Subgroups — C₂×#26019912 #36542192

Order 22784.

850 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151996	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151978	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151949	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19151945	C₂	1	1	yes	maximal, characteristic
5	C₂×#19151945	C₂	1	1	yes	maximal, characteristic
6	C₂×#19151945	C₂	1	1	yes	maximal, characteristic
7	#26019912	C₂	1	8	yes	maximal, direct product factor
8	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₃₅₆	—	2	2	no	—
10	C₂×#15668012	C₂²	1	1	yes	characteristic
11	C₂×#15668012	C₂²	1	1	yes	characteristic
12	C₂×#15668012	C₂²	1	1	yes	characteristic
13	C₂×#15668009	C₂²	1	1	yes	characteristic
14	#19151996	—	2	2	no	—
15	#19151996	C₂²	1	4	yes	complemented
16	#19151978	—	2	2	no	—
17	#19151978	C₂²	1	4	yes	complemented
18	#19151949	—	2	2	no	—
19	#19151949	C₂²	1	4	yes	complemented
20	#19151945	—	2	2	no	—
21	#19151945	—	2	2	no	—
22	#19151945	—	2	2	no	—
23	#19151945	C₂²	1	4	yes	—
24	#19151945	C₂²	1	4	yes	—
25	#19151945	C₂²	1	4	yes	—
26	C₂×#15667944	—	2	2	no	—
27	C₂×#15667937	—	2	2	no	—
28	C₂×#15667934	—	2	2	no	—
29	C₂×#15667934	—	2	2	no	—
30	C₂×#15667934	—	2	2	no	—
31	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
34	#322×C₈₉	C₂²	1	4	yes	complemented
35	C₂²×Dic₁₇₈	—	2	1	no	—
36	C₂²×Dic₁₇₈	—	2	1	no	—
37	C₂²×Dic₁₇₈	—	2	1	no	—
38	C₂²×D₃₅₆	—	2	1	no	—
39	C₂²×D₃₅₆	—	2	1	no	—
40	C₂²×D₃₅₆	—	2	1	no	—
41	C₂×Dic₃₅₆	—	2	8	no	—
42	C₂×Dic₃₅₆	—	4	8	no	—
43	#15668012	—	2	1	no	—
44	#15668012	—	2	1	no	—
45	#15668012	—	2	1	no	—
46	#15668012	—	2	1	no	—
47	#15668012	—	2	1	no	—
48	#15668012	—	2	1	no	—
49	#15668012	—	2	1	no	—
50	#15668012	—	2	1	no	—
51	#15668012	—	2	1	no	—
52	#15668012	C₂³	1	2	yes	—
53	#15668012	C₂³	1	2	yes	—
54	#15668012	C₂³	1	2	yes	—
55	#15668009	—	2	1	no	—
56	#15668009	—	2	1	no	—
57	#15668009	—	2	1	no	—
58	#15668009	C₂³	1	2	yes	—
59	C₂×C₄⋊Dic₈₉	—	2	1	no	—
60	C₂×C₄⋊Dic₈₉	—	2	1	no	—
61	C₂×C₄⋊Dic₈₉	—	2	1	no	—
62	C₂×C₄⋊Dic₈₉	—	2	1	no	—
63	C₂×C₄⋊Dic₈₉	—	2	1	no	—
64	C₂×C₄⋊Dic₈₉	—	2	1	no	—
65	#15667944	—	2	8	no	—
66	#15667937	—	2	8	no	—
67	#15667934	—	2	8	no	—
68	#15667934	—	2	8	no	—
69	#15667934	—	2	8	no	—
70	C₂²×C₈×C₈₉	D₄	1	2	yes	—
71	C₂²×C₈×C₈₉	D₄	1	2	yes	—
72	C₂²×C₈×C₈₉	D₄	1	2	yes	—
73	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₈₉	D₄	1	2	yes	—
75	C₄×C₈×C₈₉	D₄	1	2	yes	—
76	C₄×C₈×C₈₉	D₄	1	2	yes	—
77	C₄×C₈×C₈₉	C₂³	1	2	yes	—
78	C₄×C₈×C₈₉	C₂³	1	2	yes	—
79	C₄×C₈×C₈₉	C₂³	1	2	yes	—
80	C₂³×D₈₉	—	4	1	no	—
81	C₂×Dic₁₇₈	—	2	2	no	—
82	C₂×Dic₁₇₈	—	2	2	no	—
83	C₂×Dic₁₇₈	—	2	2	no	—
84	C₂×Dic₁₇₈	—	4	4	no	—
85	C₂×Dic₁₇₈	—	4	4	no	—
86	C₂×Dic₁₇₈	—	4	4	no	—
87	C₂×Dic₁₇₈	—	4	1	no	—
88	C₂×Dic₁₇₈	—	4	1	no	—
89	C₂×Dic₁₇₈	—	4	1	no	—
90	C₂×D₃₅₆	—	2	2	no	—
91	C₂×D₃₅₆	—	2	2	no	—
92	C₂×D₃₅₆	—	2	2	no	—
93	C₂×D₃₅₆	—	4	4	no	—
94	C₂×D₃₅₆	—	4	4	no	—
