Subgroups — #26031184

Order 11392.

438 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152147	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152145	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151973	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151969	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151947	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151943	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151814	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151813	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151810	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151746	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151744	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151742	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151740	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151650	C₂	1	1	yes	maximal, characteristic, complemented
15	#463×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083575	C₂²	1	1	yes	characteristic
17	C₂×#13083574	—	2	1	no	—
18	C₂×#13083573	C₂²	1	1	yes	characteristic
19	C₂×#13083563	C₂²	1	1	yes	characteristic
20	C₂×#13083562	C₂²	1	1	yes	characteristic
21	C₂×#13083561	C₂²	1	1	yes	characteristic
22	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
23	C₂×#13083559	C₂²	1	1	yes	characteristic
24	C₂×#13083559	C₂²	1	1	yes	characteristic
25	#15668011	C₂²	1	1	yes	characteristic
26	#15668010	C₂²	1	1	yes	characteristic
27	#15668008	C₂²	1	1	yes	characteristic
28	#15668008	C₂²	1	1	yes	characteristic
29	#15668000	C₂²	1	1	yes	characteristic
30	#15667998	—	2	1	no	—
31	#15667987	C₂²	1	1	yes	characteristic
32	#15667985	—	2	1	no	—
33	#15667985	—	2	1	no	—
34	#15667985	—	2	1	no	—
35	#15667983	C₂²	1	1	yes	characteristic
36	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊D₈₉	—	2	1	no	—
38	#15667972	—	2	1	no	—
39	#15667972	—	2	1	no	—
40	C₄²×D₈₉	C₂²	1	1	yes	characteristic
41	C₄⋊#10623494	C₂²	1	1	yes	characteristic
42	#15667937	C₂²	1	1	yes	characteristic
43	#15667936	C₂²	1	1	yes	characteristic
44	#15667935	C₂²	1	1	yes	characteristic
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
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
54	#124×C₈₉	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	#102×C₈₉	C₂²	1	1	yes	characteristic
59	#13083575	—	4	1	no	—
60	#13083575	—	4	1	no	—
61	#13083574	—	4	4	no	—
62	#13083573	—	4	1	no	—
63	#13083573	—	4	1	no	—
64	C₂×Dic₁₇₈	—	2	1	no	—
65	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
66	C₂×D₃₅₆	—	2	1	no	—
67	C₂×D₃₅₆	—	2	1	no	—
68	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
69	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊D₈₉	—	2	1	no	—
71	C₂×C₂²⋊D₈₉	—	2	1	no	—
72	C₂×C₂²⋊D₈₉	—	2	1	no	—
73	C₂×C₂²⋊D₈₉	—	2	1	no	—
74	C₂×C₂²⋊D₈₉	—	2	1	no	—
75	C₂×C₂²⋊D₈₉	—	2	1	no	—
76	C₂²×Dic₈₉	—	2	1	no	—
77	C₂×C₄×D₈₉	—	2	1	no	—
78	C₂×C₄×D₈₉	—	2	1	no	—
79	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
80	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
81	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
