Subgroups — #26034653

Order 11392.

1258 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152340	C₂	1	4	yes	maximal, complemented
2	#19152314	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152311	C₂	1	4	yes	maximal, complemented
4	D₄×D₃₅₆	C₂	1	2	yes	maximal, complemented
5	#19152232	C₂	1	4	yes	maximal, complemented
6	#19152167	C₂	1	4	yes	maximal, complemented
7	#19152166	C₂	1	4	yes	maximal, complemented
8	#19152164	C₂	1	4	yes	maximal, complemented
9	#19152149	C₂	1	1	yes	maximal, characteristic, complemented
10	#19152145	C₂	1	2	yes	maximal, complemented
11	#555×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13083575	C₂²	1	2	yes	complemented
13	C₂×#13083574	C₂²	1	4	yes	complemented
14	C₂²×D₃₅₆	C₂²	1	4	yes	complemented
15	C₂×D₄×D₈₉	C₂²	1	4	yes	complemented
16	C₂×D₄×D₈₉	C₂²	1	4	yes	complemented
17	#15668011	C₂²	1	2	yes	complemented
18	#15668009	C₂²	1	1	yes	characteristic
19	#15668009	C₂²	1	4	yes	complemented
20	#15668008	C₂²	1	2	yes	—
21	#15668008	C₂²	1	4	yes	complemented
22	#15668001	C₂²	1	2	yes	complemented
23	#15668001	C₂²	1	4	yes	complemented
24	#15668000	C₂²	1	8	yes	complemented
25	#15667998	C₂²	1	4	yes	complemented
26	#15667998	C₂²	1	8	yes	complemented
27	#15667987	C₂²	1	2	yes	complemented
28	#15667987	C₂²	1	4	yes	complemented
29	#15667987	C₂²	1	4	yes	complemented
30	#15667987	C₂²	1	4	yes	complemented
31	#15667986	C₂²	1	4	yes	complemented
32	#15667985	C₂²	1	4	yes	complemented
33	#15667985	C₂²	1	8	yes	complemented
34	#15667983	C₂²	1	2	yes	complemented
35	#15667982	C₂²	1	8	yes	complemented
36	#15667982	C₂²	1	8	yes	complemented
37	#15667982	C₂²	1	8	yes	complemented
38	#15667980	C₂²	1	8	yes	complemented
39	C₄×D₃₅₆	C₂²	1	2	yes	complemented
40	#15667974	C₂²	1	4	yes	complemented
41	C₄×C₂²⋊D₈₉	C₂²	1	4	yes	complemented
42	#15667970	C₂²	1	4	yes	complemented
43	#15667969	C₂²	1	4	yes	complemented
44	C₄²×D₈₉	C₂²	1	1	yes	characteristic, complemented
45	C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
46	#124×C₈₉	C₂²	1	4	yes	complemented
47	#121×C₈₉	C₂²	1	4	yes	complemented
48	C₄×D₄×C₈₉	C₂²	1	2	yes	complemented
49	#117×C₈₉	C₂²	1	4	yes	complemented
50	C₂³×D₈₉	C₂³	1	8	yes	—
51	#13083575	—	4	4	no	—
52	#13083574	—	4	8	no	—
53	C₂×Dic₁₇₈	C₂³	1	4	yes	—
54	C₂×D₃₅₆	—	4	8	no	—
55	C₂×D₃₅₆	C₂³	1	4	yes	—
56	C₂×D₃₅₆	C₂³	1	2	yes	—
57	C₂×D₃₅₆	C₂³	1	2	yes	—
58	C₂×D₃₅₆	C₂³	1	8	yes	—
59	C₂×D₃₅₆	C₂³	1	4	yes	—
60	C₂×D₃₅₆	C₂³	1	4	yes	—
61	C₂×D₃₅₆	C₂³	1	4	yes	—
62	C₂×C₂²⋊D₈₉	C₂³	1	4	yes	—
63	C₂×C₂²⋊D₈₉	C₂³	1	4	yes	—
64	C₂×C₂²⋊D₈₉	C₂³	1	8	yes	—
65	C₂×C₂²⋊D₈₉	C₂³	1	4	yes	—
66	D₄×D₈₉	—	4	8	no	—
67	D₄×D₈₉	—	4	8	no	—
68	C₂×C₄×D₈₉	C₂³	1	4	yes	—
69	C₂×C₄×D₈₉	C₂³	1	4	yes	—
70	C₂×C₄×D₈₉	C₂³	1	2	yes	—
