Subgroups — #36460070

Order 22784.

650 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151554	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151370	C₂	1	2	yes	maximal
3	#26019985	C₂	1	1	yes	maximal, characteristic
4	#26019950	C₂	1	2	yes	maximal
5	C₈₉×#1742	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	C₂²	1	1	yes	characteristic
7	C₂³×#10623494	C₂²	1	1	yes	characteristic
8	C₂×#15667971	C₂²	1	2	yes	—
9	#19151554	—	2	4	no	—
10	C₂×#15667923	C₄	1	2	yes	complemented
11	C₂×C₄×#10623494	C₄	1	2	yes	complemented
12	#19151370	—	2	4	no	—
13	#19151365	C₄	1	4	yes	—
14	#19151352	—	2	4	no	—
15	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
16	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
17	#343×C₈₉	C₄	1	4	yes	complemented
18	C₂×#13083558	—	2	2	no	—
19	C₂×#13083558	—	2	2	no	—
20	C₂×#13083558	D₄	1	2	yes	—
21	C₂×#13083558	Q₈	1	2	yes	—
22	C₂²×#10623494	—	2	1	no	—
23	C₂²×#10623494	—	2	2	no	—
24	C₂²×#10623494	—	2	1	no	—
25	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
27	C₂²×#10623494	C₂×C₄	1	2	yes	—
28	C₂²×#10623494	D₄	1	4	yes	—
29	#15667971	—	2	2	no	—
30	#15667971	—	2	2	no	—
31	#15667971	—	2	2	no	—
32	#15667971	—	2	2	no	—
33	C₂×C₄⋊Dic₈₉	—	2	2	no	—
34	C₂×C₂²⋊Dic₈₉	C₂×C₄	1	4	yes	—
35	C₂×C₄×Dic₈₉	—	2	2	no	—
36	#15667923	—	2	4	no	—
37	C₄×#10623494	—	2	4	no	—
38	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₈₉	—	2	2	no	—
40	C₂×M₁₆×C₈₉	—	2	2	no	—
41	C₂×M₁₆×C₈₉	—	2	2	no	—
42	C₂×M₁₆×C₈₉	—	2	2	no	—
43	C₂×M₁₆×C₈₉	—	2	2	no	—
44	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₈₉	—	2	2	no	—
46	#117×C₈₉	—	2	2	no	—
47	#117×C₈₉	—	2	2	no	—
48	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
51	C₂²×Dic₈₉	—	2	4	no	—
52	#13083558	—	2	4	no	—
53	#13083558	—	2	4	no	—
54	C₂×#10623494	—	2	2	no	—
55	C₂×#10623494	—	2	1	no	—
56	C₂×#10623494	—	2	1	no	—
57	C₂×#10623494	—	2	1	no	—
58	C₂×#10623494	—	2	1	no	—
59	C₂×#10623494	—	2	2	no	—
60	C₂×#10623494	—	2	4	no	—
61	C₂×#10623494	—	2	4	no	—
62	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
63	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
64	C₂×#10623494	C₄⋊C₄	1	4	yes	—
65	C₄⋊Dic₈₉	—	4	2	no	—
66	C₄⋊Dic₈₉	—	4	2	no	—
67	C₂²⋊Dic₈₉	—	2	4	no	—
68	C₂²⋊Dic₈₉	—	2	4	no	—
69	C₄×Dic₈₉	—	4	2	no	—
70	C₄×Dic₈₉	—	4	2	no	—
71	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₄²	1	4	yes	—
73	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	#41	1	2	yes	—
79	C₂²×C₄×C₈₉	#41	1	2	yes	—
80	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	#41	1	2	yes	—
82	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
83	M₁₆×C₈₉	—	2	4	no	—
84	M₁₆×C₈₉	—	2	4	no	—
85	M₁₆×C₈₉	—	4	2	no	—
86	M₁₆×C₈₉	—	4	2	no	—
87	C₂×C₈×C₈₉	—	2	4	no	—
88	C₂×C₈×C₈₉	—	4	2	no	—
89	C₂×C₈×C₈₉	—	4	2	no	—
90	C₄⋊C₄×C₈₉	—	2	4	no	—
91	C₂²⋊C₄×C₈₉	—	2	4	no	—
92	C₄²×C₈₉	—	2	2	no	—
93	C₄²×C₈₉	—	2	2	no	—
94	C₂³×C₈₉	—	4	2	no	—
95	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
96	C₂³×C₈₉	#117	1	2	yes	—
97	C₂³×C₈₉	C₄×D₄	1	2	yes	—
98	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
99	C₂³×C₈₉	C₄×Q₈	1	1	yes	characteristic
100	C₂×C₄×C₈₉	—	2	4	no	—
101	C₂×C₄×C₈₉	—	4	4	no	—
102	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
