Subgroups — #26032308

Order 11392.

376 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	#19152289	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#19151967	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151942	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151930	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151905	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151855	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151838	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151808	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151752	C₂	1	1	yes	maximal, characteristic, complemented
11	#19151752	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151671	C₂	1	1	yes	maximal, characteristic
13	#19151605	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	#441×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₈₉	C₂²	1	1	yes	characteristic
18	C₂×#13083564	C₂²	1	1	yes	characteristic
19	C₂×#13083564	C₂²	1	1	yes	characteristic
20	C₂×#13083557	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₈₉	C₂²	1	1	yes	characteristic
22	#15668012	—	2	1	no	—
23	#15668012	C₂²	1	1	yes	characteristic
24	#15668006	—	2	1	no	—
25	#15668004	C₂²	1	1	yes	characteristic
26	#15668002	—	2	1	no	—
27	#15667997	—	2	1	no	—
28	#15667992	—	2	1	no	—
29	#15667992	—	2	1	no	—
30	#15667992	C₂²	1	1	yes	characteristic
31	#15667992	C₂²	1	1	yes	characteristic
32	#15667988	C₂²	1	1	yes	characteristic
33	#15667988	C₂²	1	1	yes	characteristic
34	C₄×D₃₅₆	—	2	1	no	—
35	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
36	#15667970	C₂²	1	1	yes	characteristic
37	#15667969	—	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	#15667937	C₂²	1	1	yes	characteristic
43	#15667937	C₂²	1	1	yes	characteristic
44	#15667937	C₂²	1	1	yes	characteristic
45	#15667937	C₂²	1	1	yes	characteristic
46	#15667936	C₂²	1	1	yes	characteristic
47	#15667936	C₂²	1	1	yes	characteristic
48	#15667935	C₂²	1	1	yes	characteristic
49	#15667935	C₂²	1	1	yes	characteristic
50	#15667924	C₂²	1	1	yes	characteristic
51	#15667924	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
55	#106×C₈₉	C₂²	1	1	yes	characteristic
56	#103×C₈₉	C₂²	1	1	yes	characteristic
57	#103×C₈₉	C₂²	1	1	yes	characteristic
58	#97×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×Dic₁₇₈	—	2	1	no	—
61	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
63	Q₈×D₈₉	—	2	2	no	—
64	Q₈×D₈₉	—	2	2	no	—
65	Q₈×D₈₉	—	4	1	no	—
66	Q₈×D₈₉	—	4	1	no	—
67	C₂×D₃₅₆	D₄	1	1	yes	characteristic
68	C₂×C₄×D₈₉	—	2	1	no	—
69	C₂×C₄×D₈₉	—	2	1	no	—
70	C₂×C₄×D₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
73	#13083564	—	4	1	no	—
74	#13083564	—	4	1	no	—
75	#13083557	—	4	1	no	—
76	C₂×#10623494	C₂³	1	1	yes	characteristic
77	C₂×#10623494	C₂³	1	1	yes	characteristic
78	C₈×D₈₉	—	4	1	no	—
79	C₄⋊Dic₈₉	—	2	1	no	—
80	C₄⋊Dic₈₉	—	2	1	no	—
81	C₄⋊Dic₈₉	—	2	1	no	—
82	C₄⋊Dic₈₉	—	2	1	no	—
83	C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
84	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
86	#13083553	—	2	1	no	—
87	#13083553	—	2	1	no	—
88	#13083553	—	2	1	no	—
89	#13083553	—	2	1	no	—
90	#13083553	—	2	1	no	—
91	#13083551	—	2	1	no	—
92	#13083551	—	2	1	no	—
93	#13083551	—	2	1	no	—
94	#13083551	—	2	1	no	—
95	#13083551	—	2	1	no	—
96	#13083551	D₄	1	1	yes	characteristic
97	#13083551	D₄	1	1	yes	characteristic
98	C₄×Dic₈₉	—	2	1	no	—
