Subgroups — #36524976

Order 22784.

666 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030812	C₂	1	1	yes	maximal, characteristic
2	#26028965	C₂	1	1	yes	maximal, characteristic
3	#26019152	C₂	1	4	yes	maximal, complemented
4	#26019062	C₂	1	4	yes	maximal, complemented
5	#26019010	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018707	C₂	1	1	yes	maximal, characteristic, complemented
7	#26018702	C₂	1	1	yes	maximal, characteristic
8	C₂×#19151320	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2960	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668018	—	2	1	no	—
11	#19152148	C₂²	1	1	yes	characteristic
12	#19151994	C₄	1	1	yes	characteristic
13	#19151972	C₄	1	1	yes	characteristic
14	#19151948	C₄	1	1	yes	characteristic
15	#19151946	C₄	1	1	yes	characteristic
16	#19151829	C₄	1	4	yes	—
17	C₄×#13083561	C₂²	1	4	yes	—
18	#19151677	—	2	1	no	—
19	#19151677	C₂²	1	2	yes	—
20	C₈×D₃₅₆	—	2	1	no	—
21	C₈×D₃₅₆	C₂²	1	2	yes	—
22	#19151622	C₂²	1	1	yes	characteristic
23	#19151564	—	2	1	no	—
24	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
25	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
26	#19151350	C₂²	1	4	yes	—
27	Q₈⋊#10623494	C₂²	1	4	yes	—
28	#19151325	C₂²	1	4	yes	—
29	#19151320	C₂²	1	2	yes	—
30	#19151320	C₂²	1	2	yes	—
31	#533×C₈₉	C₂²	1	1	yes	characteristic
32	#446×C₈₉	—	2	1	no	—
33	#446×C₈₉	C₂²	1	2	yes	—
34	C₈×Q₈×C₈₉	—	2	1	no	—
35	C₈×Q₈×C₈₉	C₂²	1	2	yes	—
36	#424×C₈₉	—	2	1	no	—
37	#423×C₈₉	C₂²	1	1	yes	characteristic
38	#405×C₈₉	—	2	1	no	—
39	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
40	C₂×#13083574	—	2	1	no	—
41	C₂×#13083561	C₂×C₄	1	4	yes	—
42	#15668018	—	4	4	no	—
43	C₂×#13083558	—	2	1	no	—
44	C₂×#13083557	—	2	2	no	—
45	C₂²×#10623494	—	2	1	no	—
46	C₂²×#10623494	C₂×C₄	1	2	yes	—
47	C₂×C₈×D₈₉	—	2	2	no	—
48	#15668012	C₂×C₄	1	1	yes	characteristic
49	#15668009	C₂×C₄	1	1	yes	characteristic
50	#15668000	—	2	1	no	—
51	#15667986	—	2	1	no	—
52	#15667986	C₂×C₄	1	2	yes	—
53	C₄×D₃₅₆	—	2	1	no	—
54	C₄×D₃₅₆	C₂³	1	2	yes	—
55	#15667943	C₂×C₄	1	4	yes	—
56	C₄⋊#10623494	—	2	1	no	—
57	C₄⋊#10623494	C₂³	1	2	yes	—
58	#15667936	C₂×C₄	1	4	yes	—
59	#15667932	C₂×C₄	1	4	yes	—
60	#15667924	—	2	2	no	—
61	#15667924	—	2	2	no	—
62	C₄×#10623494	—	2	1	no	—
63	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
64	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
65	C₄×#10623494	C₂³	1	2	yes	—
66	C₄×#10623494	C₂³	1	2	yes	—
67	C₂×M₁₆×C₈₉	—	2	1	no	—
68	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
69	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
70	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
71	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
72	#125×C₈₉	—	2	1	no	—
73	C₂²⋊Q₈×C₈₉	—	2	1	no	—
74	C₂²⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
75	C₄×Q₈×C₈₉	—	2	1	no	—
76	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
77	#117×C₈₉	—	2	1	no	—
78	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₈₉	—	2	2	no	—
80	C₄⋊C₈×C₈₉	—	2	2	no	—
81	C₄⋊C₈×C₈₉	—	2	1	no	—
