Subgroups — #36448298

Order 22784.

454 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029292	C₂	1	1	yes	maximal, characteristic
2	#26021045	C₂	1	1	yes	maximal, characteristic
3	C₄×#15667928	C₂	1	1	yes	maximal, characteristic
4	#26019002	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018090	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018088	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1565	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
9	#19151677	—	2	1	no	—
10	#19151677	—	2	1	no	—
11	#19151675	—	2	1	no	—
12	#19151675	—	2	1	no	—
13	#19151651	C₂²	1	1	yes	characteristic
14	C₂×#15667928	C₄	1	1	yes	characteristic
15	C₂×#15667928	C₄	1	1	yes	characteristic
16	#19151560	—	2	2	no	—
17	C₂×#15667924	—	2	2	no	—
18	C₄×#13083558	C₂²	1	1	yes	characteristic
19	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
20	#19151366	C₄	1	1	yes	characteristic
21	#19151366	C₄	1	1	yes	characteristic
22	#19151350	—	2	1	no	—
23	#19151348	—	2	1	no	—
24	#431×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
27	#335×C₈₉	—	2	1	no	—
28	#334×C₈₉	—	2	1	no	—
29	C₂²×D₃₅₆	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄×D₈₉	D₄	1	2	yes	—
31	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
32	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
33	C₂×#13083557	—	2	2	no	—
34	C₂×#13083557	—	2	2	no	—
35	C₂²×#10623494	—	2	1	no	—
36	C₄×D₃₅₆	—	2	1	no	—
37	C₄×D₃₅₆	—	2	1	no	—
38	C₄×D₃₅₆	—	2	1	no	—
39	C₄×D₃₅₆	—	2	1	no	—
40	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂×#13083551	Q₈	1	2	yes	—
42	C₄⋊#10623494	D₄	1	1	yes	characteristic
43	C₄⋊#10623494	D₄	1	1	yes	characteristic
44	#15667928	—	2	2	no	—
45	#15667928	—	2	2	no	—
46	C₂²⋊#10623494	—	2	1	no	—
47	#15667924	—	2	2	no	—
48	#15667924	—	2	2	no	—
49	#15667924	—	4	4	no	—
50	#15667923	—	2	1	no	—
51	C₄×#10623494	—	2	1	no	—
52	C₄×#10623494	—	2	1	no	—
53	C₄×#10623494	—	2	1	no	—
54	C₄×#10623494	—	2	1	no	—
55	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
56	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₈₉	—	2	1	no	—
58	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₈₉	—	2	1	no	—
60	C₄⋊C₈×C₈₉	—	2	1	no	—
61	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
62	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
63	C₂²⋊C₈×C₈₉	—	2	1	no	—
64	#97×C₈₉	—	2	1	no	—
65	C₄×C₈×C₈₉	—	2	1	no	—
66	C₄×C₈×C₈₉	—	2	1	no	—
67	C₂³×D₈₉	C₄⋊C₄	1	2	yes	—
68	C₂×D₃₅₆	—	2	1	no	—
69	C₂×D₃₅₆	—	2	1	no	—
70	C₂×D₃₅₆	—	2	1	no	—
71	C₂×D₃₅₆	—	2	1	no	—
72	C₂×D₃₅₆	—	2	1	no	—
73	C₂×D₃₅₆	—	2	1	no	—
74	C₂²×Dic₈₉	C₄⋊C₄	1	2	yes	—
75	C₂×C₄×D₈₉	—	2	2	no	—
76	C₂×C₄×D₈₉	—	2	2	no	—
77	C₂×C₄×D₈₉	—	4	2	no	—
78	C₂×C₄×D₈₉	—	4	2	no	—
79	#13083558	—	2	1	no	—
80	#13083558	—	2	1	no	—
81	#13083558	—	2	1	no	—