99	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
101	Q₁₆×C₈₉	—	2	2	no	—
102	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₉	—	2	1	no	—
105	C₄⋊C₄×C₈₉	—	2	1	no	—
106	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
108	C₄²×C₈₉	—	2	1	no	—
109	C₄²×C₈₉	—	2	1	no	—
110	C₄²×C₈₉	C₂³	1	1	yes	characteristic
111	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₈₉	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₈₉	—	2	1	no	—
114	Q₈×C₈₉	—	2	1	no	—
115	Q₈×C₈₉	—	2	1	no	—
116	Q₈×C₈₉	#41	1	2	yes	—
117	C₂×C₄×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	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
124	C₂×Dic₈₉	—	2	1	no	—
125	C₂×Dic₈₉	—	2	1	no	—
126	C₂×Dic₈₉	—	2	1	no	—
127	C₂×Dic₈₉	—	2	1	no	—
128	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₈₉	C₂×D₄	1	1	yes	characteristic
130	D₃₅₆	—	2	2	no	—
131	C₄×D₈₉	—	2	1	no	—
132	C₄×D₈₉	—	2	1	no	—
133	C₄×D₈₉	—	2	1	no	—
134	C₄×D₈₉	—	2	1	no	—
135	C₄×D₈₉	—	4	2	no	—
136	C₄×D₈₉	—	4	2	no	—
137	C₄×D₈₉	—	4	2	no	—
138	Dic₁₇₈	—	2	1	no	—
139	Dic₁₇₈	—	2	1	no	—
140	Dic₁₇₈	—	2	1	no	—
141	Dic₁₇₈	—	2	1	no	—
142	Dic₁₇₈	—	4	2	no	—
143	Dic₁₇₈	—	4	2	no	—
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	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	—	2	1	no	—
153	C₂×D₈₉	—	2	1	no	—
154	C₂×D₈₉	#137	1	2	yes	—
155	C₄×C₈₉	—	2	2	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	4	1	no	—
159	C₄×C₈₉	—	4	1	no	—
160	C₄×C₈₉	—	4	1	no	—
161	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
162	C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
163	Dic₈₉	—	2	1	no	—
164	Dic₈₉	—	2	1	no	—
165	Dic₈₉	—	4	1	no	—
166	Dic₈₉	—	4	1	no	—
167	Dic₈₉	—	4	1	no	—
168	Dic₈₉	—	4	1	no	—
169	C₂×C₈₉	#546	1	1	yes	characteristic
170	C₂×C₈₉	C₂×#137	1	1	yes	characteristic
171	C₂×C₈₉	#578	1	1	yes	characteristic
172	D₈₉	—	2	2	no	—
173	D₈₉	—	4	1	no	—
174	#3312	—	89	1	no	maximal, Sylow
175	C₈₉	#3312	1	1	yes	characteristic, complemented, Sylow
176	#552	—	89	1	no	—
177	D₄×Q₈	—	89	1	no	—
178	#491	—	89	1	no	—
179	#485	—	89	1	no	—
180	#483	—	89	1	no	—
181	#479	—	89	1	no	—
182	#470	—	89	1	no	—
183	#467	—	89	1	no	—
184	#462	—	89	1	no	—
185	#451	—	89	1	no	—
186	#451	—	89	1	no	—
187	#441	D₈₉	1	1	yes	characteristic, complemented
188	#435	—	89	1	no	—
189	#419	—	89	1	no	—
190	C₂×#103	—	89	1	no	—
191	C₂²×Q₈	—	89	1	no	—
192	C₂²×Q₈	—	89	1	no	—
193	C₂×Q₁₆	—	89	1	no	—
194	C₂×Q₁₆	—	89	1	no	—
195	C₂×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
196	C₂×M₁₆	—	89	1	no	—
197	C₂²×C₈	—	89	1	no	—
198	C₄⋊Q₈	—	89	1	no	—
199	C₄⋊Q₈	—	178	1	no	—
200	#126	—	178	1	no	—
201	#125	—	89	1	no	—
202	#125	—	178	1	no	—
203	#123	—	178	1	no	—
204	C₂²⋊Q₈	—	89	1	no	—
205	C₂²⋊Q₈	—	89	1	no	—
206	C₂²⋊Q₈	—	89	1	no	—
207	C₂²⋊Q₈	—	89	1	no	—
208	C₂²⋊Q₈	—	178	1	no	—
209	C₂²⋊Q₈	—	178	1	no	—
210	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
211	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
212	C₄×D₄	—	89	1	no	—
213	C₄×D₄	—	178	1	no	—
214	#117	—	89	1	no	—
215	#117	—	178	1	no	—
216	C₂×C₄⋊C₄	—	89	1	no	—
217	#107	—	89	1	no	—
218	#106	—	89	1	no	—
219	#106	C₂×D₈₉	1	1	yes	characteristic
220	C₄⋊C₈	—	89	1	no	—
221	#103	—	89	1	no	—
222	#103	—	89	1	no	—
223	#103	—	89	1	no	—
224	#103	—	89	1	no	—
225	#103	—	89	1	no	—
226	#103	—	89	1	no	—
227	#103	—	89	1	no	—
228	#103	—	89	1	no	—
229	#103	C₂×D₈₉	1	1	yes	characteristic
230	#103	C₂×D₈₉	1	1	yes	characteristic
231	C₂²⋊C₈	—	89	1	no	—
232	C₂²⋊C₈	—	89	1	no	—
