Subgroups — #36711782

Order 22784.

548 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032596	C₂	1	2	yes	maximal, complemented
2	#26022526	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022431	C₂	1	2	yes	maximal, complemented
4	#26022430	C₂	1	2	yes	maximal, complemented
5	#26022387	C₂	1	2	yes	maximal, complemented
6	#26022335	C₂	1	1	yes	maximal, characteristic, complemented
7	D₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₆×C₂²⋊D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#19151424	C₂	1	1	yes	maximal, characteristic, complemented
10	#366×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2215	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152248	—	2	2	no	—
13	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
14	#19152002	C₂²	1	2	yes	—
15	#19151910	—	2	2	no	—
16	#19151909	—	2	2	no	—
17	#19151876	—	2	2	no	—
18	#19151831	C₂²	1	2	yes	—
19	D₈×Dic₈₉	C₂²	1	2	yes	—
20	C₈×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
21	C₂×#15667946	C₂²	1	2	yes	—
22	#107×D₈₉	C₂²	1	2	yes	—
23	#19151424	C₂²	1	2	yes	—
24	#19151424	C₂²	1	1	yes	characteristic
25	#19151424	C₂²	1	1	yes	characteristic
26	#19151423	C₂²	1	2	yes	—
27	#19151423	C₂²	1	2	yes	—
28	#19151416	C₂²	1	1	yes	characteristic
29	#19151399	C₂²	1	2	yes	—
30	#19151399	C₂²	1	2	yes	—
31	C₂²⋊#13083549	C₂²	1	1	yes	characteristic
32	#19151377	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
34	C₂×D₁₆×C₈₉	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	#466×C₈₉	C₂²	1	2	yes	—
37	#366×C₈₉	C₂²	1	1	yes	characteristic
38	#357×C₈₉	C₂²	1	2	yes	—
39	C₂×#13083573	—	4	2	no	—
40	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
41	#15668011	—	2	2	no	—
42	#15667994	—	4	2	no	—
43	#15667993	—	4	2	no	—
44	#15667989	—	4	2	no	—
45	#15667983	—	2	2	no	—
46	D₄×Dic₈₉	—	2	2	no	—
47	C₄×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
48	C₂×C₄⋊Dic₈₉	—	2	2	no	—
49	C₄⋊C₄×D₈₉	—	2	2	no	—
50	C₂×#13083549	C₂³	1	1	yes	characteristic
51	C₂×#13083549	C₂³	1	1	yes	characteristic
52	C₁₆×D₈₉	—	2	2	no	—
53	#15667946	—	2	2	no	—
54	#15667946	C₂³	1	2	yes	—
55	#15667946	C₂³	1	2	yes	—
56	#15667945	—	2	2	no	—
57	#15667945	—	2	2	no	—
58	#15667941	D₄	1	1	yes	characteristic
59	#15667933	—	2	2	no	—
60	#15667933	—	2	2	no	—
61	C₂²⋊#10623494	D₄	1	1	yes	characteristic
62	#15667924	D₄	1	1	yes	characteristic
63	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
64	C₂×D₈×C₈₉	C₂³	1	2	yes	—
65	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
66	#121×C₈₉	—	2	2	no	—
67	D₁₆×C₈₉	—	2	2	no	—
68	C₂×C₁₆×C₈₉	—	2	2	no	—
69	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
