Subgroups — #36469541

Order 22784.

418 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152015	C₂	1	1	yes	maximal, characteristic
2	#26021851	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151373	C₂	1	1	yes	maximal, characteristic, complemented
4	#26020087	C₂	1	1	yes	maximal, characteristic
5	#26018340	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018333	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#350×C₈₉	C₂	1	1	yes	maximal, characteristic
8	#19152015	—	2	2	no	—
9	#19152015	C₄	1	4	yes	complemented
10	C₂×#15667952	—	2	1	no	—
11	C₂×#15667952	C₂²	1	1	yes	characteristic
12	C₂²×#13083549	—	2	1	no	—
13	C₂²×#13083549	C₂²	1	1	yes	characteristic
14	C₂×#15667939	C₄	1	1	yes	characteristic
15	#19151594	C₄	1	1	yes	characteristic
16	#19151553	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
18	#19151373	—	2	2	no	—
19	#19151373	—	2	1	no	—
20	C₁₆×Dic₈₉	—	2	1	no	—
21	#19151362	C₄	1	1	yes	characteristic
22	#19151359	C₄	1	1	yes	characteristic
23	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
26	#350×C₈₉	C₄	1	4	yes	complemented
27	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
28	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
29	#15667971	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
31	#15667952	—	2	4	no	—
32	#15667952	—	2	1	no	—
33	#15667952	C₂×C₄	1	2	yes	—
34	C₂×#13083549	—	2	4	no	—
35	C₂×#13083549	—	2	1	no	—
36	C₂×#13083549	C₂×C₄	1	2	yes	—
37	C₂×#13083549	D₄	1	1	yes	characteristic
38	C₂×#13083549	D₄	1	1	yes	characteristic
39	C₂×#13083549	D₄	1	2	yes	—
40	#15667939	C₈	1	4	yes	—
41	#15667936	C₈	1	2	yes	—
42	#15667933	C₈	1	2	yes	—
43	#15667922	—	2	1	no	—
44	C₈×Dic₈₉	—	2	1	no	—
45	C₈×Dic₈₉	—	2	1	no	—
46	C₈×Dic₈₉	—	2	1	no	—
47	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
48	#131×C₈₉	—	2	1	no	—
49	#131×C₈₉	—	2	1	no	—
50	#131×C₈₉	C₂×C₄	1	2	yes	—
51	#131×C₈₉	C₂×C₄	1	2	yes	—
52	C₂×M₁₆×C₈₉	—	2	1	no	—
53	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	—	2	1	no	—
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	M₃₂×C₈₉	—	2	1	no	—
57	M₃₂×C₈₉	C₂×C₄	1	2	yes	—
58	C₂×C₁₆×C₈₉	—	2	1	no	—
59	C₂×C₁₆×C₈₉	—	2	1	no	—
60	C₂×C₁₆×C₈₉	—	2	1	no	—
61	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
62	C₂²×Dic₈₉	—	2	1	no	—
63	#13083558	—	2	1	no	—
64	#13083558	C₂×C₈	1	2	yes	—
65	C₂×#10623494	—	2	1	no	—
66	C₂×#10623494	—	2	1	no	—
67	C₂×#10623494	—	2	1	no	—
68	C₂×#10623494	C₂×C₈	1	2	yes	—
69	C₄⋊Dic₈₉	C₂×C₈	1	2	yes	—
70	C₂²⋊Dic₈₉	—	2	1	no	—
71	C₄×Dic₈₉	—	2	1	no	—
72	C₄×Dic₈₉	—	2	1	no	—
73	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
74	#13083549	—	2	2	no	—
75	#13083549	C₂²⋊C₄	1	2	yes	—
76	#13083549	C₂²⋊C₄	1	2	yes	—
77	#41×C₈₉	—	2	1	no	—
78	#41×C₈₉	—	2	1	no	—
79	#41×C₈₉	C₄²	1	2	yes	—
80	#41×C₈₉	C₂×C₈	1	2	yes	—
81	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
82	C₂×D₄×C₈₉	—	2	1	no	—
83	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	—	2	1	no	—
85	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
86	M₁₆×C₈₉	—	2	1	no	—
87	M₁₆×C₈₉	—	2	2	no	—
88	M₁₆×C₈₉	—	2	2	no	—
89	M₁₆×C₈₉	C₄²	1	2	yes	—
90	C₂×C₈×C₈₉	—	2	1	no	—
91	C₂×C₈×C₈₉	—	2	2	no	—
92	C₂×C₈×C₈₉	—	2	2	no	—
93	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂²×C₄	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	#41	1	1	yes	characteristic
