Subgroups — #26032724

Order 11392.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152294	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152252	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152007	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151923	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151917	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151917	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151912	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151899	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151889	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151851	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151834	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151711	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151640	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#15667944	C₂	1	1	yes	maximal, characteristic, complemented
15	#435×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083573	—	2	1	no	—
17	C₂×D₇₁₂	C₂²	1	1	yes	characteristic
18	#15668012	C₂²	1	1	yes	characteristic
19	#15668007	—	2	1	no	—
20	#15668004	C₂²	1	1	yes	characteristic
21	#15667996	—	2	1	no	—
22	#15667996	—	2	1	no	—
23	#15667994	—	2	1	no	—
24	#15667992	—	2	1	no	—
25	#15667986	C₂²	1	1	yes	characteristic
26	#15667986	C₂²	1	1	yes	characteristic
27	#15667986	C₂²	1	1	yes	characteristic
28	#15667986	C₂²	1	1	yes	characteristic
29	#15667984	—	2	1	no	—
30	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
31	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
32	#15667972	—	2	1	no	—
33	#15667971	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
37	#15667946	C₂²	1	1	yes	characteristic
38	#15667946	C₂²	1	1	yes	characteristic
39	#15667946	C₂²	1	1	yes	characteristic
40	#15667944	C₂²	1	1	yes	characteristic
41	#15667944	C₂²	1	1	yes	characteristic
42	#15667944	C₂²	1	1	yes	characteristic
43	#15667944	C₂²	1	1	yes	characteristic
44	#15667944	C₂²	1	1	yes	characteristic
45	#15667944	C₂²	1	1	yes	characteristic
46	#15667934	C₂²	1	1	yes	characteristic
47	#15667934	C₂²	1	1	yes	characteristic
48	#15667934	C₂²	1	1	yes	characteristic
49	#15667934	C₂²	1	1	yes	characteristic
50	#15667934	C₂²	1	1	yes	characteristic
51	#15667934	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
58	#97×C₈₉	C₂²	1	1	yes	characteristic
59	#13083573	—	4	4	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
62	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₈₉	—	2	1	no	—
64	C₂×C₂²⋊D₈₉	—	2	1	no	—
65	C₂×C₂²⋊D₈₉	—	2	1	no	—
66	C₂×C₂²⋊D₈₉	—	2	1	no	—
67	C₂²×Dic₈₉	—	2	1	no	—
68	C₂²×Dic₈₉	—	2	1	no	—
69	C₂²×Dic₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	—	2	1	no	—
72	D₇₁₂	—	2	2	no	—
73	C₄⋊Dic₈₉	—	2	1	no	—
74	C₄⋊Dic₈₉	—	2	1	no	—
75	C₄⋊Dic₈₉	—	2	1	no	—
76	C₄⋊Dic₈₉	—	2	1	no	—
77	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	C₂²⋊Dic₈₉	—	2	1	no	—
86	#13083551	—	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	C₄×Dic₈₉	—	2	1	no	—
93	C₂×D₄×C₈₉	D₄	1	1	yes	characteristic
94	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
96	M₁₆×C₈₉	—	2	2	no	—
97	C₂×C₈×C₈₉	—	2	2	no	—
98	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
103	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
105	C₄²×C₈₉	C₂³	1	1	yes	characteristic
106	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₈₉	—	2	1	no	—
109	C₂²×D₈₉	—	2	1	no	—
110	D₄×C₈₉	—	2	2	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	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₈₉	—	4	2	no	—
