Subgroups — #26033020

Order 11392.

344 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152333	C₂	1	1	yes	maximal, characteristic
2	#19151964	C₂	1	1	yes	maximal, characteristic
3	#19151960	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151955	C₂	1	1	yes	maximal, characteristic, complemented
5	#19151902	C₂	1	1	yes	maximal, characteristic, complemented
6	#19151895	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151885	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151820	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151820	C₂	1	1	yes	maximal, characteristic, complemented
10	#19151798	C₂	1	1	yes	maximal, characteristic
11	#19151729	C₂	1	1	yes	maximal, characteristic, complemented
12	#19151703	C₂	1	1	yes	maximal, characteristic, complemented
13	#19151696	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13083561	C₂	1	1	yes	maximal, characteristic, complemented
15	#557×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13083564	C₂²	1	1	yes	characteristic
17	C₂×#13083561	C₂²	1	1	yes	characteristic
18	C₂×#13083561	C₂²	1	1	yes	characteristic
19	#15668009	C₂²	1	1	yes	characteristic
20	#15668001	—	2	1	no	—
21	#15668000	C₂²	1	1	yes	characteristic
22	#15668000	C₂²	1	1	yes	characteristic
23	#15667997	—	2	1	no	—
24	#15667997	—	2	1	no	—
25	C₄×Dic₁₇₈	C₂²	1	1	yes	characteristic
26	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
27	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
28	#15667974	—	2	1	no	—
29	#15667945	C₂²	1	1	yes	characteristic
30	#15667943	C₂²	1	1	yes	characteristic
31	#15667943	C₂²	1	1	yes	characteristic
32	C₄⋊#10623494	C₂²	1	1	yes	characteristic
33	C₄⋊#10623494	C₂²	1	1	yes	characteristic
34	C₄⋊#10623494	C₂²	1	1	yes	characteristic
35	#15667936	C₂²	1	1	yes	characteristic
36	#15667935	C₂²	1	1	yes	characteristic
37	#15667935	C₂²	1	1	yes	characteristic
38	#15667935	C₂²	1	1	yes	characteristic
39	#15667932	C₂²	1	1	yes	characteristic
40	#15667932	C₂²	1	1	yes	characteristic
41	#15667932	C₂²	1	1	yes	characteristic
42	#15667932	C₂²	1	1	yes	characteristic
43	#15667932	C₂²	1	1	yes	characteristic
44	#15667932	C₂²	1	1	yes	characteristic
45	#15667923	C₂²	1	1	yes	characteristic
46	#15667923	C₂²	1	1	yes	characteristic
47	C₄×#10623494	C₂²	1	1	yes	characteristic
48	C₄⋊Q₈×C₈₉	—	2	1	no	—
49	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
50	#125×C₈₉	—	2	1	no	—
51	#125×C₈₉	—	2	1	no	—
52	#125×C₈₉	C₂²	1	1	yes	characteristic
53	#125×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₈₉	—	2	1	no	—
55	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
56	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
57	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
58	C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
60	C₂×D₃₅₆	—	2	1	no	—
61	C₂×D₃₅₆	—	2	1	no	—
62	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
63	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₈₉	—	2	1	no	—
65	C₂×C₄×D₈₉	—	2	1	no	—
66	#13083564	—	2	2	no	—
67	#13083561	—	2	2	no	—
68	#13083561	—	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	#13083553	—	2	1	no	—
75	#13083551	—	2	1	no	—
76	#13083551	—	2	1	no	—
77	#13083551	—	2	1	no	—
78	#13083551	—	2	1	no	—
79	#13083551	—	2	1	no	—
80	#13083551	—	2	1	no	—
81	C₄×Dic₈₉	—	2	1	no	—
82	C₂×Q₈×C₈₉	—	2	1	no	—
83	C₂×Q₈×C₈₉	D₄	1	1	yes	characteristic
84	C₂×Q₈×C₈₉	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₈₉	—	2	1	no	—
86	C₄⋊C₄×C₈₉	—	2	1	no	—
87	C₄⋊C₄×C₈₉	—	2	1	no	—
88	C₄⋊C₄×C₈₉	—	2	1	no	—
