Subgroups — #26033861

Order 11392.

1242 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152308	C₂	1	2	yes	maximal, complemented
2	#19152298	C₂	1	4	yes	maximal, complemented
3	#19152276	C₂	1	4	yes	maximal, complemented
4	#19152241	C₂	1	4	yes	maximal, complemented
5	#19152233	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152146	C₂	1	1	yes	maximal, characteristic, complemented
7	#19152127	C₂	1	4	yes	maximal, complemented
8	#19152106	C₂	1	4	yes	maximal, complemented
9	C₄×#13083575	C₂	1	2	yes	maximal, complemented
10	C₄×D₄×D₈₉	C₂	1	4	yes	maximal, complemented
11	#559×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13083575	C₂²	1	2	yes	—
13	C₂×#13083573	C₂²	1	2	yes	complemented
14	C₂×D₄×D₈₉	C₂²	1	4	yes	—
15	C₂²×C₄×D₈₉	C₂²	1	4	yes	complemented
16	#15668012	C₂²	1	1	yes	characteristic, complemented
17	#15668005	C₂²	1	8	yes	complemented
18	#15668004	C₂²	1	4	yes	complemented
19	#15667999	C₂²	1	2	yes	complemented
20	#15667995	C₂²	1	4	yes	complemented
21	#15667994	C₂²	1	8	yes	complemented
22	#15667990	C₂²	1	8	yes	complemented
23	#15667988	C₂²	1	4	yes	complemented
24	#15667985	C₂²	1	8	yes	complemented
25	#15667983	C₂²	1	4	yes	—
26	#15667983	C₂²	1	4	yes	complemented
27	#15667982	C₂²	1	4	yes	complemented
28	#15667980	C₂²	1	4	yes	—
29	Q₈×Dic₈₉	C₂²	1	2	yes	complemented
30	C₄×D₃₅₆	C₂²	1	2	yes	—
31	C₄×D₃₅₆	C₂²	1	4	yes	—
32	C₄×D₃₅₆	C₂²	1	4	yes	complemented
33	D₄×Dic₈₉	C₂²	1	4	yes	complemented
34	#15667974	C₂²	1	2	yes	—
35	#15667974	C₂²	1	4	yes	complemented
36	C₄×C₂²⋊D₈₉	C₂²	1	8	yes	complemented
37	#15667972	C₂²	1	8	yes	complemented
38	#15667970	C₂²	1	4	yes	complemented
39	#15667970	C₂²	1	2	yes	complemented
40	#15667968	C₂²	1	4	yes	complemented
41	C₄⋊C₄×D₈₉	C₂²	1	4	yes	complemented
42	C₂²⋊C₄×D₈₉	C₂²	1	8	yes	complemented
43	C₄²×D₈₉	C₂²	1	1	yes	characteristic, complemented
44	C₄²×D₈₉	C₂²	1	4	yes	complemented
45	#127×C₈₉	C₂²	1	1	yes	characteristic
46	#124×C₈₉	C₂²	1	4	yes	complemented
47	#123×C₈₉	C₂²	1	4	yes	complemented
48	C₄×Q₈×C₈₉	C₂²	1	2	yes	complemented
49	C₄×D₄×C₈₉	C₂²	1	4	yes	complemented
50	C₂³×D₈₉	C₂³	1	4	yes	—
51	#13083575	—	2	8	no	—
52	#13083573	—	4	4	no	—
53	C₂×Dic₁₇₈	C₂³	1	2	yes	—
54	C₂×D₃₅₆	C₂³	1	4	yes	—
55	C₂×D₃₅₆	C₂³	1	2	yes	—
56	C₂×C₂²⋊D₈₉	C₂³	1	8	yes	—
57	C₂×C₂²⋊D₈₉	C₂³	1	4	yes	—
58	D₄×D₈₉	—	2	16	no	—
59	C₂²×Dic₈₉	C₂³	1	4	yes	—
60	C₂×C₄×D₈₉	—	4	8	no	—
61	C₂×C₄×D₈₉	C₂³	1	2	yes	—
62	C₂×C₄×D₈₉	C₂³	1	4	yes	—
63	C₂×C₄×D₈₉	C₂³	1	4	yes	—
64	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₈₉	C₂³	1	4	yes	—
66	C₂×C₄×D₈₉	C₂³	1	8	yes	—
67	C₄⋊Dic₈₉	C₂³	1	2	yes	—