82	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
83	#97×C₈₉	—	2	2	no	—
84	C₄×C₈×C₈₉	—	2	2	no	—
85	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
86	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
87	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
89	#13083574	—	4	2	no	—
90	#13083574	—	4	2	no	—
91	C₂×Dic₁₇₈	—	2	1	no	—
92	C₂×D₃₅₆	—	2	1	no	—
93	C₂×D₃₅₆	—	2	1	no	—
94	C₂×D₃₅₆	C₂²×C₄	1	2	yes	—
95	C₂×C₂²⋊D₈₉	—	2	2	no	—
96	C₂×C₄×D₈₉	—	2	2	no	—
97	#13083561	—	2	8	no	—
98	#13083558	—	4	2	no	—
99	#13083557	—	4	4	no	—
100	C₂×#10623494	—	2	4	no	—
101	C₂×#10623494	—	2	2	no	—
102	C₂×#10623494	—	2	2	no	—
103	C₂×#10623494	—	4	2	no	—
104	C₂×#10623494	C₂²×C₄	1	2	yes	—
105	C₂×#10623494	C₂²×C₄	1	2	yes	—
106	C₈×D₈₉	—	4	4	no	—
107	C₄⋊Dic₈₉	—	2	1	no	—
108	C₄⋊Dic₈₉	C₂²×C₄	1	2	yes	—
109	#13083551	—	2	2	no	—
110	#13083551	—	2	2	no	—
111	C₂×Q₈×C₈₉	—	2	1	no	—
112	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
113	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
116	M₁₆×C₈₉	—	4	2	no	—
117	C₂×C₈×C₈₉	—	2	2	no	—
118	C₂×C₈×C₈₉	—	2	2	no	—
119	C₂×C₈×C₈₉	—	2	2	no	—
120	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₄⋊C₄×C₈₉	—	2	1	no	—
125	C₄⋊C₄×C₈₉	—	2	1	no	—
126	C₄⋊C₄×C₈₉	—	2	1	no	—
127	C₄⋊C₄×C₈₉	—	2	2	no	—
128	C₄⋊C₄×C₈₉	—	2	2	no	—
129	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
130	C₂²⋊C₄×C₈₉	—	2	2	no	—
131	C₄²×C₈₉	—	2	2	no	—
132	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₈₉	#41	1	1	yes	characteristic
135	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
136	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₈₉	—	2	2	no	—
138	Q₈×C₈₉	—	2	4	no	—
139	Q₈×C₈₉	—	4	2	no	—
140	C₂×C₄×C₈₉	—	2	2	no	—
141	C₂×C₄×C₈₉	—	2	2	no	—
142	C₂×C₄×C₈₉	—	2	2	no	—
143	C₂×C₄×C₈₉	—	2	2	no	—
144	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₈₉	#137	1	2	yes	—
149	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
152	C₂²⋊D₈₉	—	4	4	no	—
153	C₂×Dic₈₉	—	2	2	no	—
154	D₃₅₆	—	2	4	no	—
155	D₃₅₆	—	4	2	no	—
156	D₃₅₆	—	4	2	no	—
157	C₄×D₈₉	—	4	2	no	—
158	C₄×D₈₉	—	4	2	no	—
159	Dic₁₇₈	—	4	2	no	—
160	C₈×C₈₉	—	2	1	no	—
161	C₈×C₈₉	—	2	1	no	—
162	C₈×C₈₉	—	2	1	no	—
163	C₈×C₈₉	—	2	1	no	—
164	C₈×C₈₉	—	4	2	no	—
165	#10623494	—	2	2	no	—
166	#10623494	—	4	2	no	—
167	#10623494	—	4	2	no	—
168	#10623494	#131	1	4	yes	complemented
169	C₂²×C₈₉	—	2	1	no	—
170	C₂²×C₈₉	—	2	1	no	—
171	C₂²×C₈₉	#440	1	2	yes	—
172	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂×D₈₉	—	4	2	no	—
174	C₂×D₈₉	—	4	2	no	—
175	C₂×D₈₉	—	4	2	no	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	—	4	2	no	—
179	C₄×C₈₉	—	4	2	no	—
180	C₄×C₈₉	#440	1	2	yes	—
181	C₄×C₈₉	#444	1	2	yes	—
182	C₄×C₈₉	C₂×#131	1	2	yes	—
183	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
185	Dic₈₉	—	4	2	no	—
186	#17960	—	89	1	no	maximal, Sylow
187	C₂×C₈₉	—	2	2	no	—
188	C₂×C₈₉	#2931	1	1	yes	characteristic
189	C₂×C₈₉	C₂×#440	1	1	yes	characteristic
190	C₂×C₈₉	C₂×#444	1	1	yes	characteristic
191	D₈₉	—	8	2	no	—
192	#3167	—	89	1	no	—
193	#2960	D₈₉	1	1	yes	characteristic, complemented
