Subgroups — #36707801

Order 22784.

646 conjugacy classes of subgroups.

The subgroup lattice has 326 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	#26022121	C₂	1	2	yes	maximal, complemented
4	#26022120	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₃₂×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022110	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022033	C₂	1	1	yes	maximal, characteristic
9	#26021970	C₂	1	2	yes	maximal
10	#26021847	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₃₂×C₈₉	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	#19151425	C₂²	1	2	yes	—
28	#19151424	C₂²	1	1	yes	characteristic
29	#19151403	C₂²	1	2	yes	—
30	#19151402	C₂²	1	2	yes	—
31	#19151401	C₂²	1	2	yes	—
32	#19151374	C₂²	1	1	yes	characteristic
33	#19151373	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	#366×C₈₉	C₂²	1	1	yes	characteristic
38	#358×C₈₉	C₂²	1	2	yes	—
39	C₄×C₁₆×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	1	yes	characteristic
97	#13083553	D₈	1	1	yes	characteristic
98	#13083551	—	4	2	no	—
99	#13083551	D₈	1	1	yes	characteristic
100	#13083551	D₈	1	1	yes	characteristic
101	C₄×Dic₈₉	—	4	2	no	—
102	C₄×Dic₈₉	—	4	2	no	—
103	C₄×Dic₈₉	—	4	2	no	—
104	C₄×Dic₈₉	—	4	2	no	—
105	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
106	#13083549	—	2	1	no	—
107	#13083549	C₂²×C₄	1	2	yes	—
108	C₂×Q₈×C₈₉	—	2	2	no	—
109	Q₁₆×C₈₉	—	2	2	no	—
110	Q₁₆×C₈₉	C₂²×C₄	1	4	yes	—
111	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₈₉	—	2	2	no	—
114	C₄⋊C₄×C₈₉	—	4	2	no	—
115	C₄²×C₈₉	—	4	2	no	—
116	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₈₉	—	2	1	no	—
118	C₁₆×C₈₉	C₂²×C₄	1	2	yes	—
119	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₈₉	—	2	4	no	—
121	Q₈×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₈₉	—	4	2	no	—
128	C₂×Dic₈₉	—	4	2	no	—
129	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
132	C₄×D₈₉	—	4	8	no	—
133	C₄×D₈₉	C₄×D₄	1	2	yes	—
134	C₄×D₈₉	#135	1	2	yes	—
135	Dic₁₇₈	—	2	4	no	—
136	Dic₁₇₈	—	4	2	no	—
137	Dic₁₇₈	—	4	8	no	—
138	C₈×C₈₉	—	2	1	no	—
139	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
140	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
141	#10623494	—	2	1	no	—
142	#10623494	C₄×D₄	1	2	yes	—
143	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
144	C₂×D₈₉	—	2	1	no	—
145	C₂×D₈₉	—	2	1	no	—
146	C₂×D₈₉	C₄×D₈	1	2	yes	—
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	—	4	4	no	—
150	C₄×C₈₉	—	8	2	no	—
151	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
153	Dic₈₉	—	2	1	no	—
154	Dic₈₉	—	2	1	no	—
155	Dic₈₉	—	4	4	no	—
156	Dic₈₉	—	8	2	no	—
157	Dic₈₉	C₄×D₈	1	2	yes	—
158	#33458	—	89	1	no	maximal, Sylow
159	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
160	C₂×C₈₉	#3417	1	1	yes	characteristic
161	C₂×C₈₉	#3417	1	1	yes	characteristic
162	D₈₉	—	2	2	no	—
163	C₂²×Q₃₂	—	89	1	no	—
164	#2950	—	89	2	no	—
165	#2176	—	89	2	no	—
166	#2176	—	89	2	no	—
167	C₄×Q₃₂	—	89	1	no	—
168	C₄×Q₃₂	—	89	1	no	—
169	C₄×Q₃₂	—	89	1	no	—
170	C₄×Q₃₂	D₈₉	1	1	yes	characteristic, complemented
171	#2158	—	89	1	no	—
172	#2141	—	89	2	no	—
173	#2108	—	89	1	no	—
174	C₈₉	#33458	1	1	yes	characteristic, complemented, Sylow
175	C₂²×Q₁₆	—	89	2	no	—
176	#519	—	178	2	no	—
177	C₂×Q₃₂	—	89	2	no	—
178	C₂×Q₃₂	—	89	8	no	—
179	C₂×Q₃₂	—	89	1	no	—
180	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic
181	C₂×M₃₂	—	89	1	no	—
182	C₂²×C₁₆	—	89	1	no	—
183	#441	—	178	2	no	—
184	#441	—	178	2	no	—
185	C₄×Q₁₆	—	89	2	no	—
186	C₄×Q₁₆	—	89	2	no	—
187	C₄×Q₁₆	—	89	2	no	—
188	C₄×Q₁₆	C₂×D₈₉	1	2	yes	—
189	#427	—	89	2	no	—
190	#419	—	178	2	no	—
191	#405	—	89	1	no	—
192	#367	—	89	2	no	—
193	#366	—	89	1	no	—
194	#366	C₂×D₈₉	1	1	yes	characteristic
195	#358	—	89	2	no	—
196	#358	—	89	2	no	—
197	#358	—	89	2	no	—
198	#358	C₂×D₈₉	1	2	yes	—
199	#346	—	89	1	no	—
