Subgroups — #36707803

Order 22784.

646 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	C₂×Q₃₂×D₈₉	C₂	1	1	yes	maximal, characteristic
2	#26029115	C₂	1	2	yes	maximal
3	#26022124	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022123	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022122	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#15667958	C₂	1	2	yes	maximal, complemented
7	#26022108	C₂	1	2	yes	maximal, complemented
8	#26022035	C₂	1	1	yes	maximal, characteristic
9	#26021970	C₂	1	2	yes	maximal
10	#26021850	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2176	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₈₉	C₂²	1	2	yes	—
13	#19152074	—	2	2	no	—
14	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
15	C₂×#15667958	C₂²	1	2	yes	—
16	Q₃₂×D₈₉	C₄	1	8	yes	complemented
17	C₂×#15667951	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
19	#19151703	—	2	2	no	—
20	#19151702	—	2	2	no	—
21	C₄×Dic₃₅₆	C₂²	1	2	yes	—
22	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
23	#19151692	C₂²	1	2	yes	—
24	#19151635	C₂²	1	2	yes	—
25	#19151605	—	2	2	no	—
26	#19151552	C₂²	1	1	yes	characteristic
27	#19151426	C₂²	1	1	yes	characteristic
28	#19151423	C₂²	1	2	yes	—
29	#19151403	C₂²	1	2	yes	—
30	#19151402	C₂²	1	2	yes	—
31	#19151401	C₂²	1	2	yes	—
32	#19151373	C₂²	1	1	yes	characteristic
33	C₄×#13083549	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
37	#367×C₈₉	C₂²	1	1	yes	characteristic
38	#358×C₈₉	C₂²	1	2	yes	—
39	#346×C₈₉	C₂²	1	1	yes	characteristic
40	C₂×Q₈×D₈₉	—	2	2	no	—
41	C₂×Dic₃₅₆	C₂³	1	2	yes	—
42	C₂×#13083564	—	2	2	no	—
43	Q₁₆×D₈₉	—	2	4	no	—
44	Q₁₆×D₈₉	C₂×C₄	1	8	yes	—
45	C₂×#13083557	D₄	1	1	yes	characteristic
46	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
47	Q₈×Dic₈₉	—	2	2	no	—
48	C₄×Dic₁₇₈	—	2	2	no	—
49	C₄×Dic₁₇₈	—	4	2	no	—
50	#15667977	—	2	2	no	—
51	#15667977	—	4	2	no	—
52	#15667970	—	2	2	no	—
53	#15667970	—	4	2	no	—
54	#15667969	—	4	2	no	—
55	#15667969	D₄	1	1	yes	characteristic
56	Dic₇₁₂	C₂×C₄	1	4	yes	—
57	#15667958	C₂×C₄	1	8	yes	—
58	#15667951	—	2	2	no	—
59	C₂×#13083549	C₂³	1	1	yes	characteristic
60	C₂×#13083549	C₂³	1	1	yes	characteristic
61	C₁₆×D₈₉	C₂×C₄	1	4	yes	—
62	#15667946	C₂³	1	2	yes	—
63	#15667943	—	2	2	no	—
64	#15667937	—	2	2	no	—
65	#15667936	—	2	2	no	—
66	#15667935	—	2	2	no	—
67	#15667923	D₄	1	1	yes	characteristic
68	#15667922	D₄	1	1	yes	characteristic
69	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
71	C₄×Q₈×C₈₉	—	2	2	no	—
72	Q₃₂×C₈₉	C₂×C₄	1	4	yes	—
73	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
75	#107×C₈₉	C₂³	1	2	yes	—
76	#103×C₈₉	—	2	2	no	—
77	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₂×Dic₁₇₈	—	2	2	no	—
79	C₂×Dic₁₇₈	—	4	2	no	—
80	Q₈×D₈₉	—	2	8	no	—
81	Q₈×D₈₉	—	4	4	no	—
82	C₂×C₄×D₈₉	—	4	2	no	—
83	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
84	Dic₃₅₆	—	2	2	no	—
85	Dic₃₅₆	C₂²×C₄	1	4	yes	—
86	#13083564	—	2	8	no	—
87	#13083557	—	2	2	no	—
88	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
89	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
90	C₈×D₈₉	C₂²×C₄	1	2	yes	—
91	C₈×D₈₉	#41	1	2	yes	—
92	C₄⋊Dic₈₉	—	2	2	no	—
93	C₄⋊Dic₈₉	—	4	2	no	—
94	#13083553	—	4	2	no	—
95	#13083553	—	4	2	no	—
96	#13083553	D₈	1	2	yes	—
97	#13083551	—	4	2	no	—
98	#13083551	D₈	1	2	yes	—
