Subgroups — C₂×#26022440 #36709195

Order 22784.

638 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151636	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022440	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151425	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19151423	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151399	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151399	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂²⋊#13083549	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#466×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#15667971	—	2	1	no	—
10	C₂²×#13083549	C₂²	1	1	yes	characteristic
11	C₂²×#13083549	C₂²	1	1	yes	characteristic
12	#19151636	—	2	2	no	—
13	#19151636	C₂²	1	4	yes	complemented
14	C₂×#15667946	C₂²	1	1	yes	characteristic
15	C₂×#15667946	C₂²	1	1	yes	characteristic
16	C₂×#15667944	—	2	1	no	—
17	#19151425	C₂²	1	4	yes	complemented
18	#19151423	C₂²	1	4	yes	complemented
19	#19151399	C₂²	1	4	yes	complemented
20	#19151399	C₂²	1	4	yes	complemented
21	C₂²⋊#13083549	C₂²	1	4	yes	complemented
22	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#121×C₈₉	—	2	1	no	—
25	#466×C₈₉	—	2	2	no	—
26	#466×C₈₉	C₂²	1	4	yes	—
27	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#102×C₈₉	—	2	1	no	—
29	#15667971	—	2	4	no	—
30	#15667971	—	4	2	no	—
31	C₂×C₄⋊Dic₈₉	—	2	1	no	—
32	C₂×C₄⋊Dic₈₉	—	2	1	no	—
33	C₂×C₂²⋊Dic₈₉	—	4	1	no	—
34	C₂×C₄×Dic₈₉	—	4	1	no	—
35	C₂×#13083549	—	2	2	no	—
36	C₂×#13083549	—	2	2	no	—
37	C₂×#13083549	C₂³	1	2	yes	—
38	C₂×#13083549	C₂³	1	2	yes	—
39	#15667946	—	2	1	no	—
40	#15667946	—	2	1	no	—
41	#15667946	C₂³	1	2	yes	—
42	#15667946	C₂³	1	2	yes	—
43	#15667944	—	2	4	no	—
44	C₂²×D₄×C₈₉	—	2	1	no	—
45	C₂²×D₄×C₈₉	—	4	1	no	—
46	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
47	C₂×D₈×C₈₉	—	2	1	no	—
48	C₂×D₈×C₈₉	—	2	2	no	—
49	C₂×D₈×C₈₉	—	2	2	no	—
50	C₂×D₈×C₈₉	C₂³	1	2	yes	—
51	C₂²×C₈×C₈₉	—	2	2	no	—
52	C₂²×C₈×C₈₉	—	2	2	no	—
53	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	D₄	1	2	yes	—
55	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
56	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
57	#121×C₈₉	—	2	4	no	—
58	#121×C₈₉	—	4	2	no	—
59	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
60	C₂×C₂²⋊C₄×C₈₉	—	4	1	no	—
61	#107×C₈₉	—	2	1	no	—
62	#107×C₈₉	C₂³	1	2	yes	—
63	#102×C₈₉	—	2	4	no	—
64	C₂²×Dic₈₉	—	4	1	no	—
65	C₂²×Dic₈₉	—	4	1	no	—
66	C₄⋊Dic₈₉	—	2	2	no	—
67	C₄⋊Dic₈₉	—	2	2	no	—
68	C₄⋊Dic₈₉	—	4	1	no	—
69	C₄⋊Dic₈₉	—	4	1	no	—
70	C₂²⋊Dic₈₉	—	4	4	no	—
71	C₄×Dic₈₉	—	4	4	no	—
72	#13083549	—	2	2	no	—
73	#13083549	—	2	2	no	—
74	C₂⁴×C₈₉	—	4	1	no	—
75	C₂⁴×C₈₉	D₈	1	1	yes	characteristic
76	C₂×D₄×C₈₉	—	2	2	no	—
77	C₂×D₄×C₈₉	—	4	4	no	—
78	C₂×D₄×C₈₉	—	4	2	no	—
79	C₂×D₄×C₈₉	—	4	1	no	—
80	C₂×D₄×C₈₉	—	4	2	no	—
81	C₂×D₄×C₈₉	—	8	4	no	—
82	C₂²×C₄×C₈₉	—	2	2	no	—
83	C₂²×C₄×C₈₉	—	2	2	no	—
84	C₂²×C₄×C₈₉	—	4	1	no	—
85	C₂²×C₄×C₈₉	D₈	1	2	yes	—
86	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
89	D₈×C₈₉	—	2	2	no	—
90	D₈×C₈₉	—	2	2	no	—
91	D₈×C₈₉	—	4	2	no	—
92	C₂×C₈×C₈₉	—	2	2	no	—
93	C₂×C₈×C₈₉	—	2	4	no	—
94	C₂×C₈×C₈₉	—	2	2	no	—
95	C₂×C₈×C₈₉	—	2	2	no	—
96	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₈₉	#41	1	2	yes	—
99	C₂×C₈×C₈₉	#41	1	2	yes	—
100	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₈₉	—	2	2	no	—
102	C₄⋊C₄×C₈₉	—	4	1	no	—
103	C₂²⋊C₄×C₈₉	—	4	4	no	—
104	C₂³×C₈₉	—	2	2	no	—
105	C₂³×C₈₉	—	2	2	no	—
106	C₂³×C₈₉	—	2	2	no	—
107	C₂³×C₈₉	—	4	2	no	—
108	C₂³×C₈₉	—	8	2	no	—
