Subgroups — #36708043

Order 22784.

584 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028948	C₂	1	1	yes	maximal, characteristic
2	#26022079	C₂	1	2	yes	maximal, complemented
3	#26022076	C₂	1	2	yes	maximal, complemented
4	C₁₆×D₃₅₆	C₂	1	2	yes	maximal, complemented
5	#26022071	C₂	1	2	yes	maximal, complemented
6	#26022059	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021864	C₂	1	2	yes	maximal, complemented
8	C₂²⋊C₁₆×D₈₉	C₂	1	2	yes	maximal, complemented
9	C₈₉×#2164	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668018	C₂²	1	1	yes	characteristic
11	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
12	#19152110	C₄	1	1	yes	characteristic
13	C₂×#15667951	C₂²	1	2	yes	complemented
14	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
15	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
16	C₂×C₁₆×D₈₉	C₂²	1	2	yes	complemented
17	#19151671	C₄	1	2	yes	—
18	#19151671	C₄	1	2	yes	—
19	C₈×D₃₅₆	C₂²	1	2	yes	—
20	C₈×D₃₅₆	C₂²	1	2	yes	—
21	#19151625	C₄	1	1	yes	characteristic
22	#19151566	C₄	1	1	yes	characteristic
23	#19151561	C₄	1	1	yes	characteristic
24	C₄⋊#13083549	C₂²	1	2	yes	complemented
25	C₂²⋊#13083549	C₂²	1	2	yes	—
26	#19151377	C₂²	1	2	yes	—
27	#19151377	C₂²	1	2	yes	—
28	#19151377	C₂²	1	2	yes	complemented
29	#19151377	C₂²	1	2	yes	complemented
30	#19151374	C₂²	1	1	yes	characteristic, complemented
31	C₄×#13083549	C₂²	1	1	yes	characteristic, complemented
32	#405×C₈₉	C₂²	1	1	yes	characteristic
33	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	complemented
34	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
35	#346×C₈₉	C₂²	1	1	yes	characteristic, complemented
36	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
37	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
39	#15668018	—	4	2	no	—
40	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
41	C₂×#13083557	C₂×C₄	1	2	yes	—
42	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₈₉	—	4	2	no	—
44	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
45	C₂×C₈×D₈₉	C₂³	1	2	yes	—
46	C₂×C₈×D₈₉	C₂³	1	2	yes	—
47	#15667996	C₈	1	2	yes	—
48	#15667992	C₈	1	2	yes	—
49	#15667987	C₈	1	2	yes	—
50	#15667982	C₈	1	2	yes	—
51	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
52	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
53	#15667951	—	4	2	no	—
54	C₂×#13083549	C₂³	1	2	yes	—
55	C₂×#13083549	C₂³	1	2	yes	—
56	C₁₆×D₈₉	—	4	2	no	—
57	C₁₆×D₈₉	—	4	2	no	—
58	C₁₆×D₈₉	—	4	2	no	—
59	C₄⋊#10623494	C₂×C₄	1	2	yes	—
60	C₄⋊#10623494	C₂×C₄	1	2	yes	—
61	#15667924	C₂×C₄	1	2	yes	—
62	#15667924	C₂×C₄	1	2	yes	—
63	#15667924	C₂×C₄	1	2	yes	—
64	#15667924	C₂×C₄	1	2	yes	—
65	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
68	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
69	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
70	#97×C₈₉	C₂×C₄	1	1	yes	characteristic
71	#97×C₈₉	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	C₂³×D₈₉	C₂×C₈	1	1	yes	characteristic
