Subgroups — #26030304

Order 11392.

454 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152133	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151935	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151918	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151895	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151814	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151813	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151750	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151733	C₂	1	1	yes	maximal, characteristic
10	#19151689	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151685	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151680	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151678	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151650	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₈₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₈₉	—	2	1	no	—
18	C₂×#13083563	C₂²	1	1	yes	characteristic
19	C₂×#13083562	C₂²	1	1	yes	characteristic
20	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
21	C₂×#13083559	C₂²	1	1	yes	characteristic
22	#15668007	—	2	1	no	—
23	#15668000	C₂²	1	1	yes	characteristic, complemented
24	#15667997	—	2	1	no	—
25	#15667992	C₂²	1	1	yes	characteristic
26	#15667987	C₂²	1	1	yes	characteristic
27	#15667977	C₂²	1	1	yes	characteristic, complemented
28	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
29	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
30	#15667974	—	2	1	no	—
31	#15667974	C₂²	1	1	yes	characteristic
32	#15667974	C₂²	1	1	yes	characteristic, complemented
33	C₄×C₂²⋊D₈₉	—	2	1	no	—
34	#15667972	—	2	1	no	—
35	#15667970	—	2	1	no	—
36	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
37	C₂²⋊C₄×D₈₉	—	2	1	no	—
38	C₄²×D₈₉	C₂²	1	1	yes	characteristic
39	#15667945	C₂²	1	1	yes	characteristic, complemented
40	#15667943	C₂²	1	1	yes	characteristic, complemented
41	C₄⋊#10623494	C₂²	1	1	yes	characteristic, complemented
42	#15667937	C₂²	1	1	yes	characteristic, complemented
43	#15667935	C₂²	1	1	yes	characteristic, complemented
44	#15667934	C₂²	1	1	yes	characteristic, complemented
45	#15667932	C₂²	1	1	yes	characteristic
46	#15667932	C₂²	1	1	yes	characteristic
47	#15667932	C₂²	1	1	yes	characteristic, complemented
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	#125×C₈₉	C₂²	1	1	yes	characteristic, complemented
53	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
54	#107×C₈₉	C₂²	1	1	yes	characteristic, complemented
55	#106×C₈₉	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	#102×C₈₉	C₂²	1	1	yes	characteristic
59	C₂³×D₈₉	—	2	1	no	—
60	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
61	C₂×D₃₅₆	—	2	1	no	—
62	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
63	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₈₉	—	2	1	no	—
