Subgroups — #36711357

Order 22784.

546 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	#26032684	C₂	1	1	yes	maximal, characteristic, complemented
2	#26031973	C₂	1	1	yes	maximal, characteristic
3	#26022480	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022463	C₂	1	2	yes	maximal, complemented
5	#26022389	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022342	C₂	1	2	yes	maximal, complemented
7	#26022302	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022189	C₂	1	2	yes	maximal, complemented
9	C₁₆×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#26021956	C₂	1	2	yes	maximal, complemented
11	C₄×SD₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668031	C₂²	1	2	yes	—
13	#19152321	—	2	1	no	—
14	#19152252	—	2	1	no	—
15	C₂×#15667958	C₂²	1	2	yes	—
16	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
17	#19152003	C₂²	1	1	yes	characteristic
18	#19151978	C₂²	1	1	yes	characteristic
19	#19151938	—	2	2	no	—
20	#19151883	—	2	1	no	—
21	#19151840	C₂²	1	2	yes	—
22	#19151836	C₂²	1	2	yes	—
23	#19151803	—	2	1	no	—
24	#19151761	—	2	2	no	—
25	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
26	#19151635	C₂²	1	2	yes	—
27	#19151423	C₂²	1	2	yes	—
28	C₄⋊#13083549	C₂²	1	1	yes	characteristic
29	#19151403	C₂²	1	2	yes	—
30	#19151403	C₂²	1	2	yes	—
31	#19151401	C₂²	1	2	yes	—
32	#19151399	C₂²	1	2	yes	—
33	#19151377	C₂²	1	2	yes	—
34	C₂×SD₃₂×C₈₉	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
37	#367×C₈₉	C₂²	1	1	yes	characteristic
38	#358×C₈₉	C₂²	1	1	yes	characteristic
39	#357×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083573	—	2	2	no	—
42	#15668031	—	2	4	no	—
43	#15668031	—	4	2	no	—
44	C₂×Dic₃₅₆	—	2	1	no	—
45	C₂×Dic₃₅₆	C₂³	1	2	yes	—
46	C₂×#13083562	—	2	2	no	—
47	C₂×C₈×D₈₉	C₂³	1	2	yes	—
48	#15668012	—	2	1	no	—
49	#15668012	—	2	1	no	—
50	#15668007	—	4	2	no	—
51	#15668000	—	4	1	no	—
52	#15667996	—	4	2	no	—
53	#15667992	—	2	2	no	—
54	#15667988	—	2	2	no	—
55	#15667986	—	4	1	no	—
56	C₄×Dic₁₇₈	—	4	1	no	—
57	C₄×D₃₅₆	D₄	1	1	yes	characteristic
58	#15667970	—	2	2	no	—
59	C₂×C₄⋊Dic₈₉	—	4	1	no	—
60	#15667958	—	4	2	no	—
61	C₂×#13083549	C₂³	1	2	yes	—
62	C₁₆×D₈₉	—	4	2	no	—
63	#15667946	—	2	1	no	—
64	#15667946	C₂³	1	2	yes	—
65	#15667943	—	2	2	no	—
66	C₄⋊#10623494	D₄	1	1	yes	characteristic
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	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₄×Q₈×C₈₉	—	2	1	no	—
73	C₄×D₄×C₈₉	—	2	1	no	—
74	SD₃₂×C₈₉	—	2	2	no	—
75	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
77	#107×C₈₉	C₂³	1	1	yes	characteristic
78	#107×C₈₉	C₂³	1	1	yes	characteristic
79	#103×C₈₉	—	2	1	no	—
80	#102×C₈₉	—	2	1	no	—
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	#13083573	—	4	4	no	—
