Subgroups — #36711845

Order 22784.

480 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032675	C₂	1	1	yes	maximal, characteristic
2	#26022532	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022457	C₂	1	2	yes	maximal, complemented
4	#26022443	C₂	1	2	yes	maximal, complemented
5	#26022404	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022386	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022350	C₂	1	2	yes	maximal, complemented
8	C₄×#15667955	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
10	#26022034	C₂	1	2	yes	maximal, complemented
11	C₈₉×#3342	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152313	—	2	1	no	—
13	C₂×#15667955	C₂²	1	1	yes	characteristic
14	C₂²×#13083549	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19151923	—	2	2	no	—
17	#19151889	—	2	1	no	—
18	#19151834	C₂²	1	2	yes	—
19	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
20	#19151636	C₂²	1	2	yes	—
21	#19151425	C₂²	1	2	yes	—
22	#19151425	C₂²	1	2	yes	—
23	#19151425	C₂²	1	1	yes	characteristic
24	#19151423	C₂²	1	2	yes	—
25	#19151423	C₂²	1	2	yes	—
26	C₄⋊#13083549	C₂²	1	1	yes	characteristic
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	C₂²⋊#13083549	C₂²	1	2	yes	—
32	C₄×#13083549	C₂²	1	1	yes	characteristic
33	#556×C₈₉	—	2	1	no	—
34	#500×C₈₉	C₂²	1	1	yes	characteristic
35	#485×C₈₉	—	2	2	no	—
36	#475×C₈₉	—	2	1	no	—
37	#467×C₈₉	C₂²	1	2	yes	—
38	C₄×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
40	#427×C₈₉	C₂²	1	2	yes	—
41	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
42	#15668012	—	2	1	no	—
43	#15667998	—	4	2	no	—
44	#15667986	—	2	2	no	—
45	#15667983	—	4	1	no	—
46	C₄×D₃₅₆	—	2	1	no	—
47	#15667971	—	2	2	no	—
48	#15667970	—	4	1	no	—
49	#15667955	—	2	2	no	—
50	C₂×#13083549	—	2	4	no	—
51	C₂×#13083549	C₂³	1	1	yes	characteristic
52	C₂×#13083549	C₂³	1	2	yes	—
53	C₂×#13083549	C₂³	1	1	yes	characteristic
54	#15667946	—	2	1	no	—
55	#15667946	C₂³	1	1	yes	characteristic
56	#15667946	C₂³	1	2	yes	—
57	#15667944	—	2	2	no	—
58	#15667934	—	2	2	no	—
59	#15667934	—	2	1	no	—
60	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₈₉	—	2	1	no	—
63	#126×C₈₉	—	4	2	no	—
64	#124×C₈₉	—	4	1	no	—
65	C₂²⋊Q₈×C₈₉	—	2	2	no	—
66	C₄×Q₈×C₈₉	—	2	1	no	—
67	C₄×Q₈×C₈₉	—	4	1	no	—
68	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
69	#117×C₈₉	—	2	2	no	—
70	#107×C₈₉	—	2	1	no	—
71	#107×C₈₉	C₂³	1	1	yes	characteristic
72	#107×C₈₉	C₂³	1	2	yes	—
73	#106×C₈₉	—	2	2	no	—
74	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
75	#103×C₈₉	—	2	1	no	—
76	#103×C₈₉	—	2	2	no	—
77	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₇₈	—	4	1	no	—
80	C₂×D₃₅₆	—	2	1	no	—
81	C₂×C₂²⋊D₈₉	—	4	2	no	—
82	C₂×C₄×D₈₉	—	4	1	no	—
83	D₇₁₂	—	2	1	no	—
84	D₇₁₂	—	2	1	no	—
85	C₄⋊Dic₈₉	—	2	1	no	—
86	C₄⋊Dic₈₉	—	2	2	no	—
87	C₄⋊Dic₈₉	—	4	1	no	—
88	C₂²⋊Dic₈₉	—	4	2	no	—
89	#13083551	—	4	2	no	—
90	#13083551	—	4	1	no	—
91	C₄×Dic₈₉	—	4	2	no	—
92	#13083549	—	2	2	no	—
93	#13083549	—	2	1	no	—
94	#13083549	#41	1	2	yes	—
95	C₂×Q₈×C₈₉	—	2	1	no	—
96	C₂×Q₈×C₈₉	—	4	1	no	—
97	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
99	Q₁₆×C₈₉	—	2	1	no	—
100	Q₁₆×C₈₉	—	2	1	no	—
101	C₂×C₈×C₈₉	—	2	2	no	—
102	C₂×C₈×C₈₉	—	2	2	no	—
103	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₈₉	—	2	1	no	—
107	C₄⋊C₄×C₈₉	—	2	2	no	—
108	C₄⋊C₄×C₈₉	—	4	1	no	—
109	C₄⋊C₄×C₈₉	—	4	2	no	—
