Subgroups — #26030616

Order 11392.

372 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152228	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152168	C₂	1	1	yes	maximal, characteristic
3	#19151968	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151965	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151957	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151956	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151951	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151949	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151947	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151605	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151596	C₂	1	1	yes	maximal, characteristic
12	#19151591	C₂	1	1	yes	maximal, characteristic
13	#103×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151552	C₂	1	1	yes	maximal, characteristic, complemented
15	#488×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083573	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×#13083557	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
20	#15668012	C₂²	1	1	yes	characteristic
21	#15668012	C₂²	1	1	yes	characteristic
22	#15668011	—	2	1	no	—
23	#15668011	C₂²	1	1	yes	characteristic
24	#15668010	C₂²	1	1	yes	characteristic
25	#15668003	C₂²	1	1	yes	characteristic
26	#15668002	—	2	1	no	—
27	#15667999	C₂²	1	1	yes	characteristic
28	#15667994	—	2	1	no	—
29	#15667994	—	2	1	no	—
30	#15667992	—	2	1	no	—
31	#15667990	—	2	1	no	—
32	#15667989	—	2	1	no	—
33	#15667989	—	2	1	no	—
34	#15667970	C₂²	1	1	yes	characteristic
35	#15667969	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
37	#15667937	C₂²	1	1	yes	characteristic
38	#15667937	C₂²	1	1	yes	characteristic
39	#15667937	C₂²	1	1	yes	characteristic
40	#15667937	C₂²	1	1	yes	characteristic
41	#15667936	C₂²	1	1	yes	characteristic
42	#15667936	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	#15667935	C₂²	1	1	yes	characteristic
47	#15667933	C₂²	1	1	yes	characteristic
48	#15667933	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	#125×C₈₉	C₂²	1	1	yes	characteristic
53	#124×C₈₉	C₂²	1	1	yes	characteristic
54	#103×C₈₉	C₂²	1	1	yes	characteristic
55	#103×C₈₉	C₂²	1	1	yes	characteristic
56	#102×C₈₉	C₂²	1	1	yes	characteristic
57	#102×C₈₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
59	#13083573	—	2	2	no	—
60	#13083573	—	2	2	no	—
61	C₂×Dic₁₇₈	—	2	1	no	—
62	C₂×Dic₁₇₈	—	2	1	no	—
63	C₂×Dic₁₇₈	—	2	1	no	—
64	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
65	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
66	Q₈×D₈₉	—	2	2	no	—
67	Q₈×D₈₉	—	2	2	no	—
68	C₂×C₂²⋊D₈₉	—	2	1	no	—
69	C₂²×Dic₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	—	2	1	no	—
72	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
73	#13083557	—	2	2	no	—
74	C₂×#10623494	C₂³	1	1	yes	characteristic
75	C₂×#10623494	C₂³	1	1	yes	characteristic
76	C₈×D₈₉	—	2	2	no	—
77	C₄⋊Dic₈₉	—	2	1	no	—
78	C₄⋊Dic₈₉	—	2	1	no	—
79	C₄⋊Dic₈₉	—	2	1	no	—
80	C₄⋊Dic₈₉	—	2	1	no	—
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	#13083553	—	2	1	no	—
86	#13083553	—	2	1	no	—
87	#13083553	—	2	1	no	—
88	#13083553	D₄	1	1	yes	characteristic
89	#13083553	D₄	1	1	yes	characteristic
90	C₂²⋊Dic₈₉	—	2	1	no	—
91	C₂²⋊Dic₈₉	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	#13083551	D₄	1	1	yes	characteristic