233	#97	C₂×D₈₉	1	1	yes	characteristic
234	C₂×Q₈	—	89	1	no	—
235	C₂×Q₈	—	89	1	no	—
236	C₂×Q₈	—	356	1	no	—
237	C₂×Q₈	—	356	1	no	—
238	C₂×Q₈	—	178	1	no	—
239	C₂×Q₈	—	178	1	no	—
240	C₂×Q₈	—	178	2	no	—
241	C₂×Q₈	—	178	2	no	—
242	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
243	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
244	C₂×D₄	—	89	1	no	—
245	C₂²×C₄	—	89	1	no	—
246	C₂²×C₄	—	89	1	no	—
247	C₂²×C₄	—	178	1	no	—
248	C₂²×C₄	—	178	1	no	—
249	C₂²×C₄	—	178	1	no	—
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	356	1	no	—
252	Q₁₆	—	356	1	no	—
253	M₁₆	—	356	1	no	—
254	C₂×C₈	—	89	1	no	—
255	C₂×C₈	—	89	1	no	—
256	C₂×C₈	—	356	1	no	—
257	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	89	1	no	—
262	C₄⋊C₄	—	89	1	no	—
263	C₄⋊C₄	—	89	1	no	—
264	C₄⋊C₄	—	178	1	no	—
265	C₄⋊C₄	—	178	1	no	—
266	C₄⋊C₄	—	178	1	no	—
267	C₄⋊C₄	—	178	1	no	—
268	C₄⋊C₄	—	178	1	no	—
269	C₄⋊C₄	—	178	1	no	—
270	C₄⋊C₄	—	178	1	no	—
271	C₄⋊C₄	—	178	1	no	—
272	C₄⋊C₄	—	178	1	no	—
273	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
275	C₂²⋊C₄	—	89	1	no	—
276	C₂²⋊C₄	—	89	1	no	—
277	C₂²⋊C₄	—	178	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	C₄²	—	2	1	no	—
283	C₄²	—	2	1	no	—
284	C₄²	—	178	1	no	—
285	C₄²	C₂²×D₈₉	1	1	yes	characteristic
286	C₂³	—	89	1	no	—
287	C₂³	—	89	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	356	2	no	—
292	Q₈	—	356	2	no	—
293	Q₈	—	178	1	no	—
294	Q₈	—	178	1	no	—
295	Q₈	—	178	1	no	—
296	Q₈	—	178	1	no	—
297	Q₈	#13083574	1	2	yes	—
298	D₄	—	178	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	89	1	no	—
304	C₂×C₄	—	89	1	no	—
305	C₂×C₄	—	356	2	no	—
306	C₂×C₄	—	356	2	no	—
307	C₂×C₄	—	356	2	no	—
308	C₂×C₄	—	178	1	no	—
309	C₂×C₄	—	178	1	no	—
310	C₂×C₄	—	178	1	no	—
311	C₂×C₄	—	178	1	no	—
312	C₂×C₄	—	178	1	no	—
313	C₂×C₄	—	178	1	no	—
314	C₂×C₄	—	178	1	no	—
315	C₂×C₄	—	178	1	no	—
316	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
317	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
318	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	178	1	no	—
322	C₈	—	178	1	no	—
323	C₂²	—	89	2	no	—
324	C₂²	—	178	1	no	—
325	C₂²	—	178	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	—	2	2	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	356	1	no	—
337	C₄	—	356	1	no	—
338	C₄	—	356	1	no	—
339	C₄	—	356	1	no	—
340	C₄	—	178	1	no	—
341	C₄	—	178	1	no	—
342	C₄	C₂×#13083574	1	1	yes	characteristic
343	C₄	#15668058	1	1	yes	characteristic
344	C₂	—	356	1	no	—
345	C₂	—	178	2	no	—
346	C₂	#19152244	1	1	yes	characteristic
347	C₂	#137×D₈₉	1	1	yes	characteristic
348	C₂	#19152414	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#441×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	C₂×D₃₅₆	D₄
hypercenter	#441	D₈₉
hypocenter	C₈₉	#3312
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
3. #19151967	C₂	semidirect
4. #19151942	C₂	semidirect
5. #19151930	C₂	semidirect
6. #19151905	C₂	semidirect
7. #19151855	C₂	semidirect
8. #19151838	C₂	semidirect
9. #19151808	C₂	semidirect
10. #19151752	C₂	semidirect
11. #19151752	C₂	semidirect
13. #19151605	C₂	semidirect
14. #103×D₈₉	C₂	semidirect
15. #441×C₈₉	C₂	semidirect
175. C₈₉	#3312	semidirect
187. #441	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152289	2
Q₈×D₃₅₆	2
#19151967	2
#19151942	2
#19151930	2
#19151905	2
#19151855	2
#19151838	2
#19151808	2
#19151752	2
#19151671	2
#19151605	2
#103×D₈₉	2
#441×C₈₉	2
#3312	89