Subgroups — #36542360

Order 22784.

858 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26031369	C₂	1	2	yes	maximal, complemented
2	#26030829	C₂	1	1	yes	maximal, characteristic
3	#500×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#26030706	C₂	1	2	yes	maximal, complemented
5	#26019925	C₂	1	2	yes	maximal, complemented
6	#26019921	C₂	1	2	yes	maximal, complemented
7	#26019870	C₂	1	1	yes	maximal, characteristic, complemented
8	#26019869	C₂	1	1	yes	maximal, characteristic, complemented
9	#26019841	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#13083557	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1706	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668041	—	2	4	no	—
13	#19152146	C₂²	1	2	yes	—
14	C₄⋊Q₈×D₈₉	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19152003	C₂²	1	2	yes	—
17	#19152003	C₂²	1	1	yes	characteristic
18	#19152002	—	2	2	no	—
19	#19152001	C₂²	1	2	yes	—
20	#19152001	C₂²	1	2	yes	—
21	#19151989	—	2	4	no	—
22	#19151978	C₂²	1	2	yes	—
23	#19151977	C₂²	1	2	yes	—
24	#19151970	C₂²	1	2	yes	—
25	#19151944	C₂²	1	2	yes	—
26	#19151944	C₂²	1	2	yes	—
27	#19151910	—	2	4	no	—
28	#19151876	—	2	4	no	—
29	#19151831	—	2	2	no	—
30	D₈×Dic₈₉	—	2	2	no	—
31	#19151649	C₂²	1	1	yes	characteristic
32	#107×D₈₉	—	2	2	no	—
33	C₄×#13083557	C₂²	1	1	yes	characteristic
34	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
35	#19151320	C₂²	1	1	yes	characteristic
36	#19151319	C₂²	1	1	yes	characteristic
37	C₈×#10623494	C₂²	1	1	yes	characteristic
38	#500×C₈₉	C₂²	1	1	yes	characteristic
39	#493×C₈₉	C₂²	1	1	yes	characteristic
40	#492×C₈₉	C₂²	1	2	yes	—
41	#486×C₈₉	C₂²	1	2	yes	—
42	C₈²×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×#13083573	—	2	2	no	—
44	C₂×Q₈×D₈₉	—	2	2	no	—
45	#15668041	—	4	16	no	—
46	C₂×Dic₃₅₆	—	2	4	no	—
47	C₂×#13083564	—	2	4	no	—
48	C₂×#13083562	—	2	4	no	—
49	C₂×#13083557	D₄	1	2	yes	—
50	C₂×C₈×D₈₉	D₄	1	2	yes	—
51	#15668012	C₂³	1	2	yes	—
52	#15668012	C₂³	1	2	yes	—
53	#15668011	—	2	2	no	—
54	#15668011	—	2	2	no	—
55	#15668010	—	2	2	no	—
56	#15667999	—	2	2	no	—
57	#15667983	—	2	2	no	—
58	#15667983	—	2	2	no	—
59	D₄×Dic₈₉	—	2	2	no	—
60	D₄×Dic₈₉	—	2	2	no	—
61	C₄⋊C₄×D₈₉	—	2	2	no	—
62	C₄⋊C₄×D₈₉	—	2	2	no	—
63	C₄²×D₈₉	C₂³	1	1	yes	characteristic
64	#15667946	—	2	4	no	—
65	#15667946	—	2	2	no	—
66	#15667946	—	2	2	no	—
67	#15667945	—	2	4	no	—
68	#15667945	—	2	4	no	—
69	#15667941	D₄	1	2	yes	—
70	#15667933	—	2	4	no	—
71	#15667933	—	2	4	no	—
72	#15667924	D₄	1	2	yes	—
73	#15667922	D₄	1	2	yes	—
74	C₄×#10623494	C₂³	1	1	yes	characteristic
75	C₄×#10623494	C₂³	1	1	yes	characteristic
76	C₄×#10623494	C₂³	1	1	yes	characteristic
77	C₈×Dic₈₉	D₄	1	2	yes	—
78	C₂×SD₁₆×C₈₉	—	2	4	no	—
79	C₂×D₈×C₈₉	—	2	2	no	—
80	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