94	#13083551	D₄	1	1	yes	characteristic
95	C₄×Dic₈₉	—	2	1	no	—
96	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₂×D₄×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	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	C₂²⋊C₄×C₈₉	—	2	1	no	—
106	C₂²⋊C₄×C₈₉	—	2	1	no	—
107	C₄²×C₈₉	C₂³	1	1	yes	characteristic
108	C₂³×C₈₉	—	2	1	no	—
109	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₈₉	—	2	1	no	—
111	Q₈×C₈₉	—	2	1	no	—
112	D₄×C₈₉	—	2	1	no	—
113	D₄×C₈₉	—	2	1	no	—
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	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	C₂×Dic₈₉	—	2	1	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	C₄×D₈₉	—	4	2	no	—
130	C₄×D₈₉	—	8	1	no	—
131	C₄×D₈₉	#41	1	2	yes	—
132	C₄×D₈₉	#41	1	2	yes	—
133	Dic₁₇₈	—	2	1	no	—
134	Dic₁₇₈	—	2	1	no	—
135	Dic₁₇₈	—	2	1	no	—
136	Dic₁₇₈	—	2	1	no	—
137	Dic₁₇₈	—	4	2	no	—
138	Dic₁₇₈	—	8	1	no	—
139	Dic₁₇₈	—	8	1	no	—
140	C₈×C₈₉	—	2	1	no	—
141	C₈×C₈₉	—	2	1	no	—
142	#10623494	—	2	1	no	—
143	#10623494	—	2	1	no	—
144	C₂²×C₈₉	—	4	1	no	—
145	C₂²×C₈₉	—	4	1	no	—
146	C₂²×C₈₉	—	4	1	no	—
147	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₈₉	—	2	1	no	—
149	C₂×D₈₉	—	2	1	no	—
150	C₂×D₈₉	#124	1	2	yes	—
151	C₄×C₈₉	—	2	1	no	—
152	C₄×C₈₉	—	2	1	no	—
153	C₄×C₈₉	—	4	1	no	—
154	C₄×C₈₉	—	4	1	no	—
155	C₄×C₈₉	—	4	1	no	—
156	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
157	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
158	Dic₈₉	—	2	1	no	—
159	Dic₈₉	—	2	1	no	—
160	Dic₈₉	—	4	1	no	—
161	Dic₈₉	—	4	1	no	—
162	Dic₈₉	—	4	1	no	—
163	Dic₈₉	—	4	1	no	—
164	Dic₈₉	#124	1	2	yes	—
165	C₂×C₈₉	—	8	1	no	—
166	C₂×C₈₉	C₂×#124	1	1	yes	characteristic
167	C₂×C₈₉	#577	1	1	yes	characteristic
168	C₂×C₈₉	#578	1	1	yes	characteristic
169	D₈₉	—	2	2	no	—
170	#3137	—	89	1	no	maximal, Sylow
171	C₈₉	#3137	1	1	yes	characteristic, complemented, Sylow
172	#544	—	89	1	no	—
173	#536	—	89	1	no	—
174	#491	—	89	1	no	—
175	#491	—	89	1	no	—
176	#489	—	89	1	no	—
177	#489	—	89	1	no	—
178	#488	—	89	1	no	—
179	#488	D₈₉	1	1	yes	characteristic, complemented
180	#487	—	89	1	no	—
181	#487	—	89	1	no	—
182	#419	—	89	1	no	—
183	#417	—	89	1	no	—
184	#416	—	89	1	no	—
185	C₂×#103	—	89	1	no	—
186	#405	—	89	1	no	—
187	C₂×#41	—	89	1	no	—
188	C₂²×Q₈	—	89	1	no	—
189	C₂×M₁₆	—	89	1	no	—
190	C₂²×C₈	—	89	1	no	—
191	C₄⋊Q₈	—	89	1	no	—
192	C₄⋊Q₈	—	89	1	no	—
193	C₄⋊Q₈	—	89	1	no	—
194	C₄⋊Q₈	—	89	1	no	—
195	C₄⋊Q₈	—	178	1	no	—
196	#125	—	89	1	no	—
197	#125	—	178	1	no	—
198	#125	C₂×D₈₉	1	1	yes	characteristic
199	#124	—	89	1	no	—
200	#124	C₂×D₈₉	1	1	yes	characteristic
201	#123	—	178	1	no	—
202	#123	—	178	1	no	—
203	C₂²⋊Q₈	—	178	1	no	—
204	C₂²⋊Q₈	—	178	1	no	—
205	C₂²⋊Q₈	—	178	1	no	—
206	C₂²⋊Q₈	—	178	1	no	—
207	#117	—	89	1	no	—
208	#117	—	89	1	no	—
209	C₂×C₄⋊C₄	—	89	1	no	—
210	#103	—	89	1	no	—
211	#103	—	89	1	no	—
212	#103	—	89	1	no	—
213	#103	—	89	1	no	—
214	#103	—	89	1	no	—
215	#103	—	89	1	no	—
216	#103	—	89	1	no	—
217	#103	—	89	1	no	—
218	#103	—	89	1	no	—
219	#103	—	89	1	no	—
220	#103	C₂×D₈₉	1	1	yes	characteristic
221	#103	C₂×D₈₉	1	1	yes	characteristic
222	#102	—	89	1	no	—
223	#102	—	89	1	no	—
224	#102	C₂×D₈₉	1	1	yes	characteristic
225	#102	C₂×D₈₉	1	1	yes	characteristic
226	#97	—	89	1	no	—
227	#97	—	89	1	no	—
228	C₄×C₈	—	89	1	no	—
