Subgroups — #26030456

Order 11392.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152204	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152077	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151918	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151913	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151902	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151799	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151793	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151744	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151744	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151707	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151697	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151685	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151678	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151658	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×#13083563	C₂²	1	1	yes	characteristic
18	C₂×#13083563	C₂²	1	1	yes	characteristic
19	C₂×#13083559	C₂²	1	1	yes	characteristic
20	C₂×#13083559	C₂²	1	1	yes	characteristic
21	#15668009	C₂²	1	1	yes	characteristic
22	#15667997	—	2	1	no	—
23	#15667997	—	2	1	no	—
24	#15667987	—	2	1	no	—
25	#15667987	—	2	1	no	—
26	#15667987	C₂²	1	1	yes	characteristic
27	#15667987	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
29	#15667977	C₂²	1	1	yes	characteristic
30	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
31	#15667974	C₂²	1	1	yes	characteristic
32	#15667974	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₈₉	—	2	1	no	—
34	#15667972	—	2	1	no	—
35	#15667969	C₂²	1	1	yes	characteristic
36	#15667968	—	2	1	no	—
37	C₂×#13083553	—	2	1	no	—
38	C₄⋊C₄×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	#15667935	C₂²	1	1	yes	characteristic
43	#15667935	C₂²	1	1	yes	characteristic
44	#15667934	C₂²	1	1	yes	characteristic
45	#15667934	C₂²	1	1	yes	characteristic
46	#15667932	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	C₈×Dic₈₉	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×D₃₅₆	—	2	1	no	—
63	C₂×D₃₅₆	—	2	1	no	—
64	C₂×D₃₅₆	—	2	1	no	—
65	C₂×D₃₅₆	—	2	1	no	—
66	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
67	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₈₉	—	2	1	no	—
69	C₂²×Dic₈₉	—	2	1	no	—
70	C₂²×Dic₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	—	2	1	no	—
72	C₂×C₄×D₈₉	—	2	1	no	—
73	C₂×C₄×D₈₉	—	2	1	no	—
74	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
75	#13083563	—	2	2	no	—
76	#13083563	—	2	2	no	—
77	#13083559	—	2	2	no	—
78	#13083559	—	2	2	no	—
79	C₂×#10623494	C₂³	1	1	yes	characteristic
80	C₂×#10623494	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
82	#13083553	—	2	1	no	—
83	#13083553	—	2	1	no	—
84	#13083553	—	2	1	no	—
85	#13083553	D₄	1	1	yes	characteristic
86	#13083553	D₄	1	1	yes	characteristic
87	C₂²⋊Dic₈₉	—	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	D₄	1	1	yes	characteristic
93	#13083551	D₄	1	1	yes	characteristic
94	C₄×Dic₈₉	—	2	1	no	—
95	C₄×Dic₈₉	—	2	1	no	—
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	—	2	1	no	—
111	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
112	D₄×C₈₉	—	2	1	no	—
113	D₄×C₈₉	#41	1	2	yes	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	8	1	no	—
