Subgroups — #26032563

Order 11392.

388 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	#19152306	C₂	1	1	yes	maximal, characteristic, complemented
2	#19152250	C₂	1	1	yes	maximal, characteristic
3	#19151947	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151930	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151926	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151887	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151855	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151806	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151759	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151758	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151683	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151674	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151600	C₂	1	1	yes	maximal, characteristic, complemented
14	#19151598	C₂	1	1	yes	maximal, characteristic, complemented
15	#470×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083575	C₂²	1	1	yes	characteristic
17	C₂×#13083574	C₂²	1	1	yes	characteristic
18	C₂×#13083564	C₂²	1	1	yes	characteristic
19	C₂×#13083561	C₂²	1	1	yes	characteristic
20	C₂×#13083557	C₂²	1	1	yes	characteristic
21	#15668011	C₂²	1	1	yes	characteristic
22	#15668010	C₂²	1	1	yes	characteristic
23	#15668002	—	2	1	no	—
24	#15667997	—	2	1	no	—
25	#15667994	—	2	1	no	—
26	#15667992	—	2	1	no	—
27	#15667992	C₂²	1	1	yes	characteristic
28	#15667989	—	2	1	no	—
29	#15667988	C₂²	1	1	yes	characteristic
30	#15667985	—	2	1	no	—
31	Q₈×Dic₈₉	C₂²	1	1	yes	characteristic
32	#15667974	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
34	#15667972	—	2	1	no	—
35	C₂×C₄⋊Dic₈₉	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₈₉	—	2	1	no	—
37	C₄⋊C₄×D₈₉	C₂²	1	1	yes	characteristic
38	#15667943	C₂²	1	1	yes	characteristic
39	#15667943	C₂²	1	1	yes	characteristic
40	#15667941	C₂²	1	1	yes	characteristic
41	#15667937	C₂²	1	1	yes	characteristic
42	#15667937	C₂²	1	1	yes	characteristic
43	#15667936	C₂²	1	1	yes	characteristic
44	#15667936	C₂²	1	1	yes	characteristic
45	#15667935	C₂²	1	1	yes	characteristic
46	#15667934	C₂²	1	1	yes	characteristic
47	#15667934	C₂²	1	1	yes	characteristic
48	#15667932	C₂²	1	1	yes	characteristic
49	C₂²⋊#10623494	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
56	#106×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	#103×C₈₉	C₂²	1	1	yes	characteristic
59	#13083575	—	2	2	no	—
60	#13083575	—	4	1	no	—
61	#13083574	—	2	2	no	—
62	#13083574	—	4	1	no	—
63	C₂×Dic₁₇₈	—	2	1	no	—
64	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
65	C₂×D₃₅₆	—	2	1	no	—
66	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₉	—	2	1	no	—
68	C₂×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
69	C₂²×Dic₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	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	#13083564	—	4	1	no	—
76	#13083561	—	2	2	no	—
77	#13083557	—	4	1	no	—
78	C₂×#10623494	C₂³	1	1	yes	characteristic
79	C₂×#10623494	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₉	—	2	1	no	—
81	C₄⋊Dic₈₉	—	2	1	no	—
82	C₄⋊Dic₈₉	—	2	1	no	—
83	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
85	#13083553	—	2	1	no	—
86	#13083553	—	2	1	no	—
87	#13083553	—	2	1	no	—
88	#13083553	—	2	1	no	—
89	#13083553	—	2	1	no	—
90	#13083553	D₄	1	1	yes	characteristic
91	C₂²⋊Dic₈₉	D₄	1	1	yes	characteristic
92	#13083551	—	2	1	no	—
93	#13083551	—	2	1	no	—
94	#13083551	—	2	1	no	—
95	#13083551	D₄	1	1	yes	characteristic
96	C₄×Dic₈₉	—	2	1	no	—
97	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
98	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