121	C₂²⋊D₈₉	—	4	2	no	—
122	C₂²⋊D₈₉	—	4	2	no	—
123	C₂²⋊D₈₉	—	8	1	no	—
124	C₂×Dic₈₉	—	2	1	no	—
125	C₂×Dic₈₉	—	2	1	no	—
126	C₂×Dic₈₉	—	2	1	no	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	—	2	1	no	—
129	C₂×Dic₈₉	—	2	1	no	—
130	C₂×Dic₈₉	—	4	2	no	—
131	C₂×Dic₈₉	—	4	2	no	—
132	C₂×Dic₈₉	—	8	1	no	—
133	D₃₅₆	—	2	1	no	—
134	D₃₅₆	—	2	1	no	—
135	D₃₅₆	—	2	1	no	—
136	D₃₅₆	—	2	1	no	—
137	C₄×D₈₉	—	4	2	no	—
138	C₄×D₈₉	—	8	1	no	—
139	Dic₁₇₈	—	4	2	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	—	2	1	no	—
143	C₈×C₈₉	#41	1	2	yes	complemented
144	C₂²×C₈₉	—	2	1	no	—
145	C₂²×C₈₉	—	2	1	no	—
146	C₂²×C₈₉	—	2	1	no	—
147	C₂²×C₈₉	—	2	1	no	—
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
150	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₈₉	—	4	1	no	—
152	C₂×D₈₉	—	4	1	no	—
153	C₂×D₈₉	—	4	1	no	—
154	C₂×D₈₉	—	4	1	no	—
155	C₂×D₈₉	—	4	1	no	—
156	C₂×D₈₉	—	4	1	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	2	1	no	—
160	C₄×C₈₉	—	2	1	no	—
161	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
162	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
163	Dic₈₉	—	4	1	no	—
164	Dic₈₉	—	4	1	no	—
165	Dic₈₉	—	4	1	no	—
166	Dic₈₉	—	4	1	no	—
167	Dic₈₉	—	4	1	no	—
168	Dic₈₉	—	4	1	no	—
169	C₂×C₈₉	—	2	2	no	—
170	C₂×C₈₉	—	4	1	no	—
171	C₂×C₈₉	#547	1	1	yes	characteristic
172	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	C₂×C₈₉	#577	1	1	yes	characteristic
174	D₈₉	—	8	1	no	—
175	D₈₉	—	8	1	no	—
176	#3345	—	89	1	no	maximal, Sylow
177	C₈₉	#3345	1	1	yes	characteristic, complemented, Sylow
178	#553	—	89	1	no	—
179	#547	—	89	1	no	—
180	#501	—	89	1	no	—
181	#482	—	89	1	no	—
182	#481	—	89	1	no	—
183	#481	—	89	1	no	—
184	#480	—	89	1	no	—
185	#478	—	89	1	no	—
186	#476	—	89	1	no	—
187	#469	—	89	1	no	—
188	#466	—	89	1	no	—
189	#442	—	89	1	no	—
190	#435	D₈₉	1	1	yes	characteristic, complemented
191	#428	—	89	1	no	—
192	C₂×#106	—	89	1	no	—
193	C₂×#41	—	178	1	no	—
194	C₂×D₈	—	89	1	no	—
195	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
196	C₂²×C₈	C₂×D₈₉	1	1	yes	characteristic
197	C₄⋊Q₈	—	89	1	no	—
198	#126	—	178	1	no	—
199	#125	—	89	1	no	—
200	#123	—	178	1	no	—
201	#123	—	178	1	no	—
202	#123	—	178	1	no	—
203	C₂²⋊Q₈	—	178	1	no	—
204	#121	—	89	1	no	—
205	#121	—	89	1	no	—
206	#121	—	89	1	no	—
207	#121	—	89	1	no	—
208	#121	—	178	1	no	—
209	C₄×D₄	—	89	1	no	—
210	C₄×D₄	—	89	1	no	—
211	C₄×D₄	—	178	1	no	—
212	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
213	#117	—	89	1	no	—
214	C₂×C₄⋊C₄	—	89	1	no	—
215	C₂×C₄⋊C₄	—	89	1	no	—
216	C₂×C₄⋊C₄	—	89	1	no	—
217	#107	—	89	1	no	—
218	#107	—	89	1	no	—
219	#107	—	89	1	no	—
220	#106	—	89	1	no	—
221	#106	—	89	1	no	—
222	#106	—	89	1	no	—
223	#106	—	89	1	no	—
224	#106	—	89	1	no	—
225	#106	—	89	1	no	—
226	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
227	#102	—	89	1	no	—
228	#102	—	89	1	no	—
229	#102	—	89	1	no	—
230	#102	—	89	1	no	—
231	#102	—	89	1	no	—
232	#102	—	89	1	no	—
233	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
234	C₂²⋊C₈	C₂×D₈₉	1	1	yes	characteristic
235	#97	C₂×D₈₉	1	1	yes	characteristic
236	#41	—	356	4	no	—
237	C₂×Q₈	—	178	1	no	—
238	C₂×D₄	—	89	1	no	—
239	C₂×D₄	—	89	1	no	—
240	C₂×D₄	—	178	1	no	—
241	C₂×D₄	—	178	1	no	—
242	C₂×D₄	—	178	1	no	—
243	C₂×D₄	—	178	1	no	—
244	C₂×D₄	D₃₅₆	1	1	yes	characteristic
245	C₂²×C₄	—	178	1	no	—
246	C₂²×C₄	—	178	1	no	—
247	C₂²×C₄	—	178	1	no	—
248	C₂²×C₄	—	178	1	no	—
249	C₂²×C₄	—	178	1	no	—
250	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
251	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
