Subgroups — C₂×#26022329 #36708809

Order 22784.

790 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×#19152036	C₂	1	2	yes	maximal, complemented
2	C₂×#19151975	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151716	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022329	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19151412	C₂	1	2	yes	maximal, complemented
6	C₂×#364×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15668034	C₂²	1	1	yes	characteristic
8	C₂²×D₇₁₂	—	2	1	no	—
9	C₂²×D₇₁₂	C₂²	1	2	yes	—
10	C₂×#15668018	—	2	1	no	—
11	C₂×#15668009	—	2	1	no	—
12	#19152036	—	2	4	no	—
13	#19152036	C₂²	1	8	yes	complemented
14	C₂×#15667957	—	2	2	no	—
15	C₂×D₁₄₂₄	—	2	2	no	—
16	#19151975	—	2	2	no	—
17	#19151975	C₂²	1	4	yes	complemented
18	#19151716	—	2	2	no	—
19	#19151716	C₂²	1	4	yes	complemented
20	C₂×#15667948	C₂²	1	1	yes	characteristic
21	C₂×#15667940	—	2	1	no	—
22	#19151412	C₂²	1	8	yes	complemented
23	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	#364×C₈₉	C₂²	1	4	yes	complemented
26	C₂×#13083574	—	2	1	no	—
27	C₂²×D₃₅₆	—	2	2	no	—
28	C₂²×D₃₅₆	—	4	2	no	—
29	#15668034	—	2	1	no	—
30	#15668034	—	2	1	no	—
31	#15668034	D₄	1	2	yes	complemented
32	#15668034	C₂³	1	2	yes	—
33	C₂×Dic₃₅₆	D₄	1	1	yes	characteristic, complemented
34	C₂×#13083563	—	2	1	no	—
35	C₂×D₇₁₂	—	2	2	no	—
36	C₂×D₇₁₂	—	2	2	no	—
37	C₂×D₇₁₂	—	2	2	no	—
38	C₂×D₇₁₂	—	2	4	no	—
39	C₂×D₇₁₂	—	2	2	no	—
40	C₂×D₇₁₂	—	2	2	no	—
41	C₂×D₇₁₂	—	4	1	no	—
42	C₂×D₇₁₂	—	4	1	no	—
43	C₂×D₇₁₂	—	4	2	no	—
44	C₂×D₇₁₂	D₄	1	1	yes	characteristic, complemented
45	C₂×D₇₁₂	C₂³	1	4	yes	—
46	#15668018	—	2	4	no	—
47	#15668018	—	4	2	no	—
48	C₂×#13083558	—	2	1	no	—
49	C₂×#13083557	—	4	1	no	—
50	C₂×C₈×D₈₉	—	4	1	no	—
51	#15668009	—	2	4	no	—
52	#15668009	—	4	2	no	—
53	#15667957	—	2	8	no	—
54	D₁₄₂₄	—	2	8	no	—
55	#15667948	—	2	1	no	—
56	#15667948	C₂³	1	2	yes	—
57	#15667940	—	2	4	no	—
58	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
61	M₃₂×C₈₉	—	2	2	no	—
62	M₃₂×C₈₉	C₂³	1	4	yes	—
63	C₂×C₁₆×C₈₉	—	2	2	no	—
64	C₄×C₈×C₈₉	D₄	1	2	yes	—
65	C₄×C₈×C₈₉	C₂³	1	2	yes	—
66	C₂³×D₈₉	—	4	2	no	—
67	#13083574	—	2	2	no	—
68	#13083574	—	2	2	no	—
69	#13083574	—	4	1	no	—
70	#13083574	—	4	1	no	—
71	C₂×Dic₁₇₈	—	2	1	no	—
72	C₂×D₃₅₆	—	2	1	no	—
73	C₂×D₃₅₆	—	2	4	no	—
74	C₂×D₃₅₆	—	4	2	no	—
75	C₂×D₃₅₆	—	4	2	no	—
76	C₂×D₃₅₆	—	4	2	no	—
77	C₂×D₃₅₆	—	4	2	no	—
78	C₂×D₃₅₆	—	4	8	no	—
79	C₂×D₃₅₆	—	4	2	no	—
80	C₂×D₃₅₆	—	8	8	no	—
81	C₂×C₂²⋊D₈₉	—	4	1	no	—
82	C₂×C₄×D₈₉	—	4	1	no	—
83	Dic₃₅₆	—	2	1	no	—
84	Dic₃₅₆	C₂×D₄	1	2	yes	complemented
85	#13083563	—	2	2	no	—
86	#13083563	—	2	2	no	—
87	D₇₁₂	—	2	1	no	—
88	D₇₁₂	—	2	4	no	—
89	D₇₁₂	—	2	4	no	—
90	D₇₁₂	—	4	2	no	—
91	D₇₁₂	—	4	2	no	—
92	D₇₁₂	—	4	4	no	—