82	#13083563	—	4	1	no	—
83	#13083562	—	4	1	no	—
84	#13083561	—	4	1	no	—
85	D₇₁₂	—	4	1	no	—
86	#13083559	—	4	1	no	—
87	#13083559	—	4	1	no	—
88	C₂×#10623494	C₂³	1	1	yes	characteristic
89	C₂×#10623494	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
91	#13083553	—	2	1	no	—
92	#13083553	—	2	1	no	—
93	#13083553	—	2	1	no	—
94	C₂²⋊Dic₈₉	—	2	1	no	—
95	#13083551	—	2	1	no	—
96	#13083551	—	2	1	no	—
97	#13083551	—	2	1	no	—
98	#13083551	—	2	1	no	—
99	C₄×Dic₈₉	D₄	1	1	yes	characteristic
100	C₄×Dic₈₉	D₄	1	1	yes	characteristic
101	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
102	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
104	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
105	C₂²×C₄×C₈₉	—	2	1	no	—
106	SD₁₆×C₈₉	—	4	1	no	—
107	D₈×C₈₉	—	4	1	no	—
108	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
109	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
110	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
111	C₂²⋊C₄×C₈₉	—	2	1	no	—
112	C₂²⋊C₄×C₈₉	—	2	1	no	—
113	C₂²⋊C₄×C₈₉	—	2	1	no	—
114	C₄²×C₈₉	C₂³	1	1	yes	characteristic
115	C₂³×C₈₉	—	2	1	no	—
116	C₂³×C₈₉	—	2	1	no	—
117	C₂²×D₈₉	—	2	1	no	—
118	C₂²×D₈₉	—	2	1	no	—
119	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
120	Q₈×C₈₉	—	2	1	no	—
121	Q₈×C₈₉	—	2	1	no	—
122	D₄×C₈₉	—	2	1	no	—
123	D₄×C₈₉	—	2	1	no	—
124	D₄×C₈₉	—	2	1	no	—
125	D₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	—	2	1	no	—
128	C₂×C₄×C₈₉	—	4	2	no	—
129	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
132	C₂²⋊D₈₉	—	4	2	no	—
133	C₂²⋊D₈₉	—	4	2	no	—
134	C₂²⋊D₈₉	—	4	2	no	—
135	C₂²⋊D₈₉	—	4	2	no	—
136	C₂²⋊D₈₉	—	4	2	no	—
137	C₂²⋊D₈₉	—	4	2	no	—
138	C₂×Dic₈₉	—	2	1	no	—
139	C₂×Dic₈₉	—	2	1	no	—
140	C₂×Dic₈₉	—	4	2	no	—
141	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
142	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
143	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
144	D₃₅₆	—	2	1	no	—
145	D₃₅₆	—	2	1	no	—
146	D₃₅₆	—	2	1	no	—
147	D₃₅₆	—	2	1	no	—
148	D₃₅₆	—	4	2	no	—
149	D₃₅₆	—	4	2	no	—
150	C₄×D₈₉	—	2	1	no	—
151	C₄×D₈₉	—	2	1	no	—
152	C₄×D₈₉	—	2	2	no	—
153	C₄×D₈₉	—	2	2	no	—
154	C₄×D₈₉	—	4	2	no	—
155	C₄×D₈₉	—	4	2	no	—
156	Dic₁₇₈	—	2	1	no	—
157	Dic₁₇₈	—	2	1	no	—
158	Dic₁₇₈	—	4	2	no	—
159	C₈×C₈₉	—	2	1	no	—
160	C₈×C₈₉	—	2	1	no	—
161	#10623494	—	2	1	no	—
162	#10623494	—	2	1	no	—
163	C₂²×C₈₉	—	4	1	no	—
164	C₂²×C₈₉	—	4	1	no	—
165	C₂²×C₈₉	—	4	1	no	—
166	C₂²×C₈₉	—	4	1	no	—
167	C₂²×C₈₉	—	4	1	no	—
168	C₂²×C₈₉	—	4	1	no	—
169	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
170	C₂×D₈₉	—	2	1	no	—
171	C₂×D₈₉	—	2	1	no	—
172	C₂×D₈₉	—	2	1	no	—
173	C₂×D₈₉	—	4	1	no	—
174	C₂×D₈₉	—	4	1	no	—
175	C₂×D₈₉	—	4	1	no	—
176	C₂×D₈₉	—	4	1	no	—
177	C₂×D₈₉	—	4	1	no	—
178	C₂×D₈₉	—	4	1	no	—
179	C₄×C₈₉	—	2	1	no	—
180	C₄×C₈₉	—	2	1	no	—
181	C₄×C₈₉	—	4	1	no	—
182	C₄×C₈₉	—	4	1	no	—
183	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
184	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
185	Dic₈₉	—	2	1	no	—
186	Dic₈₉	—	4	1	no	—
187	Dic₈₉	—	4	1	no	—
188	Dic₈₉	#135	1	2	yes	—
189	Dic₈₉	AGL(1,ℤ/8)	1	2	yes	—
