Subgroups — #36453676

Order 22784.

522 conjugacy classes of subgroups.

The subgroup lattice has 352 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	#26021201	C₂	1	1	yes	maximal, characteristic, complemented
3	#26021178	C₂	1	1	yes	maximal, characteristic
4	#26020117	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151348	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151341	C₂	1	1	yes	maximal, characteristic
7	C₂×#333×C₈₉	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×C₄⋊#10623494	C₂²	1	1	yes	characteristic
12	C₂×#15667934	—	2	2	no	—
13	C₂×#15667932	—	2	1	no	—
14	C₂×#15667932	—	2	1	no	—
15	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₈₉	—	2	1	no	—
17	C₄×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
18	#19151360	—	2	2	no	—
19	#19151352	—	2	1	no	—
20	#19151348	—	2	2	no	—
21	#19151341	—	2	2	no	—
22	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
25	#333×C₈₉	—	2	2	no	—
26	C₂²×D₃₅₆	—	2	1	no	—
27	C₂²×D₃₅₆	—	2	1	no	—
28	C₂²×D₃₅₆	—	2	1	no	—
29	C₂²×#10623494	—	2	1	no	—
30	C₂²×#10623494	D₄	1	1	yes	characteristic
31	C₂²×#10623494	D₄	1	1	yes	characteristic
32	#15668009	—	2	1	no	—
33	#15668009	—	2	1	no	—
34	#15668009	—	2	1	no	—
35	#15668009	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊Dic₈₉	D₄	1	2	yes	—
37	C₂×C₄×Dic₈₉	—	2	1	no	—
38	C₄⋊#10623494	—	2	1	no	—
39	C₄⋊#10623494	—	2	1	no	—
40	#15667934	—	4	4	no	—
41	#15667932	—	2	4	no	—
42	#15667932	—	2	4	no	—
43	C₄×#10623494	C₂×C₄	1	2	yes	—
44	C₄×#10623494	D₄	1	2	yes	—
45	C₈×Dic₈₉	—	4	2	no	—
46	#15667919	—	2	1	no	—
47	C₂²×Q₈×C₈₉	—	2	1	no	—
48	C₂²×C₈×C₈₉	—	2	1	no	—
49	C₂²×C₈×C₈₉	—	2	1	no	—
50	C₄⋊Q₈×C₈₉	—	2	1	no	—
51	C₄⋊Q₈×C₈₉	—	2	1	no	—
52	C₄⋊Q₈×C₈₉	—	2	1	no	—
53	C₄⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
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	C₄⋊C₈×C₈₉	—	2	1	no	—
60	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
61	C₂³×D₈₉	—	4	1	no	—
62	C₂×D₃₅₆	—	2	2	no	—
63	C₂×D₃₅₆	—	2	2	no	—
64	C₂×D₃₅₆	—	2	2	no	—
65	C₂×D₃₅₆	—	4	2	no	—
66	C₂×D₃₅₆	—	4	1	no	—
67	C₂×D₃₅₆	—	4	1	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	C₂×D₃₅₆	—	4	1	no	—
73	C₂×D₃₅₆	—	4	1	no	—
74	C₂×D₃₅₆	—	8	2	no	—
75	C₂²×Dic₈₉	—	2	2	no	—
76	C₂×#10623494	—	2	1	no	—
77	C₂×#10623494	—	2	1	no	—
78	C₂×#10623494	—	2	1	no	—
79	C₂×#10623494	—	2	1	no	—
80	C₂×#10623494	—	4	1	no	—
81	C₂×#10623494	—	4	2	no	—
82	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
83	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
84	C₄⋊Dic₈₉	—	2	2	no	—
85	C₄⋊Dic₈₉	SD₁₆	1	4	yes	—
86	C₄×Dic₈₉	—	4	1	no	—
87	C₄×Dic₈₉	—	4	1	no	—
88	C₂×Q₈×C₈₉	—	2	2	no	—
89	C₂×Q₈×C₈₉	—	4	1	no	—
90	C₂×Q₈×C₈₉	—	8	2	no	—
91	C₂²×C₄×C₈₉	—	4	1	no	—
92	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
95	C₂×C₈×C₈₉	—	2	2	no	—
96	C₂×C₈×C₈₉	—	4	1	no	—
97	C₂×C₈×C₈₉	—	4	1	no	—
98	C₂×C₈×C₈₉	—	4	1	no	—
99	C₂×C₈×C₈₉	—	4	2	no	—
100	C₄⋊C₄×C₈₉	—	2	2	no	—
101	C₄⋊C₄×C₈₉	—	2	2	no	—
102	C₄⋊C₄×C₈₉	—	4	1	no	—
103	C₄⋊C₄×C₈₉	—	4	1	no	—
104	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₈₉	#41	1	1	yes	characteristic
108	C₂³×C₈₉	#121	1	1	yes	characteristic
109	C₂²×D₈₉	—	4	2	no	—
110	C₂²×D₈₉	—	8	1	no	—
111	C₂²×D₈₉	—	8	1	no	—
112	C₂²×D₈₉	—	8	2	no	—
113	C₂²×D₈₉	—	8	2	no	—
114	Q₈×C₈₉	—	8	2	no	—
115	Q₈×C₈₉	—	8	2	no	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	4	2	no	—
118	C₂×C₄×C₈₉	—	8	2	no	—
