Subgroups — C₂×#26019865 #36542105

Order 22784.

802 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151950	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151948	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151946	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151946	C₂	1	1	yes	maximal, characteristic
5	C₂×#19151945	C₂	1	1	yes	maximal, characteristic
6	#26019865	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₈×#10623494	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#487×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668012	C₂²	1	1	yes	characteristic
10	C₂×#15668009	C₂²	1	1	yes	characteristic
11	#19151950	—	2	2	no	—
12	#19151950	C₂²	1	4	yes	complemented
13	#19151948	—	2	2	no	—
14	#19151948	C₂²	1	4	yes	complemented
15	#19151946	—	2	2	no	—
16	#19151946	—	2	2	no	—
17	#19151946	C₂²	1	4	yes	—
18	#19151946	C₂²	1	4	yes	—
19	#19151945	—	2	2	no	—
20	#19151945	C₂²	1	4	yes	—
21	C₂×#15667936	—	2	2	no	—
22	C₂×#15667935	—	2	2	no	—
23	C₂×#15667934	—	2	2	no	—
24	C₂×#15667932	—	2	2	no	—
25	C₂×#15667932	—	2	2	no	—
26	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
27	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
28	C₈×#10623494	C₂²	1	4	yes	complemented
29	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
31	#487×C₈₉	—	2	2	no	—
32	#487×C₈₉	C₂²	1	4	yes	complemented
33	C₂×#103×C₈₉	—	2	2	no	—
34	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
35	C₂²×Dic₁₇₈	—	2	1	no	—
36	C₂²×D₃₅₆	—	2	1	no	—
37	C₂²×D₃₅₆	—	2	1	no	—
38	C₂²×D₃₅₆	—	2	1	no	—
39	C₂²×#10623494	D₄	1	2	yes	—
40	C₂²×#10623494	D₄	1	2	yes	—
41	#15668012	—	2	1	no	—
42	#15668012	—	2	1	no	—
43	#15668012	—	2	1	no	—
44	#15668012	C₂³	1	2	yes	—
45	#15668009	—	2	1	no	—
46	#15668009	—	2	1	no	—
47	#15668009	—	2	1	no	—
48	#15668009	C₂³	1	2	yes	—
49	C₂×C₄⋊Dic₈₉	—	2	1	no	—
50	C₂×C₄⋊Dic₈₉	—	2	1	no	—
51	#15667936	—	2	8	no	—
52	#15667935	—	2	8	no	—
53	#15667934	—	2	8	no	—
54	#15667932	—	2	8	no	—
55	#15667932	—	2	8	no	—
56	C₄×#10623494	D₄	1	2	yes	—
57	C₄×#10623494	D₄	1	2	yes	—
58	C₄×#10623494	C₂³	1	2	yes	—
59	C₄×#10623494	C₂³	1	2	yes	—
60	C₂²×Q₈×C₈₉	—	2	1	no	—
61	C₂²×Q₈×C₈₉	—	2	1	no	—
62	C₂²×C₈×C₈₉	D₄	1	2	yes	—
63	C₄⋊Q₈×C₈₉	—	2	1	no	—
64	C₄⋊Q₈×C₈₉	—	2	1	no	—
65	C₄⋊Q₈×C₈₉	—	2	1	no	—
66	C₄⋊Q₈×C₈₉	—	2	1	no	—
67	C₄⋊Q₈×C₈₉	—	2	1	no	—
68	C₄⋊Q₈×C₈₉	—	2	1	no	—
69	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
70	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
72	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
73	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
74	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
75	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
76	#103×C₈₉	—	2	8	no	—
77	C₄×C₈×C₈₉	D₄	1	2	yes	—
78	C₄×C₈×C₈₉	C₂³	1	2	yes	—
79	C₂³×D₈₉	—	4	1	no	—
80	C₂×Dic₁₇₈	—	2	2	no	—
81	C₂×Dic₁₇₈	—	4	4	no	—
82	C₂×Dic₁₇₈	—	4	1	no	—
83	C₂×D₃₅₆	—	2	2	no	—
