Subgroups — #36707973

Order 22784.

508 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029227	C₂	1	1	yes	maximal, characteristic
2	#26022158	C₂	1	2	yes	maximal, complemented
3	#26022082	C₂	1	2	yes	maximal, complemented
4	#26022079	C₂	1	2	yes	maximal, complemented
5	#26022075	C₂	1	2	yes	maximal, complemented
6	#26022057	C₂	1	2	yes	maximal, complemented
7	#26022055	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021847	C₂	1	2	yes	maximal, complemented
9	C₄×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#19152132	C₄	1	1	yes	characteristic
11	C₂×#15667951	C₂²	1	2	yes	—
12	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
13	#19151728	C₄	1	1	yes	characteristic
14	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
15	#19151677	C₄	1	1	yes	characteristic
16	#19151669	C₄	1	2	yes	—
17	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
18	C₈×C₂²⋊D₈₉	C₂²	1	2	yes	—
19	#19151657	C₄	1	2	yes	—
20	#19151655	C₄	1	1	yes	characteristic
21	#19151552	C₂²	1	2	yes	—
22	C₄⋊#13083549	C₂²	1	2	yes	—
23	#19151416	C₂²	1	2	yes	—
24	#19151416	C₂²	1	2	yes	—
25	C₂²⋊#13083549	C₂²	1	1	yes	characteristic
26	C₂²⋊#13083549	C₂²	1	1	yes	characteristic
27	#19151377	C₂²	1	2	yes	—
28	#19151377	C₂²	1	2	yes	—
29	#19151374	C₂²	1	2	yes	—
30	#19151373	C₂²	1	2	yes	—
31	C₁₆×Dic₈₉	C₂²	1	2	yes	—
32	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
33	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
34	#346×C₈₉	C₂²	1	2	yes	—
35	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×#13083557	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₈₉	C₂³	1	2	yes	—
39	#15668006	C₈	1	2	yes	—
40	#15668004	C₈	1	1	yes	characteristic
41	#15668000	C₈	1	1	yes	characteristic
42	#15667995	C₈	1	2	yes	—
43	#15667991	C₈	1	1	yes	characteristic
44	#15667986	C₈	1	1	yes	characteristic
45	C₄×Dic₁₇₈	C₂×C₄	1	1	yes	characteristic
46	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
47	C₄×C₂²⋊D₈₉	C₂×C₄	1	2	yes	—
48	#15667969	C₂×C₄	1	2	yes	—
49	#15667951	—	2	4	no	—
50	C₂×#13083549	C₂³	1	2	yes	—
51	C₂×#13083549	C₂³	1	2	yes	—
52	C₁₆×D₈₉	—	2	4	no	—
53	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
54	C₄⋊#10623494	C₂×C₄	1	1	yes	characteristic
55	#15667941	C₂×C₄	1	2	yes	—
56	#15667941	C₂×C₄	1	2	yes	—
57	C₂²⋊#10623494	C₂×C₄	1	2	yes	—
58	#15667924	C₂×C₄	1	2	yes	—
59	#15667924	C₂×C₄	1	2	yes	—
60	#15667923	C₂×C₄	1	2	yes	—
61	#15667922	C₂×C₄	1	2	yes	—
62	C₈×Dic₈₉	C₂³	1	2	yes	—
63	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
66	M₃₂×C₈₉	—	2	4	no	—
67	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
68	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
69	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
70	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
74	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
75	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
76	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
77	#13083557	—	2	4	no	—
78	C₂×#10623494	C₂²×C₄	1	2	yes	—
79	C₂×#10623494	C₂²×C₄	1	2	yes	—
80	C₈×D₈₉	—	2	2	no	—
81	C₈×D₈₉	—	2	2	no	—
82	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
83	C₄⋊Dic₈₉	C₂×C₈	1	1	yes	characteristic
84	#13083553	C₂×C₈	1	2	yes	—
85	#13083553	C₂×C₈	1	2	yes	—
86	#13083553	C₂×C₈	1	2	yes	—
87	#13083553	C₂×C₈	1	2	yes	—
88	C₂²⋊Dic₈₉	C₂×C₈	1	2	yes	—
89	#13083551	C₂×C₈	1	2	yes	—
90	#13083551	C₂×C₈	1	2	yes	—
91	#13083551	C₂×C₈	1	2	yes	—