95	C₂×D₃₅₆	—	4	4	no	—
96	C₂×D₃₅₆	—	4	1	no	—
97	C₂×D₃₅₆	—	4	1	no	—
98	C₂×D₃₅₆	—	4	1	no	—
99	C₂²×Dic₈₉	—	4	1	no	—
100	C₂²×Dic₈₉	—	4	1	no	—
101	C₂²×Dic₈₉	—	4	1	no	—
102	Dic₃₅₆	—	4	16	no	—
103	C₄⋊Dic₈₉	—	2	2	no	—
104	C₄⋊Dic₈₉	—	2	2	no	—
105	C₄⋊Dic₈₉	—	2	2	no	—
106	C₄⋊Dic₈₉	—	2	2	no	—
107	C₄⋊Dic₈₉	—	2	2	no	—
108	C₄⋊Dic₈₉	—	2	2	no	—
109	C₄⋊Dic₈₉	—	4	1	no	—
110	C₄⋊Dic₈₉	—	4	1	no	—
111	C₄⋊Dic₈₉	—	4	1	no	—
112	C₄⋊Dic₈₉	—	4	1	no	—
113	C₄⋊Dic₈₉	—	4	1	no	—
114	C₄⋊Dic₈₉	—	4	1	no	—
115	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₈₉	—	2	4	no	—
119	C₂×C₈×C₈₉	—	2	4	no	—
120	C₂×C₈×C₈₉	—	2	4	no	—
121	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
122	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
123	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
124	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
128	C₂³×C₈₉	#127	1	1	yes	characteristic
129	C₂²×D₈₉	—	4	2	no	—
130	C₂²×D₈₉	—	8	2	no	—
131	C₂²×D₈₉	—	8	2	no	—
132	C₂²×D₈₉	—	8	2	no	—
133	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
134	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
135	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
136	C₂×C₄×C₈₉	#135	1	4	yes	—
137	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
138	C₂×C₄×C₈₉	#137	1	4	yes	—
139	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×Dic₈₉	—	4	2	no	—
143	C₂×Dic₈₉	—	4	2	no	—
144	C₂×Dic₈₉	—	4	2	no	—
145	C₂×Dic₈₉	—	8	2	no	—
146	C₂×Dic₈₉	—	8	2	no	—
147	C₂×Dic₈₉	—	8	2	no	—
148	D₃₅₆	—	4	4	no	—
149	D₃₅₆	—	4	4	no	—
150	D₃₅₆	—	4	4	no	—
151	D₃₅₆	—	8	2	no	—
152	D₃₅₆	—	8	2	no	—
153	D₃₅₆	—	8	2	no	—
154	Dic₁₇₈	—	4	4	no	—
155	Dic₁₇₈	—	4	4	no	—
156	Dic₁₇₈	—	4	4	no	—
157	Dic₁₇₈	—	8	2	no	—
158	Dic₁₇₈	—	8	2	no	—
159	Dic₁₇₈	—	8	2	no	—
160	C₈×C₈₉	—	2	4	no	—
161	C₈×C₈₉	—	2	4	no	—
162	C₈×C₈₉	—	2	4	no	—
163	C₂²×C₈₉	#495	1	2	yes	—
164	C₂²×C₈₉	#496	1	2	yes	—
165	C₂²×C₈₉	#497	1	2	yes	—
166	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
167	C₂×D₈₉	—	8	2	no	—
168	C₂×D₈₉	—	8	2	no	—
169	C₂×D₈₉	—	8	2	no	—
170	C₂×D₈₉	—	16	4	no	—
171	C₄×C₈₉	#495	1	2	yes	—
172	C₄×C₈₉	#496	1	2	yes	—
173	C₄×C₈₉	#497	1	2	yes	—
174	C₄×C₈₉	C₂×#135	1	2	yes	—
175	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
176	C₄×C₈₉	C₂×#137	1	2	yes	—
177	Dic₈₉	—	8	2	no	—
178	Dic₈₉	—	8	2	no	—
179	Dic₈₉	—	8	2	no	—
180	C₂×#1720	—	89	1	no	maximal, Sylow
181	C₂×C₈₉	#1720	1	4	yes	complemented
182	C₂×C₈₉	C₂×#495	1	1	yes	characteristic
183	C₂×C₈₉	C₂×#496	1	1	yes	characteristic
184	C₂×C₈₉	C₂×#497	1	1	yes	characteristic
185	D₈₉	—	16	2	no	—
186	C₂×#498	—	89	1	no	—
187	C₂×#494	—	89	1	no	—
188	C₂×#487	—	89	1	no	—
189	C₂×#486	—	89	1	no	—
190	C₂×#486	—	89	1	no	—
191	C₂×#486	—	89	1	no	—
192	#1720	—	89	8	no	—
193	C₂×#322	D₈₉	1	1	yes	characteristic, complemented
194	C₈₉	C₂×#1720	1	1	yes	characteristic, complemented, Sylow
195	C₂²×Q₁₆	—	178	2	no	—
196	C₂×C₄⋊Q₈	—	89	1	no	—