81	#127×C₈₉	C₂³	1	2	yes	—
82	#107×C₈₉	—	2	2	no	—
83	#102×C₈₉	—	2	4	no	—
84	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
87	#13083573	—	4	8	no	—
88	C₂×Dic₁₇₈	—	2	2	no	—
89	C₂×Dic₁₇₈	—	2	2	no	—
90	C₂×Dic₁₇₈	—	4	2	no	—
91	Q₈×D₈₉	—	4	8	no	—
92	C₂×C₂²⋊D₈₉	—	4	2	no	—
93	C₂²×Dic₈₉	—	4	2	no	—
94	C₂×C₄×D₈₉	—	4	2	no	—
95	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
98	Dic₃₅₆	—	4	8	no	—
99	#13083564	—	4	8	no	—
100	#13083562	—	4	8	no	—
101	#13083557	D₈	1	8	yes	complemented
102	C₂×#10623494	C₂×D₄	1	2	yes	—
103	C₂×#10623494	C₂×D₄	1	2	yes	—
104	C₂×#10623494	C₂×D₄	1	2	yes	—
105	C₈×D₈₉	—	2	4	no	—
106	C₄⋊Dic₈₉	—	2	2	no	—
107	C₄⋊Dic₈₉	—	2	2	no	—
108	C₄⋊Dic₈₉	—	2	2	no	—
109	C₄⋊Dic₈₉	—	2	2	no	—
110	#13083553	—	4	2	no	—
111	#13083553	—	4	2	no	—
112	C₂²⋊Dic₈₉	—	4	2	no	—
113	C₂²⋊Dic₈₉	—	4	2	no	—
114	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
115	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
116	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×Q₈×C₈₉	—	2	2	no	—
118	C₂×D₄×C₈₉	—	2	2	no	—
119	C₂×D₄×C₈₉	—	2	2	no	—
120	C₂×D₄×C₈₉	—	2	2	no	—
121	SD₁₆×C₈₉	—	4	8	no	—
122	D₈×C₈₉	—	4	4	no	—
123	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
124	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
126	C₄⋊C₄×C₈₉	—	2	2	no	—
127	C₄⋊C₄×C₈₉	—	2	2	no	—
128	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
129	C₂³×C₈₉	—	4	2	no	—
130	C₂²×D₈₉	#127	1	1	yes	characteristic
131	Q₈×C₈₉	—	4	4	no	—
132	D₄×C₈₉	—	4	4	no	—
133	D₄×C₈₉	—	4	4	no	—
134	D₄×C₈₉	—	4	4	no	—
135	C₂×C₄×C₈₉	—	4	2	no	—
136	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₈₉	—	8	4	no	—
140	C₂×Dic₈₉	—	4	2	no	—
141	C₂×Dic₈₉	—	4	2	no	—
142	C₂×Dic₈₉	—	8	4	no	—
143	C₂×Dic₈₉	#127	1	1	yes	characteristic
144	C₂×Dic₈₉	#127	1	1	yes	characteristic
145	C₂×Dic₈₉	#127	1	1	yes	characteristic
146	C₄×D₈₉	—	2	2	no	—
147	C₄×D₈₉	—	2	2	no	—
148	C₄×D₈₉	—	8	4	no	—
149	C₄×D₈₉	C₂×D₈	1	4	yes	—
150	Dic₁₇₈	—	4	4	no	—
151	Dic₁₇₈	—	4	4	no	—
152	Dic₁₇₈	—	8	4	no	—
153	C₈×C₈₉	—	2	2	no	—
154	C₈×C₈₉	C₂×D₈	1	4	yes	—
155	C₈×C₈₉	#137	1	4	yes	complemented
156	#10623494	—	2	2	no	—
157	#10623494	—	2	2	no	—
158	#10623494	C₂×D₈	1	4	yes	—
159	C₂²×C₈₉	—	8	2	no	—
160	C₂²×C₈₉	—	8	2	no	—
161	C₂²×C₈₉	—	8	2	no	—
162	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
163	C₂×D₈₉	—	2	1	no	—
164	C₂×D₈₉	—	2	1	no	—
165	C₂×D₈₉	#493	1	2	yes	—
166	C₄×C₈₉	—	8	2	no	—
167	C₄×C₈₉	C₂²×D₈	1	2	yes	—
168	C₄×C₈₉	C₂×#137	1	2	yes	—
169	C₄×C₈₉	#578	1	2	yes	—
170	Dic₈₉	—	2	1	no	—
171	Dic₈₉	—	2	1	no	—
172	Dic₈₉	—	8	2	no	—
173	Dic₈₉	—	8	2	no	—
174	Dic₈₉	#493	1	2	yes	—
175	#19630	—	89	1	no	maximal, Sylow
176	C₂×C₈₉	—	16	2	no	—
177	C₂×C₈₉	C₂×#493	1	1	yes	characteristic
178	C₂×C₈₉	#3152	1	1	yes	characteristic