93	D₇₁₂	—	4	4	no	—
94	D₇₁₂	C₂×D₄	1	2	yes	complemented
95	#13083558	—	2	2	no	—
96	#13083558	—	4	1	no	—
97	#13083557	—	4	4	no	—
98	C₂×#10623494	—	4	1	no	—
99	C₈×D₈₉	—	4	4	no	—
100	C₂²×C₄×C₈₉	—	2	1	no	—
101	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
103	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
104	C₂×C₈×C₈₉	D₈	1	2	yes	—
105	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
110	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
112	C₄²×C₈₉	—	2	1	no	—
113	C₄²×C₈₉	C₂×D₄	1	2	yes	—
114	C₁₆×C₈₉	—	2	4	no	—
115	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
116	C₂²×D₈₉	—	4	1	no	—
117	C₂²×D₈₉	—	4	4	no	—
118	C₂²×D₈₉	—	8	8	no	—
119	C₂²×D₈₉	—	8	2	no	—
120	C₂²×D₈₉	—	8	2	no	—
121	C₂×C₄×C₈₉	—	2	4	no	—
122	C₂×C₄×C₈₉	—	2	2	no	—
123	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
124	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
129	C₂²⋊D₈₉	—	4	2	no	—
130	C₂²⋊D₈₉	—	4	2	no	—
131	C₂×Dic₈₉	—	4	1	no	—
132	D₃₅₆	—	2	2	no	—
133	D₃₅₆	—	4	4	no	—
134	D₃₅₆	—	4	4	no	—
135	D₃₅₆	—	4	1	no	—
136	D₃₅₆	—	8	16	no	—
137	D₃₅₆	—	8	2	no	—
138	D₃₅₆	—	8	2	no	—
139	C₄×D₈₉	—	4	2	no	—
140	C₄×D₈₉	—	4	2	no	—
141	Dic₁₇₈	—	2	2	no	—
142	Dic₁₇₈	—	4	1	no	—
143	C₈×C₈₉	#121	1	1	yes	characteristic
144	C₈×C₈₉	#121	1	1	yes	characteristic
145	C₈×C₈₉	C₂×D₈	1	2	yes	—
146	C₈×C₈₉	C₂×D₈	1	2	yes	—
147	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
149	#10623494	—	4	2	no	—
150	C₂²×C₈₉	—	2	2	no	—
151	C₂²×C₈₉	#459	1	1	yes	characteristic
152	C₂²×C₈₉	#459	1	1	yes	characteristic
153	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂×D₈₉	—	4	2	no	—
155	C₂×D₈₉	—	8	2	no	—
156	C₂×D₈₉	—	8	8	no	—
157	C₂×D₈₉	—	8	4	no	—
158	C₂×D₈₉	—	16	4	no	—
159	C₂×D₈₉	—	16	4	no	—
160	C₄×C₈₉	—	2	4	no	—
161	C₄×C₈₉	#459	1	1	yes	characteristic
162	C₄×C₈₉	#459	1	1	yes	characteristic
163	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
164	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
165	Dic₈₉	—	4	2	no	—
166	C₂×#2213	—	89	1	no	maximal, Sylow
167	C₂×C₈₉	—	2	1	no	—
168	C₂×C₈₉	—	2	1	no	—
169	C₂×C₈₉	#2213	1	2	yes	complemented
170	C₂×C₈₉	C₂×#459	1	1	yes	characteristic
171	D₈₉	—	8	2	no	—
172	D₈₉	—	16	4	no	—
173	C₂×#509	—	89	2	no	—
174	C₂×#493	—	89	1	no	—
175	C₂×#443	—	89	1	no	—
176	#2213	—	89	8	no	—
177	C₂×#364	D₈₉	1	1	yes	characteristic, complemented
178	C₂×#361	—	89	2	no	—
179	C₈₉	C₂×#2213	1	1	yes	characteristic, complemented, Sylow
180	C₂×#135	—	89	1	no	—
181	C₂²×D₈	—	89	2	no	—
182	C₂²×D₈	—	178	1	no	—
183	C₂×#131	—	178	1	no	—
184	C₂×#127	—	178	1	no	—
185	#509	—	89	8	no	—
186	#509	—	178	4	no	—
187	C₂×SD₃₂	—	178	2	no	—
188	C₂×D₁₆	—	178	2	no	—
189	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
190	#493	—	89	4	no	—
