Subgroups — #36447482

Order 22784.

542 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020412	C₂	1	1	yes	maximal, characteristic, complemented
2	#26020252	C₂	1	2	yes	maximal
3	C₂×#19151351	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018136	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15667946	—	2	1	no	—
7	C₂×#15667946	—	2	1	no	—
8	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
9	C₂×#15667933	—	2	2	no	—
10	#19151468	C₂²	1	2	yes	—
11	#19151360	—	2	2	no	—
12	#19151351	—	2	2	no	—
13	C₂³×C₈×C₈₉	—	2	1	no	—
14	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
15	C₈×D₄×C₈₉	—	2	2	no	—
16	C₈×D₄×C₈₉	C₄	1	4	yes	complemented
17	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊C₈×C₈₉	—	2	1	no	—
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×#10623494	—	2	2	no	—
21	C₂×C₄⋊Dic₈₉	—	2	2	no	—
22	C₂×C₄⋊Dic₈₉	—	2	2	no	—
23	C₂×C₄⋊Dic₈₉	—	2	2	no	—
24	#15667946	—	4	2	no	—
25	#15667946	—	4	2	no	—
26	C₄⋊#10623494	—	2	2	no	—
27	C₄⋊#10623494	—	2	2	no	—
28	#15667933	—	4	4	no	—
29	C₂²×D₄×C₈₉	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₈₉	—	2	1	no	—
31	C₂²×C₈×C₈₉	—	2	1	no	—
32	C₂²×C₈×C₈₉	—	2	4	no	—
33	C₂²×C₈×C₈₉	—	4	2	no	—
34	C₂²×C₈×C₈₉	—	4	2	no	—
35	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
37	C₄×D₄×C₈₉	—	2	1	no	—
38	C₄×D₄×C₈₉	—	2	1	no	—
39	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
40	C₄×D₄×C₈₉	Q₈	1	2	yes	—
41	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
42	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
43	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₈₉	—	2	1	no	—
45	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
46	C₂²⋊C₈×C₈₉	—	2	4	no	—
47	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
48	C₄×C₈×C₈₉	D₄	1	2	yes	—
49	C₂²×Dic₈₉	—	4	2	no	—
50	C₂×#10623494	—	4	2	no	—
51	C₂×#10623494	—	4	2	no	—
52	C₂×#10623494	—	4	2	no	—
53	C₄⋊Dic₈₉	—	4	2	no	—
54	C₄⋊Dic₈₉	—	4	2	no	—
55	C₄⋊Dic₈₉	—	4	2	no	—
56	C₄⋊Dic₈₉	—	4	2	no	—
57	C₄⋊Dic₈₉	—	8	2	no	—
58	C₂⁴×C₈₉	—	2	1	no	—
59	C₂×D₄×C₈₉	—	2	1	no	—
60	C₂×D₄×C₈₉	—	2	1	no	—
61	C₂×D₄×C₈₉	—	2	1	no	—
62	C₂×D₄×C₈₉	C₄⋊C₄	1	2	yes	—
63	C₂×D₄×C₈₉	D₈	1	2	yes	—
64	C₂×D₄×C₈₉	Q₁₆	1	2	yes	—
65	C₂²×C₄×C₈₉	—	2	1	no	—
66	C₂²×C₄×C₈₉	—	2	2	no	—
67	C₂²×C₄×C₈₉	—	2	2	no	—
68	C₂²×C₄×C₈₉	—	4	1	no	—
69	C₂²×C₄×C₈₉	—	4	1	no	—
70	C₂²×C₄×C₈₉	—	4	1	no	—
71	C₂²×C₄×C₈₉	—	4	1	no	—
72	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
75	C₂×C₈×C₈₉	—	2	1	no	—
76	C₂×C₈×C₈₉	—	2	1	no	—
77	C₂×C₈×C₈₉	—	2	2	no	—
78	C₂×C₈×C₈₉	—	4	1	no	—
79	C₂×C₈×C₈₉	—	4	1	no	—
80	C₂×C₈×C₈₉	—	4	8	no	—
81	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
82	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₈₉	D₈	1	2	yes	—
84	C₂×C₈×C₈₉	SD₁₆	1	2	yes	—
85	C₄⋊C₄×C₈₉	—	2	1	no	—
86	C₄⋊C₄×C₈₉	C₄⋊C₄	1	2	yes	—
87	C₂²⋊C₄×C₈₉	—	2	2	no	—
88	C₂²⋊C₄×C₈₉	—	2	2	no	—
89	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
92	C₄²×C₈₉	#41	1	1	yes	characteristic
93	C₂³×C₈₉	—	2	2	no	—
94	C₂³×C₈₉	—	2	2	no	—
95	C₂³×C₈₉	—	2	2	no	—
96	C₂³×C₈₉	—	4	1	no	—
97	C₂³×C₈₉	—	4	1	no	—
98	C₂³×C₈₉	—	4	1	no	—
99	C₂³×C₈₉	—	4	1	no	—
100	C₂³×C₈₉	#121	1	1	yes	characteristic
101	D₄×C₈₉	—	2	2	no	—
102	D₄×C₈₉	—	2	2	no	—
103	D₄×C₈₉	—	2	2	no	—
104	D₄×C₈₉	#107	1	4	yes	complemented
105	C₂×C₄×C₈₉	—	2	1	no	—
106	C₂×C₄×C₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	2	no	—
108	C₂×C₄×C₈₉	—	4	1	no	—
109	C₂×C₄×C₈₉	—	4	1	no	—
110	C₂×C₄×C₈₉	—	4	2	no	—
111	C₂×C₄×C₈₉	—	4	2	no	—
112	C₂×C₄×C₈₉	—	4	2	no	—
113	C₂×C₄×C₈₉	—	4	2	no	—
114	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
124	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