68	C₄⋊Dic₈₉	C₂³	1	4	yes	—
69	C₄⋊Dic₈₉	C₂³	1	4	yes	—
70	#13083553	C₂³	1	8	yes	—
71	#13083553	C₂³	1	8	yes	—
72	C₂²⋊Dic₈₉	C₂³	1	4	yes	—
73	C₂²⋊Dic₈₉	C₂³	1	8	yes	—
74	#13083551	C₂³	1	4	yes	—
75	#13083551	C₂³	1	4	yes	—
76	#13083551	C₂³	1	4	yes	—
77	#13083551	C₂³	1	8	yes	—
78	C₄×Dic₈₉	C₂³	1	4	yes	—
79	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
80	C₄×Dic₈₉	C₂³	1	4	yes	—
81	C₄×Dic₈₉	C₂³	1	2	yes	—
82	C₂×Q₈×C₈₉	C₂³	1	2	yes	—
83	C₂×D₄×C₈₉	C₂³	1	4	yes	—
84	C₂×D₄×C₈₉	C₂³	1	2	yes	—
85	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
86	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
87	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
88	C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
89	C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
90	C₄²×C₈₉	C₂³	1	1	yes	characteristic
91	C₄²×C₈₉	C₂³	1	4	yes	—
92	C₂³×C₈₉	C₂⁴	1	4	yes	—
93	C₂²×D₈₉	—	2	16	no	—
94	C₂²×D₈₉	—	4	4	no	—
95	C₂²×D₈₉	C₂⁴	1	1	yes	characteristic
96	C₂²×D₈₉	C₂⁴	1	4	yes	—
97	Q₈×C₈₉	—	2	4	no	—
98	D₄×C₈₉	—	2	8	no	—
99	D₄×C₈₉	—	4	2	no	—
100	C₂×C₄×C₈₉	—	4	4	no	—
101	C₂×C₄×C₈₉	C₂⁴	1	4	yes	—
102	C₂×C₄×C₈₉	C₂⁴	1	4	yes	—
103	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
105	C₂²⋊D₈₉	—	2	16	no	—
106	C₂²⋊D₈₉	—	4	4	no	—
107	C₂×Dic₈₉	—	2	8	no	—
108	C₂×Dic₈₉	C₂⁴	1	2	yes	—
109	C₂×Dic₈₉	C₂⁴	1	4	yes	—
110	C₂×Dic₈₉	C₂⁴	1	4	yes	—
111	C₂×Dic₈₉	C₂⁴	1	1	yes	characteristic
112	D₃₅₆	—	2	8	no	—
113	D₃₅₆	#41	1	8	yes	complemented
114	C₄×D₈₉	—	2	8	no	—
115	C₄×D₈₉	—	2	8	no	—
116	C₄×D₈₉	—	2	4	no	—
117	C₄×D₈₉	—	2	8	no	—
118	C₄×D₈₉	—	2	2	no	—
119	C₄×D₈₉	—	4	8	no	—
120	Dic₁₇₈	—	4	2	no	—
121	C₂²×C₈₉	—	2	8	no	—
122	C₂²×C₈₉	—	2	4	no	—
123	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
124	C₂×D₈₉	—	2	4	no	—
125	C₂×D₈₉	—	2	4	no	—
126	C₂×D₈₉	—	2	2	no	—
127	C₂×D₈₉	—	2	1	no	—
128	C₂×D₈₉	—	4	16	no	—
129	C₂×D₈₉	C₂×#41	1	8	yes	—
130	C₄×C₈₉	—	2	4	no	—
131	C₄×C₈₉	—	2	1	no	—
132	C₄×C₈₉	—	2	4	no	—
133	C₄×C₈₉	C₂×#41	1	4	yes	—
134	Dic₈₉	—	2	2	no	—
135	Dic₈₉	—	2	4	no	—
136	Dic₈₉	—	2	4	no	—
137	Dic₈₉	H(5,𝔽₂)	1	2	yes	—
138	C₂×C₈₉	—	4	4	no	—
139	C₂×C₈₉	C₂²×#41	1	2	yes	—
140	C₂×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
141	D₈₉	—	2	8	no	—
142	D₈₉	—	4	1	no	—
143	#3481	—	89	1	no	maximal, Sylow
144	C₈₉	#3481	1	1	yes	characteristic, complemented, Sylow
145	#559	D₈₉	1	1	yes	characteristic, complemented
146	#554	—	89	2	no	—
147	#553	—	89	4	no	—
148	#551	—	89	4	no	—
149	#546	—	89	4	no	—
150	D₄²	—	89	1	no	—
151	#532	—	89	1	no	—
152	#529	—	89	4	no	—
153	#525	—	89	4	no	—
154	C₄×#41	—	89	2	no	—
155	C₂×C₄×D₄	—	89	4	no	—
156	C₂×#41	—	89	2	no	—
