Subgroups — #26030816

Order 11392.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152132	C₂	1	1	yes	maximal, characteristic, complemented
2	#19151997	C₂	1	1	yes	maximal, characteristic, complemented
3	#19151980	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151955	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151952	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151837	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151832	C₂	1	1	yes	maximal, characteristic
8	#19151829	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151825	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#13083564	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#13083561	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#13083562	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#13083559	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₄×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
15	#529×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083564	C₂²	1	1	yes	characteristic
17	C₂×#13083562	C₂²	1	1	yes	characteristic
18	C₂×#13083561	C₂²	1	1	yes	characteristic
19	C₂×#13083559	C₂²	1	1	yes	characteristic
20	C₂²×#10623494	C₂²	1	1	yes	characteristic
21	C₂²×#10623494	C₂²	1	1	yes	characteristic
22	#15668006	—	2	1	no	—
23	#15668004	C₂²	1	1	yes	characteristic
24	#15668000	C₂²	1	1	yes	characteristic
25	#15667995	—	2	1	no	—
26	#15667991	C₂²	1	1	yes	characteristic
27	#15667986	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
29	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₈₉	—	2	1	no	—
31	#15667969	—	2	1	no	—
32	#15667945	C₂²	1	1	yes	characteristic
33	#15667945	C₂²	1	1	yes	characteristic
34	#15667943	C₂²	1	1	yes	characteristic
35	#15667943	C₂²	1	1	yes	characteristic
36	#15667936	C₂²	1	1	yes	characteristic
37	#15667936	C₂²	1	1	yes	characteristic
38	#15667935	C₂²	1	1	yes	characteristic
39	#15667935	C₂²	1	1	yes	characteristic
40	#15667933	C₂²	1	1	yes	characteristic
41	#15667933	C₂²	1	1	yes	characteristic
42	#15667932	C₂²	1	1	yes	characteristic
43	#15667932	C₂²	1	1	yes	characteristic
44	C₄×#10623494	C₂²	1	1	yes	characteristic
45	C₄×#10623494	C₂²	1	1	yes	characteristic
46	C₄×#10623494	C₂²	1	1	yes	characteristic
47	C₄×#10623494	C₂²	1	1	yes	characteristic
48	#126×C₈₉	—	2	1	no	—
49	#125×C₈₉	C₂²	1	1	yes	characteristic
50	#124×C₈₉	C₂²	1	1	yes	characteristic
51	#123×C₈₉	—	2	1	no	—
52	C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
53	#121×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₈₉	—	2	1	no	—
56	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
57	#117×C₈₉	—	2	1	no	—
58	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
60	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₈₉	—	2	1	no	—
62	C₂×C₄×D₈₉	—	2	1	no	—
63	#13083564	—	2	2	no	—
64	#13083562	—	2	2	no	—
65	#13083561	—	2	2	no	—
66	#13083559	—	2	2	no	—
67	C₂×#10623494	—	2	2	no	—
68	C₂×#10623494	—	2	2	no	—
69	C₂×#10623494	C₂³	1	1	yes	characteristic
70	C₂×#10623494	C₂³	1	1	yes	characteristic
71	C₂×#10623494	C₂³	1	1	yes	characteristic
72	C₂×#10623494	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
75	#13083553	—	2	1	no	—
76	#13083553	—	2	1	no	—
77	#13083553	—	2	1	no	—
78	#13083553	—	2	1	no	—
79	C₂²⋊Dic₈₉	—	2	1	no	—
80	#13083551	—	2	1	no	—
81	#13083551	—	2	1	no	—
82	#13083551	—	2	1	no	—
83	#13083551	—	2	1	no	—
84	C₄×Dic₈₉	—	2	1	no	—
85	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
86	C₂×D₄×C₈₉	—	2	1	no	—
87	C₂×D₄×C₈₉	C₂³	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	—	2	1	no	—