200	#346	—	89	1	no	—
201	C₄×C₁₆	—	89	1	no	—
202	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
203	C₂²×Q₈	—	178	2	no	—
204	C₂×Q₁₆	—	89	8	no	—
205	C₂×Q₁₆	—	89	2	no	—
206	C₂×Q₁₆	—	178	4	no	—
207	C₂×Q₁₆	—	178	2	no	—
208	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
209	C₂×M₁₆	—	89	1	no	—
210	C₂²×C₈	—	89	1	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	356	2	no	—
213	C₄×Q₈	—	356	2	no	—
214	C₄×Q₈	—	178	2	no	—
215	C₄×Q₈	—	178	2	no	—
216	C₄×Q₈	—	178	2	no	—
217	#117	—	89	1	no	—
218	#117	—	356	2	no	—
219	#117	—	356	2	no	—
220	#117	—	178	2	no	—
221	Q₃₂	—	89	8	no	—
222	Q₃₂	—	89	4	no	—
223	Q₃₂	C₄×D₈₉	1	4	yes	—
224	M₃₂	—	178	2	no	—
225	C₂×C₁₆	—	89	1	no	—
226	C₂×C₁₆	—	89	4	no	—
227	C₂×C₁₆	—	89	1	no	—
228	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
229	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
230	#107	—	89	2	no	—
231	#107	C₂²×D₈₉	1	2	yes	—
232	#106	—	178	2	no	—
233	#103	—	2	2	no	—
234	#103	—	178	2	no	—
235	#103	—	178	2	no	—
236	#103	—	178	2	no	—
237	#97	—	89	1	no	—
238	#97	—	89	1	no	—
239	C₄×C₈	—	89	1	no	—
240	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	356	2	no	—
243	C₂×Q₈	—	356	4	no	—
244	C₂×Q₈	—	178	8	no	—
245	C₂×Q₈	—	178	2	no	—
246	C₂²×C₄	—	89	1	no	—
247	C₂²×C₄	—	356	2	no	—
248	Q₁₆	—	2	2	no	—
249	Q₁₆	—	89	4	no	—
250	Q₁₆	—	178	8	no	—
251	Q₁₆	—	178	2	no	—
252	Q₁₆	C₂×C₄×D₈₉	1	4	yes	—
253	M₁₆	—	178	2	no	—
254	C₂×C₈	—	89	1	no	—
255	C₂×C₈	—	89	2	no	—
256	C₂×C₈	—	89	2	no	—
257	C₂×C₈	—	89	1	no	—
258	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
259	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	89	1	no	—
263	C₄⋊C₄	—	89	1	no	—
264	C₄⋊C₄	—	356	2	no	—
265	C₄⋊C₄	—	356	2	no	—
266	C₄⋊C₄	—	356	2	no	—
267	C₄⋊C₄	—	178	2	no	—
268	C₂²⋊C₄	—	89	1	no	—
269	C₂²⋊C₄	—	89	1	no	—
270	C₂²⋊C₄	—	356	2	no	—
271	C₄²	—	4	2	no	—
272	C₄²	—	89	1	no	—
273	C₄²	—	356	2	no	—
274	C₄²	—	356	2	no	—
275	C₄²	—	356	2	no	—
276	C₄²	—	356	2	no	—
277	C₄²	D₄×D₈₉	1	1	yes	characteristic
278	C₁₆	—	2	1	no	—
279	C₁₆	—	89	2	no	—
280	C₁₆	—	178	1	no	—
281	C₁₆	C₂×C₄×D₈₉	1	2	yes	—
282	C₂³	—	89	1	no	—
283	Q₈	—	2	4	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	356	8	no	—
286	Q₈	—	356	2	no	—
287	Q₈	—	178	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	89	1	no	—
291	C₂×C₄	—	89	2	no	—
292	C₂×C₄	—	89	2	no	—
293	C₂×C₄	—	89	1	no	—
294	C₂×C₄	—	89	1	no	—
295	C₂×C₄	—	356	8	no	—
296	C₂×C₄	—	356	2	no	—
297	C₂×C₄	—	356	2	no	—
298	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
299	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	89	2	no	—
303	C₈	—	178	1	no	—
304	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
305	C₈	#41×D₈₉	1	1	yes	characteristic
306	C₂²	—	89	2	no	—
307	C₂²	—	178	1	no	—
308	C₂²	—	178	1	no	—
309	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	4	no	—
313	C₄	—	8	2	no	—
314	C₄	—	89	2	no	—
315	C₄	—	356	4	no	—
316	C₄	—	178	1	no	—
317	C₄	—	178	1	no	—
318	C₄	—	712	2	no	—
319	C₄	C₄×D₄×D₈₉	1	1	yes	characteristic
320	C₄	#135×D₈₉	1	1	yes	characteristic
321	C₂	—	178	2	no	—
322	C₂	C₄×D₈×D₈₉	1	1	yes	characteristic
323	C₂	#26033401	1	1	yes	characteristic
324	C₂	#26033402	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₄×Q₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂²×D₈₉	C₂×D₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₄×Q₃₂	D₈₉
hypocenter	C₈₉	#33458
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
3. #26022121	C₂	semidirect
4. #26022120	C₂	semidirect
5. C₄×Dic₇₁₂	C₂	semidirect
6. Q₃₂×Dic₈₉	C₂	semidirect
7. #26022110	C₂	semidirect
11. C₄×Q₃₂×C₈₉	C₂	semidirect
16. Q₃₂×D₈₉	C₄	semidirect
170. C₄×Q₃₂	D₈₉	semidirect
174. 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
#26022121	2
#26022120	2
C₄×Dic₇₁₂	2
Q₃₂×Dic₈₉	2
#26022110	2
#26022033	2
#26021970	2
#26021847	2
C₄×Q₃₂×C₈₉	2
#33458	89