119	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
123	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
124	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
126	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
127	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
128	C₂×C₄×C₈₉	#135	1	2	yes	—
129	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
131	C₂×Dic₈₉	—	2	4	no	—
132	C₂×Dic₈₉	—	4	2	no	—
133	C₂×Dic₈₉	—	4	2	no	—
134	D₃₅₆	—	4	2	no	—
135	D₃₅₆	—	4	2	no	—
136	D₃₅₆	—	4	4	no	—
137	D₃₅₆	—	8	2	no	—
138	D₃₅₆	—	8	2	no	—
139	D₃₅₆	—	8	1	no	—
140	D₃₅₆	—	8	1	no	—
141	D₃₅₆	—	8	1	no	—
142	D₃₅₆	—	8	1	no	—
143	C₈×C₈₉	—	4	2	no	—
144	C₈×C₈₉	—	8	1	no	—
145	#10623494	—	2	2	no	—
146	#10623494	—	2	2	no	—
147	#10623494	—	8	1	no	—
148	C₂²×C₈₉	#390	1	1	yes	characteristic
149	C₂²×C₈₉	#463	1	1	yes	characteristic
150	C₂²×C₈₉	#464	1	1	yes	characteristic
151	C₂²×C₈₉	#465	1	2	yes	—
152	C₂²×C₈₉	#472	1	2	yes	—
153	C₂×D₈₉	—	8	2	no	—
154	C₂×D₈₉	—	8	2	no	—
155	C₂×D₈₉	—	8	2	no	—
156	C₂×D₈₉	—	16	2	no	—
157	C₂×D₈₉	—	16	2	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	2	1	no	—
160	C₄×C₈₉	—	8	2	no	—
161	C₄×C₈₉	#416	1	1	yes	characteristic
162	C₄×C₈₉	#417	1	1	yes	characteristic
163	C₄×C₈₉	C₄×SD₁₆	1	2	yes	—
164	C₄×C₈₉	#463	1	1	yes	characteristic
165	C₄×C₈₉	#464	1	1	yes	characteristic
166	C₄×C₈₉	#492	1	1	yes	characteristic
167	C₄×C₈₉	#495	1	1	yes	characteristic
168	Dic₈₉	—	4	4	no	—
169	#9735	—	89	1	no	maximal, Sylow
170	C₂×C₈₉	#1677	1	2	yes	—
171	C₂×C₈₉	#1678	1	2	yes	—
172	C₂×C₈₉	#1968	1	1	yes	characteristic
173	C₂×C₈₉	#1978	1	1	yes	characteristic
174	C₂×C₈₉	#1983	1	1	yes	characteristic
175	D₈₉	—	16	2	no	—
176	C₂×#486	—	89	1	no	—
177	#1987	—	89	1	no	—
178	#1982	—	89	1	no	—
179	C₈×C₄⋊C₄	—	89	1	no	—
180	C₂×#334	—	89	1	no	—
181	C₂×#333	D₈₉	1	1	yes	characteristic, complemented
182	C₂×#331	—	89	1	no	—
183	C₈₉	#9735	1	1	yes	characteristic, complemented, Sylow
184	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
185	C₂×#127	—	89	1	no	—
186	#486	—	89	4	no	—
187	#486	—	178	2	no	—
188	C₂×C₄⋊C₈	—	89	1	no	—
189	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
190	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
191	C₂×#102	—	178	2	no	—
192	C₂×#102	—	178	1	no	—
193	C₂×#102	—	178	1	no	—
194	C₂×C₄×C₈	—	89	1	no	—
195	C₂×C₄×C₈	—	178	1	no	—
196	C₄×C₄⋊C₄	—	89	1	no	—
197	#340	—	178	2	no	—
198	#336	—	178	1	no	—
199	#334	—	178	2	no	—
200	#333	—	2	2	no	—
201	#331	—	178	2	no	—
202	C₂²×Q₈	—	2	1	no	—
203	C₂²×D₄	—	178	1	no	—
204	C₂²×D₄	—	178	1	no	—
205	C₂²×D₄	—	178	1	no	—
206	C₂²×C₈	—	2	1	no	—
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	—	89	1	no	—
209	C₂²×C₈	—	89	1	no	—
210	C₂²×C₈	—	178	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	C₄×D₈₉	1	2	yes	—
215	#127	—	89	2	no	—
216	#127	—	178	1	no	—
217	#127	—	178	1	no	—
218	#127	—	178	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	89	2	no	—
222	C₂×C₄²	—	178	1	no	—
223	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	178	1	no	—
228	C₄⋊C₈	—	178	1	no	—
229	C₄⋊C₈	C₂²⋊D₈₉	1	2	yes	—
230	#102	—	356	4	no	—
231	#102	—	178	4	no	—
232	#102	—	178	4	no	—
233	C₄×C₈	—	89	2	no	—
234	C₄×C₈	—	89	2	no	—
235	C₄×C₈	—	356	2	no	—
236	#95	—	178	1	no	—
237	C₂⁴	—	356	1	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	8	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	1	no	—
246	C₂×D₄	—	356	2	no	—
247	C₂×D₄	—	356	1	no	—