65	D₄×D₈₉	—	2	2	no	—
66	D₄×D₈₉	—	2	2	no	—
67	C₂²×Dic₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	—	2	1	no	—
72	C₂×C₄×D₈₉	—	4	4	no	—
73	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
74	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
75	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
76	#13083563	—	2	2	no	—
77	#13083562	—	2	2	no	—
78	D₇₁₂	—	2	2	no	—
79	#13083559	—	2	2	no	—
80	C₂×#10623494	C₂³	1	1	yes	characteristic
81	C₂×#10623494	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₉	—	2	1	no	—
83	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
84	#13083553	—	2	1	no	—
85	#13083553	—	2	1	no	—
86	C₂²⋊Dic₈₉	—	2	1	no	—
87	#13083551	—	2	1	no	—
88	#13083551	—	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	C₄×Dic₈₉	—	2	1	no	—
95	C₄×Dic₈₉	D₄	1	1	yes	characteristic
96	C₄×Dic₈₉	D₄	1	1	yes	characteristic
97	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	—	2	1	no	—
100	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	—	2	1	no	—
103	C₄⋊C₄×C₈₉	—	2	1	no	—
104	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
107	C₂²⋊C₄×C₈₉	—	2	1	no	—
108	C₄²×C₈₉	C₂³	1	1	yes	characteristic
109	C₂³×C₈₉	—	2	1	no	—
110	C₂²×D₈₉	—	2	4	no	—
111	C₂²×D₈₉	—	2	1	no	—
112	C₂²×D₈₉	—	2	1	no	—
113	C₂²×D₈₉	—	4	2	no	—
114	C₂²×D₈₉	—	4	2	no	—
115	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
116	D₄×C₈₉	—	2	1	no	—
117	D₄×C₈₉	#41	1	2	yes	—
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	—	4	2	no	—
122	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₈₉	—	2	4	no	—
126	C₂×Dic₈₉	—	2	1	no	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	4	2	no	—
129	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
132	D₃₅₆	—	2	1	no	—
133	D₃₅₆	—	2	1	no	—
134	D₃₅₆	—	2	1	no	—
135	D₃₅₆	—	4	2	no	—
136	D₃₅₆	#41	1	2	yes	—
137	C₄×D₈₉	—	2	2	no	—
138	C₄×D₈₉	—	2	1	no	—
139	C₄×D₈₉	—	2	1	no	—
140	C₄×D₈₉	—	2	2	no	—
141	C₄×D₈₉	—	4	2	no	—
142	C₄×D₈₉	—	4	2	no	—
143	C₄×D₈₉	—	4	2	no	—
144	C₄×D₈₉	—	4	2	no	—
145	Dic₁₇₈	—	2	1	no	—
146	Dic₁₇₈	—	2	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	2	no	—
152	C₂²×C₈₉	—	4	1	no	—
153	C₂²×C₈₉	—	4	1	no	—
154	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₈₉	—	2	1	no	—
156	C₂×D₈₉	—	2	1	no	—
157	C₂×D₈₉	—	2	2	no	—
158	C₂×D₈₉	—	4	8	no	—
159	C₂×D₈₉	—	4	1	no	—
160	C₂×D₈₉	—	4	1	no	—
161	C₂×D₈₉	—	4	1	no	—
162	C₂×D₈₉	—	4	1	no	—
163	C₂×D₈₉	—	4	1	no	—
164	C₂×D₈₉	#135	1	2	yes	—
165	C₄×C₈₉	—	2	1	no	—
166	C₄×C₈₉	—	2	1	no	—
167	C₄×C₈₉	—	4	1	no	—
168	C₄×C₈₉	—	4	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	—	4	1	no	—
175	Dic₈₉	—	4	1	no	—
176	Dic₈₉	AGL(1,ℤ/8)	1	2	yes	—
177	C₂×C₈₉	—	4	2	no	—
178	C₂×C₈₉	#525	1	1	yes	characteristic
179	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
180	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	D₈₉	—	2	2	no	—
182	D₈₉	—	4	2	no	—
183	D₈₉	—	8	1	no	—
184	#3105	—	89	1	no	maximal, Sylow
