Subgroups — C₂×#26022023 #36707438

Order 22784.

658 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#108×D₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151633	C₂	1	1	yes	maximal, characteristic
4	#26022023	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19151422	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151421	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151420	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#365×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×M₁₆×D₈₉	—	2	1	no	—
10	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
11	C₂×#15667970	—	2	1	no	—
12	C₂×#15667952	C₂²	1	1	yes	characteristic
13	C₂×#15667951	C₄	1	2	yes	complemented
14	M₃₂×D₈₉	C₄	1	4	yes	complemented
15	M₃₂×D₈₉	C₂²	1	4	yes	—
16	C₂×C₁₆×D₈₉	C₄	1	2	yes	complemented
17	C₂×#15667948	C₂²	1	1	yes	characteristic
18	C₂×#15667947	—	2	1	no	—
19	#108×D₈₉	—	2	2	no	—
20	#108×D₈₉	C₂²	1	4	yes	—
21	#19151633	—	2	2	no	—
22	#19151633	C₂²	1	4	yes	—
23	C₂×#15667945	—	2	1	no	—
24	C₂×#15667944	C₂²	1	1	yes	characteristic
25	#19151422	C₂²	1	4	yes	complemented
26	#19151421	C₂²	1	4	yes	complemented
27	#19151420	C₂²	1	4	yes	complemented
28	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
31	#365×C₈₉	C₂²	1	4	yes	complemented
32	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
33	C₂×#13083558	—	2	1	no	—
34	C₂×#13083557	—	4	1	no	—
35	M₁₆×D₈₉	—	2	4	no	—
36	M₁₆×D₈₉	—	4	2	no	—
37	C₂²×#10623494	D₄	1	1	yes	characteristic
38	C₂×C₈×D₈₉	—	4	1	no	—
39	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₈₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₈×D₈₉	D₄	1	2	yes	—
43	C₂×C₈×D₈₉	Q₈	1	2	yes	—
44	C₂×C₈×D₈₉	Q₈	1	1	yes	characteristic
45	C₂×C₈×D₈₉	Q₈	1	1	yes	characteristic
46	C₂×C₈×D₈₉	C₂³	1	2	yes	—
47	#15667970	—	2	4	no	—
48	#15667970	—	4	2	no	—
49	C₂×C₄⋊Dic₈₉	—	2	1	no	—
50	C₂×#13083551	—	4	1	no	—
51	C₂×C₄×Dic₈₉	—	4	1	no	—
52	#15667952	C₂×C₄	1	2	yes	—
53	#15667952	C₂³	1	2	yes	—
54	#15667951	C₂×C₄	1	8	yes	complemented
55	C₂×#13083549	C₂×C₄	1	2	yes	—
56	C₁₆×D₈₉	C₂×C₄	1	8	yes	complemented
57	#15667948	—	2	1	no	—
58	#15667948	C₂³	1	2	yes	—
59	#15667947	—	2	4	no	—
60	#15667945	—	2	4	no	—
61	#15667944	—	2	1	no	—
62	#15667944	C₂³	1	2	yes	—
63	C₂×M₁₆×C₈₉	—	2	1	no	—
64	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
66	M₃₂×C₈₉	C₂×C₄	1	2	yes	—
67	M₃₂×C₈₉	C₂³	1	2	yes	—
68	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
69	#108×C₈₉	—	2	1	no	—
70	#108×C₈₉	C₂³	1	2	yes	—
71	#106×C₈₉	—	2	1	no	—
72	#106×C₈₉	C₂³	1	2	yes	—
73	C₂³×D₈₉	SD₁₆	1	1	yes	characteristic
74	C₂²×Dic₈₉	—	4	1	no	—
75	C₂²×Dic₈₉	SD₁₆	1	1	yes	characteristic
76	C₂×C₄×D₈₉	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×D₈₉	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×D₈₉	C₄⋊C₄	1	2	yes	—
79	C₂×C₄×D₈₉	SD₁₆	1	2	yes	—
80	C₂×C₄×D₈₉	SD₁₆	1	1	yes	characteristic
81	C₂×C₄×D₈₉	SD₁₆	1	2	yes	—
82	C₂×C₄×D₈₉	SD₁₆	1	1	yes	characteristic
83	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
84	#13083558	—	2	2	no	—
85	#13083558	—	4	1	no	—
86	#13083557	—	4	4	no	—
87	C₂×#10623494	—	4	1	no	—
88	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