229	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
230	#41	—	178	2	no	—
231	#41	—	178	2	no	—
232	C₂×Q₈	—	89	1	no	—
233	C₂×Q₈	—	89	1	no	—
234	C₂×Q₈	—	178	1	no	—
235	C₂×Q₈	—	178	1	no	—
236	C₂×Q₈	—	178	2	no	—
237	C₂×Q₈	—	178	2	no	—
238	C₂×Q₈	—	178	1	no	—
239	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
240	C₂×D₄	—	178	1	no	—
241	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
242	C₂²×C₄	—	89	1	no	—
243	C₂²×C₄	—	178	1	no	—
244	C₂²×C₄	—	178	1	no	—
245	C₂²×C₄	—	178	1	no	—
246	M₁₆	—	178	2	no	—
247	C₂×C₈	—	89	1	no	—
248	C₂×C₈	—	89	1	no	—
249	C₂×C₈	—	178	2	no	—
250	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	89	1	no	—
255	C₄⋊C₄	—	89	1	no	—
256	C₄⋊C₄	—	89	1	no	—
257	C₄⋊C₄	—	89	1	no	—
258	C₄⋊C₄	—	178	1	no	—
259	C₄⋊C₄	—	178	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	—	178	1	no	—
265	C₄⋊C₄	—	178	1	no	—
266	C₄⋊C₄	—	178	1	no	—
267	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	89	1	no	—
272	C₂²⋊C₄	—	89	1	no	—
273	C₂²⋊C₄	—	178	1	no	—
274	C₂²⋊C₄	—	178	1	no	—
275	C₂²⋊C₄	—	178	1	no	—
276	C₄²	—	89	1	no	—
277	C₄²	—	178	1	no	—
278	C₄²	C₂²×D₈₉	1	1	yes	characteristic
279	C₂³	—	2	1	no	—
280	C₂³	—	89	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	356	2	no	—
284	Q₈	—	178	1	no	—
285	Q₈	—	178	1	no	—
286	Q₈	—	178	1	no	—
287	Q₈	—	178	1	no	—
288	Q₈	—	712	1	no	—
289	Q₈	—	712	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	356	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	89	1	no	—
297	C₂×C₄	—	89	2	no	—
298	C₂×C₄	—	89	2	no	—
299	C₂×C₄	—	89	1	no	—
300	C₂×C₄	—	89	1	no	—
301	C₂×C₄	—	356	2	no	—
302	C₂×C₄	—	356	2	no	—
303	C₂×C₄	—	178	1	no	—
304	C₂×C₄	—	178	1	no	—
305	C₂×C₄	—	178	1	no	—
306	C₂×C₄	—	178	1	no	—
307	C₂×C₄	—	712	1	no	—
308	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
309	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	178	1	no	—
314	C₈	—	178	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	89	2	no	—
319	C₂²	—	178	1	no	—
320	C₂²	—	178	1	no	—
321	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	89	2	no	—
328	C₄	—	356	1	no	—
329	C₄	—	356	1	no	—
330	C₄	—	356	1	no	—
331	C₄	—	356	1	no	—
332	C₄	—	178	1	no	—
333	C₄	—	178	1	no	—
334	C₄	#41×D₈₉	1	1	yes	characteristic
335	C₄	#41×D₈₉	1	1	yes	characteristic
336	C₂	—	8	1	no	—
337	C₂	—	178	2	no	—
338	C₂	#124×D₈₉	1	1	yes	characteristic
339	C₂	#19152405	1	1	yes	characteristic
340	C₂	#19152412	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#488×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×#13083573	C₂²
hypercenter	#488	D₈₉
hypocenter	C₈₉	#3137
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #488

Complements

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

N	H ≅ G/N	type
1. #19152228	C₂	semidirect
3. #19151968	C₂	semidirect
4. #19151965	C₂	semidirect
5. #19151957	C₂	semidirect
6. #19151956	C₂	semidirect
7. #19151951	C₂	semidirect
8. #19151949	C₂	semidirect
9. #19151947	C₂	semidirect
10. #19151605	C₂	semidirect
13. #103×D₈₉	C₂	semidirect
14. #19151552	C₂	semidirect
15. #488×C₈₉	C₂	semidirect
171. C₈₉	#3137	semidirect
179. #488	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152228	2
#19152168	2
#19151968	2
#19151965	2
#19151957	2
#19151956	2
#19151951	2
#19151949	2
#19151947	2
#19151605	2
#19151596	2
#19151591	2
#103×D₈₉	2
#19151552	2
#488×C₈₉	2
#3137	89