75	#13083574	—	4	1	no	—
76	#13083574	—	4	1	no	—
77	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
78	C₂×D₃₅₆	C₂×C₈	1	2	yes	—
79	C₂×D₃₅₆	C₂×C₈	1	2	yes	—
80	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
81	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
82	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
83	C₂×C₄×D₈₉	—	4	1	no	—
84	C₂×C₄×D₈₉	—	4	1	no	—
85	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
86	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
87	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
88	#13083558	—	4	1	no	—
89	#13083557	—	4	2	no	—
90	C₂×#10623494	—	4	1	no	—
91	C₂×#10623494	C₂²×C₄	1	2	yes	—
92	C₂×#10623494	C₂²×C₄	1	2	yes	—
93	C₈×D₈₉	—	2	2	no	—
94	C₈×D₈₉	—	2	2	no	—
95	C₈×D₈₉	—	2	2	no	—
96	C₈×D₈₉	—	2	2	no	—
97	C₈×D₈₉	—	4	2	no	—
98	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
99	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
100	#13083551	C₂×C₈	1	2	yes	—
101	#13083551	C₂×C₈	1	2	yes	—
102	#13083551	C₂×C₈	1	2	yes	—
103	#13083551	C₂×C₈	1	2	yes	—
104	#13083549	—	2	2	no	—
105	#13083549	—	2	2	no	—
106	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
107	M₁₆×C₈₉	—	4	1	no	—
108	C₂×C₈×C₈₉	—	2	2	no	—
109	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
114	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
115	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
117	C₁₆×C₈₉	—	2	2	no	—
118	C₁₆×C₈₉	—	2	2	no	—
119	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
120	C₂²×D₈₉	—	2	4	no	—
121	C₂²×D₈₉	—	4	1	no	—
122	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
123	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
130	C₂²⋊D₈₉	—	4	2	no	—
131	C₂×Dic₈₉	—	4	1	no	—
132	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
133	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
134	D₃₅₆	—	2	4	no	—
135	D₃₅₆	—	2	4	no	—
136	D₃₅₆	—	4	1	no	—
137	C₄×D₈₉	—	2	2	no	—
138	C₄×D₈₉	—	2	2	no	—
139	C₄×D₈₉	—	2	2	no	—
140	C₄×D₈₉	—	2	2	no	—
141	C₄×D₈₉	—	4	2	no	—
142	Dic₁₇₈	—	4	1	no	—
143	C₈×C₈₉	—	2	1	no	—
144	C₈×C₈₉	—	2	1	no	—
145	C₈×C₈₉	—	2	1	no	—
146	C₈×C₈₉	H(5,𝔽₂)	1	2	yes	—
147	#10623494	—	2	2	no	—
148	#10623494	—	2	2	no	—
149	C₂²×C₈₉	—	2	2	no	—
150	C₂²×C₈₉	—	2	1	no	—
151	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
152	C₂×D₈₉	—	2	2	no	—
153	C₂×D₈₉	—	2	2	no	—
154	C₂×D₈₉	—	2	2	no	—
155	C₂×D₈₉	—	2	2	no	—
156	C₂×D₈₉	—	2	2	no	—
157	C₂×D₈₉	—	4	4	no	—
158	C₂×D₈₉	#504	1	4	yes	—
159	C₄×C₈₉	—	2	2	no	—
160	C₄×C₈₉	—	2	2	no	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	#518	1	1	yes	characteristic
163	C₄×C₈₉	#568	1	1	yes	characteristic
164	Dic₈₉	—	2	2	no	—
165	Dic₈₉	—	2	2	no	—
166	#33491	—	89	1	no	maximal, Sylow
167	C₂×C₈₉	—	4	1	no	—
168	C₂×C₈₉	#2972	1	1	yes	characteristic
169	C₂×C₈₉	C₂×#504	1	2	yes	—
170	D₈₉	—	2	4	no	—
171	D₈₉	—	4	2	no	—
172	#2929	—	89	1	no	—
173	#2168	—	89	2	no	—
174	#2168	—	89	2	no	—