157	C₂×#41	—	89	2	no	—
158	C₂²×D₄	—	89	4	no	—
159	C₂³×C₄	—	89	4	no	—
160	C₄⋊Q₈	—	89	1	no	—
161	#127	C₂×D₈₉	1	1	yes	characteristic
162	#126	—	89	8	no	—
163	#125	—	89	4	no	—
164	#124	—	89	2	no	—
165	#124	C₂×D₈₉	1	4	yes	complemented
166	#123	—	89	4	no	—
167	#123	—	89	8	no	—
168	#123	C₂×D₈₉	1	4	yes	complemented
169	C₂²⋊Q₈	—	89	4	no	—
170	C₂²⋊Q₈	—	89	8	no	—
171	#121	—	89	4	no	—
172	#121	—	89	4	no	—
173	#121	—	89	8	no	—
174	#120	—	89	4	no	—
175	#120	—	89	4	no	—
176	C₄×Q₈	—	89	2	no	—
177	C₄×Q₈	C₂×D₈₉	1	2	yes	complemented
178	C₄×D₄	—	89	4	no	—
179	C₄×D₄	—	89	8	no	—
180	C₄×D₄	—	89	2	no	—
181	C₄×D₄	—	89	4	no	—
182	C₄×D₄	—	89	8	no	—
183	C₄×D₄	—	89	4	no	—
184	C₄×D₄	—	89	2	no	—
185	C₄×D₄	—	89	4	no	—
186	C₄×D₄	C₂×D₈₉	1	4	yes	complemented
187	#117	—	89	4	no	—
188	#117	—	89	2	no	—
189	#117	—	89	4	no	—
190	C₂×C₄⋊C₄	—	89	4	no	—
191	C₂×C₂²⋊C₄	—	89	8	no	—
192	C₂×C₄²	—	89	4	no	—
193	C₂×C₄²	—	89	1	no	—
194	C₂⁴	—	89	4	no	—
195	#41	—	356	4	no	—
196	#41	—	178	8	no	—
197	C₂×Q₈	—	89	2	no	—
198	C₂×Q₈	C₂²×D₈₉	1	2	yes	—
199	C₂×D₄	—	89	4	no	—
200	C₂×D₄	—	89	8	no	—
201	C₂×D₄	—	89	2	no	—
202	C₂×D₄	—	89	4	no	—
203	C₂×D₄	—	178	16	no	—
204	C₂×D₄	C₂²×D₈₉	1	4	yes	—
205	C₂×D₄	C₂²×D₈₉	1	2	yes	—
206	C₂²×C₄	—	89	4	no	—
207	C₂²×C₄	—	89	8	no	—
208	C₂²×C₄	—	89	4	no	—
209	C₂²×C₄	—	89	4	no	—
210	C₂²×C₄	—	89	2	no	—
211	C₂²×C₄	—	89	4	no	—
212	C₂²×C₄	—	89	1	no	—
213	C₂²×C₄	—	356	8	no	—
214	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
215	C₄⋊C₄	—	89	4	no	—
216	C₄⋊C₄	—	89	8	no	—
217	C₄⋊C₄	—	89	2	no	—
218	C₄⋊C₄	—	89	4	no	—
219	C₄⋊C₄	—	89	8	no	—
220	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
221	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
222	C₂²⋊C₄	—	89	8	no	—
223	C₂²⋊C₄	—	89	4	no	—
224	C₂²⋊C₄	—	89	4	no	—
225	C₂²⋊C₄	—	89	8	no	—
226	C₂²⋊C₄	—	89	4	no	—
227	C₂²⋊C₄	—	89	4	no	—
228	C₂²⋊C₄	C₂²×D₈₉	1	8	yes	—
229	C₂²⋊C₄	C₂²×D₈₉	1	4	yes	—
230	C₄²	—	89	2	no	—
231	C₄²	—	89	1	no	—
232	C₄²	—	89	4	no	—
233	C₄²	—	89	4	no	—
234	C₄²	C₂²×D₈₉	1	4	yes	—
235	C₄²	C₂²×D₈₉	1	1	yes	characteristic
236	C₂³	—	89	4	no	—
237	C₂³	—	89	1	no	—
238	C₂³	—	356	4	no	—
239	C₂³	—	178	16	no	—
240	C₂³	C₂³×D₈₉	1	4	yes	—
241	Q₈	—	2	4	no	—
242	Q₈	—	356	2	no	—
243	D₄	—	2	8	no	—
244	D₄	—	4	2	no	—
245	D₄	—	89	8	no	—
246	D₄	—	356	4	no	—
247	D₄	—	178	8	no	—
248	D₄	—	178	16	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	89	1	no	—
251	C₂×C₄	—	89	2	no	—
252	C₂×C₄	—	89	4	no	—
253	C₂×C₄	—	89	4	no	—
254	C₂×C₄	—	356	8	no	—
255	C₂×C₄	—	178	8	no	—
256	C₂×C₄	—	178	2	no	—
257	C₂×C₄	—	178	8	no	—
258	C₂×C₄	—	178	8	no	—
259	C₂×C₄	—	178	8	no	—
