Subgroups — #36454231

Order 22784.

578 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151946	C₂	1	1	yes	maximal, characteristic
2	#26021058	C₂	1	1	yes	maximal, characteristic
3	#26021057	C₂	1	1	yes	maximal, characteristic
4	#26021055	C₂	1	1	yes	maximal, characteristic, complemented
5	#26020411	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151320	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668009	C₂²	1	1	yes	characteristic
9	#19151946	—	2	2	no	—
10	#19151946	C₄	1	4	yes	complemented
11	C₂×#15667945	—	2	2	no	—
12	C₂×#15667936	—	2	2	no	—
13	C₂×#15667934	—	2	2	no	—
14	C₂×#15667932	—	2	2	no	—
15	C₂×#15667932	—	2	2	no	—
16	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
17	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
18	#19151468	C₂²	1	1	yes	characteristic
19	#19151320	—	2	2	no	—
20	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	#384×C₈₉	C₂²	1	1	yes	characteristic
23	#340×C₈₉	—	2	2	no	—
24	C₂²×D₃₅₆	—	2	1	no	—
25	C₂²×D₃₅₆	—	2	1	no	—
26	C₂²×D₃₅₆	—	2	1	no	—
27	C₂²×#10623494	D₄	1	2	yes	—
28	C₂²×#10623494	D₄	1	2	yes	—
29	#15668009	—	2	1	no	—
30	#15668009	—	2	1	no	—
31	#15668009	—	2	1	no	—
32	#15668009	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊Dic₈₉	—	2	1	no	—
34	C₂×C₄⋊Dic₈₉	—	2	1	no	—
35	C₂×C₄⋊Dic₈₉	—	2	1	no	—
36	#15667945	—	4	4	no	—
37	#15667936	—	4	4	no	—
38	#15667934	—	4	4	no	—
39	#15667932	—	2	8	no	—
40	#15667932	—	4	4	no	—
41	C₄×#10623494	—	2	1	no	—
42	C₄×#10623494	—	2	1	no	—
43	C₄×#10623494	C₂×C₄	1	2	yes	—
44	C₄×#10623494	D₄	1	2	yes	—
45	C₂²×Q₈×C₈₉	—	2	1	no	—
46	C₂²×C₈×C₈₉	D₄	1	2	yes	—
47	C₄⋊Q₈×C₈₉	—	2	1	no	—
48	C₄⋊Q₈×C₈₉	—	2	1	no	—
49	C₄⋊Q₈×C₈₉	—	2	1	no	—
50	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
52	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
53	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
55	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
56	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
57	C₄×C₈×C₈₉	—	2	1	no	—
58	C₄×C₈×C₈₉	—	2	1	no	—
59	C₂³×D₈₉	—	4	1	no	—
60	C₂×D₃₅₆	—	2	2	no	—
61	C₂×D₃₅₆	—	2	2	no	—
62	C₂×D₃₅₆	—	2	2	no	—
63	C₂×D₃₅₆	—	4	2	no	—
64	C₂×D₃₅₆	—	4	2	no	—
65	C₂×D₃₅₆	—	4	2	no	—
66	C₂×D₃₅₆	—	4	2	no	—
67	C₂×D₃₅₆	—	4	2	no	—
68	C₂×D₃₅₆	—	4	2	no	—
69	C₂×D₃₅₆	—	4	1	no	—
70	C₂×D₃₅₆	—	4	1	no	—
71	C₂×D₃₅₆	—	4	1	no	—
72	C₂²×Dic₈₉	—	4	1	no	—
73	C₂×#10623494	—	2	2	no	—
74	C₂×#10623494	—	2	2	no	—
75	C₂×#10623494	—	2	2	no	—
76	C₂×#10623494	—	2	2	no	—
77	C₂×#10623494	C₂²⋊C₄	1	4	yes	—
78	C₂×#10623494	D₈	1	4	yes	—
79	C₄⋊Dic₈₉	—	4	2	no	—
80	C₄⋊Dic₈₉	—	4	2	no	—
81	C₄⋊Dic₈₉	—	4	2	no	—
82	C₂×Q₈×C₈₉	—	2	2	no	—
