Subgroups — #36709730

Order 22784.

598 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030763	C₂	1	1	yes	maximal, characteristic, complemented
2	#26030681	C₂	1	1	yes	maximal, characteristic
3	#26022485	C₂	1	4	yes	maximal, complemented
4	#26022479	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022477	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022476	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021958	C₂	1	4	yes	maximal, complemented
8	C₄×C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2241	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668035	—	2	1	no	—
11	C₂×#15668035	C₂²	1	2	yes	—
12	#19152153	—	2	1	no	—
13	#19152149	—	2	1	no	—
14	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19151999	—	2	1	no	—
17	#19151999	C₂²	1	2	yes	—
18	#19151994	—	2	1	no	—
19	#19151983	—	2	1	no	—
20	#19151983	C₂²	1	2	yes	—
21	#19151975	C₂²	1	1	yes	characteristic
22	#19151972	—	2	1	no	—
23	#19151635	—	2	1	no	—
24	#19151635	C₂²	1	2	yes	—
25	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
26	#19151402	C₂²	1	4	yes	—
27	#19151400	C₂²	1	4	yes	—
28	#19151398	C₂²	1	4	yes	—
29	C₄×#13083549	C₂²	1	1	yes	characteristic
30	C₁₆×Dic₈₉	C₂²	1	2	yes	—
31	#500×C₈₉	C₂²	1	1	yes	characteristic
32	#494×C₈₉	C₂²	1	1	yes	characteristic
33	#358×C₈₉	C₂²	1	4	yes	—
34	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×#13083575	—	2	2	no	—
36	#15668035	—	2	8	no	—
37	#15668035	—	4	4	no	—
38	C₂×#13083561	—	2	2	no	—
39	C₂×#13083561	—	2	2	no	—
40	C₂×D₇₁₂	—	2	1	no	—
41	C₂×D₇₁₂	C₂³	1	2	yes	—
42	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
43	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
44	#15668012	—	2	1	no	—
45	#15668009	—	2	1	no	—
46	#15668003	—	2	2	no	—
47	#15668001	—	2	2	no	—
48	#15667992	—	4	1	no	—
49	#15667988	—	4	1	no	—
50	#15667987	—	4	1	no	—
51	#15667987	—	4	1	no	—
52	Q₈×Dic₈₉	—	4	1	no	—
53	#15667977	—	4	1	no	—
54	#15667974	—	4	1	no	—
55	#15667974	—	4	1	no	—
56	#15667970	—	2	2	no	—
57	C₄²×D₈₉	D₄	1	1	yes	characteristic
58	C₂×#13083549	C₂³	1	2	yes	—
59	C₁₆×D₈₉	—	2	4	no	—
60	#15667946	—	2	1	no	—
61	#15667946	C₂³	1	2	yes	—
62	#15667943	—	2	2	no	—
63	#15667943	—	2	2	no	—
64	C₄×#10623494	D₄	1	1	yes	characteristic
65	C₈×Dic₈₉	D₄	1	1	yes	characteristic
66	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₈₉	—	2	1	no	—
68	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₈₉	—	2	1	no	—
70	C₄⋊Q₈×C₈₉	—	2	1	no	—
71	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
72	#107×C₈₉	—	2	1	no	—
73	#107×C₈₉	C₂³	1	2	yes	—
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	#13083575	—	4	4	no	—
76	#13083575	—	4	4	no	—
77	C₂×Dic₁₇₈	—	4	1	no	—
78	C₂×D₃₅₆	—	2	2	no	—
79	C₂×D₃₅₆	—	4	2	no	—
80	C₂×D₃₅₆	—	4	1	no	—
81	C₂×D₃₅₆	—	4	1	no	—
82	C₂×C₄×D₈₉	—	4	2	no	—
83	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
84	C₂×C₄×D₈₉	D₈	1	1	yes	characteristic
85	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
86	#13083561	—	4	4	no	—
87	#13083561	—	4	4	no	—
88	D₇₁₂	—	2	4	no	—
89	D₇₁₂	—	4	2	no	—
90	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
91	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
92	C₈×D₈₉	SD₁₆	1	4	yes	—
93	C₈×D₈₉	#41	1	4	yes	—
94	C₄⋊Dic₈₉	—	2	2	no	—
95	C₄⋊Dic₈₉	—	4	1	no	—
96	C₄⋊Dic₈₉	—	4	2	no	—
97	#13083553	—	4	2	no	—
98	#13083551	—	4	2	no	—
99	#13083551	—	4	2	no	—
100	#13083551	—	4	2	no	—
101	C₄×Dic₈₉	—	4	2	no	—
102	C₄×Dic₈₉	D₈	1	1	yes	characteristic
