Subgroups — #36707527

Order 22784.

582 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028674	C₂	1	1	yes	maximal, characteristic
2	#26022058	C₂	1	4	yes	maximal, complemented
3	#26022054	C₂	1	4	yes	maximal, complemented
4	#26022011	C₂	1	1	yes	maximal, characteristic, complemented
5	#26021867	C₂	1	1	yes	maximal, characteristic
6	#26021865	C₂	1	1	yes	maximal, characteristic
7	M₃₂×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#19151373	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#15668020	C₄	1	1	yes	characteristic
11	C₂×#15668018	C₂²	1	1	yes	characteristic
12	#19152066	C₄	1	1	yes	characteristic
13	C₂×#15667952	C₂²	1	1	yes	characteristic
14	C₂²×#13083549	C₂²	1	1	yes	characteristic
15	#19151555	C₄	1	2	yes	—
16	#19151554	C₄	1	1	yes	characteristic
17	#19151552	C₂²	1	2	yes	—
18	#19151550	C₄	1	2	yes	—
19	M₁₆×Dic₈₉	C₄	1	1	yes	characteristic
20	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
21	C₄×C₈×D₈₉	C₂²	1	2	yes	—
22	C₄⋊#13083549	C₂²	1	4	yes	—
23	#19151377	C₂²	1	4	yes	—
24	#19151377	C₂²	1	4	yes	—
25	#19151373	C₂²	1	2	yes	—
26	#19151373	C₂²	1	2	yes	—
27	#19151373	C₂²	1	2	yes	—
28	C₁₆×Dic₈₉	C₂²	1	2	yes	—
29	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
30	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
31	#405×C₈₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₁₆×C₈₉	C₂²	1	4	yes	—
33	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
34	#15668020	—	2	4	no	—
35	#15668018	—	2	2	no	—
36	#15668018	—	2	2	no	—
37	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
38	C₂×#13083557	C₂×C₄	1	2	yes	—
39	C₂×#13083557	C₂×C₄	1	2	yes	—
40	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
41	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₈₉	C₂³	1	2	yes	—
43	#15667977	C₈	1	2	yes	—
44	#15667974	C₈	1	2	yes	—
45	#15667972	C₈	1	4	yes	—
46	#15667969	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
48	C₄²×D₈₉	C₂×C₄	1	2	yes	—
49	#15667952	—	2	2	no	—
50	C₂×#13083549	—	2	2	no	—
51	C₂×#13083549	C₂³	1	2	yes	—
52	C₂×#13083549	C₂³	1	2	yes	—
53	#15667923	C₂×C₄	1	2	yes	—
54	#15667922	C₂×C₄	1	2	yes	—
55	C₄×#10623494	C₂×C₄	1	2	yes	—
56	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
57	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
58	C₈×Dic₈₉	C₂×C₄	1	1	yes	characteristic
59	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
60	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
63	#117×C₈₉	C₂×C₄	1	1	yes	characteristic
64	M₃₂×C₈₉	—	2	2	no	—
65	C₂×C₁₆×C₈₉	—	2	2	no	—
66	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
67	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
68	#97×C₈₉	C₂×C₄	1	2	yes	—
69	C₄×C₈×C₈₉	C₂³	1	2	yes	—
70	#13083574	—	2	2	no	—
71	#13083574	—	2	2	no	—
72	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
73	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
74	C₂×C₂²⋊D₈₉	C₂×C₈	1	2	yes	—
75	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
76	C₂×C₄×D₈₉	C₂×C₈	1	4	yes	—
77	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
78	#13083558	—	2	2	no	—
79	#13083557	—	2	4	no	—
80	#13083557	—	2	4	no	—
81	C₂×#10623494	M₁₆	1	4	yes	—
82	C₂×#10623494	C₂²×C₄	1	2	yes	—
83	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
84	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
85	C₈×D₈₉	—	2	2	no	—
86	C₈×D₈₉	—	2	2	no	—
87	C₈×D₈₉	—	2	4	no	—
88	#13083553	C₂×C₈	1	4	yes	—
89	#13083551	C₂×C₈	1	4	yes	—
90	C₄×Dic₈₉	C₂×C₈	1	4	yes	—
