Subgroups — C₂×#26018543 #36520030

Order 22784.

902 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151649	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151649	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₈×D₈₉	C₂	1	1	yes	maximal, characteristic
4	#26018543	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19151320	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151320	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151319	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₈₉	C₄	1	2	yes	—
10	C₂×C₄²×D₈₉	C₂²	1	1	yes	characteristic
11	#19151649	C₄	1	4	yes	—
12	#19151649	C₄	1	4	yes	—
13	#19151649	C₂²	1	4	yes	—
14	#19151649	C₂²	1	4	yes	—
15	C₂×#15667941	C₄	1	2	yes	—
16	C₂×#15667941	C₄	1	2	yes	—
17	C₂×#15667924	C₄	1	2	yes	—
18	C₂×#15667924	C₄	1	2	yes	—
19	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
20	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
21	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₈₉	C₄	1	2	yes	—
23	C₄×C₈×D₈₉	C₄	1	4	yes	—
24	C₄×C₈×D₈₉	C₂²	1	4	yes	—
25	#19151320	C₂²	1	4	yes	complemented
26	#19151320	C₂²	1	4	yes	complemented
27	#19151319	C₂²	1	4	yes	complemented
28	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
31	#322×C₈₉	C₂²	1	4	yes	complemented
32	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×#10623494	C₂×C₄	1	2	yes	—
36	C₂²×#10623494	C₂×C₄	1	2	yes	—
37	C₂²×#10623494	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₈₉	—	2	8	no	—
39	C₂×C₈×D₈₉	C₂×C₄	1	8	yes	—
40	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
43	C₄²×D₈₉	C₂×C₄	1	2	yes	—
44	C₄²×D₈₉	C₂×C₄	1	2	yes	—
45	C₄²×D₈₉	C₂×C₄	1	2	yes	—
46	C₄²×D₈₉	C₂³	1	2	yes	—
47	#15667941	C₂×C₄	1	8	yes	—
48	#15667941	C₂×C₄	1	8	yes	—
49	#15667924	C₂×C₄	1	8	yes	—
50	#15667924	C₂×C₄	1	8	yes	—
51	C₄×#10623494	C₂×C₄	1	2	yes	—
52	C₄×#10623494	C₂×C₄	1	2	yes	—
53	C₄×#10623494	C₂×C₄	1	2	yes	—
54	C₄×#10623494	C₂³	1	2	yes	—
55	C₄×#10623494	C₂³	1	2	yes	—
56	C₄×#10623494	C₂³	1	2	yes	—
57	C₈×Dic₈₉	C₂×C₄	1	8	yes	—
58	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
59	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
60	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
61	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₈₉	C₂³	1	2	yes	—
66	C₄×C₈×C₈₉	C₂³	1	2	yes	—
67	C₄×C₈×C₈₉	C₂³	1	2	yes	—
68	C₂³×D₈₉	C₄²	1	1	yes	characteristic
69	C₂²×Dic₈₉	C₄²	1	1	yes	characteristic
70	C₂²×Dic₈₉	C₄²	1	1	yes	characteristic
71	C₂²×Dic₈₉	C₄²	1	1	yes	characteristic
72	C₂×C₄×D₈₉	—	2	4	no	—
73	C₂×C₄×D₈₉	—	2	4	no	—
74	C₂×C₄×D₈₉	C₄²	1	2	yes	—
75	C₂×C₄×D₈₉	C₄²	1	2	yes	—
76	C₂×C₄×D₈₉	C₄²	1	2	yes	—
