Subgroups — #26031183

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	#19152144	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152143	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	#19151816	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151815	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151812	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151811	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151745	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151743	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151741	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151648	C₂	1	1	yes	maximal, characteristic, complemented
15	#449×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083574	C₂²	1	1	yes	characteristic
17	C₂×#13083573	—	2	1	no	—
18	C₂×#13083563	C₂²	1	1	yes	characteristic
19	C₂×#13083562	C₂²	1	1	yes	characteristic
20	C₂×#13083561	C₂²	1	1	yes	characteristic
21	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
22	C₂×#13083559	C₂²	1	1	yes	characteristic
23	C₂×#13083559	C₂²	1	1	yes	characteristic
24	#15668011	C₂²	1	1	yes	characteristic
25	#15668010	C₂²	1	1	yes	characteristic
26	#15668008	C₂²	1	1	yes	characteristic
27	#15668008	C₂²	1	1	yes	characteristic
28	#15668001	C₂²	1	1	yes	characteristic
29	#15667999	C₂²	1	1	yes	characteristic
30	#15667998	—	2	1	no	—
31	#15667986	C₂²	1	1	yes	characteristic
32	#15667984	—	2	1	no	—
33	#15667984	—	2	1	no	—
34	#15667984	—	2	1	no	—
35	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
36	#15667974	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊D₈₉	—	2	1	no	—
38	C₄×C₂²⋊D₈₉	—	2	1	no	—
39	#15667972	—	2	1	no	—
40	C₂×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
41	#15667941	C₂²	1	1	yes	characteristic
42	#15667941	C₂²	1	1	yes	characteristic
43	#15667937	C₂²	1	1	yes	characteristic
44	#15667936	C₂²	1	1	yes	characteristic
45	#15667935	C₂²	1	1	yes	characteristic
46	#15667934	C₂²	1	1	yes	characteristic
47	#15667933	C₂²	1	1	yes	characteristic
48	#15667932	C₂²	1	1	yes	characteristic
49	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
50	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
55	#121×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
59	#13083574	—	4	1	no	—
60	#13083574	—	4	1	no	—
61	#13083573	—	4	4	no	—
62	C₂×Dic₁₇₈	—	2	1	no	—
63	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
64	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
65	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₈₉	—	2	1	no	—
67	C₂×C₂²⋊D₈₉	—	2	1	no	—
68	C₂×C₂²⋊D₈₉	—	2	1	no	—
69	C₂×C₂²⋊D₈₉	—	2	1	no	—
70	C₂×C₂²⋊D₈₉	—	2	1	no	—
71	C₂×C₂²⋊D₈₉	—	2	1	no	—
72	C₂²×Dic₈₉	—	2	1	no	—
73	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
74	C₂²×Dic₈₉	D₄	1	1	yes	characteristic
75	C₂×C₄×D₈₉	—	2	1	no	—
76	C₂×C₄×D₈₉	—	2	1	no	—
77	#13083563	—	4	1	no	—
78	#13083562	—	4	1	no	—
79	#13083561	—	4	1	no	—
80	D₇₁₂	—	4	1	no	—
81	#13083559	—	4	1	no	—
82	#13083559	—	4	1	no	—
83	C₂×#10623494	C₂³	1	1	yes	characteristic
84	C₂×#10623494	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
86	#13083553	—	2	1	no	—
87	#13083553	—	2	1	no	—
88	#13083553	—	2	1	no	—
89	C₂²⋊Dic₈₉	—	2	1	no	—
90	C₂²⋊Dic₈₉	—	2	1	no	—