92	#13083551	C₂×C₈	1	2	yes	—
93	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
94	#13083549	—	2	2	no	—
95	#13083549	—	2	2	no	—
96	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	—	2	2	no	—
100	C₂×C₈×C₈₉	—	2	2	no	—
101	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
105	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
106	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
107	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
109	C₁₆×C₈₉	—	2	2	no	—
110	C₁₆×C₈₉	#41	1	4	yes	—
111	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
112	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	4	1	no	—
116	C₂×C₄×C₈₉	—	4	1	no	—
117	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
123	C₂²⋊D₈₉	—	4	2	no	—
124	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
125	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
126	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
127	D₃₅₆	—	4	1	no	—
128	C₄×D₈₉	—	2	2	no	—
129	C₄×D₈₉	—	2	2	no	—
130	Dic₁₇₈	—	4	1	no	—
131	C₈×C₈₉	—	2	1	no	—
132	C₈×C₈₉	—	2	1	no	—
133	C₈×C₈₉	#131	1	2	yes	—
134	C₈×C₈₉	C₂×#41	1	2	yes	—
135	#10623494	—	2	2	no	—
136	#10623494	—	2	2	no	—
137	C₂²×C₈₉	—	2	1	no	—
138	C₂²×C₈₉	—	2	1	no	—
139	C₂²×C₈₉	—	2	1	no	—
140	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
141	C₂×D₈₉	—	2	2	no	—
142	C₂×D₈₉	—	2	2	no	—
143	C₂×D₈₉	—	2	2	no	—
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	—	2	1	no	—
146	C₄×C₈₉	—	2	1	no	—
147	C₄×C₈₉	—	2	1	no	—
148	C₄×C₈₉	—	2	1	no	—
149	C₄×C₈₉	C₄×#41	1	1	yes	characteristic
150	C₄×C₈₉	C₂×#131	1	1	yes	characteristic
151	Dic₈₉	—	2	2	no	—
152	Dic₈₉	—	2	2	no	—
153	Dic₈₉	—	2	2	no	—
154	#33481	—	89	1	no	maximal, Sylow
155	C₂×C₈₉	—	4	1	no	—
156	C₂×C₈₉	C₈×#41	1	1	yes	characteristic
157	C₂×C₈₉	#3407	1	1	yes	characteristic
158	C₂×C₈₉	#3407	1	1	yes	characteristic
159	D₈₉	—	4	2	no	—
160	#2966	—	89	1	no	—
161	#2183	—	89	2	no	—
162	#2169	—	89	2	no	—
163	#2168	—	89	2	no	—
164	#2168	—	89	2	no	—
165	#2164	—	89	2	no	—
166	#2163	—	89	1	no	—
167	#2108	—	89	2	no	—
168	C₄×M₃₂	D₈₉	1	1	yes	characteristic, complemented
169	C₈₉	#33481	1	1	yes	characteristic, complemented, Sylow
170	#529	—	89	1	no	—
171	C₂×M₃₂	—	89	2	no	—
172	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
173	C₂²×C₁₆	—	89	2	no	—
174	#446	—	89	1	no	—
175	C₈×Q₈	—	89	1	no	—
176	#436	—	89	1	no	—
177	#435	—	89	2	no	—
178	C₈×D₄	—	89	2	no	—
179	C₈×D₄	—	89	1	no	—
180	#433	—	89	2	no	—
181	#432	—	89	1	no	—
182	#405	—	89	2	no	—
183	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
184	C₄⋊C₁₆	—	89	2	no	—
185	C₄⋊C₁₆	—	89	2	no	—
186	C₄⋊C₁₆	—	89	2	no	—
187	C₂²⋊C₁₆	—	89	1	no	—
188	C₂²⋊C₁₆	—	89	1	no	—
189	C₂²⋊C₁₆	—	89	2	no	—
190	C₂²⋊C₁₆	—	89	2	no	—
191	#346	—	89	2	no	—
192	#346	—	89	2	no	—
193	#346	C₂×D₈₉	1	2	yes	—
194	C₄×C₁₆	—	89	2	no	—
195	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
196	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
197	C₂×M₁₆	—	89	2	no	—
198	C₂²×C₈	—	89	2	no	—
199	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
201	#126	—	89	2	no	—
202	#125	—	89	1	no	—
203	#124	—	89	1	no	—
204	#123	—	89	2	no	—
205	C₂²⋊Q₈	—	89	1	no	—
206	#121	—	89	1	no	—
207	C₄×Q₈	—	89	1	no	—
208	C₄×D₄	—	89	2	no	—
209	C₄×D₄	—	89	1	no	—
210	#117	—	89	2	no	—
211	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
212	M₃₂	—	2	4	no	—
213	M₃₂	—	178	4	no	—
214	C₂×C₁₆	—	89	2	no	—
215	C₂×C₁₆	—	89	2	no	—
216	C₂×C₁₆	—	178	4	no	—
