Subgroups — #26034025

Order 11392.

1748 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668058	C₂	1	2	yes	maximal, complemented
2	#19152269	C₂	1	4	yes	maximal, complemented
3	D₄×D₃₅₆	C₂	1	4	yes	maximal, complemented
4	#19152232	C₂	1	8	yes	maximal, complemented
5	#19152166	C₂	1	4	yes	maximal, complemented
6	#19152162	C₂	1	2	yes	maximal, complemented
7	#19152149	C₂	1	2	yes	maximal, complemented
8	C₂×#15668009	C₂	1	1	yes	maximal, characteristic, complemented
9	#127×D₈₉	C₂	1	2	yes	maximal, complemented
10	C₄×#13083574	C₂	1	1	yes	maximal, characteristic, complemented
11	#532×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#15668058	—	2	16	no	—
13	C₂×#13083575	C₂²	1	4	yes	complemented
14	C₂×#13083574	C₂²	1	1	yes	characteristic
15	C₂×#13083574	C₂²	1	4	yes	complemented
16	C₂²×D₃₅₆	C₂²	1	2	yes	complemented
17	C₂²×D₃₅₆	C₂²	1	4	yes	complemented
18	C₂²×D₃₅₆	C₂²	1	4	yes	complemented
19	C₂×D₄×D₈₉	C₂²	1	4	yes	complemented
20	C₂×D₄×D₈₉	C₂²	1	8	yes	complemented
21	C₂×D₄×D₈₉	C₂²	1	8	yes	complemented
22	#15668009	C₂²	1	2	yes	complemented
23	#15668009	C₂²	1	2	yes	complemented
24	#15668009	C₂²	1	4	yes	complemented
25	#15668008	C₂²	1	4	yes	complemented
26	#15668008	C₂²	1	8	yes	complemented
27	#15668001	C₂²	1	4	yes	complemented
28	#15667987	C₂²	1	8	yes	complemented
29	#15667986	C₂²	1	8	yes	complemented
30	#15667985	C₂²	1	16	yes	complemented
31	#15667982	C₂²	1	16	yes	complemented
32	#15667980	C₂²	1	8	yes	complemented
33	C₄×Dic₁₇₈	C₂²	1	2	yes	complemented
34	C₄×D₃₅₆	C₂²	1	2	yes	complemented
35	#15667974	C₂²	1	8	yes	complemented
36	C₄×C₂²⋊D₈₉	C₂²	1	4	yes	complemented
37	#15667971	C₂²	1	1	yes	characteristic, complemented
38	#15667969	C₂²	1	2	yes	complemented
39	C₄²×D₈₉	C₂²	1	2	yes	complemented
40	C₂×#41×C₈₉	C₂²	1	2	yes	complemented
41	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
42	#127×C₈₉	C₂²	1	1	yes	characteristic, complemented
43	#124×C₈₉	C₂²	1	2	yes	complemented
44	#121×C₈₉	C₂²	1	4	yes	complemented
45	C₄×D₄×C₈₉	C₂²	1	4	yes	complemented
46	C₂×C₄²×C₈₉	C₂²	1	1	yes	characteristic, complemented
47	C₂³×D₈₉	C₂³	1	4	yes	—
48	C₂³×D₈₉	C₂³	1	8	yes	—
49	#13083575	—	2	16	no	—
50	#13083574	—	2	16	no	—
51	#13083574	D₄	1	8	yes	complemented
52	C₂×Dic₁₇₈	C₂³	1	1	yes	characteristic
53	C₂×D₃₅₆	—	2	8	no	—