175	D₄×C₁₆	—	89	2	no	—
176	D₄×C₁₆	—	89	2	no	—
177	#2164	—	89	1	no	—
178	#2164	D₈₉	1	1	yes	characteristic, complemented
179	#2113	—	89	2	no	—
180	C₂×C₂²⋊C₁₆	—	89	2	no	—
181	C₈₉	#33491	1	1	yes	characteristic, complemented, Sylow
182	C₂×#131	—	89	1	no	—
183	C₂³×C₈	—	89	1	no	—
184	#525	—	89	1	no	—
185	C₂×M₃₂	—	89	2	no	—
186	C₂²×C₁₆	—	89	2	no	—
187	C₂²×C₁₆	—	89	2	no	—
188	C₂²×C₁₆	—	89	2	no	—
189	#435	—	89	2	no	—
190	#435	—	89	2	no	—
191	C₈×D₄	—	89	2	no	—
192	C₈×D₄	—	89	2	no	—
193	#424	—	89	1	no	—
194	#408	—	89	1	no	—
195	#407	—	89	1	no	—
196	#405	C₂×D₈₉	1	1	yes	characteristic
197	C₄⋊C₁₆	—	89	2	no	—
198	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	complemented
199	C₂²⋊C₁₆	—	89	2	no	—
200	C₂²⋊C₁₆	—	89	2	no	—
201	C₂²⋊C₁₆	—	89	2	no	—
202	C₂²⋊C₁₆	—	89	2	no	—
203	C₂²⋊C₁₆	—	89	2	no	—
204	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
205	#346	—	89	1	no	—
206	#346	C₂×D₈₉	1	1	yes	characteristic, complemented
207	C₄×C₁₆	—	89	1	no	—
208	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
209	C₂×#41	—	89	1	no	—
210	C₂³×C₄	—	89	1	no	—
211	#131	—	356	2	no	—
212	C₂×M₁₆	—	89	1	no	—
213	C₂×M₁₆	—	89	2	no	—
214	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
215	C₂²×C₈	—	89	1	no	—
216	C₂²×C₈	—	89	2	no	—
217	C₂²×C₈	—	89	2	no	—
218	C₂²×C₈	—	89	2	no	—
219	C₂²×C₈	—	356	2	no	—
220	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
221	#123	—	89	2	no	—
222	C₂²⋊Q₈	—	89	2	no	—
223	#121	—	89	2	no	—
224	#120	—	89	2	no	—
225	C₄×D₄	—	89	2	no	—
226	C₄×D₄	—	89	2	no	—
227	#117	C₄×D₈₉	1	1	yes	characteristic
228	M₃₂	—	356	2	no	—
229	C₂×C₁₆	—	89	2	no	—
230	C₂×C₁₆	—	89	2	no	—
231	C₂×C₁₆	—	356	2	no	—
232	C₂×C₁₆	—	356	2	no	—
233	C₂×C₁₆	—	356	2	no	—
234	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
235	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
236	C₄⋊C₈	—	89	2	no	—
237	C₄⋊C₈	—	89	2	no	—
238	C₂²⋊C₈	—	89	2	no	—
239	C₂²⋊C₈	—	89	2	no	—
240	C₂²⋊C₈	—	89	2	no	—
241	C₂²⋊C₈	—	89	2	no	—
242	#97	C₄×D₈₉	1	1	yes	characteristic
243	#97	C₄×D₈₉	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
246	C₂⁴	—	89	1	no	—
247	#41	—	356	1	no	—
248	#41	—	356	1	no	—
249	C₂×Q₈	—	89	1	no	—
250	C₂×D₄	—	89	2	no	—
251	C₂×D₄	—	89	1	no	—
252	C₂×D₄	—	89	2	no	—
253	C₂×D₄	—	89	2	no	—
254	C₂²×C₄	—	89	1	no	—
255	C₂²×C₄	—	89	2	no	—
256	C₂²×C₄	—	89	2	no	—
257	C₂²×C₄	—	89	2	no	—
258	C₂²×C₄	—	356	1	no	—
259	C₂²×C₄	—	356	1	no	—
260	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
261	M₁₆	—	4	1	no	—
262	M₁₆	—	356	2	no	—
263	M₁₆	—	356	1	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	89	2	no	—
266	C₂×C₈	—	89	2	no	—
267	C₂×C₈	—	356	2	no	—
268	C₂×C₈	—	356	1	no	—
269	C₂×C₈	—	178	2	no	—
270	C₂×C₈	—	178	2	no	—
271	C₂×C₈	—	178	2	no	—
272	C₂×C₈	—	178	2	no	—
273	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
274	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
275	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
