Subgroups — #36712344

Order 22784.

464 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033120	C₂	1	1	yes	maximal, characteristic
2	#26022535	C₂	1	2	yes	maximal, complemented
3	#26022525	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022416	C₂	1	2	yes	maximal, complemented
5	#26022404	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022396	C₂	1	1	yes	maximal, characteristic
7	#26022386	C₂	1	2	yes	maximal
8	#26022161	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022114	C₂	1	2	yes	maximal
10	C₄×#15667955	C₂	1	1	yes	maximal, characteristic
11	Q₈×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152308	—	2	1	no	—
13	C₂×#15667955	C₂²	1	1	yes	characteristic
14	#19152003	C₂²	1	1	yes	characteristic
15	#19152000	C₂²	1	2	yes	—
16	#19151899	—	2	2	no	—
17	#19151889	—	2	1	no	—
18	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
19	C₄×D₇₁₂	C₂²	1	2	yes	—
20	#19151425	C₂²	1	2	yes	—
21	#19151425	C₂²	1	1	yes	characteristic
22	#19151423	C₂²	1	2	yes	—
23	#19151423	C₂²	1	2	yes	—
24	#19151423	C₂²	1	2	yes	—
25	C₄⋊#13083549	C₂²	1	1	yes	characteristic
26	C₄⋊#13083549	C₂²	1	2	yes	—
27	#19151402	C₂²	1	2	yes	—
28	#19151402	C₂²	1	1	yes	characteristic
29	#19151398	C₂²	1	1	yes	characteristic
30	#19151398	C₂²	1	2	yes	—
31	#19151374	C₂²	1	2	yes	—
32	C₄×#13083549	C₂²	1	1	yes	characteristic
33	Q₈²×C₈₉	—	2	1	no	—
34	#500×C₈₉	C₂²	1	1	yes	characteristic
35	#500×C₈₉	C₂²	1	2	yes	—
36	#477×C₈₉	—	2	1	no	—
37	#477×C₈₉	—	2	2	no	—
38	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
41	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
42	#15668012	—	2	1	no	—
43	#15668004	—	2	2	no	—
44	#15667990	—	4	2	no	—
45	#15667988	—	4	1	no	—
46	C₄×D₃₅₆	—	2	1	no	—
47	C₄×D₃₅₆	—	2	2	no	—
48	C₄⋊C₄×D₈₉	—	4	1	no	—
49	#15667955	Q₈	1	4	yes	complemented
50	C₂×#13083549	C₂³	1	1	yes	characteristic
51	C₂×#13083549	C₂³	1	2	yes	—
52	C₂×#13083549	C₂³	1	1	yes	characteristic
53	#15667946	—	2	1	no	—
54	#15667946	C₂³	1	2	yes	—
55	#15667946	C₂³	1	1	yes	characteristic
56	#15667944	—	2	2	no	—
57	#15667934	—	2	2	no	—
58	#15667934	—	2	1	no	—
59	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₈₉	—	2	2	no	—
61	C₄⋊Q₈×C₈₉	—	2	1	no	—
62	C₄⋊Q₈×C₈₉	—	4	2	no	—
63	C₄⋊Q₈×C₈₉	—	4	1	no	—
64	C₄×Q₈×C₈₉	—	2	2	no	—
65	C₄×Q₈×C₈₉	—	2	1	no	—
66	C₄×Q₈×C₈₉	—	4	1	no	—
67	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
68	#107×C₈₉	—	2	2	no	—
69	#107×C₈₉	—	2	1	no	—
70	#107×C₈₉	C₂³	1	1	yes	characteristic
71	#107×C₈₉	C₂³	1	2	yes	—
72	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
73	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
74	#103×C₈₉	—	2	2	no	—
75	#103×C₈₉	—	2	1	no	—
76	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₈₉	C₂³	1	2	yes	—
78	C₂×Dic₁₇₈	—	4	1	no	—
79	C₂×D₃₅₆	—	2	1	no	—
80	C₂×C₄×D₈₉	—	4	1	no	—
81	C₂×C₄×D₈₉	—	4	2	no	—
82	D₇₁₂	—	2	1	no	—
83	D₇₁₂	C₂×Q₈	1	2	yes	—
84	C₄⋊Dic₈₉	—	2	1	no	—
85	C₄⋊Dic₈₉	—	2	2	no	—
86	C₄⋊Dic₈₉	—	4	1	no	—
87	#13083553	—	4	2	no	—
88	#13083553	—	4	2	no	—
89	#13083551	—	4	2	no	—
90	#13083551	—	4	1	no	—
91	#13083549	—	2	1	no	—
92	#13083549	—	2	2	no	—
93	#13083549	C₂×Q₈	1	2	yes	—
94	C₂×Q₈×C₈₉	—	2	1	no	—
95	C₂×Q₈×C₈₉	—	4	1	no	—
96	C₂×Q₈×C₈₉	—	4	2	no	—
97	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
98	Q₁₆×C₈₉	—	2	1	no	—
99	Q₁₆×C₈₉	C₂×Q₈	1	2	yes	—