190	C₂×C₈₉	—	8	1	no	—
191	C₂×C₈₉	—	8	1	no	—
192	C₂×C₈₉	#535	1	1	yes	characteristic
193	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
194	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
195	D₈₉	—	4	1	no	—
196	D₈₉	—	8	1	no	—
197	D₈₉	—	8	1	no	—
198	#3207	—	89	1	no	maximal, Sylow
199	C₈₉	#3207	1	1	yes	characteristic, complemented, Sylow
200	#532	—	89	1	no	—
201	#532	—	89	1	no	—
202	#493	—	89	1	no	—
203	#492	—	89	1	no	—
204	#487	—	89	1	no	—
205	#486	—	89	1	no	—
206	#463	—	89	1	no	—
207	#463	—	89	1	no	—
208	#463	—	89	1	no	—
209	#463	D₈₉	1	1	yes	characteristic, complemented
210	#449	—	89	1	no	—
211	#449	—	89	1	no	—
212	#449	—	89	1	no	—
213	#449	—	89	1	no	—
214	#431	—	89	1	no	—
215	C₂×#41	—	89	1	no	—
216	C₂×#41	—	89	1	no	—
217	C₂×#41	—	178	1	no	—
218	C₂×SD₁₆	—	89	1	no	—
219	C₂×SD₁₆	—	89	1	no	—
220	C₂×SD₁₆	—	89	1	no	—
221	C₂×SD₁₆	C₂×D₈₉	1	1	yes	characteristic
222	C₂×D₈	—	89	1	no	—
223	C₂×D₈	—	89	1	no	—
224	C₂×D₈	—	89	1	no	—
225	C₂×D₈	C₂×D₈₉	1	1	yes	characteristic
226	C₄⋊Q₈	—	89	1	no	—
227	C₄⋊Q₈	—	89	1	no	—
228	#127	—	89	1	no	—
229	#127	—	89	1	no	—
230	#124	—	89	1	no	—
231	#124	—	178	1	no	—
232	#124	C₂×D₈₉	1	1	yes	characteristic
233	#121	—	89	1	no	—
234	#121	—	89	1	no	—
235	#121	—	178	1	no	—
236	#121	—	178	1	no	—
237	#121	—	178	1	no	—
238	C₄×D₄	—	89	1	no	—
239	C₄×D₄	—	178	1	no	—
240	C₄×D₄	—	178	1	no	—
241	C₄×D₄	—	178	1	no	—
242	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
243	C₂×C₄²	—	89	1	no	—
244	C₄⋊C₈	—	89	1	no	—
245	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
246	#103	—	89	1	no	—
247	#103	—	89	1	no	—
248	#103	—	89	1	no	—
249	#103	C₂×D₈₉	1	1	yes	characteristic
250	#102	—	89	1	no	—
251	#102	—	89	1	no	—
252	#102	—	89	1	no	—
253	#102	C₂×D₈₉	1	1	yes	characteristic
254	C₂²⋊C₈	—	89	1	no	—
255	C₂²⋊C₈	—	89	1	no	—
256	C₄×C₈	—	89	1	no	—
257	C₄×C₈	—	89	1	no	—
258	#41	—	356	4	no	—
259	#41	—	356	1	no	—
260	#41	—	356	1	no	—
261	#41	—	356	1	no	—
262	#41	—	356	1	no	—
263	C₂×Q₈	—	89	1	no	—
264	C₂×Q₈	—	178	1	no	—
265	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
266	C₂×D₄	—	89	1	no	—
267	C₂×D₄	—	89	1	no	—
268	C₂×D₄	—	178	1	no	—
269	C₂×D₄	—	178	1	no	—
270	C₂×D₄	—	178	1	no	—
271	C₂×D₄	—	178	1	no	—
272	C₂×D₄	—	178	1	no	—
273	C₂×D₄	—	178	1	no	—
274	C₂×D₄	—	178	1	no	—
275	C₂×D₄	—	178	1	no	—
276	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
277	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
278	C₂²×C₄	—	2	1	no	—
279	C₂²×C₄	—	89	1	no	—
280	C₂²×C₄	—	89	1	no	—
281	C₂²×C₄	—	89	1	no	—
282	C₂²×C₄	—	178	1	no	—
283	C₂²×C₄	—	178	1	no	—
284	C₂²×C₄	—	178	1	no	—
285	SD₁₆	—	4	1	no	—
286	SD₁₆	—	356	1	no	—
287	SD₁₆	—	356	1	no	—
288	SD₁₆	—	356	1	no	—
289	D₈	—	4	1	no	—
290	D₈	—	356	1	no	—
291	D₈	—	356	1	no	—
292	D₈	—	356	1	no	—
293	C₂×C₈	—	89	1	no	—
294	C₂×C₈	—	89	1	no	—
295	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
296	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
297	C₄⋊C₄	—	89	1	no	—
298	C₄⋊C₄	—	178	1	no	—
299	C₄⋊C₄	—	178	1	no	—
300	C₄⋊C₄	—	178	1	no	—
301	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
