Subgroups — #36470915

Order 22784.

384 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152014	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151419	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022010	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19151381	C₂	1	1	yes	maximal, characteristic
5	#26020089	C₂	1	1	yes	maximal, characteristic
6	#26018340	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668018	C₂²	1	1	yes	characteristic
9	#19152014	—	2	2	no	—
10	#19152014	—	2	2	no	—
11	C₂×#15667952	C₂²	1	1	yes	characteristic
12	C₂×#15667952	C₂²	1	1	yes	characteristic
13	C₂×#15667951	—	2	1	no	—
14	C₂×C₁₆×D₈₉	—	2	1	no	—
15	C₂×#15667940	C₄	1	1	yes	characteristic
16	#19151595	C₄	1	1	yes	characteristic
17	#19151553	C₂²	1	1	yes	characteristic
18	#19151419	—	2	2	no	—
19	#19151416	—	2	1	no	—
20	#19151416	—	2	1	no	—
21	#19151381	—	2	2	no	—
22	#19151363	C₄	1	1	yes	characteristic
23	#19151358	C₄	1	1	yes	characteristic
24	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
28	#15668018	—	2	1	no	—
29	#15668018	—	2	1	no	—
30	#15668018	D₄	1	2	yes	—
31	#15668018	Q₈	1	2	yes	—
32	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
33	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
34	C₂×#13083557	—	2	1	no	—
35	C₂×C₈×D₈₉	—	2	1	no	—
36	#15667971	C₂×C₄	1	1	yes	characteristic
37	#15667952	—	2	1	no	—
38	#15667952	—	2	1	no	—
39	#15667952	—	2	1	no	—
40	#15667952	D₄	1	2	yes	—
41	#15667951	—	2	4	no	—
42	C₂×#13083549	—	2	1	no	—
43	C₂×#13083549	—	2	1	no	—
44	C₁₆×D₈₉	—	2	4	no	—
45	#15667940	—	2	2	no	—
46	#15667937	C₈	1	1	yes	characteristic
47	#15667937	C₈	1	1	yes	characteristic
48	#15667934	C₈	1	1	yes	characteristic
49	#15667934	C₈	1	1	yes	characteristic
50	#15667922	—	2	1	no	—
51	C₈×Dic₈₉	—	2	1	no	—
52	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₂×C₁₆×C₈₉	—	2	1	no	—
56	C₂×C₁₆×C₈₉	—	2	1	no	—
57	C₂×C₁₆×C₈₉	—	2	1	no	—
58	C₂×C₁₆×C₈₉	—	2	1	no	—
59	C₂×C₁₆×C₈₉	—	2	1	no	—
60	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
61	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
62	C₄×C₈×C₈₉	—	2	1	no	—
63	C₄×C₈×C₈₉	—	2	1	no	—
64	#13083574	—	2	1	no	—
65	#13083574	—	2	1	no	—
66	#13083574	C₄⋊C₄	1	2	yes	—
67	#13083574	M₁₆	1	2	yes	—
68	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
69	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
70	C₂×C₂²⋊D₈₉	—	2	1	no	—
71	C₂×C₄×D₈₉	—	2	1	no	—
72	#13083558	—	2	1	no	—
73	#13083558	—	2	1	no	—
74	#13083558	—	2	1	no	—
75	#13083558	C₄⋊C₄	1	2	yes	—
76	#13083557	—	2	2	no	—
77	#13083557	—	2	2	no	—
78	C₂×#10623494	—	2	1	no	—
79	C₂×#10623494	—	2	1	no	—
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	C₂²⋊Dic₈₉	—	2	1	no	—
85	C₄×Dic₈₉	—	2	1	no	—
86	#13083549	—	2	2	no	—
87	#13083549	—	4	1	no	—
88	C₂²×C₄×C₈₉	—	2	1	no	—
89	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₈₉	—	2	1	no	—
91	C₂×C₈×C₈₉	—	2	1	no	—
92	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
98	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
99	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
100	C₄²×C₈₉	—	2	1	no	—
101	C₄²×C₈₉	—	2	1	no	—
102	C₁₆×C₈₉	—	2	1	no	—
103	C₁₆×C₈₉	—	2	1	no	—
104	C₁₆×C₈₉	—	4	1	no	—
105	C₂³×C₈₉	#131	1	1	yes	characteristic
106	C₂²×D₈₉	—	2	1	no	—
107	C₂×C₄×C₈₉	—	2	2	no	—
108	C₂×C₄×C₈₉	—	4	1	no	—
109	C₂×C₄×C₈₉	—	4	1	no	—