105	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
106	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#117	1	2	yes	—
109	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₄×Q₈	1	1	yes	characteristic
112	C₂×C₄×C₈₉	#121	1	2	yes	—
113	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₈₉	#123	1	2	yes	—
115	C₂×C₄×C₈₉	#123	1	2	yes	—
116	C₂×C₄×C₈₉	#124	1	2	yes	—
117	C₂×C₄×C₈₉	#125	1	2	yes	—
118	C₂×C₄×C₈₉	#126	1	2	yes	—
119	C₂×C₄×C₈₉	#126	1	2	yes	—
120	C₂×Dic₈₉	—	4	8	no	—
121	C₂×Dic₈₉	—	4	4	no	—
122	C₈×C₈₉	—	4	4	no	—
123	#10623494	—	2	4	no	—
124	#10623494	—	2	2	no	—
125	#10623494	—	2	2	no	—
126	C₂²×C₈₉	—	2	4	no	—
127	C₂²×C₈₉	—	2	2	no	—
128	C₂²×C₈₉	—	4	2	no	—
129	C₂²×C₈₉	—	4	1	no	—
130	C₂²×C₈₉	—	4	1	no	—
131	C₂²×C₈₉	#376	1	1	yes	characteristic
132	C₂²×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
133	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
134	C₂²×C₈₉	#387	1	2	yes	—
135	C₂²×C₈₉	#388	1	2	yes	—
136	C₄×C₈₉	—	4	4	no	—
137	C₄×C₈₉	#380	1	1	yes	characteristic
138	C₄×C₈₉	#381	1	1	yes	characteristic
139	C₄×C₈₉	#382	1	1	yes	characteristic
140	C₄×C₈₉	#383	1	1	yes	characteristic
141	C₄×C₈₉	#387	1	2	yes	—
142	C₄×C₈₉	#388	1	2	yes	—
143	Dic₈₉	—	8	4	no	—
144	#10803	—	89	1	no	maximal, Sylow
145	C₂×C₈₉	—	2	2	no	—
146	C₂×C₈₉	—	2	1	no	—
147	C₂×C₈₉	—	2	1	no	—
148	C₂×C₈₉	—	2	2	no	—
149	C₂×C₈₉	#1434	1	1	yes	characteristic
150	C₂×C₈₉	#1757	1	2	yes	—
151	C₂×#405	—	89	1	no	—
152	C₂×#344	—	89	2	no	—
153	#1742	D₈₉	1	1	yes	characteristic
154	#1740	—	89	1	no	—
155	#1729	—	89	2	no	—
156	C₈₉	#10803	1	1	yes	characteristic, complemented, Sylow
157	C₂²×M₁₆	—	89	1	no	—
158	C₂²×M₁₆	C₂×D₈₉	1	2	yes	—
159	C₂³×C₈	—	89	1	no	—
160	C₂×#117	—	89	2	no	—
161	C₂×#117	C₂×D₈₉	1	1	yes	characteristic
162	#405	—	178	4	no	—
163	C₂×#97	—	89	2	no	—
164	C₂×C₄×C₈	—	89	2	no	—
165	#344	—	178	4	no	—
166	#343	Dic₈₉	1	4	yes	complemented
167	#342	—	89	4	no	—
168	#336	—	178	4	no	—
169	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	89	2	no	—
176	C₂×M₁₆	—	89	2	no	—
177	C₂×M₁₆	—	178	2	no	—
178	C₂×M₁₆	—	178	2	no	—
179	C₂×M₁₆	C₂×Dic₈₉	1	4	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	89	1	no	—
182	C₂²×C₈	—	89	1	no	—
183	C₂²×C₈	—	89	2	no	—
184	C₂²×C₈	—	89	4	no	—
185	C₂²×C₈	—	178	1	no	—
186	C₂²×C₈	—	178	2	no	—
187	C₂²×C₈	—	178	1	no	—
188	#117	—	2	2	no	—
189	#117	—	2	2	no	—
190	#117	—	178	2	no	—
191	#117	—	178	2	no	—
192	#117	—	178	2	no	—
193	#117	—	178	2	no	—
194	C₂×C₄⋊C₄	—	178	2	no	—
195	C₂×C₄⋊C₄	C₂×Dic₈₉	1	2	yes	—
196	C₂×C₂²⋊C₄	—	89	4	no	—
197	C₂×C₂²⋊C₄	C₄×D₈₉	1	2	yes	—
198	C₂×C₄²	—	178	2	no	—
199	C₂×C₄²	C₄×D₈₉	1	2	yes	—
200	#97	—	178	4	no	—
201	C₄×C₈	—	178	4	no	—
202	C₂⁴	C₂×C₄×D₈₉	1	1	yes	characteristic
203	C₂²×C₄	—	178	4	no	—
204	C₂²×C₄	C₄×Dic₈₉	1	4	yes	—
205	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
206	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
207	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
208	C₂²×C₄	D₄×D₈₉	1	2	yes	—
209	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
210	C₂²×C₄	#13083573	1	2	yes	—
211	C₂²×C₄	#13083573	1	1	yes	characteristic
212	C₂²×C₄	#13083574	1	2	yes	—
213	C₂²×C₄	#13083574	1	2	yes	—