89	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	D₄	1	1	yes	characteristic
91	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₈₉	—	2	1	no	—
93	C₄⋊C₄×C₈₉	—	2	1	no	—
94	C₄⋊C₄×C₈₉	—	2	1	no	—
95	C₄⋊C₄×C₈₉	—	2	1	no	—
96	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₂²⋊C₄×C₈₉	—	2	1	no	—
99	C₂²⋊C₄×C₈₉	—	2	1	no	—
100	C₂²⋊C₄×C₈₉	—	2	1	no	—
101	C₂²⋊C₄×C₈₉	—	2	1	no	—
102	C₂²⋊C₄×C₈₉	—	2	1	no	—
103	C₄²×C₈₉	—	2	1	no	—
104	C₄²×C₈₉	D₄	1	1	yes	characteristic
105	C₄²×C₈₉	D₄	1	1	yes	characteristic
106	C₄²×C₈₉	C₂³	1	1	yes	characteristic
107	C₄²×C₈₉	C₂³	1	1	yes	characteristic
108	C₂³×C₈₉	—	2	1	no	—
109	C₂³×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₈₉	—	2	1	no	—
111	Q₈×C₈₉	—	2	1	no	—
112	Q₈×C₈₉	—	2	1	no	—
113	D₄×C₈₉	—	2	1	no	—
114	D₄×C₈₉	—	2	1	no	—
115	D₄×C₈₉	—	4	2	no	—
116	C₂×C₄×C₈₉	—	2	2	no	—
117	C₂×C₄×C₈₉	—	2	2	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	—	4	2	no	—
124	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₈₉	—	4	2	no	—
131	C₂×Dic₈₉	—	2	1	no	—
132	C₂×Dic₈₉	—	2	1	no	—
133	C₂×Dic₈₉	—	2	1	no	—
134	D₃₅₆	—	2	1	no	—
135	D₃₅₆	—	2	1	no	—
136	C₄×D₈₉	—	4	2	no	—
137	Dic₁₇₈	—	2	1	no	—
138	Dic₁₇₈	—	2	1	no	—
139	#10623494	—	2	1	no	—
140	#10623494	—	2	1	no	—
141	#10623494	#41	1	2	yes	—
142	#10623494	#41	1	2	yes	—
143	C₂²×C₈₉	—	2	1	no	—
144	C₂²×C₈₉	—	2	1	no	—
145	C₂²×C₈₉	—	4	1	no	—
146	C₂²×C₈₉	—	4	1	no	—
147	C₂²×C₈₉	—	4	1	no	—
148	C₂²×C₈₉	#135	1	2	yes	—
149	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₈₉	—	4	1	no	—
151	C₂×D₈₉	—	4	1	no	—
152	C₂×D₈₉	—	4	1	no	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	4	1	no	—
158	C₄×C₈₉	—	4	1	no	—
159	C₄×C₈₉	—	4	1	no	—
160	C₄×C₈₉	#135	1	2	yes	—
161	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
162	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
163	Dic₈₉	—	4	1	no	—
164	Dic₈₉	—	4	1	no	—
165	Dic₈₉	—	4	1	no	—
166	C₂×C₈₉	—	2	2	no	—
167	C₂×C₈₉	—	8	1	no	—
168	C₂×C₈₉	#532	1	1	yes	characteristic
169	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
170	C₂×C₈₉	C₂×#135	1	1	yes	characteristic
171	D₈₉	—	8	1	no	—
172	#3168	—	89	1	no	maximal, Sylow
173	C₈₉	#3168	1	1	yes	characteristic, complemented, Sylow
174	#529	—	89	1	no	—
175	#529	D₈₉	1	1	yes	characteristic, complemented
176	#499	—	89	1	no	—
177	#495	—	89	1	no	—
178	#488	—	89	1	no	—
179	#488	—	89	1	no	—
180	#467	—	89	1	no	—
181	#466	—	89	1	no	—
182	#465	—	89	1	no	—
183	#465	—	89	1	no	—
184	C₄×Q₁₆	—	89	1	no	—
185	C₄×SD₁₆	—	89	1	no	—
186	C₄×SD₁₆	—	89	1	no	—
187	C₄×D₈	—	89	1	no	—
188	C₂×C₄×C₈	—	89	1	no	—
189	C₂×Q₁₆	—	89	1	no	—
190	C₂×SD₁₆	—	89	1	no	—
191	C₂×SD₁₆	—	89	1	no	—
192	C₂×D₈	—	89	1	no	—
193	C₂²×C₈	—	89	1	no	—
194	C₂²×C₈	—	89	1	no	—
195	#126	—	2	1	no	—
196	#126	—	178	1	no	—
197	#125	—	89	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	—	2	1	no	—
202	#123	—	178	1	no	—
203	C₂²⋊Q₈	—	89	1	no	—
204	C₂²⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
205	#121	—	89	1	no	—
206	#121	C₂×D₈₉	1	1	yes	characteristic
207	C₄×Q₈	—	89	1	no	—
208	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
209	C₄×D₄	—	2	1	no	—
210	C₄×D₄	—	89	1	no	—
211	C₄×D₄	—	178	1	no	—
212	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
213	#117	—	2	1	no	—
214	#117	—	178	1	no	—
215	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic
216	#107	—	89	1	no	—
217	#107	—	89	1	no	—
218	#106	—	89	1	no	—
219	#106	—	89	1	no	—
220	#103	—	89	1	no	—
221	#103	—	89	1	no	—
222	#103	—	89	1	no	—
223	#103	—	89	1	no	—