91	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
92	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
93	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
94	#13083549	—	2	2	no	—
95	#13083549	—	2	2	no	—
96	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
97	M₁₆×C₈₉	—	2	2	no	—
98	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
99	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	#41	1	2	yes	—
101	C₂×C₈×C₈₉	#41	1	2	yes	—
102	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
104	C₂²⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
105	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
106	C₁₆×C₈₉	—	2	2	no	—
107	C₁₆×C₈₉	—	2	2	no	—
108	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
109	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₈₉	#117	1	2	yes	—
111	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
112	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
114	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
115	C₂²⋊D₈₉	—	2	4	no	—
116	C₂×Dic₈₉	—	4	2	no	—
117	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
118	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
119	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
120	D₃₅₆	—	2	2	no	—
121	C₄×D₈₉	—	2	2	no	—
122	C₄×D₈₉	—	2	2	no	—
123	C₄×D₈₉	—	4	4	no	—
124	Dic₁₇₈	—	2	2	no	—
125	C₈×C₈₉	—	2	2	no	—
126	C₈×C₈₉	#117	1	2	yes	—
127	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
128	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
129	#10623494	—	2	2	no	—
130	#10623494	C₂×M₁₆	1	2	yes	—
131	#10623494	C₂×M₁₆	1	2	yes	—
132	C₂²×C₈₉	—	2	2	no	—
133	C₂²×C₈₉	#431	1	2	yes	—
134	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
135	C₂×D₈₉	—	2	4	no	—
136	C₂×D₈₉	—	2	2	no	—
137	C₄×C₈₉	—	2	4	no	—
138	C₄×C₈₉	#431	1	2	yes	—
139	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
140	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
141	Dic₈₉	—	2	2	no	—
142	Dic₈₉	—	2	2	no	—
143	Dic₈₉	—	2	2	no	—
144	#33421	—	89	1	no	maximal, Sylow
145	C₂×C₈₉	—	2	2	no	—
146	C₂×C₈₉	C₂×#431	1	1	yes	characteristic
147	C₂×C₈₉	#3407	1	2	yes	—
148	D₈₉	—	4	2	no	—
149	C₄×#131	—	89	1	no	—
150	#2164	—	89	4	no	—
151	#2163	—	89	4	no	—
152	#2151	—	89	1	no	—
153	#2151	D₈₉	1	1	yes	characteristic, complemented
154	#2113	—	89	1	no	—
155	#2113	—	89	1	no	—
156	C₄×M₃₂	—	89	1	no	—
157	C₂×#346	—	89	1	no	—
158	C₈₉	#33421	1	1	yes	characteristic, complemented, Sylow
159	C₂×#131	—	89	1	no	—
160	C₂×#131	—	89	1	no	—
161	C₄×#41	—	89	1	no	—
162	C₂×M₃₂	—	89	1	no	—
163	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
164	C₂²×C₁₆	—	89	1	no	—
165	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
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	1	no	—
171	C₄×M₁₆	—	89	2	no	—
172	C₂×C₄×C₈	—	89	1	no	—
173	C₂×C₄×C₈	—	89	2	no	—
174	C₄⋊C₁₆	—	89	4	no	—
175	C₄⋊C₁₆	C₂×D₈₉	1	4	yes	—
176	C₂²⋊C₁₆	—	89	4	no	—
177	C₂²⋊C₁₆	—	89	4	no	—
178	#346	—	89	2	no	—
179	#346	—	89	2	no	—
180	#346	—	89	2	no	—
181	C₄×C₁₆	—	89	2	no	—
182	C₂×#41	—	89	1	no	—
183	#131	—	178	4	no	—
184	#131	—	178	2	no	—
185	#131	—	178	2	no	—
186	C₂×M₁₆	—	89	1	no	—
187	C₂×M₁₆	—	89	2	no	—
188	C₂×M₁₆	—	89	2	no	—
189	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
190	C₂²×C₈	—	89	1	no	—
191	C₂²×C₈	—	89	2	no	—
192	C₂²×C₈	—	89	2	no	—
193	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
194	C₄×Q₈	—	89	2	no	—
195	C₄×D₄	—	89	2	no	—
196	C₄×D₄	—	89	4	no	—