118	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₈₉	—	2	4	no	—
122	C₂×Dic₈₉	—	2	4	no	—
123	C₂×Dic₈₉	—	2	1	no	—
124	C₂×Dic₈₉	—	2	1	no	—
125	C₂×Dic₈₉	—	8	1	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	—	2	1	no	—
132	D₃₅₆	—	2	1	no	—
133	D₃₅₆	—	8	1	no	—
134	D₃₅₆	—	8	1	no	—
135	D₃₅₆	—	8	1	no	—
136	D₃₅₆	—	8	1	no	—
137	C₄×D₈₉	—	2	1	no	—
138	C₄×D₈₉	—	2	1	no	—
139	C₄×D₈₉	—	8	1	no	—
140	C₄×D₈₉	—	8	1	no	—
141	C₄×D₈₉	—	8	1	no	—
142	Dic₁₇₈	—	2	1	no	—
143	Dic₁₇₈	#41	1	2	yes	—
144	C₈×C₈₉	—	2	1	no	—
145	C₈×C₈₉	—	2	1	no	—
146	#10623494	—	2	1	no	—
147	#10623494	—	2	1	no	—
148	C₂²×C₈₉	—	2	2	no	—
149	C₂²×C₈₉	—	4	1	no	—
150	C₂²×C₈₉	—	4	1	no	—
151	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₈₉	—	2	1	no	—
153	C₂×D₈₉	—	2	1	no	—
154	C₂×D₈₉	—	2	1	no	—
155	C₂×D₈₉	—	4	1	no	—
156	C₂×D₈₉	—	4	1	no	—
157	C₂×D₈₉	—	4	1	no	—
158	C₂×D₈₉	—	4	1	no	—
159	C₂×D₈₉	—	4	1	no	—
160	C₂×D₈₉	—	4	1	no	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	—	2	1	no	—
163	C₄×C₈₉	—	4	1	no	—
164	C₄×C₈₉	—	4	1	no	—
165	C₄×C₈₉	—	4	1	no	—
166	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
167	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
168	Dic₈₉	—	2	2	no	—
169	Dic₈₉	—	2	1	no	—
170	Dic₈₉	—	2	1	no	—
171	Dic₈₉	—	2	1	no	—
172	Dic₈₉	—	4	1	no	—
173	C₂×C₈₉	—	4	2	no	—
174	C₂×C₈₉	#525	1	1	yes	characteristic
175	C₂×C₈₉	#576	1	1	yes	characteristic
176	C₂×C₈₉	#577	1	1	yes	characteristic
177	D₈₉	—	4	1	no	—
178	D₈₉	—	8	1	no	—
179	D₈₉	—	8	1	no	—
180	#3119	—	89	1	no	maximal, Sylow
181	C₈₉	#3119	1	1	yes	characteristic, complemented, Sylow
182	#540	—	89	1	no	—
183	#520	—	89	1	no	—
184	#481	—	89	1	no	—
185	#480	—	89	1	no	—
186	#479	—	89	1	no	—
187	#478	D₈₉	1	1	yes	characteristic, complemented
188	#460	—	89	1	no	—
189	#459	—	89	1	no	—
190	#449	—	89	1	no	—
191	#449	—	89	1	no	—
192	#442	—	89	1	no	—
193	#440	—	89	1	no	—
194	C₄×SD₁₆	—	89	1	no	—
195	C₄×D₈	—	89	1	no	—
196	#433	—	89	1	no	—
197	C₂×#41	—	89	1	no	—
198	C₂×SD₁₆	—	89	1	no	—
199	C₂×SD₁₆	—	89	1	no	—
200	C₂×D₈	—	89	1	no	—
201	C₂×D₈	—	89	1	no	—
202	#127	—	89	1	no	—
203	#125	C₂×D₈₉	1	1	yes	characteristic
204	#123	—	178	1	no	—
205	#123	—	178	1	no	—
206	#121	—	89	1	no	—
207	#121	—	89	1	no	—
208	#121	—	178	1	no	—
209	#121	—	178	1	no	—
210	C₄×Q₈	—	89	1	no	—
211	C₄×Q₈	—	89	1	no	—
212	C₄×D₄	—	89	1	no	—
213	C₄×D₄	—	89	1	no	—
214	C₄×D₄	—	89	1	no	—
215	C₄×D₄	—	178	1	no	—
216	C₄×D₄	—	178	1	no	—
217	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
218	#117	—	89	1	no	—
219	#117	—	178	1	no	—
220	C₂×C₄⋊C₄	—	89	1	no	—
221	C₂×C₄⋊C₄	—	178	1	no	—
222	#107	—	89	1	no	—
223	#107	C₂×D₈₉	1	1	yes	characteristic
224	#106	—	89	1	no	—
225	#106	C₂×D₈₉	1	1	yes	characteristic
226	C₄⋊C₈	—	89	1	no	—
227	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
228	#103	—	89	1	no	—
229	#103	—	89	1	no	—
230	#102	—	89	1	no	—
231	#102	—	89	1	no	—
232	#102	—	89	1	no	—
233	#102	—	89	1	no	—
234	#102	C₂×D₈₉	1	1	yes	characteristic
235	#102	C₂×D₈₉	1	1	yes	characteristic
236	C₂²⋊C₈	—	89	1	no	—
237	C₂²⋊C₈	—	89	1	no	—
238	#97	—	89	1	no	—
239	C₄×C₈	—	89	1	no	—
240	#41	—	178	2	no	—
241	#41	—	178	2	no	—
242	C₂×Q₈	—	89	1	no	—
243	C₂×D₄	—	89	1	no	—
244	C₂×D₄	—	89	1	no	—
245	C₂×D₄	—	178	1	no	—
246	C₂×D₄	—	178	1	no	—
247	C₂×D₄	—	178	1	no	—
248	C₂×D₄	—	178	1	no	—
249	C₂×D₄	—	178	1	no	—
250	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
