Subgroups — C₂×#26022091 #36707659

Order 22784.

654 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152016	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19152015	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151713	C₂	1	1	yes	maximal, characteristic
4	#26022091	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19151418	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151381	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151373	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#350×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668032	C₄	1	1	yes	characteristic
10	C₂×#15668018	C₂²	1	1	yes	characteristic
11	#19152016	—	2	2	no	—
12	#19152016	C₂²	1	4	yes	complemented
13	#19152015	—	2	2	no	—
14	#19152015	C₂²	1	4	yes	complemented
15	C₂×#15667952	—	2	1	no	—
16	C₂×#15667952	C₂²	1	1	yes	characteristic
17	C₂×#15667951	—	2	1	no	—
18	C₂²×#13083549	—	2	1	no	—
19	C₂²×#13083549	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×D₈₉	—	2	1	no	—
21	#19151713	C₄	1	4	yes	—
22	#19151713	C₂²	1	4	yes	—
23	C₂×#15667947	C₄	1	1	yes	characteristic
24	C₂×#15667939	C₄	1	1	yes	characteristic
25	C₂×#15667938	C₄	1	1	yes	characteristic
26	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
27	#19151418	C₂²	1	4	yes	complemented
28	#19151381	C₂²	1	4	yes	complemented
29	#19151373	C₂²	1	4	yes	complemented
30	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
32	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
33	#350×C₈₉	C₂²	1	4	yes	complemented
34	C₂×#13083574	C₂×C₄	1	1	yes	characteristic
35	#15668032	C₈	1	4	yes	—
36	#15668032	C₂×C₄	1	4	yes	—
37	C₂×#13083564	C₈	1	1	yes	characteristic
38	C₂×#13083562	C₈	1	1	yes	characteristic
39	C₂×#13083561	C₈	1	1	yes	characteristic
40	C₂×#13083559	C₈	1	1	yes	characteristic
41	#15668018	—	2	1	no	—
42	#15668018	—	2	1	no	—
43	#15668018	C₂×C₄	1	2	yes	—
44	#15668018	C₂³	1	2	yes	—
45	C₂×#13083558	C₂×C₄	1	1	yes	characteristic
46	C₂×#13083557	—	2	1	no	—
47	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₈₉	—	2	1	no	—
49	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
50	#15667952	—	2	4	no	—
51	#15667952	—	2	1	no	—
52	#15667952	C₂³	1	2	yes	—
53	#15667951	—	2	4	no	—
54	C₂×#13083549	—	2	4	no	—
55	C₂×#13083549	—	2	1	no	—
56	C₂×#13083549	D₄	1	1	yes	characteristic
57	C₂×#13083549	D₄	1	1	yes	characteristic
58	C₂×#13083549	D₄	1	2	yes	—
59	C₂×#13083549	C₂³	1	2	yes	—
60	C₁₆×D₈₉	—	2	4	no	—
61	#15667947	C₂×C₄	1	4	yes	—
62	#15667939	C₂×C₄	1	4	yes	—
63	#15667938	C₂×C₄	1	4	yes	—
64	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
65	C₈×Dic₈₉	C₂³	1	2	yes	—
66	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
67	#131×C₈₉	—	2	1	no	—
68	#131×C₈₉	—	2	1	no	—
69	#131×C₈₉	C₂×C₄	1	2	yes	—
70	#131×C₈₉	C₂³	1	2	yes	—
71	C₂×M₁₆×C₈₉	—	2	1	no	—
72	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
73	C₂²×C₈×C₈₉	—	2	1	no	—
74	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	M₃₂×C₈₉	—	2	1	no	—
76	M₃₂×C₈₉	C₂³	1	2	yes	—
77	C₂×C₁₆×C₈₉	—	2	1	no	—
78	C₂×C₁₆×C₈₉	—	2	1	no	—
79	C₂×C₁₆×C₈₉	—	2	1	no	—
80	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
81	#13083574	—	2	1	no	—
82	#13083574	—	2	1	no	—
83	#13083574	C₂×C₈	1	2	yes	—
84	#13083574	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₇₈	C₂×C₈	1	1	yes	characteristic
