Subgroups — #36551935

Order 22784.

908 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028670	C₂	1	2	yes	maximal
2	#26020007	C₂	1	8	yes	maximal, complemented
3	C₂×#19151356	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151355	C₂	1	1	yes	maximal, characteristic
5	C₂×#19151354	C₂	1	2	yes	maximal
6	C₈₉×#2873	C₂	1	1	yes	maximal, characteristic
7	C₂²×#13083558	C₂²	1	2	yes	—
8	C₂×#15667971	C₂²	1	2	yes	—
9	#19151556	C₄	1	8	yes	complemented
10	#19151553	C₂²	1	8	yes	—
11	M₁₆×Dic₈₉	C₄	1	8	yes	complemented
12	#19151356	C₂²	1	4	yes	—
13	#19151355	C₂²	1	4	yes	—
14	#19151354	C₂²	1	8	yes	—
15	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
18	#405×C₈₉	C₄	1	4	yes	complemented
19	#405×C₈₉	C₂²	1	4	yes	—
20	C₄×M₁₆×C₈₉	C₄	1	4	yes	complemented
21	C₂×#13083558	—	2	2	no	—
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₂×#13083558	C₂×C₄	1	4	yes	—
26	C₂×#13083558	C₂³	1	4	yes	—
27	C₂²×#10623494	C₂×C₄	1	4	yes	—
28	#15667971	—	2	2	no	—
29	#15667971	—	2	2	no	—
30	#15667971	C₂×C₄	1	4	yes	—
31	#15667971	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₈₉	—	2	2	no	—
33	C₂×C₂²⋊Dic₈₉	—	2	2	no	—
34	C₂×C₄×Dic₈₉	C₂×C₄	1	4	yes	—
35	#15667922	C₂×C₄	1	16	yes	complemented
36	C₈×Dic₈₉	C₂×C₄	1	16	yes	complemented
37	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₈₉	—	2	2	no	—
39	C₂×M₁₆×C₈₉	—	2	2	no	—
40	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
41	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
42	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
43	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
44	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
45	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
46	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
48	#117×C₈₉	—	2	1	no	—
49	#117×C₈₉	—	2	1	no	—
50	#117×C₈₉	C₂×C₄	1	2	yes	—
51	#117×C₈₉	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
53	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
54	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
55	#97×C₈₉	C₂×C₄	1	8	yes	complemented
56	C₄×C₈×C₈₉	C₂×C₄	1	8	yes	complemented
57	C₂²×Dic₈₉	—	2	4	no	—
58	#13083558	—	2	4	no	—
59	#13083558	—	2	4	no	—
60	#13083558	—	2	4	no	—
61	#13083558	—	2	4	no	—
62	C₂×#10623494	—	2	4	no	—
63	C₂×#10623494	—	2	4	no	—
64	C₂×#10623494	C₂²×C₄	1	8	yes	—
65	C₄⋊Dic₈₉	—	2	4	no	—
66	C₄⋊Dic₈₉	—	2	4	no	—
67	C₂²⋊Dic₈₉	—	2	4	no	—
68	C₂²⋊Dic₈₉	—	2	4	no	—
69	C₄×Dic₈₉	—	2	4	no	—
70	C₄×Dic₈₉	C₂²×C₄	1	8	yes	—
71	C₂⁴×C₈₉	#41	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	—	2	2	no	—
73	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	#41	1	2	yes	—
79	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	#41	1	2	yes	—
82	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
83	M₁₆×C₈₉	—	2	4	no	—
84	M₁₆×C₈₉	—	2	4	no	—
85	M₁₆×C₈₉	C₄²	1	4	yes	—
86	M₁₆×C₈₉	C₄²	1	8	yes	—
87	M₁₆×C₈₉	C₄²	1	4	yes	—
88	C₂×C₈×C₈₉	—	2	2	no	—
89	C₂×C₈×C₈₉	—	2	2	no	—
90	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₈₉	C₂²×C₄	1	8	yes	—
92	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
93	C₄⋊C₄×C₈₉	—	2	2	no	—
94	C₄⋊C₄×C₈₉	—	2	2	no	—
95	C₂²⋊C₄×C₈₉	—	2	2	no	—
96	C₂²⋊C₄×C₈₉	—	2	2	no	—
97	C₄²×C₈₉	—	2	2	no	—
98	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
99	C₂³×C₈₉	—	8	1	no	—
100	C₂³×C₈₉	#117	1	2	yes	—
101	C₂³×C₈₉	#117	1	2	yes	—
102	C₂³×C₈₉	#117	1	1	yes	characteristic
103	C₂³×C₈₉	#117	1	1	yes	characteristic
104	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₈₉	—	2	4	no	—
106	C₂×C₄×C₈₉	—	8	2	no	—
107	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#117	1	2	yes	—