82	#13083558	M₁₆	1	2	yes	complemented
83	#13083557	—	8	2	no	—
84	#13083557	—	8	2	no	—
85	C₂×#10623494	—	2	1	no	—
86	C₂×#10623494	—	2	2	no	—
87	C₂×#10623494	—	2	1	no	—
88	C₂×#10623494	—	2	1	no	—
89	C₂×#10623494	—	2	1	no	—
90	C₂×#10623494	—	4	1	no	—
91	C₂×#10623494	—	4	1	no	—
92	C₄⋊Dic₈₉	—	2	1	no	—
93	C₄⋊Dic₈₉	—	2	1	no	—
94	#13083551	—	2	2	no	—
95	#13083551	—	2	2	no	—
96	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
99	M₁₆×C₈₉	—	2	1	no	—
100	M₁₆×C₈₉	—	2	1	no	—
101	M₁₆×C₈₉	—	2	1	no	—
102	M₁₆×C₈₉	M₁₆	1	2	yes	complemented
103	C₂×C₈×C₈₉	—	2	1	no	—
104	C₂×C₈×C₈₉	—	2	1	no	—
105	C₂×C₈×C₈₉	—	2	1	no	—
106	C₂×C₈×C₈₉	—	2	1	no	—
107	C₂×C₈×C₈₉	—	4	2	no	—
108	C₂×C₈×C₈₉	—	4	1	no	—
109	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₈₉	#41	1	1	yes	characteristic
112	C₄²×C₈₉	#41	1	1	yes	characteristic
113	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×D₈₉	—	2	2	no	—
115	C₂²×D₈₉	—	2	2	no	—
116	C₂²×D₈₉	—	2	2	no	—
117	C₂²×D₈₉	—	4	2	no	—
118	C₂²×D₈₉	—	4	2	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	4	1	no	—
122	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
130	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
131	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
132	C₂×Dic₈₉	—	2	2	no	—
133	C₂×Dic₈₉	—	4	2	no	—
134	D₃₅₆	—	4	1	no	—
135	D₃₅₆	—	4	1	no	—
136	D₃₅₆	—	4	1	no	—
137	D₃₅₆	—	4	1	no	—
138	C₄×D₈₉	—	4	2	no	—
139	C₄×D₈₉	—	4	2	no	—
140	C₄×D₈₉	—	8	2	no	—
141	C₈×C₈₉	—	2	2	no	—
142	C₈×C₈₉	—	4	1	no	—
143	C₈×C₈₉	—	4	1	no	—
144	C₈×C₈₉	—	4	1	no	—
145	C₈×C₈₉	—	8	1	no	—
146	#10623494	—	2	2	no	—
147	#10623494	—	4	2	no	—
148	#10623494	—	4	1	no	—
149	#10623494	—	4	1	no	—
150	#10623494	—	4	1	no	—
151	C₂²×C₈₉	—	2	1	no	—
152	C₂²×C₈₉	—	2	1	no	—
153	C₂²×C₈₉	#385	1	1	yes	characteristic
154	C₂²×C₈₉	#423	1	1	yes	characteristic
155	C₂²×C₈₉	#424	1	1	yes	characteristic
156	C₂×D₈₉	—	4	4	no	—
157	C₂×D₈₉	—	4	2	no	—
158	C₂×D₈₉	—	4	2	no	—
159	C₂×D₈₉	—	4	2	no	—
160	C₂×D₈₉	—	8	2	no	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	—	2	1	no	—
164	C₄×C₈₉	#435	1	1	yes	characteristic
165	C₄×C₈₉	#435	1	1	yes	characteristic
166	C₄×C₈₉	#436	1	1	yes	characteristic
167	C₄×C₈₉	#436	1	1	yes	characteristic
168	C₄×C₈₉	#453	1	1	yes	characteristic
169	C₄×C₈₉	#455	1	1	yes	characteristic
170	Dic₈₉	—	4	2	no	—
171	#8837	—	89	1	no	maximal, Sylow
172	C₂×C₈₉	—	2	1	no	—
173	C₂×C₈₉	—	2	1	no	—
174	C₂×C₈₉	#1851	1	1	yes	characteristic
175	C₂×C₈₉	#1856	1	1	yes	characteristic
176	C₂×C₈₉	#1858	1	1	yes	characteristic
177	D₈₉	—	8	2	no	—
178	#2975	—	89	1	no	—
179	#1954	—	89	1	no	—
180	C₄×#100	—	89	1	no	—
181	#1565	—	89	1	no	—
182	#1565	D₈₉	1	1	yes	characteristic, complemented
183	#1278	—	89	1	no	—
184	#1278	—	89	1	no	—
185	C₈₉	#8837	1	1	yes	characteristic, complemented, Sylow
