Subgroups — #36708048

Order 22784.

564 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028946	C₂	1	1	yes	maximal, characteristic
2	#26022082	C₂	1	2	yes	maximal, complemented
3	#26022081	C₂	1	2	yes	maximal, complemented
4	#26022080	C₂	1	2	yes	maximal, complemented
5	#26022075	C₂	1	2	yes	maximal, complemented
6	#26022056	C₂	1	2	yes	maximal, complemented
7	#26021864	C₂	1	2	yes	maximal, complemented
8	#26021863	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2112	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668018	C₂²	1	1	yes	characteristic
11	#19152109	C₄	1	1	yes	characteristic
12	C₂×#15667952	C₂²	1	2	yes	—
13	C₂×#15667951	C₂²	1	2	yes	complemented
14	C₂×C₁₆×D₈₉	C₂²	1	2	yes	complemented
15	#19151669	C₄	1	2	yes	—
16	#19151669	C₄	1	2	yes	—
17	C₈×C₂²⋊D₈₉	C₂²	1	2	yes	—
18	C₈×C₂²⋊D₈₉	C₂²	1	2	yes	—
19	#19151624	C₄	1	1	yes	characteristic
20	#19151564	C₄	1	1	yes	characteristic
21	#19151562	C₄	1	1	yes	characteristic
22	#19151553	C₂²	1	1	yes	characteristic
23	#19151416	C₂²	1	2	yes	complemented
24	#19151416	C₂²	1	2	yes	complemented
25	C₂²⋊#13083549	C₂²	1	2	yes	—
26	C₂²⋊#13083549	C₂²	1	1	yes	characteristic
27	C₂²⋊#13083549	C₂²	1	1	yes	characteristic
28	#19151377	C₂²	1	2	yes	complemented
29	#19151377	C₂²	1	2	yes	complemented
30	#19151373	C₂²	1	2	yes	complemented
31	C₁₆×Dic₈₉	C₂²	1	2	yes	complemented
32	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
34	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
35	C₂²⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	C₂²⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
38	#15668018	—	4	2	no	—
39	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
40	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
41	C₂×#13083557	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₈₉	C₂³	1	2	yes	—
43	#15667995	C₈	1	2	yes	—
44	#15667991	C₈	1	1	yes	characteristic
45	#15667991	C₈	1	1	yes	characteristic
46	#15667986	C₈	1	1	yes	characteristic
47	#15667986	C₈	1	1	yes	characteristic
48	#15667981	C₈	1	2	yes	—
49	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
50	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
51	#15667971	C₂×C₄	1	1	yes	characteristic
52	#15667952	—	4	2	no	—
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	#15667941	C₂×C₄	1	2	yes	—
58	#15667941	C₂×C₄	1	2	yes	—
59	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
60	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
61	#15667924	C₂×C₄	1	2	yes	—
62	#15667924	C₂×C₄	1	2	yes	—
63	#15667922	C₂×C₄	1	2	yes	—
64	C₈×Dic₈₉	C₂³	1	2	yes	—
65	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₈₉	—	4	2	no	—
67	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
70	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	M₃₂×C₈₉	—	4	2	no	—
74	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
75	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
76	#13083574	—	4	1	no	—
77	#13083574	—	4	1	no	—
78	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
79	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
80	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
81	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
82	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