302	C₂²⋊C₄	—	2	1	no	—
303	C₂²⋊C₄	—	2	1	no	—
304	C₂²⋊C₄	—	2	1	no	—
305	C₂²⋊C₄	—	178	1	no	—
306	C₂²⋊C₄	—	178	1	no	—
307	C₂²⋊C₄	—	178	1	no	—
308	C₂²⋊C₄	—	178	1	no	—
309	C₂²⋊C₄	—	178	1	no	—
310	C₄²	—	89	1	no	—
311	C₄²	—	89	1	no	—
312	C₄²	—	89	1	no	—
313	C₄²	C₂²×D₈₉	1	1	yes	characteristic
314	C₂³	—	2	1	no	—
315	C₂³	—	2	1	no	—
316	C₂³	—	89	1	no	—
317	C₂³	—	178	1	no	—
318	C₂³	—	178	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	2	1	no	—
321	Q₈	—	356	2	no	—
322	Q₈	—	178	1	no	—
323	Q₈	—	178	1	no	—
324	D₄	—	2	1	no	—
325	D₄	—	2	1	no	—
326	D₄	—	2	1	no	—
327	D₄	—	2	1	no	—
328	D₄	—	356	2	no	—
329	D₄	—	356	2	no	—
330	D₄	—	356	2	no	—
331	D₄	—	356	2	no	—
332	D₄	—	356	2	no	—
333	D₄	—	356	2	no	—
334	D₄	—	356	2	no	—
335	D₄	—	356	2	no	—
336	D₄	—	178	1	no	—
337	D₄	—	178	1	no	—
338	D₄	—	178	1	no	—
339	D₄	—	178	1	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	4	2	no	—
343	C₂×C₄	—	89	1	no	—
344	C₂×C₄	—	89	1	no	—
345	C₂×C₄	—	89	1	no	—
346	C₂×C₄	—	356	2	no	—
347	C₂×C₄	—	356	2	no	—
348	C₂×C₄	—	356	2	no	—
349	C₂×C₄	—	178	2	no	—
350	C₂×C₄	—	178	2	no	—
351	C₂×C₄	—	178	1	no	—
352	C₂×C₄	—	178	1	no	—
353	C₂×C₄	—	178	1	no	—
354	C₂×C₄	—	178	1	no	—
355	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
356	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
357	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	178	1	no	—
361	C₈	—	178	1	no	—
362	C₂²	—	4	1	no	—
363	C₂²	—	4	1	no	—
364	C₂²	—	4	1	no	—
365	C₂²	—	4	1	no	—
366	C₂²	—	4	1	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	356	1	no	—
369	C₂²	—	356	1	no	—
370	C₂²	—	356	1	no	—
371	C₂²	—	356	1	no	—
372	C₂²	—	356	1	no	—
373	C₂²	—	356	1	no	—
374	C₂²	—	178	1	no	—
375	C₂²	—	178	1	no	—
376	C₂²	—	178	1	no	—
377	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	4	1	no	—
381	C₄	—	4	1	no	—
382	C₄	—	89	2	no	—
383	C₄	—	89	2	no	—
384	C₄	—	356	1	no	—
385	C₄	—	356	1	no	—
386	C₄	—	178	1	no	—
387	C₄	#15668057	1	1	yes	characteristic
388	C₄	#15668058	1	1	yes	characteristic
389	C₂	—	8	1	no	—
390	C₂	—	8	1	no	—
391	C₂	—	356	1	no	—
392	C₂	—	712	1	no	—
393	C₂	—	712	1	no	—
394	C₂	#19152164	1	1	yes	characteristic
395	C₂	#135×D₈₉	1	1	yes	characteristic
396	C₂	AGL(1,ℤ/8)×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#463×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#463	D₈₉
hypocenter	C₈₉	#3207
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #19152147	C₂	semidirect
2. #19152145	C₂	semidirect
3. #19151973	C₂	semidirect
4. #19151969	C₂	semidirect
5. #19151947	C₂	semidirect
6. #19151943	C₂	semidirect
7. #19151814	C₂	semidirect
8. #19151813	C₂	semidirect
9. #19151810	C₂	semidirect
10. #19151746	C₂	semidirect
11. #19151744	C₂	semidirect
12. #19151742	C₂	semidirect
13. #19151740	C₂	semidirect
14. #19151650	C₂	semidirect
15. #463×C₈₉	C₂	semidirect
199. C₈₉	#3207	semidirect
209. #463	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152147	2
#19152145	2
#19151973	2
#19151969	2
#19151947	2
#19151943	2
#19151814	2
#19151813	2
#19151810	2
#19151746	2
#19151744	2
#19151742	2
#19151740	2
#19151650	2
#463×C₈₉	2
#3207	89