Subgroups — C₂×#26022473 #36709320

Order 22784.

552 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151857	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151645	C₂	1	1	yes	maximal, characteristic
3	#26022473	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151420	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151415	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151413	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151379	C₂	1	1	yes	maximal, characteristic
8	C₂×#365×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
10	C₂²×#13083557	C₂²	1	1	yes	characteristic
11	C₂×M₁₆×D₈₉	—	2	1	no	—
12	C₂×#15667992	—	2	1	no	—
13	C₂×#15667952	C₂²	1	1	yes	characteristic
14	#19151857	—	2	2	no	—
15	#19151857	C₂²	1	4	yes	—
16	#19151645	—	2	2	no	—
17	#19151645	C₂²	1	4	yes	—
18	C₂×#15667948	C₂²	1	1	yes	characteristic
19	C₂×#15667947	—	2	1	no	—
20	C₂×#15667935	—	2	1	no	—
21	#19151420	C₂²	1	4	yes	complemented
22	#19151415	C₂²	1	4	yes	complemented
23	#19151413	C₂²	1	4	yes	complemented
24	#19151379	C₂²	1	4	yes	—
25	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
28	#365×C₈₉	C₂²	1	4	yes	complemented
29	C₂²×Dic₁₇₈	—	2	1	no	—
30	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
31	C₂×Dic₃₅₆	—	2	1	no	—
32	C₂×Dic₃₅₆	—	2	1	no	—
33	C₂×Dic₃₅₆	—	2	1	no	—
34	C₂×Dic₃₅₆	—	2	1	no	—
35	C₂×Dic₃₅₆	—	2	1	no	—
36	C₂×Dic₃₅₆	C₂³	1	2	yes	—
37	C₂×#13083558	—	2	1	no	—
38	C₂×#13083557	—	2	1	no	—
39	C₂×#13083557	—	2	1	no	—
40	C₂×#13083557	—	2	1	no	—
41	C₂×#13083557	—	2	1	no	—
42	C₂×#13083557	—	4	1	no	—
43	C₂×#13083557	D₄	1	2	yes	—
44	C₂×#13083557	C₂³	1	2	yes	—
45	M₁₆×D₈₉	—	2	4	no	—
46	M₁₆×D₈₉	—	4	2	no	—
47	C₂²×#10623494	D₄	1	1	yes	characteristic
48	C₂×C₈×D₈₉	—	4	1	no	—
49	#15667992	—	2	4	no	—
50	#15667992	—	4	2	no	—
51	C₂×C₄⋊Dic₈₉	—	4	1	no	—
52	C₂×#13083551	—	4	1	no	—
53	#15667952	—	2	1	no	—
54	#15667952	C₂³	1	2	yes	—
55	C₂×#13083549	—	2	1	no	—
56	#15667948	—	2	1	no	—
57	#15667948	C₂³	1	2	yes	—
58	#15667947	—	2	4	no	—
59	#15667935	—	2	4	no	—
60	C₂×M₁₆×C₈₉	—	2	1	no	—
61	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
63	M₃₂×C₈₉	—	2	1	no	—
64	M₃₂×C₈₉	C₂³	1	2	yes	—
65	C₂×C₁₆×C₈₉	—	2	1	no	—
66	#108×C₈₉	—	2	1	no	—
67	#108×C₈₉	C₂³	1	2	yes	—
68	#106×C₈₉	—	2	1	no	—
69	#106×C₈₉	C₂³	1	2	yes	—
70	C₂³×D₈₉	SD₁₆	1	1	yes	characteristic
71	C₂×Dic₁₇₈	—	2	2	no	—
72	C₂×Dic₁₇₈	—	4	1	no	—
73	C₂×Dic₁₇₈	—	4	1	no	—
74	C₂×Dic₁₇₈	—	4	1	no	—
75	C₂×Dic₁₇₈	—	4	1	no	—
76	C₂×Dic₁₇₈	—	4	1	no	—
77	C₂²×Dic₈₉	—	4	1	no	—
78	C₂²×Dic₈₉	SD₁₆	1	1	yes	characteristic
79	C₂×C₄×D₈₉	—	2	1	no	—
80	C₂×C₄×D₈₉	—	2	1	no	—
81	C₂×C₄×D₈₉	—	2	1	no	—
82	C₂×C₄×D₈₉	—	2	1	no	—
83	C₂×C₄×D₈₉	SD₁₆	1	2	yes	—
84	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
85	Dic₃₅₆	—	2	2	no	—
86	Dic₃₅₆	—	2	2	no	—
87	Dic₃₅₆	—	4	1	no	—