54	C₂×D₃₅₆	—	2	16	no	—
55	C₂×D₃₅₆	—	4	8	no	—
56	C₂×D₃₅₆	C₂³	1	8	yes	—
57	C₂×D₃₅₆	C₂³	1	8	yes	—
58	C₂×D₃₅₆	C₂³	1	4	yes	—
59	C₂×D₃₅₆	C₂³	1	4	yes	—
60	C₂×D₃₅₆	C₂³	1	8	yes	—
61	C₂×D₃₅₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₈₉	C₂³	1	8	yes	—
63	C₂×C₂²⋊D₈₉	C₂³	1	16	yes	—
64	C₂×C₂²⋊D₈₉	C₂³	1	2	yes	—
65	D₄×D₈₉	—	2	32	no	—
66	D₄×D₈₉	—	2	16	no	—
67	D₄×D₈₉	—	4	16	no	—
68	C₂×C₄×D₈₉	C₂³	1	4	yes	—
69	C₂×C₄×D₈₉	C₂³	1	8	yes	—
70	C₂×C₄×D₈₉	C₂³	1	2	yes	—
71	C₄⋊Dic₈₉	C₂³	1	2	yes	—
72	#13083553	C₂³	1	4	yes	—
73	C₂²⋊Dic₈₉	C₂³	1	2	yes	—
74	#13083551	C₂³	1	4	yes	—
75	#13083551	C₂³	1	16	yes	—
76	C₄×Dic₈₉	C₂³	1	2	yes	—
77	C₄×Dic₈₉	C₂³	1	4	yes	—
78	#41×C₈₉	—	2	8	no	—
79	C₂×Q₈×C₈₉	C₂³	1	2	yes	—
80	C₂×D₄×C₈₉	C₂³	1	4	yes	—
81	C₂×D₄×C₈₉	C₂³	1	4	yes	—
82	C₂×D₄×C₈₉	C₂³	1	2	yes	—
83	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
84	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
85	C₂²×C₄×C₈₉	C₂³	1	1	yes	characteristic
86	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
87	C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
88	C₄²×C₈₉	C₂³	1	1	yes	characteristic
89	C₄²×C₈₉	C₂³	1	2	yes	—
90	C₄²×C₈₉	C₂³	1	1	yes	characteristic
91	C₂³×C₈₉	C₂⁴	1	4	yes	—
92	C₂³×C₈₉	C₂⁴	1	1	yes	characteristic
93	C₂²×D₈₉	—	2	8	no	—
94	C₂²×D₈₉	—	2	16	no	—
95	C₂²×D₈₉	—	4	16	no	—
96	C₂²×D₈₉	—	4	8	no	—
97	C₂²×D₈₉	C₂⁴	1	8	yes	—
98	C₂²×D₈₉	C₂⁴	1	2	yes	—
99	Q₈×C₈₉	—	2	4	no	—
100	D₄×C₈₉	—	2	8	no	—
101	D₄×C₈₉	—	2	4	no	—
102	D₄×C₈₉	—	2	8	no	—
103	C₂×C₄×C₈₉	—	2	8	no	—
104	C₂×C₄×C₈₉	—	2	4	no	—
105	C₂×C₄×C₈₉	C₂×D₄	1	4	yes	complemented
106	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
107	C₂×C₄×C₈₉	C₂⁴	1	4	yes	—
108	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
111	C₂²⋊D₈₉	—	2	16	no	—
112	C₂²⋊D₈₉	—	4	16	no	—
113	C₂²⋊D₈₉	C₂×D₄	1	8	yes	complemented
114	C₂×Dic₈₉	C₂⁴	1	4	yes	—
115	C₂×Dic₈₉	C₂⁴	1	2	yes	—
116	D₃₅₆	—	2	16	no	—
117	D₃₅₆	—	2	8	no	—
118	D₃₅₆	—	2	8	no	—
119	D₃₅₆	—	2	16	no	—
120	D₃₅₆	—	4	8	no	—
121	D₃₅₆	C₂×D₄	1	4	yes	complemented
122	C₄×D₈₉	—	2	8	no	—
123	C₄×D₈₉	—	2	16	no	—
124	C₄×D₈₉	C₂×D₄	1	8	yes	complemented
125	Dic₁₇₈	C₂×D₄	1	4	yes	complemented
126	C₂²×C₈₉	—	2	2	no	—
127	C₂²×C₈₉	—	2	4	no	—
128	C₂²×C₈₉	—	2	8	no	—
129	C₂²×C₈₉	C₂²×D₄	1	2	yes	—
130	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