260	C₂×C₄	—	178	4	no	—
261	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
262	C₂×C₄	C₂³×D₈₉	1	4	yes	—
263	C₂×C₄	C₂³×D₈₉	1	4	yes	—
264	C₂×C₄	C₂³×D₈₉	1	2	yes	—
265	C₂²	—	2	8	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	89	8	no	—
268	C₂²	—	356	16	no	—
269	C₂²	—	178	2	no	—
270	C₂²	—	178	1	no	—
271	C₂²	—	178	4	no	—
272	C₂²	—	178	4	no	—
273	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
274	C₄	—	2	4	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	1	no	—
277	C₄	—	89	2	no	—
278	C₄	—	178	2	no	—
279	C₄	—	178	4	no	—
280	C₄	—	178	4	no	—
281	C₄	#41×D₈₉	1	4	yes	—
282	C₂	—	4	4	no	—
283	C₂	—	356	1	no	—
284	C₂	—	178	8	no	—
285	C₂	C₂×#41×D₈₉	1	2	yes	—
286	C₂	H(5,𝔽₂)×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₉
commutator, socle, cosocle	C₂²×C₈₉	C₂⁵
Fitting	#559×C₈₉	C₂
hypercenter	#559	D₈₉
hypocenter	C₈₉	#3481
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #559

Complements

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

N	H ≅ G/N	type
1. #19152308	C₂	semidirect
2. #19152298	C₂	semidirect
3. #19152276	C₂	semidirect
4. #19152241	C₂	semidirect
5. #19152233	C₂	semidirect
6. #19152146	C₂	semidirect
7. #19152127	C₂	semidirect
8. #19152106	C₂	semidirect
9. C₄×#13083575	C₂	semidirect
10. C₄×D₄×D₈₉	C₂	semidirect
11. #559×C₈₉	C₂	semidirect
13. C₂×#13083573	C₂²	semidirect
15. C₂²×C₄×D₈₉	C₂²	semidirect
16. #15668012	C₂²	semidirect
17. #15668005	C₂²	semidirect
18. #15668004	C₂²	semidirect
19. #15667999	C₂²	semidirect
20. #15667995	C₂²	semidirect
21. #15667994	C₂²	semidirect
22. #15667990	C₂²	semidirect
23. #15667988	C₂²	semidirect
24. #15667985	C₂²	semidirect
26. #15667983	C₂²	semidirect
27. #15667982	C₂²	semidirect
29. Q₈×Dic₈₉	C₂²	semidirect
32. C₄×D₃₅₆	C₂²	semidirect
33. D₄×Dic₈₉	C₂²	semidirect
35. #15667974	C₂²	semidirect
36. C₄×C₂²⋊D₈₉	C₂²	semidirect
37. #15667972	C₂²	semidirect
38. #15667970	C₂²	semidirect
39. #15667970	C₂²	semidirect
40. #15667968	C₂²	semidirect
41. C₄⋊C₄×D₈₉	C₂²	semidirect
42. C₂²⋊C₄×D₈₉	C₂²	semidirect
43. C₄²×D₈₉	C₂²	semidirect
44. C₄²×D₈₉	C₂²	semidirect
46. #124×C₈₉	C₂²	semidirect
47. #123×C₈₉	C₂²	semidirect
48. C₄×Q₈×C₈₉	C₂²	semidirect
49. C₄×D₄×C₈₉	C₂²	semidirect
113. D₃₅₆	#41	semidirect
144. C₈₉	#3481	semidirect
145. #559	D₈₉	semidirect
165. #124	C₂×D₈₉	semidirect
168. #123	C₂×D₈₉	semidirect
177. C₄×Q₈	C₂×D₈₉	semidirect
186. C₄×D₄	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152308	2
#19152298	2
#19152276	2
#19152241	2
#19152233	2
#19152146	2
#19152127	2
#19152106	2
C₄×#13083575	2
C₄×D₄×D₈₉	2
#559×C₈₉	2
#3481	89