197	C₂×C₄⋊Q₈	—	89	1	no	—
198	C₂×C₄⋊Q₈	—	89	1	no	—
199	C₂×#127	—	89	1	no	—
200	#498	—	89	4	no	—
201	#498	—	178	2	no	—
202	#494	—	89	4	no	—
203	#494	—	178	2	no	—
204	#487	—	89	4	no	—
205	#487	—	178	2	no	—
206	#486	—	89	4	no	—
207	#486	—	89	4	no	—
208	#486	—	89	4	no	—
209	#486	—	178	2	no	—
210	#486	—	178	2	no	—
211	#486	—	178	2	no	—
212	C₂×#106	—	178	2	no	—
213	C₂×#103	—	178	2	no	—
214	C₂×#102	—	178	2	no	—
215	C₂×#102	—	178	2	no	—
216	C₂×#102	—	178	2	no	—
217	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
218	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
219	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
220	#322	C₂×D₈₉	1	4	yes	complemented
221	C₂²×Q₈	—	178	1	no	—
222	C₂²×Q₈	—	178	1	no	—
223	C₂²×Q₈	—	178	1	no	—
224	C₂²×D₄	—	178	1	no	—
225	C₂²×D₄	—	178	1	no	—
226	C₂²×D₄	—	178	1	no	—
227	C₂×Q₁₆	—	356	8	no	—
228	C₂×Q₁₆	—	178	8	no	—
229	C₂²×C₈	D₃₅₆	1	2	yes	—
230	C₂²×C₈	D₃₅₆	1	2	yes	—
231	C₂²×C₈	D₃₅₆	1	2	yes	—
232	C₄⋊Q₈	—	89	2	no	—
233	C₄⋊Q₈	—	89	2	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	—	178	1	no	—
239	C₄⋊Q₈	—	178	1	no	—
240	C₄⋊Q₈	—	178	1	no	—
241	C₄⋊Q₈	—	178	1	no	—
242	C₄⋊Q₈	—	178	1	no	—
243	C₄⋊Q₈	—	178	1	no	—
244	#127	—	89	2	no	—
245	#127	—	178	1	no	—
246	#127	—	178	1	no	—
247	#127	—	178	1	no	—
248	C₂×C₄⋊C₄	—	178	1	no	—
249	C₂×C₄⋊C₄	—	178	1	no	—
250	C₂×C₄⋊C₄	—	178	1	no	—
251	C₂×C₄⋊C₄	—	178	1	no	—
252	C₂×C₄⋊C₄	—	178	1	no	—
253	C₂×C₄⋊C₄	—	178	1	no	—
254	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
255	#106	—	178	8	no	—
256	#103	—	178	8	no	—
257	#102	—	178	8	no	—
258	#102	—	178	8	no	—
259	#102	—	178	8	no	—
260	C₄×C₈	D₃₅₆	1	2	yes	—
261	C₄×C₈	D₃₅₆	1	2	yes	—
262	C₄×C₈	D₃₅₆	1	2	yes	—
263	C₄×C₈	C₂²×D₈₉	1	2	yes	—
264	C₄×C₈	C₂²×D₈₉	1	2	yes	—
265	C₄×C₈	C₂²×D₈₉	1	2	yes	—
266	C₂⁴	—	356	1	no	—
267	C₂×Q₈	—	356	1	no	—
268	C₂×Q₈	—	356	1	no	—
269	C₂×Q₈	—	356	1	no	—
270	C₂×Q₈	—	356	4	no	—
271	C₂×Q₈	—	356	4	no	—
272	C₂×Q₈	—	356	4	no	—
273	C₂×Q₈	—	178	2	no	—
274	C₂×Q₈	—	178	2	no	—
275	C₂×Q₈	—	178	2	no	—
276	C₂×D₄	—	356	1	no	—
277	C₂×D₄	—	356	1	no	—
278	C₂×D₄	—	356	1	no	—
279	C₂×D₄	—	356	4	no	—
280	C₂×D₄	—	356	4	no	—
281	C₂×D₄	—	356	4	no	—
282	C₂×D₄	—	178	2	no	—
283	C₂×D₄	—	178	2	no	—
284	C₂×D₄	—	178	2	no	—
285	C₂²×C₄	—	356	1	no	—
286	C₂²×C₄	—	356	1	no	—
287	C₂²×C₄	—	356	1	no	—
288	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
289	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
290	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
291	Q₁₆	—	356	16	no	—
292	C₂×C₈	—	2	4	no	—
293	C₂×C₈	—	2	4	no	—
294	C₂×C₈	—	2	4	no	—
295	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
296	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
297	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
298	C₄⋊C₄	—	356	1	no	—
299	C₄⋊C₄	—	356	1	no	—
300	C₄⋊C₄	—	356	1	no	—
301	C₄⋊C₄	—	356	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	C₄⋊C₄	—	178	2	no	—