251	C₂²×C₄	—	2	1	no	—
252	C₂²×C₄	—	89	1	no	—
253	C₂²×C₄	—	178	1	no	—
254	C₂²×C₄	—	178	1	no	—
255	C₂²×C₄	—	178	1	no	—
256	C₂²×C₄	—	178	1	no	—
257	C₂²×C₄	—	178	1	no	—
258	SD₁₆	—	178	2	no	—
259	SD₁₆	—	178	2	no	—
260	D₈	—	178	2	no	—
261	D₈	—	178	2	no	—
262	C₂×C₈	—	89	1	no	—
263	C₂×C₈	—	89	1	no	—
264	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
265	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	89	1	no	—
269	C₄⋊C₄	—	89	1	no	—
270	C₄⋊C₄	—	89	1	no	—
271	C₄⋊C₄	—	178	1	no	—
272	C₄⋊C₄	—	178	1	no	—
273	C₄⋊C₄	—	178	1	no	—
274	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
275	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	89	1	no	—
279	C₂²⋊C₄	—	89	1	no	—
280	C₂²⋊C₄	—	178	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	C₄²	—	89	1	no	—
286	C₄²	—	178	1	no	—
287	C₄²	—	178	1	no	—
288	C₄²	C₂²×D₈₉	1	1	yes	characteristic
289	C₂³	—	2	1	no	—
290	C₂³	—	89	1	no	—
291	C₂³	—	178	1	no	—
292	C₂³	—	178	1	no	—
293	Q₈	—	89	2	no	—
294	Q₈	—	178	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	178	4	no	—
297	D₄	—	178	1	no	—
298	D₄	—	178	1	no	—
299	D₄	—	178	1	no	—
300	D₄	—	178	1	no	—
301	D₄	—	712	1	no	—
302	D₄	—	712	1	no	—
303	D₄	—	712	1	no	—
304	D₄	—	712	1	no	—
305	D₄	#13083575	1	2	yes	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	8	1	no	—
310	C₂×C₄	—	89	1	no	—
311	C₂×C₄	—	89	1	no	—
312	C₂×C₄	—	89	1	no	—
313	C₂×C₄	—	178	1	no	—
314	C₂×C₄	—	178	4	no	—
315	C₂×C₄	—	178	1	no	—
316	C₂×C₄	—	178	1	no	—
317	C₂×C₄	—	178	1	no	—
318	C₂×C₄	—	712	1	no	—
319	C₂×C₄	—	712	1	no	—
320	C₂×C₄	—	712	1	no	—
321	C₂×C₄	—	712	1	no	—
322	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
323	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
324	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	2	1	no	—
327	C₈	—	178	1	no	—
328	C₈	—	178	1	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	356	1	no	—
333	C₂²	—	356	1	no	—
334	C₂²	—	356	1	no	—
335	C₂²	—	356	1	no	—
336	C₂²	—	356	1	no	—
337	C₂²	—	356	1	no	—
338	C₂²	—	178	1	no	—
339	C₂²	—	178	1	no	—
340	C₂²	—	178	1	no	—
341	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	356	1	no	—
348	C₄	—	178	1	no	—
349	C₄	—	178	1	no	—
350	C₄	—	178	1	no	—
351	C₄	—	178	2	no	—
352	C₄	#41×D₈₉	1	1	yes	characteristic
353	C₄	C₂×#13083575	1	1	yes	characteristic
354	C₂	—	4	2	no	—
355	C₂	—	356	1	no	—
356	C₂	—	712	1	no	—
357	C₂	—	712	1	no	—
358	C₂	#19152111	1	1	yes	characteristic
359	C₂	#19152402	1	1	yes	characteristic
360	C₂	#19152405	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
hypercenter	#478	D₈₉
hypocenter	C₈₉	#3119
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. #19152204	C₂	semidirect
2. #19152077	C₂	semidirect
3. #19151918	C₂	semidirect
4. #19151913	C₂	semidirect
5. #19151902	C₂	semidirect
6. #19151799	C₂	semidirect
7. #19151793	C₂	semidirect
8. #19151744	C₂	semidirect
9. #19151744	C₂	semidirect
10. #19151707	C₂	semidirect
11. #19151697	C₂	semidirect
12. #19151685	C₂	semidirect
13. #19151678	C₂	semidirect
14. #19151658	C₂	semidirect
15. #478×C₈₉	C₂	semidirect
181. C₈₉	#3119	semidirect
187. #478	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152204	2
#19152077	2
#19151918	2
#19151913	2
#19151902	2
#19151799	2
#19151793	2
#19151744	2
#19151707	2
#19151697	2
#19151685	2
#19151678	2
#19151658	2
#478×C₈₉	2
#3119	89