Subgroups — #26030366

Order 11392.

392 conjugacy classes of subgroups.

The subgroup lattice has 350 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×#13083573	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151941	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151924	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151902	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151821	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151813	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151761	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151744	C₂	1	1	yes	maximal, characteristic
10	#19151707	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151704	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151697	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151695	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151654	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083573	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₅₆	C₂²	1	1	yes	characteristic
18	C₂×#13083563	C₂²	1	1	yes	characteristic
19	C₂×#13083562	C₂²	1	1	yes	characteristic
20	C₂×#13083559	C₂²	1	1	yes	characteristic
21	#15668007	—	2	1	no	—
22	#15668000	C₂²	1	1	yes	characteristic
23	#15667997	—	2	1	no	—
24	#15667992	C₂²	1	1	yes	characteristic
25	#15667987	C₂²	1	1	yes	characteristic
26	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
27	#15667977	C₂²	1	1	yes	characteristic
28	#15667977	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	C₄×C₂²⋊D₈₉	—	2	1	no	—
33	#15667972	—	2	1	no	—
34	#15667970	—	2	1	no	—
35	#15667970	C₂²	1	1	yes	characteristic
36	#15667968	—	2	1	no	—
37	C₂×C₄×Dic₈₉	—	2	1	no	—
38	C₄²×D₈₉	C₂²	1	1	yes	characteristic
39	#15667945	C₂²	1	1	yes	characteristic
40	#15667943	C₂²	1	1	yes	characteristic
41	C₄⋊#10623494	C₂²	1	1	yes	characteristic
42	#15667937	C₂²	1	1	yes	characteristic
43	#15667935	C₂²	1	1	yes	characteristic
44	#15667935	C₂²	1	1	yes	characteristic
45	#15667935	C₂²	1	1	yes	characteristic
46	#15667934	C₂²	1	1	yes	characteristic
47	#15667932	C₂²	1	1	yes	characteristic
48	#15667924	C₂²	1	1	yes	characteristic
49	#15667924	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	#15667922	C₂²	1	1	yes	characteristic
52	#125×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
54	#107×C₈₉	C₂²	1	1	yes	characteristic
55	#106×C₈₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	#102×C₈₉	C₂²	1	1	yes	characteristic
59	#13083573	—	2	2	no	—
60	#13083573	—	2	2	no	—
61	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	C₂×D₃₅₆	—	2	1	no	—
64	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₈₉	—	2	1	no	—
66	C₂²×Dic₈₉	—	2	1	no	—
67	C₂²×Dic₈₉	—	2	1	no	—
68	C₂×C₄×D₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
70	C₂×C₄×D₈₉	D₄	1	1	yes	characteristic
71	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
72	Dic₃₅₆	—	2	2	no	—
73	#13083563	—	2	2	no	—
74	#13083562	—	2	2	no	—
75	#13083559	—	2	2	no	—
76	C₂×#10623494	C₂³	1	1	yes	characteristic
77	C₂×#10623494	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₈₉	—	2	1	no	—
79	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
80	#13083553	—	2	1	no	—
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	C₂²⋊Dic₈₉	—	2	1	no	—
84	#13083551	—	2	1	no	—
85	#13083551	—	2	1	no	—
86	#13083551	—	2	1	no	—
87	#13083551	—	2	1	no	—
88	#13083551	—	2	1	no	—
89	#13083551	—	2	1	no	—
90	C₄×Dic₈₉	—	2	1	no	—
91	C₄×Dic₈₉	—	2	1	no	—