83	C₂×Q₈×C₈₉	—	4	1	no	—
84	C₂×Q₈×C₈₉	—	4	2	no	—
85	C₂×Q₈×C₈₉	—	4	2	no	—
86	C₂²×C₄×C₈₉	—	4	1	no	—
87	C₂²×C₄×C₈₉	—	4	1	no	—
88	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	—	2	2	no	—
92	C₂×C₈×C₈₉	—	2	2	no	—
93	C₂×C₈×C₈₉	—	2	2	no	—
94	C₄⋊C₄×C₈₉	—	2	2	no	—
95	C₄⋊C₄×C₈₉	—	2	2	no	—
96	C₄⋊C₄×C₈₉	—	4	1	no	—
97	C₄⋊C₄×C₈₉	—	4	1	no	—
98	C₄⋊C₄×C₈₉	—	4	2	no	—
99	C₄⋊C₄×C₈₉	—	4	2	no	—
100	C₄⋊C₄×C₈₉	—	4	2	no	—
101	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₈₉	#41	1	1	yes	characteristic
104	C₄²×C₈₉	#41	1	1	yes	characteristic
105	C₂³×C₈₉	#127	1	1	yes	characteristic
106	C₂²×D₈₉	—	4	2	no	—
107	C₂²×D₈₉	—	8	1	no	—
108	C₂²×D₈₉	—	8	1	no	—
109	C₂²×D₈₉	—	8	1	no	—
110	C₂²×D₈₉	—	8	1	no	—
111	C₂²×D₈₉	—	8	1	no	—
112	C₂²×D₈₉	—	8	1	no	—
113	Q₈×C₈₉	—	4	4	no	—
114	Q₈×C₈₉	—	8	2	no	—
115	C₂×C₄×C₈₉	—	4	2	no	—
116	C₂×C₄×C₈₉	—	8	1	no	—
117	C₂×C₄×C₈₉	—	8	1	no	—
118	C₂×C₄×C₈₉	—	8	1	no	—
119	C₂×C₄×C₈₉	—	8	1	no	—
120	C₂×C₄×C₈₉	—	8	1	no	—
121	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
125	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
126	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
128	C₂×C₄×C₈₉	#135	1	2	yes	—
129	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
130	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₈₉	#137	1	2	yes	—
132	C₂×C₄×C₈₉	#137	1	2	yes	—
133	C₂×Dic₈₉	—	8	1	no	—
134	C₂×Dic₈₉	—	8	1	no	—
135	C₂×Dic₈₉	—	8	1	no	—
136	D₃₅₆	—	4	4	no	—
137	D₃₅₆	—	4	4	no	—
138	D₃₅₆	—	4	4	no	—
139	D₃₅₆	—	8	2	no	—
140	D₃₅₆	—	8	2	no	—
141	D₃₅₆	—	8	2	no	—
142	C₈×C₈₉	—	4	2	no	—
143	#10623494	—	2	4	no	—
144	#10623494	—	2	4	no	—
145	C₂²×C₈₉	#390	1	1	yes	characteristic
146	C₂²×C₈₉	#493	1	1	yes	characteristic
147	C₂²×C₈₉	#495	1	1	yes	characteristic
148	C₂²×C₈₉	#496	1	1	yes	characteristic
149	C₂²×C₈₉	#496	1	1	yes	characteristic
150	C₂²×C₈₉	#497	1	1	yes	characteristic
151	C₂²×C₈₉	#497	1	1	yes	characteristic
152	C₂×D₈₉	—	8	2	no	—
153	C₂×D₈₉	—	8	2	no	—
154	C₂×D₈₉	—	8	2	no	—
155	C₂×D₈₉	—	16	1	no	—
156	C₂×D₈₉	—	16	1	no	—
157	C₂×D₈₉	—	16	1	no	—
158	C₂×D₈₉	—	16	1	no	—
159	C₄×C₈₉	—	8	2	no	—
160	C₄×C₈₉	—	16	1	no	—
161	C₄×C₈₉	#417	1	2	yes	—
162	C₄×C₈₉	C₄×D₈	1	2	yes	—
163	C₄×C₈₉	#440	1	2	yes	—
164	C₄×C₈₉	#466	1	2	yes	—
165	C₄×C₈₉	#468	1	2	yes	—
166	C₄×C₈₉	#489	1	2	yes	—
167	Dic₈₉	—	16	1	no	—
168	#9830	—	89	1	no	maximal, Sylow
169	C₂×C₈₉	#1719	1	1	yes	characteristic
170	C₂×C₈₉	#1720	1	1	yes	characteristic
171	C₂×C₈₉	#1721	1	1	yes	characteristic
172	C₂×C₈₉	#1722	1	1	yes	characteristic
173	C₂×C₈₉	#1959	1	1	yes	characteristic
174	C₂×C₈₉	#1970	1	1	yes	characteristic
175	C₂×C₈₉	#1973	1	1	yes	characteristic
