Subgroups — #36707564

Order 22784.

524 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	#26028668	C₂	1	1	yes	maximal, characteristic
2	#26022066	C₂	1	4	yes	maximal, complemented
3	#26022062	C₂	1	4	yes	maximal, complemented
4	#26022017	C₂	1	2	yes	maximal, complemented
5	C₂×#19151379	C₂	1	2	yes	maximal
6	#26021857	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2110	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#13083557	C₂²	1	1	yes	characteristic
9	C₂×M₁₆×D₈₉	C₄	1	1	yes	characteristic
10	#117×D₈₉	C₄	1	1	yes	characteristic
11	C₂×#15667952	C₂²	1	2	yes	—
12	#19151556	C₄	1	1	yes	characteristic
13	#19151555	C₂²	1	2	yes	—
14	#19151554	C₄	1	1	yes	characteristic
15	#19151553	C₂²	1	1	yes	characteristic
16	#19151552	C₄	1	2	yes	—
17	#19151550	C₂²	1	2	yes	—
18	C₄×#13083557	C₄	1	2	yes	—
19	#19151418	C₂²	1	8	yes	complemented
20	#19151379	C₂²	1	4	yes	—
21	#19151379	C₂²	1	4	yes	—
22	#19151376	C₂²	1	4	yes	complemented
23	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
24	#405×C₈₉	C₂²	1	1	yes	characteristic
25	#347×C₈₉	C₂²	1	4	yes	complemented
26	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
27	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
28	C₂×#13083557	—	2	2	no	—
29	C₂×#13083557	—	2	2	no	—
30	C₂×#13083557	C₂×C₄	1	1	yes	characteristic
31	C₂×#13083557	C₂×C₄	1	2	yes	—
32	C₂×#13083557	C₂×C₄	1	1	yes	characteristic
33	C₂×#13083557	C₂³	1	1	yes	characteristic
34	C₂×#13083557	C₂³	1	1	yes	characteristic
35	M₁₆×D₈₉	—	2	4	no	—
36	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
38	#15667971	C₂×C₄	1	1	yes	characteristic
39	#15667970	—	2	1	no	—
40	#15667969	C₂×C₄	1	2	yes	—
41	#15667968	—	2	1	no	—
42	C₄⋊C₄×D₈₉	—	2	1	no	—
43	C₂²⋊C₄×D₈₉	—	2	1	no	—
44	C₄²×D₈₉	C₂×C₄	1	2	yes	—
45	#15667952	—	2	2	no	—
46	#15667952	C₂³	1	4	yes	—
47	C₂×#13083549	—	2	2	no	—
48	#15667923	C₂×C₄	1	2	yes	—
49	#15667922	C₂×C₄	1	2	yes	—
50	#15667922	C₂×C₄	1	2	yes	—
51	C₄×#10623494	C₂×C₄	1	2	yes	—
52	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
53	C₈×Dic₈₉	C₂³	1	2	yes	—
54	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
57	M₃₂×C₈₉	—	2	2	no	—
58	M₃₂×C₈₉	C₂³	1	4	yes	—
59	C₂×C₁₆×C₈₉	—	2	2	no	—
60	#97×C₈₉	C₂³	1	2	yes	—
61	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
62	C₂³×D₈₉	M₁₆	1	1	yes	characteristic
63	C₂²×Dic₈₉	M₁₆	1	1	yes	characteristic
64	C₂×C₄×D₈₉	—	2	2	no	—
65	C₂×C₄×D₈₉	—	2	2	no	—
66	C₂×C₄×D₈₉	—	2	2	no	—
67	C₂×C₄×D₈₉	M₁₆	1	1	yes	characteristic
68	C₂×C₄×D₈₉	M₁₆	1	1	yes	characteristic
69	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
71	#13083558	M₁₆	1	4	yes	—
72	#13083557	—	2	1	no	—
73	#13083557	—	2	1	no	—
74	#13083557	—	2	1	no	—
75	#13083557	—	2	1	no	—
76	#13083557	—	2	2	no	—
77	#13083557	—	2	2	no	—
78	#13083557	—	2	4	no	—
79	C₂×#10623494	—	2	2	no	—
80	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
81	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
82	C₂×#10623494	C₂²×C₄	1	2	yes	—
83	C₈×D₈₉	—	2	4	no	—
84	C₄⋊Dic₈₉	—	2	1	no	—
85	#13083553	—	2	2	no	—
86	C₂²⋊Dic₈₉	—	2	1	no	—
87	#13083551	—	2	2	no	—
88	C₄×Dic₈₉	—	2	2	no	—
89	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
90	#13083549	—	2	4	no	—
91	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	M₁₆×C₈₉	—	2	2	no	—
93	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	#41	1	2	yes	—