125	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
126	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
128	C₂×Dic₈₉	—	8	2	no	—
129	C₂×Dic₈₉	—	8	2	no	—
130	C₂×Dic₈₉	—	8	2	no	—
131	C₈×C₈₉	—	2	2	no	—
132	C₈×C₈₉	—	4	2	no	—
133	C₈×C₈₉	#102	1	4	yes	complemented
134	#10623494	—	8	2	no	—
135	C₂²×C₈₉	—	2	4	no	—
136	C₂²×C₈₉	—	4	2	no	—
137	C₂²×C₈₉	—	4	2	no	—
138	C₂²×C₈₉	—	4	2	no	—
139	C₂²×C₈₉	—	4	2	no	—
140	C₂²×C₈₉	—	4	2	no	—
141	C₂²×C₈₉	—	4	2	no	—
142	C₂²×C₈₉	#380	1	1	yes	characteristic
143	C₂²×C₈₉	#459	1	1	yes	characteristic
144	C₂²×C₈₉	#461	1	1	yes	characteristic
145	C₂²×C₈₉	#466	1	1	yes	characteristic
146	C₂²×C₈₉	#467	1	1	yes	characteristic
147	C₂²×C₈₉	#472	1	1	yes	characteristic
148	C₂²×C₈₉	#473	1	1	yes	characteristic
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	—	4	2	no	—
152	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
153	C₄×C₈₉	#417	1	1	yes	characteristic
154	C₄×C₈₉	C₂×#107	1	1	yes	characteristic
155	C₄×C₈₉	#427	1	1	yes	characteristic
156	C₄×C₈₉	#466	1	1	yes	characteristic
157	C₄×C₈₉	#467	1	1	yes	characteristic
158	C₄×C₈₉	#474	1	1	yes	characteristic
159	C₄×C₈₉	#476	1	1	yes	characteristic
160	Dic₈₉	—	16	2	no	—
161	#8644	—	89	1	no	maximal, Sylow
162	C₂×C₈₉	—	4	4	no	—
163	C₂×C₈₉	#1667	1	1	yes	characteristic
164	C₂×C₈₉	#1669	1	1	yes	characteristic
165	C₂×C₈₉	#1673	1	1	yes	characteristic
166	C₂×C₈₉	#1675	1	1	yes	characteristic
167	C₂×C₈₉	#1802	1	1	yes	characteristic
168	C₂×C₈₉	#1808	1	1	yes	characteristic
169	C₂×C₈₉	#1816	1	1	yes	characteristic
170	C₂×C₈×D₄	D₈₉	1	1	yes	characteristic
171	#1845	—	89	1	no	—
172	#1802	—	89	2	no	—
173	C₂×#335	—	89	1	no	—
174	#1296	—	89	2	no	—
175	C₈₉	#8644	1	1	yes	characteristic, complemented, Sylow
176	C₂³×C₈	—	2	1	no	—
177	C₂×C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
178	C₈×D₄	—	2	2	no	—
179	C₈×D₄	Dic₈₉	1	4	yes	complemented
180	C₂×#107	—	178	1	no	—
181	C₂×#107	—	178	1	no	—
182	C₂×C₄⋊C₈	—	89	2	no	—
183	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
184	C₂×#102	—	178	2	no	—
185	C₂×C₂²⋊C₈	—	2	1	no	—
186	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
187	#384	—	89	2	no	—
188	#340	—	178	2	no	—
189	#335	—	178	2	no	—
190	C₂²×D₄	D₃₅₆	1	1	yes	characteristic
191	C₂³×C₄	—	2	1	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	2	4	no	—
194	C₂²×C₈	—	4	2	no	—
195	C₂²×C₈	—	4	2	no	—
196	C₂²×C₈	—	178	2	no	—
197	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
198	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	—	2	1	no	—
201	C₄×D₄	Dic₁₇₈	1	2	yes	—
202	C₄×D₄	C₂×Dic₈₉	1	2	yes	—
203	C₂×C₄⋊C₄	—	178	2	no	—
204	C₂×C₄⋊C₄	—	178	2	no	—
205	C₂×C₄⋊C₄	—	178	2	no	—
206	C₂×C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
207	C₂×C₂²⋊C₄	—	2	1	no	—
208	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
209	#107	—	356	2	no	—
210	#107	—	356	2	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	178	2	no	—
213	C₄⋊C₈	—	178	2	no	—
214	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
215	#102	—	356	4	no	—
216	C₂²⋊C₈	—	2	4	no	—
217	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
218	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
219	C₂⁴	—	2	1	no	—
220	C₂×D₄	—	2	1	no	—
221	C₂×D₄	—	2	1	no	—
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	C₄⋊Dic₈₉	1	2	yes	—
224	C₂×D₄	D₇₁₂	1	2	yes	—
225	C₂×D₄	Dic₃₅₆	1	2	yes	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	—	356	2	no	—
234	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
236	C₂²×C₄	#13083574	1	1	yes	characteristic
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	8	no	—
243	C₂×C₈	—	356	2	no	—
244	C₂×C₈	—	356	2	no	—
245	C₂×C₈	—	356	2	no	—
