Subgroups — #36552647

Order 22784.

1078 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151651	C₂	1	1	yes	maximal, characteristic
2	C₄×#15667971	C₂	1	1	yes	maximal, characteristic
3	#26020119	C₂	1	4	yes	maximal, complemented
4	#26020028	C₂	1	4	yes	maximal
5	#26019997	C₂	1	2	yes	maximal, complemented
6	C₄×#15667922	C₂	1	2	yes	maximal, complemented
7	C₂²×C₄×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#15667971	C₄	1	2	yes	—
9	#19151651	C₄	1	8	yes	complemented
10	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
11	C₂×C₂²⋊#10623494	C₂²	1	2	yes	—
12	C₂×#15667922	C₂²	1	8	yes	—
13	C₂×C₄×#10623494	C₂²	1	2	yes	—
14	C₄×C₄⋊Dic₈₉	C₄	1	2	yes	—
15	C₄×C₄⋊Dic₈₉	C₂²	1	2	yes	—
16	C₄×C₂²⋊Dic₈₉	C₄	1	2	yes	—
17	C₄×C₂²⋊Dic₈₉	C₂²	1	2	yes	—
18	#19151444	C₄	1	2	yes	—
19	C₄²×Dic₈₉	C₂²	1	2	yes	—
20	#19151352	C₂²	1	8	yes	—
21	C₂³×C₈×C₈₉	C₂²	1	2	yes	—
22	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₈₉	C₄	1	8	yes	complemented
24	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂²×#10623494	C₂³	1	4	yes	—
28	#15667971	—	2	8	no	—
29	C₂×C₄⋊Dic₈₉	C₂×C₄	1	4	yes	—
30	C₂×C₂²⋊Dic₈₉	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₈₉	C₂×C₄	1	4	yes	—
32	C₂×C₄×Dic₈₉	C₂×C₄	1	4	yes	—
33	C₂×C₄×Dic₈₉	C₂³	1	4	yes	—
34	C₄⋊#10623494	C₂×C₄	1	8	yes	—
35	C₂²⋊#10623494	C₂×C₄	1	8	yes	—
36	#15667922	—	2	16	no	—
37	C₄×#10623494	C₂×C₄	1	8	yes	—
38	#15667919	C₂×C₄	1	4	yes	—
39	#15667919	C₂×C₄	1	4	yes	—
40	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	—	2	8	no	—
44	C₂²×C₈×C₈₉	C₂×C₄	1	8	yes	—
45	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
46	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
47	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
54	C₄×C₈×C₈₉	C₂×C₄	1	8	yes	—
55	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
56	C₂²×Dic₈₉	C₂²×C₄	1	8	yes	—
57	C₂×#10623494	—	2	8	no	—
58	C₂×#10623494	C₂²×C₄	1	8	yes	—
59	C₄⋊Dic₈₉	—	2	8	no	—
60	C₂²⋊Dic₈₉	—	2	8	no	—
61	C₄×Dic₈₉	—	2	8	no	—
62	C₄×Dic₈₉	—	2	8	no	—
63	C₄×Dic₈₉	—	2	8	no	—
64	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₉	—	2	4	no	—
66	C₂²×C₄×C₈₉	—	2	4	no	—
67	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
68	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
69	C₂²×C₄×C₈₉	M₁₆	1	8	yes	—
70	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₈₉	—	2	4	no	—
79	C₂×C₈×C₈₉	—	2	16	no	—
80	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₈₉	#41	1	8	yes	—
83	C₄²×C₈₉	C₄²	1	4	yes	—
84	C₄²×C₈₉	C₄²	1	4	yes	—
85	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
86	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
87	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
88	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
89	C₂³×C₈₉	—	2	2	no	—
90	C₂³×C₈₉	—	2	2	no	—
91	C₂³×C₈₉	C₂×C₄²	1	2	yes	—
92	C₂³×C₈₉	C₂×M₁₆	1	4	yes	—
93	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₈₉	—	2	2	no	—
95	C₂×C₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	—	2	4	no	—
97	C₂×C₄×C₈₉	—	2	4	no	—
98	C₂×C₄×C₈₉	—	2	8	no	—
99	C₂×C₄×C₈₉	#97	1	16	yes	complemented
100	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
101	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
103	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