89	C₂×#10623494	C₄⋊C₄	1	2	yes	—
90	C₂×#10623494	C₄⋊C₄	1	1	yes	characteristic
91	C₂×#10623494	C₂×D₄	1	2	yes	—
92	C₈×D₈₉	—	4	4	no	—
93	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
94	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
95	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
96	C₈×D₈₉	C₄⋊C₄	1	2	yes	—
97	C₈×D₈₉	C₂²×C₄	1	2	yes	—
98	C₈×D₈₉	C₂²×C₄	1	2	yes	—
99	C₈×D₈₉	C₂×Q₈	1	2	yes	—
100	C₈×D₈₉	C₂×Q₈	1	2	yes	—
101	C₄⋊Dic₈₉	—	2	2	no	—
102	C₄⋊Dic₈₉	—	4	1	no	—
103	#13083551	—	4	4	no	—
104	C₄×Dic₈₉	—	4	4	no	—
105	#13083549	C₂²×C₄	1	4	yes	—
106	C₂²×C₄×C₈₉	—	4	1	no	—
107	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
108	M₁₆×C₈₉	—	2	2	no	—
109	M₁₆×C₈₉	—	4	1	no	—
110	C₂×C₈×C₈₉	—	4	1	no	—
111	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
115	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
116	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
117	C₄⋊C₄×C₈₉	—	2	2	no	—
118	C₄⋊C₄×C₈₉	—	4	1	no	—
119	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
120	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
121	C₂²×D₈₉	—	4	2	no	—
122	C₂²×D₈₉	#106	1	1	yes	characteristic
123	C₂²×D₈₉	#106	1	1	yes	characteristic
124	C₂²×D₈₉	#106	1	2	yes	—
125	C₂²×D₈₉	C₂×SD₁₆	1	2	yes	—
126	C₂×C₄×C₈₉	—	4	2	no	—
127	C₂×C₄×C₈₉	—	8	2	no	—
128	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
134	C₂×Dic₈₉	—	4	2	no	—
135	C₂×Dic₈₉	—	8	2	no	—
136	C₂×Dic₈₉	#106	1	2	yes	—
137	C₂×Dic₈₉	#106	1	1	yes	characteristic
138	C₂×Dic₈₉	#106	1	1	yes	characteristic
139	C₂×Dic₈₉	C₂×SD₁₆	1	2	yes	—
140	C₄×D₈₉	#106	1	2	yes	—
141	C₄×D₈₉	#106	1	2	yes	—
142	C₄×D₈₉	#106	1	2	yes	—
143	C₄×D₈₉	#106	1	2	yes	—
144	C₄×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
145	C₄×D₈₉	C₂×C₄⋊C₄	1	2	yes	—
146	C₄×D₈₉	C₂×SD₁₆	1	2	yes	—
147	C₄×D₈₉	C₂×SD₁₆	1	2	yes	—
148	C₈×C₈₉	—	4	2	no	—
149	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
150	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
152	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
153	#10623494	—	4	2	no	—
154	#10623494	C₂×C₄⋊C₄	1	2	yes	—
155	#10623494	C₂×C₄⋊C₄	1	2	yes	—
156	C₂²×C₈₉	—	2	2	no	—
157	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
158	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
159	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
160	C₂×D₈₉	—	2	2	no	—
161	C₂×D₈₉	—	2	2	no	—
162	C₂×D₈₉	—	4	2	no	—
163	C₂×D₈₉	—	4	2	no	—
164	C₂×D₈₉	C₂×#106	1	2	yes	—
165	C₂×D₈₉	C₂×#106	1	2	yes	—
166	C₄×C₈₉	—	8	2	no	—
167	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
168	C₄×C₈₉	C₂×#106	1	1	yes	characteristic
169	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
170	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
171	Dic₈₉	—	8	2	no	—
172	Dic₈₉	C₂×#106	1	2	yes	—
173	Dic₈₉	C₂×#106	1	2	yes	—
174	C₂×#2155	—	89	1	no	maximal, Sylow
175	C₂×C₈₉	—	2	1	no	—
176	C₂×C₈₉	—	2	1	no	—
177	C₂×C₈₉	#2155	1	2	yes	complemented
178	C₂×C₈₉	C₂²×#106	1	1	yes	characteristic
179	D₈₉	—	2	2	no	—
180	D₈₉	—	2	2	no	—
181	C₂²×M₃₂	—	89	1	no	—
182	C₂²×#108	—	89	1	no	—
183	C₂×#427	—	89	1	no	—
184	#2155	—	89	8	no	—
185	C₂×#365	—	89	1	no	—
