Subgroups — #36464337

Order 22784.

430 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×D₄×D₈₉	C₂	1	1	yes	maximal, characteristic
2	#26022050	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022049	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018208	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊#13083549	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018206	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×D₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₈₉	—	2	1	no	—
9	C₄×D₄×D₈₉	C₄	1	1	yes	characteristic
10	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
11	D₄×#10623494	C₄	1	1	yes	characteristic
12	#19151663	C₄	1	1	yes	characteristic
13	C₈×C₂²⋊D₈₉	—	2	1	no	—
14	#19151661	—	2	1	no	—
15	C₄⋊C₈×D₈₉	C₄	1	1	yes	characteristic
16	C₂²⋊C₈×D₈₉	—	2	1	no	—
17	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
18	C₄⋊#13083549	C₂²	1	1	yes	characteristic, complemented
19	#19151416	—	2	1	no	—
20	#19151377	—	2	1	no	—
21	#19151377	—	2	1	no	—
22	C₄×#13083549	C₂²	1	1	yes	characteristic, complemented
23	C₁₆×Dic₈₉	—	2	1	no	—
24	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
25	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
26	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
27	C₂×D₄×D₈₉	C₈	1	1	yes	characteristic
28	C₂²×C₄×D₈₉	—	2	1	no	—
29	C₂²×#10623494	—	2	1	no	—
30	C₂×C₈×D₈₉	—	2	1	no	—
31	C₂×C₈×D₈₉	—	2	1	no	—
32	C₂×C₈×D₈₉	—	2	1	no	—
33	C₂×C₈×D₈₉	—	4	2	no	—
34	C₂×C₈×D₈₉	—	4	2	no	—
35	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
36	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
37	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
38	D₄×Dic₈₉	C₈	1	1	yes	characteristic
39	#15667974	C₈	1	1	yes	characteristic
40	C₄×C₂²⋊D₈₉	—	2	1	no	—
41	#15667972	—	2	1	no	—
42	C₄⋊C₄×D₈₉	C₈	1	1	yes	characteristic
43	C₂²⋊C₄×D₈₉	—	2	1	no	—
44	C₄²×D₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂×#13083549	—	2	1	no	—
46	C₂×#13083549	—	2	1	no	—
47	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
48	#15667941	—	2	1	no	—
49	C₂²⋊#10623494	—	2	1	no	—
50	#15667924	—	2	1	no	—
51	#15667924	—	2	1	no	—
52	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
53	C₈×Dic₈₉	D₄	1	1	yes	characteristic
54	C₈×Dic₈₉	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	—	2	1	no	—
56	C₄×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₂×C₁₆×C₈₉	—	2	1	no	—
58	C₂×C₁₆×C₈₉	—	2	1	no	—
59	C₄⋊C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
60	C₂²⋊C₈×C₈₉	—	2	1	no	—
61	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₂³×D₈₉	—	2	1	no	—
63	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
64	C₂×C₂²⋊D₈₉	—	2	1	no	—
65	D₄×D₈₉	—	2	2	no	—
66	D₄×D₈₉	C₁₆	1	4	yes	complemented
67	C₂²×Dic₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	—	4	2	no	—
72	C₂×C₄×D₈₉	—	4	2	no	—
