Subgroups — #36445481

Order 22784.

436 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029227	C₂	1	1	yes	maximal, characteristic
2	#26018838	C₂	1	1	yes	maximal, characteristic
3	#26018736	C₂	1	1	yes	maximal, characteristic
4	#26018541	C₂	1	1	yes	maximal, characteristic, complemented
5	#26018151	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018145	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1726	C₂	1	1	yes	maximal, characteristic, complemented
8	#19152132	C₂²	1	1	yes	characteristic
9	#19151728	C₄	1	1	yes	characteristic, complemented
10	C₈×Dic₁₇₈	C₄	1	1	yes	characteristic, complemented
11	#19151677	C₄	1	1	yes	characteristic, complemented
12	#19151669	—	2	1	no	—
13	C₈×D₃₅₆	C₄	1	1	yes	characteristic, complemented
14	C₈×C₂²⋊D₈₉	—	2	1	no	—
15	#19151657	—	2	1	no	—
16	#19151655	C₂²	1	1	yes	characteristic
17	#19151651	C₂²	1	1	yes	characteristic
18	#19151552	—	2	1	no	—
19	C₄×#13083558	C₂²	1	1	yes	characteristic
20	#19151339	C₄	1	2	yes	complemented
21	#19151337	C₄	1	2	yes	complemented
22	#19151334	C₄	1	2	yes	—
23	#19151331	C₄	1	2	yes	—
24	#19151320	—	2	1	no	—
25	#19151319	—	2	1	no	—
26	C₂×#97×C₈₉	—	2	1	no	—
27	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₈₉	—	2	1	no	—
29	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
30	#376×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#13083558	—	2	1	no	—
32	C₂×#13083557	—	2	1	no	—
33	C₂×C₈×D₈₉	—	2	1	no	—
34	#15668006	—	2	1	no	—
35	#15668004	C₂×C₄	1	1	yes	characteristic
36	#15668000	C₂×C₄	1	1	yes	characteristic
37	#15667995	—	2	1	no	—
38	#15667991	C₂×C₄	1	1	yes	characteristic
39	#15667986	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
41	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
42	C₄×C₂²⋊D₈₉	—	2	1	no	—
43	#15667969	—	2	1	no	—
44	C₄⋊#10623494	—	2	1	no	—
45	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
46	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
47	#15667941	—	2	1	no	—
48	#15667941	—	2	1	no	—
49	C₂²⋊#10623494	—	2	1	no	—
50	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
51	#15667924	—	2	1	no	—
52	#15667924	—	2	1	no	—
53	#15667923	—	2	1	no	—
54	#15667923	C₂×C₄	1	2	yes	—
55	#15667922	—	2	1	no	—
56	C₄×#10623494	—	2	1	no	—
57	C₄×#10623494	—	2	1	no	—
58	C₈×Dic₈₉	—	2	1	no	—
59	C₂²×C₈×C₈₉	—	2	1	no	—
60	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₈₉	—	2	1	no	—
63	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
64	#97×C₈₉	—	2	4	no	—
65	#97×C₈₉	—	2	1	no	—
66	#97×C₈₉	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₈₉	—	2	4	no	—
68	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
71	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
72	#95×C₈₉	—	2	1	no	—
73	#95×C₈₉	C₂×C₄	1	2	yes	—
74	C₂×Dic₁₇₈	C₄²	1	1	yes	characteristic
75	C₂×D₃₅₆	C₄²	1	1	yes	characteristic
76	C₂×C₂²⋊D₈₉	—	2	1	no	—
77	C₂×C₄×D₈₉	—	2	1	no	—
78	#13083558	—	4	2	no	—
79	#13083557	—	4	2	no	—
80	C₂×#10623494	—	2	1	no	—
81	C₂×#10623494	—	2	1	no	—
82	C₂×#10623494	—	2	2	no	—
83	C₂×#10623494	—	2	2	no	—
84	C₈×D₈₉	—	4	2	no	—
85	C₄⋊Dic₈₉	C₄²	1	1	yes	characteristic
86	C₄⋊Dic₈₉	C₄²	1	1	yes	characteristic
87	#13083553	—	2	1	no	—
88	#13083553	—	2	1	no	—
89	#13083553	—	2	1	no	—
90	#13083553	—	2	1	no	—
91	C₂²⋊Dic₈₉	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	#13083551	—	2	1	no	—
94	#13083551	—	2	1	no	—
95	#13083551	—	2	1	no	—
96	C₄×Dic₈₉	—	2	1	no	—
97	C₂²×C₄×C₈₉	—	2	2	no	—
98	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
101	C₂×C₈×C₈₉	—	2	2	no	—