91	C₂²⋊Dic₈₉	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	#13083551	—	2	1	no	—
94	#13083551	—	2	1	no	—
95	#13083551	—	2	1	no	—
96	C₄×Dic₈₉	D₄	1	1	yes	characteristic
97	C₄×Dic₈₉	D₄	1	1	yes	characteristic
98	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
99	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
100	#41×C₈₉	—	4	1	no	—
101	#41×C₈₉	—	4	1	no	—
102	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₂×D₄×C₈₉	—	2	1	no	—
104	C₂×D₄×C₈₉	—	2	1	no	—
105	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
106	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
107	C₂²×C₄×C₈₉	—	2	1	no	—
108	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
109	SD₁₆×C₈₉	—	4	1	no	—
110	D₈×C₈₉	—	4	1	no	—
111	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
113	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
114	C₂²⋊C₄×C₈₉	—	2	1	no	—
115	C₂³×C₈₉	—	2	1	no	—
116	C₂³×C₈₉	—	2	1	no	—
117	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂²×D₈₉	—	2	1	no	—
119	C₂²×D₈₉	—	2	1	no	—
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	D₄×C₈₉	—	4	2	no	—
127	D₄×C₈₉	—	4	2	no	—
128	C₂×C₄×C₈₉	—	2	1	no	—
129	C₂×C₄×C₈₉	—	2	1	no	—
130	C₂×C₄×C₈₉	—	2	1	no	—
131	C₂×C₄×C₈₉	—	2	1	no	—
132	C₂×C₄×C₈₉	—	4	2	no	—
133	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
135	C₂²⋊D₈₉	—	4	2	no	—
136	C₂²⋊D₈₉	—	4	2	no	—
137	C₂²⋊D₈₉	—	4	2	no	—
138	C₂²⋊D₈₉	—	4	2	no	—
139	C₂²⋊D₈₉	—	4	2	no	—
140	C₂²⋊D₈₉	—	4	2	no	—
141	C₂×Dic₈₉	—	2	2	no	—
142	C₂×Dic₈₉	—	2	2	no	—
143	C₂×Dic₈₉	—	2	1	no	—
144	C₂×Dic₈₉	—	2	1	no	—
145	C₂×Dic₈₉	—	4	2	no	—
146	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
147	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
148	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
149	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
150	D₃₅₆	—	2	1	no	—
151	D₃₅₆	—	2	1	no	—
152	D₃₅₆	—	2	1	no	—
153	D₃₅₆	—	2	1	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	—	2	1	no	—
164	C₂²×C₈₉	—	2	1	no	—
165	C₂²×C₈₉	—	2	1	no	—
166	C₂²×C₈₉	—	4	1	no	—
167	C₂²×C₈₉	—	4	1	no	—
168	C₂²×C₈₉	—	4	1	no	—
169	C₂²×C₈₉	—	4	1	no	—
170	C₂²×C₈₉	—	4	1	no	—
171	C₂²×C₈₉	—	4	1	no	—
172	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
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	—	4	1	no	—
181	C₄×C₈₉	—	4	1	no	—
182	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
183	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
184	Dic₈₉	—	2	1	no	—
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	—	4	1	no	—
191	C₂×C₈₉	—	8	1	no	—
192	C₂×C₈₉	—	8	1	no	—
193	C₂×C₈₉	#535	1	1	yes	characteristic
194	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
195	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
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	—	89	1	no	—
210	#449	—	89	1	no	—
211	#449	—	89	1	no	—
212	#449	—	89	1	no	—
213	#449	D₈₉	1	1	yes	characteristic, complemented
214	#431	—	89	1	no	—
215	C₂×#41	—	89	1	no	—
216	C₂×#41	—	178	1	no	—
217	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
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	—	89	1	no	—
232	#124	—	178	1	no	—
233	#121	—	89	1	no	—
234	#121	—	178	1	no	—
235	#121	—	178	1	no	—
236	#121	—	178	1	no	—
237	#121	C₂×D₈₉	1	1	yes	characteristic
