Subgroups — C₂×#26019876 #36542143

Order 22784.

914 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152001	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151994	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151972	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151971	C₂	1	1	yes	maximal, characteristic
5	C₂×#19151946	C₂	1	1	yes	maximal, characteristic
6	#26019876	C₂	1	8	yes	maximal, direct product factor
7	C₂×#19151320	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#487×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#13083563	—	2	2	no	—
10	C₂²×#13083561	—	2	2	no	—
11	C₂×#15668012	C₂²	1	1	yes	characteristic
12	C₂×#15668009	C₂²	1	1	yes	characteristic
13	#19152001	—	2	2	no	—
14	#19152001	C₂²	1	4	yes	complemented
15	#19151994	—	2	2	no	—
16	#19151994	C₂²	1	4	yes	complemented
17	#19151972	—	2	2	no	—
18	#19151972	C₂²	1	4	yes	—
19	#19151971	—	2	2	no	—
20	#19151971	C₂²	1	4	yes	—
21	#19151946	—	2	2	no	—
22	#19151946	C₂²	1	4	yes	—
23	C₂×#15667945	—	2	2	no	—
24	C₂×#15667943	—	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	#19151320	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	C₂×#13083563	—	2	8	no	—
40	C₂×#13083563	—	4	8	no	—
41	C₂×#13083561	—	2	8	no	—
42	C₂×#13083561	—	4	8	no	—
43	C₂²×#10623494	D₄	1	2	yes	—
44	C₂²×#10623494	D₄	1	2	yes	—
45	#15668012	—	2	1	no	—
46	#15668012	—	2	1	no	—
47	#15668012	—	2	1	no	—
48	#15668012	C₂³	1	2	yes	—
49	#15668009	—	2	1	no	—
50	#15668009	—	2	1	no	—
51	#15668009	—	2	1	no	—
52	#15668009	C₂³	1	2	yes	—
53	C₂×C₄⋊Dic₈₉	—	2	1	no	—
54	C₂×C₄⋊Dic₈₉	—	2	1	no	—
55	#15667945	—	2	8	no	—
56	#15667943	—	2	8	no	—
57	#15667932	—	2	8	no	—
58	C₄×#10623494	D₄	1	2	yes	—
59	C₄×#10623494	D₄	1	2	yes	—
60	C₄×#10623494	C₂³	1	2	yes	—
61	C₄×#10623494	C₂³	1	2	yes	—
62	C₂²×Q₈×C₈₉	—	2	1	no	—
63	C₂²×Q₈×C₈₉	—	2	1	no	—
64	C₂²×C₈×C₈₉	D₄	1	2	yes	—
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	—	2	1	no	—
70	C₄⋊Q₈×C₈₉	—	2	1	no	—
71	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
72	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
74	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
75	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
76	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
77	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
78	#103×C₈₉	—	2	8	no	—
79	C₄×C₈×C₈₉	D₄	1	2	yes	—
80	C₄×C₈×C₈₉	C₂³	1	2	yes	—
81	C₂³×D₈₉	—	4	1	no	—
82	C₂×Dic₁₇₈	—	2	2	no	—
83	C₂×Dic₁₇₈	—	4	4	no	—
84	C₂×Dic₁₇₈	—	4	1	no	—
85	C₂×D₃₅₆	—	2	2	no	—
86	C₂×D₃₅₆	—	2	2	no	—
87	C₂×D₃₅₆	—	2	2	no	—
88	C₂×D₃₅₆	—	4	4	no	—
89	C₂×D₃₅₆	—	4	4	no	—
90	C₂×D₃₅₆	—	4	4	no	—
91	C₂×D₃₅₆	—	4	1	no	—
92	C₂×D₃₅₆	—	4	1	no	—
93	C₂×D₃₅₆	—	4	1	no	—
94	C₂²×Dic₈₉	—	4	1	no	—
95	#13083563	—	4	16	no	—
96	#13083561	—	4	16	no	—
97	C₂×#10623494	—	2	4	no	—