99	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
101	Q₁₆×C₈₉	—	4	1	no	—
102	M₁₆×C₈₉	—	4	1	no	—
103	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₉	—	2	1	no	—
106	C₄⋊C₄×C₈₉	—	2	1	no	—
107	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
109	C₂²⋊C₄×C₈₉	—	2	1	no	—
110	C₂²⋊C₄×C₈₉	—	2	1	no	—
111	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₈₉	—	2	1	no	—
113	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₈₉	—	2	1	no	—
115	Q₈×C₈₉	—	2	1	no	—
116	Q₈×C₈₉	—	2	1	no	—
117	Q₈×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	2	1	no	—
123	C₂×C₄×C₈₉	—	2	1	no	—
124	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
126	C₂²⋊D₈₉	—	2	2	no	—
127	C₂²⋊D₈₉	—	4	2	no	—
128	C₂×Dic₈₉	—	2	1	no	—
129	C₂×Dic₈₉	—	2	1	no	—
130	C₂×Dic₈₉	—	2	1	no	—
131	C₂×Dic₈₉	—	8	1	no	—
132	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
135	D₃₅₆	—	2	1	no	—
136	D₃₅₆	—	4	2	no	—
137	D₃₅₆	#41	1	2	yes	—
138	C₄×D₈₉	—	2	1	no	—
139	C₄×D₈₉	—	2	1	no	—
140	C₄×D₈₉	—	4	2	no	—
141	C₄×D₈₉	—	4	2	no	—
142	C₄×D₈₉	—	4	2	no	—
143	C₄×D₈₉	—	8	1	no	—
144	Dic₁₇₈	—	2	1	no	—
145	Dic₁₇₈	—	2	1	no	—
146	Dic₁₇₈	—	8	1	no	—
147	C₈×C₈₉	—	2	1	no	—
148	C₈×C₈₉	—	2	1	no	—
149	#10623494	—	2	1	no	—
150	#10623494	—	2	1	no	—
151	C₂²×C₈₉	—	2	1	no	—
152	C₂²×C₈₉	—	2	1	no	—
153	C₂²×C₈₉	—	2	1	no	—
154	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₈₉	—	2	1	no	—
156	C₂×D₈₉	—	2	2	no	—
157	C₂×D₈₉	—	2	1	no	—
158	C₂×D₈₉	—	2	1	no	—
159	C₂×D₈₉	—	4	1	no	—
160	C₂×D₈₉	—	4	1	no	—
161	C₄×C₈₉	—	2	1	no	—
162	C₄×C₈₉	—	4	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	H(5,𝔽₂)	1	1	yes	characteristic
168	Dic₈₉	—	2	1	no	—
169	Dic₈₉	—	2	1	no	—
170	Dic₈₉	—	4	1	no	—
171	Dic₈₉	—	4	1	no	—
172	Dic₈₉	—	4	1	no	—
173	Dic₈₉	#137	1	2	yes	—
174	C₂×C₈₉	—	4	1	no	—
175	C₂×C₈₉	#546	1	1	yes	characteristic
176	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
177	C₂×C₈₉	#577	1	1	yes	characteristic
178	D₈₉	—	4	1	no	—
179	D₈₉	—	4	2	no	—
180	#3331	—	89	1	no	maximal, Sylow
181	C₈₉	#3331	1	1	yes	characteristic, complemented, Sylow
182	#554	—	89	1	no	—
183	#547	—	89	1	no	—
184	#487	—	89	1	no	—
185	#483	—	89	1	no	—
186	#483	—	89	1	no	—
187	#476	—	89	1	no	—
188	#470	—	89	1	no	—
189	#470	D₈₉	1	1	yes	characteristic, complemented
190	#462	—	89	1	no	—
191	#452	—	89	1	no	—
192	#452	—	89	1	no	—
193	C₄×SD₁₆	—	89	1	no	—
194	#436	—	89	1	no	—
195	#417	—	89	1	no	—
196	#417	—	89	1	no	—
197	C₂×#41	—	89	1	no	—
198	C₂×#41	—	89	1	no	—
199	C₂×Q₁₆	—	89	1	no	—
200	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
201	C₂×SD₁₆	—	89	1	no	—
202	C₂×M₁₆	—	89	1	no	—
203	C₂×M₁₆	C₂×D₈₉	1	1	yes	characteristic
204	C₄⋊Q₈	—	89	1	no	—
205	C₄⋊Q₈	—	89	1	no	—
206	#125	—	178	1	no	—
207	#123	—	178	1	no	—
208	#123	—	178	1	no	—
209	C₂²⋊Q₈	—	89	1	no	—
210	C₂²⋊Q₈	—	89	1	no	—
211	C₂²⋊Q₈	—	178	1	no	—
212	C₂²⋊Q₈	—	178	1	no	—
213	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
214	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
215	#121	—	178	1	no	—
216	C₄×Q₈	—	89	1	no	—
217	C₄×D₄	—	89	1	no	—
218	C₄×D₄	—	89	1	no	—
219	C₄×D₄	—	178	1	no	—
220	C₂×C₄⋊C₄	—	89	1	no	—
221	C₂×C₄⋊C₄	—	89	1	no	—
222	C₂×C₄⋊C₄	—	178	1	no	—
223	#106	—	89	1	no	—
224	#106	—	89	1	no	—
225	#106	C₂×D₈₉	1	1	yes	characteristic
226	C₄⋊C₈	—	89	1	no	—
227	#103	—	89	1	no	—
228	#103	—	89	1	no	—
229	#103	—	89	1	no	—
230	#103	—	89	1	no	—
231	#103	—	89	1	no	—
232	#103	C₂×D₈₉	1	1	yes	characteristic
233	#103	C₂×D₈₉	1	1	yes	characteristic
234	#102	—	89	1	no	—
235	#102	—	89	1	no	—
236	#102	—	89	1	no	—
237	C₂²⋊C₈	—	89	1	no	—
238	C₂²⋊C₈	—	89	1	no	—
239	C₄×C₈	—	89	1	no	—
240	#41	—	356	1	no	—
241	#41	—	356	1	no	—
242	#41	—	178	2	no	—
243	#41	—	178	2	no	—
244	C₂×Q₈	—	89	1	no	—
245	C₂×Q₈	—	178	1	no	—