71	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₈₉	C₂³	1	2	yes	—
73	#13083553	C₂³	1	4	yes	—
74	C₂²⋊Dic₈₉	C₂³	1	4	yes	—
75	#13083551	C₂³	1	8	yes	—
76	#13083551	C₂³	1	8	yes	—
77	#13083551	C₂³	1	4	yes	—
78	#13083551	C₂³	1	8	yes	—
79	#13083551	C₂³	1	4	yes	—
80	C₄×Dic₈₉	C₂³	1	4	yes	—
81	C₄×Dic₈₉	C₂³	1	2	yes	—
82	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
83	C₂×Q₈×C₈₉	C₂³	1	2	yes	—
84	C₂×D₄×C₈₉	C₂³	1	2	yes	—
85	C₂×D₄×C₈₉	C₂³	1	4	yes	—
86	C₂²×C₄×C₈₉	C₂³	1	4	yes	—
87	C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
88	C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
89	C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
90	C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
91	C₄²×C₈₉	C₂³	1	4	yes	—
92	C₄²×C₈₉	C₂³	1	1	yes	characteristic
93	C₂³×C₈₉	C₂⁴	1	4	yes	—
94	C₂²×D₈₉	—	4	8	no	—
95	C₂²×D₈₉	—	4	8	no	—
96	C₂²×D₈₉	—	4	8	no	—
97	C₂²×D₈₉	C₂⁴	1	4	yes	—
98	C₂²×D₈₉	C₂⁴	1	4	yes	—
99	C₂²×D₈₉	C₂⁴	1	1	yes	characteristic
100	Q₈×C₈₉	—	4	2	no	—
101	D₄×C₈₉	—	2	4	no	—
102	D₄×C₈₉	—	4	4	no	—
103	C₂×C₄×C₈₉	—	2	8	no	—
104	C₂×C₄×C₈₉	C₂⁴	1	4	yes	—
105	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂⁴	1	4	yes	—
107	C₂×C₄×C₈₉	C₂⁴	1	2	yes	—
108	C₂²⋊D₈₉	—	2	8	no	—
109	C₂²⋊D₈₉	—	4	8	no	—
110	C₂²⋊D₈₉	—	4	4	no	—
111	C₂²⋊D₈₉	—	4	4	no	—
112	C₂×Dic₈₉	C₂⁴	1	2	yes	—
113	C₂×Dic₈₉	C₂⁴	1	4	yes	—
114	C₂×Dic₈₉	C₂⁴	1	1	yes	characteristic
115	D₃₅₆	—	2	8	no	—
116	D₃₅₆	—	2	4	no	—
117	D₃₅₆	—	4	4	no	—
118	D₃₅₆	—	4	8	no	—
119	D₃₅₆	—	4	4	no	—
120	D₃₅₆	—	4	4	no	—
121	D₃₅₆	—	4	2	no	—
122	C₄×D₈₉	—	2	8	no	—
123	C₄×D₈₉	—	2	4	no	—
124	C₄×D₈₉	—	2	8	no	—
125	C₄×D₈₉	—	2	2	no	—
126	Dic₁₇₈	—	4	4	no	—
127	C₂²×C₈₉	—	2	4	no	—
128	C₂²×C₈₉	—	2	4	no	—
129	C₂²×C₈₉	—	2	4	no	—
130	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
131	C₂×D₈₉	—	2	1	no	—
132	C₂×D₈₉	—	2	2	no	—
133	C₂×D₈₉	—	2	4	no	—
134	C₂×D₈₉	—	2	4	no	—
135	C₂×D₈₉	—	2	4	no	—
136	C₂×D₈₉	—	2	8	no	—
137	C₂×D₈₉	—	2	4	no	—
138	C₂×D₈₉	—	8	16	no	—
139	C₄×C₈₉	—	2	4	no	—
140	C₄×C₈₉	—	2	4	no	—
141	C₄×C₈₉	—	2	1	no	—
142	C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
143	Dic₈₉	—	2	2	no	—
144	Dic₈₉	—	2	4	no	—
145	Dic₈₉	H(5,𝔽₂)	1	2	yes	—
146	C₂×C₈₉	—	4	4	no	—
147	C₂×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
149	D₈₉	—	4	1	no	—
150	D₈₉	—	4	4	no	—
151	D₈₉	—	4	4	no	—
152	#3549	—	89	1	no	maximal, Sylow
153	C₈₉	#3549	1	1	yes	characteristic, complemented, Sylow
154	H(3,𝔽₄)	—	89	4	no	—
155	#555	—	89	1	no	—
156	#555	—	89	4	no	—
157	#555	D₈₉	1	1	yes	characteristic, complemented
158	D₄²	—	89	4	no	—
159	D₄²	—	89	2	no	—
160	#535	—	89	4	no	—
161	#535	—	89	4	no	—
162	#535	—	89	4	no	—
163	#532	—	89	1	no	—
164	#532	—	89	2	no	—
165	C₂×#41	—	89	2	no	—
166	C₂×#41	—	89	4	no	—