110	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
116	C₂²⋊D₈₉	—	2	2	no	—
117	C₂²⋊D₈₉	—	2	2	no	—
118	C₂×Dic₈₉	—	2	1	no	—
119	C₂×Dic₈₉	—	2	1	no	—
120	C₂×Dic₈₉	—	2	1	no	—
121	D₃₅₆	—	2	1	no	—
122	D₃₅₆	C₄⋊C₈	1	2	yes	—
123	C₄×D₈₉	—	2	2	no	—
124	C₄×D₈₉	—	2	2	no	—
125	Dic₁₇₈	—	2	1	no	—
126	Dic₁₇₈	C₄⋊C₈	1	2	yes	—
127	C₈×C₈₉	—	2	1	no	—
128	C₈×C₈₉	—	2	1	no	—
129	C₈×C₈₉	#120	1	1	yes	characteristic
130	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
131	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
132	C₈×C₈₉	#123	1	1	yes	characteristic
133	#10623494	—	2	2	no	—
134	#10623494	—	4	1	no	—
135	C₂²×C₈₉	—	2	1	no	—
136	C₂²×C₈₉	—	2	1	no	—
137	C₂²×C₈₉	#424	1	1	yes	characteristic
138	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
139	C₂²×C₈₉	#435	1	1	yes	characteristic
140	C₂×D₈₉	—	2	2	no	—
141	C₂×D₈₉	—	4	1	no	—
142	C₂×D₈₉	—	4	1	no	—
143	C₄×C₈₉	—	4	2	no	—
144	C₄×C₈₉	#385	1	1	yes	characteristic
145	C₄×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
146	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
147	C₄×C₈₉	#435	1	1	yes	characteristic
148	Dic₈₉	—	2	2	no	—
149	Dic₈₉	—	4	1	no	—
150	Dic₈₉	—	4	1	no	—
151	#12917	—	89	1	no	maximal, Sylow
152	C₂×C₈₉	—	2	1	no	—
153	C₂×C₈₉	—	2	1	no	—
154	C₂×C₈₉	#1850	1	1	yes	characteristic
155	C₂×C₈₉	#2170	1	1	yes	characteristic
156	C₂×C₈₉	#2171	1	1	yes	characteristic
157	D₈₉	—	4	2	no	—
158	C₂×#504	—	89	1	no	—
159	C₂×#364	—	89	1	no	—
160	#2151	—	89	1	no	—
161	C₂×#350	—	89	1	no	—
162	#1764	—	89	1	no	—
163	#1379	—	89	1	no	—
164	#1374	D₈₉	1	1	yes	characteristic, complemented
165	C₈₉	#12917	1	1	yes	characteristic, complemented, Sylow
166	C₂×#131	—	89	1	no	—
167	#504	—	178	2	no	—
168	#504	—	178	2	no	—
169	C₂×M₃₂	—	89	1	no	—
170	C₂×M₃₂	—	89	1	no	—
171	C₂×M₃₂	—	178	1	no	—
172	C₂²×C₁₆	—	178	1	no	—
173	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
174	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
175	C₂×#104	—	89	1	no	—
176	#416	—	89	1	no	—
177	#405	—	89	1	no	—
178	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
179	#364	—	178	2	no	—
180	C₄⋊C₁₆	—	178	1	no	—
181	C₄⋊C₁₆	—	178	1	no	—
182	#350	—	178	2	no	—
183	#341	—	89	1	no	—
184	#339	—	89	1	no	—
185	C₂×#41	—	89	1	no	—
186	#131	—	89	2	no	—
187	#131	—	89	2	no	—
188	#131	—	178	1	no	—
189	#131	—	178	1	no	—
190	C₂×M₁₆	—	89	1	no	—
191	C₂×M₁₆	—	89	1	no	—
192	C₂×M₁₆	—	178	1	no	—
193	C₂²×C₈	—	178	1	no	—
194	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
195	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
196	#117	—	89	1	no	—
197	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
198	M₃₂	—	89	2	no	—
199	M₃₂	—	178	4	no	—
200	M₃₂	—	178	1	no	—
201	M₃₂	—	178	1	no	—
202	M₃₂	—	178	1	no	—
203	C₂×C₁₆	—	2	1	no	—
204	C₂×C₁₆	—	2	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	178	4	no	—
209	C₂×C₁₆	—	178	1	no	—
210	C₂×C₁₆	—	178	1	no	—
211	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
212	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
213	#104	—	178	2	no	—
214	#103	—	89	1	no	—
215	#103	—	89	1	no	—
216	#102	—	89	1	no	—
217	#102	—	89	1	no	—
218	#97	—	178	1	no	—
219	C₄×C₈	—	2	1	no	—
220	C₄×C₈	—	2	1	no	—
221	C₄×C₈	—	178	1	no	—
222	#41	—	89	2	no	—
223	#41	—	89	2	no	—
224	#41	—	178	1	no	—
225	#41	—	178	1	no	—
226	C₂×Q₈	—	89	1	no	—
