Subgroups — #26030489

Order 11392.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152201	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152073	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151923	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151919	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151900	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151896	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151791	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151791	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151732	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151710	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151707	C₂	1	1	yes	maximal, characteristic, complemented
12	D₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151678	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151656	C₂	1	1	yes	maximal, characteristic, complemented
15	#447×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083559	C₂²	1	1	yes	characteristic
17	C₂×#13083559	C₂²	1	1	yes	characteristic
18	#15668008	C₂²	1	1	yes	characteristic, complemented
19	#15668003	C₂²	1	1	yes	characteristic, complemented
20	#15667996	—	2	1	no	—
21	#15667994	—	2	1	no	—
22	#15667986	C₂²	1	1	yes	characteristic
23	#15667984	—	2	1	no	—
24	#15667983	—	2	1	no	—
25	D₄×Dic₈₉	—	2	1	no	—
26	D₄×Dic₈₉	—	2	1	no	—
27	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
28	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
29	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
30	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
31	D₄×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
32	#15667974	C₂²	1	1	yes	characteristic, complemented
33	#15667971	C₂²	1	1	yes	characteristic
34	#15667971	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₈₉	—	2	1	no	—
36	C₂×C₂²⋊Dic₈₉	—	2	1	no	—
37	#15667946	C₂²	1	1	yes	characteristic, complemented
38	#15667945	C₂²	1	1	yes	characteristic, complemented
39	#15667944	C₂²	1	1	yes	characteristic, complemented
40	#15667943	C₂²	1	1	yes	characteristic, complemented
41	#15667941	C₂²	1	1	yes	characteristic, complemented
42	#15667941	C₂²	1	1	yes	characteristic, complemented
43	#15667934	C₂²	1	1	yes	characteristic, complemented
44	#15667934	C₂²	1	1	yes	characteristic, complemented
45	#15667933	C₂²	1	1	yes	characteristic
46	#15667933	C₂²	1	1	yes	characteristic
47	#15667933	C₂²	1	1	yes	characteristic, complemented
48	#15667933	C₂²	1	1	yes	characteristic, complemented
49	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
52	C₂²×D₄×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
55	#121×C₈₉	C₂²	1	1	yes	characteristic
56	#102×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₈₉	—	2	1	no	—
61	C₂×C₂²⋊D₈₉	—	2	1	no	—
62	C₂×C₂²⋊D₈₉	—	2	1	no	—
63	C₂²×Dic₈₉	—	2	1	no	—
64	C₂²×Dic₈₉	—	2	1	no	—
65	C₂²×Dic₈₉	—	2	1	no	—
66	C₂²×Dic₈₉	—	2	1	no	—
67	C₂²×Dic₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	2	1	no	—