103	C₄×Dic₈₉	D₈	1	1	yes	characteristic
104	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
105	#13083549	—	2	2	no	—
106	C₂×Q₈×C₈₉	—	2	2	no	—
107	C₂×Q₈×C₈₉	—	4	1	no	—
108	Q₁₆×C₈₉	—	2	4	no	—
109	Q₁₆×C₈₉	—	4	2	no	—
110	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
112	C₄⋊C₄×C₈₉	—	2	2	no	—
113	C₄⋊C₄×C₈₉	—	4	1	no	—
114	C₄⋊C₄×C₈₉	—	4	1	no	—
115	C₄⋊C₄×C₈₉	—	4	1	no	—
116	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₈₉	—	2	2	no	—
118	C₂²×D₈₉	—	4	2	no	—
119	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₈₉	—	4	2	no	—
121	Q₈×C₈₉	—	4	2	no	—
122	Q₈×C₈₉	—	8	2	no	—
123	C₂×C₄×C₈₉	—	4	2	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	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
132	D₃₅₆	—	4	2	no	—
133	D₃₅₆	—	4	2	no	—
134	D₃₅₆	—	8	4	no	—
135	D₃₅₆	—	8	2	no	—
136	D₃₅₆	—	8	2	no	—
137	C₄×D₈₉	—	2	2	no	—
138	C₄×D₈₉	—	2	2	no	—
139	C₄×D₈₉	—	8	4	no	—
140	C₄×D₈₉	#124	1	2	yes	—
141	C₄×D₈₉	C₂×SD₁₆	1	2	yes	—
142	Dic₁₇₈	—	8	2	no	—
143	C₈×C₈₉	C₂×SD₁₆	1	2	yes	—
144	C₈×C₈₉	C₂×#41	1	2	yes	—
145	#10623494	#124	1	2	yes	—
146	#10623494	C₂×SD₁₆	1	2	yes	—
147	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₈₉	—	2	1	no	—
149	C₂×D₈₉	—	2	1	no	—
150	C₂×D₈₉	—	8	2	no	—
151	C₂×D₈₉	—	8	2	no	—
152	C₂×D₈₉	—	8	2	no	—
153	C₂×D₈₉	#486	1	2	yes	—
154	C₄×C₈₉	—	8	2	no	—
155	C₄×C₈₉	—	8	2	no	—
156	C₄×C₈₉	#508	1	2	yes	—
157	C₄×C₈₉	#508	1	2	yes	—
158	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
159	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
160	Dic₈₉	—	2	1	no	—
161	Dic₈₉	—	2	1	no	—
162	Dic₈₉	—	8	2	no	—
163	Dic₈₉	#486	1	2	yes	—
164	#33697	—	89	1	no	maximal, Sylow
165	C₂×C₈₉	C₂×#486	1	1	yes	characteristic
166	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
167	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
168	D₈₉	—	2	2	no	—
169	D₈₉	—	16	2	no	—
170	#3160	—	89	1	no	—
171	#3147	—	89	1	no	—
172	#2242	—	89	4	no	—
173	#2241	—	89	1	no	—
174	#2241	D₈₉	1	1	yes	characteristic, complemented
175	#2240	—	89	1	no	—
176	#2240	—	89	1	no	—
177	#2138	—	89	4	no	—
178	C₂×C₄×C₁₆	—	89	1	no	—
179	C₈₉	#33697	1	1	yes	characteristic, complemented, Sylow
180	C₂×#135	—	89	2	no	—
181	C₂×#135	—	178	1	no	—
182	#533	—	178	1	no	—
183	#532	—	178	1	no	—
184	C₂²×C₁₆	—	89	2	no	—
185	#500	—	89	1	no	—
186	#500	C₂×D₈₉	1	1	yes	characteristic
187	#499	—	89	2	no	—
188	#499	—	178	1	no	—
189	#498	—	178	1	no	—
190	#495	—	89	2	no	—
191	#495	—	178	1	no	—
192	#494	C₂×D₈₉	1	1	yes	characteristic
193	#493	—	89	1	no	—
194	#492	—	178	1	no	—
195	#427	—	89	2	no	—
196	#427	—	178	1	no	—
197	C₂×C₄×C₈	—	89	1	no	—
198	#358	—	89	4	no	—
199	#358	C₂×D₈₉	1	4	yes	—
200	#357	—	89	4	no	—
201	#357	—	89	4	no	—
202	C₄×C₁₆	—	89	2	no	—
203	C₄×C₁₆	—	89	1	no	—
204	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
205	C₂×#41	—	178	2	no	—
206	#135	—	356	4	no	—
207	#135	—	178	8	no	—
208	C₂×Q₁₆	—	2	1	no	—
209	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
210	C₂×SD₁₆	—	178	2	no	—
211	C₂×SD₁₆	—	178	2	no	—
212	C₂×D₈	—	89	2	no	—
213	C₂×D₈	—	178	1	no	—
214	C₂²×C₈	—	89	1	no	—
215	C₂²×C₈	—	89	1	no	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	178	1	no	—
219	#127	—	178	1	no	—
220	#125	—	178	2	no	—
221	#124	—	178	2	no	—
222	C₂²⋊Q₈	—	356	1	no	—
223	C₂²⋊Q₈	—	356	1	no	—
224	#121	—	356	1	no	—
225	#121	—	356	1	no	—
226	C₄×Q₈	—	356	1	no	—
227	C₄×Q₈	—	356	1	no	—
228	C₄×D₄	—	356	1	no	—
229	C₄×D₄	—	356	1	no	—