224	#102	—	89	1	no	—
225	#102	—	89	1	no	—
226	#102	—	89	1	no	—
227	#102	—	89	1	no	—
228	C₄×C₈	—	89	1	no	—
229	C₄×C₈	—	89	1	no	—
230	C₄×C₈	—	89	1	no	—
231	C₄×C₈	—	89	1	no	—
232	C₂×Q₈	—	89	1	no	—
233	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	89	1	no	—
236	C₂×D₄	—	178	1	no	—
237	C₂×D₄	C₂²×D₈₉	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	178	1	no	—
240	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
241	C₂²×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
242	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
243	Q₁₆	—	178	2	no	—
244	SD₁₆	—	178	2	no	—
245	SD₁₆	—	178	2	no	—
246	D₈	—	178	2	no	—
247	C₂×C₈	—	89	1	no	—
248	C₂×C₈	—	89	1	no	—
249	C₂×C₈	—	89	1	no	—
250	C₂×C₈	—	89	1	no	—
251	C₂×C₈	—	178	2	no	—
252	C₂×C₈	—	178	2	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	89	1	no	—
258	C₄⋊C₄	—	89	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	C₂²×D₈₉	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	178	1	no	—
271	C₂²⋊C₄	—	178	1	no	—
272	C₂²⋊C₄	—	178	1	no	—
273	C₂²⋊C₄	—	178	1	no	—
274	C₂²⋊C₄	—	178	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	178	1	no	—
277	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
278	C₄²	C₂²⋊D₈₉	1	1	yes	characteristic
279	C₄²	C₂²×D₈₉	1	1	yes	characteristic
280	C₄²	C₂²×D₈₉	1	1	yes	characteristic
281	C₂³	—	2	1	no	—
282	C₂³	—	178	1	no	—
283	C₂³	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	178	1	no	—
287	Q₈	—	178	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	4	2	no	—
291	D₄	—	356	2	no	—
292	D₄	—	178	1	no	—
293	D₄	—	178	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	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	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
307	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
308	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
309	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
310	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
312	C₈	—	89	2	no	—
313	C₈	—	89	2	no	—
314	C₈	—	178	1	no	—
315	C₈	—	178	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	356	1	no	—
322	C₂²	—	356	1	no	—
323	C₂²	—	356	1	no	—
324	C₂²	#15668032	1	2	yes	—
325	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	356	1	no	—
334	C₄	—	356	1	no	—
335	C₄	—	356	1	no	—
336	C₄	#15668032	1	2	yes	—
337	C₄	#41×D₈₉	1	1	yes	characteristic
338	C₄	#41×D₈₉	1	1	yes	characteristic
339	C₂	—	2	2	no	—
340	C₂	—	8	1	no	—
341	C₂	—	712	1	no	—
342	C₂	#19152144	1	1	yes	characteristic
343	C₂	C₂×#15668032	1	1	yes	characteristic
344	C₂	C₂×#15668032	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#529×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#19151832	C₂
hypercenter	#529	D₈₉
hypocenter	C₈₉	#3168
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. #19152132	C₂	semidirect
2. #19151997	C₂	semidirect
3. #19151980	C₂	semidirect
4. #19151955	C₂	semidirect
5. #19151952	C₂	semidirect
6. #19151837	C₂	semidirect
8. #19151829	C₂	semidirect
9. #19151825	C₂	semidirect
10. C₄×#13083564	C₂	semidirect
11. C₄×#13083561	C₂	semidirect
12. C₄×#13083562	C₂	semidirect
13. C₄×#13083559	C₂	semidirect
14. C₂×C₄×#10623494	C₂	semidirect
15. #529×C₈₉	C₂	semidirect
173. C₈₉	#3168	semidirect
175. #529	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152132	2
#19151997	2
#19151980	2
#19151955	2
#19151952	2
#19151837	2
#19151832	2
#19151829	2
#19151825	2
C₄×#13083564	2
C₄×#13083561	2
C₄×#13083562	2
C₄×#13083559	2
C₂×C₄×#10623494	2
#529×C₈₉	2
#3168	89