Subgroups — #26032954

Order 11392.

396 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152307	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152268	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151962	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151961	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151943	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151899	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151896	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151886	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151791	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151791	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151727	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151710	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151708	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#459×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083573	—	2	1	no	—
17	C₂×#13083559	C₂²	1	1	yes	characteristic
18	#15668011	C₂²	1	1	yes	characteristic
19	#15668008	C₂²	1	1	yes	characteristic
20	#15668008	C₂²	1	1	yes	characteristic
21	#15668004	C₂²	1	1	yes	characteristic
22	#15668003	C₂²	1	1	yes	characteristic
23	#15667991	—	2	1	no	—
24	#15667989	—	2	1	no	—
25	#15667989	—	2	1	no	—
26	#15667984	—	2	1	no	—
27	#15667984	—	2	1	no	—
28	#15667983	—	2	1	no	—
29	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
30	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
31	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
32	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
33	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
34	D₄×Dic₈₉	C₂²	1	1	yes	characteristic
35	C₂×#13083553	—	2	1	no	—
36	#15667946	C₂²	1	1	yes	characteristic
37	#15667944	C₂²	1	1	yes	characteristic
38	#15667943	C₂²	1	1	yes	characteristic
39	#15667941	C₂²	1	1	yes	characteristic
40	#15667941	C₂²	1	1	yes	characteristic
41	#15667934	C₂²	1	1	yes	characteristic
42	#15667934	C₂²	1	1	yes	characteristic
43	#15667933	C₂²	1	1	yes	characteristic
44	#15667933	C₂²	1	1	yes	characteristic
45	#15667933	C₂²	1	1	yes	characteristic
46	#15667933	C₂²	1	1	yes	characteristic
47	#15667933	C₂²	1	1	yes	characteristic
48	#15667932	C₂²	1	1	yes	characteristic
49	#15667923	C₂²	1	1	yes	characteristic
50	#15667922	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₈₉	C₂²	1	1	yes	characteristic
54	#127×C₈₉	C₂²	1	1	yes	characteristic
55	C₄×D₄×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	—	4	4	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	D₄	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	C₂²×Dic₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	—	2	1	no	—
72	#13083559	—	2	2	no	—
73	C₂×#10623494	C₂³	1	1	yes	characteristic
74	C₂×#10623494	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₈₉	—	2	1	no	—
76	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
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	#13083553	—	2	1	no	—
81	#13083553	—	2	1	no	—
82	#13083553	—	2	1	no	—
83	#13083553	—	2	1	no	—
84	#13083553	D₄	1	1	yes	characteristic
85	#13083553	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₈₉	—	2	1	no	—
87	C₂²⋊Dic₈₉	—	2	1	no	—
88	C₂²⋊Dic₈₉	—	2	1	no	—
89	C₂²⋊Dic₈₉	—	2	1	no	—
90	#13083551	—	2	1	no	—
91	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
92	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₈₉	—	2	1	no	—
94	C₂×D₄×C₈₉	—	2	1	no	—