77	C₂×C₄×D₈₉	M₁₆	1	8	yes	—
78	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
79	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
80	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
81	C₂×#10623494	—	2	4	no	—
82	C₂×#10623494	—	2	4	no	—
83	C₂×#10623494	—	2	4	no	—
84	C₂×#10623494	C₂²×C₄	1	4	yes	—
85	C₂×#10623494	C₂²×C₄	1	4	yes	—
86	C₂×#10623494	C₂²×C₄	1	4	yes	—
87	C₈×D₈₉	—	2	16	no	—
88	C₄×Dic₈₉	C₄²	1	2	yes	—
89	C₄×Dic₈₉	C₄²	1	2	yes	—
90	C₄×Dic₈₉	C₄²	1	2	yes	—
91	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
92	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
93	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₈₉	—	2	4	no	—
98	C₂×C₈×C₈₉	—	2	4	no	—
99	C₂×C₈×C₈₉	—	2	4	no	—
100	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
101	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
102	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
103	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
107	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
108	C₂²×D₈₉	—	2	2	no	—
109	C₂²×D₈₉	—	2	2	no	—
110	C₂²×D₈₉	#97	1	4	yes	—
111	C₂²×D₈₉	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
116	C₂×C₄×C₈₉	#131	1	4	yes	—
117	C₂×C₄×C₈₉	#131	1	4	yes	—
118	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
121	C₂×Dic₈₉	—	2	2	no	—
122	C₂×Dic₈₉	—	2	2	no	—
123	C₂×Dic₈₉	#97	1	4	yes	—
124	C₂×Dic₈₉	C₂×C₄²	1	2	yes	—
125	C₂×Dic₈₉	C₂×C₄²	1	2	yes	—
126	C₂×Dic₈₉	C₂×C₄²	1	2	yes	—
127	C₄×D₈₉	—	2	4	no	—
128	C₄×D₈₉	—	2	4	no	—
129	C₄×D₈₉	—	2	4	no	—
130	C₄×D₈₉	—	2	4	no	—
131	C₄×D₈₉	#97	1	8	yes	—
132	C₄×D₈₉	C₂×M₁₆	1	8	yes	—
133	C₈×C₈₉	—	2	4	no	—
134	C₈×C₈₉	—	2	4	no	—
135	C₈×C₈₉	—	2	4	no	—
136	#10623494	—	2	4	no	—
137	#10623494	—	2	4	no	—
138	#10623494	—	2	4	no	—
139	C₂²×C₈₉	C₂×#97	1	2	yes	—
140	C₂²×C₈₉	#405	1	2	yes	—
141	C₂²×C₈₉	#405	1	2	yes	—
142	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
143	C₂×D₈₉	—	2	2	no	—
144	C₂×D₈₉	—	2	2	no	—
145	C₂×D₈₉	—	2	8	no	—
146	C₂×D₈₉	C₂×#97	1	4	yes	—
147	C₄×C₈₉	C₂×#97	1	2	yes	—
148	C₄×C₈₉	#405	1	2	yes	—
149	C₄×C₈₉	#405	1	2	yes	—
150	C₄×C₈₉	C₂²×M₁₆	1	2	yes	—
151	C₄×C₈₉	C₂×#131	1	2	yes	—
152	C₄×C₈₉	C₂×#131	1	2	yes	—
153	Dic₈₉	—	2	2	no	—
154	Dic₈₉	—	2	2	no	—
155	Dic₈₉	C₂×#97	1	4	yes	—
156	C₂×#1452	—	89	1	no	maximal, Sylow
157	C₂×C₈₉	#1452	1	4	yes	complemented
158	C₂×C₈₉	C₂²×#97	1	1	yes	characteristic
159	C₂×C₈₉	C₂×#405	1	1	yes	characteristic
160	C₂×C₈₉	C₂×#405	1	1	yes	characteristic
161	D₈₉	—	2	4	no	—
162	C₂×#431	—	89	1	no	—
163	C₂×#431	—	89	1	no	—
164	C₂²×C₄×C₈	—	89	1	no	—
165	#1452	—	89	8	no	—
166	C₂×#322	—	89	1	no	—
167	C₂×#322	—	89	1	no	—
168	C₂×#322	—	89	1	no	—
