Subgroups — #26024913

Order 11392.

454 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19151523	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151514	C₂	1	1	yes	maximal, characteristic
3	#19151514	C₂	1	1	yes	maximal, characteristic
4	#19151497	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151490	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151488	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151488	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151488	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151484	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151475	C₂	1	1	yes	maximal, characteristic
11	#19151465	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151463	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²⋊C₄×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²⋊C₄×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#396×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
17	C₂³×Dic₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₂²⋊Dic₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Dic₈₉	C₂²	1	1	yes	characteristic
23	C₂×#13083551	C₂²	1	1	yes	characteristic
24	C₂×#13083551	C₂²	1	1	yes	characteristic
25	C₂×#13083551	C₂²	1	1	yes	characteristic
26	C₂×#13083551	C₂²	1	1	yes	characteristic
27	C₂×#13083551	C₂²	1	1	yes	characteristic
28	C₂×#13083551	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
31	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
33	#15667919	C₂²	1	1	yes	characteristic
34	#15667919	C₂²	1	1	yes	characteristic
35	#15667919	C₂²	1	1	yes	characteristic
36	#15667919	C₂²	1	1	yes	characteristic
37	#15667919	C₂²	1	1	yes	characteristic
38	#15667919	C₂²	1	1	yes	characteristic
39	#15667919	C₂²	1	1	yes	characteristic
40	#15667919	C₂²	1	1	yes	characteristic
41	#15667919	C₂²	1	1	yes	characteristic
42	#15667919	C₂²	1	1	yes	characteristic
43	#15667919	C₂²	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
50	#95×C₈₉	C₂²	1	1	yes	characteristic
51	C₂³×D₈₉	C₂³	1	1	yes	characteristic
52	C₂×C₂²⋊D₈₉	—	2	2	no	—
53	C₂×C₂²⋊D₈₉	—	4	1	no	—
54	C₂×C₂²⋊D₈₉	—	4	1	no	—
55	C₂²×Dic₈₉	—	2	2	no	—
56	C₂²×Dic₈₉	—	2	2	no	—
57	C₂²×Dic₈₉	—	4	1	no	—
58	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
59	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
60	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
61	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
62	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
63	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
64	C₂²×Dic₈₉	C₂³	1	1	yes	characteristic
65	C₄⋊Dic₈₉	—	4	1	no	—
66	C₄⋊Dic₈₉	—	4	1	no	—
67	C₄⋊Dic₈₉	—	4	1	no	—
68	C₂²⋊Dic₈₉	—	2	2	no	—
69	C₂²⋊Dic₈₉	—	2	2	no	—
70	#13083551	—	2	2	no	—
71	#13083551	—	2	2	no	—
72	#13083551	—	2	2	no	—