95	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	—	2	1	no	—
99	D₈×C₈₉	—	2	2	no	—
100	D₈×C₈₉	—	2	2	no	—
101	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₉	—	2	1	no	—
105	C₄²×C₈₉	C₂³	1	1	yes	characteristic
106	C₂³×C₈₉	—	2	1	no	—
107	C₂³×C₈₉	—	2	1	no	—
108	C₂³×C₈₉	—	2	1	no	—
109	C₂²×D₈₉	—	2	1	no	—
110	D₄×C₈₉	—	2	1	no	—
111	D₄×C₈₉	—	2	1	no	—
112	D₄×C₈₉	—	2	1	no	—
113	D₄×C₈₉	—	2	1	no	—
114	D₄×C₈₉	—	2	1	no	—
115	D₄×C₈₉	—	4	2	no	—
116	D₄×C₈₉	—	8	1	no	—
117	D₄×C₈₉	#41	1	2	yes	complemented
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	8	1	no	—
120	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₈₉	—	4	2	no	—
124	C₂²⋊D₈₉	—	4	2	no	—
125	C₂²⋊D₈₉	—	8	1	no	—
126	C₂²⋊D₈₉	—	8	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	—	4	2	no	—
133	C₂×Dic₈₉	—	8	1	no	—
134	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
138	D₃₅₆	—	2	1	no	—
139	D₃₅₆	—	2	1	no	—
140	C₄×D₈₉	—	4	2	no	—
141	Dic₁₇₈	—	2	2	no	—
142	Dic₁₇₈	—	8	1	no	—
143	C₈×C₈₉	—	2	1	no	—
144	C₈×C₈₉	—	2	1	no	—
145	#10623494	—	2	1	no	—
146	#10623494	—	2	1	no	—
147	C₂²×C₈₉	—	2	2	no	—
148	C₂²×C₈₉	—	4	1	no	—
149	C₂²×C₈₉	—	4	1	no	—
150	C₂²×C₈₉	—	4	1	no	—
151	C₂²×C₈₉	—	4	1	no	—
152	C₂²×C₈₉	—	4	1	no	—
153	C₂²×C₈₉	—	4	1	no	—
154	C₂²×C₈₉	—	4	1	no	—
155	C₂²×C₈₉	—	4	1	no	—
156	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
157	C₂×D₈₉	—	4	1	no	—
158	C₂×D₈₉	—	4	1	no	—
159	C₂×D₈₉	—	4	1	no	—
160	C₄×C₈₉	—	2	1	no	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	—	4	1	no	—
163	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
164	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
165	Dic₈₉	—	2	1	no	—
166	Dic₈₉	—	2	1	no	—
167	Dic₈₉	—	2	1	no	—
168	Dic₈₉	—	4	1	no	—
169	Dic₈₉	—	4	1	no	—
170	Dic₈₉	—	4	1	no	—
171	Dic₈₉	AGL(1,ℤ/8)	1	2	yes	—
172	C₂×C₈₉	—	4	2	no	—
173	C₂×C₈₉	—	8	1	no	—
174	C₂×C₈₉	—	8	1	no	—
175	C₂×C₈₉	#548	1	1	yes	characteristic
176	C₂×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₂×C₈₉	#576	1	1	yes	characteristic
178	D₈₉	—	8	1	no	—
179	#3366	—	89	1	no	maximal, Sylow
180	C₈₉	#3366	1	1	yes	characteristic, complemented, Sylow
181	#554	—	89	1	no	—
182	#550	—	89	1	no	—
183	#490	—	89	1	no	—
184	#490	—	89	1	no	—
185	#486	—	89	1	no	—
186	#478	—	89	1	no	—
187	#478	—	89	1	no	—
188	#476	—	89	1	no	—
189	#459	—	89	1	no	—
190	#459	—	89	1	no	—
191	#459	D₈₉	1	1	yes	characteristic, complemented
192	#446	—	89	1	no	—
193	#442	—	89	1	no	—
194	#442	—	89	1	no	—
195	C₄×D₈	—	89	1	no	—
196	C₂×#41	—	178	1	no	—
197	C₂×D₈	—	89	1	no	—
198	C₂×D₈	C₂×D₈₉	1	1	yes	characteristic
199	C₂×D₈	C₂×D₈₉	1	1	yes	characteristic
200	C₄⋊Q₈	—	89	1	no	—
201	#127	—	89	1	no	—
202	#127	—	89	1	no	—
203	#127	C₂×D₈₉	1	1	yes	characteristic
204	#125	—	89	1	no	—
205	#125	—	89	1	no	—
206	C₂²⋊Q₈	—	178	1	no	—
207	C₂²⋊Q₈	—	178	1	no	—
208	C₂²⋊Q₈	—	178	1	no	—
209	#121	—	178	1	no	—
210	#121	—	178	1	no	—
211	#121	—	178	1	no	—
212	C₄×Q₈	—	89	1	no	—
213	C₄×D₄	—	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	C₂×D₈₉	1	1	yes	characteristic
219	C₂×C₄⋊C₄	—	178	1	no	—
220	#107	—	89	1	no	—
221	#106	—	89	1	no	—
222	#106	—	89	1	no	—
223	C₄⋊C₈	—	89	1	no	—
224	C₄⋊C₈	—	89	1	no	—
225	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
226	#102	—	89	1	no	—
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	#102	—	89	1	no	—
234	#102	C₂×D₈₉	1	1	yes	characteristic
235	#102	C₂×D₈₉	1	1	yes	characteristic
236	#97	—	89	1	no	—
237	#97	—	89	1	no	—
238	C₄×C₈	—	89	1	no	—
239	#41	—	356	4	no	—
240	C₂×Q₈	—	89	1	no	—
