Subgroups — #26031221

Order 11392.

410 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152135	C₂	1	1	yes	maximal, characteristic, complemented
2	#124×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151983	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151979	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151954	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151951	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151820	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151814	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151810	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151752	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151747	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151738	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151733	C₂	1	1	yes	maximal, characteristic
14	#19151650	C₂	1	1	yes	maximal, characteristic, complemented
15	#464×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
17	C₂×D₄×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×#13083564	C₂²	1	1	yes	characteristic, complemented
19	C₂×#13083563	C₂²	1	1	yes	characteristic, complemented
20	C₂×#13083561	C₂²	1	1	yes	characteristic
21	C₂×#13083561	C₂²	1	1	yes	characteristic, complemented
22	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
23	C₂×#13083559	C₂²	1	1	yes	characteristic
24	#15668007	—	2	1	no	—
25	#15668003	C₂²	1	1	yes	characteristic, complemented
26	#15668001	C₂²	1	1	yes	characteristic
27	#15668001	C₂²	1	1	yes	characteristic, complemented
28	#15668000	C₂²	1	1	yes	characteristic
29	#15667999	C₂²	1	1	yes	characteristic
30	#15667998	—	2	1	no	—
31	#15667998	—	2	1	no	—
32	#15667997	—	2	1	no	—
33	#15667988	C₂²	1	1	yes	characteristic
34	#15667987	C₂²	1	1	yes	characteristic
35	C₄×D₃₅₆	—	2	1	no	—
36	C₄×D₃₅₆	C₂²	1	1	yes	characteristic, complemented
37	#15667969	—	2	1	no	—
38	C₂²⋊C₄×D₈₉	—	2	1	no	—
39	C₂²⋊C₄×D₈₉	—	2	1	no	—
40	C₄²×D₈₉	C₂²	1	1	yes	characteristic
41	C₄⋊#10623494	C₂²	1	1	yes	characteristic, complemented
42	#15667937	C₂²	1	1	yes	characteristic, complemented
43	#15667936	C₂²	1	1	yes	characteristic, complemented
44	#15667935	C₂²	1	1	yes	characteristic, complemented
45	#15667934	C₂²	1	1	yes	characteristic
46	#15667933	C₂²	1	1	yes	characteristic
47	#15667932	C₂²	1	1	yes	characteristic
48	#15667924	C₂²	1	1	yes	characteristic
49	#15667924	C₂²	1	1	yes	characteristic
50	C₈×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
52	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
53	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	#124×C₈₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
57	#103×C₈₉	C₂²	1	1	yes	characteristic, complemented
58	#102×C₈₉	C₂²	1	1	yes	characteristic
59	C₂³×D₈₉	—	2	1	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	Q₈×D₈₉	—	4	1	no	—
63	Q₈×D₈₉	—	4	1	no	—
64	C₂×D₃₅₆	—	2	1	no	—
65	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
66	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₉	—	2	1	no	—
68	D₄×D₈₉	—	4	1	no	—
69	D₄×D₈₉	—	4	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	—	2	1	no	—
72	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