73	C₂×C₄×D₈₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₄×D₈₉	C₂×C₈	1	1	yes	characteristic
75	C₂×C₄×D₈₉	M₁₆	1	1	yes	characteristic
76	C₂×#10623494	—	2	1	no	—
77	C₂×#10623494	—	4	2	no	—
78	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
80	C₈×D₈₉	—	2	1	no	—
81	C₈×D₈₉	—	2	1	no	—
82	C₈×D₈₉	—	4	1	no	—
83	C₈×D₈₉	—	4	2	no	—
84	C₈×D₈₉	—	4	2	no	—
85	C₈×D₈₉	—	4	1	no	—
86	C₈×D₈₉	—	4	1	no	—
87	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
88	#13083553	—	2	1	no	—
89	C₂²⋊Dic₈₉	—	2	1	no	—
90	#13083551	—	2	1	no	—
91	#13083551	—	2	1	no	—
92	C₄×Dic₈₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₄×Dic₈₉	C₂×C₈	1	1	yes	characteristic
94	C₄×Dic₈₉	M₁₆	1	1	yes	characteristic
95	#13083549	—	4	1	no	—
96	#13083549	—	4	1	no	—
97	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	—	2	1	no	—
99	C₂×C₈×C₈₉	—	2	1	no	—
100	C₂×C₈×C₈₉	—	4	1	no	—
101	C₂×C₈×C₈₉	—	4	1	no	—
102	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₄⋊C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
105	C₂²⋊C₄×C₈₉	—	2	1	no	—
106	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
107	C₁₆×C₈₉	—	4	1	no	—
108	C₁₆×C₈₉	—	4	1	no	—
109	C₂³×C₈₉	—	2	1	no	—
110	C₂²×D₈₉	—	2	1	no	—
111	C₂²×D₈₉	—	2	4	no	—
112	C₂²×D₈₉	—	4	2	no	—
113	C₂²×D₈₉	—	4	2	no	—
114	C₂²×D₈₉	C₂²⋊C₈	1	1	yes	characteristic
115	D₄×C₈₉	—	2	1	no	—
116	D₄×C₈₉	C₂×C₁₆	1	2	yes	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	4	1	no	—
119	C₂×C₄×C₈₉	—	4	1	no	—
120	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
123	C₂²⋊D₈₉	—	2	4	no	—
124	C₂×Dic₈₉	—	2	1	no	—
125	C₂×Dic₈₉	—	4	2	no	—
126	C₂×Dic₈₉	C₂²⋊C₈	1	1	yes	characteristic
127	C₂×Dic₈₉	C₂²⋊C₈	1	1	yes	characteristic
128	C₂×Dic₈₉	C₂²⋊C₈	1	1	yes	characteristic
129	D₃₅₆	—	2	1	no	—
130	D₃₅₆	C₂×C₁₆	1	2	yes	—
131	C₄×D₈₉	—	2	1	no	—
132	C₄×D₈₉	—	2	1	no	—
133	C₄×D₈₉	—	4	1	no	—
134	C₄×D₈₉	—	4	2	no	—
135	C₄×D₈₉	—	4	2	no	—
136	C₄×D₈₉	—	4	1	no	—
137	C₄×D₈₉	—	4	1	no	—
138	C₄×D₈₉	C₂×C₁₆	1	2	yes	—
139	C₄×D₈₉	M₃₂	1	2	yes	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	4	1	no	—
142	C₈×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₈×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
144	#10623494	—	2	1	no	—
145	#10623494	—	2	1	no	—
146	#10623494	—	4	1	no	—
147	C₂²×C₈₉	—	2	2	no	—
148	C₂²×C₈₉	—	4	1	no	—
149	C₂²×C₈₉	—	4	1	no	—
150	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
151	C₂×D₈₉	—	2	1	no	—
152	C₂×D₈₉	—	2	1	no	—
153	C₂×D₈₉	—	2	2	no	—
154	C₂×D₈₉	—	4	8	no	—
155	C₂×D₈₉	—	4	1	no	—
156	C₂×D₈₉	—	4	1	no	—
157	C₂×D₈₉	C₂²⋊C₁₆	1	2	yes	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	4	1	no	—
160	C₄×C₈₉	#418	1	1	yes	characteristic
161	C₄×C₈₉	#421	1	1	yes	characteristic
162	C₄×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
163	C₄×C₈₉	C₂×M₃₂	1	1	yes	characteristic
164	Dic₈₉	—	2	1	no	—
165	Dic₈₉	—	2	1	no	—
166	Dic₈₉	—	4	1	no	—
167	Dic₈₉	C₂²⋊C₁₆	1	2	yes	—
168	#11629	—	89	1	no	maximal, Sylow