276	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
277	C₄⋊C₄	—	89	2	no	—
278	C₄⋊C₄	—	89	2	no	—
279	C₄⋊C₄	C₈×D₈₉	1	2	yes	—
280	C₂²⋊C₄	—	89	2	no	—
281	C₂²⋊C₄	—	89	2	no	—
282	C₂²⋊C₄	—	89	2	no	—
283	C₂²⋊C₄	—	89	2	no	—
284	C₂²⋊C₄	C₈×D₈₉	1	2	yes	—
285	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
286	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
287	C₁₆	—	2	2	no	—
288	C₁₆	—	2	2	no	—
289	C₁₆	—	178	2	no	—
290	C₁₆	—	178	2	no	—
291	C₂³	—	89	2	no	—
292	C₂³	—	89	2	no	—
293	C₂³	—	356	1	no	—
294	C₂³	—	178	4	no	—
295	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
296	Q₈	—	356	1	no	—
297	D₄	—	356	1	no	—
298	D₄	—	356	2	no	—
299	D₄	—	178	4	no	—
300	D₄	—	178	4	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	89	2	no	—
304	C₂×C₄	—	89	2	no	—
305	C₂×C₄	—	356	2	no	—
306	C₂×C₄	—	356	1	no	—
307	C₂×C₄	—	178	2	no	—
308	C₂×C₄	—	178	2	no	—
309	C₂×C₄	—	178	2	no	—
310	C₂×C₄	—	178	2	no	—
311	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
312	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
313	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
314	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	178	2	no	—
319	C₈	—	178	2	no	—
320	C₈	#15668058	1	2	yes	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	89	4	no	—
324	C₂²	—	356	4	no	—
325	C₂²	—	178	2	no	—
326	C₂²	—	178	2	no	—
327	C₂²	—	178	2	no	—
328	C₂²	—	178	2	no	—
329	C₂²	—	178	2	no	—
330	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
331	C₄	—	2	2	no	—
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	178	2	no	—
335	C₄	—	178	2	no	—
336	C₄	#19152072	1	1	yes	characteristic
337	C₄	#19152364	1	1	yes	characteristic
338	C₂	—	4	1	no	—
339	C₂	—	356	2	no	—
340	C₂	—	178	4	no	—
341	C₂	#26029275	1	1	yes	characteristic
342	C₂	#504×D₈₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2164	C₂
prime radical	#19152110	C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2164	D₈₉
hypocenter	C₈₉	#33491
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2164

Complements

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

N	H ≅ G/N	type
2. #26022079	C₂	semidirect
3. #26022076	C₂	semidirect
4. C₁₆×D₃₅₆	C₂	semidirect
5. #26022071	C₂	semidirect
6. #26022059	C₂	semidirect
7. #26021864	C₂	semidirect
8. C₂²⋊C₁₆×D₈₉	C₂	semidirect
9. C₈₉×#2164	C₂	semidirect
13. C₂×#15667951	C₂²	semidirect
16. C₂×C₁₆×D₈₉	C₂²	semidirect
24. C₄⋊#13083549	C₂²	semidirect
28. #19151377	C₂²	semidirect
29. #19151377	C₂²	semidirect
30. #19151374	C₂²	semidirect
31. C₄×#13083549	C₂²	semidirect
33. C₄⋊C₁₆×C₈₉	C₂²	semidirect
35. #346×C₈₉	C₂²	semidirect
36. C₄×C₁₆×C₈₉	C₂²	semidirect
178. #2164	D₈₉	semidirect
181. C₈₉	#33491	semidirect
198. C₄⋊C₁₆	C₂×D₈₉	semidirect
206. #346	C₂×D₈₉	semidirect
208. 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
#26028948	2
#26022079	2
#26022076	2
C₁₆×D₃₅₆	2
#26022071	2
#26022059	2
#26021864	2
C₂²⋊C₁₆×D₈₉	2
C₈₉×#2164	2
#33491	89