73	#13083551	—	2	2	no	—
74	#13083551	—	4	1	no	—
75	#13083551	—	4	1	no	—
76	C₄×Dic₈₉	—	2	2	no	—
77	C₄×Dic₈₉	—	2	2	no	—
78	C₄×Dic₈₉	—	4	1	no	—
79	C₄×Dic₈₉	—	4	1	no	—
80	C₂⁴×C₈₉	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
87	C₄⋊C₄×C₈₉	—	2	2	no	—
88	C₄⋊C₄×C₈₉	—	2	2	no	—
89	C₄⋊C₄×C₈₉	—	4	1	no	—
90	C₂²⋊C₄×C₈₉	—	2	2	no	—
91	C₂²⋊C₄×C₈₉	—	2	2	no	—
92	C₂²⋊C₄×C₈₉	—	4	1	no	—
93	C₂³×C₈₉	—	2	1	no	—
94	C₂³×C₈₉	—	2	1	no	—
95	C₂³×C₈₉	—	2	1	no	—
96	C₂³×C₈₉	—	2	1	no	—
97	C₂³×C₈₉	—	2	1	no	—
98	C₂³×C₈₉	—	2	1	no	—
99	C₂³×C₈₉	#41	1	2	yes	—
100	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
101	C₂²×D₈₉	—	2	1	no	—
102	C₂²×D₈₉	—	2	1	no	—
103	C₂²×D₈₉	—	2	1	no	—
104	C₂²×D₈₉	—	2	1	no	—
105	C₂²×D₈₉	—	2	1	no	—
106	C₂²×D₈₉	—	2	1	no	—
107	C₂²×D₈₉	—	2	1	no	—
108	C₂×C₄×C₈₉	—	2	1	no	—
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	2	1	no	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	#41	1	2	yes	—
125	C₂×C₄×C₈₉	#41	1	2	yes	—
126	C₂²⋊D₈₉	—	4	4	no	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	2	1	no	—
129	C₂×Dic₈₉	—	2	1	no	—
130	C₂×Dic₈₉	—	2	1	no	—
131	C₂×Dic₈₉	—	2	1	no	—
132	C₂×Dic₈₉	—	2	1	no	—
133	C₂×Dic₈₉	—	2	1	no	—
134	C₂×Dic₈₉	—	2	1	no	—
135	C₂×Dic₈₉	—	2	1	no	—
136	C₂×Dic₈₉	—	2	1	no	—
137	C₂×Dic₈₉	—	2	1	no	—
138	C₂×Dic₈₉	—	2	1	no	—
139	C₂×Dic₈₉	—	2	1	no	—
140	C₂×Dic₈₉	—	2	1	no	—
141	C₂×Dic₈₉	—	2	1	no	—
142	C₂×Dic₈₉	—	2	1	no	—
143	C₂×Dic₈₉	—	2	1	no	—
144	C₂×Dic₈₉	—	2	1	no	—
145	C₂×Dic₈₉	—	2	1	no	—
146	C₂×Dic₈₉	—	4	4	no	—
147	C₂×Dic₈₉	#41	1	2	yes	—
148	C₂×Dic₈₉	#41	1	2	yes	—
149	C₂²×C₈₉	—	2	2	no	—
150	C₂²×C₈₉	—	2	2	no	—
151	C₂²×C₈₉	—	2	2	no	—
152	C₂²×C₈₉	—	4	1	no	—
153	C₂²×C₈₉	—	4	1	no	—
154	C₂²×C₈₉	—	4	1	no	—
155	C₂²×C₈₉	—	4	1	no	—
156	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
157	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
158	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
159	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
160	C₂²×C₈₉	C₂×#41	1	1	yes	characteristic
161	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂²×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
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	—	4	1	no	—
170	C₄×C₈₉	—	2	2	no	—
171	C₄×C₈₉	—	2	2	no	—
172	C₄×C₈₉	—	4	1	no	—
173	C₄×C₈₉	—	4	1	no	—
174	C₄×C₈₉	—	4	1	no	—
175	C₄×C₈₉	—	4	1	no	—
176	Dic₈₉	—	2	2	no	—
177	Dic₈₉	—	2	2	no	—
178	Dic₈₉	—	4	1	no	—
179	Dic₈₉	—	4	1	no	—
180	Dic₈₉	—	4	1	no	—
181	Dic₈₉	—	4	1	no	—
182	Dic₈₉	—	4	1	no	—
183	C₂×C₈₉	—	4	2	no	—
184	C₂×C₈₉	#529	1	1	yes	characteristic
185	C₂×C₈₉	#529	1	1	yes	characteristic
186	C₂×C₈₉	#538	1	1	yes	characteristic
187	C₂×C₈₉	#551	1	1	yes	characteristic
188	C₂×C₈₉	#553	1	1	yes	characteristic
189	C₂×C₈₉	#553	1	1	yes	characteristic
190	C₂×C₈₉	#555	1	1	yes	characteristic
191	D₈₉	—	8	1	no	—