186	C₂×C₄×D₄	—	89	1	no	—
187	#436	—	178	1	no	—
188	#436	—	178	1	no	—
189	#436	—	178	1	no	—
190	#436	—	178	1	no	—
191	#431	—	89	1	no	—
192	#431	C₂×D₈₉	1	1	yes	characteristic
193	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
194	C₂×#100	—	89	1	no	—
195	C₂×#100	—	89	1	no	—
196	#407	—	178	2	no	—
197	C₂×C₂²⋊C₈	—	178	2	no	—
198	C₄×M₁₆	—	89	1	no	—
199	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
200	C₂×C₄×C₈	—	89	1	no	—
201	#343	—	89	1	no	—
202	#343	—	89	1	no	—
203	#335	—	2	1	no	—
204	#335	—	178	1	no	—
205	#334	—	2	1	no	—
206	#334	—	178	1	no	—
207	C₂²×D₄	—	89	1	no	—
208	C₂³×C₄	—	89	2	no	—
209	C₂×M₁₆	—	89	1	no	—
210	C₂×M₁₆	—	89	1	no	—
211	C₂×M₁₆	—	178	2	no	—
212	C₂×M₁₆	—	178	2	no	—
213	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
214	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
215	C₂²×C₈	—	2	1	no	—
216	C₂²×C₈	—	178	1	no	—
217	C₄×D₄	—	178	1	no	—
218	C₄×D₄	—	178	1	no	—
219	C₄×D₄	—	178	1	no	—
220	C₄×D₄	—	178	1	no	—
221	C₂×C₄⋊C₄	—	89	1	no	—
222	C₂×C₂²⋊C₄	—	89	2	no	—
223	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	89	1	no	—
227	C₄⋊C₈	—	89	1	no	—
228	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
229	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
230	#100	—	178	2	no	—
231	#100	—	178	2	no	—
232	C₂²⋊C₈	—	2	1	no	—
233	C₂²⋊C₈	—	356	4	no	—
234	C₂²⋊C₈	—	178	2	no	—
235	C₂²⋊C₈	—	178	2	no	—
236	C₂²⋊C₈	—	178	1	no	—
237	#97	—	2	1	no	—
238	#97	—	178	1	no	—
239	C₄×C₈	—	2	1	no	—
240	C₄×C₈	—	2	1	no	—
241	C₄×C₈	—	178	1	no	—
242	C₄×C₈	—	178	1	no	—
243	C₄×C₈	—	178	1	no	—
244	C₄×C₈	—	178	1	no	—
245	C₂⁴	—	89	2	no	—
246	C₂×D₄	—	178	1	no	—
247	C₂×D₄	—	178	1	no	—
248	C₂×D₄	—	178	1	no	—
249	C₂×D₄	—	178	1	no	—
250	C₂×D₄	—	178	1	no	—
251	C₂×D₄	—	178	1	no	—
252	C₂²×C₄	—	89	2	no	—
253	C₂²×C₄	—	356	2	no	—
254	C₂²×C₄	—	356	2	no	—
255	C₂²×C₄	—	178	2	no	—
256	C₂²×C₄	—	178	2	no	—
257	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
258	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
259	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
260	M₁₆	—	2	1	no	—
261	M₁₆	—	2	1	no	—
262	M₁₆	—	2	1	no	—
263	M₁₆	—	89	2	no	—
264	M₁₆	—	178	1	no	—
265	M₁₆	—	178	1	no	—
266	M₁₆	—	178	1	no	—
267	M₁₆	—	712	2	no	—
268	M₁₆	—	712	2	no	—
269	M₁₆	#13083557	1	2	yes	complemented
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	1	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	4	2	no	—
275	C₂×C₈	—	4	1	no	—
276	C₂×C₈	—	356	1	no	—
277	C₂×C₈	—	356	1	no	—
278	C₂×C₈	—	178	2	no	—
279	C₂×C₈	—	178	1	no	—
280	C₂×C₈	—	178	1	no	—
281	C₂×C₈	—	178	1	no	—
282	C₂×C₈	—	178	1	no	—
283	C₄⋊C₄	—	178	1	no	—
284	C₄⋊C₄	—	178	1	no	—
285	C₂²⋊C₄	—	178	2	no	—
286	C₂²⋊C₄	—	178	2	no	—
287	C₄²	D₄×D₈₉	1	1	yes	characteristic
288	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