83	#13083573	—	8	2	no	—
84	C₂×Dic₁₇₈	—	2	2	no	—
85	C₂×Dic₁₇₈	—	4	1	no	—
86	C₂×D₃₅₆	D₈	1	1	yes	characteristic
87	C₂×C₂²⋊D₈₉	—	4	2	no	—
88	C₂²×Dic₈₉	—	4	2	no	—
89	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
90	Dic₃₅₆	—	2	2	no	—
91	Dic₃₅₆	—	2	2	no	—
92	Dic₃₅₆	—	8	1	no	—
93	#13083562	—	4	4	no	—
94	C₂×#10623494	C₂×D₄	1	2	yes	—
95	C₈×D₈₉	—	2	2	no	—
96	C₈×D₈₉	—	2	2	no	—
97	C₄⋊Dic₈₉	—	2	2	no	—
98	C₄⋊Dic₈₉	—	4	1	no	—
99	C₄⋊Dic₈₉	—	4	1	no	—
100	C₄⋊Dic₈₉	—	4	1	no	—
101	C₄⋊Dic₈₉	—	4	2	no	—
102	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
103	#13083553	—	4	2	no	—
104	#13083551	—	4	2	no	—
105	#13083551	—	4	2	no	—
106	#13083551	—	4	2	no	—
107	#13083551	D₈	1	2	yes	—
108	C₄×Dic₈₉	—	4	2	no	—
109	#13083549	—	2	2	no	—
110	C₂×Q₈×C₈₉	—	2	1	no	—
111	C₂×D₄×C₈₉	—	2	1	no	—
112	C₂²×C₄×C₈₉	—	4	1	no	—
113	Q₁₆×C₈₉	—	2	1	no	—
114	Q₁₆×C₈₉	—	2	1	no	—
115	D₈×C₈₉	—	2	1	no	—
116	D₈×C₈₉	#41	1	2	yes	—
117	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
119	C₄⋊C₄×C₈₉	—	2	1	no	—
120	C₄⋊C₄×C₈₉	—	2	1	no	—
121	C₄⋊C₄×C₈₉	—	4	1	no	—
122	C₂²⋊C₄×C₈₉	—	4	1	no	—
123	C₄²×C₈₉	—	4	1	no	—
124	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₁₆×C₈₉	—	2	1	no	—
126	C₁₆×C₈₉	—	2	1	no	—
127	C₂³×C₈₉	—	4	1	no	—
128	C₂²×D₈₉	C₂×D₈	1	2	yes	—
129	Q₈×C₈₉	—	4	1	no	—
130	Q₈×C₈₉	—	4	1	no	—
131	D₄×C₈₉	—	2	2	no	—
132	D₄×C₈₉	—	4	1	no	—
133	C₂×C₄×C₈₉	—	4	1	no	—
134	C₂×C₄×C₈₉	—	4	1	no	—
135	C₂×C₄×C₈₉	—	4	1	no	—
136	C₂×C₄×C₈₉	—	8	2	no	—
137	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
140	C₂²⋊D₈₉	—	8	4	no	—
141	C₂×Dic₈₉	—	4	2	no	—
142	C₂×Dic₈₉	—	4	2	no	—
143	C₂×Dic₈₉	—	8	4	no	—
144	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
145	D₃₅₆	—	2	2	no	—
146	C₄×D₈₉	—	2	2	no	—
147	C₄×D₈₉	—	2	2	no	—
148	Dic₁₇₈	—	4	2	no	—
149	Dic₁₇₈	—	4	2	no	—
150	Dic₁₇₈	—	8	2	no	—
151	C₈×C₈₉	—	2	1	no	—
152	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
153	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
154	#10623494	—	2	2	no	—
155	C₂²×C₈₉	—	4	2	no	—
156	C₂²×C₈₉	—	8	1	no	—
157	C₂²×C₈₉	—	8	1	no	—
158	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
159	C₂×D₈₉	—	2	2	no	—
160	C₂×D₈₉	—	2	2	no	—
161	C₂×D₈₉	—	2	2	no	—
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	—	8	1	no	—
164	C₄×C₈₉	—	8	1	no	—
165	C₄×C₈₉	—	8	1	no	—
166	C₄×C₈₉	#508	1	2	yes	—
167	C₄×C₈₉	#546	1	1	yes	characteristic
168	C₄×C₈₉	#577	1	1	yes	characteristic
169	Dic₈₉	—	2	2	no	—
170	Dic₈₉	—	8	2	no	—
171	Dic₈₉	—	8	2	no	—
172	#33869	—	89	1	no	maximal, Sylow
173	C₂×C₈₉	—	8	2	no	—
174	C₂×C₈₉	#3294	1	1	yes	characteristic
175	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
176	C₂×C₈₉	#3417	1	1	yes	characteristic
177	D₈₉	—	4	2	no	—
178	#3342	—	89	1	no	—
179	#3283	—	89	1	no	—
180	#2241	—	89	1	no	—