96	C₂×C₈×C₈₉	#41	1	2	yes	—
97	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
98	C₄⋊C₄×C₈₉	—	2	1	no	—
99	C₂²⋊C₄×C₈₉	—	2	1	no	—
100	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
101	C₁₆×C₈₉	—	2	4	no	—
102	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
103	C₂²×D₈₉	—	2	2	no	—
104	C₂²×D₈₉	—	8	1	no	—
105	C₂²×D₈₉	C₂×M₁₆	1	1	yes	characteristic
106	C₂²×D₈₉	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₈₉	—	2	2	no	—
108	C₂×C₄×C₈₉	#117	1	2	yes	—
109	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
112	C₂×Dic₈₉	—	2	2	no	—
113	C₂×Dic₈₉	—	2	2	no	—
114	C₂×Dic₈₉	C₂×M₁₆	1	1	yes	characteristic
115	C₂×Dic₈₉	C₂×M₁₆	1	1	yes	characteristic
116	C₄×D₈₉	—	2	1	no	—
117	C₄×D₈₉	—	2	1	no	—
118	C₄×D₈₉	—	2	1	no	—
119	C₄×D₈₉	—	2	1	no	—
120	C₄×D₈₉	—	2	2	no	—
121	C₄×D₈₉	—	2	2	no	—
122	C₄×D₈₉	—	8	2	no	—
123	C₈×C₈₉	—	2	2	no	—
124	C₈×C₈₉	#117	1	1	yes	characteristic
125	C₈×C₈₉	#117	1	1	yes	characteristic
126	C₈×C₈₉	C₂×#41	1	2	yes	—
127	#10623494	—	2	1	no	—
128	#10623494	—	2	1	no	—
129	#10623494	C₂×M₁₆	1	4	yes	—
130	C₂²×C₈₉	—	4	1	no	—
131	C₂²×C₈₉	#432	1	2	yes	—
132	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
133	C₂×D₈₉	—	2	1	no	—
134	C₂×D₈₉	—	2	1	no	—
135	C₂×D₈₉	—	4	1	no	—
136	C₂×D₈₉	—	4	1	no	—
137	C₂×D₈₉	—	8	2	no	—
138	C₄×C₈₉	—	4	2	no	—
139	C₄×C₈₉	#432	1	2	yes	—
140	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
141	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
142	Dic₈₉	—	2	1	no	—
143	Dic₈₉	—	2	1	no	—
144	Dic₈₉	—	4	2	no	—
145	#33427	—	89	1	no	maximal, Sylow
146	C₂×C₈₉	—	2	1	no	—
147	C₂×C₈₉	—	2	2	no	—
148	C₂×C₈₉	C₂×#432	1	1	yes	characteristic
149	D₈₉	—	4	1	no	—
150	D₈₉	—	4	1	no	—
151	#2873	—	89	1	no	—
152	#2166	—	89	4	no	—
153	#2165	—	89	4	no	—
154	#2153	—	89	2	no	—
155	C₂×#349	—	89	2	no	—
156	#2110	—	89	1	no	—
157	#2110	D₈₉	1	1	yes	characteristic, complemented
158	C₈₉	#33427	1	1	yes	characteristic, complemented, Sylow
159	C₂²×M₁₆	—	89	1	no	—
160	C₂²×M₁₆	—	89	1	no	—
161	C₂×#117	—	89	1	no	—
162	C₂×M₃₂	—	89	2	no	—
163	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
164	#405	—	89	1	no	—
165	#405	—	89	1	no	—
166	#405	—	89	1	no	—
167	#405	—	89	2	no	—
168	#405	—	89	2	no	—
169	#405	C₂×D₈₉	1	1	yes	characteristic
170	C₄×M₁₆	—	89	2	no	—
171	C₄×M₁₆	—	89	2	no	—
172	#364	—	89	8	no	—
173	#349	—	89	4	no	—
174	#349	—	89	4	no	—
175	#347	—	89	4	no	—
176	#347	C₂×D₈₉	1	4	yes	complemented
177	C₂³×C₄	—	89	1	no	—
178	C₂×M₁₆	—	89	1	no	—
179	C₂×M₁₆	—	89	2	no	—
180	C₂×M₁₆	—	89	1	no	—
181	C₂×M₁₆	—	89	1	no	—
182	C₂×M₁₆	—	89	1	no	—
183	C₂×M₁₆	—	89	1	no	—
184	C₂×M₁₆	—	178	4	no	—
185	C₂×M₁₆	—	178	2	no	—
186	C₂×M₁₆	—	178	2	no	—
187	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
188	C₂²×C₈	—	89	1	no	—
189	C₂²×C₈	—	89	2	no	—
190	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
191	#117	—	89	1	no	—
192	#117	—	89	2	no	—
193	#117	—	178	1	no	—
194	#117	—	178	1	no	—
195	#117	C₄×D₈₉	1	1	yes	characteristic
196	C₂×C₄⋊C₄	—	178	1	no	—
197	C₂×C₂²⋊C₄	—	178	1	no	—
198	C₂×C₄²	—	89	2	no	—
199	M₃₂	—	2	2	no	—
200	M₃₂	—	89	4	no	—
201	M₃₂	—	178	2	no	—
202	M₃₂	C₂²×D₈₉	1	4	yes	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	178	2	no	—
205	#97	—	89	2	no	—
206	#97	—	89	2	no	—
207	#97	—	89	2	no	—
208	#97	C₂²×D₈₉	1	2	yes	—