104	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
106	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
107	C₂×C₄×C₈₉	#131	1	4	yes	—
108	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
111	C₂×Dic₈₉	—	2	8	no	—
112	C₂×Dic₈₉	—	2	8	no	—
113	C₂×Dic₈₉	—	2	8	no	—
114	C₈×C₈₉	—	2	4	no	—
115	C₈×C₈₉	#117	1	8	yes	complemented
116	#10623494	—	2	8	no	—
117	C₂²×C₈₉	—	2	2	no	—
118	C₂²×C₈₉	—	2	8	no	—
119	C₂²×C₈₉	—	2	2	no	—
120	C₂²×C₈₉	C₂×#97	1	4	yes	—
121	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
122	C₂²×C₈₉	C₄×#41	1	2	yes	—
123	C₂²×C₈₉	C₂²×M₁₆	1	2	yes	—
124	C₂²×C₈₉	C₂×#131	1	2	yes	—
125	C₄×C₈₉	—	2	2	no	—
126	C₄×C₈₉	—	2	2	no	—
127	C₄×C₈₉	C₂×#97	1	4	yes	—
128	C₄×C₈₉	C₂×#97	1	4	yes	—
129	C₄×C₈₉	#405	1	4	yes	—
130	C₄×C₈₉	C₂×#117	1	2	yes	—
131	C₄×C₈₉	C₄×#41	1	2	yes	—
132	Dic₈₉	—	4	8	no	—
133	#20473	—	89	1	no	maximal, Sylow
134	C₂×C₈₉	—	2	4	no	—
135	C₂×C₈₉	C₄×#117	1	1	yes	characteristic
136	C₂×C₈₉	C₂²×#97	1	1	yes	characteristic
137	C₂×C₈₉	C₂×#405	1	1	yes	characteristic
138	C₂×C₈₉	#2965	1	4	yes	—
139	C₂×#431	—	89	1	no	—
140	C₂²×C₄×C₈	D₈₉	1	1	yes	characteristic
141	C₄×#117	—	89	1	no	—
142	#1770	—	89	4	no	—
143	#1752	—	89	4	no	—
144	#1744	—	89	2	no	—
145	C₄×#97	—	89	2	no	—
146	C₈₉	#20473	1	1	yes	characteristic, complemented, Sylow
147	C₂³×C₈	C₂×D₈₉	1	2	yes	—
148	C₂×#117	—	89	2	no	—
149	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
150	#431	—	89	8	no	—
151	C₂×C₄⋊C₈	—	89	2	no	—
152	C₂×C₂²⋊C₈	—	89	2	no	—
153	C₂×#97	—	89	8	no	—
154	C₂×C₄×C₈	—	89	2	no	—
155	C₂×C₄×C₈	Dic₈₉	1	8	yes	complemented
156	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
157	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
159	C₄×C₄⋊C₄	—	89	2	no	—
160	C₄×C₄⋊C₄	—	89	2	no	—
161	C₄×C₂²⋊C₄	—	89	2	no	—
162	C₄×C₂²⋊C₄	—	89	2	no	—
163	#376	—	89	2	no	—
164	C₄³	—	89	2	no	—
165	#336	—	89	8	no	—
166	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
167	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
168	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
169	C₂²×C₈	—	2	8	no	—
170	C₂²×C₈	—	89	4	no	—
171	C₂²×C₈	C₂×Dic₈₉	1	8	yes	—
172	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
173	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
174	#117	—	178	8	no	—
175	C₂×C₄⋊C₄	—	89	4	no	—
176	C₂×C₂²⋊C₄	—	89	4	no	—
177	C₂×C₄²	—	89	4	no	—
178	C₂×C₄²	—	89	4	no	—
179	C₂×C₄²	—	89	4	no	—
180	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
181	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
182	C₂×C₄²	C₄×D₈₉	1	4	yes	—
183	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
184	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
185	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
186	C₄⋊C₈	—	89	8	no	—
187	C₂²⋊C₈	—	89	8	no	—
188	#97	—	178	16	no	—
189	C₄×C₈	—	89	8	no	—
190	C₄×C₈	C₂×Dic₈₉	1	8	yes	—
191	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
192	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
193	#95	—	89	4	no	—
194	#95	—	89	4	no	—
195	C₂⁴	C₂×C₄×D₈₉	1	1	yes	characteristic
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	89	8	no	—
199	C₂²×C₄	C₄×Dic₈₉	1	4	yes	—
200	C₂²×C₄	C₄×Dic₈₉	1	4	yes	—
201	C₂²×C₄	#13083557	1	8	yes	—