131	C₂×D₈₉	—	2	4	no	—
132	C₂×D₈₉	—	2	8	no	—
133	C₂×D₈₉	—	2	16	no	—
134	C₂×D₈₉	—	8	16	no	—
135	C₂×D₈₉	—	8	8	no	—
136	C₂×D₈₉	C₂²×D₄	1	4	yes	—
137	C₄×C₈₉	—	2	4	no	—
138	C₄×C₈₉	—	2	2	no	—
139	C₄×C₈₉	C₂²×D₄	1	2	yes	—
140	C₄×C₈₉	C₂²×D₄	1	2	yes	—
141	C₄×C₈₉	H(5,𝔽₂)	1	4	yes	complemented
142	Dic₈₉	—	2	4	no	—
143	Dic₈₉	C₂²×D₄	1	4	yes	—
144	C₂×C₈₉	—	2	2	no	—
145	C₂×C₈₉	—	4	4	no	—
146	C₂×C₈₉	C₂³×D₄	1	1	yes	characteristic
147	C₂×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
148	D₈₉	—	2	4	no	—
149	D₈₉	—	4	8	no	—
150	#3498	—	89	1	no	maximal, Sylow
151	C₈₉	#3498	1	1	yes	characteristic, complemented, Sylow
152	C₂×H(5,𝔽₂)	—	89	2	no	—
153	#550	—	89	4	no	—
154	D₄²	—	89	8	no	—
155	D₄²	—	89	4	no	—
156	#535	—	89	2	no	—
157	#535	—	89	4	no	—
158	#532	—	89	2	no	—
159	#532	D₈₉	1	1	yes	characteristic, complemented
160	C₂×#127	—	89	2	no	—
161	C₂×#127	—	89	1	no	—
162	C₄×#41	—	89	1	no	—
163	H(5,𝔽₂)	—	178	16	no	—
164	C₂×#41	—	89	4	no	—
165	C₂×#41	—	89	1	no	—
166	C₂×#41	—	89	4	no	—
167	C₂×#41	C₂×D₈₉	1	2	yes	complemented
168	C₂²×D₄	—	89	8	no	—
169	C₂²×D₄	—	89	4	no	—
170	C₂²×D₄	—	89	8	no	—
171	C₂²×D₄	—	89	4	no	—
172	C₂²×D₄	—	89	2	no	—
173	C₂²×D₄	—	89	4	no	—
174	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic, complemented
175	#127	—	89	4	no	—
176	#127	—	89	8	no	—
177	#127	—	89	2	no	—
178	#127	—	89	2	no	—
179	#127	—	89	4	no	—
180	#127	C₂×D₈₉	1	1	yes	characteristic, complemented
181	#124	—	89	4	no	—
182	#124	C₂×D₈₉	1	2	yes	complemented
183	#121	—	89	8	no	—
184	#121	—	89	8	no	—
185	#121	—	89	16	no	—
186	#121	C₂×D₈₉	1	4	yes	complemented
187	#120	—	89	8	no	—
188	#120	—	89	16	no	—
189	C₄×Q₈	—	89	2	no	—
190	C₄×D₄	—	89	4	no	—
191	C₄×D₄	—	89	8	no	—
192	C₄×D₄	—	89	2	no	—
193	C₄×D₄	C₂×D₈₉	1	4	yes	complemented
194	#117	—	89	1	no	—
195	#117	—	89	2	no	—
196	C₂×C₄²	—	89	2	no	—
197	C₂×C₄²	C₂×D₈₉	1	1	yes	characteristic, complemented
198	C₂⁴	—	89	8	no	—
199	C₂⁴	—	89	4	no	—
200	#41	—	2	8	no	—
201	#41	—	89	8	no	—
202	#41	—	178	16	no	—
203	#41	—	178	16	no	—
204	C₂×Q₈	—	89	1	no	—
205	C₂×Q₈	C₂²×D₈₉	1	2	yes	—
206	C₂×D₄	—	89	2	no	—
207	C₂×D₄	—	89	16	no	—
208	C₂×D₄	—	89	8	no	—
209	C₂×D₄	—	89	1	no	—
210	C₂×D₄	—	89	8	no	—
211	C₂×D₄	—	89	8	no	—
212	C₂×D₄	—	89	8	no	—
213	C₂×D₄	—	89	4	no	—
214	C₂×D₄	—	89	4	no	—
215	C₂×D₄	—	356	8	no	—