73	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
74	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
75	#13083564	—	4	1	no	—
76	#13083563	—	4	1	no	—
77	#13083561	—	4	1	no	—
78	#13083561	—	4	1	no	—
79	D₇₁₂	—	4	1	no	—
80	#13083559	—	4	1	no	—
81	C₂×#10623494	C₂³	1	1	yes	characteristic
82	C₂×#10623494	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₈₉	—	2	1	no	—
84	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
85	#13083553	—	2	1	no	—
86	#13083553	—	2	1	no	—
87	C₂²⋊Dic₈₉	—	2	1	no	—
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	#13083551	—	2	1	no	—
91	#13083551	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	#13083551	—	2	1	no	—
94	#13083551	—	2	1	no	—
95	#13083551	—	2	1	no	—
96	#13083551	—	2	1	no	—
97	#13083551	—	2	1	no	—
98	C₄×Dic₈₉	D₄	1	1	yes	characteristic
99	C₄×Dic₈₉	D₄	1	1	yes	characteristic
100	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
101	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
104	Q₁₆×C₈₉	—	4	1	no	—
105	SD₁₆×C₈₉	—	4	1	no	—
106	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	—	2	1	no	—
109	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₈₉	—	2	1	no	—
111	C₂²⋊C₄×C₈₉	—	2	1	no	—
112	C₄²×C₈₉	—	2	1	no	—
113	C₄²×C₈₉	C₂³	1	1	yes	characteristic
114	C₂³×C₈₉	—	2	1	no	—
115	C₂²×D₈₉	—	2	1	no	—
116	C₂²×D₈₉	—	2	1	no	—
117	C₂²×D₈₉	—	4	2	no	—
118	C₂²×D₈₉	—	4	2	no	—
119	C₂²×D₈₉	—	4	2	no	—
120	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
121	Q₈×C₈₉	—	2	1	no	—
122	Q₈×C₈₉	—	2	1	no	—
123	Q₈×C₈₉	—	2	1	no	—
124	Q₈×C₈₉	—	2	1	no	—
125	D₄×C₈₉	—	2	1	no	—
126	D₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	—	2	1	no	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
133	C₂²⋊D₈₉	—	4	2	no	—
134	C₂×Dic₈₉	—	2	1	no	—
135	C₂×Dic₈₉	—	2	1	no	—
136	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
138	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
139	D₃₅₆	—	2	1	no	—
140	D₃₅₆	—	2	1	no	—
141	D₃₅₆	—	2	1	no	—
142	D₃₅₆	—	2	1	no	—
143	D₃₅₆	—	4	2	no	—
144	C₄×D₈₉	—	2	2	no	—
145	C₄×D₈₉	—	2	1	no	—
146	C₄×D₈₉	—	2	1	no	—
147	C₄×D₈₉	—	4	1	no	—
148	C₄×D₈₉	—	4	2	no	—
149	C₄×D₈₉	—	4	2	no	—
150	Dic₁₇₈	—	2	1	no	—
151	Dic₁₇₈	—	2	1	no	—
152	Dic₁₇₈	—	4	2	no	—
153	C₈×C₈₉	—	2	1	no	—
154	C₈×C₈₉	—	2	1	no	—
155	#10623494	—	2	1	no	—
156	#10623494	—	2	1	no	—
157	C₂²×C₈₉	—	4	1	no	—
158	C₂²×C₈₉	—	4	1	no	—
159	C₂²×C₈₉	—	4	1	no	—
160	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
161	C₂×D₈₉	—	2	1	no	—
162	C₂×D₈₉	—	2	1	no	—
163	C₂×D₈₉	—	4	1	no	—
164	C₂×D₈₉	—	4	1	no	—
165	C₂×D₈₉	—	4	1	no	—
166	C₂×D₈₉	—	4	1	no	—
167	C₂×D₈₉	—	4	1	no	—
168	C₂×D₈₉	—	4	1	no	—
169	C₂×D₈₉	—	8	4	no	—
170	C₂×D₈₉	#135	1	2	yes	—
171	C₄×C₈₉	—	2	1	no	—
172	C₄×C₈₉	—	2	1	no	—
173	C₄×C₈₉	—	4	1	no	—
174	C₄×C₈₉	—	4	1	no	—
175	C₄×C₈₉	—	4	1	no	—
176	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
177	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
178	Dic₈₉	—	2	1	no	—
179	Dic₈₉	—	2	1	no	—
180	Dic₈₉	—	2	1	no	—
181	Dic₈₉	—	4	1	no	—
182	Dic₈₉	—	4	1	no	—
183	C₂×C₈₉	—	8	1	no	—
184	C₂×C₈₉	#535	1	1	yes	characteristic
185	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
186	C₂×C₈₉	#575	1	1	yes	characteristic
187	D₈₉	—	2	2	no	—