102	C₂×C₈×C₈₉	—	2	2	no	—
103	C₂×C₈×C₈₉	—	2	2	no	—
104	C₂×C₈×C₈₉	—	2	2	no	—
105	C₂×C₈×C₈₉	—	2	2	no	—
106	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
110	C₄²×C₈₉	—	2	2	no	—
111	C₄²×C₈₉	—	2	2	no	—
112	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₈₉	#41	1	1	yes	characteristic
116	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	4	no	—
122	C₂×C₄×C₈₉	—	4	1	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	4	2	no	—
125	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
131	C₂²⋊D₈₉	—	4	2	no	—
132	C₂×Dic₈₉	—	2	1	no	—
133	C₂×Dic₈₉	—	2	1	no	—
134	C₂×Dic₈₉	—	2	1	no	—
135	D₃₅₆	—	4	1	no	—
136	C₄×D₈₉	—	4	1	no	—
137	C₄×D₈₉	—	4	1	no	—
138	Dic₁₇₈	—	4	1	no	—
139	C₈×C₈₉	—	2	4	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	#104	1	2	yes	—
143	C₈×C₈₉	#104	1	2	yes	—
144	#10623494	—	4	2	no	—
145	#10623494	—	4	2	no	—
146	#10623494	—	4	1	no	—
147	#10623494	—	4	1	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₈₉	#444	1	2	yes	—
152	C₂×D₈₉	—	4	1	no	—
153	C₂×D₈₉	—	4	1	no	—
154	C₂×D₈₉	—	4	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	2	1	no	—
160	C₄×C₈₉	—	4	4	no	—
161	C₄×C₈₉	C₂×#104	1	1	yes	characteristic
162	C₄×C₈₉	C₂×#104	1	1	yes	characteristic
163	Dic₈₉	—	4	1	no	—
164	Dic₈₉	—	4	1	no	—
165	Dic₈₉	—	4	1	no	—
166	#8249	—	89	1	no	maximal, Sylow
167	C₂×C₈₉	—	2	2	no	—
168	C₂×C₈₉	#1757	1	1	yes	characteristic
169	C₂×C₈₉	C₄×#104	1	1	yes	characteristic
170	C₂×C₈₉	#1765	1	1	yes	characteristic
171	D₈₉	—	8	1	no	—
172	#2966	—	89	1	no	—
173	#1726	D₈₉	1	1	yes	characteristic, complemented
174	#1524	—	89	1	no	—
175	#1500	—	89	1	no	—
176	#1451	—	89	1	no	—
177	#1302	—	89	1	no	—
178	#1300	—	89	1	no	—
179	C₈₉	#8249	1	1	yes	characteristic, complemented, Sylow
180	#529	—	89	1	no	—
181	#446	—	89	1	no	—
182	C₈×Q₈	—	89	1	no	—
183	#436	—	89	1	no	—
184	#435	—	178	1	no	—
185	C₈×D₄	—	89	1	no	—
186	C₈×D₄	—	178	1	no	—
187	#433	—	178	1	no	—
188	#432	—	89	1	no	—
189	#431	—	89	1	no	—
190	#405	—	178	1	no	—
191	C₄×M₁₆	—	89	1	no	—
192	C₂×#97	—	2	1	no	—
193	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
194	C₂×C₄×C₈	—	2	1	no	—
195	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
196	#376	C₂×D₈₉	1	1	yes	characteristic
197	#330	—	89	2	no	—
198	#329	—	89	2	no	—
199	#328	—	89	2	no	—
200	#327	—	89	2	no	—
201	#322	—	178	1	no	—
202	#322	—	178	1	no	—
203	C₂×M₁₆	—	178	1	no	—
204	C₂×M₁₆	—	178	1	no	—
205	C₂²×C₈	—	2	1	no	—
206	C₂²×C₈	—	178	1	no	—
207	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
208	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
209	#126	—	178	1	no	—
210	#125	—	89	1	no	—
211	#124	—	89	1	no	—
212	#123	—	178	1	no	—
213	C₂²⋊Q₈	—	89	1	no	—
214	#121	—	89	1	no	—
215	C₄×Q₈	—	89	1	no	—
216	C₄×D₄	—	89	1	no	—
217	C₄×D₄	—	178	1	no	—
218	#117	—	178	1	no	—
219	C₂×C₄²	—	2	1	no	—
220	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
221	C₄⋊C₈	—	89	1	no	—
222	C₄⋊C₈	—	89	1	no	—
223	C₄⋊C₈	—	178	1	no	—
224	C₄⋊C₈	—	178	1	no	—
225	C₄⋊C₈	—	178	1	no	—
226	C₂²⋊C₈	—	89	2	no	—
227	C₂²⋊C₈	—	178	1	no	—
228	C₂²⋊C₈	—	178	1	no	—
229	C₂²⋊C₈	—	178	1	no	—
230	#97	—	2	4	no	—
231	#97	—	2	1	no	—
232	#97	—	89	2	no	—
233	#97	—	178	1	no	—
234	#97	—	178	1	no	—
235	#97	C₄×D₈₉	1	2	yes	—
236	C₄×C₈	—	2	4	no	—
237	C₄×C₈	—	178	1	no	—
238	C₄×C₈	—	178	1	no	—
239	C₄×C₈	—	178	1	no	—
240	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
241	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
242	C₄×C₈	D₃₅₆	1	1	yes	characteristic
243	C₄×C₈	D₃₅₆	1	1	yes	characteristic