214	C₂²×C₄	#13083575	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	4	2	no	—
219	M₁₆	—	178	4	no	—
220	M₁₆	—	178	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	4	2	no	—
224	C₂×C₈	—	89	2	no	—
225	C₂×C₈	—	89	2	no	—
226	C₂×C₈	—	89	4	no	—
227	C₂×C₈	—	178	1	no	—
228	C₂×C₈	—	178	1	no	—
229	C₂×C₈	—	178	2	no	—
230	C₂×C₈	—	178	1	no	—
231	C₂×C₈	—	178	1	no	—
232	C₂×C₈	—	178	4	no	—
233	C₂×C₈	—	178	4	no	—
234	C₂×C₈	—	178	2	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	356	2	no	—
237	C₄⋊C₄	—	356	2	no	—
238	C₂²⋊C₄	—	2	4	no	—
239	C₂²⋊C₄	—	178	4	no	—
240	C₂²⋊C₄	—	178	4	no	—
241	C₄²	—	2	2	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	356	2	no	—
244	C₄²	—	356	2	no	—
245	C₂³	—	4	2	no	—
246	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
247	C₂³	#15667969	1	2	yes	—
248	C₂³	#15667972	1	2	yes	—
249	C₂³	#15667974	1	1	yes	characteristic
250	C₂³	#15667977	1	1	yes	characteristic
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	356	8	no	—
254	C₂×C₄	—	356	4	no	—
255	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
256	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
257	C₂×C₄	#15667968	1	1	yes	characteristic
258	C₂×C₄	#15667969	1	1	yes	characteristic
259	C₂×C₄	#15667969	1	1	yes	characteristic
260	C₂×C₄	#15667970	1	1	yes	characteristic
261	C₂×C₄	#15667971	1	2	yes	—
262	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
263	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
264	C₂×C₄	Q₈×Dic₈₉	1	1	yes	characteristic
265	C₂×C₄	#15667985	1	2	yes	—
266	C₂×C₄	#15667990	1	2	yes	—
267	C₂×C₄	#15667994	1	2	yes	—
268	C₂×C₄	#15667997	1	2	yes	—
269	C₂×C₄	#15667998	1	2	yes	—
270	C₂×C₄	#15668002	1	2	yes	—
271	C₂×C₄	#15668005	1	2	yes	—
272	C₂×C₄	#15668007	1	2	yes	—
273	C₈	—	4	4	no	—
274	C₈	—	178	4	no	—
275	C₈	—	178	2	no	—
276	C₈	—	178	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	#19151444	1	1	yes	characteristic
283	C₂²	C₂²⋊C₄×Dic₈₉	1	1	yes	characteristic
284	C₂²	C₄⋊C₄×Dic₈₉	1	1	yes	characteristic
285	C₂²	#19151480	1	2	yes	—
286	C₂²	#19151487	1	2	yes	—
287	C₄	—	4	4	no	—
288	C₄	—	712	4	no	—
289	C₄	#19151457	1	1	yes	characteristic
290	C₄	#19151460	1	1	yes	characteristic
291	C₄	#19151462	1	1	yes	characteristic
292	C₄	#19151465	1	1	yes	characteristic
293	C₄	#19151484	1	2	yes	—
294	C₄	#19151491	1	2	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	2	1	no	—
297	C₂	—	2	1	no	—
298	C₂	—	2	2	no	—
299	C₂	#26018477	1	1	yes	characteristic
300	C₂	#26020055	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₈₉
commutator	C₂³×C₈₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₈₉×#1742	C₂
socle	C₂²×C₈₉	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₈₉	C₂²×C₄
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	#1742	D₈₉
hypocenter	C₈₉	#10803
LCS	G ▷ C₂³×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1742

Complements

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

N	H ≅ G/N	type
10. C₂×#15667923	C₄	semidirect
11. C₂×C₄×#10623494	C₄	semidirect
17. #343×C₈₉	C₄	semidirect
156. C₈₉	#10803	semidirect
166. #343	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151554	2
C₂×#19151370	2
#26019985	2
#26019950	2
C₈₉×#1742	2
#10803	89