Subgroups — #26028988

Order 11392.

544 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×D₈₉	C₂	1	2	yes	maximal, complemented
2	C₄×D₄×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#19151676	C₂	1	1	yes	maximal, characteristic, complemented
4	#19151672	C₂	1	2	yes	maximal, complemented
5	#19151667	C₂	1	2	yes	maximal, complemented
6	C₈×D₃₅₆	C₂	1	1	yes	maximal, characteristic, complemented
7	#19151663	C₂	1	1	yes	maximal, characteristic, complemented
8	#19151620	C₂	1	1	yes	maximal, characteristic, complemented
9	#19151560	C₂	1	2	yes	maximal, complemented
10	C₄×#13083557	C₂	1	1	yes	maximal, characteristic, complemented
11	#436×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₈₉	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₈₉	C₂²	1	2	yes	—
14	C₂×#13083558	C₂²	1	2	yes	complemented
15	C₂×#13083557	C₂²	1	2	yes	—
16	C₂×#13083557	C₂²	1	1	yes	characteristic
17	C₂×#13083557	C₂²	1	1	yes	characteristic
18	M₁₆×D₈₉	—	2	8	no	—
19	C₂×C₈×D₈₉	C₂²	1	2	yes	complemented
20	C₂×C₈×D₈₉	C₂²	1	2	yes	complemented
21	C₄×D₃₅₆	C₂²	1	1	yes	characteristic
22	D₄×Dic₈₉	C₄	1	1	yes	characteristic
23	#15667974	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊D₈₉	C₂²	1	2	yes	—
25	#15667972	C₄	1	2	yes	—
26	C₄⋊C₄×D₈₉	C₄	1	1	yes	characteristic
27	C₂²⋊C₄×D₈₉	C₄	1	2	yes	—
28	C₄²×D₈₉	C₂²	1	1	yes	characteristic
29	C₄⋊#10623494	C₂²	1	1	yes	characteristic, complemented
30	#15667941	C₂²	1	2	yes	complemented
31	C₂²⋊#10623494	C₂²	1	2	yes	complemented
32	#15667924	C₂²	1	2	yes	complemented
33	#15667924	C₂²	1	2	yes	complemented
34	#15667922	C₂²	1	1	yes	characteristic, complemented
35	#15667922	C₂²	1	1	yes	characteristic, complemented
36	C₄×#10623494	C₂²	1	1	yes	characteristic, complemented
37	C₂×M₁₆×C₈₉	C₂²	1	2	yes	complemented
38	C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
40	C₂²⋊C₈×C₈₉	C₂²	1	2	yes	complemented
41	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
42	C₂³×D₈₉	C₂×C₄	1	2	yes	—
43	C₂×D₃₅₆	C₂×C₄	1	1	yes	characteristic
44	C₂×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
45	D₄×D₈₉	—	2	4	no	—
46	C₂²×Dic₈₉	C₂×C₄	1	2	yes	—
47	C₂×C₄×D₈₉	—	2	4	no	—
48	C₂×C₄×D₈₉	—	2	4	no	—
49	C₂×C₄×D₈₉	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₈₉	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₈₉	C₂³	1	2	yes	—
54	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
55	#13083558	—	2	4	no	—
56	#13083557	—	2	4	no	—
57	#13083557	—	2	2	no	—
58	#13083557	D₄	1	4	yes	complemented
59	C₂×#10623494	C₂³	1	2	yes	—
60	C₂×#10623494	C₂³	1	1	yes	characteristic
61	C₂×#10623494	C₂³	1	1	yes	characteristic