71	#107×C₈₉	C₂³	1	2	yes	—
72	#102×C₈₉	—	2	2	no	—
73	#13083573	—	8	8	no	—
74	C₂×Dic₁₇₈	—	4	2	no	—
75	C₂×C₂²⋊D₈₉	—	4	2	no	—
76	C₂×C₂²⋊D₈₉	D₈	1	1	yes	characteristic
77	C₂²×Dic₈₉	—	4	2	no	—
78	C₂²×Dic₈₉	—	4	2	no	—
79	C₂×C₄×D₈₉	—	4	2	no	—
80	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
82	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
83	C₈×D₈₉	—	2	1	no	—
84	C₈×D₈₉	—	2	1	no	—
85	C₄⋊Dic₈₉	—	2	2	no	—
86	C₄⋊Dic₈₉	—	2	2	no	—
87	C₄⋊Dic₈₉	—	4	2	no	—
88	#13083553	—	4	2	no	—
89	#13083553	—	4	2	no	—
90	#13083553	D₈	1	1	yes	characteristic
91	C₂²⋊Dic₈₉	—	4	2	no	—
92	C₂²⋊Dic₈₉	—	4	2	no	—
93	C₂²⋊Dic₈₉	D₈	1	1	yes	characteristic
94	#13083551	D₈	1	1	yes	characteristic
95	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
96	#13083549	—	2	1	no	—
97	#13083549	—	2	1	no	—
98	C₂×D₄×C₈₉	—	2	2	no	—
99	C₂×D₄×C₈₉	—	4	2	no	—
100	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
101	D₈×C₈₉	—	2	2	no	—
102	D₈×C₈₉	—	2	2	no	—
103	C₂×C₈×C₈₉	—	2	1	no	—
104	C₂×C₈×C₈₉	—	2	1	no	—
105	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
107	C₄⋊C₄×C₈₉	—	2	2	no	—
108	C₂²⋊C₄×C₈₉	—	4	2	no	—
109	C₁₆×C₈₉	—	2	1	no	—
110	C₁₆×C₈₉	#41	1	2	yes	complemented
111	C₂³×C₈₉	—	4	2	no	—
112	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
113	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
114	D₄×C₈₉	—	4	2	no	—
115	D₄×C₈₉	—	4	2	no	—
116	D₄×C₈₉	—	8	4	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
122	C₂²⋊D₈₉	—	8	4	no	—
123	C₂²⋊D₈₉	D₁₆	1	4	yes	complemented
124	C₂×Dic₈₉	—	4	2	no	—
125	C₂×Dic₈₉	—	4	2	no	—
126	C₂×Dic₈₉	—	8	4	no	—
127	C₂×Dic₈₉	—	8	4	no	—
128	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
129	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
131	C₄×D₈₉	—	2	1	no	—
132	C₄×D₈₉	—	2	1	no	—
133	C₄×D₈₉	—	8	4	no	—
134	Dic₁₇₈	—	8	4	no	—
135	C₈×C₈₉	—	2	1	no	—
136	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
137	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
138	#10623494	—	2	1	no	—
139	#10623494	—	2	1	no	—
140	C₂²×C₈₉	—	2	1	no	—
141	C₂²×C₈₉	—	2	1	no	—
142	C₂²×C₈₉	—	8	2	no	—
143	C₂²×C₈₉	—	8	2	no	—
144	C₂²×C₈₉	—	8	2	no	—
145	C₂²×C₈₉	C₂×D₁₆	1	2	yes	—
146	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₈₉	—	2	1	no	—
148	C₂×D₈₉	—	2	1	no	—
149	C₂×D₈₉	C₂×D₁₆	1	2	yes	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	—	8	2	no	—
152	C₄×C₈₉	#547	1	1	yes	characteristic
153	C₄×C₈₉	#578	1	1	yes	characteristic
154	Dic₈₉	—	2	1	no	—
155	Dic₈₉	—	2	1	no	—
156	Dic₈₉	—	8	2	no	—
157	Dic₈₉	—	8	2	no	—
158	Dic₈₉	C₂×D₁₆	1	2	yes	—
159	#33901	—	89	1	no	maximal, Sylow
160	C₂×C₈₉	—	2	2	no	—
161	C₂×C₈₉	—	16	2	no	—