110	C₄⋊C₄×C₈₉	—	4	1	no	—
111	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
112	C₂²⋊C₄×C₈₉	—	4	2	no	—
113	C₂²⋊C₄×C₈₉	—	4	2	no	—
114	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
115	C₄²×C₈₉	—	4	2	no	—
116	C₄²×C₈₉	—	4	1	no	—
117	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂³×C₈₉	C₂×D₈	1	2	yes	—
119	C₂²×D₈₉	—	4	1	no	—
120	Q₈×C₈₉	—	4	1	no	—
121	Q₈×C₈₉	—	4	1	no	—
122	Q₈×C₈₉	—	8	2	no	—
123	D₄×C₈₉	—	2	2	no	—
124	C₂×C₄×C₈₉	—	2	2	no	—
125	C₂×C₄×C₈₉	—	2	2	no	—
126	C₂×C₄×C₈₉	—	4	1	no	—
127	C₂×C₄×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	—	4	1	no	—
129	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
133	C₂²⋊D₈₉	—	16	2	no	—
134	C₂×Dic₈₉	—	4	1	no	—
135	C₂×Dic₈₉	—	4	2	no	—
136	D₃₅₆	—	4	1	no	—
137	D₃₅₆	—	4	1	no	—
138	C₄×D₈₉	—	8	2	no	—
139	Dic₁₇₈	—	16	1	no	—
140	C₈×C₈₉	—	2	2	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
143	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
144	C₂²×C₈₉	—	2	2	no	—
145	C₂²×C₈₉	—	2	2	no	—
146	C₂²×C₈₉	—	2	2	no	—
147	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
148	C₂×D₈₉	—	8	1	no	—
149	C₂×D₈₉	—	8	1	no	—
150	C₂×D₈₉	—	8	1	no	—
151	C₄×C₈₉	—	2	2	no	—
152	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	—	8	1	no	—
154	C₄×C₈₉	—	8	2	no	—
155	C₄×C₈₉	—	8	1	no	—
156	C₄×C₈₉	#508	1	2	yes	—
157	C₄×C₈₉	#547	1	1	yes	characteristic
158	C₄×C₈₉	#578	1	1	yes	characteristic
159	Dic₈₉	—	8	1	no	—
160	Dic₈₉	—	8	2	no	—
161	#33907	—	89	1	no	maximal, Sylow
162	C₂×C₈₉	—	4	2	no	—
163	C₂×C₈₉	#3334	1	1	yes	characteristic
164	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
165	C₂×C₈₉	#3416	1	1	yes	characteristic
166	D₈₉	—	16	1	no	—
167	#3342	D₈₉	1	1	yes	characteristic, complemented
168	#3341	—	89	1	no	—
169	#2254	—	89	1	no	—
170	#2237	—	89	2	no	—
171	#2234	—	89	2	no	—
172	#2227	—	89	1	no	—
173	#2224	—	89	1	no	—
174	#2217	—	89	2	no	—
175	C₄×SD₃₂	—	89	1	no	—
176	D₄×C₁₆	—	89	1	no	—
177	#2158	—	89	2	no	—
178	C₈₉	#33907	1	1	yes	characteristic, complemented, Sylow
179	#556	—	2	1	no	—
180	#555	—	178	1	no	—
181	C₂×SD₃₂	—	89	1	no	—
182	C₂²×C₁₆	—	89	2	no	—
183	#500	—	89	1	no	—
184	#500	C₂×D₈₉	1	1	yes	characteristic
185	#485	—	2	2	no	—
186	#482	—	178	2	no	—
187	#476	—	178	1	no	—
188	#475	—	2	1	no	—
189	#467	C₂×D₈₉	1	2	yes	—
190	#466	—	89	2	no	—
191	C₄×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
192	C₄×D₈	—	89	1	no	—
193	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
194	#427	—	89	2	no	—
195	#427	C₂×D₈₉	1	2	yes	—
196	#367	—	89	1	no	—
197	#367	—	89	2	no	—
198	#367	—	89	2	no	—
199	#366	—	89	2	no	—
200	#366	—	89	2	no	—
201	C₄⋊C₁₆	—	89	1	no	—
202	#358	—	89	1	no	—
203	#358	—	89	2	no	—
204	#357	—	89	1	no	—
205	#357	—	89	2	no	—
206	C₂²⋊C₁₆	—	89	2	no	—
207	C₄×C₁₆	—	89	1	no	—
208	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
209	C₂×D₈	—	89	1	no	—
210	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	178	1	no	—
213	#126	—	4	2	no	—
214	#124	—	4	1	no	—
215	#124	—	356	2	no	—
216	C₂²⋊Q₈	—	2	2	no	—
217	#121	—	356	1	no	—
218	#121	—	178	2	no	—
219	C₄×Q₈	—	2	1	no	—
220	C₄×Q₈	—	4	1	no	—
221	C₄×D₄	—	178	1	no	—
222	C₄×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
223	#117	—	2	2	no	—
224	#117	—	356	1	no	—
225	#117	—	178	2	no	—
226	SD₃₂	—	178	2	no	—
227	C₂×C₁₆	—	89	1	no	—
228	C₂×C₁₆	—	89	1	no	—
229	C₂×C₁₆	—	89	2	no	—
230	C₂×C₁₆	—	178	4	no	—
231	#107	—	2	1	no	—