88	Dic₃₅₆	—	4	1	no	—
89	#13083558	—	2	2	no	—
90	#13083558	—	4	1	no	—
91	#13083557	—	2	2	no	—
92	#13083557	—	2	2	no	—
93	#13083557	—	2	2	no	—
94	#13083557	—	2	2	no	—
95	#13083557	—	4	4	no	—
96	C₂×#10623494	—	2	1	no	—
97	C₂×#10623494	—	2	1	no	—
98	C₂×#10623494	—	4	1	no	—
99	C₂×#10623494	C₂×D₄	1	2	yes	—
100	C₈×D₈₉	—	4	4	no	—
101	C₄⋊Dic₈₉	—	4	4	no	—
102	#13083551	—	4	4	no	—
103	#13083549	—	2	2	no	—
104	C₂²×C₄×C₈₉	—	4	1	no	—
105	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
106	M₁₆×C₈₉	—	2	2	no	—
107	M₁₆×C₈₉	—	4	1	no	—
108	C₂×C₈×C₈₉	—	4	1	no	—
109	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
111	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
112	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
113	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₈₉	—	2	2	no	—
116	C₄⋊C₄×C₈₉	—	4	1	no	—
117	C₁₆×C₈₉	—	2	2	no	—
118	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
119	C₂²×D₈₉	—	2	1	no	—
120	C₂²×D₈₉	—	2	1	no	—
121	C₂²×D₈₉	—	4	2	no	—
122	C₂²×D₈₉	C₂×SD₁₆	1	2	yes	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	8	2	no	—
125	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
126	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
129	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×Dic₈₉	—	2	1	no	—
132	C₂×Dic₈₉	—	2	1	no	—
133	C₂×Dic₈₉	—	4	2	no	—
134	C₂×Dic₈₉	—	8	1	no	—
135	C₂×Dic₈₉	—	8	1	no	—
136	C₂×Dic₈₉	C₂×SD₁₆	1	2	yes	—
137	C₄×D₈₉	—	2	2	no	—
138	C₄×D₈₉	—	2	2	no	—
139	C₄×D₈₉	—	2	2	no	—
140	C₄×D₈₉	—	2	2	no	—
141	Dic₁₇₈	—	4	2	no	—
142	Dic₁₇₈	—	4	2	no	—
143	Dic₁₇₈	—	8	1	no	—
144	Dic₁₇₈	—	8	1	no	—
145	C₈×C₈₉	—	4	2	no	—
146	C₈×C₈₉	#123	1	1	yes	characteristic
147	C₈×C₈₉	#123	1	1	yes	characteristic
148	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
149	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
150	#10623494	—	2	2	no	—
151	#10623494	—	4	2	no	—
152	C₂²×C₈₉	—	2	2	no	—
153	C₂²×C₈₉	#481	1	1	yes	characteristic
154	C₂²×C₈₉	#481	1	1	yes	characteristic
155	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
156	C₂×D₈₉	—	2	2	no	—
157	C₂×D₈₉	—	4	2	no	—
158	C₂×D₈₉	—	4	2	no	—
159	C₂×D₈₉	—	8	1	no	—
160	C₄×C₈₉	—	8	2	no	—
161	C₄×C₈₉	#481	1	1	yes	characteristic
162	C₄×C₈₉	#481	1	1	yes	characteristic
163	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
164	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
165	Dic₈₉	—	2	2	no	—
166	Dic₈₉	—	8	2	no	—
167	C₂×#2239	—	89	1	no	maximal, Sylow
168	C₂×C₈₉	—	2	1	no	—
169	C₂×C₈₉	—	2	1	no	—
170	C₂×C₈₉	#2239	1	2	yes	complemented
171	C₂×C₈₉	C₂×#481	1	1	yes	characteristic
172	D₈₉	—	4	2	no	—
173	C₂×#470	—	89	1	no	—
174	C₂×#430	—	89	1	no	—
175	#2239	—	89	8	no	—
176	C₂×#365	—	89	1	no	—
177	C₂×#365	D₈₉	1	1	yes	characteristic, complemented
178	C₂×#362	—	89	1	no	—
179	C₂×#362	—	89	1	no	—