98	C₂×#10623494	SD₁₆	1	8	yes	complemented
99	C₂×#10623494	C₂×D₄	1	4	yes	—
100	C₂×#10623494	C₂×D₄	1	4	yes	—
101	C₄⋊Dic₈₉	—	2	2	no	—
102	C₄⋊Dic₈₉	—	2	2	no	—
103	C₄⋊Dic₈₉	—	4	1	no	—
104	C₄⋊Dic₈₉	—	4	1	no	—
105	C₂×Q₈×C₈₉	—	2	2	no	—
106	C₂×Q₈×C₈₉	—	2	2	no	—
107	C₂×Q₈×C₈₉	—	4	1	no	—
108	C₂×Q₈×C₈₉	—	4	1	no	—
109	C₂×Q₈×C₈₉	—	4	4	no	—
110	C₂×Q₈×C₈₉	—	4	4	no	—
111	C₂²×C₄×C₈₉	—	4	1	no	—
112	C₂²×C₄×C₈₉	—	4	1	no	—
113	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₈₉	—	2	4	no	—
117	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
118	C₄⋊C₄×C₈₉	—	2	2	no	—
119	C₄⋊C₄×C₈₉	—	2	2	no	—
120	C₄⋊C₄×C₈₉	—	2	2	no	—
121	C₄⋊C₄×C₈₉	—	2	2	no	—
122	C₄⋊C₄×C₈₉	—	4	1	no	—
123	C₄⋊C₄×C₈₉	—	4	1	no	—
124	C₄⋊C₄×C₈₉	—	4	1	no	—
125	C₄⋊C₄×C₈₉	—	4	1	no	—
126	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
130	C₂³×C₈₉	#127	1	1	yes	characteristic
131	C₂²×D₈₉	—	4	2	no	—
132	C₂²×D₈₉	—	8	2	no	—
133	C₂²×D₈₉	—	8	2	no	—
134	C₂²×D₈₉	—	8	2	no	—
135	Q₈×C₈₉	—	4	4	no	—
136	Q₈×C₈₉	—	4	4	no	—
137	Q₈×C₈₉	—	8	2	no	—
138	Q₈×C₈₉	—	8	2	no	—
139	C₂×C₄×C₈₉	—	4	2	no	—
140	C₂×C₄×C₈₉	—	4	2	no	—
141	C₂×C₄×C₈₉	—	8	2	no	—
142	C₂×C₄×C₈₉	—	8	2	no	—
143	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
144	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
145	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
147	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
148	C₂×C₄×C₈₉	#137	1	4	yes	—
149	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×Dic₈₉	—	4	2	no	—
153	C₂×Dic₈₉	—	8	2	no	—
154	D₃₅₆	—	4	4	no	—
155	D₃₅₆	—	4	4	no	—
156	D₃₅₆	—	4	4	no	—
157	D₃₅₆	—	8	2	no	—
158	D₃₅₆	—	8	2	no	—
159	D₃₅₆	—	8	2	no	—
160	Dic₁₇₈	—	4	4	no	—
161	Dic₁₇₈	—	8	2	no	—
162	C₈×C₈₉	—	2	4	no	—
163	#10623494	—	2	4	no	—
164	#10623494	C₂×SD₁₆	1	8	yes	complemented
165	C₂²×C₈₉	#492	1	2	yes	—
166	C₂²×C₈₉	#496	1	2	yes	—
167	C₂²×C₈₉	#497	1	2	yes	—
168	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
169	C₂×D₈₉	—	8	2	no	—
170	C₂×D₈₉	—	8	2	no	—
171	C₂×D₈₉	—	8	2	no	—
172	C₂×D₈₉	—	16	4	no	—
173	C₄×C₈₉	—	8	2	no	—
174	C₄×C₈₉	—	8	2	no	—
175	C₄×C₈₉	#492	1	2	yes	—
176	C₄×C₈₉	#496	1	2	yes	—
177	C₄×C₈₉	#497	1	2	yes	—
178	C₄×C₈₉	C₂²×SD₁₆	1	2	yes	—
179	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
180	C₄×C₈₉	C₂×#137	1	2	yes	—
181	Dic₈₉	—	8	2	no	—
182	C₂×#1714	—	89	1	no	maximal, Sylow
183	C₂×C₈₉	#1714	1	4	yes	complemented
184	C₂×C₈₉	C₂×#492	1	1	yes	characteristic
185	C₂×C₈₉	C₂×#496	1	1	yes	characteristic
186	C₂×C₈₉	C₂×#497	1	1	yes	characteristic
187	D₈₉	—	16	2	no	—
188	C₂×#500	—	89	1	no	—
189	C₂×#498	—	89	1	no	—
190	C₂×#492	—	89	1	no	—
191	C₂×#492	—	89	1	no	—
192	C₂×#487	D₈₉	1	1	yes	characteristic, complemented
193	C₂×#486	—	89	1	no	—
194	#1714	—	89	8	no	—
195	C₂×#322	—	89	1	no	—
196	C₈₉	C₂×#1714	1	1	yes	characteristic, complemented, Sylow
197	C₂²×SD₁₆	—	178	2	no	—