167	C₂²×D₄	—	89	4	no	—
168	C₂²×D₄	—	89	4	no	—
169	C₂²×D₄	—	89	4	no	—
170	C₄⋊Q₈	—	89	2	no	—
171	C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic, complemented
172	#127	—	89	2	no	—
173	#127	—	89	4	no	—
174	#127	—	89	1	no	—
175	#127	—	89	4	no	—
176	#124	—	89	2	no	—
177	#124	—	89	4	no	—
178	#124	—	89	4	no	—
179	#124	—	89	8	no	—
180	#124	—	89	8	no	—
181	#124	C₂×D₈₉	1	4	yes	complemented
182	#121	—	89	2	no	—
183	#121	—	89	4	no	—
184	#121	—	89	4	no	—
185	#121	—	89	2	no	—
186	#121	—	89	4	no	—
187	#121	—	89	4	no	—
188	#121	—	89	4	no	—
189	#121	—	89	8	no	—
190	#121	C₂×D₈₉	1	4	yes	complemented
191	#120	—	89	8	no	—
192	#120	—	89	8	no	—
193	#120	—	89	8	no	—
194	#120	—	89	8	no	—
195	C₄×D₄	—	89	4	no	—
196	C₄×D₄	—	89	4	no	—
197	C₄×D₄	—	89	2	no	—
198	C₄×D₄	C₂×D₈₉	1	2	yes	complemented
199	#117	—	89	4	no	—
200	#117	—	89	4	no	—
201	#117	C₂×D₈₉	1	4	yes	complemented
202	C₂×C₄²	—	89	1	no	—
203	C₂⁴	—	89	8	no	—
204	#41	—	356	8	no	—
205	#41	—	356	4	no	—
206	C₂×Q₈	—	89	4	no	—
207	C₂×Q₈	C₂²×D₈₉	1	2	yes	—
208	C₂×D₄	—	89	4	no	—
209	C₂×D₄	—	89	8	no	—
210	C₂×D₄	—	89	4	no	—
211	C₂×D₄	—	89	4	no	—
212	C₂×D₄	—	89	2	no	—
213	C₂×D₄	—	89	4	no	—
214	C₂×D₄	—	89	8	no	—
215	C₂×D₄	—	89	4	no	—
216	C₂×D₄	—	89	4	no	—
217	C₂×D₄	—	89	4	no	—
218	C₂×D₄	—	89	2	no	—
219	C₂×D₄	—	356	8	no	—
220	C₂×D₄	—	356	8	no	—
221	C₂×D₄	—	356	8	no	—
222	C₂×D₄	C₂²×D₈₉	1	2	yes	—
223	C₂×D₄	C₂²×D₈₉	1	4	yes	—
224	C₂²×C₄	—	89	1	no	—
225	C₂²×C₄	—	89	2	no	—
226	C₂²×C₄	—	89	4	no	—
227	C₂²×C₄	—	89	4	no	—
228	C₂²×C₄	C₂²×D₈₉	1	4	yes	—
229	C₄⋊C₄	—	89	4	no	—
230	C₄⋊C₄	—	89	2	no	—
231	C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
232	C₄⋊C₄	C₂²×D₈₉	1	2	yes	—
233	C₂²⋊C₄	—	89	4	no	—
234	C₂²⋊C₄	—	89	8	no	—
235	C₂²⋊C₄	—	89	8	no	—
236	C₂²⋊C₄	—	89	8	no	—
237	C₂²⋊C₄	—	89	4	no	—
238	C₂²⋊C₄	—	89	4	no	—
239	C₂²⋊C₄	C₂²×D₈₉	1	8	yes	—
240	C₂²⋊C₄	C₂²×D₈₉	1	4	yes	—
241	C₄²	—	89	2	no	—
242	C₄²	—	89	1	no	—
243	C₄²	—	89	4	no	—
244	C₄²	C₂²×D₈₉	1	1	yes	characteristic
245	C₄²	C₂²×D₈₉	1	4	yes	—
246	C₂³	—	89	4	no	—
247	C₂³	—	89	4	no	—
248	C₂³	—	89	1	no	—
249	C₂³	—	356	8	no	—
250	C₂³	—	356	8	no	—
251	C₂³	—	356	8	no	—
252	C₂³	C₂³×D₈₉	1	4	yes	—
253	Q₈	—	4	2	no	—
254	Q₈	—	356	4	no	—
255	D₄	—	2	4	no	—
256	D₄	—	4	4	no	—
257	D₄	—	356	4	no	—
258	D₄	—	356	8	no	—
259	D₄	—	356	4	no	—
260	D₄	—	356	2	no	—
261	D₄	—	356	4	no	—
262	D₄	—	356	8	no	—
263	D₄	—	356	4	no	—
264	D₄	—	356	4	no	—
265	D₄	—	178	8	no	—
266	D₄	—	178	8	no	—
267	D₄	—	178	4	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	89	1	no	—
270	C₂×C₄	—	89	2	no	—
271	C₂×C₄	—	89	4	no	—
272	C₂×C₄	—	178	8	no	—
273	C₂×C₄	—	178	2	no	—
274	C₂×C₄	—	178	8	no	—
275	C₂×C₄	—	178	4	no	—
276	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
277	C₂×C₄	C₂³×D₈₉	1	4	yes	—