84	C₂×D₃₅₆	—	2	2	no	—
85	C₂×D₃₅₆	—	2	2	no	—
86	C₂×D₃₅₆	—	4	4	no	—
87	C₂×D₃₅₆	—	4	4	no	—
88	C₂×D₃₅₆	—	4	4	no	—
89	C₂×D₃₅₆	—	4	1	no	—
90	C₂×D₃₅₆	—	4	1	no	—
91	C₂×D₃₅₆	—	4	1	no	—
92	C₂²×Dic₈₉	—	4	1	no	—
93	C₂×#10623494	—	2	4	no	—
94	C₂×#10623494	—	2	4	no	—
95	C₂×#10623494	C₂×D₄	1	4	yes	—
96	C₂×#10623494	C₂×D₄	1	4	yes	—
97	C₄⋊Dic₈₉	—	2	2	no	—
98	C₄⋊Dic₈₉	—	2	2	no	—
99	C₄⋊Dic₈₉	—	4	1	no	—
100	C₄⋊Dic₈₉	—	4	1	no	—
101	C₂×Q₈×C₈₉	—	2	2	no	—
102	C₂×Q₈×C₈₉	—	2	2	no	—
103	C₂×Q₈×C₈₉	—	4	1	no	—
104	C₂×Q₈×C₈₉	—	4	1	no	—
105	C₂×Q₈×C₈₉	—	4	4	no	—
106	C₂×Q₈×C₈₉	—	4	4	no	—
107	C₂²×C₄×C₈₉	—	4	1	no	—
108	C₂²×C₄×C₈₉	—	4	1	no	—
109	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₈₉	—	2	4	no	—
113	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
114	C₄⋊C₄×C₈₉	—	2	2	no	—
115	C₄⋊C₄×C₈₉	—	2	2	no	—
116	C₄⋊C₄×C₈₉	—	2	2	no	—
117	C₄⋊C₄×C₈₉	—	2	2	no	—
118	C₄⋊C₄×C₈₉	—	4	1	no	—
119	C₄⋊C₄×C₈₉	—	4	1	no	—
120	C₄⋊C₄×C₈₉	—	4	1	no	—
121	C₄⋊C₄×C₈₉	—	4	1	no	—
122	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
126	C₂³×C₈₉	#127	1	1	yes	characteristic
127	C₂²×D₈₉	—	4	2	no	—
128	C₂²×D₈₉	—	8	2	no	—
129	C₂²×D₈₉	—	8	2	no	—
130	C₂²×D₈₉	—	8	2	no	—
131	Q₈×C₈₉	—	4	4	no	—
132	Q₈×C₈₉	—	4	4	no	—
133	Q₈×C₈₉	—	8	2	no	—
134	Q₈×C₈₉	—	8	2	no	—
135	C₂×C₄×C₈₉	—	4	2	no	—
136	C₂×C₄×C₈₉	—	4	2	no	—
137	C₂×C₄×C₈₉	—	8	2	no	—
138	C₂×C₄×C₈₉	—	8	2	no	—
139	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
140	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
141	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
142	C₂×C₄×C₈₉	#135	1	4	yes	—
143	C₂×C₄×C₈₉	#135	1	4	yes	—
144	C₂×C₄×C₈₉	#135	1	4	yes	—
145	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×Dic₈₉	—	4	2	no	—
149	C₂×Dic₈₉	—	8	2	no	—
150	D₃₅₆	—	4	4	no	—
151	D₃₅₆	—	4	4	no	—
152	D₃₅₆	—	4	4	no	—
153	D₃₅₆	—	8	2	no	—
154	D₃₅₆	—	8	2	no	—
155	D₃₅₆	—	8	2	no	—
156	Dic₁₇₈	—	4	4	no	—
157	Dic₁₇₈	—	8	2	no	—
158	C₈×C₈₉	—	2	4	no	—
159	#10623494	—	2	4	no	—
160	#10623494	—	2	4	no	—
161	C₂²×C₈₉	#495	1	2	yes	—
162	C₂²×C₈₉	#495	1	2	yes	—
163	C₂²×C₈₉	#495	1	2	yes	—
164	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
165	C₂×D₈₉	—	8	2	no	—
166	C₂×D₈₉	—	8	2	no	—
167	C₂×D₈₉	—	8	2	no	—
168	C₂×D₈₉	—	16	4	no	—
169	C₄×C₈₉	—	8	2	no	—
170	C₄×C₈₉	—	8	2	no	—
171	C₄×C₈₉	#495	1	2	yes	—
172	C₄×C₈₉	#495	1	2	yes	—
173	C₄×C₈₉	#495	1	2	yes	—
174	C₄×C₈₉	C₂×#135	1	2	yes	—
175	C₄×C₈₉	C₂×#135	1	2	yes	—
176	C₄×C₈₉	C₂×#135	1	2	yes	—
177	Dic₈₉	—	8	2	no	—
178	C₂×#1711	—	89	1	no	maximal, Sylow
179	C₂×C₈₉	#1711	1	4	yes	complemented
180	C₂×C₈₉	C₂×#495	1	1	yes	characteristic
181	C₂×C₈₉	C₂×#495	1	1	yes	characteristic
182	C₂×C₈₉	C₂×#495	1	1	yes	characteristic
183	D₈₉	—	16	2	no	—
184	C₂×#487	—	89	1	no	—
185	C₂×#487	—	89	1	no	—
186	C₂×#487	D₈₉	1	1	yes	characteristic, complemented
187	C₂×#486	—	89	1	no	—
188	C₂×#486	—	89	1	no	—
189	C₂×#486	—	89	1	no	—
190	#1711	—	89	8	no	—