192	#2518	—	89	1	no	maximal, Sylow
193	C₈₉	#2518	1	1	yes	characteristic, complemented, Sylow
194	#397	—	89	1	no	—
195	#396	D₈₉	1	1	yes	characteristic, complemented
196	#394	—	89	1	no	—
197	#394	—	89	1	no	—
198	#389	—	89	1	no	—
199	#388	—	89	1	no	—
200	#388	—	89	1	no	—
201	#388	—	89	1	no	—
202	#388	—	89	1	no	—
203	#387	—	89	1	no	—
204	#386	—	89	1	no	—
205	#383	—	89	1	no	—
206	#382	—	89	1	no	—
207	C₄×C₂²⋊C₄	—	89	1	no	—
208	C₄×C₂²⋊C₄	—	89	1	no	—
209	C₂²×D₄	—	89	1	no	—
210	C₂³×C₄	—	89	1	no	—
211	C₂×C₄⋊C₄	—	89	1	no	—
212	C₂×C₄⋊C₄	—	89	1	no	—
213	C₂×C₄⋊C₄	—	89	1	no	—
214	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
215	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
216	C₂×C₄⋊C₄	C₂×D₈₉	1	1	yes	characteristic
217	C₂×C₂²⋊C₄	—	89	1	no	—
218	C₂×C₂²⋊C₄	—	89	1	no	—
219	C₂×C₂²⋊C₄	—	89	1	no	—
220	C₂×C₂²⋊C₄	—	89	1	no	—
221	C₂×C₂²⋊C₄	—	89	1	no	—
222	C₂×C₂²⋊C₄	—	89	1	no	—
223	C₂×C₂²⋊C₄	—	89	1	no	—
224	C₂×C₂²⋊C₄	—	89	1	no	—
225	C₂×C₂²⋊C₄	C₂×D₈₉	1	1	yes	characteristic
226	C₂×C₂²⋊C₄	C₂×D₈₉	1	1	yes	characteristic
227	C₂×C₂²⋊C₄	C₂×D₈₉	1	1	yes	characteristic
228	C₂×C₄²	—	89	1	no	—
229	C₂×C₄²	—	89	1	no	—
230	C₂×C₄²	—	89	1	no	—
231	C₂×C₄²	—	89	1	no	—
232	#95	—	89	1	no	—
233	#95	—	89	1	no	—
234	#95	—	89	1	no	—
235	#95	—	89	1	no	—
236	#95	—	89	1	no	—
237	#95	—	89	1	no	—
238	#95	—	89	1	no	—
239	#95	—	89	1	no	—
240	#95	—	89	1	no	—
241	#95	—	89	1	no	—
242	#95	—	89	1	no	—
243	#95	C₂×D₈₉	1	1	yes	characteristic
244	C₂⁴	—	89	1	no	—
245	C₂⁴	C₂²×D₈₉	1	1	yes	characteristic
246	C₂×D₄	—	356	1	no	—
247	C₂×D₄	—	356	1	no	—
248	C₂×D₄	—	178	2	no	—
249	C₂²×C₄	—	89	1	no	—
250	C₂²×C₄	—	89	1	no	—
251	C₂²×C₄	—	89	1	no	—
252	C₂²×C₄	—	89	1	no	—
253	C₂²×C₄	—	89	1	no	—
254	C₂²×C₄	—	89	1	no	—
255	C₂²×C₄	—	89	1	no	—
256	C₂²×C₄	—	356	1	no	—
257	C₂²×C₄	—	178	2	no	—
258	C₂²×C₄	—	178	2	no	—
259	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
260	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
261	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
262	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
263	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
264	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	356	1	no	—
269	C₄⋊C₄	—	356	1	no	—
270	C₄⋊C₄	—	356	1	no	—
271	C₂²⋊C₄	—	2	2	no	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	356	1	no	—
275	C₂²⋊C₄	—	356	1	no	—
276	C₂²⋊C₄	—	178	2	no	—
277	C₂²⋊C₄	—	178	2	no	—
278	C₂²⋊C₄	—	178	2	no	—
279	C₂²⋊C₄	—	178	2	no	—
280	C₂²⋊C₄	—	178	2	no	—
281	C₂²⋊C₄	—	178	2	no	—
282	C₄²	—	356	1	no	—
283	C₄²	—	356	1	no	—
284	C₄²	—	178	2	no	—
285	C₄²	—	178	2	no	—
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	2	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	2	1	no	—
292	C₂³	—	178	1	no	—
293	C₂³	—	178	1	no	—
294	C₂³	—	178	1	no	—
295	C₂³	—	178	1	no	—
296	C₂³	—	178	1	no	—