197	#117	—	89	2	no	—
198	#117	C₄×D₈₉	1	1	yes	characteristic
199	C₂×C₄²	—	89	1	no	—
200	C₂×C₄²	—	89	2	no	—
201	M₃₂	—	2	2	no	—
202	M₃₂	—	178	2	no	—
203	C₂×C₁₆	—	2	2	no	—
204	C₂×C₁₆	—	89	2	no	—
205	C₂×C₁₆	—	89	2	no	—
206	C₂×C₁₆	—	178	2	no	—
207	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
208	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
209	#97	—	89	2	no	—
210	#97	—	89	2	no	—
211	#97	C₄×D₈₉	1	2	yes	—
212	C₄×C₈	—	89	1	no	—
213	C₄×C₈	—	89	2	no	—
214	C₄×C₈	—	89	1	no	—
215	C₄×C₈	—	89	1	no	—
216	C₄×C₈	—	89	1	no	—
217	C₄×C₈	—	89	2	no	—
218	C₄×C₈	C₂²×D₈₉	1	2	yes	—
219	#41	—	178	2	no	—
220	#41	—	178	2	no	—
221	C₂×Q₈	—	89	1	no	—
222	C₂×D₄	—	89	2	no	—
223	C₂×D₄	—	89	1	no	—
224	C₂²×C₄	—	89	2	no	—
225	C₂²×C₄	—	89	4	no	—
226	C₂²×C₄	—	89	2	no	—
227	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
228	M₁₆	—	2	2	no	—
229	M₁₆	—	178	4	no	—
230	M₁₆	—	178	2	no	—
231	M₁₆	—	178	4	no	—
232	C₂×C₈	—	89	4	no	—
233	C₂×C₈	—	89	1	no	—
234	C₂×C₈	—	89	1	no	—
235	C₂×C₈	—	89	2	no	—
236	C₂×C₈	—	178	4	no	—
237	C₂×C₈	—	178	2	no	—
238	C₂×C₈	—	178	2	no	—
239	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
240	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
241	C₂×C₈	#13083573	1	2	yes	—
242	C₂×C₈	#13083575	1	2	yes	—
243	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
244	C₄⋊C₄	—	89	4	no	—
245	C₄⋊C₄	C₈×D₈₉	1	2	yes	—
246	C₂²⋊C₄	—	89	4	no	—
247	C₂²⋊C₄	C₈×D₈₉	1	2	yes	—
248	C₄²	—	89	2	no	—
249	C₄²	—	89	4	no	—
250	C₄²	—	89	1	no	—
251	C₄²	—	89	1	no	—
252	C₄²	C₂×C₄×D₈₉	1	2	yes	—
253	C₁₆	—	2	2	no	—
254	C₁₆	—	2	2	no	—
255	C₁₆	—	178	2	no	—
256	C₁₆	—	178	2	no	—
257	C₂³	—	89	2	no	—
258	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
259	Q₈	—	178	2	no	—
260	D₄	—	178	4	no	—
261	D₄	—	178	2	no	—
262	C₂×C₄	—	89	2	no	—
263	C₂×C₄	—	89	2	no	—
264	C₂×C₄	—	89	2	no	—
265	C₂×C₄	—	356	4	no	—
266	C₂×C₄	—	356	2	no	—
267	C₂×C₄	—	178	2	no	—
268	C₂×C₄	—	178	2	no	—
269	C₂×C₄	#15667970	1	2	yes	—
270	C₂×C₄	C₂×C₈×D₈₉	1	4	yes	—
271	C₂×C₄	C₂×C₈×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	2	no	—
276	C₈	—	89	2	no	—
277	C₈	—	178	2	no	—
278	C₈	#15667970	1	2	yes	—
279	C₈	C₂×#13083573	1	1	yes	characteristic
280	C₈	C₂×#13083575	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	178	2	no	—
283	C₂²	—	178	4	no	—
284	C₂²	#19151650	1	2	yes	—
285	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
286	C₄	—	2	4	no	—
287	C₄	—	178	2	no	—
288	C₄	—	178	2	no	—
289	C₄	—	178	2	no	—
290	C₄	#19151650	1	2	yes	—
291	C₄	C₂×#15667970	1	1	yes	characteristic
292	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	356	2	no	—
295	C₂	C₂×#19151650	1	1	yes	characteristic
296	C₂	#26033353	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₈₉
commutator, socle	C₂²×C₈₉	C₂³×C₈
Fitting	C₈₉×#2151	C₂
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2151	D₈₉
hypocenter	C₈₉	#33421
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ #2151

Complements

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

N	H ≅ G/N	type
2. #26022058	C₂	semidirect
3. #26022054	C₂	semidirect
4. #26022011	C₂	semidirect
7. M₃₂×Dic₈₉	C₂	semidirect
8. C₂×#19151373	C₂	semidirect
9. C₈₉×#2151	C₂	semidirect
153. #2151	D₈₉	semidirect
158. C₈₉	#33421	semidirect

Maximal subgroups

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

subgroup	index
#26028674	2
#26022058	2
#26022054	2
#26022011	2
#26021867	2
#26021865	2
M₃₂×Dic₈₉	2
C₂×#19151373	2
C₈₉×#2151	2
#33421	89