62	C₈×D₈₉	—	2	4	no	—
63	C₈×D₈₉	—	2	4	no	—
64	C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
65	#13083553	C₂×C₄	1	2	yes	—
66	C₂²⋊Dic₈₉	C₂×C₄	1	2	yes	—
67	#13083551	C₂×C₄	1	2	yes	—
68	#13083551	C₂×C₄	1	2	yes	—
69	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₈₉	C₂³	1	1	yes	characteristic
72	C₂×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂³	1	2	yes	—
74	M₁₆×C₈₉	—	2	4	no	—
75	C₂×C₈×C₈₉	C₂³	1	2	yes	—
76	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
79	C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
80	C₄²×C₈₉	C₂³	1	1	yes	characteristic
81	C₂³×C₈₉	C₂²×C₄	1	2	yes	—
82	C₂²×D₈₉	—	2	4	no	—
83	C₂²×D₈₉	—	2	4	no	—
84	C₂²×D₈₉	—	4	2	no	—
85	C₂²×D₈₉	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₈₉	C₂²×C₄	1	2	yes	—
87	D₄×C₈₉	—	2	2	no	—
88	C₂×C₄×C₈₉	—	2	2	no	—
89	C₂×C₄×C₈₉	—	2	2	no	—
90	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₈₉	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
94	C₂²⋊D₈₉	—	2	4	no	—
95	C₂×Dic₈₉	—	2	4	no	—
96	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₈₉	C₂²×C₄	1	2	yes	—
99	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
100	D₃₅₆	M₁₆	1	4	yes	complemented
101	C₄×D₈₉	—	2	4	no	—
102	C₄×D₈₉	—	2	2	no	—
103	C₄×D₈₉	—	2	2	no	—
104	C₄×D₈₉	—	2	2	no	—
105	C₄×D₈₉	—	2	4	no	—
106	C₄×D₈₉	—	4	1	no	—
107	C₄×D₈₉	C₂×D₄	1	2	yes	—
108	C₄×D₈₉	#41	1	2	yes	—
109	C₈×C₈₉	—	2	2	no	—
110	C₈×C₈₉	—	2	1	no	—
111	C₈×C₈₉	C₂×D₄	1	2	yes	complemented
112	#10623494	—	2	2	no	—
113	#10623494	—	2	1	no	—
114	#10623494	C₂×D₄	1	2	yes	complemented
115	C₂²×C₈₉	—	2	2	no	—
116	C₂²×C₈₉	—	2	2	no	—
117	C₂²×C₈₉	—	2	2	no	—
118	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
119	C₂×D₈₉	—	2	2	no	—
120	C₂×D₈₉	—	2	1	no	—
121	C₂×D₈₉	—	2	1	no	—
122	C₂×D₈₉	—	2	2	no	—
123	C₂×D₈₉	—	4	8	no	—
124	C₂×D₈₉	C₄×D₄	1	2	yes	—
125	C₂×D₈₉	C₂×M₁₆	1	4	yes	complemented
126	C₄×C₈₉	—	2	2	no	—
127	C₄×C₈₉	—	2	1	no	—
128	C₄×C₈₉	C₂×M₁₆	1	2	yes	complemented
129	C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
130	C₄×C₈₉	C₂×#41	1	1	yes	characteristic
131	Dic₈₉	—	2	2	no	—
132	Dic₈₉	—	2	1	no	—
133	Dic₈₉	—	2	1	no	—
134	Dic₈₉	C₄×D₄	1	2	yes	—
135	C₂×C₈₉	—	4	2	no	—
136	C₂×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
137	C₂×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
138	C₂×C₈₉	#568	1	1	yes	characteristic
139	D₈₉	—	2	2	no	—
140	D₈₉	—	2	4	no	—
141	D₄×M₁₆	—	89	1	no	maximal, Sylow
142	C₈₉	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
143	C₂²×M₁₆	—	89	2	no	—
144	C₂×C₄×D₄	—	89	1	no	—
145	#436	—	89	1	no	—
146	#436	D₈₉	1	1	yes	characteristic, complemented