209	C₄×C₈	—	89	2	no	—
210	C₄×C₈	—	89	2	no	—
211	C₄×C₈	—	89	2	no	—
212	C₄×C₈	C₄×D₈₉	1	2	yes	—
213	C₂⁴	—	89	1	no	—
214	C₂²×C₄	—	89	1	no	—
215	C₂²×C₄	—	89	1	no	—
216	C₂²×C₄	—	89	1	no	—
217	C₂²×C₄	—	89	1	no	—
218	C₂²×C₄	—	89	1	no	—
219	C₂²×C₄	—	178	2	no	—
220	C₂²×C₄	—	178	2	no	—
221	C₂²×C₄	—	178	2	no	—
222	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
223	M₁₆	—	2	2	no	—
224	M₁₆	—	89	4	no	—
225	M₁₆	—	178	4	no	—
226	M₁₆	—	178	1	no	—
227	M₁₆	—	178	1	no	—
228	M₁₆	—	178	1	no	—
229	M₁₆	—	178	1	no	—
230	M₁₆	—	178	2	no	—
231	M₁₆	—	178	2	no	—
232	C₂×C₈	—	89	2	no	—
233	C₂×C₈	—	89	1	no	—
234	C₂×C₈	—	89	1	no	—
235	C₂×C₈	—	178	4	no	—
236	C₂×C₈	—	178	2	no	—
237	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
238	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
239	C₂×C₈	#13083574	1	2	yes	—
240	C₂×C₈	#13083574	1	2	yes	—
241	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	178	2	no	—
244	C₄⋊C₄	—	178	1	no	—
245	C₂²⋊C₄	—	2	1	no	—
246	C₂²⋊C₄	—	178	2	no	—
247	C₂²⋊C₄	—	178	1	no	—
248	C₄²	—	89	2	no	—
249	C₄²	—	178	2	no	—
250	C₄²	C₂×C₄×D₈₉	1	2	yes	—
251	C₁₆	—	2	4	no	—
252	C₁₆	—	178	4	no	—
253	C₂³	—	89	1	no	—
254	C₂³	—	89	1	no	—
255	C₂³	—	178	2	no	—
256	C₂³	—	712	1	no	—
257	C₂³	M₁₆×D₈₉	1	1	yes	characteristic
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	89	1	no	—
260	C₂×C₄	—	89	1	no	—
261	C₂×C₄	—	178	2	no	—
262	C₂×C₄	—	178	2	no	—
263	C₂×C₄	—	178	2	no	—
264	C₂×C₄	—	178	2	no	—
265	C₂×C₄	—	178	1	no	—
266	C₂×C₄	—	178	1	no	—
267	C₂×C₄	—	178	1	no	—
268	C₂×C₄	—	178	1	no	—
269	C₂×C₄	—	712	2	no	—
270	C₂×C₄	#15667969	1	2	yes	—
271	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
272	C₂×C₄	M₁₆×D₈₉	1	2	yes	—
273	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
274	C₈	—	2	2	no	—
275	C₈	—	89	4	no	—
276	C₈	—	178	1	no	—
277	C₈	—	178	1	no	—
278	C₈	#15667969	1	1	yes	characteristic
279	C₈	#15667969	1	1	yes	characteristic
280	C₈	C₂×#13083574	1	2	yes	—
281	C₂²	—	4	1	no	—
282	C₂²	—	356	1	no	—
283	C₂²	—	356	1	no	—
284	C₂²	—	178	1	no	—
285	C₂²	—	178	1	no	—
286	C₂²	—	712	2	no	—
287	C₂²	#19151653	1	2	yes	—
288	C₂²	C₂×M₁₆×D₈₉	1	1	yes	characteristic
289	C₄	—	4	2	no	—
290	C₄	—	356	2	no	—
291	C₄	—	178	1	no	—
292	C₄	—	178	1	no	—
293	C₄	#19151653	1	2	yes	—
294	C₄	C₂×#15667969	1	1	yes	characteristic
295	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
296	C₂	—	2	1	no	—
297	C₂	—	2	2	no	—
298	C₂	—	356	1	no	—
299	C₂	—	356	1	no	—
300	C₂	C₂×#19151653	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#15667969
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2110	C₂
socle	C₂×C₈₉	C₂×#432
prime radical	C₂³×D₈₉	M₁₆
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2110	D₈₉
hypocenter	C₈₉	#33427
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ #405 ◁ #2110

Complements

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

N	H ≅ G/N	type
2. #26022066	C₂	semidirect
3. #26022062	C₂	semidirect
4. #26022017	C₂	semidirect
6. #26021857	C₂	semidirect
7. C₈₉×#2110	C₂	semidirect
19. #19151418	C₂²	semidirect
22. #19151376	C₂²	semidirect
25. #347×C₈₉	C₂²	semidirect
157. #2110	D₈₉	semidirect
158. C₈₉	#33427	semidirect
176. #347	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26028668	2
#26022066	2
#26022062	2
#26022017	2
C₂×#19151379	2
#26021857	2
C₈₉×#2110	2
#33427	89