230	#117	—	178	2	no	—
231	C₂×C₄²	—	89	1	no	—
232	C₂×C₁₆	—	89	2	no	—
233	C₂×C₁₆	—	178	4	no	—
234	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
235	#107	—	2	1	no	—
236	#107	—	89	2	no	—
237	#107	—	178	1	no	—
238	#107	C₂²×D₈₉	1	2	yes	—
239	#106	—	178	2	no	—
240	#106	—	178	2	no	—
241	C₄×C₈	—	89	1	no	—
242	C₄×C₈	—	89	1	no	—
243	C₄×C₈	—	89	1	no	—
244	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
245	#41	—	356	4	no	—
246	#41	—	356	4	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	356	1	no	—
250	C₂×D₄	—	356	2	no	—
251	C₂×D₄	—	356	1	no	—
252	C₂×D₄	—	356	1	no	—
253	C₂×D₄	—	178	2	no	—
254	C₂²×C₄	—	89	1	no	—
255	C₂²×C₄	—	89	1	no	—
256	C₂²×C₄	—	89	1	no	—
257	C₂²×C₄	—	356	2	no	—
258	Q₁₆	—	2	4	no	—
259	Q₁₆	—	4	2	no	—
260	SD₁₆	—	356	4	no	—
261	SD₁₆	—	356	4	no	—
262	D₈	—	356	2	no	—
263	D₈	—	178	4	no	—
264	C₂×C₈	—	89	1	no	—
265	C₂×C₈	—	89	1	no	—
266	C₂×C₈	—	89	4	no	—
267	C₂×C₈	—	89	4	no	—
268	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
269	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
270	C₄⋊C₄	—	2	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	356	2	no	—
275	C₄⋊C₄	—	356	2	no	—
276	C₄⋊C₄	—	356	1	no	—
277	C₄⋊C₄	—	178	2	no	—
278	C₂²⋊C₄	—	356	2	no	—
279	C₂²⋊C₄	—	356	2	no	—
280	C₂²⋊C₄	—	356	2	no	—
281	C₄²	—	89	1	no	—
282	C₄²	—	89	1	no	—
283	C₄²	—	89	1	no	—
284	C₄²	—	356	2	no	—
285	C₄²	D₄×D₈₉	1	1	yes	characteristic
286	C₁₆	—	2	2	no	—
287	C₁₆	—	178	2	no	—
288	C₂³	—	89	1	no	—
289	C₂³	—	356	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	8	2	no	—
293	Q₈	—	712	2	no	—
294	D₄	—	356	2	no	—
295	D₄	—	356	2	no	—
296	D₄	—	712	4	no	—
297	D₄	—	712	2	no	—
298	D₄	—	712	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	89	1	no	—
302	C₂×C₄	—	89	2	no	—
303	C₂×C₄	—	89	2	no	—
304	C₂×C₄	—	89	1	no	—
305	C₂×C₄	—	89	1	no	—
306	C₂×C₄	—	356	2	no	—
307	C₂×C₄	—	178	2	no	—
308	C₂×C₄	—	178	2	no	—
309	C₂×C₄	—	712	4	no	—
310	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
311	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
312	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
313	C₈	—	89	2	no	—
314	C₈	—	89	2	no	—
315	C₈	SD₁₆×D₈₉	1	2	yes	—
316	C₈	#41×D₈₉	1	2	yes	—
317	C₂²	—	89	2	no	—
318	C₂²	—	178	1	no	—
319	C₂²	—	178	1	no	—
320	C₂²	—	712	2	no	—
321	C₂²	—	712	2	no	—
322	C₂²	—	712	2	no	—
323	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	—	89	2	no	—
327	C₄	—	178	1	no	—
328	C₄	—	178	1	no	—
329	C₄	—	712	2	no	—
330	C₄	#19152029	1	2	yes	—
331	C₄	#19152031	1	2	yes	—
332	C₄	#124×D₈₉	1	1	yes	characteristic
333	C₄	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
334	C₂	—	1424	2	no	—
335	C₂	—	178	2	no	—
336	C₂	#486×D₈₉	1	1	yes	characteristic
337	C₂	C₂×#19152029	1	1	yes	characteristic
338	C₂	C₂×#19152031	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	#26030681	C₂
hypercenter	#2241	D₈₉
hypocenter	C₈₉	#33697
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
1. #26030763	C₂	semidirect
3. #26022485	C₂	semidirect
4. #26022479	C₂	semidirect
5. #26022477	C₂	semidirect
6. #26022476	C₂	semidirect
7. #26021958	C₂	semidirect
8. C₄×C₁₆×D₈₉	C₂	semidirect
9. C₈₉×#2241	C₂	semidirect
174. #2241	D₈₉	semidirect
179. C₈₉	#33697	semidirect

Maximal subgroups

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

subgroup	index
#26030763	2
#26030681	2
#26022485	2
#26022479	2
#26022477	2
#26022476	2
#26021958	2
C₄×C₁₆×D₈₉	2
C₈₉×#2241	2
#33697	89