185	C₈₉	#3105	1	1	yes	characteristic, complemented, Sylow
186	#529	—	89	1	no	—
187	C₂×C₄×D₄	—	89	1	no	—
188	#484	—	89	1	no	—
189	#481	—	89	1	no	—
190	#478	—	89	1	no	—
191	#478	D₈₉	1	1	yes	characteristic, complemented
192	#463	—	89	1	no	—
193	#463	—	89	1	no	—
194	#450	—	89	1	no	—
195	#447	—	89	1	no	—
196	C₄×SD₁₆	—	89	1	no	—
197	C₄×SD₁₆	—	89	1	no	—
198	C₄×D₈	—	89	1	no	—
199	C₄×D₈	—	89	1	no	—
200	#431	—	89	1	no	—
201	C₂²×D₄	—	89	1	no	—
202	C₂³×C₄	—	178	1	no	—
203	C₂×SD₁₆	—	89	1	no	—
204	C₂×SD₁₆	—	89	1	no	—
205	C₂×D₈	—	89	1	no	—
206	C₂×D₈	—	89	1	no	—
207	#126	—	178	1	no	—
208	#125	C₂×D₈₉	1	1	yes	characteristic, complemented
209	#124	—	89	1	no	—
210	#123	—	178	1	no	—
211	C₂²⋊Q₈	—	89	1	no	—
212	#121	—	89	1	no	—
213	C₄×Q₈	—	89	1	no	—
214	C₄×D₄	—	89	1	no	—
215	C₄×D₄	—	89	1	no	—
216	C₄×D₄	—	89	1	no	—
217	C₄×D₄	—	89	1	no	—
218	C₄×D₄	—	178	1	no	—
219	C₄×D₄	—	178	1	no	—
220	C₄×D₄	—	178	1	no	—
221	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
222	#117	—	178	1	no	—
223	C₂×C₄⋊C₄	—	89	1	no	—
224	C₂×C₂²⋊C₄	—	178	1	no	—
225	C₂×C₄²	—	89	1	no	—
226	#107	—	89	1	no	—
227	#107	C₂×D₈₉	1	1	yes	characteristic, complemented
228	#106	—	89	1	no	—
229	#106	C₂×D₈₉	1	1	yes	characteristic, complemented
230	C₄⋊C₈	—	89	1	no	—
231	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
232	#103	—	89	1	no	—
233	#103	—	89	1	no	—
234	#102	—	89	1	no	—
235	#102	—	89	1	no	—
236	#102	—	89	1	no	—
237	#102	—	89	1	no	—
238	#102	C₂×D₈₉	1	1	yes	characteristic
239	#102	C₂×D₈₉	1	1	yes	characteristic
240	C₂²⋊C₈	—	89	1	no	—
241	C₂²⋊C₈	—	89	1	no	—
242	C₄×C₈	—	89	1	no	—
243	C₄×C₈	—	89	1	no	—
244	C₂⁴	—	178	1	no	—
245	C₂×Q₈	—	89	1	no	—
246	C₂×D₄	—	89	1	no	—
247	C₂×D₄	—	89	1	no	—
248	C₂×D₄	—	178	1	no	—
249	C₂×D₄	—	178	1	no	—
250	C₂×D₄	—	178	2	no	—
251	C₂×D₄	—	178	2	no	—
252	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	89	1	no	—
255	C₂²×C₄	—	89	1	no	—
256	C₂²×C₄	—	89	1	no	—
257	C₂²×C₄	—	356	4	no	—
258	C₂²×C₄	—	178	1	no	—
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	SD₁₆	—	178	2	no	—
264	SD₁₆	—	178	2	no	—
265	D₈	—	178	2	no	—
266	D₈	—	178	2	no	—
267	C₂×C₈	—	89	1	no	—
268	C₂×C₈	—	89	1	no	—
269	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	89	1	no	—
274	C₄⋊C₄	—	178	1	no	—
275	C₄⋊C₄	—	178	1	no	—
276	C₄⋊C₄	—	178	1	no	—
277	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
278	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
279	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	178	1	no	—
282	C₂²⋊C₄	—	178	1	no	—
283	C₂²⋊C₄	—	178	1	no	—
284	C₂²⋊C₄	—	178	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	C₄²	—	89	1	no	—
290	C₄²	—	89	1	no	—
291	C₄²	—	89	1	no	—
292	C₄²	—	178	1	no	—
293	C₄²	C₂²×D₈₉	1	1	yes	characteristic
294	C₂³	—	2	1	no	—
295	C₂³	—	89	1	no	—
296	C₂³	—	356	2	no	—
297	C₂³	—	356	2	no	—
298	C₂³	—	178	4	no	—
299	C₂³	—	178	1	no	—