83	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
84	#13083558	—	4	1	no	—
85	#13083558	—	4	1	no	—
86	#13083557	—	4	2	no	—
87	C₂×#10623494	C₂²×C₄	1	2	yes	—
88	C₂×#10623494	C₂²×C₄	1	2	yes	—
89	C₈×D₈₉	—	2	2	no	—
90	C₈×D₈₉	—	2	2	no	—
91	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
92	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
93	#13083553	C₂×C₈	1	2	yes	—
94	#13083553	C₂×C₈	1	2	yes	—
95	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
96	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
97	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
98	#13083551	C₂×C₈	1	2	yes	—
99	#13083551	C₂×C₈	1	2	yes	—
100	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
101	#13083549	—	2	2	no	—
102	#13083549	—	2	2	no	—
103	C₂⁴×C₈₉	C₂×C₈	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	—	4	1	no	—
105	C₂²×C₄×C₈₉	—	4	1	no	—
106	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
107	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
108	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
109	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
110	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
111	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	—	2	2	no	—
113	C₂×C₈×C₈₉	—	2	2	no	—
114	C₂×C₈×C₈₉	—	2	2	no	—
115	C₂×C₈×C₈₉	—	4	1	no	—
116	C₂×C₈×C₈₉	—	4	1	no	—
117	C₂×C₈×C₈₉	—	4	1	no	—
118	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₁₆×C₈₉	—	2	2	no	—
123	C₁₆×C₈₉	—	2	2	no	—
124	C₂³×C₈₉	—	4	1	no	—
125	C₂³×C₈₉	—	4	1	no	—
126	C₂³×C₈₉	—	4	1	no	—
127	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
128	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
129	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
130	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₈₉	—	2	1	no	—
132	C₂×C₄×C₈₉	—	2	1	no	—
133	C₂×C₄×C₈₉	—	2	1	no	—
134	C₂×C₄×C₈₉	—	2	1	no	—
135	C₂×C₄×C₈₉	—	2	1	no	—
136	C₂×C₄×C₈₉	—	2	1	no	—
137	C₂×C₄×C₈₉	—	4	1	no	—
138	C₂×C₄×C₈₉	—	4	1	no	—
139	C₂×C₄×C₈₉	—	4	1	no	—
140	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
144	C₂²⋊D₈₉	—	4	2	no	—
145	C₂²⋊D₈₉	—	4	2	no	—
146	C₂²⋊D₈₉	—	4	2	no	—
147	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
148	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
149	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
150	D₃₅₆	—	4	1	no	—
151	C₄×D₈₉	—	2	2	no	—
152	C₄×D₈₉	—	2	2	no	—
153	Dic₁₇₈	—	4	1	no	—
154	C₈×C₈₉	—	2	1	no	—
155	C₈×C₈₉	—	2	1	no	—
156	C₈×C₈₉	—	2	1	no	—
157	C₈×C₈₉	H(5,𝔽₂)	1	2	yes	—
158	#10623494	—	2	2	no	—
159	#10623494	—	2	2	no	—
160	C₂²×C₈₉	—	2	1	no	—
161	C₂²×C₈₉	—	2	1	no	—
162	C₂²×C₈₉	—	2	1	no	—
163	C₂²×C₈₉	—	2	1	no	—
164	C₂²×C₈₉	—	2	1	no	—
165	C₂²×C₈₉	—	2	1	no	—
166	C₂²×C₈₉	—	2	1	no	—
167	C₂²×C₈₉	—	2	1	no	—
168	C₂²×C₈₉	—	2	1	no	—
169	C₂²×C₈₉	—	4	4	no	—
170	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
171	C₂×D₈₉	—	2	2	no	—
172	C₂×D₈₉	—	2	2	no	—
173	C₂×D₈₉	—	2	2	no	—
174	C₄×C₈₉	—	2	1	no	—
175	C₄×C₈₉	—	2	1	no	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	#518	1	1	yes	characteristic
178	C₄×C₈₉	#568	1	1	yes	characteristic
179	Dic₈₉	—	2	2	no	—
180	Dic₈₉	—	2	2	no	—
181	Dic₈₉	—	2	2	no	—
182	#33492	—	89	1	no	maximal, Sylow
183	C₂×C₈₉	—	4	1	no	—