147	#435	—	89	2	no	—
148	#435	—	89	2	no	—
149	C₈×D₄	—	89	1	no	—
150	C₈×D₄	—	89	1	no	—
151	#423	—	89	1	no	—
152	#407	—	89	2	no	—
153	C₄×M₁₆	—	89	1	no	—
154	C₂²×D₄	—	89	1	no	—
155	C₂³×C₄	—	89	2	no	—
156	C₂×M₁₆	—	89	2	no	—
157	C₂×M₁₆	—	89	1	no	—
158	C₂×M₁₆	—	89	2	no	—
159	C₂×M₁₆	—	89	1	no	—
160	C₂×M₁₆	—	178	8	no	—
161	C₂×M₁₆	C₂×D₈₉	1	2	yes	complemented
162	C₂²×C₈	—	89	2	no	—
163	C₂²×C₈	—	89	2	no	—
164	C₄×D₄	—	89	1	no	—
165	C₄×D₄	—	89	2	no	—
166	C₄×D₄	—	89	1	no	—
167	C₄×D₄	—	89	2	no	—
168	C₄×D₄	—	89	1	no	—
169	C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	89	1	no	—
171	C₂×C₂²⋊C₄	—	89	2	no	—
172	C₂×C₄²	—	89	1	no	—
173	C₄⋊C₈	—	89	2	no	—
174	C₄⋊C₈	—	89	1	no	—
175	C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
176	C₂²⋊C₈	—	89	2	no	—
177	C₂²⋊C₈	—	89	2	no	—
178	C₂²⋊C₈	—	89	2	no	—
179	C₂²⋊C₈	C₂×D₈₉	1	2	yes	complemented
180	#97	—	89	1	no	—
181	#97	—	89	1	no	—
182	C₄×C₈	—	89	1	no	—
183	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic, complemented
184	C₂⁴	—	89	2	no	—
185	C₂×D₄	—	89	2	no	—
186	C₂×D₄	—	89	1	no	—
187	C₂×D₄	—	178	4	no	—
188	C₂×D₄	C₄×D₈₉	1	1	yes	characteristic
189	C₂²×C₄	—	89	1	no	—
190	C₂²×C₄	—	89	2	no	—
191	C₂²×C₄	—	89	1	no	—
192	C₂²×C₄	—	89	1	no	—
193	C₂²×C₄	—	89	2	no	—
194	C₂²×C₄	—	89	2	no	—
195	C₂²×C₄	—	89	2	no	—
196	C₂²×C₄	—	178	4	no	—
197	C₂²×C₄	—	178	4	no	—
198	C₂²×C₄	C₂²×D₈₉	1	2	yes	—
199	M₁₆	—	2	4	no	—
200	M₁₆	—	89	4	no	—
201	M₁₆	—	178	4	no	—
202	M₁₆	—	178	2	no	—
203	M₁₆	—	178	4	no	—
204	C₂×C₈	—	89	1	no	—
205	C₂×C₈	—	89	1	no	—
206	C₂×C₈	—	89	2	no	—
207	C₂×C₈	—	178	4	no	—
208	C₂×C₈	—	178	4	no	—
209	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
210	C₂×C₈	C₂²×D₈₉	1	2	yes	—
211	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
212	C₄⋊C₄	—	89	2	no	—
213	C₄⋊C₄	—	89	1	no	—
214	C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
215	C₂²⋊C₄	—	89	2	no	—
216	C₂²⋊C₄	—	89	2	no	—
217	C₂²⋊C₄	—	89	2	no	—
218	C₂²⋊C₄	C₄×D₈₉	1	2	yes	—
219	C₄²	—	89	1	no	—
220	C₄²	—	89	1	no	—
221	C₄²	—	89	1	no	—
222	C₄²	C₂²×D₈₉	1	1	yes	characteristic
223	C₂³	—	89	2	no	—
224	C₂³	—	89	1	no	—
225	C₂³	—	356	2	no	—
226	C₂³	—	178	4	no	—
227	C₂³	—	178	4	no	—
228	C₂³	C₂×C₄×D₈₉	1	2	yes	—
229	D₄	—	2	2	no	—
230	D₄	—	89	4	no	—
231	D₄	—	178	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	89	1	no	—
235	C₂×C₄	—	89	2	no	—
236	C₂×C₄	—	89	2	no	—
237	C₂×C₄	—	89	1	no	—
238	C₂×C₄	—	89	1	no	—
239	C₂×C₄	—	89	2	no	—
240	C₂×C₄	—	356	1	no	—
241	C₂×C₄	—	178	4	no	—
242	C₂×C₄	—	178	4	no	—
243	C₂×C₄	—	178	4	no	—
244	C₂×C₄	—	178	2	no	—
245	C₂×C₄	—	178	2	no	—
246	C₂×C₄	—	178	2	no	—