248	C₂×D₄	—	356	1	no	—
249	C₂×D₄	—	356	1	no	—
250	C₂×D₄	—	178	2	no	—
251	C₂×D₄	—	178	2	no	—
252	C₂×D₄	—	178	2	no	—
253	C₂×D₄	—	712	2	no	—
254	C₂²×C₄	—	4	1	no	—
255	C₂²×C₄	—	178	2	no	—
256	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
257	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
258	C₂²×C₄	#13083574	1	1	yes	characteristic
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	4	2	no	—
264	C₂×C₈	—	89	2	no	—
265	C₂×C₈	—	89	2	no	—
266	C₂×C₈	—	356	1	no	—
267	C₂×C₈	—	356	2	no	—
268	C₂×C₈	—	178	1	no	—
269	C₂×C₈	—	178	1	no	—
270	C₂×C₈	—	178	1	no	—
271	C₂×C₈	—	178	1	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	89	4	no	—
277	C₄⋊C₄	—	178	2	no	—
278	C₄²	—	356	1	no	—
279	C₄²	—	356	1	no	—
280	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
281	C₄²	D₄×D₈₉	1	1	yes	characteristic
282	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
283	C₄²	#13083573	1	1	yes	characteristic
284	C₂³	—	356	2	no	—
285	C₂³	—	712	1	no	—
286	C₂³	—	712	1	no	—
287	C₂³	—	712	2	no	—
288	C₂³	—	712	2	no	—
289	C₂³	#15667985	1	1	yes	characteristic
290	Q₈	—	8	2	no	—
291	Q₈	—	8	2	no	—
292	D₄	—	356	4	no	—
293	D₄	—	356	2	no	—
294	D₄	—	356	2	no	—
295	D₄	—	712	2	no	—
296	D₄	—	712	2	no	—
297	D₄	—	712	1	no	—
298	D₄	—	712	1	no	—
299	D₄	—	712	1	no	—
300	D₄	—	712	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	8	2	no	—
304	C₂×C₄	—	356	2	no	—
305	C₂×C₄	—	356	2	no	—
306	C₂×C₄	—	178	4	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	#15667998	1	1	yes	characteristic
312	C₂×C₄	#15668008	1	1	yes	characteristic
313	C₂×C₄	SD₁₆×D₈₉	1	2	yes	—
314	C₂×C₄	#15668033	1	1	yes	characteristic
315	C₂×C₄	#15668035	1	2	yes	—
316	C₂×C₄	#15668035	1	1	yes	characteristic
317	C₂×C₄	#15668039	1	1	yes	characteristic
318	C₂×C₄	#15668044	1	1	yes	characteristic
319	C₈	—	4	2	no	—
320	C₈	—	8	1	no	—
321	C₈	—	178	2	no	—
322	C₈	—	178	2	no	—
323	C₈	—	712	1	no	—
324	C₂²	—	1424	2	no	—
325	C₂²	—	1424	2	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	#19151816	1	1	yes	characteristic
331	C₂²	#19151823	1	1	yes	characteristic
332	C₂²	#19151830	1	2	yes	—
333	C₂²	#19151867	1	2	yes	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	356	4	no	—
338	C₄	#19151593	1	1	yes	characteristic
339	C₄	#19151598	1	1	yes	characteristic
340	C₄	#19151683	1	2	yes	—
341	C₄	#19151815	1	1	yes	characteristic
342	C₄	#19151822	1	1	yes	characteristic
343	C₄	#19151969	1	1	yes	characteristic
344	C₄	#19151983	1	1	yes	characteristic
345	C₂	—	1424	2	no	—
346	C₂	#26019664	1	2	yes	—
347	C₂	#26019671	1	2	yes	—
348	C₂	#26021103	1	1	yes	characteristic
349	C₂	#26021159	1	1	yes	characteristic
350	C₂	#26021181	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667985
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#333×C₈₉	C₂
socle	C₂³×C₈₉	#121
prime radical	C₂×#15668009	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#333	D₈₉
hypocenter	C₈₉	#9735
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#333

Complements

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

N	H ≅ G/N	type
2. #26021201	C₂	semidirect
4. #26020117	C₂	semidirect
5. C₂×#19151348	C₂	semidirect
7. C₂×#333×C₈₉	C₂	semidirect
10. #19151946	C₄	semidirect
181. C₂×#333	D₈₉	semidirect
183. C₈₉	#9735	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
#26021201	2
#26021178	2
#26020117	2
C₂×#19151348	2
C₂×#19151341	2
C₂×#333×C₈₉	2
#9735	89