179	C₂×C₈₉	#3156	1	1	yes	characteristic
180	D₈₉	—	2	2	no	—
181	#3226	—	89	2	no	—
182	#3169	—	89	1	no	—
183	C₂×#500	—	89	1	no	—
184	#3152	—	89	2	no	—
185	#1722	—	89	2	no	—
186	#1722	—	89	2	no	—
187	#1713	—	89	1	no	—
188	#1713	—	89	1	no	—
189	#1706	—	89	1	no	—
190	#1706	D₈₉	1	1	yes	characteristic, complemented
191	C₈×M₁₆	—	89	1	no	—
192	C₈₉	#19630	1	1	yes	characteristic, complemented, Sylow
193	C₂×#137	—	178	4	no	—
194	#532	—	89	2	no	—
195	C₂×C₄⋊Q₈	—	89	2	no	—
196	#500	—	89	2	no	—
197	#500	—	89	2	no	—
198	#500	—	89	1	no	—
199	#500	—	89	1	no	—
200	#500	—	89	2	no	—
201	#500	—	178	2	no	—
202	#500	C₂×D₈₉	1	1	yes	characteristic
203	#497	—	178	4	no	—
204	#494	—	89	2	no	—
205	#494	—	89	2	no	—
206	#493	C₂×D₈₉	1	1	yes	characteristic
207	#492	—	89	2	no	—
208	#492	C₂×D₈₉	1	2	yes	—
209	#486	—	89	2	no	—
210	#486	—	89	2	no	—
211	#486	C₂×D₈₉	1	2	yes	—
212	#480	—	178	4	no	—
213	#474	—	178	4	no	—
214	#466	—	178	2	no	—
215	C₄×D₈	—	178	2	no	—
216	#431	—	89	1	no	—
217	C₂×#107	—	178	2	no	—
218	C₄×M₁₆	—	89	1	no	—
219	C₂×C₄×C₈	—	89	1	no	—
220	#322	—	89	1	no	—
221	#322	—	89	1	no	—
222	C₈²	—	89	1	no	—
223	C₈²	C₂×D₈₉	1	1	yes	characteristic
224	C₂×#41	—	178	2	no	—
225	C₂²×Q₈	—	178	2	no	—
226	#137	—	356	16	no	—
227	C₂×Q₁₆	—	178	4	no	—
228	C₂×Q₁₆	—	178	4	no	—
229	C₂×SD₁₆	—	2	4	no	—
230	C₂×SD₁₆	—	178	4	no	—
231	C₂×D₈	—	2	2	no	—
232	C₂×M₁₆	—	89	2	no	—
233	C₂²×C₈	—	89	2	no	—
234	C₄⋊Q₈	—	89	2	no	—
235	C₄⋊Q₈	—	89	2	no	—
236	C₄⋊Q₈	—	178	2	no	—
237	C₄⋊Q₈	—	178	2	no	—
238	C₄⋊Q₈	—	178	2	no	—
239	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
240	#127	C₂²×D₈₉	1	2	yes	—
241	#124	—	178	2	no	—
242	#121	—	178	2	no	—
243	#121	—	178	2	no	—
244	C₄×D₄	—	178	2	no	—
245	C₄×D₄	—	178	2	no	—
246	C₂×C₄⋊C₄	—	178	2	no	—
247	C₂×C₄⋊C₄	—	178	2	no	—
248	C₂×C₄²	—	89	1	no	—
249	#107	—	2	2	no	—
250	#107	—	178	4	no	—
251	#107	—	178	4	no	—
252	#107	—	178	2	no	—
253	#107	—	178	2	no	—
254	#107	—	178	4	no	—
255	C₄⋊C₈	—	89	2	no	—
256	#102	—	2	4	no	—
257	#102	—	178	4	no	—
258	#102	—	178	4	no	—
259	C₂²⋊C₈	—	89	2	no	—
260	#97	—	89	2	no	—
261	C₄×C₈	—	89	2	no	—
262	C₄×C₈	—	89	1	no	—
263	C₄×C₈	—	89	1	no	—
264	C₄×C₈	—	89	1	no	—
265	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
266	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
267	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
268	#41	—	356	8	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	356	2	no	—
271	C₂×Q₈	—	356	8	no	—
272	C₂×Q₈	—	178	2	no	—
273	C₂×Q₈	—	178	2	no	—
274	C₂×D₄	—	2	2	no	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	356	2	no	—
278	C₂²×C₄	—	89	1	no	—
279	C₂²×C₄	—	89	1	no	—
280	C₂²×C₄	—	89	1	no	—
281	C₂²×C₄	—	356	2	no	—
282	C₂²×C₄	—	356	2	no	—
283	Q₁₆	—	356	8	no	—
284	Q₁₆	—	356	8	no	—
285	SD₁₆	—	4	8	no	—
286	SD₁₆	—	356	8	no	—