216	C₂×D₄	—	356	16	no	—
217	C₂×D₄	—	178	16	no	—
218	C₂×D₄	—	178	16	no	—
219	C₂×D₄	—	178	32	no	—
220	C₂×D₄	—	178	8	no	—
221	C₂×D₄	C₂²×D₈₉	1	4	yes	—
222	C₂×D₄	C₂²×D₈₉	1	4	yes	—
223	C₂×D₄	C₂²×D₈₉	1	2	yes	—
224	C₂²×C₄	—	89	2	no	—
225	C₂²×C₄	—	89	4	no	—
226	C₂²×C₄	—	89	8	no	—
227	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
228	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
229	C₂²×C₄	C₂²×D₈₉	1	1	yes	characteristic
230	C₄⋊C₄	—	89	4	no	—
231	C₄⋊C₄	—	89	2	no	—
232	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
233	C₂²⋊C₄	—	89	2	no	—
234	C₂²⋊C₄	—	89	4	no	—
235	C₂²⋊C₄	—	89	16	no	—
236	C₂²⋊C₄	C₂²×D₈₉	1	8	yes	—
237	C₄²	—	89	2	no	—
238	C₄²	—	89	4	no	—
239	C₄²	C₂²×D₈₉	1	1	yes	characteristic
240	C₄²	C₂²×D₈₉	1	1	yes	characteristic
241	C₄²	C₂²×D₈₉	1	2	yes	—
242	C₂³	—	89	2	no	—
243	C₂³	—	89	8	no	—
244	C₂³	—	356	8	no	—
245	C₂³	—	356	16	no	—
246	C₂³	—	178	16	no	—
247	C₂³	—	178	8	no	—
248	C₂³	C₂³×D₈₉	1	4	yes	—
249	C₂³	C₂³×D₈₉	1	1	yes	characteristic
250	Q₈	—	2	4	no	—
251	Q₈	—	89	4	no	—
252	D₄	—	2	8	no	—
253	D₄	—	2	4	no	—
254	D₄	—	2	8	no	—
255	D₄	—	89	8	no	—
256	D₄	—	89	4	no	—
257	D₄	—	356	8	no	—
258	D₄	—	356	16	no	—
259	D₄	—	178	16	no	—
260	D₄	—	178	16	no	—
261	D₄	—	178	8	no	—
262	D₄	—	178	8	no	—
263	D₄	—	178	16	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	89	2	no	—
267	C₂×C₄	—	89	8	no	—
268	C₂×C₄	—	89	4	no	—
269	C₂×C₄	—	178	16	no	—
270	C₂×C₄	—	178	8	no	—
271	C₂×C₄	D₄×D₈₉	1	4	yes	complemented
272	C₂×C₄	C₂³×D₈₉	1	2	yes	—
273	C₂×C₄	C₂³×D₈₉	1	4	yes	—
274	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
275	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
276	C₂×C₄	C₂³×D₈₉	1	2	yes	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	8	no	—
280	C₂²	—	89	4	no	—
281	C₂²	—	178	8	no	—
282	C₂²	—	178	16	no	—
283	C₂²	—	178	4	no	—
284	C₂²	—	712	8	no	—
285	C₂²	—	712	16	no	—
286	C₂²	C₂×D₄×D₈₉	1	2	yes	—
287	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
288	C₄	—	2	4	no	—
289	C₄	—	2	2	no	—
290	C₄	—	89	4	no	—
291	C₄	—	178	4	no	—
292	C₄	C₂×D₄×D₈₉	1	2	yes	—
293	C₄	C₂×D₄×D₈₉	1	2	yes	—
294	C₄	#15668058	1	4	yes	complemented
295	C₂	—	2	2	no	—
296	C₂	—	4	4	no	—
297	C₂	—	356	8	no	—
298	C₂	—	178	4	no	—
299	C₂	C₂²×D₄×D₈₉	1	1	yes	characteristic
300	C₂	C₂×#15668058	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₉