69	#13083559	—	2	2	no	—
70	#13083559	—	2	2	no	—
71	C₂×#10623494	C₂³	1	1	yes	characteristic
72	C₂×#10623494	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₉	—	2	1	no	—
74	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
75	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
76	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
79	#13083553	—	2	1	no	—
80	C₂²⋊Dic₈₉	—	2	1	no	—
81	C₂²⋊Dic₈₉	—	2	1	no	—
82	C₂²⋊Dic₈₉	—	2	1	no	—
83	C₂²⋊Dic₈₉	—	2	1	no	—
84	C₂²⋊Dic₈₉	—	2	1	no	—
85	C₂²⋊Dic₈₉	—	2	1	no	—
86	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
88	#13083551	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	C₄×Dic₈₉	—	2	1	no	—
91	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
92	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
93	C₂⁴×C₈₉	—	2	1	no	—
94	C₂×D₄×C₈₉	—	2	2	no	—
95	C₂×D₄×C₈₉	—	2	2	no	—
96	C₂×D₄×C₈₉	—	2	1	no	—
97	C₂×D₄×C₈₉	—	2	1	no	—
98	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
102	D₈×C₈₉	—	2	2	no	—
103	D₈×C₈₉	—	2	2	no	—
104	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₈₉	—	2	1	no	—
108	C₂³×C₈₉	—	2	4	no	—
109	C₂³×C₈₉	—	2	1	no	—
110	C₂³×C₈₉	—	2	1	no	—
111	C₂³×C₈₉	—	2	1	no	—
112	C₂³×C₈₉	—	8	1	no	—
113	C₂³×C₈₉	—	8	1	no	—
114	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₈₉	—	2	1	no	—
116	D₄×C₈₉	—	2	4	no	—
117	D₄×C₈₉	—	2	1	no	—
118	D₄×C₈₉	—	2	1	no	—
119	D₄×C₈₉	—	2	1	no	—
120	D₄×C₈₉	—	2	1	no	—
121	D₄×C₈₉	—	8	1	no	—
122	D₄×C₈₉	#41	1	2	yes	—
123	D₄×C₈₉	#41	1	2	yes	—
124	C₂×C₄×C₈₉	—	2	1	no	—
125	C₂×C₄×C₈₉	—	2	1	no	—
126	C₂×C₄×C₈₉	—	2	1	no	—
127	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₈₉	—	8	1	no	—
130	C₂²⋊D₈₉	—	8	1	no	—
131	C₂²⋊D₈₉	—	8	1	no	—
132	C₂×Dic₈₉	—	2	1	no	—
133	C₂×Dic₈₉	—	2	1	no	—
134	C₂×Dic₈₉	—	2	1	no	—
135	C₂×Dic₈₉	—	8	1	no	—
136	C₂×Dic₈₉	—	8	1	no	—
137	C₂×Dic₈₉	—	8	1	no	—
138	C₂×Dic₈₉	—	8	1	no	—
139	C₂×Dic₈₉	—	8	1	no	—
140	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
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	C₄×D₈₉	—	8	1	no	—
147	C₈×C₈₉	—	2	1	no	—
148	C₈×C₈₉	—	2	1	no	—
149	#10623494	—	2	1	no	—
150	#10623494	—	2	1	no	—
151	C₂²×C₈₉	—	2	1	no	—
152	C₂²×C₈₉	—	2	2	no	—
153	C₂²×C₈₉	—	2	2	no	—
154	C₂²×C₈₉	—	2	1	no	—
155	C₂²×C₈₉	—	2	1	no	—
156	C₂²×C₈₉	—	4	1	no	—
157	C₂²×C₈₉	—	4	1	no	—
158	C₂²×C₈₉	—	4	1	no	—
159	C₂²×C₈₉	—	4	1	no	—
160	C₂²×C₈₉	—	4	1	no	—
161	C₂²×C₈₉	—	4	1	no	—
162	C₂²×C₈₉	—	4	1	no	—
163	C₂²×C₈₉	—	8	4	no	—
164	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
165	C₂×D₈₉	—	4	1	no	—
166	C₂×D₈₉	—	4	1	no	—
167	C₂×D₈₉	—	4	1	no	—
168	C₄×C₈₉	—	2	1	no	—
169	C₄×C₈₉	—	4	1	no	—
170	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
171	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
172	Dic₈₉	—	2	1	no	—
173	Dic₈₉	—	2	1	no	—
174	Dic₈₉	—	2	1	no	—
175	Dic₈₉	—	2	1	no	—
176	Dic₈₉	—	4	1	no	—
177	Dic₈₉	—	4	1	no	—
178	Dic₈₉	—	4	1	no	—
179	C₂×C₈₉	—	4	1	no	—
180	C₂×C₈₉	—	4	2	no	—
181	C₂×C₈₉	—	4	2	no	—
182	C₂×C₈₉	—	8	1	no	—
183	C₂×C₈₉	#525	1	1	yes	characteristic