217	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
218	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
219	C₄⋊C₈	—	89	2	no	—
220	C₄⋊C₈	—	89	2	no	—
221	C₄⋊C₈	—	89	1	no	—
222	C₄⋊C₈	—	89	1	no	—
223	C₂²⋊C₈	—	89	2	no	—
224	C₂²⋊C₈	—	89	2	no	—
225	C₂²⋊C₈	—	89	2	no	—
226	#97	—	89	2	no	—
227	#97	—	89	2	no	—
228	C₄×C₈	—	89	2	no	—
229	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
230	C₄×C₈	C₄×D₈₉	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
233	C₂×Q₈	—	89	1	no	—
234	C₂×D₄	—	89	2	no	—
235	C₂×D₄	—	89	1	no	—
236	C₂²×C₄	—	89	2	no	—
237	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
238	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
240	M₁₆	—	178	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	89	2	no	—
244	C₂×C₈	—	89	2	no	—
245	C₂×C₈	—	178	2	no	—
246	C₂×C₈	—	178	2	no	—
247	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
249	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
250	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
251	C₄⋊C₄	—	89	2	no	—
252	C₄⋊C₄	—	89	2	no	—
253	C₄⋊C₄	—	89	1	no	—
254	C₄⋊C₄	—	89	2	no	—
255	C₄⋊C₄	—	89	1	no	—
256	C₄⋊C₄	—	89	2	no	—
257	C₂²⋊C₄	—	89	2	no	—
258	C₂²⋊C₄	—	89	2	no	—
259	C₂²⋊C₄	—	89	2	no	—
260	C₂²⋊C₄	—	89	2	no	—
261	C₂²⋊C₄	—	89	2	no	—
262	C₄²	—	89	2	no	—
263	C₄²	C₈×D₈₉	1	1	yes	characteristic
264	C₄²	C₈×D₈₉	1	1	yes	characteristic
265	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
266	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
267	C₁₆	—	2	2	no	—
268	C₁₆	—	178	2	no	—
269	C₁₆	—	178	2	no	—
270	C₁₆	#13083574	1	4	yes	—
271	C₂³	—	89	2	no	—
272	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
273	Q₈	—	356	1	no	—
274	D₄	—	356	1	no	—
275	D₄	—	356	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	89	2	no	—
281	C₂×C₄	—	89	2	no	—
282	C₂×C₄	—	89	2	no	—
283	C₂×C₄	—	178	2	no	—
284	C₂×C₄	—	178	2	no	—
285	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
286	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
287	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
288	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
289	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	178	2	no	—
294	C₈	—	178	2	no	—
295	C₈	#15668018	1	2	yes	—
296	C₈	C₂×#13083574	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	178	2	no	—
301	C₂²	—	178	2	no	—
302	C₂²	—	178	2	no	—
303	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	178	2	no	—
310	C₄	—	178	2	no	—
311	C₄	—	178	2	no	—
312	C₄	C₄×#13083574	1	1	yes	characteristic
313	C₄	C₂×#15668018	1	1	yes	characteristic
314	C₂	—	4	1	no	—
315	C₂	—	356	2	no	—
316	C₂	C₈×#13083574	1	1	yes	characteristic
317	C₂	#26033354	1	1	yes	characteristic
318	C₂	#26033354	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₄×M₃₂×C₈₉	C₂
prime radical	#15667986	C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₄×M₃₂	D₈₉
hypocenter	C₈₉	#33481
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. #26022158	C₂	semidirect
3. #26022082	C₂	semidirect
4. #26022079	C₂	semidirect
5. #26022075	C₂	semidirect
6. #26022057	C₂	semidirect
7. #26022055	C₂	semidirect
8. #26021847	C₂	semidirect
9. C₄×M₃₂×C₈₉	C₂	semidirect
168. C₄×M₃₂	D₈₉	semidirect
169. C₈₉	#33481	semidirect

Maximal subgroups

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

subgroup	index
#26029227	2
#26022158	2
#26022082	2
#26022079	2
#26022075	2
#26022057	2
#26022055	2
#26021847	2
C₄×M₃₂×C₈₉	2
#33481	89