194	#2931	—	89	1	no	—
195	#1592	—	89	4	no	—
196	SD₁₆⋊C₈	—	89	4	no	—
197	#1567	—	89	1	no	—
198	#1493	—	89	1	no	—
199	#1492	—	89	1	no	—
200	C₂×#322	—	89	1	no	—
201	C₈₉	#17960	1	1	yes	characteristic, complemented, Sylow
202	C₂×#131	—	178	1	no	—
203	#533	C₂×D₈₉	1	1	yes	characteristic
204	#532	—	89	1	no	—
205	#498	—	89	1	no	—
206	#492	—	89	1	no	—
207	#487	—	89	1	no	—
208	#486	—	89	1	no	—
209	#465	—	89	4	no	—
210	#446	—	2	1	no	—
211	#446	C₂×D₈₉	1	2	yes	—
212	C₈×Q₈	—	2	1	no	—
213	C₈×Q₈	C₂×D₈₉	1	2	yes	—
214	C₄×SD₁₆	—	89	4	no	—
215	#436	—	89	2	no	—
216	#436	—	178	1	no	—
217	C₈×D₄	—	89	2	no	—
218	C₈×D₄	—	178	1	no	—
219	#424	—	2	1	no	—
220	#423	—	89	1	no	—
221	#423	C₂×D₈₉	1	1	yes	characteristic
222	#408	—	178	1	no	—
223	#405	—	2	1	no	—
224	C₂×C₄×C₈	—	89	1	no	—
225	C₂×C₄×C₈	—	89	1	no	—
226	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
227	#335	—	89	4	no	—
228	Q₈⋊C₈	—	89	4	no	—
229	D₄⋊C₈	—	89	4	no	—
230	#322	—	89	2	no	—
231	#322	—	89	2	no	—
232	C₂×#41	—	178	1	no	—
233	C₂×SD₁₆	—	89	4	no	—
234	#131	—	356	4	no	—
235	C₂×M₁₆	—	2	1	no	—
236	C₂×M₁₆	—	178	1	no	—
237	C₂×M₁₆	—	178	2	no	—
238	C₂²×C₈	—	89	2	no	—
239	C₂²×C₈	—	178	1	no	—
240	C₂²×C₈	—	178	2	no	—
241	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
242	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
243	C₄⋊Q₈	—	89	1	no	—
244	C₄⋊Q₈	C₄×D₈₉	1	1	yes	characteristic
245	C₄⋊Q₈	C₄×D₈₉	1	1	yes	characteristic
246	#127	—	89	1	no	—
247	#125	—	2	1	no	—
248	#124	—	178	1	no	—
249	C₂²⋊Q₈	—	2	1	no	—
250	C₂²⋊Q₈	C₄×D₈₉	1	2	yes	—
251	#121	—	89	2	no	—
252	#121	—	178	1	no	—
253	C₄×Q₈	—	2	1	no	—
254	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
255	C₄×D₄	—	89	2	no	—
256	C₄×D₄	—	178	1	no	—
257	#117	—	2	1	no	—
258	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
259	#106	—	89	4	no	—
260	C₄⋊C₈	—	2	2	no	—
261	C₄⋊C₈	—	2	2	no	—
262	C₄⋊C₈	—	2	1	no	—
263	C₄⋊C₈	—	89	2	no	—
264	C₄⋊C₈	—	178	1	no	—
265	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
266	#103	—	89	4	no	—
267	#102	—	89	4	no	—
268	C₂²⋊C₈	—	178	2	no	—
269	C₂²⋊C₈	—	178	2	no	—
270	#97	—	2	2	no	—
271	C₄×C₈	—	2	2	no	—
272	C₄×C₈	—	89	1	no	—
273	C₄×C₈	—	89	1	no	—
274	C₄×C₈	—	89	2	no	—
275	C₄×C₈	—	89	2	no	—
276	C₄×C₈	—	178	1	no	—
277	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
278	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
280	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
281	#41	—	356	2	no	—
282	#41	—	356	2	no	—
283	C₂×Q₈	—	2	1	no	—
284	C₂×Q₈	—	178	1	no	—
285	C₂×Q₈	C₂×C₄×D₈₉	1	2	yes	—
286	C₂×D₄	—	89	2	no	—
287	C₂×D₄	—	178	2	no	—
288	C₂×D₄	—	178	1	no	—
289	C₂×D₄	—	178	1	no	—
290	C₂²×C₄	—	178	2	no	—
291	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
292	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
293	C₂²×C₄	#13083574	1	1	yes	characteristic
294	SD₁₆	—	178	8	no	—
295	M₁₆	—	4	2	no	—
296	M₁₆	—	356	4	no	—
297	M₁₆	—	356	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	2	2	no	—
300	C₂×C₈	—	2	2	no	—
301	C₂×C₈	—	89	2	no	—
302	C₂×C₈	—	89	2	no	—