307	C₄⋊C₄	—	178	2	no	—
308	C₄⋊C₄	—	178	2	no	—
309	C₄⋊C₄	—	178	2	no	—
310	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
311	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
312	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
313	C₄²	C₂³×D₈₉	1	1	yes	characteristic
314	C₂³	—	356	2	no	—
315	C₂³	—	712	2	no	—
316	C₂³	—	712	2	no	—
317	C₂³	—	712	2	no	—
318	C₂³	#15668009	1	1	yes	characteristic
319	Q₈	—	356	4	no	—
320	Q₈	—	356	4	no	—
321	Q₈	—	356	4	no	—
322	Q₈	—	712	2	no	—
323	Q₈	—	712	2	no	—
324	Q₈	—	712	2	no	—
325	D₄	—	356	4	no	—
326	D₄	—	356	4	no	—
327	D₄	—	356	4	no	—
328	D₄	—	712	2	no	—
329	D₄	—	712	2	no	—
330	D₄	—	712	2	no	—
331	C₂×C₄	—	356	2	no	—
332	C₂×C₄	—	356	2	no	—
333	C₂×C₄	—	356	2	no	—
334	C₂×C₄	—	712	2	no	—
335	C₂×C₄	—	712	2	no	—
336	C₂×C₄	—	712	2	no	—
337	C₂×C₄	#15668009	1	1	yes	characteristic
338	C₂×C₄	#15668009	1	1	yes	characteristic
339	C₂×C₄	#15668009	1	1	yes	characteristic
340	C₂×C₄	#15668034	1	4	yes	—
341	C₂×C₄	#15668040	1	4	yes	—
342	C₂×C₄	#15668045	1	4	yes	—
343	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
344	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
345	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
346	C₈	—	2	4	no	—
347	C₈	—	2	4	no	—
348	C₈	—	2	4	no	—
349	C₂²	—	1424	4	no	—
350	C₂²	—	712	2	no	—
351	C₂²	—	712	2	no	—
352	C₂²	—	712	2	no	—
353	C₂²	#19151982	1	2	yes	—
354	C₂²	#19151987	1	2	yes	—
355	C₂²	#19151992	1	2	yes	—
356	C₂²	C₂×#15668009	1	1	yes	characteristic
357	C₄	—	712	2	no	—
358	C₄	—	712	2	no	—
359	C₄	—	712	2	no	—
360	C₄	#19151982	1	2	yes	—
361	C₄	#19151987	1	2	yes	—
362	C₄	#19151992	1	2	yes	—
363	C₄	C₂×#15668034	1	2	yes	—
364	C₄	C₂×#15668040	1	2	yes	—
365	C₄	C₂×#15668045	1	2	yes	—
366	C₂	—	1424	2	no	—
367	C₂	#26019912	1	4	yes	direct product factor
368	C₂	C₂×#19151982	1	1	yes	characteristic
369	C₂	C₂×#19151987	1	1	yes	characteristic
370	C₂	C₂×#19151992	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15668009
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle	C₂³×C₈₉	#127
prime radical	C₂×#15668009	C₂²
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	C₂×#1720
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
1. C₂×#19151996	C₂	semidirect
2. C₂×#19151978	C₂	semidirect
3. C₂×#19151949	C₂	semidirect
7. #26019912	C₂	direct (N × H)
8. C₂×#322×C₈₉	C₂	semidirect
15. #19151996	C₂²	semidirect
17. #19151978	C₂²	semidirect
19. #19151949	C₂²	semidirect
34. #322×C₈₉	C₂²	semidirect
181. C₂×C₈₉	#1720	semidirect
193. C₂×#322	D₈₉	semidirect
194. C₈₉	C₂×#1720	semidirect
220. #322	C₂×D₈₉	semidirect
367. C₂	#26019912	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×#19151996	2
C₂×#19151978	2
C₂×#19151949	2
C₂×#19151945	2
#26019912	2
C₂×#322×C₈₉	2
C₂×#1720	89

Subgroups — C2×#26019912 #36542192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019912 #36542192