238	C₄×D₄	—	89	1	no	—
239	C₄×D₄	—	89	1	no	—
240	C₄×D₄	—	178	1	no	—
241	C₄×D₄	—	178	1	no	—
242	C₄×D₄	—	178	1	no	—
243	C₂×C₄²	—	89	1	no	—
244	C₄⋊C₈	—	89	1	no	—
245	C₄⋊C₈	—	89	1	no	—
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	C₂×D₈₉	1	1	yes	characteristic
256	C₄×C₈	—	89	1	no	—
257	C₄×C₈	—	89	1	no	—
258	#41	—	4	1	no	—
259	#41	—	4	1	no	—
260	#41	—	356	1	no	—
261	#41	—	356	1	no	—
262	#41	—	356	4	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	—	2	1	no	—
267	C₂×D₄	—	2	1	no	—
268	C₂×D₄	—	89	1	no	—
269	C₂×D₄	—	89	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	—	178	1	no	—
282	C₂²×C₄	—	178	1	no	—
283	C₂²×C₄	—	178	1	no	—
284	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
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	—	178	1	no	—
304	C₂²⋊C₄	—	178	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	—	89	1	no	—
314	C₂³	—	2	1	no	—
315	C₂³	—	2	1	no	—
316	C₂³	—	178	1	no	—
317	C₂³	—	178	1	no	—
318	C₂³	D₄×D₈₉	1	1	yes	characteristic
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	—	4	2	no	—
329	D₄	—	4	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	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	89	1	no	—
346	C₂×C₄	—	89	1	no	—
347	C₂×C₄	—	89	1	no	—
348	C₂×C₄	—	89	1	no	—
349	C₂×C₄	—	356	2	no	—
350	C₂×C₄	—	356	2	no	—
351	C₂×C₄	—	356	2	no	—
352	C₂×C₄	—	178	2	no	—
353	C₂×C₄	—	178	2	no	—
354	C₂×C₄	—	178	1	no	—
355	C₂×C₄	—	178	1	no	—
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	—	2	1	no	—
363	C₂²	—	2	1	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	4	1	no	—
366	C₂²	—	4	1	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	4	1	no	—
370	C₂²	—	4	1	no	—
371	C₂²	—	356	1	no	—
372	C₂²	—	356	1	no	—
373	C₂²	—	356	1	no	—
374	C₂²	—	356	1	no	—
375	C₂²	—	356	1	no	—
376	C₂²	—	356	1	no	—
377	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
378	C₄	—	2	1	no	—
379	C₄	—	4	1	no	—
380	C₄	—	4	1	no	—
381	C₄	—	89	2	no	—
382	C₄	—	89	2	no	—
383	C₄	—	356	1	no	—
384	C₄	—	356	1	no	—
385	C₄	—	178	1	no	—
386	C₄	—	178	1	no	—
387	C₄	#15668057	1	1	yes	characteristic
388	C₄	#15668057	1	1	yes	characteristic
389	C₂	—	4	1	no	—
390	C₂	—	8	1	no	—
391	C₂	—	8	1	no	—
392	C₂	—	712	1	no	—
393	C₂	—	712	1	no	—
394	C₂	#19152163	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	#449×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#449	D₈₉
hypocenter	C₈₉	#3207
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #449

Complements

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

N	H ≅ G/N	type
1. #19152144	C₂	semidirect
2. #19152143	C₂	semidirect
3. #19151973	C₂	semidirect
4. #19151969	C₂	semidirect
5. #19151947	C₂	semidirect
6. #19151943	C₂	semidirect
7. #19151816	C₂	semidirect
8. #19151815	C₂	semidirect
9. #19151812	C₂	semidirect
10. #19151811	C₂	semidirect
11. #19151745	C₂	semidirect
12. #19151743	C₂	semidirect
13. #19151741	C₂	semidirect
14. #19151648	C₂	semidirect
15. #449×C₈₉	C₂	semidirect
199. C₈₉	#3207	semidirect
213. #449	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152144	2
#19152143	2
#19151973	2
#19151969	2
#19151947	2
#19151943	2
#19151816	2
#19151815	2
#19151812	2
#19151811	2
#19151745	2
#19151743	2
#19151741	2
#19151648	2
#449×C₈₉	2
#3207	89