227	C₂×D₄	—	89	1	no	—
228	C₂×D₄	—	178	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	178	1	no	—
231	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
232	M₁₆	—	89	2	no	—
233	M₁₆	—	178	1	no	—
234	M₁₆	—	178	1	no	—
235	M₁₆	—	178	1	no	—
236	M₁₆	—	178	2	no	—
237	M₁₆	—	178	2	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	178	1	no	—
241	C₂×C₈	—	178	1	no	—
242	C₂×C₈	—	178	2	no	—
243	C₂×C₈	—	178	2	no	—
244	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
245	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
246	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
247	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
248	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
249	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
250	C₂×C₈	#13083573	1	1	yes	characteristic
251	C₂×C₈	#13083574	1	1	yes	characteristic
252	C₄⋊C₄	—	89	1	no	—
253	C₄⋊C₄	—	89	1	no	—
254	C₂²⋊C₄	—	178	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	178	1	no	—
258	C₁₆	—	2	1	no	—
259	C₁₆	—	2	1	no	—
260	C₁₆	—	4	1	no	—
261	C₁₆	—	356	1	no	—
262	C₁₆	—	178	2	no	—
263	C₂³	—	178	1	no	—
264	C₂³	#15668018	1	1	yes	characteristic
265	Q₈	—	89	2	no	—
266	Q₈	—	178	1	no	—
267	D₄	—	89	2	no	—
268	D₄	—	178	2	no	—
269	D₄	—	178	2	no	—
270	D₄	—	178	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	178	1	no	—
275	C₂×C₄	—	178	1	no	—
276	C₂×C₄	—	178	1	no	—
277	C₂×C₄	—	178	2	no	—
278	C₂×C₄	—	178	2	no	—
279	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
280	C₂×C₄	#15667972	1	1	yes	characteristic
281	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
282	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
283	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
284	C₂×C₄	#15668020	1	1	yes	characteristic
285	C₈	—	2	1	no	—
286	C₈	—	2	1	no	—
287	C₈	—	356	1	no	—
288	C₈	—	178	2	no	—
289	C₈	#15667982	1	1	yes	characteristic
290	C₈	#15667990	1	1	yes	characteristic
291	C₈	#15667992	1	1	yes	characteristic
292	C₈	#15667994	1	1	yes	characteristic
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	356	1	no	—
296	C₂²	—	356	1	no	—
297	C₂²	—	178	2	no	—
298	C₂²	#19151623	1	1	yes	characteristic
299	C₂²	C₈×D₃₅₆	1	1	yes	characteristic
300	C₂²	#19151672	1	1	yes	characteristic
301	C₄	—	4	2	no	—
302	C₄	—	356	1	no	—
303	C₄	—	356	1	no	—
304	C₄	—	178	2	no	—
305	C₄	#19151472	1	1	yes	characteristic
306	C₄	C₄⋊C₈×D₈₉	1	1	yes	characteristic
307	C₄	#19151661	1	1	yes	characteristic
308	C₄	#19151671	1	1	yes	characteristic
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	—	356	2	no	—
312	C₂	#26020431	1	1	yes	characteristic
313	C₂	#26022090	1	1	yes	characteristic
314	C₂	#26022095	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1374	C₂
socle	C₂²×C₈₉	C₈×D₄
prime radical	C₂×#13083574	C₂×C₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1374	D₈₉
hypocenter	C₈₉	#12917
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ #1374

Complements

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

N	H ≅ G/N	type
2. C₂×#19151419	C₂	semidirect
3. #26022010	C₂	semidirect
6. #26018340	C₂	semidirect
7. C₈₉×#1374	C₂	semidirect
164. #1374	D₈₉	semidirect
165. C₈₉	#12917	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152014	2
C₂×#19151419	2
#26022010	2
C₂×#19151381	2
#26020089	2
#26018340	2
C₈₉×#1374	2
#12917	89