98	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
99	C₁₆×C₈₉	—	2	2	no	—
100	C₁₆×C₈₉	—	2	2	no	—
101	C₂³×C₈₉	—	2	1	no	—
102	C₂³×C₈₉	#131	1	1	yes	characteristic
103	Q₈×C₈₉	—	2	1	no	—
104	Q₈×C₈₉	C₄×C₈	1	2	yes	—
105	D₄×C₈₉	—	2	2	no	—
106	D₄×C₈₉	—	2	2	no	—
107	D₄×C₈₉	—	2	1	no	—
108	D₄×C₈₉	C₄×C₈	1	2	yes	—
109	C₂×C₄×C₈₉	—	2	1	no	—
110	C₂×C₄×C₈₉	—	2	2	no	—
111	C₂×C₄×C₈₉	—	2	2	no	—
112	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
118	C₂×Dic₈₉	—	2	2	no	—
119	C₂×Dic₈₉	—	2	2	no	—
120	C₂×Dic₈₉	—	4	2	no	—
121	C₈×C₈₉	—	2	2	no	—
122	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₈×C₈₉	#117	1	1	yes	characteristic
124	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
125	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
126	#10623494	—	2	2	no	—
127	#10623494	—	2	2	no	—
128	C₂²×C₈₉	—	2	2	no	—
129	C₂²×C₈₉	—	2	2	no	—
130	C₂²×C₈₉	—	4	2	no	—
131	C₂²×C₈₉	#405	1	1	yes	characteristic
132	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
133	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
134	C₄×C₈₉	—	2	2	no	—
135	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₈₉	C₂×C₄×C₈	1	1	yes	characteristic
137	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
138	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
139	Dic₈₉	—	4	2	no	—
140	Dic₈₉	—	4	2	no	—
141	#12615	—	89	1	no	maximal, Sylow
142	C₂×C₈₉	—	2	1	no	—
143	C₂×C₈₉	—	2	1	no	—
144	C₂×C₈₉	—	4	2	no	—
145	C₂×C₈₉	C₈×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₈₉	#2171	1	2	yes	—
147	C₂×#504	—	89	1	no	—
148	C₂×#350	D₈₉	1	1	yes	characteristic
149	#2108	—	89	1	no	—
150	C₂×#346	—	89	1	no	—
151	#1764	—	89	1	no	—
152	#1379	—	89	1	no	—
153	#1376	—	89	1	no	—
154	C₈₉	#12615	1	1	yes	characteristic, complemented, Sylow
155	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
156	#504	—	89	4	no	—
157	#504	—	178	2	no	—
158	C₂×M₃₂	—	89	1	no	—
159	C₂×M₃₂	—	178	1	no	—
160	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
161	C₂²×C₁₆	—	89	1	no	—
162	C₂²×C₁₆	—	178	1	no	—
163	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
164	C₂×#104	—	89	1	no	—
165	#416	—	89	1	no	—
166	#405	—	89	1	no	—
167	C₂×C₄×C₈	—	89	1	no	—
168	#350	Dic₈₉	1	4	yes	complemented
169	#346	—	178	2	no	—
170	#346	—	178	1	no	—
171	C₄×C₁₆	—	178	1	no	—
172	#341	—	89	1	no	—
173	#339	—	89	1	no	—
174	C₂×#41	C₄×D₈₉	1	1	yes	characteristic
175	#131	—	2	1	no	—
176	#131	—	2	1	no	—
177	#131	C₄×D₈₉	1	2	yes	—
178	#131	C₂×Dic₈₉	1	2	yes	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	89	1	no	—
181	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	—	89	1	no	—
184	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
185	#117	—	89	1	no	—
186	C₂×C₄²	—	89	1	no	—
187	M₃₂	—	2	1	no	—
188	M₃₂	—	89	2	no	—
189	M₃₂	—	178	4	no	—
190	M₃₂	—	178	1	no	—
191	M₃₂	C₂×Dic₈₉	1	2	yes	—
192	C₂×C₁₆	—	2	1	no	—
193	C₂×C₁₆	—	2	1	no	—
194	C₂×C₁₆	—	2	1	no	—
195	C₂×C₁₆	—	89	2	no	—
196	C₂×C₁₆	—	89	2	no	—
197	C₂×C₁₆	—	89	1	no	—
198	C₂×C₁₆	—	89	1	no	—
199	C₂×C₁₆	—	178	4	no	—
200	C₂×C₁₆	—	178	1	no	—
201	C₂×C₁₆	C₂×Dic₈₉	1	2	yes	—
202	#104	—	89	4	no	—
203	#103	—	89	2	no	—
204	#102	—	89	2	no	—
205	#97	—	178	1	no	—
206	C₄×C₈	—	178	1	no	—
207	C₄×C₈	—	178	1	no	—
208	C₄×C₈	—	178	1	no	—