commutator, socle, cosocle	C₂²×C₈₉	C₂⁵
Fitting	#532×C₈₉	C₂
hypercenter	#532	D₈₉
hypocenter	C₈₉	#3498
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #532

Complements

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

N	H ≅ G/N	type
1. C₂×#15668058	C₂	semidirect
2. #19152269	C₂	semidirect
3. D₄×D₃₅₆	C₂	semidirect
4. #19152232	C₂	semidirect
5. #19152166	C₂	semidirect
6. #19152162	C₂	semidirect
7. #19152149	C₂	semidirect
8. C₂×#15668009	C₂	semidirect
9. #127×D₈₉	C₂	semidirect
10. C₄×#13083574	C₂	semidirect
11. #532×C₈₉	C₂	semidirect
13. C₂×#13083575	C₂²	semidirect
15. C₂×#13083574	C₂²	semidirect
16. C₂²×D₃₅₆	C₂²	semidirect
17. C₂²×D₃₅₆	C₂²	semidirect
18. C₂²×D₃₅₆	C₂²	semidirect
19. C₂×D₄×D₈₉	C₂²	semidirect
20. C₂×D₄×D₈₉	C₂²	semidirect
21. C₂×D₄×D₈₉	C₂²	semidirect
22. #15668009	C₂²	semidirect
23. #15668009	C₂²	semidirect
24. #15668009	C₂²	semidirect
25. #15668008	C₂²	semidirect
26. #15668008	C₂²	semidirect
27. #15668001	C₂²	semidirect
28. #15667987	C₂²	semidirect
29. #15667986	C₂²	semidirect
30. #15667985	C₂²	semidirect
31. #15667982	C₂²	semidirect
32. #15667980	C₂²	semidirect
33. C₄×Dic₁₇₈	C₂²	semidirect
34. C₄×D₃₅₆	C₂²	semidirect
35. #15667974	C₂²	semidirect
36. C₄×C₂²⋊D₈₉	C₂²	semidirect
37. #15667971	C₂²	semidirect
38. #15667969	C₂²	semidirect
39. C₄²×D₈₉	C₂²	semidirect
40. C₂×#41×C₈₉	C₂²	semidirect
41. C₄⋊Q₈×C₈₉	C₂²	semidirect
42. #127×C₈₉	C₂²	semidirect
43. #124×C₈₉	C₂²	semidirect
44. #121×C₈₉	C₂²	semidirect
45. C₄×D₄×C₈₉	C₂²	semidirect
46. C₂×C₄²×C₈₉	C₂²	semidirect
51. #13083574	D₄	semidirect
105. C₂×C₄×C₈₉	C₂×D₄	semidirect
113. C₂²⋊D₈₉	C₂×D₄	semidirect
121. D₃₅₆	C₂×D₄	semidirect
124. C₄×D₈₉	C₂×D₄	semidirect
125. Dic₁₇₈	C₂×D₄	semidirect
141. C₄×C₈₉	H(5,𝔽₂)	semidirect
151. C₈₉	#3498	semidirect
159. #532	D₈₉	semidirect
167. C₂×#41	C₂×D₈₉	semidirect
174. C₄⋊Q₈	C₂×D₈₉	semidirect
180. #127	C₂×D₈₉	semidirect
182. #124	C₂×D₈₉	semidirect
186. #121	C₂×D₈₉	semidirect
193. C₄×D₄	C₂×D₈₉	semidirect
197. C₂×C₄²	C₂×D₈₉	semidirect
271. C₂×C₄	D₄×D₈₉	semidirect
294. C₄	#15668058	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668058	2
#19152269	2
D₄×D₃₅₆	2
#19152232	2
#19152166	2
#19152162	2
#19152149	2
C₂×#15668009	2
#127×D₈₉	2
C₄×#13083574	2
#532×C₈₉	2
#3498	89