109	C₂³×C₈₉	—	8	2	no	—
110	C₂³×C₈₉	—	8	2	no	—
111	C₂³×C₈₉	C₂×D₈	1	2	yes	—
112	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
113	D₄×C₈₉	—	4	2	no	—
114	D₄×C₈₉	—	4	2	no	—
115	D₄×C₈₉	—	8	8	no	—
116	D₄×C₈₉	—	8	2	no	—
117	C₂×C₄×C₈₉	—	2	2	no	—
118	C₂×C₄×C₈₉	—	2	2	no	—
119	C₂×C₄×C₈₉	—	2	2	no	—
120	C₂×C₄×C₈₉	—	2	4	no	—
121	C₂×C₄×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	—	8	2	no	—
123	C₂×C₄×C₈₉	#123	1	2	yes	—
124	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
125	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	#137	1	2	yes	—
127	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
128	C₂×Dic₈₉	—	4	2	no	—
129	C₂×Dic₈₉	—	4	2	no	—
130	C₂×Dic₈₉	—	8	2	no	—
131	C₂×Dic₈₉	—	8	2	no	—
132	C₈×C₈₉	—	2	2	no	—
133	C₈×C₈₉	#123	1	2	yes	—
134	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
135	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
136	C₂²×C₈₉	—	2	2	no	—
137	C₂²×C₈₉	—	2	2	no	—
138	C₂²×C₈₉	—	2	2	no	—
139	C₂²×C₈₉	—	4	4	no	—
140	C₂²×C₈₉	—	8	2	no	—
141	C₂²×C₈₉	—	8	2	no	—
142	C₂²×C₈₉	—	8	2	no	—
143	C₂²×C₈₉	—	16	4	no	—
144	C₂²×C₈₉	#480	1	2	yes	—
145	C₂²×C₈₉	#508	1	2	yes	—
146	C₂²×C₈₉	#509	1	2	yes	—
147	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
148	C₄×C₈₉	—	2	2	no	—
149	C₄×C₈₉	—	8	2	no	—
150	C₄×C₈₉	#480	1	2	yes	—
151	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
152	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
153	Dic₈₉	—	8	2	no	—
154	Dic₈₉	—	8	2	no	—
155	C₂×#2234	—	89	1	no	maximal, Sylow
156	C₂×C₈₉	—	4	2	no	—
157	C₂×C₈₉	—	16	2	no	—
158	C₂×C₈₉	#2234	1	4	yes	complemented
159	C₂×C₈₉	C₂×#480	1	1	yes	characteristic
160	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
161	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
162	C₂×#466	D₈₉	1	1	yes	characteristic
163	C₂×#427	—	89	1	no	—
164	#2234	—	89	8	no	—
165	C₂×#367	—	89	1	no	—
166	C₂×#366	—	89	1	no	—
167	C₂×#357	—	89	1	no	—
168	C₂×#357	—	89	1	no	—
169	C₂×C₂²⋊C₁₆	—	89	1	no	—
170	C₈₉	C₂×#2234	1	1	yes	characteristic, complemented, Sylow
171	C₂²×D₈	C₂×D₈₉	1	1	yes	characteristic
172	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
173	C₂×#121	—	2	1	no	—
174	C₂×#117	—	178	1	no	—
175	C₂²×C₁₆	—	89	1	no	—
176	C₂²×C₁₆	—	89	1	no	—
177	#466	—	2	2	no	—
178	#466	C₂×D₈₉	1	4	yes	—
179	#427	—	89	4	no	—
180	#427	—	178	2	no	—
181	C₂×#107	—	89	1	no	—
182	C₂×#107	—	89	1	no	—
183	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
184	C₂×#106	—	178	1	no	—
185	C₂×#102	—	2	1	no	—
186	#367	—	89	4	no	—
187	#366	—	89	4	no	—
188	#357	—	89	4	no	—
189	#357	—	89	4	no	—
190	C₂²⋊C₁₆	—	89	4	no	—
191	C₂²×D₄	—	2	1	no	—
192	C₂²×D₄	—	4	1	no	—
193	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	—	2	2	no	—
196	C₂×D₈	—	2	2	no	—
197	C₂×D₈	C₂²×D₈₉	1	2	yes	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
201	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
202	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
203	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
204	#121	—	2	4	no	—
205	#121	—	4	2	no	—
206	#117	—	356	2	no	—
207	#117	—	178	4	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	178	1	no	—
210	C₂×C₄⋊C₄	—	178	1	no	—
211	C₂×C₂²⋊C₄	—	4	1	no	—
212	C₂×C₂²⋊C₄	—	356	1	no	—
213	C₂×C₄²	—	356	1	no	—
214	C₂×C₁₆	—	89	2	no	—
215	C₂×C₁₆	—	89	2	no	—
216	C₂×C₁₆	—	178	2	no	—
217	C₂×C₁₆	—	178	2	no	—
218	#107	—	2	1	no	—
219	#107	—	89	2	no	—
220	#107	—	89	2	no	—
221	#107	—	178	1	no	—
222	#107	—	178	1	no	—
223	#107	C₂²×D₈₉	1	2	yes	—