89	C₄⋊C₄×C₈₉	—	2	1	no	—
90	C₄⋊C₄×C₈₉	—	2	1	no	—
91	C₄⋊C₄×C₈₉	—	2	1	no	—
92	C₄⋊C₄×C₈₉	—	2	1	no	—
93	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
94	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
95	C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
96	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₄²×C₈₉	—	2	1	no	—
100	C₄²×C₈₉	—	2	1	no	—
101	C₄²×C₈₉	—	2	1	no	—
102	C₄²×C₈₉	C₂³	1	1	yes	characteristic
103	C₄²×C₈₉	C₂³	1	1	yes	characteristic
104	C₄²×C₈₉	C₂³	1	1	yes	characteristic
105	C₂²×D₈₉	—	2	1	no	—
106	C₂²×D₈₉	—	2	1	no	—
107	Q₈×C₈₉	—	2	1	no	—
108	Q₈×C₈₉	—	4	1	no	—
109	Q₈×C₈₉	—	4	2	no	—
110	Q₈×C₈₉	#41	1	2	yes	—
111	C₂×C₄×C₈₉	—	2	1	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₈₉	—	2	1	no	—
123	C₂×Dic₈₉	—	2	1	no	—
124	D₃₅₆	—	2	1	no	—
125	D₃₅₆	—	2	1	no	—
126	D₃₅₆	—	2	1	no	—
127	D₃₅₆	—	2	1	no	—
128	D₃₅₆	—	4	2	no	—
129	D₃₅₆	—	8	1	no	—
130	C₄×D₈₉	—	8	1	no	—
131	C₄×D₈₉	—	8	1	no	—
132	Dic₁₇₈	—	2	1	no	—
133	Dic₁₇₈	—	2	1	no	—
134	#10623494	—	2	1	no	—
135	#10623494	—	2	1	no	—
136	#10623494	—	2	1	no	—
137	#10623494	—	2	1	no	—
138	C₂²×C₈₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₈₉	—	4	1	no	—
140	C₂×D₈₉	—	4	1	no	—
141	C₂×D₈₉	—	4	1	no	—
142	C₂×D₈₉	—	4	1	no	—
143	C₂×D₈₉	—	4	1	no	—
144	C₂×D₈₉	—	4	1	no	—
145	C₄×C₈₉	—	2	2	no	—
146	C₄×C₈₉	—	2	1	no	—
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	2	1	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	—	2	1	no	—
152	C₄×C₈₉	—	4	1	no	—
153	C₄×C₈₉	—	4	1	no	—
154	C₄×C₈₉	—	4	1	no	—
155	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
156	C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
157	Dic₈₉	—	4	1	no	—
158	Dic₈₉	—	4	1	no	—
159	C₂×C₈₉	#548	1	1	yes	characteristic
160	C₂×C₈₉	#575	1	1	yes	characteristic
161	C₂×C₈₉	#577	1	1	yes	characteristic
162	D₈₉	—	8	1	no	—
163	D₈₉	—	8	1	no	—
164	#3371	—	89	1	no	maximal, Sylow
165	C₈₉	#3371	1	1	yes	characteristic, complemented, Sylow
166	#559	—	89	1	no	—
167	#557	D₈₉	1	1	yes	characteristic, complemented
168	#490	—	89	1	no	—
169	#489	—	89	1	no	—
170	#488	—	89	1	no	—
171	#479	—	89	1	no	—
172	#478	—	89	1	no	—
173	#476	—	89	1	no	—
174	#464	—	89	1	no	—
175	#464	—	89	1	no	—
176	#460	—	89	1	no	—
177	#446	—	89	1	no	—
178	#441	—	89	1	no	—
179	#440	—	89	1	no	—
180	C₄×SD₁₆	—	89	1	no	—
181	C₂×Q₁₆	—	89	1	no	—
182	C₂×SD₁₆	—	89	1	no	—
183	C₂×SD₁₆	—	89	1	no	—
184	C₄⋊Q₈	—	2	1	no	—
185	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
186	#127	—	89	1	no	—
187	#125	—	2	1	no	—
188	#125	—	2	1	no	—
189	#125	C₂×D₈₉	1	1	yes	characteristic
190	#125	C₂×D₈₉	1	1	yes	characteristic
191	#124	—	89	1	no	—
192	#124	—	89	1	no	—
193	#124	—	178	1	no	—
194	#123	—	178	1	no	—
195	#123	—	178	1	no	—
196	C₄×Q₈	—	2	1	no	—
197	C₄×Q₈	—	89	1	no	—
198	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
199	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
200	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
201	C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
202	C₄×D₄	—	89	1	no	—
203	C₄×D₄	—	89	1	no	—
204	C₄×D₄	—	178	1	no	—
205	#107	—	89	1	no	—
206	#106	—	89	1	no	—
207	#106	—	89	1	no	—
208	C₄⋊C₈	—	89	1	no	—
209	C₄⋊C₈	—	89	1	no	—
210	C₄⋊C₈	—	89	1	no	—
211	#103	—	89	1	no	—
212	#103	—	89	1	no	—
213	#103	—	89	1	no	—
214	#103	—	89	1	no	—
215	#102	—	89	1	no	—
216	#102	—	89	1	no	—
217	#102	—	89	1	no	—
218	#102	—	89	1	no	—
219	#102	—	89	1	no	—
220	#102	—	89	1	no	—
221	#97	—	89	1	no	—
222	#97	—	89	1	no	—
223	C₄×C₈	—	89	1	no	—