99	C₄×Dic₈₉	—	4	2	no	—
100	C₄×Dic₈₉	—	4	2	no	—
101	C₄×Dic₈₉	—	4	2	no	—
102	C₄×Dic₈₉	—	4	2	no	—
103	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
104	#13083549	—	2	1	no	—
105	#13083549	C₂²×C₄	1	2	yes	—
106	C₂×Q₈×C₈₉	—	2	2	no	—
107	Q₁₆×C₈₉	—	2	2	no	—
108	Q₁₆×C₈₉	C₂²×C₄	1	4	yes	—
109	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₈₉	—	2	2	no	—
112	C₄⋊C₄×C₈₉	—	4	2	no	—
113	C₄²×C₈₉	—	4	2	no	—
114	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₁₆×C₈₉	—	2	1	no	—
116	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
117	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
118	Q₈×C₈₉	—	2	4	no	—
119	Q₈×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	—	4	2	no	—
121	C₂×C₄×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₈₉	—	4	2	no	—
125	C₂×Dic₈₉	—	4	2	no	—
126	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
127	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
128	C₄×D₈₉	—	4	8	no	—
129	C₄×D₈₉	C₄×D₄	1	2	yes	—
130	C₄×D₈₉	#135	1	2	yes	—
131	Dic₁₇₈	—	2	4	no	—
132	Dic₁₇₈	—	4	2	no	—
133	Dic₁₇₈	—	4	8	no	—
134	C₈×C₈₉	—	2	1	no	—
135	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
136	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
137	#10623494	—	2	1	no	—
138	#10623494	C₄×D₄	1	2	yes	—
139	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
140	C₂×D₈₉	—	2	2	no	—
141	C₂×D₈₉	C₄×D₈	1	2	yes	—
142	C₄×C₈₉	—	2	2	no	—
143	C₄×C₈₉	—	4	4	no	—
144	C₄×C₈₉	—	8	2	no	—
145	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
146	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
147	Dic₈₉	—	2	2	no	—
148	Dic₈₉	—	4	4	no	—
149	Dic₈₉	—	8	2	no	—
150	Dic₈₉	C₄×D₈	1	2	yes	—
151	#33458	—	89	1	no	maximal, Sylow
152	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
153	C₂×C₈₉	#3417	1	2	yes	—
154	D₈₉	—	2	2	no	—
155	C₂²×Q₃₂	—	89	1	no	—
156	#2950	—	89	2	no	—
157	#2176	—	89	1	no	—
158	#2176	—	89	1	no	—
159	#2176	—	89	1	no	—
160	#2176	D₈₉	1	1	yes	characteristic, complemented
161	C₄×Q₃₂	—	89	2	no	—
162	C₄×Q₃₂	—	89	2	no	—
163	#2158	—	89	1	no	—
164	#2141	—	89	2	no	—
165	#2108	—	89	1	no	—
166	C₈₉	#33458	1	1	yes	characteristic, complemented, Sylow
167	C₂²×Q₁₆	—	89	2	no	—
168	#519	—	178	2	no	—
169	C₂×Q₃₂	—	89	2	no	—
170	C₂×Q₃₂	—	89	8	no	—
171	C₂×Q₃₂	—	89	1	no	—
172	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic
173	C₂×M₃₂	—	89	1	no	—
174	C₂²×C₁₆	—	89	1	no	—
175	#441	—	178	2	no	—
176	#441	—	178	2	no	—
177	C₄×Q₁₆	—	89	2	no	—
178	C₄×Q₁₆	—	89	2	no	—
179	C₄×Q₁₆	—	89	2	no	—
180	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
181	#427	—	89	2	no	—
182	#419	—	178	2	no	—
183	#405	—	89	1	no	—
184	#367	—	89	1	no	—
185	#367	C₂×D₈₉	1	1	yes	characteristic
186	#366	—	89	2	no	—
187	#358	—	89	2	no	—
188	#358	—	89	2	no	—
189	#358	—	89	2	no	—
190	#358	C₂×D₈₉	1	2	yes	—
191	#346	—	89	1	no	—
192	#346	C₂×D₈₉	1	1	yes	characteristic
193	C₄×C₁₆	—	89	1	no	—
194	C₄×C₁₆	—	89	1	no	—
195	C₂²×Q₈	—	178	2	no	—
196	C₂×Q₁₆	—	89	8	no	—
197	C₂×Q₁₆	—	89	2	no	—
198	C₂×Q₁₆	—	178	4	no	—
199	C₂×Q₁₆	—	178	2	no	—
200	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
201	C₂×M₁₆	—	89	1	no	—
202	C₂²×C₈	—	89	1	no	—
203	C₄×Q₈	—	2	2	no	—
204	C₄×Q₈	—	356	2	no	—
205	C₄×Q₈	—	356	2	no	—
206	C₄×Q₈	—	178	2	no	—
207	C₄×Q₈	—	178	2	no	—
208	C₄×Q₈	—	178	2	no	—
209	#117	—	89	1	no	—