186	C₂×#365	—	89	1	no	—
187	C₂×#365	—	89	1	no	—
188	C₂×#365	D₈₉	1	1	yes	characteristic, complemented
189	C₈₉	C₂×#2155	1	1	yes	characteristic, complemented, Sylow
190	C₂²×M₁₆	—	178	1	no	—
191	C₂³×C₈	—	89	1	no	—
192	C₂×#117	—	178	1	no	—
193	C₂×M₃₂	—	89	1	no	—
194	C₂×M₃₂	—	89	4	no	—
195	C₂×M₃₂	—	89	4	no	—
196	C₂×M₃₂	—	89	2	no	—
197	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
198	C₂²×C₁₆	—	89	2	no	—
199	C₂×#108	—	89	1	no	—
200	C₂×#108	—	89	4	no	—
201	C₂×#108	—	178	2	no	—
202	C₂×#108	—	178	1	no	—
203	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
204	#427	—	89	4	no	—
205	#427	—	178	2	no	—
206	C₂×#107	—	178	1	no	—
207	C₂×#106	—	89	1	no	—
208	C₂×#106	C₂×D₈₉	1	1	yes	characteristic
209	#365	—	89	4	no	—
210	#365	—	89	4	no	—
211	#365	—	89	4	no	—
212	#365	C₂×D₈₉	1	4	yes	complemented
213	C₂³×C₄	—	89	1	no	—
214	C₂×M₁₆	—	2	1	no	—
215	C₂×M₁₆	—	356	1	no	—
216	C₂×M₁₆	—	356	2	no	—
217	C₂×M₁₆	—	178	4	no	—
218	C₂×M₁₆	—	178	1	no	—
219	C₂²×C₈	—	89	1	no	—
220	C₂²×C₈	—	89	1	no	—
221	C₂²×C₈	—	89	2	no	—
222	C₂²×C₈	—	89	1	no	—
223	C₂²×C₈	—	89	2	no	—
224	C₂²×C₈	—	89	2	no	—
225	C₂²×C₈	—	89	1	no	—
226	C₂²×C₈	—	89	1	no	—
227	C₂²×C₈	—	89	2	no	—
228	C₂²×C₈	—	356	1	no	—
229	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
230	#117	—	356	2	no	—
231	#117	—	178	4	no	—
232	C₂×C₄⋊C₄	—	2	1	no	—
233	C₂×C₄⋊C₄	—	178	1	no	—
234	C₂×C₂²⋊C₄	—	356	1	no	—
235	C₂×C₄²	—	356	1	no	—
236	M₃₂	—	89	2	no	—
237	M₃₂	—	89	2	no	—
238	M₃₂	—	89	8	no	—
239	M₃₂	C₄×D₈₉	1	2	yes	—
240	M₃₂	C₂²×D₈₉	1	2	yes	—
241	C₂×C₁₆	—	89	2	no	—
242	C₂×C₁₆	—	89	8	no	—
243	C₂×C₁₆	C₄×D₈₉	1	2	yes	—
244	#108	—	2	1	no	—
245	#108	—	89	2	no	—
246	#108	—	178	4	no	—
247	#108	—	178	1	no	—
248	#108	C₂²×D₈₉	1	2	yes	—
249	#107	—	178	4	no	—
250	#106	—	2	1	no	—
251	#106	—	89	2	no	—
252	#106	—	178	1	no	—
253	#106	C₂²×D₈₉	1	2	yes	—
254	C₂⁴	—	89	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	89	1	no	—
257	C₂²×C₄	—	89	2	no	—
258	C₂²×C₄	—	89	1	no	—
259	C₂²×C₄	—	89	2	no	—
260	C₂²×C₄	—	89	1	no	—
261	C₂²×C₄	—	89	2	no	—
262	C₂²×C₄	—	89	2	no	—
263	C₂²×C₄	—	89	1	no	—
264	C₂²×C₄	—	89	1	no	—
265	C₂²×C₄	—	356	1	no	—
266	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
267	M₁₆	—	2	2	no	—
268	M₁₆	—	4	1	no	—
269	M₁₆	—	356	4	no	—
270	M₁₆	—	356	1	no	—
271	M₁₆	—	178	2	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	89	2	no	—
274	C₂×C₈	—	89	2	no	—
275	C₂×C₈	—	89	1	no	—
276	C₂×C₈	—	89	1	no	—
277	C₂×C₈	—	89	2	no	—
278	C₂×C₈	—	89	2	no	—
279	C₂×C₈	—	89	2	no	—
280	C₂×C₈	—	89	2	no	—
281	C₂×C₈	—	89	2	no	—
282	C₂×C₈	—	89	2	no	—
283	C₂×C₈	—	89	2	no	—
284	C₂×C₈	—	89	2	no	—
285	C₂×C₈	—	356	4	no	—
286	C₂×C₈	—	356	1	no	—
287	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
288	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
289	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
290	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
291	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
292	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	356	1	no	—
296	C₄⋊C₄	—	178	2	no	—
297	C₂²⋊C₄	—	356	4	no	—
298	C₄²	—	356	4	no	—
299	C₁₆	—	89	4	no	—
300	C₁₆	C₂×C₄×D₈₉	1	4	yes	—