181	#2237	—	89	2	no	—
182	#2224	—	89	1	no	—
183	#2216	—	89	2	no	—
184	#2209	—	89	1	no	—
185	#2189	—	89	2	no	—
186	C₄×SD₃₂	D₈₉	1	1	yes	characteristic, complemented
187	D₄×C₁₆	—	89	1	no	—
188	#2138	—	89	2	no	—
189	C₈₉	#33869	1	1	yes	characteristic, complemented, Sylow
190	C₂×#135	—	89	2	no	—
191	#556	—	178	1	no	—
192	#547	—	178	1	no	—
193	C₂×Q₃₂	—	89	2	no	—
194	C₂×SD₃₂	C₂×D₈₉	1	1	yes	characteristic
195	C₂²×C₁₆	—	89	2	no	—
196	#500	—	89	1	no	—
197	#494	—	89	1	no	—
198	#485	—	178	2	no	—
199	#475	—	178	1	no	—
200	#467	—	89	2	no	—
201	#467	—	89	2	no	—
202	#461	—	178	1	no	—
203	#452	—	178	2	no	—
204	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
205	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
206	C₈×D₄	—	89	1	no	—
207	#427	—	89	2	no	—
208	#367	C₂×D₈₉	1	1	yes	characteristic
209	#366	—	89	2	no	—
210	C₄⋊C₁₆	—	89	1	no	—
211	#358	—	89	2	no	—
212	#358	—	89	2	no	—
213	#358	—	89	2	no	—
214	#358	C₂×D₈₉	1	1	yes	characteristic
215	#357	—	89	2	no	—
216	#357	C₂×D₈₉	1	1	yes	characteristic
217	C₂²⋊C₁₆	—	89	2	no	—
218	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
219	C₂×#41	—	178	2	no	—
220	#135	—	356	2	no	—
221	#135	—	178	4	no	—
222	C₂×Q₁₆	—	89	2	no	—
223	C₂×Q₁₆	—	178	1	no	—
224	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
225	C₂×SD₁₆	—	178	2	no	—
226	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
227	C₂²×C₈	—	89	2	no	—
228	C₄⋊Q₈	—	178	1	no	—
229	C₄⋊Q₈	—	178	1	no	—
230	#126	—	356	2	no	—
231	#124	—	356	1	no	—
232	#123	—	356	2	no	—
233	C₂²⋊Q₈	—	178	2	no	—
234	C₂²⋊Q₈	—	178	2	no	—
235	#121	—	356	1	no	—
236	C₄×Q₈	—	2	1	no	—
237	C₄×Q₈	—	356	1	no	—
238	C₄×D₄	—	2	1	no	—
239	C₄×D₄	—	89	1	no	—
240	#117	—	178	2	no	—
241	C₂×C₄⋊C₄	—	356	1	no	—
242	Q₃₂	—	356	2	no	—
243	SD₃₂	—	2	2	no	—
244	C₂×C₁₆	—	89	2	no	—
245	C₂×C₁₆	—	356	2	no	—
246	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
247	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
248	#107	—	89	2	no	—
249	#107	—	178	1	no	—
250	#107	C₂²×D₈₉	1	1	yes	characteristic
251	#107	C₂²×D₈₉	1	1	yes	characteristic
252	#106	—	178	2	no	—
253	C₄⋊C₈	—	89	1	no	—
254	#103	—	2	1	no	—
255	#103	—	178	2	no	—
256	#103	—	178	2	no	—
257	#102	—	2	1	no	—
258	C₂²⋊C₈	—	89	2	no	—
259	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
260	#41	—	356	4	no	—
261	#41	—	712	2	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	356	1	no	—
264	C₂×Q₈	—	178	2	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	89	1	no	—
267	C₂×D₄	—	356	2	no	—
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	89	2	no	—
270	C₂²×C₄	—	356	2	no	—
271	Q₁₆	—	2	1	no	—
272	Q₁₆	—	2	1	no	—
273	Q₁₆	—	178	2	no	—
274	Q₁₆	—	178	2	no	—
275	Q₁₆	—	712	1	no	—
276	SD₁₆	—	356	4	no	—
277	D₈	—	2	1	no	—
278	D₈	#13083574	1	2	yes	—