210	#117	—	356	2	no	—
211	#117	—	356	2	no	—
212	#117	—	178	2	no	—
213	Q₃₂	—	89	8	no	—
214	Q₃₂	—	89	4	no	—
215	Q₃₂	C₄×D₈₉	1	4	yes	—
216	M₃₂	—	178	2	no	—
217	C₂×C₁₆	—	89	1	no	—
218	C₂×C₁₆	—	89	4	no	—
219	C₂×C₁₆	—	89	1	no	—
220	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
221	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
222	#107	—	89	2	no	—
223	#107	C₂²×D₈₉	1	2	yes	—
224	#106	—	178	2	no	—
225	#103	—	2	2	no	—
226	#103	—	178	2	no	—
227	#103	—	178	2	no	—
228	#103	—	178	2	no	—
229	#97	—	89	1	no	—
230	#97	—	89	1	no	—
231	C₄×C₈	—	89	1	no	—
232	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	—	356	2	no	—
235	C₂×Q₈	—	356	4	no	—
236	C₂×Q₈	—	178	8	no	—
237	C₂×Q₈	—	178	2	no	—
238	C₂²×C₄	—	89	1	no	—
239	C₂²×C₄	—	356	2	no	—
240	Q₁₆	—	2	2	no	—
241	Q₁₆	—	89	4	no	—
242	Q₁₆	—	178	8	no	—
243	Q₁₆	—	178	2	no	—
244	Q₁₆	C₂×C₄×D₈₉	1	4	yes	—
245	M₁₆	—	178	2	no	—
246	C₂×C₈	—	89	1	no	—
247	C₂×C₈	—	89	2	no	—
248	C₂×C₈	—	89	2	no	—
249	C₂×C₈	—	89	1	no	—
250	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
251	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	4	2	no	—
254	C₄⋊C₄	—	89	2	no	—
255	C₄⋊C₄	—	356	2	no	—
256	C₄⋊C₄	—	356	2	no	—
257	C₄⋊C₄	—	356	2	no	—
258	C₄⋊C₄	—	178	2	no	—
259	C₂²⋊C₄	—	89	2	no	—
260	C₂²⋊C₄	—	356	2	no	—
261	C₄²	—	4	2	no	—
262	C₄²	—	89	1	no	—
263	C₄²	—	356	2	no	—
264	C₄²	—	356	2	no	—
265	C₄²	—	356	2	no	—
266	C₄²	—	356	2	no	—
267	C₄²	D₄×D₈₉	1	1	yes	characteristic
268	C₁₆	—	2	1	no	—
269	C₁₆	—	89	2	no	—
270	C₁₆	—	178	1	no	—
271	C₁₆	C₂×C₄×D₈₉	1	2	yes	—
272	C₂³	—	89	1	no	—
273	Q₈	—	2	4	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	356	8	no	—
276	Q₈	—	356	2	no	—
277	Q₈	—	178	4	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	89	1	no	—
281	C₂×C₄	—	89	2	no	—
282	C₂×C₄	—	89	2	no	—
283	C₂×C₄	—	89	2	no	—
284	C₂×C₄	—	356	8	no	—
285	C₂×C₄	—	356	2	no	—
286	C₂×C₄	—	356	2	no	—
287	C₂×C₄	D₈×D₈₉	1	2	yes	—
288	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	89	2	no	—
291	C₈	—	178	1	no	—
292	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
293	C₈	#41×D₈₉	1	1	yes	characteristic
294	C₂²	—	89	2	no	—
295	C₂²	—	178	2	no	—
296	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
297	C₄	—	2	2	no	—
298	C₄	—	4	4	no	—
299	C₄	—	8	2	no	—
300	C₄	—	89	2	no	—
301	C₄	—	356	4	no	—
302	C₄	—	178	2	no	—
303	C₄	—	712	2	no	—
304	C₄	C₄×D₄×D₈₉	1	1	yes	characteristic
305	C₄	#135×D₈₉	1	1	yes	characteristic
306	C₂	—	178	2	no	—
307	C₂	C₄×D₈×D₈₉	1	1	yes	characteristic
308	C₂	#26033404	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2176	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂²×D₈₉	C₂×D₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2176	D₈₉
hypocenter	C₈₉	#33458
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

Complements

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

N	H ≅ G/N	type
3. #26022124	C₂	semidirect
4. #26022123	C₂	semidirect
5. #26022122	C₂	semidirect
6. C₄×#15667958	C₂	semidirect
7. #26022108	C₂	semidirect
11. C₈₉×#2176	C₂	semidirect
16. Q₃₂×D₈₉	C₄	semidirect
160. #2176	D₈₉	semidirect
166. C₈₉	#33458	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×D₈₉	2
#26029115	2
#26022124	2
#26022123	2
#26022122	2
C₄×#15667958	2
#26022108	2
#26022035	2
#26021970	2
#26021850	2
C₈₉×#2176	2
#33458	89