184	C₂×C₈₉	#576	1	1	yes	characteristic
185	C₂×C₈₉	#576	1	1	yes	characteristic
186	D₈₉	—	8	1	no	—
187	#3122	—	89	1	no	maximal, Sylow
188	C₈₉	#3122	1	1	yes	characteristic, complemented, Sylow
189	#540	—	89	1	no	—
190	#518	—	89	1	no	—
191	#482	—	89	1	no	—
192	#482	—	89	1	no	—
193	#478	—	89	1	no	—
194	#478	—	89	1	no	—
195	#459	—	89	1	no	—
196	#459	—	89	1	no	—
197	#447	—	89	1	no	—
198	#447	D₈₉	1	1	yes	characteristic, complemented
199	#442	—	89	1	no	—
200	#442	—	89	1	no	—
201	C₄×D₈	—	89	1	no	—
202	C₄×D₈	—	89	1	no	—
203	#433	—	89	1	no	—
204	C₂²×D₄	C₂×D₈₉	1	1	yes	characteristic
205	C₂×D₈	—	89	1	no	—
206	C₂×D₈	—	89	1	no	—
207	C₂×D₈	C₂×D₈₉	1	1	yes	characteristic
208	C₂×D₈	C₂×D₈₉	1	1	yes	characteristic
209	#127	—	89	1	no	—
210	#125	—	89	1	no	—
211	#123	—	178	1	no	—
212	#123	—	178	1	no	—
213	#121	—	89	1	no	—
214	#121	—	178	1	no	—
215	#121	—	178	1	no	—
216	#121	C₂×D₈₉	1	1	yes	characteristic
217	C₄×D₄	—	89	1	no	—
218	C₄×D₄	—	89	1	no	—
219	C₄×D₄	—	89	1	no	—
220	C₄×D₄	—	89	1	no	—
221	C₄×D₄	—	89	1	no	—
222	C₄×D₄	—	89	1	no	—
223	C₄×D₄	—	178	1	no	—
224	C₄×D₄	—	178	1	no	—
225	#117	—	89	1	no	—
226	#117	—	89	1	no	—
227	C₂×C₂²⋊C₄	—	178	1	no	—
228	C₂×C₂²⋊C₄	—	178	1	no	—
229	#107	—	89	1	no	—
230	#107	—	89	1	no	—
231	#106	—	89	1	no	—
232	#106	—	89	1	no	—
233	C₄⋊C₈	—	89	1	no	—
234	C₄⋊C₈	—	89	1	no	—
235	#102	—	89	1	no	—
236	#102	—	89	1	no	—
237	#102	—	89	1	no	—
238	#102	—	89	1	no	—
239	#102	—	89	1	no	—
240	#102	—	89	1	no	—
241	#102	C₂×D₈₉	1	1	yes	characteristic
242	#102	C₂×D₈₉	1	1	yes	characteristic
243	C₂²⋊C₈	—	89	1	no	—
244	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
245	#97	—	89	1	no	—
246	C₄×C₈	—	89	1	no	—
247	C₂⁴	—	2	1	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	89	1	no	—
253	C₂×D₄	—	178	1	no	—
254	C₂×D₄	—	178	1	no	—
255	C₂×D₄	—	178	1	no	—
256	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
257	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
258	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
259	C₂²×C₄	—	178	1	no	—
260	C₂²×C₄	—	178	1	no	—
261	C₂²×C₄	—	178	1	no	—
262	C₂²×C₄	—	178	1	no	—
263	C₂²×C₄	—	178	1	no	—
264	C₂²×C₄	—	178	1	no	—
265	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
266	D₈	—	2	2	no	—
267	D₈	—	2	2	no	—
268	D₈	—	178	2	no	—
269	D₈	—	178	2	no	—
270	C₂×C₈	—	89	1	no	—
271	C₂×C₈	—	89	1	no	—
272	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
273	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
274	C₄⋊C₄	—	89	1	no	—
275	C₄⋊C₄	—	89	1	no	—
276	C₄⋊C₄	—	89	1	no	—
277	C₄⋊C₄	—	89	1	no	—
278	C₄⋊C₄	—	89	1	no	—
279	C₄⋊C₄	—	178	1	no	—
280	C₄⋊C₄	—	178	1	no	—
281	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
282	C₂²⋊C₄	—	2	1	no	—
283	C₂²⋊C₄	—	89	1	no	—
284	C₂²⋊C₄	—	89	1	no	—
285	C₂²⋊C₄	—	178	1	no	—
286	C₂²⋊C₄	—	178	1	no	—
287	C₂²⋊C₄	—	178	1	no	—
288	C₂²⋊C₄	—	178	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	C₄²	—	89	1	no	—
294	C₄²	—	89	1	no	—
295	C₄²	—	178	1	no	—
296	C₂³	—	2	4	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	8	1	no	—