86	C₂×D₃₅₆	C₂×C₈	1	1	yes	characteristic
87	C₂×C₂²⋊D₈₉	—	2	1	no	—
88	C₂²×Dic₈₉	—	2	1	no	—
89	C₂×C₄×D₈₉	—	2	1	no	—
90	#13083564	C₂×C₈	1	4	yes	—
91	#13083562	C₂×C₈	1	4	yes	—
92	#13083561	C₂×C₈	1	4	yes	—
93	#13083559	C₂×C₈	1	4	yes	—
94	#13083558	—	2	1	no	—
95	#13083558	C₂²×C₄	1	2	yes	—
96	#13083557	—	2	2	no	—
97	#13083557	—	2	2	no	—
98	C₂×#10623494	—	2	1	no	—
99	C₂×#10623494	C₂×C₈	1	1	yes	characteristic
100	C₂×#10623494	C₂×C₈	1	1	yes	characteristic
101	C₂×#10623494	C₂×C₈	1	2	yes	—
102	C₂×#10623494	C₂²×C₄	1	2	yes	—
103	C₈×D₈₉	—	2	2	no	—
104	C₈×D₈₉	—	2	2	no	—
105	C₄×Dic₈₉	—	2	1	no	—
106	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
107	#13083549	—	2	2	no	—
108	#13083549	C₂×D₄	1	2	yes	—
109	#13083549	C₂×D₄	1	2	yes	—
110	#41×C₈₉	—	2	1	no	—
111	#41×C₈₉	—	2	1	no	—
112	#41×C₈₉	C₂×C₈	1	2	yes	—
113	#41×C₈₉	C₂²×C₄	1	2	yes	—
114	C₂×Q₈×C₈₉	C₂×C₈	1	1	yes	characteristic
115	C₂×D₄×C₈₉	—	2	1	no	—
116	C₂×D₄×C₈₉	C₂×C₈	1	1	yes	characteristic
117	C₂²×C₄×C₈₉	—	2	1	no	—
118	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
119	M₁₆×C₈₉	—	2	1	no	—
120	M₁₆×C₈₉	—	2	2	no	—
121	M₁₆×C₈₉	—	2	2	no	—
122	M₁₆×C₈₉	C₂²×C₄	1	2	yes	—
123	C₂×C₈×C₈₉	—	2	1	no	—
124	C₂×C₈×C₈₉	—	2	2	no	—
125	C₂×C₈×C₈₉	—	2	2	no	—
126	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
130	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
131	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
132	C₁₆×C₈₉	—	2	2	no	—
133	C₁₆×C₈₉	—	2	2	no	—
134	C₂³×C₈₉	—	2	1	no	—
135	C₂³×C₈₉	#131	1	1	yes	characteristic
136	C₂²×D₈₉	—	2	1	no	—
137	Q₈×C₈₉	—	2	1	no	—
138	Q₈×C₈₉	C₂²×C₈	1	2	yes	—
139	D₄×C₈₉	—	2	2	no	—
140	D₄×C₈₉	—	2	2	no	—
141	D₄×C₈₉	—	2	1	no	—
142	D₄×C₈₉	C₂²×C₈	1	2	yes	—
143	C₂×C₄×C₈₉	—	2	1	no	—
144	C₂×C₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	—	2	2	no	—
146	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
149	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
150	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
151	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
152	C₂²⋊D₈₉	—	2	2	no	—
153	C₂²⋊D₈₉	—	2	2	no	—
154	C₂×Dic₈₉	—	2	1	no	—
155	C₂×Dic₈₉	—	2	2	no	—
156	C₂×Dic₈₉	—	4	2	no	—
157	D₃₅₆	—	2	1	no	—
158	D₃₅₆	C₂²×C₈	1	2	yes	—
159	C₄×D₈₉	—	2	2	no	—
160	C₄×D₈₉	—	2	2	no	—
161	Dic₁₇₈	—	2	1	no	—
162	Dic₁₇₈	C₂²×C₈	1	2	yes	—
163	C₈×C₈₉	—	2	2	no	—
164	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
165	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
166	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
167	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
168	#10623494	—	2	2	no	—
169	#10623494	C₂²×C₈	1	2	yes	—
170	#10623494	C₂²×C₈	1	2	yes	—
171	C₂²×C₈₉	—	2	2	no	—
172	C₂²×C₈₉	—	2	2	no	—
173	C₂²×C₈₉	—	4	2	no	—
174	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
175	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
176	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
177	C₂×D₈₉	—	2	2	no	—
178	C₂×D₈₉	—	4	2	no	—
179	C₄×C₈₉	—	2	2	no	—
180	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
181	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
182	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
183	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
184	Dic₈₉	—	2	2	no	—
185	Dic₈₉	—	4	2	no	—
186	C₂×#2171	—	89	1	no	maximal, Sylow