169	C₂×#322	D₈₉	1	1	yes	characteristic, complemented
170	C₈₉	C₂×#1452	1	1	yes	characteristic, complemented, Sylow
171	C₂³×C₈	—	89	2	no	—
172	C₂²×C₄²	—	89	1	no	—
173	#431	—	89	4	no	—
174	#431	—	89	4	no	—
175	#431	—	89	4	no	—
176	#431	—	89	4	no	—
177	C₂×C₄⋊C₈	—	89	2	no	—
178	C₂×C₄⋊C₈	—	89	2	no	—
179	C₂×C₂²⋊C₈	—	89	2	no	—
180	C₂×C₂²⋊C₈	—	89	2	no	—
181	C₂×C₄×C₈	—	89	2	no	—
182	C₂×C₄×C₈	—	89	1	no	—
183	C₂×C₄×C₈	—	89	1	no	—
184	C₂×C₄×C₈	—	89	1	no	—
185	C₂×C₄×C₈	—	89	4	no	—
186	C₂×C₄×C₈	—	89	4	no	—
187	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
188	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
189	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
190	#322	—	89	4	no	—
191	#322	—	89	4	no	—
192	#322	—	89	4	no	—
193	#322	C₂×D₈₉	1	4	yes	complemented
194	C₂³×C₄	—	89	1	no	—
195	C₂³×C₄	—	89	1	no	—
196	C₂³×C₄	—	89	1	no	—
197	C₂²×C₈	—	89	2	no	—
198	C₂²×C₈	—	89	2	no	—
199	C₂²×C₈	—	89	2	no	—
200	C₂²×C₈	—	89	8	no	—
201	C₂²×C₈	—	178	8	no	—
202	C₂²×C₈	C₄×D₈₉	1	2	yes	—
203	C₂²×C₈	C₄×D₈₉	1	2	yes	—
204	C₂²×C₈	C₄×D₈₉	1	2	yes	—
205	C₂×C₄²	—	89	1	no	—
206	C₂×C₄²	—	89	1	no	—
207	C₂×C₄²	—	89	1	no	—
208	C₂×C₄²	—	89	2	no	—
209	C₂×C₄²	—	89	2	no	—
210	C₂×C₄²	—	89	2	no	—
211	C₂×C₄²	—	89	2	no	—
212	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
213	C₄⋊C₈	—	89	8	no	—
214	C₄⋊C₈	—	89	8	no	—
215	C₂²⋊C₈	—	89	8	no	—
216	C₂²⋊C₈	—	89	8	no	—
217	C₄×C₈	—	89	8	no	—
218	C₄×C₈	—	89	2	no	—
219	C₄×C₈	—	89	2	no	—
220	C₄×C₈	—	89	2	no	—
221	C₄×C₈	—	89	2	no	—
222	C₄×C₈	—	89	2	no	—
223	C₄×C₈	—	89	2	no	—
224	C₄×C₈	C₄×D₈₉	1	2	yes	—
225	C₄×C₈	C₄×D₈₉	1	2	yes	—
226	C₄×C₈	C₄×D₈₉	1	2	yes	—
227	C₄×C₈	C₂²×D₈₉	1	2	yes	—
228	C₄×C₈	C₂²×D₈₉	1	2	yes	—
229	C₄×C₈	C₂²×D₈₉	1	2	yes	—
230	C₂⁴	—	89	1	no	—
231	C₂²×C₄	—	89	1	no	—
232	C₂²×C₄	—	89	8	no	—
233	C₂²×C₄	—	89	2	no	—
234	C₂²×C₄	—	89	2	no	—
235	C₂²×C₄	—	89	1	no	—
236	C₂²×C₄	—	89	1	no	—
237	C₂²×C₄	—	89	2	no	—
238	C₂²×C₄	—	89	2	no	—
239	C₂²×C₄	—	89	2	no	—
240	C₂²×C₄	—	89	2	no	—
241	C₂²×C₄	—	178	4	no	—
242	C₂²×C₄	—	178	4	no	—
243	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	4	no	—
249	C₂×C₈	—	89	4	no	—
250	C₂×C₈	—	89	4	no	—
251	C₂×C₈	—	89	4	no	—
252	C₂×C₈	—	178	16	no	—
253	C₂×C₈	—	178	4	no	—
254	C₂×C₈	—	178	4	no	—
255	C₂×C₈	—	178	4	no	—
256	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
257	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
258	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
259	C₄²	—	89	2	no	—
260	C₄²	—	89	2	no	—
261	C₄²	—	89	2	no	—
262	C₄²	—	89	2	no	—
263	C₄²	—	89	2	no	—
264	C₄²	—	89	2	no	—
265	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