303	C₂×C₈	—	356	4	no	—
304	C₂×C₈	—	356	2	no	—
305	C₂×C₈	—	178	4	no	—
306	C₂×C₈	—	178	2	no	—
307	C₂×C₈	—	178	2	no	—
308	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
309	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
310	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
311	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	89	2	no	—
318	C₄⋊C₄	—	178	1	no	—
319	C₄⋊C₄	C₂×C₄×D₈₉	1	2	yes	—
320	C₂²⋊C₄	—	2	2	no	—
321	C₂²⋊C₄	—	178	2	no	—
322	C₂²⋊C₄	—	178	2	no	—
323	C₄²	—	2	2	no	—
324	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
325	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
326	C₄²	#13083574	1	1	yes	characteristic
327	C₄²	C₂³×D₈₉	1	1	yes	characteristic
328	C₂³	—	178	2	no	—
329	C₂³	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
330	Q₈	—	2	4	no	—
331	Q₈	—	4	2	no	—
332	Q₈	—	356	2	no	—
333	D₄	—	356	4	no	—
334	D₄	—	356	2	no	—
335	D₄	—	356	2	no	—
336	D₄	—	178	4	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	356	2	no	—
342	C₂×C₄	—	356	2	no	—
343	C₂×C₄	—	178	2	no	—
344	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
345	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
346	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
347	C₂×C₄	#15668037	1	2	yes	—
348	C₂×C₄	#15668043	1	2	yes	—
349	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
350	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
351	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
352	C₈	—	2	1	no	—
353	C₈	—	2	1	no	—
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	4	2	no	—
357	C₈	—	89	4	no	—
358	C₈	—	356	2	no	—
359	C₈	—	356	2	no	—
360	C₈	—	178	2	no	—
361	C₂²	—	2	1	no	—
362	C₂²	—	2	1	no	—
363	C₂²	—	356	2	no	—
364	C₂²	—	356	2	no	—
365	C₂²	—	356	2	no	—
366	C₂²	#19151696	1	2	yes	—
367	C₂²	C₂×C₄×C₂²⋊D₈₉	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	—	356	2	no	—
373	C₄	#19151696	1	2	yes	—
374	C₄	#19151718	1	2	yes	—
375	C₄	#131×D₈₉	1	2	yes	—
376	C₄	C₂×#15668037	1	1	yes	characteristic
377	C₄	C₂×#15668043	1	1	yes	characteristic
378	C₂	—	2	2	no	—
379	C₂	—	712	2	no	—
380	C₂	#26028962	1	1	yes	characteristic
381	C₂	C₂×#19151696	1	1	yes	characteristic
382	C₂	C₂×#19151718	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#2960	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	#19152148	C₂²
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#2960	D₈₉
hypocenter	C₈₉	#17960
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #2960

Complements

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

N	H ≅ G/N	type
3. #26019152	C₂	semidirect
4. #26019062	C₂	semidirect
5. #26019010	C₂	semidirect
6. #26018707	C₂	semidirect
8. C₂×#19151320	C₂	semidirect
9. C₈₉×#2960	C₂	semidirect
168. #10623494	#131	semidirect
193. #2960	D₈₉	semidirect
201. C₈₉	#17960	semidirect

Maximal subgroups

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

subgroup	index
#26030812	2
#26028965	2
#26019152	2
#26019062	2
#26019010	2
#26018707	2
#26018702	2
C₂×#19151320	2
C₈₉×#2960	2
#17960	89