188	D₈₉	—	8	1	no	—
189	D₈₉	—	8	1	no	—
190	#3211	—	89	1	no	maximal, Sylow
191	C₈₉	#3211	1	1	yes	characteristic, complemented, Sylow
192	#529	—	89	1	no	—
193	C₂×#124	—	89	1	no	—
194	#495	—	89	1	no	—
195	#495	—	89	1	no	—
196	#488	—	89	1	no	—
197	#488	—	89	1	no	—
198	#464	—	89	1	no	—
199	#464	D₈₉	1	1	yes	characteristic, complemented
200	#463	—	89	1	no	—
201	#463	—	89	1	no	—
202	#451	—	89	1	no	—
203	#450	—	89	1	no	—
204	#448	—	89	1	no	—
205	#447	—	89	1	no	—
206	#431	—	89	1	no	—
207	C₂²×Q₈	—	89	1	no	—
208	C₂²×D₄	—	89	1	no	—
209	C₂×Q₁₆	—	89	1	no	—
210	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
211	C₂×SD₁₆	—	89	1	no	—
212	C₂×SD₁₆	—	89	1	no	—
213	C₂×SD₁₆	—	89	1	no	—
214	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
215	C₂×D₈	—	89	1	no	—
216	C₂×D₈	—	89	1	no	—
217	#126	—	178	1	no	—
218	#125	—	89	1	no	—
219	#124	—	89	1	no	—
220	#124	—	89	1	no	—
221	#124	—	89	1	no	—
222	#124	—	89	1	no	—
223	#124	—	178	1	no	—
224	#124	—	178	1	no	—
225	#124	C₂×D₈₉	1	1	yes	characteristic
226	#123	—	178	1	no	—
227	C₂²⋊Q₈	—	89	1	no	—
228	#121	—	89	1	no	—
229	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic, complemented
230	C₄×D₄	—	89	1	no	—
231	C₄×D₄	—	178	1	no	—
232	#117	—	178	1	no	—
233	C₂×C₂²⋊C₄	—	178	1	no	—
234	C₂×C₂²⋊C₄	—	178	1	no	—
235	C₂×C₄²	—	89	1	no	—
236	C₄⋊C₈	—	89	1	no	—
237	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
238	#103	—	89	1	no	—
239	#103	—	89	1	no	—
240	#103	—	89	1	no	—
241	#103	C₂×D₈₉	1	1	yes	characteristic, complemented
242	#102	—	89	1	no	—
243	#102	—	89	1	no	—
244	#102	—	89	1	no	—
245	#102	C₂×D₈₉	1	1	yes	characteristic
246	C₂²⋊C₈	—	89	1	no	—
247	C₂²⋊C₈	—	89	1	no	—
248	C₄×C₈	—	89	1	no	—
249	C₄×C₈	—	89	1	no	—
250	C₂⁴	—	178	1	no	—
251	C₂×Q₈	—	89	1	no	—
252	C₂×Q₈	—	356	1	no	—
253	C₂×Q₈	—	356	1	no	—
254	C₂×Q₈	—	178	1	no	—
255	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
256	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
257	C₂×D₄	—	89	1	no	—
258	C₂×D₄	—	89	1	no	—
259	C₂×D₄	—	356	1	no	—
260	C₂×D₄	—	356	1	no	—
261	C₂×D₄	—	178	1	no	—
262	C₂×D₄	—	178	1	no	—
263	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
264	C₂²×C₄	—	89	1	no	—
265	C₂²×C₄	—	89	1	no	—
266	C₂²×C₄	—	89	1	no	—
267	C₂²×C₄	—	178	1	no	—
268	C₂²×C₄	—	178	1	no	—
269	Q₁₆	—	4	1	no	—
270	Q₁₆	—	356	1	no	—
271	SD₁₆	—	4	1	no	—
272	SD₁₆	—	356	1	no	—
273	SD₁₆	—	356	1	no	—
274	SD₁₆	—	356	1	no	—
275	D₈	—	356	1	no	—
276	D₈	—	356	1	no	—
277	C₂×C₈	—	89	1	no	—
278	C₂×C₈	—	89	1	no	—
279	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
280	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	89	1	no	—
283	C₄⋊C₄	—	178	1	no	—
284	C₄⋊C₄	—	178	1	no	—
285	C₄⋊C₄	—	178	1	no	—
286	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
287	C₂²⋊C₄	—	2	1	no	—
288	C₂²⋊C₄	—	2	1	no	—
289	C₂²⋊C₄	—	178	1	no	—
290	C₂²⋊C₄	—	178	1	no	—
291	C₂²⋊C₄	—	178	1	no	—
292	C₂²⋊C₄	—	178	1	no	—
293	C₂²⋊C₄	—	178	1	no	—
294	C₂²⋊C₄	—	178	1	no	—
295	C₂²⋊C₄	—	178	1	no	—
296	C₂²⋊C₄	—	178	1	no	—
297	C₂²⋊C₄	—	178	1	no	—
298	C₂²⋊C₄	—	178	1	no	—
299	C₂²⋊C₄	—	178	1	no	—
300	C₄²	—	2	1	no	—
301	C₄²	—	89	1	no	—
302	C₄²	—	89	1	no	—
303	C₄²	—	89	1	no	—
304	C₄²	C₂²×D₈₉	1	1	yes	characteristic
305	C₂³	—	2	1	no	—