198	C₂²×SD₁₆	—	178	2	no	—
199	C₂×C₄⋊Q₈	—	89	1	no	—
200	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
201	C₂×C₄⋊Q₈	C₂×D₈₉	1	1	yes	characteristic
202	C₂×#127	—	89	1	no	—
203	#500	—	89	4	no	—
204	#500	—	178	2	no	—
205	#498	—	89	4	no	—
206	#498	—	178	2	no	—
207	#492	—	89	4	no	—
208	#492	—	89	4	no	—
209	#492	—	178	2	no	—
210	#492	—	178	2	no	—
211	#487	—	2	2	no	—
212	#487	C₂×D₈₉	1	4	yes	complemented
213	#486	—	89	4	no	—
214	#486	—	178	2	no	—
215	C₂×#107	—	178	2	no	—
216	C₂×#106	—	178	2	no	—
217	C₂×#103	—	2	2	no	—
218	C₂×#102	—	178	2	no	—
219	C₂×C₄×C₈	—	89	1	no	—
220	C₂×C₄×C₈	—	89	1	no	—
221	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
222	#322	—	89	4	no	—
223	C₂²×Q₈	—	2	1	no	—
224	C₂²×Q₈	—	2	1	no	—
225	C₂²×Q₈	—	178	1	no	—
226	C₂²×D₄	—	178	1	no	—
227	C₂²×D₄	—	178	1	no	—
228	C₂²×D₄	—	178	1	no	—
229	C₂×SD₁₆	—	356	8	no	—
230	C₂×SD₁₆	—	356	8	no	—
231	C₂×SD₁₆	—	178	8	no	—
232	C₂×SD₁₆	—	178	8	no	—
233	C₂²×C₈	—	89	2	no	—
234	C₂²×C₈	—	89	2	no	—
235	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
236	C₄⋊Q₈	—	2	1	no	—
237	C₄⋊Q₈	—	2	1	no	—
238	C₄⋊Q₈	—	2	1	no	—
239	C₄⋊Q₈	—	2	1	no	—
240	C₄⋊Q₈	—	2	1	no	—
241	C₄⋊Q₈	—	2	1	no	—
242	C₄⋊Q₈	—	89	2	no	—
243	C₄⋊Q₈	—	178	1	no	—
244	C₄⋊Q₈	—	178	1	no	—
245	C₄⋊Q₈	—	178	1	no	—
246	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
247	C₄⋊Q₈	C₂²×D₈₉	1	2	yes	—
248	#127	—	89	2	no	—
249	#127	—	178	1	no	—
250	#127	—	178	1	no	—
251	#127	—	178	1	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	2	1	no	—
255	C₂×C₄⋊C₄	—	2	1	no	—
256	C₂×C₄⋊C₄	—	178	1	no	—
257	C₂×C₄⋊C₄	—	178	1	no	—
258	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
259	#107	—	178	8	no	—
260	#106	—	178	8	no	—
261	#103	—	2	8	no	—
262	#102	—	178	8	no	—
263	C₄×C₈	—	89	2	no	—
264	C₄×C₈	—	89	2	no	—
265	C₄×C₈	—	89	2	no	—
266	C₄×C₈	—	89	2	no	—
267	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
268	C₄×C₈	C₂²×D₈₉	1	2	yes	—
269	C₂⁴	—	356	1	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	4	1	no	—
273	C₂×Q₈	—	4	1	no	—
274	C₂×Q₈	—	4	4	no	—
275	C₂×Q₈	—	4	4	no	—
276	C₂×Q₈	—	356	1	no	—
277	C₂×Q₈	—	356	4	no	—
278	C₂×Q₈	—	178	2	no	—
279	C₂×D₄	—	356	1	no	—
280	C₂×D₄	—	356	1	no	—
281	C₂×D₄	—	356	1	no	—
282	C₂×D₄	—	356	4	no	—
283	C₂×D₄	—	356	4	no	—
284	C₂×D₄	—	356	4	no	—
285	C₂×D₄	—	178	2	no	—
286	C₂×D₄	—	178	2	no	—
287	C₂×D₄	—	178	2	no	—
288	C₂²×C₄	—	4	1	no	—
289	C₂²×C₄	—	4	1	no	—
290	C₂²×C₄	—	356	1	no	—
291	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
292	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
293	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
294	SD₁₆	—	356	16	no	—
295	SD₁₆	—	356	16	no	—
296	C₂×C₈	—	2	4	no	—
297	C₂×C₈	—	89	8	no	—
298	C₂×C₈	—	89	4	no	—
299	C₂×C₈	—	89	4	no	—
300	C₂×C₈	—	178	4	no	—
301	C₂×C₈	C₂×C₂²⋊D₈₉	1	4	yes	—
302	C₄⋊C₄	—	2	2	no	—