301	C₂³	—	8	1	no	—
302	C₂³	—	178	1	no	—
303	C₂³	D₄×D₈₉	1	1	yes	characteristic
304	D₄	—	2	4	no	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	2	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	8	1	no	—
310	D₄	—	178	1	no	—
311	D₄	—	178	1	no	—
312	D₄	—	712	1	no	—
313	D₄	—	712	1	no	—
314	D₄	—	712	1	no	—
315	D₄	#13083573	1	2	yes	—
316	D₄	#13083573	1	2	yes	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	89	1	no	—
321	C₂×C₄	—	89	1	no	—
322	C₂×C₄	—	89	1	no	—
323	C₂×C₄	—	89	1	no	—
324	C₂×C₄	—	178	1	no	—
325	C₂×C₄	—	178	1	no	—
326	C₂×C₄	—	178	1	no	—
327	C₂×C₄	—	712	1	no	—
328	C₂×C₄	—	712	1	no	—
329	C₂×C₄	—	712	1	no	—
330	C₂×C₄	—	712	1	no	—
331	C₂×C₄	—	712	1	no	—
332	C₂×C₄	—	712	1	no	—
333	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
334	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	178	1	no	—
338	C₈	—	178	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	2	no	—
343	C₂²	—	2	2	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	8	4	no	—
352	C₂²	—	356	1	no	—
353	C₂²	—	356	1	no	—
354	C₂²	—	356	1	no	—
355	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	356	1	no	—
359	C₄	—	356	1	no	—
360	C₄	—	356	1	no	—
361	C₄	—	178	1	no	—
362	C₄	—	178	1	no	—
363	C₄	—	178	1	no	—
364	C₄	—	178	1	no	—
365	C₄	C₂×#13083573	1	1	yes	characteristic
366	C₄	C₂×#13083573	1	1	yes	characteristic
367	C₂	—	4	1	no	—
368	C₂	—	4	2	no	—
369	C₂	—	4	2	no	—
370	C₂	—	8	1	no	—
371	C₂	—	712	1	no	—
372	C₂	#19152104	1	1	yes	characteristic
373	C₂	#19152399	1	1	yes	characteristic
374	C₂	#19152402	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#447×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#447	D₈₉
hypocenter	C₈₉	#3122
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #447

Complements

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

N	H ≅ G/N	type
1. #19152201	C₂	semidirect
2. #19152073	C₂	semidirect
3. #19151923	C₂	semidirect
4. #19151919	C₂	semidirect
5. #19151900	C₂	semidirect
6. #19151896	C₂	semidirect
7. #19151791	C₂	semidirect
8. #19151791	C₂	semidirect
9. #19151732	C₂	semidirect
10. #19151710	C₂	semidirect
11. #19151707	C₂	semidirect
12. D₈×Dic₈₉	C₂	semidirect
13. #19151678	C₂	semidirect
14. #19151656	C₂	semidirect
15. #447×C₈₉	C₂	semidirect
18. #15668008	C₂²	semidirect
19. #15668003	C₂²	semidirect
31. D₄×Dic₈₉	C₂²	semidirect
32. #15667974	C₂²	semidirect
37. #15667946	C₂²	semidirect
38. #15667945	C₂²	semidirect
39. #15667944	C₂²	semidirect
40. #15667943	C₂²	semidirect
41. #15667941	C₂²	semidirect
42. #15667941	C₂²	semidirect
43. #15667934	C₂²	semidirect
44. #15667934	C₂²	semidirect
47. #15667933	C₂²	semidirect
48. #15667933	C₂²	semidirect
50. #15667922	C₂²	semidirect
51. C₈×Dic₈₉	C₂²	semidirect
188. C₈₉	#3122	semidirect
198. #447	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152201	2
#19152073	2
#19151923	2
#19151919	2
#19151900	2
#19151896	2
#19151791	2
#19151732	2
#19151710	2
#19151707	2
D₈×Dic₈₉	2
#19151678	2
#19151656	2
#447×C₈₉	2
#3122	89