92	C₄×Dic₈₉	—	2	1	no	—
93	C₄×Dic₈₉	—	2	1	no	—
94	C₄×Dic₈₉	D₄	1	1	yes	characteristic
95	C₄×Dic₈₉	D₄	1	1	yes	characteristic
96	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	—	2	1	no	—
99	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	—	2	1	no	—
102	C₄⋊C₄×C₈₉	—	2	1	no	—
103	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₈₉	—	2	1	no	—
107	C₄²×C₈₉	C₂³	1	1	yes	characteristic
108	C₂³×C₈₉	—	2	1	no	—
109	C₂²×D₈₉	—	2	1	no	—
110	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
111	D₄×C₈₉	—	2	1	no	—
112	D₄×C₈₉	#41	1	2	yes	—
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	—	4	2	no	—
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	—	2	4	no	—
121	C₂×Dic₈₉	—	2	4	no	—
122	C₂×Dic₈₉	—	2	1	no	—
123	C₂×Dic₈₉	—	2	1	no	—
124	C₂×Dic₈₉	—	2	1	no	—
125	C₂×Dic₈₉	—	4	2	no	—
126	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
129	D₃₅₆	—	2	1	no	—
130	D₃₅₆	—	2	1	no	—
131	D₃₅₆	—	4	2	no	—
132	C₄×D₈₉	—	2	1	no	—
133	C₄×D₈₉	—	2	1	no	—
134	C₄×D₈₉	—	4	1	no	—
135	C₄×D₈₉	—	4	1	no	—
136	C₄×D₈₉	—	4	2	no	—
137	Dic₁₇₈	—	2	1	no	—
138	Dic₁₇₈	—	2	1	no	—
139	Dic₁₇₈	—	2	1	no	—
140	Dic₁₇₈	#41	1	2	yes	—
141	C₈×C₈₉	—	2	1	no	—
142	C₈×C₈₉	—	2	1	no	—
143	#10623494	—	2	1	no	—
144	#10623494	—	2	1	no	—
145	C₂²×C₈₉	—	2	2	no	—
146	C₂²×C₈₉	—	4	1	no	—
147	C₂²×C₈₉	—	4	1	no	—
148	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₈₉	—	2	1	no	—
150	C₂×D₈₉	—	2	1	no	—
151	C₂×D₈₉	—	2	1	no	—
152	C₂×D₈₉	—	4	1	no	—
153	C₂×D₈₉	—	4	1	no	—
154	C₂×D₈₉	—	4	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	4	1	no	—
158	C₄×C₈₉	—	4	1	no	—
159	C₄×C₈₉	—	4	1	no	—
160	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
161	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
162	Dic₈₉	—	2	2	no	—
163	Dic₈₉	—	2	1	no	—
164	Dic₈₉	—	2	1	no	—
165	Dic₈₉	—	2	1	no	—
166	Dic₈₉	—	4	1	no	—
167	Dic₈₉	—	4	1	no	—
168	C₂×C₈₉	—	4	2	no	—
169	C₂×C₈₉	#525	1	1	yes	characteristic
170	C₂×C₈₉	#575	1	1	yes	characteristic
171	C₂×C₈₉	#575	1	1	yes	characteristic
172	D₈₉	—	4	1	no	—
173	D₈₉	—	8	1	no	—
174	#3111	—	89	1	no	maximal, Sylow
175	C₈₉	#3111	1	1	yes	characteristic, complemented, Sylow
176	#529	—	89	1	no	—
177	C₄×#41	—	89	1	no	—
178	#485	—	89	1	no	—
179	#482	—	89	1	no	—
180	#479	—	89	1	no	—
181	#478	D₈₉	1	1	yes	characteristic, complemented
182	#464	—	89	1	no	—
183	#463	—	89	1	no	—
184	#452	—	89	1	no	—
185	#449	—	89	1	no	—
186	#442	—	89	1	no	—
187	#441	—	89	1	no	—
188	#440	—	89	1	no	—
189	#440	—	89	1	no	—
190	#432	—	89	1	no	—
191	C₂×#41	—	89	1	no	—
192	C₂×Q₁₆	—	89	1	no	—
193	C₂×SD₁₆	—	89	1	no	—
194	C₂×SD₁₆	—	89	1	no	—
195	C₂×D₈	—	89	1	no	—
196	#126	—	178	1	no	—
197	#125	C₂×D₈₉	1	1	yes	characteristic
198	#124	—	89	1	no	—
199	#123	—	178	1	no	—
200	C₂²⋊Q₈	—	89	1	no	—
201	#121	—	89	1	no	—
202	C₄×Q₈	—	89	1	no	—
203	C₄×Q₈	—	89	1	no	—
204	C₄×Q₈	—	89	1	no	—
205	C₄×D₄	—	89	1	no	—
206	C₄×D₄	—	89	1	no	—
207	C₄×D₄	—	178	1	no	—
208	C₄×D₄	—	178	1	no	—
209	C₄×D₄	—	178	1	no	—
210	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
211	#117	—	89	1	no	—
212	#117	—	178	1	no	—
213	#117	—	178	1	no	—
214	C₂×C₄²	—	89	1	no	—
215	C₂×C₄²	—	178	1	no	—
216	#107	—	89	1	no	—
217	#107	C₂×D₈₉	1	1	yes	characteristic
218	#106	—	89	1	no	—
219	#106	C₂×D₈₉	1	1	yes	characteristic
220	C₄⋊C₈	—	89	1	no	—
221	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
222	#103	—	89	1	no	—
223	#103	—	89	1	no	—
224	#103	—	89	1	no	—
225	#103	—	89	1	no	—
226	#102	—	89	1	no	—
227	#102	—	89	1	no	—