100	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	—	2	2	no	—
104	C₄⋊C₄×C₈₉	—	2	1	no	—
105	C₄⋊C₄×C₈₉	—	4	1	no	—
106	C₄⋊C₄×C₈₉	—	4	2	no	—
107	C₄⋊C₄×C₈₉	—	4	2	no	—
108	C₄⋊C₄×C₈₉	—	4	1	no	—
109	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
110	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
111	C₄²×C₈₉	—	4	2	no	—
112	C₄²×C₈₉	—	4	1	no	—
113	C₄²×C₈₉	C₂×D₄	1	2	yes	—
114	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₈₉	—	4	1	no	—
116	Q₈×C₈₉	—	2	2	no	—
117	Q₈×C₈₉	—	2	2	no	—
118	Q₈×C₈₉	—	4	1	no	—
119	Q₈×C₈₉	—	16	1	no	—
120	Q₈×C₈₉	—	16	2	no	—
121	C₂×C₄×C₈₉	—	4	1	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	—	4	1	no	—
124	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
129	C₂×Dic₈₉	—	4	1	no	—
130	C₂×Dic₈₉	—	4	2	no	—
131	D₃₅₆	—	2	2	no	—
132	D₃₅₆	—	4	1	no	—
133	C₄×D₈₉	—	8	2	no	—
134	C₄×D₈₉	—	16	2	no	—
135	Dic₁₇₈	—	16	1	no	—
136	C₈×C₈₉	—	2	2	no	—
137	C₈×C₈₉	—	2	1	no	—
138	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
139	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
140	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
141	C₂×D₈₉	—	4	2	no	—
142	C₂×D₈₉	—	8	1	no	—
143	C₂×D₈₉	—	8	1	no	—
144	C₄×C₈₉	—	2	2	no	—
145	C₄×C₈₉	—	2	1	no	—
146	C₄×C₈₉	—	2	2	no	—
147	C₄×C₈₉	—	4	2	no	—
148	C₄×C₈₉	—	8	2	no	—
149	C₄×C₈₉	—	8	1	no	—
150	C₄×C₈₉	#510	1	2	yes	—
151	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
152	C₄×C₈₉	#578	1	1	yes	characteristic
153	Dic₈₉	—	8	1	no	—
154	Dic₈₉	—	8	2	no	—
155	#33952	—	89	1	no	maximal, Sylow
156	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
157	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
158	C₂×C₈₉	#3415	1	1	yes	characteristic
159	D₈₉	—	8	2	no	—
160	Q₈×Q₁₆	D₈₉	1	1	yes	characteristic, complemented
161	#3380	—	89	1	no	—
162	#2255	—	89	2	no	—
163	#2252	—	89	1	no	—
164	#2229	—	89	2	no	—
165	#2227	—	89	1	no	—
166	#2226	—	89	1	no	—
167	#2224	—	89	2	no	—
168	#2183	—	89	1	no	—
169	#2175	—	89	2	no	—
170	C₄×SD₃₂	—	89	1	no	—
171	C₈₉	#33952	1	1	yes	characteristic, complemented, Sylow
172	Q₈²	—	2	1	no	—
173	#554	—	178	1	no	—
174	C₂×SD₃₂	—	89	1	no	—
175	#500	—	89	1	no	—
176	#500	C₂×D₈₉	1	2	yes	—
177	#500	C₂×D₈₉	1	1	yes	characteristic
178	#499	—	89	2	no	—
179	#478	—	178	2	no	—
180	#477	—	2	1	no	—
181	#477	—	2	2	no	—
182	#476	—	178	1	no	—
183	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
184	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
185	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
186	C₄×D₈	—	89	1	no	—
187	C₄×D₈	—	89	2	no	—
188	#367	—	89	1	no	—
189	#367	—	89	2	no	—
190	#366	—	89	2	no	—
191	#366	—	89	2	no	—
192	#366	—	89	2	no	—
193	C₄⋊C₁₆	—	89	2	no	—
194	C₄⋊C₁₆	—	89	1	no	—
195	#358	—	89	1	no	—
196	#358	—	89	2	no	—
197	#357	—	89	1	no	—
198	#357	—	89	2	no	—
199	#346	—	89	2	no	—
200	C₄×C₁₆	—	89	1	no	—
201	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
202	C₂×D₈	—	89	1	no	—
203	C₄⋊Q₈	—	2	2	no	—
204	C₄⋊Q₈	—	2	1	no	—
205	C₄⋊Q₈	—	4	1	no	—
206	C₄⋊Q₈	—	4	2	no	—
207	C₄⋊Q₈	—	178	1	no	—
208	#125	—	178	2	no	—
209	C₂²⋊Q₈	—	356	1	no	—
210	C₂²⋊Q₈	—	356	2	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	2	1	no	—
213	C₄×Q₈	—	4	1	no	—
214	C₄×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
215	C₄×D₄	—	178	1	no	—
216	C₄×D₄	—	178	2	no	—
217	C₂×C₄⋊C₄	—	356	1	no	—