184	C₂×C₈₉	—	4	1	no	—
185	C₂×C₈₉	—	4	1	no	—
186	C₂×C₈₉	#2972	1	1	yes	characteristic
187	C₂×C₈₉	#3407	1	1	yes	characteristic
188	C₂×C₈₉	#3407	1	1	yes	characteristic
189	D₈₉	—	4	2	no	—
190	#2929	—	89	1	no	—
191	#2169	—	89	2	no	—
192	#2169	—	89	2	no	—
193	#2168	—	89	2	no	—
194	#2168	—	89	2	no	—
195	#2164	—	89	2	no	—
196	#2113	—	89	2	no	—
197	#2112	—	89	1	no	—
198	#2112	D₈₉	1	1	yes	characteristic, complemented
199	C₈₉	#33492	1	1	yes	characteristic, complemented, Sylow
200	C₂×#131	—	89	1	no	—
201	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
202	#525	—	89	1	no	—
203	C₂×M₃₂	—	89	2	no	—
204	C₂×M₃₂	—	89	2	no	—
205	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
206	C₂²×C₁₆	—	89	2	no	—
207	#435	—	89	2	no	—
208	#435	—	89	2	no	—
209	C₈×D₄	—	89	2	no	—
210	C₈×D₄	—	89	2	no	—
211	#424	—	89	1	no	—
212	#408	—	89	1	no	—
213	#407	—	89	1	no	—
214	#405	—	89	1	no	—
215	C₄⋊C₁₆	—	89	2	no	—
216	C₄⋊C₁₆	—	89	2	no	—
217	C₂²⋊C₁₆	—	89	2	no	—
218	C₂²⋊C₁₆	—	89	1	no	—
219	C₂²⋊C₁₆	—	89	1	no	—
220	C₂²⋊C₁₆	—	89	2	no	—
221	C₂²⋊C₁₆	—	89	2	no	—
222	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
223	C₂²⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
224	C₂²⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
225	#346	—	89	2	no	—
226	C₄×C₁₆	—	89	2	no	—
227	C₂×#41	—	89	1	no	—
228	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
229	#131	—	356	2	no	—
230	C₂×M₁₆	—	89	1	no	—
231	C₂×M₁₆	—	89	1	no	—
232	C₂×M₁₆	—	89	2	no	—
233	C₂²×C₈	—	4	2	no	—
234	C₂²×C₈	—	89	2	no	—
235	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
236	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
237	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
238	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
239	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
240	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
241	#123	—	89	2	no	—
242	C₂²⋊Q₈	—	89	1	no	—
243	C₂²⋊Q₈	—	89	1	no	—
244	#121	—	89	1	no	—
245	#121	—	89	1	no	—
246	#120	—	89	2	no	—
247	C₄×D₄	—	89	2	no	—
248	C₄×D₄	—	89	2	no	—
249	#117	—	89	1	no	—
250	M₃₂	—	4	2	no	—
251	M₃₂	—	356	2	no	—
252	M₃₂	—	356	2	no	—
253	C₂×C₁₆	—	89	2	no	—
254	C₂×C₁₆	—	89	2	no	—
255	C₂×C₁₆	—	356	2	no	—
256	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
257	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
258	C₄⋊C₈	—	89	2	no	—
259	C₄⋊C₈	—	89	2	no	—
260	C₂²⋊C₈	—	89	2	no	—
261	C₂²⋊C₈	—	89	2	no	—
262	C₂²⋊C₈	—	89	2	no	—
263	C₂²⋊C₈	—	89	2	no	—
264	#97	—	89	2	no	—
265	C₄×C₈	—	89	2	no	—
266	C₂⁴	C₈×D₈₉	1	1	yes	characteristic
267	#41	—	356	1	no	—
268	#41	—	356	1	no	—
269	C₂×Q₈	—	89	1	no	—
270	C₂×D₄	—	89	2	no	—
271	C₂×D₄	—	89	2	no	—
272	C₂×D₄	—	89	2	no	—
273	C₂×D₄	—	89	1	no	—
274	C₂²×C₄	—	4	1	no	—
275	C₂²×C₄	—	4	1	no	—
276	C₂²×C₄	—	89	2	no	—
277	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
278	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
279	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
280	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
281	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
282	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
283	M₁₆	—	356	2	no	—
284	M₁₆	—	356	1	no	—
285	M₁₆	—	356	1	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	4	1	no	—
