Subgroups — #36712353

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	#26033135	C₂	1	1	yes	maximal, characteristic, complemented
2	#26033121	C₂	1	1	yes	maximal, characteristic
3	#26022536	C₂	1	2	yes	maximal, complemented
4	#26022415	C₂	1	2	yes	maximal, complemented
5	#26022401	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022396	C₂	1	1	yes	maximal, characteristic
7	#26022388	C₂	1	2	yes	maximal
8	#26022160	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022115	C₂	1	2	yes	maximal
10	C₄×#15667957	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2252	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152328	—	2	1	no	—
13	#19152309	—	2	1	no	—
14	C₂×#15667957	C₂²	1	1	yes	characteristic
15	#19152002	C₂²	1	2	yes	—
16	#19151999	C₂²	1	2	yes	—
17	#19151901	—	2	2	no	—
18	#19151892	—	2	2	no	—
19	#19151890	—	2	1	no	—
20	#19151885	—	2	1	no	—
21	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
22	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
23	#19151692	C₂²	1	2	yes	—
24	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
25	#19151680	C₂²	1	2	yes	—
26	#19151426	C₂²	1	1	yes	characteristic
27	#19151425	C₂²	1	2	yes	—
28	#19151423	C₂²	1	2	yes	—
29	C₄⋊#13083549	C₂²	1	1	yes	characteristic
30	#19151416	C₂²	1	2	yes	—
31	#19151403	C₂²	1	1	yes	characteristic
32	#19151401	C₂²	1	2	yes	—
33	#19151400	C₂²	1	1	yes	characteristic
34	#19151398	C₂²	1	2	yes	—
35	#19151373	C₂²	1	2	yes	—
36	#500×C₈₉	C₂²	1	1	yes	characteristic
37	#500×C₈₉	C₂²	1	1	yes	characteristic
38	#366×C₈₉	C₂²	1	2	yes	—
39	#366×C₈₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
42	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
43	#15668011	—	2	2	no	—
44	#15668011	—	4	1	no	—
45	#15668010	—	4	2	no	—
46	#15668003	—	2	2	no	—
47	#15667992	—	4	1	no	—
48	#15667988	—	4	2	no	—
49	C₄×Dic₁₇₈	—	2	1	no	—
50	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
51	#15667977	—	2	2	no	—
52	#15667977	—	4	1	no	—
53	C₄×D₃₅₆	—	2	1	no	—
54	#15667974	—	2	2	no	—
55	C₄⋊C₄×D₈₉	—	4	1	no	—
56	#15667957	Q₈	1	4	yes	complemented
57	C₂×#13083549	C₂³	1	2	yes	—
58	#15667946	C₂³	1	1	yes	characteristic
59	#15667946	C₂³	1	1	yes	characteristic
60	#15667945	—	2	2	no	—
61	#15667943	—	2	2	no	—
62	C₄⋊#10623494	D₄	1	1	yes	characteristic
63	#15667941	D₄	1	2	yes	—
64	#15667937	—	2	1	no	—
65	#15667935	—	2	2	no	—
66	#15667934	—	2	1	no	—
67	#15667932	—	2	2	no	—
68	C₈×Dic₈₉	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₈₉	—	2	1	no	—
70	C₄⋊Q₈×C₈₉	—	2	1	no	—
71	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
72	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
73	#107×C₈₉	—	2	1	no	—
74	#107×C₈₉	—	2	1	no	—
75	#107×C₈₉	C₂³	1	2	yes	—
76	#107×C₈₉	C₂³	1	2	yes	—
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₂×Dic₁₇₈	—	2	1	no	—
79	C₂×Dic₁₇₈	—	4	2	no	—
80	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
81	C₂×D₃₅₆	—	2	1	no	—
82	C₂×C₄×D₈₉	—	4	1	no	—
83	C₂×C₄×D₈₉	—	4	2	no	—
84	Dic₃₅₆	—	2	1	no	—
85	Dic₃₅₆	C₂×Q₈	1	2	yes	—
86	D₇₁₂	—	2	1	no	—
87	D₇₁₂	C₂×Q₈	1	2	yes	—