111	C₂×C₄×C₈₉	#117	1	2	yes	—
112	C₂×C₄×C₈₉	#117	1	2	yes	—
113	C₂×C₄×C₈₉	#117	1	2	yes	—
114	C₂×C₄×C₈₉	#117	1	2	yes	—
115	C₂×C₄×C₈₉	#117	1	2	yes	—
116	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
118	C₂×C₄×C₈₉	#117	1	2	yes	—
119	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
120	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
122	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
125	C₂×Dic₈₉	—	2	8	no	—
126	C₂×Dic₈₉	—	8	4	no	—
127	C₈×C₈₉	—	2	4	no	—
128	C₈×C₈₉	C₂×C₄²	1	8	yes	—
129	#10623494	—	2	8	no	—
130	C₂²×C₈₉	—	4	2	no	—
131	C₂²×C₈₉	—	4	1	no	—
132	C₂²×C₈₉	—	4	2	no	—
133	C₂²×C₈₉	—	4	1	no	—
134	C₂²×C₈₉	—	4	1	no	—
135	C₂²×C₈₉	#376	1	1	yes	characteristic
136	C₂²×C₈₉	#376	1	2	yes	—
137	C₂²×C₈₉	#376	1	1	yes	characteristic
138	C₂²×C₈₉	C₂×#117	1	2	yes	—
139	C₂²×C₈₉	C₂×#117	1	1	yes	characteristic
140	C₄×C₈₉	—	4	4	no	—
141	C₄×C₈₉	#376	1	1	yes	characteristic
142	C₄×C₈₉	#376	1	2	yes	—
143	C₄×C₈₉	#376	1	1	yes	characteristic
144	C₄×C₈₉	C₂²×C₄²	1	1	yes	characteristic
145	C₄×C₈₉	C₂×#117	1	2	yes	—
146	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
147	Dic₈₉	—	4	8	no	—
148	#20370	—	89	1	no	maximal, Sylow
149	C₂×C₈₉	—	2	2	no	—
150	C₂×C₈₉	—	2	1	no	—
151	C₂×C₈₉	—	2	1	no	—
152	C₂×C₈₉	—	2	2	no	—
153	C₂×C₈₉	—	2	1	no	—
154	C₂×C₈₉	C₂×#376	1	1	yes	characteristic
155	#2873	—	89	2	no	—
156	#2873	D₈₉	1	1	yes	characteristic
157	#1747	—	89	8	no	—
158	C₂×#338	—	89	1	no	—
159	C₂×H(3,ℤ/4)	—	89	1	no	—
160	C₂×H(3,ℤ/4)	—	89	2	no	—
161	C₈₉	#20370	1	1	yes	characteristic, complemented, Sylow
162	C₂²×M₁₆	—	89	2	no	—
163	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
164	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
165	C₂×#117	—	89	2	no	—
166	C₂×#117	C₂×D₈₉	1	1	yes	characteristic
167	#405	—	89	8	no	—
168	#405	—	89	8	no	—
169	#405	Dic₈₉	1	4	yes	complemented
170	#405	C₂×D₈₉	1	4	yes	—
171	C₄×M₁₆	—	89	8	no	—
172	C₄×M₁₆	Dic₈₉	1	4	yes	complemented
173	#338	—	89	4	no	—
174	H(3,ℤ/4)	—	89	4	no	—
175	H(3,ℤ/4)	—	89	8	no	—
176	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	89	4	no	—
180	C₂×M₁₆	—	89	4	no	—
181	C₂×M₁₆	—	178	2	no	—
182	C₂×M₁₆	—	178	2	no	—
183	C₂×M₁₆	—	178	2	no	—
184	C₂×M₁₆	—	178	2	no	—
185	C₂×M₁₆	C₄×D₈₉	1	4	yes	—
186	C₂×M₁₆	C₄×D₈₉	1	4	yes	—
187	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
188	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
189	C₂×M₁₆	C₂×Dic₈₉	1	2	yes	—
190	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
191	C₂²×C₈	—	89	4	no	—
192	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
193	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
194	#117	—	2	1	no	—
195	#117	—	2	1	no	—
196	#117	—	89	4	no	—
197	#117	—	89	4	no	—
198	#117	—	178	2	no	—
199	#117	—	178	2	no	—
200	#117	C₂×Dic₈₉	1	2	yes	—
201	#117	C₂²×D₈₉	1	2	yes	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	—	178	2	no	—
204	C₂×C₂²⋊C₄	—	2	1	no	—
205	C₂×C₂²⋊C₄	—	178	2	no	—
206	C₂×C₄²	—	89	4	no	—
207	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
208	#97	—	89	16	no	—
209	#97	C₂×Dic₈₉	1	8	yes	complemented
210	C₄×C₈	—	89	16	no	—
211	C₄×C₈	C₂×Dic₈₉	1	8	yes	complemented
212	C₂⁴	#13083574	1	1	yes	characteristic
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	—	178	4	no	—
215	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
216	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
217	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
218	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
219	C₂²×C₄	#13083574	1	1	yes	characteristic
220	C₂²×C₄	#13083574	1	1	yes	characteristic