290	C₂×C₈	—	4	1	no	—
291	C₂×C₈	—	4	1	no	—
292	C₂×C₈	—	89	2	no	—
293	C₂×C₈	—	89	2	no	—
294	C₂×C₈	—	178	2	no	—
295	C₂×C₈	—	178	2	no	—
296	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
298	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
299	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
300	C₄⋊C₄	—	89	1	no	—
301	C₄⋊C₄	—	89	1	no	—
302	C₄⋊C₄	—	89	2	no	—
303	C₄⋊C₄	—	89	2	no	—
304	C₂²⋊C₄	—	89	2	no	—
305	C₂²⋊C₄	—	89	2	no	—
306	C₂²⋊C₄	—	89	2	no	—
307	C₂²⋊C₄	—	89	2	no	—
308	C₂²⋊C₄	—	89	2	no	—
309	C₄²	—	89	2	no	—
310	C₁₆	—	2	2	no	—
311	C₁₆	—	2	2	no	—
312	C₁₆	—	178	2	no	—
313	C₁₆	—	178	2	no	—
314	C₂³	—	4	1	no	—
315	C₂³	—	4	1	no	—
316	C₂³	—	4	1	no	—
317	C₂³	—	89	2	no	—
318	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
319	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
320	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
321	Q₈	—	356	1	no	—
322	D₄	—	356	1	no	—
323	D₄	—	356	2	no	—
324	D₄	—	356	2	no	—
325	D₄	—	356	2	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	89	2	no	—
336	C₂×C₄	—	89	2	no	—
337	C₂×C₄	—	89	2	no	—
338	C₂×C₄	—	178	2	no	—
339	C₂×C₄	—	178	2	no	—
340	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
341	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
342	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
343	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	2	1	no	—
347	C₈	—	178	2	no	—
348	C₈	—	178	2	no	—
349	C₈	#15668057	1	2	yes	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	2	1	no	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	2	1	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	4	4	no	—
360	C₂²	—	178	2	no	—
361	C₂²	—	178	2	no	—
362	C₂²	—	178	2	no	—
363	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
364	C₄	—	2	1	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	178	2	no	—
368	C₄	—	178	2	no	—
369	C₄	—	178	2	no	—
370	C₄	#19152071	1	1	yes	characteristic
371	C₄	#19152364	1	1	yes	characteristic
372	C₂	—	4	1	no	—
373	C₂	—	4	1	no	—
374	C₂	—	4	1	no	—
375	C₂	—	356	2	no	—
376	C₂	#26029274	1	1	yes	characteristic
377	C₂	#26033354	1	1	yes	characteristic
378	C₂	#26033354	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2112	C₂
prime radical	#19152109	C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2112	D₈₉
hypocenter	C₈₉	#33492
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2112

Complements

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

N	H ≅ G/N	type
2. #26022082	C₂	semidirect
3. #26022081	C₂	semidirect
4. #26022080	C₂	semidirect
5. #26022075	C₂	semidirect
6. #26022056	C₂	semidirect
7. #26021864	C₂	semidirect
8. #26021863	C₂	semidirect
9. C₈₉×#2112	C₂	semidirect
13. C₂×#15667951	C₂²	semidirect
14. C₂×C₁₆×D₈₉	C₂²	semidirect
23. #19151416	C₂²	semidirect
24. #19151416	C₂²	semidirect
28. #19151377	C₂²	semidirect
29. #19151377	C₂²	semidirect
30. #19151373	C₂²	semidirect
31. C₁₆×Dic₈₉	C₂²	semidirect
198. #2112	D₈₉	semidirect
199. C₈₉	#33492	semidirect

Maximal subgroups

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

subgroup	index
#26028946	2
#26022082	2
#26022081	2
#26022080	2
#26022075	2
#26022056	2
#26021864	2
#26021863	2
C₈₉×#2112	2
#33492	89