247	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₈₉	1	2	yes	—
249	C₂×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
250	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	2	1	no	—
253	C₈	—	89	2	no	—
254	C₈	—	178	2	no	—
255	C₈	—	178	1	no	—
256	C₈	D₄×D₈₉	1	2	yes	complemented
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	89	2	no	—
261	C₂²	—	89	4	no	—
262	C₂²	—	356	8	no	—
263	C₂²	—	178	2	no	—
264	C₂²	—	178	1	no	—
265	C₂²	—	178	1	no	—
266	C₂²	—	178	2	no	—
267	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	2	1	no	—
270	C₄	—	89	2	no	—
271	C₄	—	178	2	no	—
272	C₄	—	178	1	no	—
273	C₄	—	178	1	no	—
274	C₄	M₁₆×D₈₉	1	2	yes	complemented
275	C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
276	C₄	#41×D₈₉	1	1	yes	characteristic
277	C₂	—	4	2	no	—
278	C₂	—	178	2	no	—
279	C₂	—	178	4	no	—
280	C₂	C₄×D₄×D₈₉	1	1	yes	characteristic
281	C₂	C₂×M₁₆×D₈₉	1	1	yes	characteristic
282	C₂	#19152363	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₄
Fitting	#436×C₈₉	C₂
prime radical	C₂×D₄×D₈₉	C₄
cosocle	C₂×C₄×C₈₉	C₂⁴
hypercenter	#436	D₈₉
hypocenter	C₈₉	D₄×M₁₆
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ #436

Complements

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

N	H ≅ G/N	type
1. C₂×M₁₆×D₈₉	C₂	semidirect
3. #19151676	C₂	semidirect
4. #19151672	C₂	semidirect
5. #19151667	C₂	semidirect
6. C₈×D₃₅₆	C₂	semidirect
7. #19151663	C₂	semidirect
8. #19151620	C₂	semidirect
9. #19151560	C₂	semidirect
10. C₄×#13083557	C₂	semidirect
11. #436×C₈₉	C₂	semidirect
14. C₂×#13083558	C₂²	semidirect
19. C₂×C₈×D₈₉	C₂²	semidirect
20. C₂×C₈×D₈₉	C₂²	semidirect
29. C₄⋊#10623494	C₂²	semidirect
30. #15667941	C₂²	semidirect
31. C₂²⋊#10623494	C₂²	semidirect
32. #15667924	C₂²	semidirect
33. #15667924	C₂²	semidirect
34. #15667922	C₂²	semidirect
35. #15667922	C₂²	semidirect
36. C₄×#10623494	C₂²	semidirect
37. C₂×M₁₆×C₈₉	C₂²	semidirect
39. C₄⋊C₈×C₈₉	C₂²	semidirect
40. C₂²⋊C₈×C₈₉	C₂²	semidirect
41. C₄×C₈×C₈₉	C₂²	semidirect
58. #13083557	D₄	semidirect
100. D₃₅₆	M₁₆	semidirect
111. C₈×C₈₉	C₂×D₄	semidirect
114. #10623494	C₂×D₄	semidirect
125. C₂×D₈₉	C₂×M₁₆	semidirect
128. C₄×C₈₉	C₂×M₁₆	semidirect
142. C₈₉	D₄×M₁₆	semidirect
146. #436	D₈₉	semidirect
161. C₂×M₁₆	C₂×D₈₉	semidirect
175. C₄⋊C₈	C₂×D₈₉	semidirect
179. C₂²⋊C₈	C₂×D₈₉	semidirect
183. C₄×C₈	C₂×D₈₉	semidirect
256. C₈	D₄×D₈₉	semidirect
274. C₄	M₁₆×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×D₈₉	2
C₄×D₄×D₈₉	2
#19151676	2
#19151672	2
#19151667	2
C₈×D₃₅₆	2
#19151663	2
#19151620	2
#19151560	2
C₄×#13083557	2
#436×C₈₉	2
D₄×M₁₆	89