289	C₄²	#13083574	1	1	yes	characteristic
290	C₄²	#13083575	1	1	yes	characteristic
291	C₂³	—	356	2	no	—
292	C₂³	—	356	2	no	—
293	C₂³	—	178	2	no	—
294	C₂³	—	178	2	no	—
295	C₂³	—	178	2	no	—
296	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
297	D₄	—	356	1	no	—
298	D₄	—	356	1	no	—
299	D₄	—	356	1	no	—
300	D₄	—	356	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	356	2	no	—
305	C₂×C₄	—	356	2	no	—
306	C₂×C₄	—	356	2	no	—
307	C₂×C₄	—	178	2	no	—
308	C₂×C₄	—	712	2	no	—
309	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
310	C₂×C₄	#15667974	1	1	yes	characteristic
311	C₂×C₄	#15667980	1	1	yes	characteristic
312	C₂×C₄	#15667988	1	1	yes	characteristic
313	C₂×C₄	#15667990	1	1	yes	characteristic
314	C₂×C₄	#15667997	1	1	yes	characteristic
315	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
316	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
317	C₂×C₄	#15668018	1	1	yes	characteristic
318	C₂×C₄	#15668020	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	4	1	no	—
321	C₈	—	4	1	no	—
322	C₈	—	4	1	no	—
323	C₈	—	8	1	no	—
324	C₈	—	356	2	no	—
325	C₈	—	356	1	no	—
326	C₈	—	356	1	no	—
327	C₈	—	356	1	no	—
328	C₈	—	178	2	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	356	4	no	—
332	C₂²	—	356	2	no	—
333	C₂²	—	356	2	no	—
334	C₂²	—	356	2	no	—
335	C₂²	—	712	2	no	—
336	C₂²	#19151469	1	1	yes	characteristic
337	C₂²	#19151620	1	1	yes	characteristic
338	C₂²	#19151623	1	1	yes	characteristic
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	356	2	no	—
343	C₄	#19151668	1	1	yes	characteristic
344	C₄	#19151669	1	1	yes	characteristic
345	C₄	#19151674	1	1	yes	characteristic
346	C₄	#19151675	1	1	yes	characteristic
347	C₄	#19151764	1	1	yes	characteristic
348	C₄	#19151774	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	712	2	no	—
352	C₂	#26020433	1	1	yes	characteristic
353	C₂	#26020460	1	1	yes	characteristic
354	C₂	#26020470	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667980
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1565	C₂
socle	C₂²×C₈₉	#385
prime radical	C₂²×D₃₅₆	C₂×C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1565	D₈₉
hypocenter	C₈₉	#8837
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1565

Complements

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

N	H ≅ G/N	type
4. #26019002	C₂	semidirect
5. #26018090	C₂	semidirect
6. #26018088	C₂	semidirect
7. C₈₉×#1565	C₂	semidirect
82. #13083558	M₁₆	semidirect
102. M₁₆×C₈₉	M₁₆	semidirect
182. #1565	D₈₉	semidirect
185. C₈₉	#8837	semidirect
269. M₁₆	#13083557	semidirect

Maximal subgroups

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

subgroup	index
#26029292	2
#26021045	2
C₄×#15667928	2
#26019002	2
#26018090	2
#26018088	2
C₈₉×#1565	2
#8837	89