	C₂²×D₈₉	—	8	4	no	—
116	Q₈×C₈₉	—	4	4	no	—
117	D₄×C₈₉	—	4	4	no	—
118	D₄×C₈₉	—	4	4	no	—
119	D₄×C₈₉	—	4	4	no	—
120	D₄×C₈₉	—	8	4	no	—
121	C₂×C₄×C₈₉	—	2	2	no	—
122	C₂×C₄×C₈₉	—	2	2	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	8	4	no	—
125	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
126	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
127	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×D₈	1	4	yes	—
129	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
132	D₃₅₆	—	4	8	no	—
133	D₃₅₆	—	4	8	no	—
134	D₃₅₆	—	4	8	no	—
135	D₃₅₆	—	8	4	no	—
136	D₃₅₆	—	8	4	no	—
137	D₃₅₆	—	8	4	no	—
138	C₈×C₈₉	—	2	2	no	—
139	C₈×C₈₉	AGL(1,ℤ/8)	1	4	yes	complemented
140	#10623494	—	2	4	no	—
141	#10623494	C₂×D₈	1	8	yes	complemented
142	C₂²×C₈₉	—	2	1	no	—
143	C₂²×C₈₉	—	2	1	no	—
144	C₂²×C₈₉	—	8	2	no	—
145	C₂²×C₈₉	—	8	2	no	—
146	C₂²×C₈₉	—	8	2	no	—
147	C₂²×C₈₉	#493	1	2	yes	—
148	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
149	C₂×D₈₉	—	8	4	no	—
150	C₂×D₈₉	—	8	4	no	—
151	C₂×D₈₉	—	8	4	no	—
152	C₂×D₈₉	—	16	8	no	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	8	2	no	—
156	C₄×C₈₉	#493	1	2	yes	—
157	C₄×C₈₉	C₂²×D₈	1	2	yes	—
158	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
159	C₄×C₈₉	#576	1	2	yes	—
160	#19629	—	89	1	no	maximal, Sylow
161	C₂×C₈₉	—	2	2	no	—
162	C₂×C₈₉	—	16	2	no	—
163	C₂×C₈₉	C₂×#493	1	1	yes	characteristic
164	C₂×C₈₉	#3151	1	1	yes	characteristic
165	C₂×C₈₉	#3155	1	1	yes	characteristic
166	D₈₉	—	16	4	no	—
167	#3205	—	89	2	no	—
168	#3169	D₈₉	1	1	yes	characteristic, complemented
169	#3151	—	89	2	no	—
170	C₂×#493	—	89	1	no	—
171	#1721	—	89	4	no	—
172	#1712	—	89	2	no	—
173	#1706	—	89	2	no	—
174	C₈×M₁₆	—	89	1	no	—
175	C₈₉	#19629	1	1	yes	characteristic, complemented, Sylow
176	C₂×AGL(1,ℤ/8)	—	178	4	no	—
177	C₂²×D₈	—	178	2	no	—
178	#532	C₂×D₈₉	1	2	yes	—
179	C₂×#127	—	89	2	no	—
180	#500	C₂×D₈₉	1	1	yes	characteristic, complemented
181	#496	—	178	4	no	—
182	#493	—	89	4	no	—
183	#493	—	89	4	no	—
184	#493	—	89	2	no	—
185	#493	—	89	2	no	—
186	#493	—	178	2	no	—
187	#493	C₂×D₈₉	1	1	yes	characteristic, complemented
188	#492	—	89	4	no	—
189	#486	—	89	4	no	—
190	#486	C₂×D₈₉	1	2	yes	complemented
191	#466	—	2	2	no	—
192	#459	—	178	4	no	—
193	#447	—	178	4	no	—
194	C₄×D₈	—	2	2	no	—
195	#431	—	89	1	no	—
196	C₄×M₁₆	—	89	1	no	—
197	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
198	#322	—	89	2	no	—
199	C₈²	—	89	2	no	—
200	C₂×#41	—	2	2	no	—
201	C₂²×D₄	—	178	2	no	—
202	C₂²×D₄	—	178	2	no	—
203	C₂²×D₄	—	178	2	no	—
204	AGL(1,ℤ/8)	—	356	16	no	—
205	C₂×SD₁₆	—	178	8	no	—
206	C₂×D₈	—	2	2	no	—
207	C₂×D₈	—	356	8	no	—
208	C₂×D₈	—	178	8	no	—
209	C₂×D₈	—	178	8	no	—
210	C₂×D₈	—	178	4	no	—
211	C₂×D₈	—	178	4	no	—
212	C₂×M₁₆	—	89	2	no	—
213	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
214	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
215	#127	—	89	4	no	—
216	#127	—	178	2	no	—
217	#127	—	178	2	no	—
218	#127	—	178	2	no	—
219	#127	C₂²×D₈₉	1	2	yes	—
220	#124	—	2	2	no	—
221	#121	—	2	2	no	—
222	#121	—	2	2	no	—
223	C₄×D₄	—	2	2	no	—
224	C₄×D₄	—	2	2	no	—
225	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
226	#107	—	2	2	no	—
227	C₄⋊C₈	—	89	2	no	—
228	#102	—	2	4	no	—
229	#102	—	178	8	no	—
230	C₂²⋊C₈	—	89	2	no	—
231	#97	—	89	2	no	—
232	C₄×C₈	—	89	2	no	—
233	C₄×C₈	—	89	2	no	—
234	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
235	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
237	C₂⁴	—	356	2	no	—
238	#41	—	4	8	no	—
239	C₂×Q₈	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	356	8	no	—
245	C₂×D₄	—	356	2	no	—
246	C₂×D₄	—	356	2	no	—
247	C₂×D₄	—	356	2	no	—
248	C₂×D₄	—	356	8	no	—
249	C₂×D₄	—	178	4	no	—
250	C₂×D₄	—	178	4	no	—
251	C₂×D₄	—	178	4	no	—
252	C₂×D₄	—	712	4	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
255	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
257	SD₁₆	—	356	16	no	—
258	D₈	—	4	4	no	—
259	D₈	—	356	8	no	—
260	D₈	—	356	16	no	—
261	D₈	—	356	8	no	—
262	D₈	—	712	8	no	—
263	M₁₆	—	89	8	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	89	4	no	—
266	C₂×C₈	—	89	4	no	—
267	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
268	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₄²	D₄×D₈₉	1	1	yes	characteristic
274	C₄²	D₄×D₈₉	1	1	yes	characteristic
275	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
276	C₄²	C₂³×D₈₉	1	1	yes	characteristic
277	C₂³	—	4	2	no	—
278	C₂³	—	356	4	no	—
279	C₂³	—	712	4	no	—
280	C₂³	—	712	4	no	—
281	C₂³	—	712	4	no	—
282	C₂³	#15668008	1	1	yes	characteristic
283	Q₈	—	4	4	no	—
284	D₄	—	4	4	no	—
285	D₄	—	4	4	no	—
286	D₄	—	4	4	no	—
287	D₄	—	8	4	no	—
288	D₄	—	356	8	no	—
289	D₄	—	356	8	no	—
290	D₄	—	356	8	no	—
291	D₄	—	712	4	no	—
292	D₄	—	712	4	no	—
293	D₄	—	712	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	#15668008	1	1	yes	characteristic
299	C₂×C₄	#15668008	1	1	yes	characteristic
300	C₂×C₄	#15668008	1	1	yes	characteristic
301	C₂×C₄	D₈×D₈₉	1	4	yes	—
302	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
303	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
305	C₈	—	2	2	no	—
306	C₈	—	89	8	no	—
307	C₈	—	178	4	no	—
308	C₈	#15668037	1	4	yes	complemented
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	1424	8	no	—
315	C₂²	—	712	4	no	—
316	C₂²	—	712	4	no	—
317	C₂²	—	712	4	no	—
318	C₂²	#19151973	1	2	yes	—
319	C₂²	C₂×#15668008	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	8	2	no	—
323	C₄	#19151973	1	2	yes	—
324	C₄	C₂×D₈×D₈₉	1	2	yes	—
325	C₄	C₂×#15668037	1	2	yes	—
326	C₄	#19152400	1	2	yes	—
327	C₂	—	2	2	no	—
328	C₂	—	16	2	no	—
329	C₂	—	1424	4	no	—
330	C₂	C₂×#19151973	1	1	yes	characteristic
331	C₂	#26030705	1	1	yes	characteristic
332	C₂	#26030739	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#15668008
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#3169	C₂
socle	C₂²×C₈₉	C₂×#127
hypercenter	#3169	D₈₉
hypocenter	C₈₉	#19629
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3169