176	D₈₉	—	16	2	no	—
177	C₂×#486	—	89	1	no	—
178	#1990	D₈₉	1	1	yes	characteristic, complemented
179	#1957	—	89	1	no	—
180	#1957	—	89	1	no	—
181	#1956	—	89	1	no	—
182	#1845	—	89	1	no	—
183	C₂×#322	—	89	1	no	—
184	C₈₉	#9830	1	1	yes	characteristic, complemented, Sylow
185	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
186	C₂×#127	—	89	1	no	—
187	#486	—	89	4	no	—
188	#486	—	178	2	no	—
189	C₂×#107	—	178	2	no	—
190	C₂×#103	—	178	2	no	—
191	C₂×#102	—	178	2	no	—
192	C₂×#102	—	178	2	no	—
193	C₂×#102	—	178	2	no	—
194	C₂×C₄×C₈	—	89	1	no	—
195	C₂×C₄×C₈	—	89	1	no	—
196	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
197	#384	—	89	1	no	—
198	#384	C₂×D₈₉	1	1	yes	characteristic
199	#340	—	2	2	no	—
200	#322	—	178	2	no	—
201	C₂²×Q₈	—	2	1	no	—
202	C₂²×D₄	—	178	1	no	—
203	C₂²×D₄	—	178	1	no	—
204	C₂²×D₄	—	178	1	no	—
205	C₂²×C₈	—	89	2	no	—
206	C₂²×C₈	—	89	2	no	—
207	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	C₄⋊Q₈	C₄×D₈₉	1	2	yes	—
212	#127	—	89	2	no	—
213	#127	—	178	1	no	—
214	#127	—	178	1	no	—
215	#127	—	178	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	178	1	no	—
222	C₂×C₄⋊C₄	—	178	1	no	—
223	C₂×C₄⋊C₄	—	178	1	no	—
224	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
225	#107	—	356	4	no	—
226	#103	—	356	4	no	—
227	#102	—	356	4	no	—
228	#102	—	356	4	no	—
229	#102	—	178	8	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	—	89	2	no	—
233	C₄×C₈	—	89	2	no	—
234	C₄×C₈	—	178	1	no	—
235	C₄×C₈	—	178	1	no	—
236	C₂⁴	—	356	1	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	4	2	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×D₄	—	356	1	no	—
242	C₂×D₄	—	356	1	no	—
243	C₂×D₄	—	356	1	no	—
244	C₂×D₄	—	356	2	no	—
245	C₂×D₄	—	356	2	no	—
246	C₂×D₄	—	356	2	no	—
247	C₂×D₄	—	356	2	no	—
248	C₂×D₄	—	356	2	no	—
249	C₂×D₄	—	356	2	no	—
250	C₂×D₄	—	178	2	no	—
251	C₂×D₄	—	178	2	no	—
252	C₂×D₄	—	178	2	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	356	1	no	—
256	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
257	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	89	4	no	—
263	C₂×C₈	—	89	4	no	—
264	C₂×C₈	—	178	2	no	—
265	C₂×C₈	—	178	2	no	—
266	C₂×C₈	—	178	2	no	—
267	C₂×C₈	—	178	2	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	356	2	no	—
276	C₄⋊C₄	—	356	2	no	—
277	C₄⋊C₄	—	356	2	no	—
278	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
279	C₄²	D₄×D₈₉	1	1	yes	characteristic
280	C₄²	#13083573	1	1	yes	characteristic
281	C₄²	#13083574	1	1	yes	characteristic
282	C₂³	—	356	2	no	—
283	C₂³	—	712	1	no	—
284	C₂³	—	712	1	no	—
285	C₂³	—	712	1	no	—
286	C₂³	—	712	1	no	—