300	C₂³	—	178	1	no	—
301	Q₈	—	178	1	no	—
302	Q₈	—	178	1	no	—
303	D₄	—	2	1	no	—
304	D₄	—	89	2	no	—
305	D₄	—	356	2	no	—
306	D₄	—	178	4	no	—
307	D₄	—	178	1	no	—
308	D₄	—	178	1	no	—
309	D₄	—	178	1	no	—
310	D₄	#13083575	1	2	yes	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	89	1	no	—
316	C₂×C₄	—	89	1	no	—
317	C₂×C₄	—	89	1	no	—
318	C₂×C₄	—	356	2	no	—
319	C₂×C₄	—	356	2	no	—
320	C₂×C₄	—	356	2	no	—
321	C₂×C₄	—	356	2	no	—
322	C₂×C₄	—	356	2	no	—
323	C₂×C₄	—	178	2	no	—
324	C₂×C₄	—	178	2	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	D₄×D₈₉	1	1	yes	characteristic
330	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
331	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	178	1	no	—
335	C₈	—	178	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	89	2	no	—
340	C₂²	—	356	8	no	—
341	C₂²	—	356	1	no	—
342	C₂²	—	356	1	no	—
343	C₂²	—	356	1	no	—
344	C₂²	—	356	1	no	—
345	C₂²	—	356	1	no	—
346	C₂²	—	178	2	no	—
347	C₂²	—	178	1	no	—
348	C₂²	—	178	1	no	—
349	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	89	2	no	—
356	C₄	—	356	1	no	—
357	C₄	—	356	1	no	—
358	C₄	—	178	1	no	—
359	C₄	—	178	1	no	—
360	C₄	#41×D₈₉	1	1	yes	characteristic
361	C₄	C₂×#13083575	1	1	yes	characteristic
362	C₂	—	4	2	no	—
363	C₂	—	356	2	no	—
364	C₂	—	178	2	no	—
365	C₂	—	712	1	no	—
366	C₂	#19152111	1	1	yes	characteristic
367	C₂	#135×D₈₉	1	1	yes	characteristic
368	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	#478×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19151733	C₂
hypercenter	#478	D₈₉
hypocenter	C₈₉	#3105
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #19152133	C₂	semidirect
2. C₄×D₄×D₈₉	C₂	semidirect
3. #19151935	C₂	semidirect
4. #19151918	C₂	semidirect
5. #19151895	C₂	semidirect
6. #19151814	C₂	semidirect
7. #19151813	C₂	semidirect
8. #19151750	C₂	semidirect
10. #19151689	C₂	semidirect
11. #19151685	C₂	semidirect
12. #19151680	C₂	semidirect
13. #19151678	C₂	semidirect
14. #19151650	C₂	semidirect
15. #478×C₈₉	C₂	semidirect
23. #15668000	C₂²	semidirect
27. #15667977	C₂²	semidirect
32. #15667974	C₂²	semidirect
39. #15667945	C₂²	semidirect
40. #15667943	C₂²	semidirect
41. C₄⋊#10623494	C₂²	semidirect
42. #15667937	C₂²	semidirect
43. #15667935	C₂²	semidirect
44. #15667934	C₂²	semidirect
47. #15667932	C₂²	semidirect
50. C₈×Dic₈₉	C₂²	semidirect
51. C₈×Dic₈₉	C₂²	semidirect
52. #125×C₈₉	C₂²	semidirect
54. #107×C₈₉	C₂²	semidirect
55. #106×C₈₉	C₂²	semidirect
56. C₄⋊C₈×C₈₉	C₂²	semidirect
185. C₈₉	#3105	semidirect
191. #478	D₈₉	semidirect
208. #125	C₂×D₈₉	semidirect
227. #107	C₂×D₈₉	semidirect
229. #106	C₂×D₈₉	semidirect
231. C₄⋊C₈	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152133	2
C₄×D₄×D₈₉	2
#19151935	2
#19151918	2
#19151895	2
#19151814	2
#19151813	2
#19151750	2
#19151733	2
#19151689	2
#19151685	2
#19151680	2
#19151678	2
#19151650	2
#478×C₈₉	2
#3105	89