191	#493	—	178	2	no	—
192	#443	—	89	4	no	—
193	#443	—	178	2	no	—
194	C₂×#108	—	89	1	no	—
195	C₂×#104	—	178	1	no	—
196	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
197	#364	C₂×D₈₉	1	4	yes	complemented
198	#361	—	89	8	no	—
199	C₂×#41	—	178	1	no	—
200	C₂²×D₄	—	356	2	no	—
201	C₂²×D₄	—	178	2	no	—
202	#135	—	89	2	no	—
203	#135	—	89	2	no	—
204	#135	—	178	1	no	—
205	#135	—	178	1	no	—
206	C₂×Q₁₆	—	89	1	no	—
207	C₂×SD₁₆	—	178	1	no	—
208	C₂×D₈	—	89	1	no	—
209	C₂×D₈	—	89	4	no	—
210	C₂×D₈	—	356	1	no	—
211	C₂×D₈	—	356	1	no	—
212	C₂×D₈	—	356	2	no	—
213	C₂×D₈	—	178	2	no	—
214	C₂×D₈	—	178	2	no	—
215	C₂×D₈	—	178	2	no	—
216	C₂×D₈	—	178	4	no	—
217	C₂×D₈	—	178	2	no	—
218	C₂×D₈	—	178	2	no	—
219	#131	—	356	2	no	—
220	#131	—	178	4	no	—
221	C₂×M₁₆	—	356	1	no	—
222	C₂×M₁₆	—	178	1	no	—
223	C₂²×C₈	—	356	1	no	—
224	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
225	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
226	#127	—	356	2	no	—
227	#127	—	178	4	no	—
228	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
229	SD₃₂	—	178	8	no	—
230	D₁₆	—	178	8	no	—
231	M₃₂	—	2	2	no	—
232	M₃₂	C₂²×D₈₉	1	4	yes	—
233	C₂×C₁₆	—	2	2	no	—
234	#108	—	89	2	no	—
235	#108	—	178	1	no	—
236	#104	—	178	4	no	—
237	C₄×C₈	D₃₅₆	1	2	yes	—
238	C₄×C₈	C₂²×D₈₉	1	2	yes	—
239	C₂⁴	—	356	2	no	—
240	#41	—	356	1	no	—
241	#41	—	356	1	no	—
242	#41	—	178	2	no	—
243	#41	—	178	2	no	—
244	C₂×Q₈	—	178	1	no	—
245	C₂×D₄	—	356	8	no	—
246	C₂×D₄	—	356	1	no	—
247	C₂×D₄	—	356	2	no	—
248	C₂×D₄	—	356	2	no	—
249	C₂×D₄	—	356	2	no	—
250	C₂×D₄	—	356	2	no	—
251	C₂×D₄	—	356	2	no	—
252	C₂×D₄	—	178	1	no	—
253	C₂×D₄	—	178	4	no	—
254	C₂×D₄	—	712	8	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	356	1	no	—
257	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
258	Q₁₆	—	89	2	no	—
259	Q₁₆	—	178	1	no	—
260	SD₁₆	—	178	2	no	—
261	SD₁₆	—	178	2	no	—
262	D₈	—	89	2	no	—
263	D₈	—	356	2	no	—
264	D₈	—	356	2	no	—
265	D₈	—	356	4	no	—
266	D₈	—	356	4	no	—
267	D₈	—	178	1	no	—
268	D₈	—	178	4	no	—
269	D₈	—	178	4	no	—
270	M₁₆	—	356	1	no	—
271	M₁₆	—	356	4	no	—
272	M₁₆	—	178	2	no	—
273	C₂×C₈	—	356	1	no	—
274	C₂×C₈	—	356	4	no	—
275	C₂×C₈	D₇₁₂	1	1	yes	characteristic
276	C₂×C₈	D₇₁₂	1	1	yes	characteristic
277	C₂×C₈	D₇₁₂	1	2	yes	—
278	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
279	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
280	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
281	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
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	C₄²	—	2	1	no	—
286	C₄²	C₂×D₃₅₆	1	2	yes	—
287	C₁₆	—	2	4	no	—
288	C₂³	—	356	1	no	—
289	C₂³	—	356	4	no	—
290	C₂³	—	712	8	no	—
291	C₂³	—	712	2	no	—
292	C₂³	—	712	2	no	—
293	C₂³	#15668040	1	1	yes	characteristic