241	C₂×Q₈	—	178	1	no	—
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	2	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	C₂²×D₈₉	1	1	yes	characteristic
250	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
251	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
252	C₂²×C₄	—	2	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	D₈	—	2	2	no	—
259	D₈	—	2	2	no	—
260	D₈	—	178	2	no	—
261	C₂×C₈	—	89	1	no	—
262	C₂×C₈	—	89	1	no	—
263	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
265	C₄⋊C₄	—	89	1	no	—
266	C₄⋊C₄	—	89	1	no	—
267	C₄⋊C₄	—	89	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	—	178	1	no	—
275	C₄⋊C₄	—	178	1	no	—
276	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	178	1	no	—
279	C₂²⋊C₄	—	178	1	no	—
280	C₂²⋊C₄	—	178	1	no	—
281	C₂²⋊C₄	—	178	1	no	—
282	C₂²⋊C₄	—	178	1	no	—
283	C₄²	—	89	1	no	—
284	C₄²	—	89	1	no	—
285	C₄²	C₂²×D₈₉	1	1	yes	characteristic
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	178	1	no	—
290	Q₈	—	178	2	no	—
291	Q₈	—	712	1	no	—
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	2	1	no	—
297	D₄	—	4	2	no	—
298	D₄	—	8	1	no	—
299	D₄	—	356	2	no	—
300	D₄	—	356	2	no	—
301	D₄	—	178	1	no	—
302	D₄	—	178	1	no	—
303	D₄	—	712	1	no	—
304	D₄	—	712	1	no	—
305	D₄	#13083573	1	2	yes	complemented
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	8	1	no	—
308	C₂×C₄	—	89	1	no	—
309	C₂×C₄	—	89	1	no	—
310	C₂×C₄	—	89	1	no	—
311	C₂×C₄	—	89	1	no	—
312	C₂×C₄	—	356	2	no	—
313	C₂×C₄	—	356	2	no	—
314	C₂×C₄	—	356	2	no	—
315	C₂×C₄	—	356	2	no	—
316	C₂×C₄	—	178	1	no	—
317	C₂×C₄	—	178	1	no	—
318	C₂×C₄	—	178	1	no	—
319	C₂×C₄	—	712	1	no	—
320	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
321	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	178	1	no	—
326	C₈	—	178	1	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	356	1	no	—
337	C₂²	—	356	1	no	—
338	C₂²	—	356	1	no	—
339	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	89	2	no	—
344	C₄	—	356	1	no	—
345	C₄	—	356	1	no	—
346	C₄	—	356	1	no	—
347	C₄	—	178	1	no	—
348	C₄	—	178	1	no	—
349	C₄	—	178	1	no	—
350	C₄	C₂×#13083573	1	1	yes	characteristic
351	C₄	#15668059	1	1	yes	characteristic
352	C₂	—	4	2	no	—
353	C₂	—	8	1	no	—
354	C₂	—	8	1	no	—
355	C₂	—	712	1	no	—
356	C₂	#19152254	1	1	yes	characteristic
357	C₂	AGL(1,ℤ/8)×D₈₉	1	1	yes	characteristic
358	C₂	#19152399	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#459×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
hypercenter	#459	D₈₉
hypocenter	C₈₉	#3366
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #459

Complements

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

N	H ≅ G/N	type
1. #19152307	C₂	semidirect
2. #19152268	C₂	semidirect
3. #19151962	C₂	semidirect
4. #19151961	C₂	semidirect
5. #19151943	C₂	semidirect
6. #19151899	C₂	semidirect
7. #19151896	C₂	semidirect
8. #19151886	C₂	semidirect
9. #19151791	C₂	semidirect
10. #19151791	C₂	semidirect
11. #19151727	C₂	semidirect
12. #19151710	C₂	semidirect
13. #19151708	C₂	semidirect
14. D₈×Dic₈₉	C₂	semidirect
15. #459×C₈₉	C₂	semidirect
117. D₄×C₈₉	#41	semidirect
180. C₈₉	#3366	semidirect
191. #459	D₈₉	semidirect
305. D₄	#13083573	semidirect

Maximal subgroups

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

subgroup	index
#19152307	2
#19152268	2
#19151962	2
#19151961	2
#19151943	2
#19151899	2
#19151896	2
#19151886	2
#19151791	2
#19151727	2
#19151710	2
#19151708	2
D₈×Dic₈₉	2
#459×C₈₉	2
#3366	89