191	C₂×C₈²	—	89	1	no	—
192	C₈₉	C₂×#1711	1	1	yes	characteristic, complemented, Sylow
193	C₂×C₄⋊Q₈	—	89	1	no	—
194	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
195	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
196	C₂×#127	—	89	1	no	—
197	#487	—	2	2	no	—
198	#487	—	89	4	no	—
199	#487	—	89	4	no	—
200	#487	—	178	2	no	—
201	#487	—	178	2	no	—
202	#487	C₂×D₈₉	1	4	yes	complemented
203	#486	—	89	4	no	—
204	#486	—	89	4	no	—
205	#486	—	89	4	no	—
206	#486	—	178	2	no	—
207	#486	—	178	2	no	—
208	#486	—	178	2	no	—
209	C₂×#103	—	2	2	no	—
210	C₂×#103	—	178	2	no	—
211	C₂×#103	—	178	2	no	—
212	C₂×#102	—	178	2	no	—
213	C₂×#102	—	178	2	no	—
214	C₂×#102	—	178	2	no	—
215	C₂×C₄×C₈	—	89	1	no	—
216	C₂×C₄×C₈	—	89	1	no	—
217	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
218	C₈²	—	89	4	no	—
219	C₂²×Q₈	—	2	1	no	—
220	C₂²×Q₈	—	2	1	no	—
221	C₂²×Q₈	—	178	1	no	—
222	C₂²×D₄	—	178	1	no	—
223	C₂²×D₄	—	178	1	no	—
224	C₂²×D₄	—	178	1	no	—
225	C₂²×C₈	—	89	2	no	—
226	C₂²×C₈	—	89	2	no	—
227	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	C₄⋊Q₈	—	2	1	no	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	89	2	no	—
235	C₄⋊Q₈	—	178	1	no	—
236	C₄⋊Q₈	—	178	1	no	—
237	C₄⋊Q₈	—	178	1	no	—
238	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
239	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
240	#127	—	89	2	no	—
241	#127	—	178	1	no	—
242	#127	—	178	1	no	—
243	#127	—	178	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	2	1	no	—
248	C₂×C₄⋊C₄	—	178	1	no	—
249	C₂×C₄⋊C₄	—	178	1	no	—
250	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
251	#103	—	2	8	no	—
252	#103	—	178	8	no	—
253	#103	—	178	8	no	—
254	#102	—	178	8	no	—
255	#102	—	178	8	no	—
256	#102	—	178	8	no	—
257	C₄×C₈	—	89	2	no	—
258	C₄×C₈	—	89	2	no	—
259	C₄×C₈	—	89	2	no	—
260	C₄×C₈	—	89	2	no	—
261	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
262	C₄×C₈	C₂²×D₈₉	1	2	yes	—
263	C₂⁴	—	356	1	no	—
264	C₂×Q₈	—	2	2	no	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	4	1	no	—
267	C₂×Q₈	—	4	1	no	—
268	C₂×Q₈	—	4	4	no	—
269	C₂×Q₈	—	4	4	no	—
270	C₂×Q₈	—	356	1	no	—
271	C₂×Q₈	—	356	4	no	—
272	C₂×Q₈	—	178	2	no	—
273	C₂×D₄	—	356	1	no	—
274	C₂×D₄	—	356	1	no	—
275	C₂×D₄	—	356	1	no	—
276	C₂×D₄	—	356	4	no	—
277	C₂×D₄	—	356	4	no	—
278	C₂×D₄	—	356	4	no	—
279	C₂×D₄	—	178	2	no	—
280	C₂×D₄	—	178	2	no	—
281	C₂×D₄	—	178	2	no	—
282	C₂²×C₄	—	4	1	no	—
283	C₂²×C₄	—	4	1	no	—
284	C₂²×C₄	—	356	1	no	—
285	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
286	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
287	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
288	C₂×C₈	—	2	4	no	—
289	C₂×C₈	—	89	4	no	—
290	C₂×C₈	—	89	4	no	—
291	C₂×C₈	—	178	4	no	—
292	C₂×C₈	—	178	4	no	—
293	C₂×C₈	C₂×C₂²⋊D₈₉	1	4	yes	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	4	1	no	—
299	C₄⋊C₄	—	4	1	no	—
300	C₄⋊C₄	—	4	1	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	356	1	no	—