287	D₈	—	4	4	no	—
288	M₁₆	—	89	8	no	—
289	C₂×C₈	—	89	2	no	—
290	C₂×C₈	—	89	2	no	—
291	C₂×C₈	—	89	2	no	—
292	C₂×C₈	—	178	4	no	—
293	C₂×C₈	D₄×D₈₉	1	2	yes	—
294	C₂×C₈	D₄×D₈₉	1	2	yes	—
295	C₂×C₈	D₄×D₈₉	1	2	yes	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	356	2	no	—
299	C₄⋊C₄	—	356	2	no	—
300	C₄⋊C₄	—	178	2	no	—
301	C₄⋊C₄	—	178	2	no	—
302	C₄⋊C₄	—	178	2	no	—
303	C₄⋊C₄	—	178	2	no	—
304	C₂²⋊C₄	—	356	2	no	—
305	C₂²⋊C₄	—	356	2	no	—
306	C₄²	—	89	1	no	—
307	C₄²	—	89	1	no	—
308	C₄²	—	89	1	no	—
309	C₄²	C₂³×D₈₉	1	1	yes	characteristic
310	C₂³	—	4	2	no	—
311	C₂³	—	89	1	no	—
312	Q₈	—	4	4	no	—
313	Q₈	—	356	4	no	—
314	Q₈	—	356	4	no	—
315	Q₈	—	712	4	no	—
316	D₄	—	4	4	no	—
317	D₄	—	4	4	no	—
318	D₄	—	4	4	no	—
319	D₄	—	712	4	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	89	1	no	—
322	C₂×C₄	—	89	4	no	—
323	C₂×C₄	—	89	1	no	—
324	C₂×C₄	—	89	1	no	—
325	C₂×C₄	—	356	2	no	—
326	C₂×C₄	—	356	2	no	—
327	C₂×C₄	—	178	2	no	—
328	C₂×C₄	—	178	2	no	—
329	C₂×C₄	—	712	4	no	—
330	C₂×C₄	—	712	4	no	—
331	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
332	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	89	4	no	—
336	C₈	—	178	2	no	—
337	C₈	—	178	2	no	—
338	C₈	D₈×D₈₉	1	4	yes	—
339	C₈	#15668041	1	4	yes	complemented
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	89	2	no	—
344	C₂²	—	178	1	no	—
345	C₂²	—	178	1	no	—
346	C₂²	#127×D₈₉	1	1	yes	characteristic
347	C₄	—	8	2	no	—
348	C₄	—	89	2	no	—
349	C₄	—	178	1	no	—
350	C₄	—	178	1	no	—
351	C₄	—	712	2	no	—
352	C₄	—	712	2	no	—
353	C₄	C₂×D₈×D₈₉	1	2	yes	—
354	C₄	C₂×#15668041	1	2	yes	—
355	C₄	#19152412	1	2	yes	—
356	C₂	—	16	2	no	—
357	C₂	—	178	2	no	—
358	C₂	#493×D₈₉	1	1	yes	characteristic
359	C₂	#26030706	1	1	yes	characteristic
360	C₂	#26030742	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₈₉
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1706	C₂
socle	C₂²×C₈₉	C₂×#127
prime radical	#26030829	C₂
hypercenter	#1706	D₈₉
hypocenter	C₈₉	#19630
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1706

Complements

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

N	H ≅ G/N	type
1. #26031369	C₂	semidirect
3. #500×D₈₉	C₂	semidirect
4. #26030706	C₂	semidirect
5. #26019925	C₂	semidirect
6. #26019921	C₂	semidirect
7. #26019870	C₂	semidirect
8. #26019869	C₂	semidirect
9. #26019841	C₂	semidirect
10. C₈×#13083557	C₂	semidirect
11. C₈₉×#1706	C₂	semidirect
101. #13083557	D₈	semidirect
155. C₈×C₈₉	#137	semidirect
190. #1706	D₈₉	semidirect
192. C₈₉	#19630	semidirect
339. C₈	#15668041	semidirect

Maximal subgroups

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

subgroup	index
#26031369	2
#26030829	2
#500×D₈₉	2
#26030706	2
#26019925	2
#26019921	2
#26019870	2
#26019869	2
#26019841	2
C₈×#13083557	2
C₈₉×#1706	2
#19630	89