209	#41	—	2	1	no	—
210	#41	—	2	1	no	—
211	#41	C₄×Dic₈₉	1	2	yes	—
212	#41	C₈×D₈₉	1	2	yes	—
213	C₂×Q₈	C₈×D₈₉	1	1	yes	characteristic
214	C₂×D₄	—	2	1	no	—
215	C₂×D₄	C₈×D₈₉	1	1	yes	characteristic
216	C₂²×C₄	—	2	1	no	—
217	C₂²×C₄	—	178	1	no	—
218	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
219	M₁₆	—	2	1	no	—
220	M₁₆	—	2	2	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	89	2	no	—
223	M₁₆	—	178	1	no	—
224	M₁₆	C₄×Dic₈₉	1	2	yes	—
225	C₂×C₈	—	2	1	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	89	2	no	—
229	C₂×C₈	—	178	1	no	—
230	C₂×C₈	—	178	1	no	—
231	C₂×C₈	—	178	1	no	—
232	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
233	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
234	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
235	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
236	C₂×C₈	#13083573	1	1	yes	characteristic
237	C₂×C₈	#13083573	1	1	yes	characteristic
238	C₄⋊C₄	—	89	2	no	—
239	C₂²⋊C₄	—	178	1	no	—
240	C₄²	—	89	2	no	—
241	C₄²	—	178	1	no	—
242	C₄²	—	178	1	no	—
243	C₁₆	—	2	2	no	—
244	C₁₆	—	2	2	no	—
245	C₁₆	—	89	2	no	—
246	C₁₆	—	89	2	no	—
247	C₁₆	—	178	2	no	—
248	C₂³	—	2	1	no	—
249	C₂³	#15668020	1	1	yes	characteristic
250	Q₈	—	2	1	no	—
251	Q₈	C₈×Dic₈₉	1	2	yes	—
252	D₄	—	2	2	no	—
253	D₄	—	2	2	no	—
254	D₄	—	2	1	no	—
255	D₄	C₈×Dic₈₉	1	2	yes	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	356	2	no	—
260	C₂×C₄	—	178	2	no	—
261	C₂×C₄	—	178	2	no	—
262	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
263	C₂×C₄	#15667972	1	1	yes	characteristic
264	C₂×C₄	#15667972	1	1	yes	characteristic
265	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
266	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
267	C₂×C₄	#15668020	1	1	yes	characteristic
268	C₈	—	2	2	no	—
269	C₈	—	178	2	no	—
270	C₈	—	178	2	no	—
271	C₈	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
272	C₈	#15667968	1	1	yes	characteristic
273	C₈	D₄×Dic₈₉	1	1	yes	characteristic
274	C₈	D₄×Dic₈₉	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	#19151556	1	1	yes	characteristic
279	C₂²	#19151661	1	1	yes	characteristic
280	C₂²	#19151661	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	356	2	no	—
283	C₄	—	356	2	no	—
284	C₄	C₂²⋊C₄×Dic₈₉	1	1	yes	characteristic
285	C₄	C₂×C₈×Dic₈₉	1	1	yes	characteristic
286	C₄	#19151661	1	1	yes	characteristic
287	C₄	#19151661	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	4	2	no	—
291	C₂	#26020026	1	1	yes	characteristic
292	C₂	#26022091	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₈₉
commutator	C₄×C₈₉	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#350×C₈₉	C₂
socle	C₂²×C₈₉	C₈×D₄
prime radical	C₂×#41×C₈₉	C₂×C₄
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#350	D₈₉
hypocenter	C₈₉	#12615
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#350

Complements

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

N	H ≅ G/N	type
2. #26021851	C₂	semidirect
3. C₂×#19151373	C₂	semidirect
5. #26018340	C₂	semidirect
6. #26018333	C₂	semidirect
9. #19152015	C₄	semidirect
26. #350×C₈₉	C₄	semidirect
154. C₈₉	#12615	semidirect
168. #350	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152015	2
#26021851	2
C₂×#19151373	2
#26020087	2
#26018340	2
#26018333	2
C₂×#350×C₈₉	2
#12615	89