279	C₂×C₈	—	89	2	no	—
280	C₂×C₈	—	178	2	no	—
281	C₂×C₈	—	178	2	no	—
282	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
283	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	89	1	no	—
288	C₄⋊C₄	—	356	2	no	—
289	C₄⋊C₄	—	356	2	no	—
290	C₄⋊C₄	—	356	1	no	—
291	C₄⋊C₄	—	356	1	no	—
292	C₄⋊C₄	—	356	1	no	—
293	C₄⋊C₄	—	178	2	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₂²⋊C₄	—	89	2	no	—
296	C₂²⋊C₄	—	356	2	no	—
297	C₂²⋊C₄	—	356	2	no	—
298	C₂²⋊C₄	—	356	2	no	—
299	C₄²	—	4	1	no	—
300	C₄²	—	356	2	no	—
301	C₄²	D₄×D₈₉	1	1	yes	characteristic
302	C₁₆	—	2	1	no	—
303	C₁₆	—	2	1	no	—
304	C₁₆	—	178	2	no	—
305	C₂³	—	4	1	no	—
306	C₂³	—	89	2	no	—
307	Q₈	—	4	1	no	—
308	Q₈	—	4	1	no	—
309	Q₈	—	356	2	no	—
310	Q₈	—	356	2	no	—
311	Q₈	—	712	2	no	—
312	D₄	—	2	2	no	—
313	D₄	—	4	1	no	—
314	D₄	—	178	2	no	—
315	D₄	—	712	4	no	—
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	8	2	no	—
320	C₂×C₄	—	89	2	no	—
321	C₂×C₄	—	356	2	no	—
322	C₂×C₄	—	356	2	no	—
323	C₂×C₄	—	178	2	no	—
324	C₂×C₄	—	178	2	no	—
325	C₂×C₄	—	712	4	no	—
326	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
327	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
329	C₈	—	2	1	no	—
330	C₈	—	178	2	no	—
331	C₈	C₂×#13083574	1	1	yes	characteristic
332	C₈	#15668058	1	1	yes	characteristic
333	C₂²	—	4	2	no	—
334	C₂²	—	8	1	no	—
335	C₂²	—	8	1	no	—
336	C₂²	—	178	2	no	—
337	C₂²	—	178	2	no	—
338	C₂²	—	178	2	no	—
339	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	8	1	no	—
342	C₄	—	8	1	no	—
343	C₄	—	8	1	no	—
344	C₄	—	178	2	no	—
345	C₄	—	712	2	no	—
346	C₄	—	712	2	no	—
347	C₄	#19152029	1	2	yes	—
348	C₄	#19152244	1	1	yes	characteristic
349	C₄	#19152405	1	1	yes	characteristic
350	C₂	—	8	2	no	—
351	C₂	—	356	2	no	—
352	C₂	#26032067	1	1	yes	characteristic
353	C₂	C₂×#19152029	1	1	yes	characteristic
354	C₂	#26033402	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₄×SD₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26031973	C₂
hypercenter	C₄×SD₃₂	D₈₉
hypocenter	C₈₉	#33869
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×SD₃₂

Complements

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

N	H ≅ G/N	type
1. #26032684	C₂	semidirect
3. #26022480	C₂	semidirect
4. #26022463	C₂	semidirect
5. #26022389	C₂	semidirect
6. #26022342	C₂	semidirect
7. #26022302	C₂	semidirect
8. #26022189	C₂	semidirect
9. C₁₆×D₃₅₆	C₂	semidirect
10. #26021956	C₂	semidirect
11. C₄×SD₃₂×C₈₉	C₂	semidirect
186. C₄×SD₃₂	D₈₉	semidirect
189. C₈₉	#33869	semidirect

Maximal subgroups

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

subgroup	index
#26032684	2
#26031973	2
#26022480	2
#26022463	2
#26022389	2
#26022342	2
#26022302	2
#26022189	2
C₁₆×D₃₅₆	2
#26021956	2
C₄×SD₃₂×C₈₉	2
#33869	89