303	C₄⋊C₄	—	356	1	no	—
304	C₄⋊C₄	—	178	2	no	—
305	C₄⋊C₄	—	178	2	no	—
306	C₄²	D₄×D₈₉	1	1	yes	characteristic
307	C₄²	D₄×D₈₉	1	1	yes	characteristic
308	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
309	C₄²	C₂³×D₈₉	1	1	yes	characteristic
310	C₂³	—	356	2	no	—
311	C₂³	—	712	2	no	—
312	C₂³	—	712	2	no	—
313	C₂³	—	712	2	no	—
314	C₂³	#15668008	1	1	yes	characteristic
315	Q₈	—	4	4	no	—
316	Q₈	—	4	4	no	—
317	Q₈	—	8	2	no	—
318	Q₈	—	8	2	no	—
319	Q₈	—	356	4	no	—
320	Q₈	—	712	2	no	—
321	D₄	—	356	4	no	—
322	D₄	—	356	4	no	—
323	D₄	—	356	4	no	—
324	D₄	—	712	2	no	—
325	D₄	—	712	2	no	—
326	D₄	—	712	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	2	no	—
330	C₂×C₄	—	8	2	no	—
331	C₂×C₄	—	356	2	no	—
332	C₂×C₄	—	712	2	no	—
333	C₂×C₄	#15668008	1	1	yes	characteristic
334	C₂×C₄	#15668008	1	1	yes	characteristic
335	C₂×C₄	#15668008	1	1	yes	characteristic
336	C₂×C₄	#15668032	1	4	yes	—
337	C₂×C₄	#15668033	1	4	yes	—
338	C₂×C₄	#15668035	1	4	yes	—
339	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
341	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
342	C₈	—	2	4	no	—
343	C₈	—	178	4	no	—
344	C₈	—	178	4	no	—
345	C₂²	—	1424	4	no	—
346	C₂²	—	712	2	no	—
347	C₂²	—	712	2	no	—
348	C₂²	—	712	2	no	—
349	C₂²	#19151979	1	2	yes	—
350	C₂²	#19151980	1	2	yes	—
351	C₂²	#19151983	1	2	yes	—
352	C₂²	C₂×#15668008	1	1	yes	characteristic
353	C₄	—	8	2	no	—
354	C₄	—	8	2	no	—
355	C₄	—	712	2	no	—
356	C₄	#19151979	1	2	yes	—
357	C₄	#19151980	1	2	yes	—
358	C₄	#19151983	1	2	yes	—
359	C₄	C₂×#15668032	1	2	yes	—
360	C₄	C₂×#15668033	1	2	yes	—
361	C₄	C₂×#15668035	1	2	yes	—
362	C₂	—	1424	2	no	—
363	C₂	#26019865	1	4	yes	direct product factor
364	C₂	C₂×#19151979	1	1	yes	characteristic
365	C₂	C₂×#19151980	1	1	yes	characteristic
366	C₂	C₂×#19151983	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15668008
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#487×C₈₉	C₂
socle	C₂³×C₈₉	#127
prime radical	C₂×#15668009	C₂²
hypercenter	C₂×#487	D₈₉
hypocenter	C₈₉	C₂×#1711
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#487

Complements

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

N	H ≅ G/N	type
1. C₂×#19151950	C₂	semidirect
2. C₂×#19151948	C₂	semidirect
6. #26019865	C₂	direct (N × H)
7. C₂×C₈×#10623494	C₂	semidirect
8. C₂×#487×C₈₉	C₂	semidirect
12. #19151950	C₂²	semidirect
14. #19151948	C₂²	semidirect
28. C₈×#10623494	C₂²	semidirect
32. #487×C₈₉	C₂²	semidirect
179. C₂×C₈₉	#1711	semidirect
186. C₂×#487	D₈₉	semidirect
192. C₈₉	C₂×#1711	semidirect
202. #487	C₂×D₈₉	semidirect
363. C₂	#26019865	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×#19151950	2
C₂×#19151948	2
C₂×#19151946	2
C₂×#19151945	2
#26019865	2
C₂×C₈×#10623494	2
C₂×#487×C₈₉	2
C₂×#1711	89

Subgroups — C2×#26019865 #36542105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019865 #36542105