228	#102	C₂×D₈₉	1	1	yes	characteristic
229	#102	C₂×D₈₉	1	1	yes	characteristic
230	C₂²⋊C₈	—	89	1	no	—
231	C₂²⋊C₈	—	89	1	no	—
232	#97	—	89	1	no	—
233	#97	—	89	1	no	—
234	#41	—	178	2	no	—
235	#41	—	178	2	no	—
236	C₂×Q₈	—	89	1	no	—
237	C₂×Q₈	—	89	1	no	—
238	C₂×D₄	—	89	1	no	—
239	C₂×D₄	—	178	1	no	—
240	C₂×D₄	—	178	1	no	—
241	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	89	1	no	—
244	C₂²×C₄	—	89	1	no	—
245	C₂²×C₄	—	89	1	no	—
246	C₂²×C₄	—	178	1	no	—
247	C₂²×C₄	—	178	1	no	—
248	C₂²×C₄	—	178	1	no	—
249	Q₁₆	—	178	2	no	—
250	SD₁₆	—	178	2	no	—
251	SD₁₆	—	178	2	no	—
252	D₈	—	178	2	no	—
253	C₂×C₈	—	89	1	no	—
254	C₂×C₈	—	89	1	no	—
255	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	89	1	no	—
260	C₄⋊C₄	—	178	1	no	—
261	C₄⋊C₄	—	178	1	no	—
262	C₄⋊C₄	—	178	1	no	—
263	C₄⋊C₄	—	178	1	no	—
264	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
266	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	178	1	no	—
269	C₂²⋊C₄	—	178	1	no	—
270	C₂²⋊C₄	—	178	1	no	—
271	C₂²⋊C₄	—	178	1	no	—
272	C₂²⋊C₄	—	178	1	no	—
273	C₂²⋊C₄	—	178	1	no	—
274	C₂²⋊C₄	—	178	1	no	—
275	C₄²	—	89	1	no	—
276	C₄²	—	89	1	no	—
277	C₄²	—	89	1	no	—
278	C₄²	—	178	1	no	—
279	C₄²	—	178	1	no	—
280	C₄²	—	178	1	no	—
281	C₄²	—	178	1	no	—
282	C₄²	C₂²×D₈₉	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	89	1	no	—
285	C₂³	—	178	1	no	—
286	Q₈	—	89	2	no	—
287	Q₈	—	178	1	no	—
288	Q₈	—	178	1	no	—
289	Q₈	—	178	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	356	2	no	—
292	D₄	—	178	4	no	—
293	D₄	—	178	1	no	—
294	D₄	—	178	1	no	—
295	D₄	#13083575	1	2	yes	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	89	1	no	—
301	C₂×C₄	—	89	1	no	—
302	C₂×C₄	—	89	1	no	—
303	C₂×C₄	—	356	1	no	—
304	C₂×C₄	—	356	1	no	—
305	C₂×C₄	—	356	2	no	—
306	C₂×C₄	—	356	2	no	—
307	C₂×C₄	—	178	1	no	—
308	C₂×C₄	—	178	1	no	—
309	C₂×C₄	—	178	4	no	—
310	C₂×C₄	—	178	1	no	—
311	C₂×C₄	—	178	1	no	—
312	C₂×C₄	—	178	1	no	—
313	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
314	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	178	1	no	—
319	C₈	—	178	1	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	356	1	no	—
324	C₂²	—	356	1	no	—
325	C₂²	—	356	1	no	—
326	C₂²	—	178	1	no	—
327	C₂²	—	178	1	no	—
328	C₂²	—	178	1	no	—
329	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	356	1	no	—
336	C₄	—	356	1	no	—
337	C₄	—	178	1	no	—
338	C₄	—	178	1	no	—
339	C₄	—	178	1	no	—
340	C₄	—	178	2	no	—
341	C₄	#41×D₈₉	1	1	yes	characteristic
342	C₄	C₂×#13083575	1	1	yes	characteristic
343	C₂	—	4	2	no	—
344	C₂	—	356	1	no	—
345	C₂	—	712	1	no	—
346	C₂	#19152111	1	1	yes	characteristic
347	C₂	#19152394	1	1	yes	characteristic
348	C₂	#19152395	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	#19151744	C₂
hypercenter	#478	D₈₉
hypocenter	C₈₉	#3111
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×#13083573	C₂	semidirect
3. #19151941	C₂	semidirect
4. #19151924	C₂	semidirect
5. #19151902	C₂	semidirect
6. #19151821	C₂	semidirect
7. #19151813	C₂	semidirect
8. #19151761	C₂	semidirect
10. #19151707	C₂	semidirect
11. #19151704	C₂	semidirect
12. #19151697	C₂	semidirect
13. #19151695	C₂	semidirect
14. #19151654	C₂	semidirect
15. #478×C₈₉	C₂	semidirect
175. C₈₉	#3111	semidirect
181. #478	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×#13083573	2
#19151941	2
#19151924	2
#19151902	2
#19151821	2
#19151813	2
#19151761	2
#19151744	2
#19151707	2
#19151704	2
#19151697	2
#19151695	2
#19151654	2
#478×C₈₉	2
#3111	89