246	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
247	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
248	C₂×C₈	#13083559	1	2	yes	—
249	C₂×C₈	#13083562	1	2	yes	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	356	2	no	—
252	C₄⋊C₄	—	356	2	no	—
253	C₄⋊C₄	—	356	2	no	—
254	C₄⋊C₄	—	356	2	no	—
255	C₄⋊C₄	—	712	2	no	—
256	C₄⋊C₄	C₄⋊Dic₈₉	1	2	yes	—
257	C₂²⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
260	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
261	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
262	C₄²	#13083574	1	1	yes	characteristic
263	C₂³	—	2	2	no	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	4	1	no	—
267	C₂³	—	4	1	no	—
268	C₂³	—	4	1	no	—
269	C₂³	—	4	1	no	—
270	C₂³	#15667986	1	1	yes	characteristic
271	D₄	—	2	2	no	—
272	D₄	—	2	2	no	—
273	D₄	—	2	2	no	—
274	D₄	#15667946	1	4	yes	complemented
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	712	2	no	—
285	C₂×C₄	—	712	2	no	—
286	C₂×C₄	—	712	2	no	—
287	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
289	C₂×C₄	#15667971	1	1	yes	characteristic
290	C₂×C₄	#15667986	1	1	yes	characteristic
291	C₂×C₄	#15667991	1	1	yes	characteristic
292	C₂×C₄	#15667991	1	1	yes	characteristic
293	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
294	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
295	C₂×C₄	C₂×#13083562	1	1	yes	characteristic
296	C₂×C₄	C₂×Dic₃₅₆	1	1	yes	characteristic
297	C₂×C₄	#15668034	1	1	yes	characteristic
298	C₂×C₄	#15668034	1	1	yes	characteristic
299	C₂×C₄	#15668037	1	1	yes	characteristic
300	C₂×C₄	#15668042	1	1	yes	characteristic
301	C₈	—	2	2	no	—
302	C₈	—	4	2	no	—
303	C₈	—	712	2	no	—
304	C₈	#15667933	1	4	yes	complemented
305	C₂²	—	2	4	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	—	4	2	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	4	2	no	—
312	C₂²	#19151458	1	1	yes	characteristic
313	C₂²	#19151793	1	1	yes	characteristic
314	C₂²	#19151803	1	1	yes	characteristic
315	C₂²	#19151834	1	1	yes	characteristic
316	C₂²	#19151839	1	1	yes	characteristic
317	C₂²	#19151868	1	1	yes	characteristic
318	C₂²	#19151873	1	1	yes	characteristic
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	2	no	—
322	C₄	—	1424	2	no	—
323	C₄	C₂×#15667933	1	1	yes	characteristic
324	C₄	#19151599	1	1	yes	characteristic
325	C₄	C₂×#15667946	1	1	yes	characteristic
326	C₄	#19151636	1	1	yes	characteristic
327	C₄	#19151834	1	1	yes	characteristic
328	C₄	#19151839	1	1	yes	characteristic
329	C₄	#19151878	1	1	yes	characteristic
330	C₄	#19151888	1	1	yes	characteristic
331	C₂	—	4	4	no	—
332	C₂	#26019607	1	1	yes	characteristic
333	C₂	#26019617	1	1	yes	characteristic
334	C₂	#26019641	1	1	yes	characteristic
335	C₂	#26019651	1	1	yes	characteristic
336	C₂	#26020252	1	1	yes	characteristic
337	C₂	#26020281	1	1	yes	characteristic
338	C₂	#26020316	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667986
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×C₈×D₄×C₈₉	C₂
socle	C₂³×C₈₉	#121
prime radical	C₂²×D₄×C₈₉	D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×C₈×D₄	D₈₉
hypocenter	C₈₉	#8644
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. #26020412	C₂	semidirect
3. C₂×#19151351	C₂	semidirect
4. #26018136	C₂	semidirect
16. C₈×D₄×C₈₉	C₄	semidirect
104. D₄×C₈₉	#107	semidirect
133. C₈×C₈₉	#102	semidirect
175. C₈₉	#8644	semidirect
179. C₈×D₄	Dic₈₉	semidirect
274. D₄	#15667946	semidirect
304. C₈	#15667933	semidirect

Maximal subgroups

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

subgroup	index
#26020412	2
#26020252	2
C₂×#19151351	2
#26018136	2
C₂×C₈×D₄×C₈₉	2
#8644	89