224	#106	—	178	4	no	—
225	#102	—	2	4	no	—
226	C₂⁴	—	4	1	no	—
227	C₂⁴	#13083559	1	1	yes	characteristic
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	—	4	2	no	—
231	C₂×D₄	—	4	1	no	—
232	C₂×D₄	—	4	2	no	—
233	C₂×D₄	—	8	4	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	356	1	no	—
238	C₂²×C₄	—	356	1	no	—
239	C₂²×C₄	#13083559	1	2	yes	—
240	C₂²×C₄	#13083559	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
242	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
243	D₈	—	2	2	no	—
244	D₈	—	2	2	no	—
245	D₈	—	4	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
251	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
252	C₂×C₈	#13083573	1	2	yes	—
253	C₂×C₈	#13083573	1	2	yes	—
254	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	356	1	no	—
258	C₄⋊C₄	—	356	1	no	—
259	C₄⋊C₄	—	178	2	no	—
260	C₄⋊C₄	—	178	2	no	—
261	C₂²⋊C₄	—	4	4	no	—
262	C₂²⋊C₄	—	356	4	no	—
263	C₄²	—	356	4	no	—
264	C₁₆	—	178	2	no	—
265	C₁₆	—	178	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	2	no	—
268	C₂³	—	2	2	no	—
269	C₂³	—	4	2	no	—
270	C₂³	—	8	2	no	—
271	C₂³	—	8	2	no	—
272	C₂³	—	8	2	no	—
273	C₂³	C₂×#13083559	1	2	yes	—
274	C₂³	C₂×#13083559	1	1	yes	characteristic
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	8	8	no	—
278	D₄	—	8	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	8	2	no	—
285	C₂×C₄	—	356	2	no	—
286	C₂×C₄	—	356	2	no	—
287	C₂×C₄	—	712	2	no	—
288	C₂×C₄	—	712	2	no	—
289	C₂×C₄	#15667993	1	2	yes	—
290	C₂×C₄	C₂×#13083559	1	2	yes	—
291	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
292	C₂×C₄	#15668042	1	2	yes	—
293	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	#15667993	1	2	yes	—
296	C₈	C₂×#13083573	1	1	yes	characteristic
297	C₈	C₂×#13083573	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	4	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	16	4	no	—
306	C₂²	#19151909	1	2	yes	—
307	C₂²	#19152028	1	2	yes	—
308	C₂²	#19152034	1	2	yes	—
309	C₂²	C₂²×#13083559	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	8	2	no	—
312	C₄	—	712	2	no	—
313	C₄	—	712	2	no	—
314	C₄	#19151909	1	2	yes	—
315	C₄	C₂×#15667993	1	1	yes	characteristic
316	C₄	C₂×#15668042	1	1	yes	characteristic
317	C₂	—	4	2	no	—
318	C₂	—	16	2	no	—
319	C₂	#26022440	1	4	yes	direct product factor
320	C₂	C₂×#19151909	1	1	yes	characteristic
321	C₂	C₂×#19152028	1	1	yes	characteristic
322	C₂	C₂×#19152034	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting, prime radical	C₂×#466×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
hypercenter	C₂×#466	D₈₉
hypocenter	C₈₉	C₂×#2234
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×#466

Complements

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

N	H ≅ G/N	type
1. C₂×#19151636	C₂	semidirect
2. #26022440	C₂	direct (N × H)
3. C₂×#19151425	C₂	semidirect
4. C₂×#19151423	C₂	semidirect
5. C₂×#19151399	C₂	semidirect
6. C₂×#19151399	C₂	semidirect
7. C₂×C₂²⋊#13083549	C₂	semidirect
13. #19151636	C₂²	semidirect
17. #19151425	C₂²	semidirect
18. #19151423	C₂²	semidirect
19. #19151399	C₂²	semidirect
20. #19151399	C₂²	semidirect
21. C₂²⋊#13083549	C₂²	semidirect
158. C₂×C₈₉	#2234	semidirect
170. C₈₉	C₂×#2234	semidirect
319. C₂	#26022440	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×#19151636	2
#26022440	2
C₂×#19151425	2
C₂×#19151423	2
C₂×#19151399	2
C₂×C₂²⋊#13083549	2
C₂×#466×C₈₉	2
C₂×#2234	89

Subgroups — C2×#26022440 #36709195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022440 #36709195