287	C₂³	—	712	1	no	—
288	C₂³	—	712	1	no	—
289	C₂³	#15668008	1	1	yes	characteristic
290	Q₈	—	4	4	no	—
291	Q₈	—	8	2	no	—
292	D₄	—	356	4	no	—
293	D₄	—	356	4	no	—
294	D₄	—	356	4	no	—
295	D₄	—	712	2	no	—
296	D₄	—	712	2	no	—
297	D₄	—	712	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	712	1	no	—
305	C₂×C₄	—	712	1	no	—
306	C₂×C₄	—	712	1	no	—
307	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
308	C₂×C₄	#15667972	1	1	yes	characteristic
309	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
310	C₂×C₄	#15667984	1	1	yes	characteristic
311	C₂×C₄	#15667985	1	1	yes	characteristic
312	C₂×C₄	#15667998	1	1	yes	characteristic
313	C₂×C₄	D₈×D₈₉	1	2	yes	—
314	C₂×C₄	#15668033	1	2	yes	—
315	C₂×C₄	#15668037	1	2	yes	—
316	C₂×C₄	#15668039	1	2	yes	—
317	C₂×C₄	#15668043	1	2	yes	—
318	C₂×C₄	#15668044	1	2	yes	—
319	C₈	—	4	2	no	—
320	C₈	—	178	4	no	—
321	C₈	—	178	4	no	—
322	C₂²	—	1424	1	no	—
323	C₂²	—	1424	1	no	—
324	C₂²	—	1424	1	no	—
325	C₂²	—	1424	1	no	—
326	C₂²	—	712	2	no	—
327	C₂²	—	712	2	no	—
328	C₂²	—	712	2	no	—
329	C₂²	#19151498	1	1	yes	characteristic
330	C₂²	#19151973	1	1	yes	characteristic
331	C₂²	#19151979	1	1	yes	characteristic
332	C₂²	#19151984	1	1	yes	characteristic
333	C₂²	#19151988	1	1	yes	characteristic
334	C₂²	#19151990	1	1	yes	characteristic
335	C₂²	#19151991	1	1	yes	characteristic
336	C₄	—	8	2	no	—
337	C₄	—	16	1	no	—
338	C₄	—	1424	1	no	—
339	C₄	#19151598	1	2	yes	—
340	C₄	#19151678	1	2	yes	—
341	C₄	#19151696	1	2	yes	—
342	C₄	#19151833	1	2	yes	—
343	C₄	#19151846	1	2	yes	—
344	C₄	#19151959	1	2	yes	—
345	C₂	—	1424	2	no	—
346	C₂	#26019907	1	1	yes	characteristic
347	C₂	#26019910	1	1	yes	characteristic
348	C₂	#26019920	1	1	yes	characteristic
349	C₂	#26019922	1	1	yes	characteristic
350	C₂	#26021065	1	1	yes	characteristic
351	C₂	#26021115	1	1	yes	characteristic
352	C₂	#26021136	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15668008
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1990	C₂
socle	C₂³×C₈₉	#127
prime radical	C₂×#15668009	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1990	D₈₉
hypocenter	C₈₉	#9830
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1990

Complements

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

N	H ≅ G/N	type
4. #26021055	C₂	semidirect
5. #26020411	C₂	semidirect
6. C₂×#19151320	C₂	semidirect
7. C₈₉×#1990	C₂	semidirect
10. #19151946	C₄	semidirect
178. #1990	D₈₉	semidirect
184. C₈₉	#9830	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151946	2
#26021058	2
#26021057	2
#26021055	2
#26020411	2
C₂×#19151320	2
C₈₉×#1990	2
#9830	89