169	C₂×C₈₉	—	4	2	no	—
170	C₂×C₈₉	#1478	1	1	yes	characteristic
171	C₂×C₈₉	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
172	C₂×C₈₉	#2117	1	1	yes	characteristic
173	D₈₉	—	2	2	no	—
174	D₈₉	—	4	2	no	—
175	C₂×C₈×D₄	—	89	1	no	—
176	#2162	—	89	1	no	—
177	#2162	—	89	1	no	—
178	D₄⋊C₁₆	—	89	1	no	—
179	D₄⋊C₁₆	—	89	1	no	—
180	D₄⋊C₁₆	—	89	1	no	—
181	D₄⋊C₁₆	D₈₉	1	1	yes	characteristic, complemented
182	C₈₉	#11629	1	1	yes	characteristic, complemented, Sylow
183	C₂³×C₈	—	178	1	no	—
184	C₂×C₄×D₄	—	89	1	no	—
185	C₈×D₄	—	89	1	no	—
186	C₈×D₄	—	89	1	no	—
187	C₈×D₄	—	89	1	no	—
188	C₈×D₄	—	178	1	no	—
189	C₈×D₄	—	178	1	no	—
190	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
191	C₂×C₄⋊C₈	—	89	1	no	—
192	C₂×C₂²⋊C₈	—	178	1	no	—
193	C₂×C₄×C₈	—	89	1	no	—
194	C₄⋊C₁₆	—	89	1	no	—
195	C₄⋊C₁₆	—	178	1	no	—
196	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
197	C₂²⋊C₁₆	—	178	1	no	—
198	C₂²⋊C₁₆	—	178	1	no	—
199	C₄×C₁₆	—	89	1	no	—
200	C₄×C₁₆	—	178	1	no	—
201	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
202	C₂²×D₄	—	89	1	no	—
203	C₂³×C₄	—	178	1	no	—
204	C₂²×C₈	—	2	1	no	—
205	C₂²×C₈	—	89	1	no	—
206	C₂²×C₈	—	89	1	no	—
207	C₂²×C₈	—	356	2	no	—
208	C₂²×C₈	—	356	2	no	—
209	C₂²×C₈	—	178	1	no	—
210	C₂²×C₈	—	178	1	no	—
211	C₂²×C₈	—	178	1	no	—
212	C₂²×C₈	—	178	1	no	—
213	C₄×D₄	—	89	1	no	—
214	C₄×D₄	—	89	1	no	—
215	C₄×D₄	—	89	1	no	—
216	C₄×D₄	—	178	1	no	—
217	C₄×D₄	—	178	1	no	—
218	C₄×D₄	C₄×D₈₉	1	1	yes	characteristic
219	C₂×C₄⋊C₄	—	89	1	no	—
220	C₂×C₂²⋊C₄	—	178	1	no	—
221	C₂×C₄²	—	89	1	no	—
222	C₂×C₁₆	—	2	1	no	—
223	C₂×C₁₆	—	2	1	no	—
224	C₂×C₁₆	—	178	1	no	—
225	C₂×C₁₆	—	178	1	no	—
226	C₄⋊C₈	—	89	1	no	—
227	C₄⋊C₈	—	178	1	no	—
228	C₄⋊C₈	C₄×D₈₉	1	1	yes	characteristic
229	C₂²⋊C₈	—	2	1	no	—
230	C₂²⋊C₈	—	178	1	no	—
231	C₂²⋊C₈	—	178	1	no	—
232	C₂²⋊C₈	—	178	1	no	—
233	C₄×C₈	—	89	1	no	—
234	C₄×C₈	—	89	1	no	—
235	C₄×C₈	—	89	1	no	—
236	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
237	C₂⁴	—	178	1	no	—
238	C₂×D₄	—	89	4	no	—
239	C₂×D₄	—	89	1	no	—
240	C₂×D₄	—	178	2	no	—
241	C₂×D₄	—	178	1	no	—
242	C₂×D₄	C₈×D₈₉	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	89	1	no	—
245	C₂²×C₄	—	89	1	no	—
246	C₂²×C₄	—	89	1	no	—
247	C₂²×C₄	—	356	2	no	—
248	C₂²×C₄	—	356	2	no	—
249	C₂²×C₄	—	178	1	no	—
250	C₂²×C₄	—	178	1	no	—
251	C₂²×C₄	—	178	1	no	—
252	C₂²×C₄	—	178	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	89	1	no	—
257	C₂×C₈	—	89	1	no	—
258	C₂×C₈	—	356	1	no	—
259	C₂×C₈	—	356	2	no	—
260	C₂×C₈	—	356	2	no	—
261	C₂×C₈	—	356	2	no	—
262	C₂×C₈	—	356	1	no	—
263	C₂×C₈	—	356	1	no	—
264	C₂×C₈	—	178	1	no	—
265	C₂×C₈	—	178	1	no	—
266	C₂×C₈	—	178	1	no	—
267	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
268	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
269	C₄⋊C₄	—	89	1	no	—
270	C₄⋊C₄	—	178	1	no	—
271	C₄⋊C₄	C₈×D₈₉	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	178	1	no	—