297	C₂³	—	178	1	no	—
298	C₂³	—	178	1	no	—
299	C₂³	#13083575	1	2	yes	—
300	C₂³	C₂³×D₈₉	1	1	yes	characteristic
301	D₄	—	356	4	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	89	2	no	—
319	C₂×C₄	—	89	2	no	—
320	C₂×C₄	—	356	4	no	—
321	C₂×C₄	—	178	1	no	—
322	C₂×C₄	—	178	1	no	—
323	C₂×C₄	—	178	1	no	—
324	C₂×C₄	—	178	1	no	—
325	C₂×C₄	—	178	1	no	—
326	C₂×C₄	—	178	1	no	—
327	C₂×C₄	—	178	1	no	—
328	C₂×C₄	—	178	1	no	—
329	C₂×C₄	—	178	1	no	—
330	C₂×C₄	—	178	1	no	—
331	C₂×C₄	—	178	1	no	—
332	C₂×C₄	—	178	1	no	—
333	C₂×C₄	—	178	1	no	—
334	C₂×C₄	—	178	1	no	—
335	C₂×C₄	—	178	1	no	—
336	C₂×C₄	—	178	1	no	—
337	C₂×C₄	—	178	1	no	—
338	C₂×C₄	—	178	1	no	—
339	C₂×C₄	—	178	1	no	—
340	C₂×C₄	#13083573	1	2	yes	—
341	C₂×C₄	#13083573	1	2	yes	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	2	2	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	356	1	no	—
350	C₂²	—	356	1	no	—
351	C₂²	—	356	1	no	—
352	C₂²	—	356	1	no	—
353	C₂²	—	356	1	no	—
354	C₂²	—	356	1	no	—
355	C₂²	—	356	1	no	—
356	C₂²	#41×D₈₉	1	1	yes	characteristic
357	C₂²	#41×D₈₉	1	1	yes	characteristic
358	C₂²	C₂×#13083573	1	1	yes	characteristic
359	C₂²	C₂×#13083573	1	1	yes	characteristic
360	C₂²	C₂×#13083575	1	1	yes	characteristic
361	C₂²	#15668057	1	1	yes	characteristic
362	C₂²	#15668058	1	1	yes	characteristic
363	C₄	—	2	2	no	—
364	C₄	—	2	2	no	—
365	C₄	—	4	1	no	—
366	C₄	—	4	1	no	—
367	C₄	—	4	1	no	—
368	C₄	—	4	1	no	—
369	C₄	—	356	1	no	—
370	C₄	—	356	1	no	—
371	C₄	—	356	1	no	—
372	C₄	—	356	1	no	—
373	C₄	—	356	1	no	—
374	C₄	—	178	2	no	—
375	C₄	—	178	2	no	—
376	C₂	—	4	2	no	—
377	C₂	—	712	1	no	—
378	C₂	#19152134	1	1	yes	characteristic
379	C₂	#19152134	1	1	yes	characteristic
380	C₂	#19152188	1	1	yes	characteristic
381	C₂	#19152273	1	1	yes	characteristic
382	C₂	#19152291	1	1	yes	characteristic
383	C₂	#19152300	1	1	yes	characteristic
384	C₂	#19152312	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₈₉
commutator, socle, cosocle	C₂³×C₈₉	C₂⁴
Fitting	#396×C₈₉	C₂
prime radical	C₂²×C₂²⋊D₈₉	C₂²
hypercenter	#396	D₈₉
hypocenter	C₈₉	#2518
LCS	G ▷ C₂³×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ #396

Complements

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

N	H ≅ G/N	type
1. #19151523	C₂	semidirect
4. #19151497	C₂	semidirect
5. #19151490	C₂	semidirect
6. #19151488	C₂	semidirect
7. #19151488	C₂	semidirect
8. #19151488	C₂	semidirect
9. #19151484	C₂	semidirect
11. #19151465	C₂	semidirect
12. #19151463	C₂	semidirect
13. C₂²⋊C₄×Dic₈₉	C₂	semidirect
14. C₂²⋊C₄×Dic₈₉	C₂	semidirect
15. #396×C₈₉	C₂	semidirect
193. C₈₉	#2518	semidirect
195. #396	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19151523	2
#19151514	2
#19151497	2
#19151490	2
#19151488	2
#19151484	2
#19151475	2
#19151465	2
#19151463	2
C₂²⋊C₄×Dic₈₉	2
#396×C₈₉	2
#2518	89