221	C₂²×C₄	#13083574	1	1	yes	characteristic
222	C₂²×C₄	#13083574	1	2	yes	—
223	C₂²×C₄	#13083574	1	2	yes	—
224	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	178	4	no	—
228	M₁₆	—	178	4	no	—
229	M₁₆	—	178	4	no	—
230	M₁₆	—	178	4	no	—
231	M₁₆	C₄×Dic₈₉	1	4	yes	—
232	M₁₆	C₄×Dic₈₉	1	8	yes	—
233	M₁₆	C₄×Dic₈₉	1	4	yes	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	89	8	no	—
237	C₂×C₈	—	178	4	no	—
238	C₂×C₈	—	178	4	no	—
239	C₂×C₈	C₂×C₄×D₈₉	1	8	yes	—
240	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
241	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	178	4	no	—
245	C₄⋊C₄	—	178	4	no	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	178	4	no	—
249	C₂²⋊C₄	—	178	4	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	89	8	no	—
252	C₄²	—	178	4	no	—
253	C₄²	C₂²×Dic₈₉	1	4	yes	—
254	C₂³	—	8	1	no	—
255	C₂³	#15667969	1	2	yes	—
256	C₂³	#15667969	1	2	yes	—
257	C₂³	#15667971	1	1	yes	characteristic
258	C₂³	#15667971	1	1	yes	characteristic
259	C₂³	C₂×#13083574	1	1	yes	characteristic
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	8	2	no	—
262	C₂×C₄	—	178	8	no	—
263	C₂×C₄	—	712	4	no	—
264	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
266	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
267	C₂×C₄	#15667969	1	2	yes	—
268	C₂×C₄	#15667969	1	2	yes	—
269	C₂×C₄	#15667969	1	2	yes	—
270	C₂×C₄	#15667969	1	2	yes	—
271	C₂×C₄	#15667969	1	2	yes	—
272	C₂×C₄	#15667969	1	2	yes	—
273	C₂×C₄	#15667971	1	1	yes	characteristic
274	C₂×C₄	#15667971	1	1	yes	characteristic
275	C₂×C₄	#15667971	1	1	yes	characteristic
276	C₂×C₄	#15667971	1	1	yes	characteristic
277	C₂×C₄	#15667971	1	2	yes	—
278	C₂×C₄	C₂²×C₄×D₈₉	1	2	yes	—
279	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
280	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
281	C₂×C₄	C₂×#13083574	1	2	yes	—
282	C₈	—	2	4	no	—
283	C₈	—	178	8	no	—
284	C₈	C₂×C₄×Dic₈₉	1	8	yes	—
285	C₂²	—	4	1	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	1	no	—
290	C₂²	#19151444	1	1	yes	characteristic
291	C₂²	#19151444	1	2	yes	—
292	C₂²	#19151444	1	1	yes	characteristic
293	C₂²	C₂×#15667969	1	2	yes	—
294	C₂²	C₂×#15667971	1	1	yes	characteristic
295	C₄	—	4	4	no	—
296	C₄	—	356	8	no	—
297	C₄	#19151444	1	1	yes	characteristic
298	C₄	#19151444	1	2	yes	—
299	C₄	#19151444	1	1	yes	characteristic
300	C₄	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
301	C₄	C₂×#15667969	1	2	yes	—
302	C₄	C₂×#15667971	1	1	yes	characteristic
303	C₂	—	2	1	no	—
304	C₂	—	2	2	no	—
305	C₂	—	2	2	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	1	no	—
308	C₂	C₂×#19151444	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₄×Dic₈₉
commutator	C₄×C₈₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#2873	C₂
socle	C₂×C₈₉	C₂×#376
prime radical	C₂⁴×C₈₉	#41
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	#2873	D₈₉
hypocenter	C₈₉	#20370
LCS	G ▷ C₄×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ #2873

Complements

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

N	H ≅ G/N	type
2. #26020007	C₂	semidirect
9. #19151556	C₄	semidirect
11. M₁₆×Dic₈₉	C₄	semidirect
18. #405×C₈₉	C₄	semidirect
20. C₄×M₁₆×C₈₉	C₄	semidirect
35. #15667922	C₂×C₄	semidirect
36. C₈×Dic₈₉	C₂×C₄	semidirect
55. #97×C₈₉	C₂×C₄	semidirect
56. C₄×C₈×C₈₉	C₂×C₄	semidirect
161. C₈₉	#20370	semidirect
169. #405	Dic₈₉	semidirect
172. C₄×M₁₆	Dic₈₉	semidirect
209. #97	C₂×Dic₈₉	semidirect
211. C₄×C₈	C₂×Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26028670	2
#26020007	2
C₂×#19151356	2
C₂×#19151355	2
C₂×#19151354	2
C₈₉×#2873	2
#20370	89