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	2	2	no	—
306	C₄⋊C₄	—	4	1	no	—
307	C₄⋊C₄	—	4	1	no	—
308	C₄⋊C₄	—	4	1	no	—
309	C₄⋊C₄	—	4	1	no	—
310	C₄⋊C₄	—	356	1	no	—
311	C₄⋊C₄	—	356	1	no	—
312	C₄⋊C₄	—	178	2	no	—
313	C₄⋊C₄	—	178	2	no	—
314	C₄²	D₄×D₈₉	1	1	yes	characteristic
315	C₄²	D₄×D₈₉	1	1	yes	characteristic
316	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
317	C₄²	C₂³×D₈₉	1	1	yes	characteristic
318	C₂³	—	356	2	no	—
319	C₂³	—	712	2	no	—
320	C₂³	—	712	2	no	—
321	C₂³	—	712	2	no	—
322	C₂³	#15668008	1	1	yes	characteristic
323	Q₈	—	4	4	no	—
324	Q₈	—	4	4	no	—
325	Q₈	—	8	2	no	—
326	Q₈	—	8	2	no	—
327	Q₈	—	356	4	no	—
328	Q₈	—	712	2	no	—
329	D₄	—	356	4	no	—
330	D₄	—	356	4	no	—
331	D₄	—	356	4	no	—
332	D₄	—	712	2	no	—
333	D₄	—	712	2	no	—
334	D₄	—	712	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	8	2	no	—
338	C₂×C₄	—	8	2	no	—
339	C₂×C₄	—	356	2	no	—
340	C₂×C₄	—	712	2	no	—
341	C₂×C₄	#15668008	1	1	yes	characteristic
342	C₂×C₄	#15668008	1	1	yes	characteristic
343	C₂×C₄	#15668008	1	1	yes	characteristic
344	C₂×C₄	SD₁₆×D₈₉	1	4	yes	—
345	C₂×C₄	#15668037	1	4	yes	—
346	C₂×C₄	#15668044	1	4	yes	—
347	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
349	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
350	C₈	—	2	4	no	—
351	C₈	—	89	8	no	—
352	C₈	—	178	4	no	—
353	C₂²	—	1424	4	no	—
354	C₂²	—	712	2	no	—
355	C₂²	—	712	2	no	—
356	C₂²	—	712	2	no	—
357	C₂²	#19151969	1	2	yes	—
358	C₂²	#19151988	1	2	yes	—
359	C₂²	#19151991	1	2	yes	—
360	C₂²	C₂×#15668008	1	1	yes	characteristic
361	C₄	—	8	2	no	—
362	C₄	—	8	2	no	—
363	C₄	—	712	2	no	—
364	C₄	#19151969	1	2	yes	—
365	C₄	#19151988	1	2	yes	—
366	C₄	#19151991	1	2	yes	—
367	C₄	C₂×SD₁₆×D₈₉	1	2	yes	—
368	C₄	C₂×#15668037	1	2	yes	—
369	C₄	C₂×#15668044	1	2	yes	—
370	C₂	—	1424	2	no	—
371	C₂	#26019876	1	4	yes	direct product factor
372	C₂	C₂×#19151969	1	1	yes	characteristic
373	C₂	C₂×#19151988	1	1	yes	characteristic
374	C₂	C₂×#19151991	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×#1714
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×#19152001	C₂	semidirect
2. C₂×#19151994	C₂	semidirect
6. #26019876	C₂	direct (N × H)
7. C₂×#19151320	C₂	semidirect
8. C₂×#487×C₈₉	C₂	semidirect
14. #19152001	C₂²	semidirect
16. #19151994	C₂²	semidirect
28. #19151320	C₂²	semidirect
32. #487×C₈₉	C₂²	semidirect
98. C₂×#10623494	SD₁₆	semidirect
164. #10623494	C₂×SD₁₆	semidirect
183. C₂×C₈₉	#1714	semidirect
192. C₂×#487	D₈₉	semidirect
196. C₈₉	C₂×#1714	semidirect
212. #487	C₂×D₈₉	semidirect
371. C₂	#26019876	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×#19152001	2
C₂×#19151994	2
C₂×#19151972	2
C₂×#19151971	2
C₂×#19151946	2
#26019876	2
C₂×#19151320	2
C₂×#487×C₈₉	2
C₂×#1714	89

Subgroups — C2×#26019876 #36542143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019876 #36542143