306	C₂³	—	89	1	no	—
307	C₂³	—	356	2	no	—
308	C₂³	—	356	2	no	—
309	C₂³	—	356	2	no	—
310	C₂³	—	178	1	no	—
311	C₂³	—	178	1	no	—
312	Q₈	—	2	1	no	—
313	Q₈	—	2	1	no	—
314	Q₈	—	2	1	no	—
315	Q₈	—	2	1	no	—
316	Q₈	—	356	2	no	—
317	Q₈	—	178	1	no	—
318	Q₈	—	178	1	no	—
319	D₄	—	2	1	no	—
320	D₄	—	2	1	no	—
321	D₄	—	356	2	no	—
322	D₄	—	356	2	no	—
323	D₄	—	178	1	no	—
324	D₄	—	178	1	no	—
325	D₄	—	178	1	no	—
326	D₄	—	178	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	89	1	no	—
331	C₂×C₄	—	89	1	no	—
332	C₂×C₄	—	89	1	no	—
333	C₂×C₄	—	356	2	no	—
334	C₂×C₄	—	356	1	no	—
335	C₂×C₄	—	356	2	no	—
336	C₂×C₄	—	178	2	no	—
337	C₂×C₄	—	178	1	no	—
338	C₂×C₄	—	178	1	no	—
339	C₂×C₄	—	178	1	no	—
340	C₂×C₄	—	178	1	no	—
341	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
342	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
343	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
344	C₈	—	2	1	no	—
345	C₈	—	2	1	no	—
346	C₈	—	178	1	no	—
347	C₈	—	178	1	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	89	2	no	—
352	C₂²	—	356	1	no	—
353	C₂²	—	356	1	no	—
354	C₂²	—	356	1	no	—
355	C₂²	—	356	1	no	—
356	C₂²	—	356	1	no	—
357	C₂²	—	356	1	no	—
358	C₂²	—	178	1	no	—
359	C₂²	—	178	1	no	—
360	C₂²	—	712	4	no	—
361	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	4	1	no	—
365	C₄	—	4	1	no	—
366	C₄	—	4	1	no	—
367	C₄	—	356	1	no	—
368	C₄	—	356	1	no	—
369	C₄	—	178	1	no	—
370	C₄	—	178	1	no	—
371	C₄	—	178	1	no	—
372	C₄	#15668057	1	1	yes	characteristic
373	C₄	#15668058	1	1	yes	characteristic
374	C₂	—	8	1	no	—
375	C₂	—	178	2	no	—
376	C₂	—	712	1	no	—
377	C₂	—	712	1	no	—
378	C₂	#19152164	1	1	yes	characteristic
379	C₂	#135×D₈₉	1	1	yes	characteristic
380	C₂	#19152397	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#464×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19151733	C₂
hypercenter	#464	D₈₉
hypocenter	C₈₉	#3211
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
1. #19152135	C₂	semidirect
2. #124×D₈₉	C₂	semidirect
3. #19151983	C₂	semidirect
4. #19151979	C₂	semidirect
5. #19151954	C₂	semidirect
6. #19151951	C₂	semidirect
7. #19151820	C₂	semidirect
8. #19151814	C₂	semidirect
9. #19151810	C₂	semidirect
10. #19151752	C₂	semidirect
11. #19151747	C₂	semidirect
12. #19151738	C₂	semidirect
14. #19151650	C₂	semidirect
15. #464×C₈₉	C₂	semidirect
18. C₂×#13083564	C₂²	semidirect
19. C₂×#13083563	C₂²	semidirect
21. C₂×#13083561	C₂²	semidirect
25. #15668003	C₂²	semidirect
27. #15668001	C₂²	semidirect
36. C₄×D₃₅₆	C₂²	semidirect
41. C₄⋊#10623494	C₂²	semidirect
42. #15667937	C₂²	semidirect
43. #15667936	C₂²	semidirect
44. #15667935	C₂²	semidirect
50. C₈×Dic₈₉	C₂²	semidirect
51. C₈×Dic₈₉	C₂²	semidirect
52. C₂×Q₁₆×C₈₉	C₂²	semidirect
55. C₄×Q₈×C₈₉	C₂²	semidirect
56. C₄⋊C₈×C₈₉	C₂²	semidirect
57. #103×C₈₉	C₂²	semidirect
191. C₈₉	#3211	semidirect
199. #464	D₈₉	semidirect
210. C₂×Q₁₆	C₂×D₈₉	semidirect
229. C₄×Q₈	C₂×D₈₉	semidirect
237. C₄⋊C₈	C₂×D₈₉	semidirect
241. #103	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152135	2
#124×D₈₉	2
#19151983	2
#19151979	2
#19151954	2
#19151951	2
#19151820	2
#19151814	2
#19151810	2
#19151752	2
#19151747	2
#19151738	2
#19151733	2
#19151650	2
#464×C₈₉	2
#3211	89