88	C₂×#10623494	C₂×D₄	1	2	yes	—
89	C₄⋊Dic₈₉	—	2	1	no	—
90	C₄⋊Dic₈₉	—	2	1	no	—
91	C₄⋊Dic₈₉	—	4	2	no	—
92	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
93	#13083553	—	4	1	no	—
94	#13083553	—	4	2	no	—
95	#13083553	—	4	2	no	—
96	#13083553	—	4	2	no	—
97	#13083553	D₈	1	2	yes	—
98	#13083551	—	4	1	no	—
99	#13083551	—	4	2	no	—
100	C₄×Dic₈₉	—	4	1	no	—
101	C₄×Dic₈₉	—	4	2	no	—
102	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
103	#13083549	—	2	2	no	—
104	C₂×Q₈×C₈₉	—	4	1	no	—
105	C₂×Q₈×C₈₉	—	4	1	no	—
106	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	—	2	2	no	—
109	C₄⋊C₄×C₈₉	—	2	2	no	—
110	C₄⋊C₄×C₈₉	—	4	1	no	—
111	C₄⋊C₄×C₈₉	—	4	1	no	—
112	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₁₆×C₈₉	—	2	1	no	—
114	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
115	C₂²×D₈₉	—	4	1	no	—
116	Q₈×C₈₉	—	16	1	no	—
117	Q₈×C₈₉	—	16	1	no	—
118	C₂×C₄×C₈₉	—	4	2	no	—
119	C₂×C₄×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₈₉	—	4	1	no	—
124	C₂×Dic₈₉	—	4	1	no	—
125	C₂×Dic₈₉	—	4	1	no	—
126	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
127	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
128	D₃₅₆	—	2	2	no	—
129	D₃₅₆	—	4	1	no	—
130	C₄×D₈₉	—	8	2	no	—
131	C₄×D₈₉	—	16	2	no	—
132	Dic₁₇₈	—	2	2	no	—
133	Dic₁₇₈	—	2	2	no	—
134	Dic₁₇₈	—	4	1	no	—
135	Dic₁₇₈	—	16	2	no	—
136	C₈×C₈₉	—	2	1	no	—
137	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
138	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
139	#10623494	—	2	2	no	—
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	1	no	—
145	C₄×C₈₉	—	8	2	no	—
146	C₄×C₈₉	—	8	2	no	—
147	C₄×C₈₉	#510	1	2	yes	—
148	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
149	C₄×C₈₉	#578	1	1	yes	characteristic
150	Dic₈₉	—	2	2	no	—
151	Dic₈₉	—	2	2	no	—
152	Dic₈₉	—	4	2	no	—
153	Dic₈₉	—	8	1	no	—
154	Dic₈₉	—	8	1	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	—	89	1	no	—
161	#3380	—	89	1	no	—
162	#2255	—	89	2	no	—
163	#2252	D₈₉	1	1	yes	characteristic, complemented
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	—	178	1	no	—
173	#554	—	178	1	no	—
174	C₂×SD₃₂	—	89	1	no	—
175	#500	—	89	2	no	—
176	#500	C₂×D₈₉	1	1	yes	characteristic
177	#500	C₂×D₈₉	1	1	yes	characteristic
178	#499	—	89	2	no	—
179	#478	—	178	2	no	—
180	#477	—	178	2	no	—
181	#477	—	178	1	no	—
182	#476	—	178	1	no	—
183	C₈×Q₈	—	89	1	no	—
184	C₄×Q₁₆	—	89	2	no	—
185	C₄×Q₁₆	—	89	1	no	—
186	C₄×D₈	—	89	2	no	—
187	C₄×D₈	—	89	1	no	—
188	#367	—	89	1	no	—
189	#367	—	89	2	no	—
190	#366	—	89	2	no	—
191	#366	C₂×D₈₉	1	2	yes	—
192	#366	C₂×D₈₉	1	2	yes	—
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	C₂×D₈₉	1	1	yes	characteristic
201	C₂×Q₁₆	—	89	1	no	—
202	C₂×D₈	—	89	1	no	—
203	C₄⋊Q₈	—	2	1	no	—
204	C₄⋊Q₈	—	2	1	no	—
205	C₄⋊Q₈	—	356	1	no	—
206	C₄⋊Q₈	—	356	2	no	—
207	C₄⋊Q₈	—	178	2	no	—
208	#125	—	178	2	no	—
209	C₂²⋊Q₈	—	356	1	no	—
210	C₂²⋊Q₈	—	356	2	no	—
211	C₄×Q₈	—	89	1	no	—
212	C₄×Q₈	—	356	1	no	—
213	C₄×Q₈	—	178	2	no	—
214	C₄×Q₈	—	178	1	no	—
215	C₄×D₄	—	178	2	no	—
216	C₄×D₄	—	178	1	no	—
217	C₂×C₄⋊C₄	—	356	1	no	—
218	SD₃₂	—	89	4	no	—