187	C₂×C₈₉	—	2	1	no	—
188	C₂×C₈₉	—	2	1	no	—
189	C₂×C₈₉	—	4	2	no	—
190	C₂×C₈₉	#2171	1	2	yes	complemented
191	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
192	D₈₉	—	4	2	no	—
193	C₂×#504	—	89	1	no	—
194	C₂×#504	—	89	1	no	—
195	C₂×#443	—	89	1	no	—
196	#2171	—	89	8	no	—
197	C₂×#364	—	89	1	no	—
198	C₂×#350	—	89	1	no	—
199	C₂×#350	D₈₉	1	1	yes	characteristic, complemented
200	C₂×#346	—	89	1	no	—
201	C₈₉	C₂×#2171	1	1	yes	characteristic, complemented, Sylow
202	C₂×#135	—	89	1	no	—
203	C₂×#131	—	89	1	no	—
204	C₂×#131	C₂×D₈₉	1	1	yes	characteristic
205	#504	—	89	4	no	—
206	#504	—	89	4	no	—
207	#504	—	178	2	no	—
208	#504	—	178	2	no	—
209	C₂×M₃₂	—	89	1	no	—
210	C₂×M₃₂	—	178	1	no	—
211	C₂×M₃₂	—	178	1	no	—
212	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
213	C₂²×C₁₆	—	89	1	no	—
214	C₂²×C₁₆	—	178	1	no	—
215	C₂²×C₁₆	—	178	1	no	—
216	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
217	#443	—	89	4	no	—
218	#443	—	89	4	no	—
219	C₂×#108	—	89	1	no	—
220	C₂×#104	—	89	1	no	—
221	C₂×#104	—	89	1	no	—
222	C₂×C₄×C₈	—	89	1	no	—
223	#364	—	89	4	no	—
224	#350	—	89	4	no	—
225	#350	C₂×D₈₉	1	4	yes	complemented
226	#346	—	89	4	no	—
227	C₂×#41	—	89	1	no	—
228	C₂×#41	C₄×D₈₉	1	1	yes	characteristic
229	#135	—	89	4	no	—
230	#135	—	89	4	no	—
231	C₂×Q₁₆	—	89	1	no	—
232	C₂×SD₁₆	—	89	1	no	—
233	C₂×SD₁₆	—	89	1	no	—
234	C₂×D₈	—	89	1	no	—
235	#131	—	2	1	no	—
236	#131	—	2	1	no	—
237	#131	—	89	2	no	—
238	#131	—	89	2	no	—
239	#131	—	178	1	no	—
240	#131	—	178	1	no	—
241	#131	C₄×D₈₉	1	2	yes	—
242	#131	C₂²×D₈₉	1	2	yes	—
243	C₂×M₁₆	—	2	1	no	—
244	C₂×M₁₆	—	89	1	no	—
245	C₂×M₁₆	—	178	1	no	—
246	C₂×M₁₆	C₄×D₈₉	1	1	yes	characteristic
247	C₂²×C₈	—	2	1	no	—
248	C₂²×C₈	—	89	1	no	—
249	C₂²×C₈	—	178	1	no	—
250	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
251	C₂×C₄²	—	89	1	no	—
252	M₃₂	—	2	1	no	—
253	M₃₂	—	89	2	no	—
254	M₃₂	—	178	4	no	—
255	M₃₂	—	178	4	no	—
256	M₃₂	—	178	1	no	—
257	M₃₂	C₂²×D₈₉	1	2	yes	—
258	C₂×C₁₆	—	2	1	no	—
259	C₂×C₁₆	—	2	1	no	—
260	C₂×C₁₆	—	2	1	no	—
261	C₂×C₁₆	—	89	2	no	—
262	C₂×C₁₆	—	89	1	no	—
263	C₂×C₁₆	—	89	1	no	—
264	C₂×C₁₆	—	89	2	no	—
265	C₂×C₁₆	—	178	4	no	—
266	C₂×C₁₆	—	178	4	no	—
267	C₂×C₁₆	—	178	1	no	—
268	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
269	#108	—	89	4	no	—
270	#104	—	89	4	no	—
271	#104	—	89	4	no	—
272	C₄×C₈	—	89	2	no	—
273	C₄×C₈	—	89	2	no	—
274	#41	—	2	1	no	—
275	#41	—	2	1	no	—
276	#41	—	89	2	no	—
277	#41	—	89	2	no	—
278	#41	—	178	1	no	—
279	#41	—	178	1	no	—
280	#41	C₈×D₈₉	1	2	yes	—
281	#41	C₂×C₄×D₈₉	1	2	yes	—
282	C₂×Q₈	—	89	1	no	—
283	C₂×Q₈	C₈×D₈₉	1	1	yes	characteristic
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	89	1	no	—
286	C₂×D₄	—	178	1	no	—
287	C₂×D₄	C₈×D₈₉	1	1	yes	characteristic
288	C₂²×C₄	—	2	1	no	—
289	C₂²×C₄	—	178	1	no	—
290	C₂²×C₄	—	178	1	no	—
291	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
292	Q₁₆	—	89	4	no	—
293	SD₁₆	—	89	4	no	—
294	SD₁₆	—	89	4	no	—
295	D₈	—	89	4	no	—
296	M₁₆	—	2	1	no	—
297	M₁₆	—	2	2	no	—
298	M₁₆	—	2	2	no	—
299	M₁₆	—	89	2	no	—
300	M₁₆	—	178	1	no	—
301	M₁₆	—	178	2	no	—
302	M₁₆	—	178	2	no	—
303	M₁₆	C₂×C₄×D₈₉	1	2	yes	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	89	1	no	—
308	C₂×C₈	—	89	1	no	—
309	C₂×C₈	—	89	2	no	—
310	C₂×C₈	—	89	2	no	—
311	C₂×C₈	—	178	1	no	—
312	C₂×C₈	—	178	2	no	—