232	#107	—	89	1	no	—
233	#107	—	89	2	no	—
234	#107	—	178	1	no	—
235	#107	C₂²×D₈₉	1	1	yes	characteristic
236	#107	C₂²×D₈₉	1	2	yes	—
237	#106	—	2	2	no	—
238	#106	—	178	2	no	—
239	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
240	#103	—	2	1	no	—
241	#103	—	2	2	no	—
242	#102	—	178	1	no	—
243	#102	—	178	2	no	—
244	C₂²⋊C₈	C₂²⋊D₈₉	1	2	yes	—
245	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×Q₈	—	356	1	no	—
249	C₂×D₄	—	356	2	no	—
250	C₂×D₄	—	178	1	no	—
251	C₂×D₄	#13083559	1	1	yes	characteristic
252	C₂²×C₄	—	356	1	no	—
253	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
254	Q₁₆	—	2	1	no	—
255	Q₁₆	—	2	1	no	—
256	D₈	—	178	1	no	—
257	D₈	—	178	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
261	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
262	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	2	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	356	1	no	—
269	C₄⋊C₄	—	178	1	no	—
270	C₄⋊C₄	—	178	2	no	—
271	C₄⋊C₄	#13083559	1	1	yes	characteristic
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	356	2	no	—
275	C₂²⋊C₄	—	356	2	no	—
276	C₂²⋊C₄	—	356	1	no	—
277	C₂²⋊C₄	#13083559	1	2	yes	—
278	C₄²	—	4	2	no	—
279	C₄²	—	4	1	no	—
280	C₄²	—	356	2	no	—
281	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
282	C₁₆	—	89	2	no	—
283	C₁₆	—	178	2	no	—
284	C₁₆	—	178	1	no	—
285	C₂³	—	356	1	no	—
286	C₂³	C₂×#13083559	1	2	yes	—
287	Q₈	—	4	1	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	8	2	no	—
290	Q₈	—	1424	1	no	—
291	D₄	—	2	2	no	—
292	D₄	—	356	1	no	—
293	D₄	—	356	1	no	—
294	D₄	—	1424	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	356	1	no	—
301	C₂×C₄	—	356	2	no	—
302	C₂×C₄	—	712	2	no	—
303	C₂×C₄	C₂×#13083559	1	2	yes	—
304	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
305	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	2	no	—
309	C₈	#41×D₈₉	1	1	yes	characteristic
310	C₈	#15668059	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	712	1	no	—
315	C₂²	—	712	1	no	—
316	C₂²	—	712	1	no	—
317	C₂²	C₂²×#13083559	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	1	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	1	no	—
323	C₄	—	712	1	no	—
324	C₄	—	712	2	no	—
325	C₄	#19152028	1	2	yes	—
326	C₄	#19152247	1	1	yes	characteristic
327	C₄	#19152413	1	1	yes	characteristic
328	C₂	—	4	2	no	—
329	C₂	—	1424	1	no	—
330	C₂	#26032595	1	1	yes	characteristic
331	C₂	C₂×#19152028	1	1	yes	characteristic
332	C₂	#26033396	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3342	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26032675	C₂
hypercenter	#3342	D₈₉
hypocenter	C₈₉	#33907
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3342

Complements

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

N	H ≅ G/N	type
2. #26022532	C₂	semidirect
3. #26022457	C₂	semidirect
4. #26022443	C₂	semidirect
5. #26022404	C₂	semidirect
6. #26022386	C₂	semidirect
7. #26022350	C₂	semidirect
8. C₄×#15667955	C₂	semidirect
9. D₄×#13083549	C₂	semidirect
10. #26022034	C₂	semidirect
11. C₈₉×#3342	C₂	semidirect
167. #3342	D₈₉	semidirect
178. C₈₉	#33907	semidirect

Maximal subgroups

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

subgroup	index
#26032675	2
#26022532	2
#26022457	2
#26022443	2
#26022404	2
#26022386	2
#26022350	2
C₄×#15667955	2
D₄×#13083549	2
#26022034	2
C₈₉×#3342	2
#33907	89