278	C₂×C₄	C₂³×D₈₉	1	4	yes	—
279	C₂×C₄	C₂³×D₈₉	1	2	yes	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	178	1	no	—
284	C₂²	—	178	4	no	—
285	C₂²	—	178	4	no	—
286	C₂²	—	178	2	no	—
287	C₂²	—	178	4	no	—
288	C₂²	—	178	4	no	—
289	C₂²	—	178	8	no	—
290	C₂²	—	712	16	no	—
291	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
292	C₄	—	2	4	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	4	no	—
295	C₄	—	89	2	no	—
296	C₄	—	178	4	no	—
297	C₄	—	178	2	no	—
298	C₄	#15668058	1	4	yes	—
299	C₂	—	4	4	no	—
300	C₂	—	356	1	no	—
301	C₂	—	356	4	no	—
302	C₂	—	356	4	no	—
303	C₂	H(5,𝔽₂)×D₈₉	1	1	yes	characteristic
304	C₂	C₂×#15668058	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₉
commutator, socle, cosocle	C₂²×C₈₉	C₂⁵
Fitting	#555×C₈₉	C₂
hypercenter	#555	D₈₉
hypocenter	C₈₉	#3549
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #555

Complements

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

N	H ≅ G/N	type
1. #19152340	C₂	semidirect
2. #19152314	C₂	semidirect
3. #19152311	C₂	semidirect
4. D₄×D₃₅₆	C₂	semidirect
5. #19152232	C₂	semidirect
6. #19152167	C₂	semidirect
7. #19152166	C₂	semidirect
8. #19152164	C₂	semidirect
9. #19152149	C₂	semidirect
10. #19152145	C₂	semidirect
11. #555×C₈₉	C₂	semidirect
12. C₂×#13083575	C₂²	semidirect
13. C₂×#13083574	C₂²	semidirect
14. C₂²×D₃₅₆	C₂²	semidirect
15. C₂×D₄×D₈₉	C₂²	semidirect
16. C₂×D₄×D₈₉	C₂²	semidirect
17. #15668011	C₂²	semidirect
19. #15668009	C₂²	semidirect
21. #15668008	C₂²	semidirect
22. #15668001	C₂²	semidirect
23. #15668001	C₂²	semidirect
24. #15668000	C₂²	semidirect
25. #15667998	C₂²	semidirect
26. #15667998	C₂²	semidirect
27. #15667987	C₂²	semidirect
28. #15667987	C₂²	semidirect
29. #15667987	C₂²	semidirect
30. #15667987	C₂²	semidirect
31. #15667986	C₂²	semidirect
32. #15667985	C₂²	semidirect
33. #15667985	C₂²	semidirect
34. #15667983	C₂²	semidirect
35. #15667982	C₂²	semidirect
36. #15667982	C₂²	semidirect
37. #15667982	C₂²	semidirect
38. #15667980	C₂²	semidirect
39. C₄×D₃₅₆	C₂²	semidirect
40. #15667974	C₂²	semidirect
41. C₄×C₂²⋊D₈₉	C₂²	semidirect
42. #15667970	C₂²	semidirect
43. #15667969	C₂²	semidirect
44. C₄²×D₈₉	C₂²	semidirect
45. C₄⋊Q₈×C₈₉	C₂²	semidirect
46. #124×C₈₉	C₂²	semidirect
47. #121×C₈₉	C₂²	semidirect
48. C₄×D₄×C₈₉	C₂²	semidirect
49. #117×C₈₉	C₂²	semidirect
153. C₈₉	#3549	semidirect
157. #555	D₈₉	semidirect
171. C₄⋊Q₈	C₂×D₈₉	semidirect
181. #124	C₂×D₈₉	semidirect
190. #121	C₂×D₈₉	semidirect
198. C₄×D₄	C₂×D₈₉	semidirect
201. #117	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#19152340	2
#19152314	2
#19152311	2
D₄×D₃₅₆	2
#19152232	2
#19152167	2
#19152166	2
#19152164	2
#19152149	2
#19152145	2
#555×C₈₉	2
#3549	89