294	Q₈	—	356	1	no	—
295	Q₈	—	178	2	no	—
296	D₄	—	356	2	no	—
297	D₄	—	356	2	no	—
298	D₄	—	356	1	no	—
299	D₄	—	356	4	no	—
300	D₄	—	356	4	no	—
301	D₄	—	178	2	no	—
302	D₄	—	712	16	no	—
303	D₄	—	712	2	no	—
304	D₄	—	712	2	no	—
305	C₂×C₄	—	2	4	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	356	1	no	—
308	C₂×C₄	—	356	2	no	—
309	C₂×C₄	—	356	2	no	—
310	C₂×C₄	#15667987	1	1	yes	characteristic
311	C₂×C₄	#15667987	1	1	yes	characteristic
312	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
313	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
314	C₂×C₄	#15668040	1	1	yes	characteristic
315	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
316	C₈	—	356	2	no	—
317	C₈	#15667987	1	1	yes	characteristic
318	C₈	#15667987	1	1	yes	characteristic
319	C₈	C₂×D₇₁₂	1	2	yes	—
320	C₈	C₂×D₇₁₂	1	2	yes	—
321	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
322	C₈	C₂×#13083575	1	1	yes	characteristic
323	C₂²	—	2	2	no	—
324	C₂²	—	356	2	no	—
325	C₂²	—	1424	4	no	—
326	C₂²	—	1424	4	no	—
327	C₂²	—	712	8	no	—
328	C₂²	—	712	4	no	—
329	C₂²	—	712	2	no	—
330	C₂²	#19151794	1	1	yes	characteristic
331	C₂²	#19151794	1	1	yes	characteristic
332	C₂²	C₂×#15668040	1	1	yes	characteristic
333	C₄	—	2	4	no	—
334	C₄	—	356	2	no	—
335	C₄	#19151794	1	1	yes	characteristic
336	C₄	#19151794	1	1	yes	characteristic
337	C₄	C₂×#15667987	1	1	yes	characteristic
338	C₄	C₂²×D₇₁₂	1	1	yes	characteristic
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	—	1424	4	no	—
342	C₂	—	712	2	no	—
343	C₂	#26022329	1	2	yes	direct product factor
344	C₂	C₂×#19151794	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151794
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#364×C₈₉	C₂
socle	C₂²×C₈₉	#459
hypercenter	C₂×#364	D₈₉
hypocenter	C₈₉	C₂×#2213
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
1. C₂×#19152036	C₂	semidirect
2. C₂×#19151975	C₂	semidirect
3. C₂×#19151716	C₂	semidirect
4. #26022329	C₂	direct (N × H)
5. C₂×#19151412	C₂	semidirect
6. C₂×#364×C₈₉	C₂	semidirect
13. #19152036	C₂²	semidirect
17. #19151975	C₂²	semidirect
19. #19151716	C₂²	semidirect
22. #19151412	C₂²	semidirect
25. #364×C₈₉	C₂²	semidirect
31. #15668034	D₄	semidirect
33. C₂×Dic₃₅₆	D₄	semidirect
44. C₂×D₇₁₂	D₄	semidirect
84. Dic₃₅₆	C₂×D₄	semidirect
94. D₇₁₂	C₂×D₄	semidirect
169. C₂×C₈₉	#2213	semidirect
177. C₂×#364	D₈₉	semidirect
179. C₈₉	C₂×#2213	semidirect
197. #364	C₂×D₈₉	semidirect
343. C₂	#26022329	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×#19152036	2
C₂×#19151975	2
C₂×#19151716	2
#26022329	2
C₂×#19151412	2
C₂×#364×C₈₉	2
C₂×#2213	89

Subgroups — C2×#26022329 #36708809

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022329 #36708809