180	C₂×#349	—	89	1	no	—
181	C₈₉	C₂×#2239	1	1	yes	characteristic, complemented, Sylow
182	C₂²×Q₁₆	—	89	1	no	—
183	C₂²×M₁₆	—	89	1	no	—
184	C₂²×M₁₆	—	178	1	no	—
185	C₂×C₂²⋊Q₈	—	178	1	no	—
186	C₂×M₃₂	—	89	1	no	—
187	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
188	#470	—	89	4	no	—
189	#470	—	178	2	no	—
190	#430	—	89	4	no	—
191	#430	—	178	2	no	—
192	C₂×#108	—	89	1	no	—
193	C₂×#108	—	178	1	no	—
194	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
195	C₂×#106	C₂×D₈₉	1	1	yes	characteristic
196	C₂×#103	—	178	1	no	—
197	#365	—	89	4	no	—
198	#365	C₂×D₈₉	1	4	yes	complemented
199	#362	—	89	4	no	—
200	#362	—	89	4	no	—
201	#349	—	89	4	no	—
202	C₂²×Q₈	—	178	1	no	—
203	C₂³×C₄	—	89	1	no	—
204	C₂×Q₁₆	—	89	2	no	—
205	C₂×Q₁₆	—	178	1	no	—
206	C₂×Q₁₆	—	178	1	no	—
207	C₂×Q₁₆	—	178	1	no	—
208	C₂×Q₁₆	—	178	1	no	—
209	C₂×Q₁₆	—	178	1	no	—
210	C₂×M₁₆	—	2	1	no	—
211	C₂×M₁₆	—	89	2	no	—
212	C₂×M₁₆	—	89	2	no	—
213	C₂×M₁₆	—	356	1	no	—
214	C₂×M₁₆	—	356	2	no	—
215	C₂×M₁₆	—	178	4	no	—
216	C₂×M₁₆	—	178	1	no	—
217	C₂×M₁₆	—	178	1	no	—
218	C₂×M₁₆	—	178	1	no	—
219	C₂×M₁₆	—	178	1	no	—
220	C₂×M₁₆	—	178	1	no	—
221	C₂²×C₈	—	89	1	no	—
222	C₂²×C₈	—	356	1	no	—
223	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
224	C₂²⋊Q₈	—	356	2	no	—
225	C₂²⋊Q₈	—	178	4	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	—	356	1	no	—
228	C₂×C₂²⋊C₄	—	356	1	no	—
229	M₃₂	—	2	1	no	—
230	M₃₂	—	89	2	no	—
231	M₃₂	—	178	1	no	—
232	M₃₂	C₂²×D₈₉	1	2	yes	—
233	C₂×C₁₆	—	2	1	no	—
234	C₂×C₁₆	—	178	1	no	—
235	#108	—	2	1	no	—
236	#108	—	89	2	no	—
237	#108	—	178	4	no	—
238	#108	—	178	1	no	—
239	#108	C₂²×D₈₉	1	2	yes	—
240	#106	—	2	1	no	—
241	#106	C₂²×D₈₉	1	2	yes	—
242	#103	—	178	4	no	—
243	C₂⁴	—	89	1	no	—
244	C₂×Q₈	—	356	1	no	—
245	C₂×Q₈	—	356	1	no	—
246	C₂×Q₈	—	356	1	no	—
247	C₂×Q₈	—	356	1	no	—
248	C₂×Q₈	—	356	1	no	—
249	C₂×Q₈	—	178	2	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	89	1	no	—
252	C₂²×C₄	—	89	2	no	—
253	C₂²×C₄	—	89	2	no	—
254	C₂²×C₄	—	356	1	no	—
255	C₂²×C₄	—	178	1	no	—
256	C₂²×C₄	—	178	1	no	—
257	C₂²×C₄	—	178	1	no	—
258	C₂²×C₄	—	178	1	no	—
259	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
260	Q₁₆	—	356	1	no	—
261	Q₁₆	—	356	1	no	—
262	Q₁₆	—	178	2	no	—
263	Q₁₆	—	178	2	no	—
264	M₁₆	—	2	2	no	—
265	M₁₆	—	4	1	no	—
266	M₁₆	—	356	4	no	—
267	M₁₆	—	356	1	no	—
268	M₁₆	—	178	2	no	—
269	M₁₆	—	178	2	no	—
270	M₁₆	—	178	2	no	—
271	M₁₆	—	178	2	no	—
272	M₁₆	—	178	2	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	89	2	no	—
275	C₂×C₈	—	356	4	no	—
276	C₂×C₈	—	356	1	no	—
277	C₂×C₈	—	178	1	no	—
278	C₂×C₈	—	178	1	no	—
279	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
280	C₂×C₈	#13083574	1	1	yes	characteristic
281	C₂×C₈	#13083574	1	1	yes	characteristic
282	C₂×C₈	#13083575	1	1	yes	characteristic