219	C₂×C₁₆	—	89	2	no	—
220	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
221	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
222	#107	—	2	1	no	—
223	#107	—	2	1	no	—
224	#107	—	89	1	no	—
225	#107	—	89	1	no	—
226	#107	—	178	2	no	—
227	#107	C₂²×D₈₉	1	2	yes	—
228	#107	C₂²×D₈₉	1	2	yes	—
229	#106	—	178	2	no	—
230	C₄⋊C₈	—	89	2	no	—
231	C₄⋊C₈	—	89	1	no	—
232	#103	—	178	2	no	—
233	#103	—	178	1	no	—
234	#102	—	178	2	no	—
235	#102	—	178	1	no	—
236	C₄×C₈	—	89	2	no	—
237	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	89	1	no	—
241	C₂×Q₈	—	356	2	no	—
242	C₂×Q₈	—	178	1	no	—
243	C₂×D₄	—	178	1	no	—
244	C₂²×C₄	—	356	2	no	—
245	C₂²×C₄	—	356	1	no	—
246	Q₁₆	—	89	2	no	—
247	Q₁₆	—	178	1	no	—
248	D₈	—	89	2	no	—
249	D₈	—	178	1	no	—
250	C₂×C₈	—	89	2	no	—
251	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
252	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	89	1	no	—
258	C₄⋊C₄	—	89	2	no	—
259	C₄⋊C₄	—	356	2	no	—
260	C₄⋊C₄	—	356	2	no	—
261	C₄⋊C₄	—	356	2	no	—
262	C₄⋊C₄	—	356	2	no	—
263	C₄⋊C₄	—	356	1	no	—
264	C₄⋊C₄	—	178	1	no	—
265	C₄⋊C₄	—	178	1	no	—
266	C₂²⋊C₄	—	356	2	no	—
267	C₂²⋊C₄	—	356	1	no	—
268	C₄²	—	89	2	no	—
269	C₄²	—	356	2	no	—
270	C₄²	—	356	1	no	—
271	C₄²	D₄×D₈₉	1	1	yes	characteristic
272	C₁₆	—	2	1	no	—
273	C₁₆	—	178	2	no	—
274	C₁₆	Q₈×D₈₉	1	2	yes	—
275	C₂³	—	356	1	no	—
276	Q₈	—	16	1	no	—
277	Q₈	—	16	1	no	—
278	Q₈	—	356	1	no	—
279	Q₈	—	1424	2	no	—
280	Q₈	—	178	2	no	—
281	Q₈	—	178	2	no	—
282	D₄	—	356	1	no	—
283	D₄	—	178	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	89	2	no	—
287	C₂×C₄	—	89	2	no	—
288	C₂×C₄	—	356	1	no	—
289	C₂×C₄	—	356	1	no	—
290	C₂×C₄	—	356	1	no	—
291	C₂×C₄	—	1424	2	no	—
292	C₂×C₄	—	712	2	no	—
293	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
294	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	178	2	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	C₂×D₈×D₈₉	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	8	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	356	2	no	—
308	C₄	—	178	2	no	—
309	C₄	—	178	2	no	—
310	C₄	—	712	1	no	—
311	C₄	—	712	1	no	—
312	C₄	#19152040	1	2	yes	—
313	C₄	#19152263	1	1	yes	characteristic
314	C₄	#19152411	1	1	yes	characteristic
315	C₂	—	712	2	no	—
316	C₂	#26033104	1	1	yes	characteristic
317	C₂	C₂×#19152040	1	1	yes	characteristic
318	C₂	#26033389	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×#2252	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₇₁₂	C₂³
hypercenter	#2252	D₈₉
hypocenter	C₈₉	#33952
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

Complements

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

N	H ≅ G/N	type
1. #26033135	C₂	semidirect
3. #26022536	C₂	semidirect
4. #26022415	C₂	semidirect
5. #26022401	C₂	semidirect
8. #26022160	C₂	semidirect
11. C₈₉×#2252	C₂	semidirect
56. #15667957	Q₈	semidirect
163. #2252	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
#26033135	2
#26033121	2
#26022536	2
#26022415	2
#26022401	2
#26022396	2
#26022388	2
#26022160	2
#26022115	2
C₄×#15667957	2
C₈₉×#2252	2
#33952	89