246	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
247	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
248	C₂×D₄	—	89	1	no	—
249	C₂×D₄	—	89	1	no	—
250	C₂×D₄	—	178	1	no	—
251	C₂×D₄	—	178	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	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
259	Q₁₆	—	4	1	no	—
260	Q₁₆	—	356	1	no	—
261	SD₁₆	—	178	2	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	—	356	1	no	—
264	C₂×C₈	—	89	1	no	—
265	C₂×C₈	—	89	1	no	—
266	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
267	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	89	1	no	—
271	C₄⋊C₄	—	89	1	no	—
272	C₄⋊C₄	—	89	1	no	—
273	C₄⋊C₄	—	178	1	no	—
274	C₄⋊C₄	—	178	1	no	—
275	C₄⋊C₄	—	178	1	no	—
276	C₄⋊C₄	—	178	1	no	—
277	C₄⋊C₄	—	178	1	no	—
278	C₄⋊C₄	—	178	1	no	—
279	C₄⋊C₄	—	178	1	no	—
280	C₄⋊C₄	—	178	1	no	—
281	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
282	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
283	C₂²⋊C₄	—	2	1	no	—
284	C₂²⋊C₄	—	2	1	no	—
285	C₂²⋊C₄	—	89	1	no	—
286	C₂²⋊C₄	—	89	1	no	—
287	C₂²⋊C₄	—	178	1	no	—
288	C₂²⋊C₄	—	178	1	no	—
289	C₂²⋊C₄	—	178	1	no	—
290	C₄²	—	89	1	no	—
291	C₄²	—	178	1	no	—
292	C₂³	—	89	1	no	—
293	C₂³	—	178	1	no	—
294	C₂³	D₄×D₈₉	1	1	yes	characteristic
295	Q₈	—	2	1	no	—
296	Q₈	—	2	1	no	—
297	Q₈	—	2	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	178	1	no	—
300	Q₈	—	178	1	no	—
301	Q₈	—	712	1	no	—
302	D₄	—	89	2	no	—
303	D₄	—	356	2	no	—
304	D₄	—	356	2	no	—
305	D₄	—	178	2	no	—
306	D₄	—	178	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	89	1	no	—
314	C₂×C₄	—	89	1	no	—
315	C₂×C₄	—	89	1	no	—
316	C₂×C₄	—	356	2	no	—
317	C₂×C₄	—	356	2	no	—
318	C₂×C₄	—	356	2	no	—
319	C₂×C₄	—	178	1	no	—
320	C₂×C₄	—	178	1	no	—
321	C₂×C₄	—	178	1	no	—
322	C₂×C₄	—	178	1	no	—
323	C₂×C₄	—	178	1	no	—
324	C₂×C₄	—	712	1	no	—
325	C₂×C₄	—	712	1	no	—
326	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
327	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	178	1	no	—
331	C₈	—	178	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	356	1	no	—
336	C₂²	—	356	1	no	—
337	C₂²	—	178	1	no	—
338	C₂²	—	178	1	no	—
339	C₂²	—	178	2	no	—
340	C₂²	—	178	1	no	—
341	C₂²	C₂×D₄×D₈₉	1	1	yes	characteristic
342	C₄	—	2	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	4	1	no	—
346	C₄	—	4	1	no	—
347	C₄	—	89	2	no	—
348	C₄	—	356	1	no	—
349	C₄	—	356	1	no	—
350	C₄	—	356	1	no	—
351	C₄	—	178	1	no	—
352	C₄	—	178	1	no	—
353	C₄	#41×D₈₉	1	1	yes	characteristic
354	C₄	#15668057	1	1	yes	characteristic
355	C₂	—	4	1	no	—
356	C₂	—	356	2	no	—
357	C₂	—	356	1	no	—
358	C₂	#19152237	1	1	yes	characteristic
359	C₂	#137×D₈₉	1	1	yes	characteristic
360	C₂	#19152407	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#470×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19152250	C₂
hypercenter	#470	D₈₉
hypocenter	C₈₉	#3331
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

Complements

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

N	H ≅ G/N	type
1. #19152306	C₂	semidirect
3. #19151947	C₂	semidirect
4. #19151930	C₂	semidirect
5. #19151926	C₂	semidirect
6. #19151887	C₂	semidirect
7. #19151855	C₂	semidirect
8. #19151806	C₂	semidirect
9. #19151759	C₂	semidirect
10. #19151758	C₂	semidirect
11. #19151683	C₂	semidirect
12. #19151674	C₂	semidirect
13. #19151600	C₂	semidirect
14. #19151598	C₂	semidirect
15. #470×C₈₉	C₂	semidirect
181. C₈₉	#3331	semidirect
189. #470	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152306	2
#19152250	2
#19151947	2
#19151930	2
#19151926	2
#19151887	2
#19151855	2
#19151806	2
#19151759	2
#19151758	2
#19151683	2
#19151674	2
#19151600	2
#19151598	2
#470×C₈₉	2
#3331	89