266	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
267	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
268	C₄²	C₂³×D₈₉	1	1	yes	characteristic
269	C₂³	—	89	2	no	—
270	C₂³	—	89	4	no	—
271	C₂³	—	178	2	no	—
272	C₂³	—	178	2	no	—
273	C₂³	C₄²×D₈₉	1	1	yes	characteristic
274	C₂×C₄	—	89	2	no	—
275	C₂×C₄	—	89	4	no	—
276	C₂×C₄	—	89	8	no	—
277	C₂×C₄	—	89	8	no	—
278	C₂×C₄	—	89	2	no	—
279	C₂×C₄	—	89	2	no	—
280	C₂×C₄	—	178	2	no	—
281	C₂×C₄	—	178	2	no	—
282	C₂×C₄	—	178	4	no	—
283	C₂×C₄	—	178	4	no	—
284	C₂×C₄	—	178	4	no	—
285	C₂×C₄	—	178	4	no	—
286	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
287	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
288	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
289	C₂×C₄	M₁₆×D₈₉	1	4	yes	—
290	C₂×C₄	#15668018	1	4	yes	—
291	C₂×C₄	#15668020	1	4	yes	—
292	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
293	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
294	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	2	4	no	—
297	C₈	—	2	4	no	—
298	C₈	—	178	4	no	—
299	C₈	—	178	4	no	—
300	C₈	—	178	4	no	—
301	C₂²	—	89	4	no	—
302	C₂²	—	178	2	no	—
303	C₂²	—	178	2	no	—
304	C₂²	—	178	8	no	—
305	C₂²	#97×D₈₉	1	2	yes	—
306	C₂²	#19151552	1	2	yes	—
307	C₂²	#19151555	1	2	yes	—
308	C₂²	C₂×C₄²×D₈₉	1	1	yes	characteristic
309	C₄	—	89	4	no	—
310	C₄	—	178	2	no	—
311	C₄	—	178	2	no	—
312	C₄	#97×D₈₉	1	2	yes	—
313	C₄	#19151552	1	2	yes	—
314	C₄	#19151555	1	2	yes	—
315	C₄	C₂×M₁₆×D₈₉	1	2	yes	—
316	C₄	C₂×#15668018	1	2	yes	—
317	C₄	C₂×#15668020	1	2	yes	—
318	C₂	—	178	4	no	—
319	C₂	#26018543	1	4	yes	direct product factor
320	C₂	C₂×#97×D₈₉	1	1	yes	characteristic
321	C₂	C₂×#19151552	1	1	yes	characteristic
322	C₂	C₂×#19151555	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂²×C₄²
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₄²
prime radical	C₂³×D₈₉	C₄²
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	C₂×#1452
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
4. #26018543	C₂	direct (N × H)
5. C₂×#19151320	C₂	semidirect
6. C₂×#19151320	C₂	semidirect
7. C₂×#19151319	C₂	semidirect
8. C₂×#322×C₈₉	C₂	semidirect
25. #19151320	C₂²	semidirect
26. #19151320	C₂²	semidirect
27. #19151319	C₂²	semidirect
31. #322×C₈₉	C₂²	semidirect
157. C₂×C₈₉	#1452	semidirect
169. C₂×#322	D₈₉	semidirect
170. C₈₉	C₂×#1452	semidirect
193. #322	C₂×D₈₉	semidirect
319. C₂	#26018543	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19151649	2
C₂×C₄×C₈×D₈₉	2
#26018543	2
C₂×#19151320	2
C₂×#19151319	2
C₂×#322×C₈₉	2
C₂×#1452	89

Subgroups — C2×#26018543 #36520030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018543 #36520030