244	#95	—	2	1	no	—
245	#95	C₄×D₈₉	1	2	yes	—
246	C₂×Q₈	—	89	1	no	—
247	C₂×D₄	—	89	1	no	—
248	C₂×D₄	—	178	1	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	178	1	no	—
251	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
253	C₂²×C₄	#13083574	1	1	yes	characteristic
254	M₁₆	—	356	2	no	—
255	M₁₆	—	356	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	356	2	no	—
262	C₂×C₈	—	178	2	no	—
263	C₂×C₈	—	178	2	no	—
264	C₂×C₈	—	178	1	no	—
265	C₂×C₈	—	178	1	no	—
266	C₂×C₈	#13083551	1	1	yes	characteristic
267	C₂×C₈	#13083551	1	1	yes	characteristic
268	C₂×C₈	#13083551	1	1	yes	characteristic
269	C₂×C₈	#13083551	1	1	yes	characteristic
270	C₄⋊C₄	—	89	1	no	—
271	C₄⋊C₄	—	89	1	no	—
272	C₄⋊C₄	—	178	1	no	—
273	C₄⋊C₄	—	178	1	no	—
274	C₄⋊C₄	—	178	1	no	—
275	C₄⋊C₄	—	178	1	no	—
276	C₂²⋊C₄	—	178	1	no	—
277	C₂²⋊C₄	—	178	1	no	—
278	C₂²⋊C₄	—	178	1	no	—
279	C₂²⋊C₄	—	178	1	no	—
280	C₂²⋊C₄	—	178	1	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	178	1	no	—
284	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
286	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
287	C₄²	#13083574	1	1	yes	characteristic
288	C₂³	—	178	1	no	—
289	C₂³	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
290	Q₈	—	356	1	no	—
291	D₄	—	356	1	no	—
292	D₄	—	356	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	356	1	no	—
301	C₂×C₄	—	356	1	no	—
302	C₂×C₄	—	178	1	no	—
303	C₂×C₄	—	178	1	no	—
304	C₂×C₄	—	178	1	no	—
305	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
306	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
307	C₂×C₄	#15667969	1	1	yes	characteristic
308	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
309	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
310	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
311	C₈	—	2	4	no	—
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	356	2	no	—
315	C₈	—	356	2	no	—
316	C₈	—	356	1	no	—
317	C₈	—	356	1	no	—
318	C₈	#15667940	1	2	yes	—
319	C₈	#15667940	1	2	yes	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	356	1	no	—
323	C₂²	—	356	1	no	—
324	C₂²	—	356	1	no	—
325	C₂²	C₄×#13083551	1	1	yes	characteristic
326	C₂²	#19151718	1	2	yes	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	4	no	—
333	C₄	—	356	1	no	—
334	C₄	—	356	1	no	—
335	C₄	—	356	1	no	—
336	C₄	C₂×#15667940	1	1	yes	characteristic
337	C₄	C₂×#15667940	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	712	1	no	—
340	C₂	#26020053	1	1	yes	characteristic
341	C₂	C₄×#15667940	1	1	yes	characteristic
342	C₂	#26020098	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#13083551
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1726	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	#15667986	C₂×C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1726	D₈₉
hypocenter	C₈₉	#8249
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1726

Complements

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

N	H ≅ G/N	type
4. #26018541	C₂	semidirect
5. #26018151	C₂	semidirect
6. #26018145	C₂	semidirect
7. C₈₉×#1726	C₂	semidirect
9. #19151728	C₄	semidirect
10. C₈×Dic₁₇₈	C₄	semidirect
11. #19151677	C₄	semidirect
13. C₈×D₃₅₆	C₄	semidirect
20. #19151339	C₄	semidirect
21. #19151337	C₄	semidirect
173. #1726	D₈₉	semidirect
179. C₈₉	#8249	semidirect

Maximal subgroups

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

subgroup	index
#26029227	2
#26018838	2
#26018736	2
#26018541	2
#26018151	2
#26018145	2
C₈₉×#1726	2
#8249	89