218	SD₃₂	—	89	4	no	—
219	C₂×C₁₆	—	89	1	no	—
220	C₂×C₁₆	—	89	1	no	—
221	C₂×C₁₆	—	89	2	no	—
222	#107	—	2	2	no	—
223	#107	—	2	1	no	—
224	#107	—	89	2	no	—
225	#107	—	89	1	no	—
226	#107	—	178	1	no	—
227	#107	C₂²×D₈₉	1	1	yes	characteristic
228	#107	C₂²×D₈₉	1	2	yes	—
229	#106	—	178	2	no	—
230	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
231	C₄⋊C₈	C₂²⋊D₈₉	1	2	yes	—
232	#103	—	2	2	no	—
233	#103	—	2	1	no	—
234	#102	—	178	2	no	—
235	#102	—	178	1	no	—
236	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₈₉	1	2	yes	—
238	C₂×Q₈	—	2	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	—	356	1	no	—
242	C₂×Q₈	#13083559	1	1	yes	characteristic
243	C₂×D₄	—	178	1	no	—
244	C₂²×C₄	—	356	2	no	—
245	C₂²×C₄	—	356	1	no	—
246	Q₁₆	—	2	1	no	—
247	Q₁₆	Q₈×D₈₉	1	2	yes	—
248	D₈	—	89	2	no	—
249	D₈	—	178	1	no	—
250	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
251	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
252	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	356	2	no	—
260	C₄⋊C₄	—	356	1	no	—
261	C₄⋊C₄	—	356	2	no	—
262	C₄⋊C₄	—	178	1	no	—
263	C₄⋊C₄	—	178	2	no	—
264	C₄⋊C₄	#13083559	1	1	yes	characteristic
265	C₄⋊C₄	#13083559	1	2	yes	—
266	C₂²⋊C₄	—	356	1	no	—
267	C₂²⋊C₄	—	356	2	no	—
268	C₄²	—	4	2	no	—
269	C₄²	—	4	1	no	—
270	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
271	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
272	C₁₆	—	89	2	no	—
273	C₁₆	—	178	1	no	—
274	C₁₆	—	178	2	no	—
275	C₂³	—	356	1	no	—
276	Q₈	—	2	2	no	—
277	Q₈	—	2	2	no	—
278	Q₈	—	4	1	no	—
279	Q₈	—	16	2	no	—
280	Q₈	—	16	1	no	—
281	Q₈	—	1424	1	no	—
282	D₄	—	356	1	no	—
283	D₄	—	178	2	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	356	1	no	—
288	C₂×C₄	—	356	2	no	—
289	C₂×C₄	—	1424	2	no	—
290	C₂×C₄	—	712	2	no	—
291	C₂×C₄	C₂×#13083559	1	2	yes	—
292	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
293	C₂×C₄	C₂×#13083559	1	2	yes	—
294	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
295	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	2	1	no	—
298	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
299	C₈	#15668060	1	1	yes	characteristic
300	C₂²	—	356	2	no	—
301	C₂²	—	712	1	no	—
302	C₂²	—	712	1	no	—
303	C₂²	C₂²×#13083559	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	2	no	—
308	C₄	—	8	1	no	—
309	C₄	—	8	2	no	—
310	C₄	—	712	1	no	—
311	C₄	—	712	2	no	—
312	C₄	#19152039	1	2	yes	—
313	C₄	Q₈×C₂²⋊D₈₉	1	1	yes	characteristic
314	C₄	#19152413	1	1	yes	characteristic
315	C₂	—	712	2	no	—
316	C₂	Q₈×#13083559	1	1	yes	characteristic
317	C₂	C₂×#19152039	1	1	yes	characteristic
318	C₂	#26033391	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	Q₈×Q₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₇₁₂	C₂³
hypercenter	Q₈×Q₁₆	D₈₉
hypocenter	C₈₉	#33952
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #26022535	C₂	semidirect
3. #26022525	C₂	semidirect
4. #26022416	C₂	semidirect
5. #26022404	C₂	semidirect
8. #26022161	C₂	semidirect
11. Q₈×Q₁₆×C₈₉	C₂	semidirect
49. #15667955	Q₈	semidirect
160. Q₈×Q₁₆	D₈₉	semidirect
171. C₈₉	#33952	semidirect

Maximal subgroups

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

subgroup	index
#26033120	2
#26022535	2
#26022525	2
#26022416	2
#26022404	2
#26022396	2
#26022386	2
#26022161	2
#26022114	2
C₄×#15667955	2
Q₈×Q₁₆×C₈₉	2
#33952	89