162	C₂×C₈₉	#3334	1	1	yes	characteristic
163	C₂×C₈₉	C₂²×D₁₆	1	1	yes	characteristic
164	C₂×C₈₉	#3417	1	1	yes	characteristic
165	D₈₉	—	2	2	no	—
166	#3334	—	89	2	no	—
167	#2252	—	89	1	no	—
168	#2232	—	89	2	no	—
169	#2232	—	89	2	no	—
170	#2224	—	89	2	no	—
171	#2215	—	89	1	no	—
172	#2215	D₈₉	1	1	yes	characteristic, complemented
173	C₄×D₁₆	—	89	1	no	—
174	D₄×C₁₆	—	89	1	no	—
175	C₂×#366	—	89	1	no	—
176	C₂×#366	—	89	1	no	—
177	C₈₉	#33901	1	1	yes	characteristic, complemented, Sylow
178	#547	—	178	2	no	—
179	C₂×D₁₆	C₂×D₈₉	1	1	yes	characteristic
180	C₂²×C₁₆	—	89	1	no	—
181	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
182	#500	—	89	2	no	—
183	#480	—	178	2	no	—
184	#480	—	178	2	no	—
185	#474	—	178	2	no	—
186	#466	—	89	2	no	—
187	#466	C₂×D₈₉	1	2	yes	—
188	C₄×D₈	—	89	2	no	—
189	C₈×D₄	—	89	1	no	—
190	C₂×#107	—	89	2	no	—
191	C₂×#107	—	89	2	no	—
192	#366	—	89	1	no	—
193	#366	—	89	1	no	—
194	#366	—	89	2	no	—
195	#366	—	89	2	no	—
196	#366	—	89	2	no	—
197	#366	C₂×D₈₉	1	1	yes	characteristic
198	C₄⋊C₁₆	—	89	1	no	—
199	#357	—	89	2	no	—
200	#357	—	89	2	no	—
201	#357	C₂×D₈₉	1	2	yes	—
202	C₂²⋊C₁₆	—	89	1	no	—
203	C₂²⋊C₁₆	—	89	1	no	—
204	C₄×C₁₆	—	89	1	no	—
205	C₂×#41	—	356	2	no	—
206	C₂×D₈	C₂²×D₈₉	1	2	yes	—
207	C₂²×C₈	—	89	1	no	—
208	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
209	C₄⋊Q₈	—	178	2	no	—
210	#123	—	356	2	no	—
211	#123	—	356	2	no	—
212	C₂²⋊Q₈	—	356	2	no	—
213	#121	—	2	2	no	—
214	#121	—	178	2	no	—
215	C₄×D₄	—	89	1	no	—
216	C₄×D₄	—	178	2	no	—
217	C₂×C₄⋊C₄	—	178	2	no	—
218	C₂×C₄⋊C₄	—	178	2	no	—
219	D₁₆	—	2	2	no	—
220	C₂×C₁₆	—	2	2	no	—
221	C₂×C₁₆	—	89	1	no	—
222	C₂×C₁₆	—	89	1	no	—
223	C₂×C₁₆	—	178	2	no	—
224	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
225	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
226	#107	—	89	2	no	—
227	#107	—	89	2	no	—
228	#107	—	178	2	no	—
229	#107	—	178	2	no	—
230	#107	—	178	2	no	—
231	#107	C₂²×D₈₉	1	2	yes	—
232	C₄⋊C₈	—	89	1	no	—
233	#102	—	2	2	no	—
234	#102	—	178	2	no	—
235	#102	—	178	2	no	—
236	C₂²⋊C₈	—	89	1	no	—
237	C₂²⋊C₈	—	89	1	no	—
238	C₄×C₈	—	89	1	no	—
239	#41	—	712	8	no	—
240	C₂×Q₈	—	356	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	89	1	no	—
244	C₂×D₄	—	356	2	no	—
245	C₂²×C₄	—	89	1	no	—
246	C₂²×C₄	—	356	2	no	—
247	C₂²×C₄	—	356	2	no	—
248	C₂²×C₄	—	356	2	no	—
249	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
250	D₈	—	2	2	no	—
251	D₈	—	2	2	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	89	1	no	—
255	C₂×C₈	—	89	1	no	—
256	C₂×C₈	—	178	1	no	—
257	C₂×C₈	—	178	1	no	—
258	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