313	C₂×C₈	—	178	2	no	—
314	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
315	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
316	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
317	C₂×C₈	#13083573	1	1	yes	characteristic
318	C₂×C₈	#13083573	1	1	yes	characteristic
319	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
320	C₄²	—	89	2	no	—
321	C₄²	—	178	1	no	—
322	C₁₆	—	2	2	no	—
323	C₁₆	—	2	2	no	—
324	C₁₆	—	89	2	no	—
325	C₁₆	—	89	2	no	—
326	C₁₆	—	178	2	no	—
327	C₂³	—	2	1	no	—
328	C₂³	—	178	1	no	—
329	C₂³	#15668020	1	1	yes	characteristic
330	Q₈	—	2	1	no	—
331	Q₈	—	89	2	no	—
332	Q₈	—	178	1	no	—
333	Q₈	C₂×C₈×D₈₉	1	2	yes	—
334	D₄	—	2	2	no	—
335	D₄	—	2	2	no	—
336	D₄	—	2	1	no	—
337	D₄	—	89	2	no	—
338	D₄	—	178	2	no	—
339	D₄	—	178	2	no	—
340	D₄	—	178	1	no	—
341	D₄	C₂×C₈×D₈₉	1	2	yes	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	356	2	no	—
346	C₂×C₄	—	178	2	no	—
347	C₂×C₄	—	178	1	no	—
348	C₂×C₄	—	178	2	no	—
349	C₂×C₄	—	178	2	no	—
350	C₂×C₄	#15667972	1	1	yes	characteristic
351	C₂×C₄	#15667972	1	1	yes	characteristic
352	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
353	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
354	C₂×C₄	#15668020	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
356	C₈	—	2	2	no	—
357	C₈	—	89	2	no	—
358	C₈	—	89	2	no	—
359	C₈	—	178	2	no	—
360	C₈	#15667972	1	1	yes	characteristic
361	C₈	#15667972	1	1	yes	characteristic
362	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
363	C₈	C₂×#13083573	1	1	yes	characteristic
364	C₂²	—	2	2	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	356	2	no	—
368	C₂²	—	178	2	no	—
369	C₂²	#19151661	1	1	yes	characteristic
370	C₂²	#19151661	1	1	yes	characteristic
371	C₂²	C₂×#15668020	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	356	2	no	—
374	C₄	—	178	2	no	—
375	C₄	#19151661	1	1	yes	characteristic
376	C₄	#19151661	1	1	yes	characteristic
377	C₄	C₂×#15667972	1	1	yes	characteristic
378	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	—	4	2	no	—
382	C₂	—	356	2	no	—
383	C₂	#26022091	1	2	yes	direct product factor
384	C₂	C₂×#19151661	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#350×C₈₉	C₂
socle	C₂²×C₈₉	C₈×D₄
prime radical	C₂×#15668032	C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#350	D₈₉
hypocenter	C₈₉	C₂×#2171
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#350

Complements

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

N	H ≅ G/N	type
1. C₂×#19152016	C₂	semidirect
2. C₂×#19152015	C₂	semidirect
4. #26022091	C₂	direct (N × H)
5. C₂×#19151418	C₂	semidirect
6. C₂×#19151381	C₂	semidirect
7. C₂×#19151373	C₂	semidirect
8. C₂×#350×C₈₉	C₂	semidirect
12. #19152016	C₂²	semidirect
14. #19152015	C₂²	semidirect
27. #19151418	C₂²	semidirect
28. #19151381	C₂²	semidirect
29. #19151373	C₂²	semidirect
33. #350×C₈₉	C₂²	semidirect
190. C₂×C₈₉	#2171	semidirect
199. C₂×#350	D₈₉	semidirect
201. C₈₉	C₂×#2171	semidirect
225. #350	C₂×D₈₉	semidirect
383. C₂	#26022091	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×#19152016	2
C₂×#19152015	2
C₂×#19151713	2
#26022091	2
C₂×#19151418	2
C₂×#19151381	2
C₂×#19151373	2
C₂×#350×C₈₉	2
C₂×#2171	89

Subgroups — C2×#26022091 #36707659

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022091 #36707659