274	C₂²⋊C₄	—	178	1	no	—
275	C₂²⋊C₄	—	178	1	no	—
276	C₄²	—	89	1	no	—
277	C₄²	—	89	1	no	—
278	C₄²	—	89	1	no	—
279	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
280	C₁₆	—	4	1	no	—
281	C₁₆	—	4	1	no	—
282	C₁₆	—	356	1	no	—
283	C₁₆	—	356	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	89	1	no	—
286	C₂³	—	356	2	no	—
287	C₂³	—	356	2	no	—
288	C₂³	—	178	4	no	—
289	C₂³	—	178	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	89	2	no	—
292	D₄	—	178	4	no	—
293	D₄	—	178	1	no	—
294	D₄	C₁₆×D₈₉	1	2	yes	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	89	1	no	—
299	C₂×C₄	—	89	2	no	—
300	C₂×C₄	—	89	2	no	—
301	C₂×C₄	—	89	1	no	—
302	C₂×C₄	—	89	1	no	—
303	C₂×C₄	—	356	1	no	—
304	C₂×C₄	—	356	2	no	—
305	C₂×C₄	—	356	2	no	—
306	C₂×C₄	—	356	2	no	—
307	C₂×C₄	—	356	1	no	—
308	C₂×C₄	—	356	1	no	—
309	C₂×C₄	—	178	1	no	—
310	C₂×C₄	—	178	1	no	—
311	C₂×C₄	—	178	1	no	—
312	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
313	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
314	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	356	1	no	—
318	C₈	—	178	1	no	—
319	C₈	—	178	1	no	—
320	C₈	#15668036	1	1	yes	characteristic
321	C₈	#15668039	1	1	yes	characteristic
322	C₂²	—	2	2	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	89	2	no	—
326	C₂²	—	356	8	no	—
327	C₂²	—	356	1	no	—
328	C₂²	—	356	1	no	—
329	C₂²	—	178	2	no	—
330	C₂²	—	178	1	no	—
331	C₂²	—	178	1	no	—
332	C₂²	C₂²⋊C₈×D₈₉	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	89	2	no	—
336	C₄	—	356	1	no	—
337	C₄	—	178	1	no	—
338	C₄	—	178	1	no	—
339	C₄	#19151601	1	1	yes	characteristic
340	C₄	#19151613	1	1	yes	characteristic
341	C₄	C₂×C₁₆×D₈₉	1	1	yes	characteristic
342	C₄	M₃₂×D₈₉	1	1	yes	characteristic
343	C₂	—	4	2	no	—
344	C₂	—	356	2	no	—
345	C₂	—	178	2	no	—
346	C₂	#26018641	1	1	yes	characteristic
347	C₂	C₂²⋊C₁₆×D₈₉	1	1	yes	characteristic
348	C₂	#26021878	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₈₉
commutator	C₄×C₈₉	C₂²×C₁₆
Frattini	C₄×C₈	C₂²×D₈₉
Fitting	C₈₉×D₄⋊C₁₆	C₂
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂×D₄×D₈₉	C₈
cosocle	C₄×C₈×C₈₉	C₂³
hypercenter	D₄⋊C₁₆	D₈₉
hypocenter	C₈₉	#11629
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ D₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
2. #26022050	C₂	semidirect
3. #26022049	C₂	semidirect
4. #26018208	C₂	semidirect
5. D₄⋊#13083549	C₂	semidirect
6. #26018206	C₂	semidirect
7. C₈₉×D₄⋊C₁₆	C₂	semidirect
18. C₄⋊#13083549	C₂²	semidirect
22. C₄×#13083549	C₂²	semidirect
25. C₄⋊C₁₆×C₈₉	C₂²	semidirect
26. C₄×C₁₆×C₈₉	C₂²	semidirect
66. D₄×D₈₉	C₁₆	semidirect
181. D₄⋊C₁₆	D₈₉	semidirect
182. C₈₉	#11629	semidirect
196. C₄⋊C₁₆	C₂×D₈₉	semidirect
201. C₄×C₁₆	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈×D₄×D₈₉	2
#26022050	2
#26022049	2
#26018208	2
D₄⋊#13083549	2
#26018206	2
C₈₉×D₄⋊C₁₆	2
#11629	89