259	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	89	1	no	—
262	C₄⋊C₄	—	356	2	no	—
263	C₄⋊C₄	—	356	2	no	—
264	C₄⋊C₄	—	356	2	no	—
265	C₄⋊C₄	—	178	2	no	—
266	C₄⋊C₄	—	178	2	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	89	1	no	—
269	C₂²⋊C₄	—	89	1	no	—
270	C₂²⋊C₄	—	356	2	no	—
271	C₂²⋊C₄	—	356	2	no	—
272	C₄²	—	89	1	no	—
273	C₁₆	—	2	1	no	—
274	C₁₆	—	178	1	no	—
275	C₁₆	—	178	1	no	—
276	C₁₆	#13083573	1	2	yes	complemented
277	C₂³	—	4	2	no	—
278	C₂³	—	89	1	no	—
279	C₂³	D₈×D₈₉	1	1	yes	characteristic
280	Q₈	—	712	4	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	8	4	no	—
284	D₄	—	89	4	no	—
285	D₄	—	712	4	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	89	1	no	—
290	C₂×C₄	—	89	1	no	—
291	C₂×C₄	—	89	1	no	—
292	C₂×C₄	—	356	2	no	—
293	C₂×C₄	—	356	2	no	—
294	C₂×C₄	—	178	1	no	—
295	C₂×C₄	—	178	1	no	—
296	C₂×C₄	—	712	4	no	—
297	C₂×C₄	—	712	4	no	—
298	C₂×C₄	—	712	4	no	—
299	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	178	1	no	—
303	C₈	—	178	1	no	—
304	C₈	C₂×#13083573	1	1	yes	characteristic
305	C₈	#15668059	1	1	yes	characteristic
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	89	2	no	—
312	C₂²	—	178	1	no	—
313	C₂²	—	178	1	no	—
314	C₂²	D₁₆×D₈₉	1	2	yes	—
315	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	8	2	no	—
318	C₄	—	89	2	no	—
319	C₄	—	178	1	no	—
320	C₄	—	178	1	no	—
321	C₄	—	712	2	no	—
322	C₄	—	712	2	no	—
323	C₄	#19152248	1	1	yes	characteristic
324	C₄	#19152412	1	1	yes	characteristic
325	C₂	—	2	2	no	—
326	C₂	—	16	2	no	—
327	C₂	—	178	2	no	—
328	C₂	#26032596	1	1	yes	characteristic
329	C₂	C₂×D₁₆×D₈₉	1	1	yes	characteristic
330	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	C₈₉×#2215	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	#2215	D₈₉
hypocenter	C₈₉	#33901
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2215

Complements

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

N	H ≅ G/N	type
1. #26032596	C₂	semidirect
2. #26022526	C₂	semidirect
3. #26022431	C₂	semidirect
4. #26022430	C₂	semidirect
5. #26022387	C₂	semidirect
6. #26022335	C₂	semidirect
7. D₁₆×Dic₈₉	C₂	semidirect
8. C₁₆×C₂²⋊D₈₉	C₂	semidirect
9. C₂×#19151424	C₂	semidirect
10. #366×D₈₉	C₂	semidirect
11. C₈₉×#2215	C₂	semidirect
110. C₁₆×C₈₉	#41	semidirect
123. C₂²⋊D₈₉	D₁₆	semidirect
172. #2215	D₈₉	semidirect
177. C₈₉	#33901	semidirect
276. C₁₆	#13083573	semidirect

Maximal subgroups

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

subgroup	index
#26032596	2
#26022526	2
#26022431	2
#26022430	2
#26022387	2
#26022335	2
D₁₆×Dic₈₉	2
C₁₆×C₂²⋊D₈₉	2
C₂×#19151424	2
#366×D₈₉	2
C₈₉×#2215	2
#33901	89