202	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
203	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
204	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
206	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
208	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
209	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	2	16	no	—
212	C₂×C₈	—	89	8	no	—
213	C₂×C₈	—	178	8	no	—
214	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
215	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
216	C₂×C₈	#13083574	1	8	yes	—
217	C₄⋊C₄	—	178	8	no	—
218	C₂²⋊C₄	—	178	8	no	—
219	C₄²	—	178	8	no	—
220	C₄²	—	178	8	no	—
221	C₄²	—	178	8	no	—
222	C₄²	C₄×Dic₈₉	1	4	yes	—
223	C₄²	C₄×Dic₈₉	1	4	yes	—
224	C₄²	C₂×C₄×D₈₉	1	2	yes	—
225	C₄²	C₂×C₄×D₈₉	1	2	yes	—
226	C₄²	C₂²×Dic₈₉	1	2	yes	—
227	C₄²	C₂²×Dic₈₉	1	2	yes	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	C₂×C₄×Dic₈₉	1	2	yes	—
231	C₂³	C₂×#13083557	1	4	yes	—
232	C₂³	C₂²×C₄×D₈₉	1	1	yes	characteristic
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	8	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	178	8	no	—
239	C₂×C₄	—	178	8	no	—
240	C₂×C₄	—	178	8	no	—
241	C₂×C₄	#15667922	1	16	yes	complemented
242	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
243	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
244	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
245	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
246	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
247	C₂×C₄	C₂×#13083557	1	4	yes	—
248	C₂×C₄	C₂×#13083557	1	4	yes	—
249	C₂×C₄	#15668018	1	4	yes	—
250	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
251	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
252	C₂×C₄	C₂×#13083574	1	4	yes	—
253	C₈	—	2	4	no	—
254	C₈	—	178	8	no	—
255	C₈	#15667971	1	8	yes	complemented
256	C₂²	—	2	2	no	—
257	C₂²	—	2	8	no	—
258	C₂²	—	2	2	no	—
259	C₂²	C₂×#15667922	1	4	yes	—
260	C₂²	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
261	C₂²	C₄×#13083574	1	2	yes	—
262	C₂²	C₂²×#13083557	1	2	yes	—
263	C₂²	C₂×#15668018	1	2	yes	—
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	356	8	no	—
267	C₄	C₂×#15667922	1	4	yes	—
268	C₄	C₂×#15667922	1	4	yes	—
269	C₄	#19151553	1	4	yes	—
270	C₄	C₂×#15667971	1	2	yes	—
271	C₄	C₄×#13083574	1	2	yes	—
272	C₂	—	2	4	no	—
273	C₂	C₄×#15667971	1	1	yes	characteristic
274	C₂	C₂²×#15667922	1	1	yes	characteristic
275	C₂	C₂×#19151553	1	1	yes	characteristic
276	C₂	#26029218	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂²×C₄×C₈×C₈₉	C₂
socle	C₂³×C₈₉	C₂³×C₄
prime radical	C₂⁴×C₈₉	C₂²×C₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂²×C₄×C₈	D₈₉
hypocenter	C₈₉	#20473
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #26020119	C₂	semidirect
5. #26019997	C₂	semidirect
6. C₄×#15667922	C₂	semidirect
9. #19151651	C₄	semidirect
23. C₂×C₄×C₈×C₈₉	C₄	semidirect
99. C₂×C₄×C₈₉	#97	semidirect
115. C₈×C₈₉	#117	semidirect
146. C₈₉	#20473	semidirect
155. C₂×C₄×C₈	Dic₈₉	semidirect
241. C₂×C₄	#15667922	semidirect
255. C₈	#15667971	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151651	2
C₄×#15667971	2
#26020119	2
#26020028	2
#26019997	2
C₄×#15667922	2
C₂²×C₄×C₈×C₈₉	2
#20473	89