283	C₂×C₈	#13083575	1	1	yes	characteristic
284	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	356	4	no	—
288	C₂²⋊C₄	—	356	4	no	—
289	C₁₆	—	2	2	no	—
290	C₁₆	—	178	2	no	—
291	C₂³	—	89	2	no	—
292	C₂³	—	356	2	no	—
293	C₂³	—	178	1	no	—
294	C₂³	—	178	1	no	—
295	C₂³	SD₁₆×D₈₉	1	1	yes	characteristic
296	Q₈	—	356	2	no	—
297	Q₈	—	356	2	no	—
298	Q₈	—	712	1	no	—
299	Q₈	—	712	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	8	2	no	—
302	C₂×C₄	—	89	2	no	—
303	C₂×C₄	—	356	2	no	—
304	C₂×C₄	—	178	1	no	—
305	C₂×C₄	—	178	1	no	—
306	C₂×C₄	—	178	2	no	—
307	C₂×C₄	—	178	2	no	—
308	C₂×C₄	—	178	2	no	—
309	C₂×C₄	—	178	2	no	—
310	C₂×C₄	—	712	1	no	—
311	C₂×C₄	—	712	1	no	—
312	C₂×C₄	#15667997	1	1	yes	characteristic
313	C₂×C₄	#15667997	1	1	yes	characteristic
314	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
315	C₂×C₄	#15668039	1	1	yes	characteristic
316	C₂×C₄	#15668039	1	1	yes	characteristic
317	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
318	C₈	—	4	2	no	—
319	C₈	—	356	2	no	—
320	C₈	—	178	2	no	—
321	C₈	#15667997	1	1	yes	characteristic
322	C₈	#15667997	1	1	yes	characteristic
323	C₈	C₂×#13083574	1	1	yes	characteristic
324	C₈	C₂×#13083575	1	1	yes	characteristic
325	C₂²	—	2	2	no	—
326	C₂²	—	356	2	no	—
327	C₂²	—	356	2	no	—
328	C₂²	—	178	2	no	—
329	C₂²	—	712	1	no	—
330	C₂²	#19151918	1	1	yes	characteristic
331	C₂²	#19151918	1	1	yes	characteristic
332	C₂²	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
333	C₄	—	8	2	no	—
334	C₄	—	178	2	no	—
335	C₄	—	712	2	no	—
336	C₄	#19151918	1	1	yes	characteristic
337	C₄	#19151918	1	1	yes	characteristic
338	C₄	C₂×#15667997	1	1	yes	characteristic
339	C₄	C₂×#15668039	1	1	yes	characteristic
340	C₂	—	2	1	no	—
341	C₂	—	2	1	no	—
342	C₂	—	356	2	no	—
343	C₂	#26022473	1	2	yes	direct product factor
344	C₂	C₂×#19151918	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151918
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#365×C₈₉	C₂
socle	C₂²×C₈₉	#481
prime radical	C₂³×D₈₉	SD₁₆
hypercenter	C₂×#365	D₈₉
hypocenter	C₈₉	C₂×#2239
LCS	G ▷ C₂×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
3. #26022473	C₂	direct (N × H)
4. C₂×#19151420	C₂	semidirect
5. C₂×#19151415	C₂	semidirect
6. C₂×#19151413	C₂	semidirect
8. C₂×#365×C₈₉	C₂	semidirect
21. #19151420	C₂²	semidirect
22. #19151415	C₂²	semidirect
23. #19151413	C₂²	semidirect
28. #365×C₈₉	C₂²	semidirect
170. C₂×C₈₉	#2239	semidirect
177. C₂×#365	D₈₉	semidirect
181. C₈₉	C₂×#2239	semidirect
198. #365	C₂×D₈₉	semidirect
343. C₂	#26022473	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×#19151857	2
C₂×#19151645	2
#26022473	2
C₂×#19151420	2
C₂×#19151415	2
C₂×#19151413	2
C₂×#19151379	2
C₂×#365×C₈₉	2
C₂×#2239	89

Subgroups — C2×#26022473 #36709320

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022473 #36709320