252	D₈	—	178	2	no	—
253	M₁₆	—	2	2	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
257	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
259	C₄⋊C₄	—	89	1	no	—
260	C₄⋊C₄	—	89	1	no	—
261	C₄⋊C₄	—	89	1	no	—
262	C₄⋊C₄	—	89	1	no	—
263	C₄⋊C₄	—	89	1	no	—
264	C₄⋊C₄	—	89	1	no	—
265	C₄⋊C₄	—	178	1	no	—
266	C₄⋊C₄	—	178	1	no	—
267	C₄⋊C₄	—	178	1	no	—
268	C₄⋊C₄	—	178	1	no	—
269	C₄⋊C₄	—	178	1	no	—
270	C₄⋊C₄	—	178	1	no	—
271	C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
272	C₂²⋊C₄	—	178	1	no	—
273	C₂²⋊C₄	—	178	1	no	—
274	C₂²⋊C₄	—	178	1	no	—
275	C₂²⋊C₄	—	178	1	no	—
276	C₂²⋊C₄	—	178	1	no	—
277	C₂²⋊C₄	—	178	1	no	—
278	C₂²⋊C₄	—	178	1	no	—
279	C₂²⋊C₄	D₃₅₆	1	1	yes	characteristic
280	C₂²⋊C₄	D₃₅₆	1	1	yes	characteristic
281	C₄²	—	178	1	no	—
282	C₄²	C₂²×D₈₉	1	1	yes	characteristic
283	C₂³	—	178	1	no	—
284	C₂³	—	178	1	no	—
285	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
286	C₂³	C₂×D₃₅₆	1	1	yes	characteristic
287	Q₈	—	356	2	no	—
288	D₄	—	2	2	no	—
289	D₄	—	356	2	no	—
290	D₄	—	356	2	no	—
291	D₄	—	356	2	no	—
292	D₄	—	178	1	no	—
293	D₄	—	178	1	no	—
294	D₄	—	178	1	no	—
295	D₄	—	178	1	no	—
296	D₄	—	712	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	356	2	no	—
302	C₂×C₄	—	356	2	no	—
303	C₂×C₄	—	356	2	no	—
304	C₂×C₄	—	178	1	no	—
305	C₂×C₄	—	178	1	no	—
306	C₂×C₄	—	178	1	no	—
307	C₂×C₄	—	178	1	no	—
308	C₂×C₄	—	178	1	no	—
309	C₂×C₄	—	178	1	no	—
310	C₂×C₄	—	712	1	no	—
311	C₂×C₄	—	712	1	no	—
312	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
313	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
314	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
315	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	#13083573	1	2	yes	complemented
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	356	1	no	—
327	C₂²	—	356	1	no	—
328	C₂²	—	356	1	no	—
329	C₂²	—	356	1	no	—
330	C₂²	—	356	1	no	—
331	C₂²	—	356	1	no	—
332	C₂²	#15668040	1	2	yes	—
333	C₂²	C₂²×D₃₅₆	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	356	1	no	—
339	C₄	—	356	1	no	—
340	C₄	—	356	1	no	—
341	C₄	—	356	1	no	—
342	C₄	—	356	1	no	—
343	C₄	—	356	1	no	—
344	C₄	C₂×#13083573	1	1	yes	characteristic
345	C₄	#15668059	1	1	yes	characteristic
346	C₂	—	2	2	no	—
347	C₂	—	4	1	no	—
348	C₂	—	712	1	no	—
349	C₂	—	712	1	no	—
350	C₂	#19152252	1	1	yes	characteristic
351	C₂	C₂×#15668040	1	1	yes	characteristic
352	C₂	#19152410	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₃₅₆
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#435×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#435	D₈₉
hypocenter	C₈₉	#3345
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #435

Complements

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

N	H ≅ G/N	type
1. #19152294	C₂	semidirect
2. #19152252	C₂	semidirect
3. #19152007	C₂	semidirect
4. #19151923	C₂	semidirect
5. #19151917	C₂	semidirect
6. #19151917	C₂	semidirect
7. #19151912	C₂	semidirect
8. #19151899	C₂	semidirect
9. #19151889	C₂	semidirect
10. #19151851	C₂	semidirect
11. #19151834	C₂	semidirect
12. #19151711	C₂	semidirect
13. #19151640	C₂	semidirect
14. C₂×#15667944	C₂	semidirect
15. #435×C₈₉	C₂	semidirect
143. C₈×C₈₉	#41	semidirect
177. C₈₉	#3345	semidirect
190. #435	D₈₉	semidirect
320. C₈	#13083573	semidirect

Maximal subgroups

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

subgroup	index
#19152294	2
#19152252	2
#19152007	2
#19151923	2
#19151917	2
#19151912	2
#19151899	2
#19151889	2
#19151851	2
#19151834	2
#19151711	2
#19151640	2
C₂×#15667944	2
#435×C₈₉	2
#3345	89