301	C₂³	—	89	2	no	—
302	C₂³	—	89	2	no	—
303	C₂³	—	89	1	no	—
304	C₂³	—	89	1	no	—
305	C₂³	—	356	2	no	—
306	C₂³	SD₁₆×D₈₉	1	1	yes	characteristic
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	2	no	—
309	C₂×C₄	—	89	2	no	—
310	C₂×C₄	—	89	1	no	—
311	C₂×C₄	—	89	1	no	—
312	C₂×C₄	—	89	2	no	—
313	C₂×C₄	—	89	2	no	—
314	C₂×C₄	—	89	2	no	—
315	C₂×C₄	—	89	2	no	—
316	C₂×C₄	—	89	2	no	—
317	C₂×C₄	—	89	2	no	—
318	C₂×C₄	—	89	2	no	—
319	C₂×C₄	—	89	2	no	—
320	C₂×C₄	—	89	2	no	—
321	C₂×C₄	—	356	2	no	—
322	C₂×C₄	—	712	2	no	—
323	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
324	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
325	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
326	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
327	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
329	C₈	—	4	2	no	—
330	C₈	—	89	2	no	—
331	C₈	—	89	2	no	—
332	C₈	—	356	2	no	—
333	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
334	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
335	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
336	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
337	C₂²	—	2	2	no	—
338	C₂²	—	89	2	no	—
339	C₂²	—	89	2	no	—
340	C₂²	—	356	2	no	—
341	C₂²	—	356	2	no	—
342	C₂²	—	178	2	no	—
343	C₂²	—	178	2	no	—
344	C₂²	#106×D₈₉	1	1	yes	characteristic
345	C₂²	#106×D₈₉	1	1	yes	characteristic
346	C₂²	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
347	C₄	—	8	2	no	—
348	C₄	—	89	2	no	—
349	C₄	—	89	2	no	—
350	C₄	—	712	2	no	—
351	C₄	#106×D₈₉	1	1	yes	characteristic
352	C₄	#106×D₈₉	1	1	yes	characteristic
353	C₄	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
354	C₄	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
355	C₂	—	2	1	no	—
356	C₂	—	2	1	no	—
357	C₂	—	178	2	no	—
358	C₂	—	178	2	no	—
359	C₂	#26022023	1	2	yes	direct product factor
360	C₂	C₂×#106×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#106×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#365×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#106
prime radical	C₂³×D₈₉	SD₁₆
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#365	D₈₉
hypocenter	C₈₉	C₂×#2155
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

Complements

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

N	H ≅ G/N	type
4. #26022023	C₂	direct (N × H)
5. C₂×#19151422	C₂	semidirect
6. C₂×#19151421	C₂	semidirect
7. C₂×#19151420	C₂	semidirect
8. C₂×#365×C₈₉	C₂	semidirect
13. C₂×#15667951	C₄	semidirect
14. M₃₂×D₈₉	C₄	semidirect
16. C₂×C₁₆×D₈₉	C₄	semidirect
25. #19151422	C₂²	semidirect
26. #19151421	C₂²	semidirect
27. #19151420	C₂²	semidirect
31. #365×C₈₉	C₂²	semidirect
54. #15667951	C₂×C₄	semidirect
56. C₁₆×D₈₉	C₂×C₄	semidirect
177. C₂×C₈₉	#2155	semidirect
188. C₂×#365	D₈₉	semidirect
189. C₈₉	C₂×#2155	semidirect
212. #365	C₂×D₈₉	semidirect
359. C₂	#26022023	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×M₃₂×D₈₉	2
C₂×#108×D₈₉	2
C₂×#19151633	2
#26022023	2
C₂×#19151422	2
C₂×#19151421	2
C₂×#19151420	2
C₂×#365×C₈₉	2
C₂×#2155	89

Subgroups — C2×#26022023 #36707438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022023 #36707438