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	89	1	no	—
226	C₂×Q₈	C₂²⋊D₈₉	1	1	yes	characteristic
227	C₂×Q₈	C₂²×D₈₉	1	1	yes	characteristic
228	C₂×D₄	—	89	1	no	—
229	C₂×D₄	—	89	1	no	—
230	C₂×D₄	—	178	1	no	—
231	C₂×D₄	—	178	1	no	—
232	C₂²×C₄	—	178	1	no	—
233	C₂²×C₄	—	178	1	no	—
234	Q₁₆	—	178	2	no	—
235	SD₁₆	—	178	2	no	—
236	SD₁₆	—	178	2	no	—
237	C₂×C₈	—	89	1	no	—
238	C₂×C₈	—	89	1	no	—
239	C₂×C₈	—	89	1	no	—
240	C₂×C₈	—	89	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	—	2	1	no	—
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	89	1	no	—
250	C₄⋊C₄	—	178	1	no	—
251	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
252	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
253	C₄⋊C₄	C₂²⋊D₈₉	1	1	yes	characteristic
254	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₈₉	1	1	yes	characteristic
257	C₂²⋊C₄	—	178	1	no	—
258	C₂²⋊C₄	—	178	1	no	—
259	C₂²⋊C₄	—	178	1	no	—
260	C₂²⋊C₄	—	178	1	no	—
261	C₂²⋊C₄	—	178	1	no	—
262	C₂²⋊C₄	—	178	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	178	1	no	—
267	C₄²	C₂²×D₈₉	1	1	yes	characteristic
268	C₄²	C₂²×D₈₉	1	1	yes	characteristic
269	C₄²	C₂²×D₈₉	1	1	yes	characteristic
270	C₂³	—	178	1	no	—
271	C₂³	—	178	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	4	1	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	178	1	no	—
276	Q₈	—	178	1	no	—
277	Q₈	#13083575	1	2	yes	—
278	D₄	—	356	2	no	—
279	D₄	—	178	1	no	—
280	D₄	—	178	1	no	—
281	D₄	—	178	1	no	—
282	D₄	—	178	1	no	—
283	D₄	—	712	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	178	1	no	—
289	C₂×C₄	—	178	1	no	—
290	C₂×C₄	—	712	1	no	—
291	C₂×C₄	—	712	1	no	—
292	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
293	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
294	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
295	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
296	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
299	C₈	—	178	1	no	—
300	C₈	—	178	1	no	—
301	C₈	—	178	1	no	—
302	C₈	—	178	1	no	—
303	C₂²	—	356	1	no	—
304	C₂²	—	356	1	no	—
305	C₂²	—	356	1	no	—
306	C₂²	—	356	1	no	—
307	C₂²	—	356	1	no	—
308	C₂²	—	356	1	no	—
309	C₂²	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	356	1	no	—
321	C₄	—	356	1	no	—
322	C₄	C₂×#13083575	1	1	yes	characteristic
323	C₄	#15668060	1	1	yes	characteristic
324	C₂	—	712	1	no	—
325	C₂	—	712	1	no	—
326	C₂	#19152259	1	1	yes	characteristic
327	C₂	#19152396	1	1	yes	characteristic
328	C₂	#19152406	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₉
commutator, cosocle	C₂×C₄×C₈₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₉
Fitting	#557×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₄
prime radical	#15668009	C₂²
hypercenter	#557	D₈₉
hypocenter	C₈₉	#3371
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

Complements

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

N	H ≅ G/N	type
3. #19151960	C₂	semidirect
4. #19151955	C₂	semidirect
5. #19151902	C₂	semidirect
6. #19151895	C₂	semidirect
7. #19151885	C₂	semidirect
8. #19151820	C₂	semidirect
9. #19151820	C₂	semidirect
11. #19151729	C₂	semidirect
12. #19151703	C₂	semidirect
13. #19151696	C₂	semidirect
14. C₄×#13083561	C₂	semidirect
15. #557×C₈₉	C₂	semidirect
165. C₈₉	#3371	semidirect
167. #557	D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152333	2
#19151964	2
#19151960	2
#19151955	2
#19151902	2
#19151895	2
#19151885	2
#19151820	2
#19151798	2
#19151729	2
#19151703	2
#19151696	2
C₄×#13083561	2
#557×C₈₉	2
#3371	89