Complements

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

N	H ≅ G/N	type
1. #26031167	C₂	semidirect
2. #26030705	C₂	semidirect
3. C₂×#19151975	C₂	semidirect
4. #26019920	C₂	semidirect
5. #26019866	C₂	semidirect
6. #26019845	C₂	semidirect
7. C₈×#13083558	C₂	semidirect
8. C₈₉×#3169	C₂	semidirect
16. #19151974	C₂²	semidirect
17. #19151974	C₂²	semidirect
18. #19151972	C₂²	semidirect
19. #19151946	C₂²	semidirect
24. #19151320	C₂²	semidirect
25. C₈×#10623494	C₂²	semidirect
27. #500×C₈₉	C₂²	semidirect
28. #493×C₈₉	C₂²	semidirect
29. #486×C₈₉	C₂²	semidirect
85. #13083558	D₈	semidirect
139. C₈×C₈₉	AGL(1,ℤ/8)	semidirect
141. #10623494	C₂×D₈	semidirect
168. #3169	D₈₉	semidirect
175. C₈₉	#19629	semidirect
180. #500	C₂×D₈₉	semidirect
187. #493	C₂×D₈₉	semidirect
190. #486	C₂×D₈₉	semidirect
308. C₈	#15668037	semidirect

Maximal subgroups

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

subgroup	index
#26031167	2
#26030705	2
C₂×#19151975	2
#26019920	2
#26019866	2
#26019845	2
C₈×#13083558	2
C₈₉×#3169	2
#19629	89