Subgroups — C₂×#26022037 #36707459

Order 22784.

718 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667951	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151635	C₂	1	1	yes	maximal, characteristic
3	C₂×#107×D₈₉	C₂	1	1	yes	maximal, characteristic
4	#26022037	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19151426	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151425	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151423	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#366×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
10	C₂×#15667970	—	2	1	no	—
11	C₂×C₄⋊C₄×D₈₉	—	2	1	no	—
12	C₂×#15667951	C₄	1	8	yes	complemented
13	C₂×#15667951	C₂²	1	4	yes	—
14	C₂²×#13083549	C₂²	1	1	yes	characteristic
15	#19151635	—	2	2	no	—
16	#19151635	C₂²	1	4	yes	—
17	C₂×#15667946	C₂²	1	1	yes	characteristic
18	C₂×#15667946	C₂²	1	1	yes	characteristic
19	C₂×#15667945	—	2	1	no	—
20	#107×D₈₉	—	2	2	no	—
21	#107×D₈₉	C₂²	1	4	yes	—
22	C₂×#15667943	—	2	1	no	—
23	#19151426	C₂²	1	4	yes	complemented
24	#19151425	C₂²	1	4	yes	complemented
25	#19151423	C₂²	1	4	yes	complemented
26	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
29	#366×C₈₉	C₂²	1	4	yes	complemented
30	C₂²×C₄×D₈₉	—	4	1	no	—
31	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
32	C₂²×#10623494	D₄	1	1	yes	characteristic
33	C₂×C₈×D₈₉	C₂×C₄	1	4	yes	—
34	C₂×C₈×D₈₉	D₄	1	2	yes	—
35	C₂×C₈×D₈₉	Q₈	1	4	yes	—
36	C₂×C₈×D₈₉	C₂³	1	2	yes	—
37	#15667970	—	2	4	no	—
38	#15667970	—	4	2	no	—
39	C₂×C₄⋊Dic₈₉	—	2	1	no	—
40	C₂×C₄⋊Dic₈₉	—	2	1	no	—
41	C₂×#13083553	—	4	1	no	—
42	C₄⋊C₄×D₈₉	—	2	4	no	—
43	C₄⋊C₄×D₈₉	—	4	2	no	—
44	C₂×#13083551	—	4	1	no	—
45	C₂×C₄×Dic₈₉	—	4	1	no	—
46	#15667951	C₂×C₄	1	16	yes	complemented
47	C₂×#13083549	C₂×C₄	1	4	yes	complemented
48	C₂×#13083549	C₂³	1	2	yes	—
49	#15667946	—	2	1	no	—
50	#15667946	—	2	1	no	—
51	#15667946	C₂³	1	2	yes	—
52	#15667946	C₂³	1	2	yes	—
53	#15667945	—	2	4	no	—
54	#15667943	—	2	4	no	—
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
57	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
58	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
59	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
60	#107×C₈₉	—	2	1	no	—
61	#107×C₈₉	—	2	1	no	—
62	#107×C₈₉	C₂³	1	2	yes	—
63	#107×C₈₉	C₂³	1	2	yes	—
64	C₂³×D₈₉	D₈	1	1	yes	characteristic
65	C₂²×Dic₈₉	—	4	1	no	—
66	C₂²×Dic₈₉	—	4	1	no	—
67	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
68	C₂×C₄×D₈₉	—	4	4	no	—
69	C₂×C₄×D₈₉	—	8	4	no	—
70	C₂×C₄×D₈₉	C₄⋊C₄	1	4	yes	—
71	C₂×C₄×D₈₉	D₈	1	2	yes	—
72	C₂×C₄×D₈₉	Q₁₆	1	4	yes	—
73	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
74	C₂×#10623494	C₄⋊C₄	1	4	yes	—
75	C₂×#10623494	C₂×D₄	1	2	yes	—
76	C₈×D₈₉	C₄⋊C₄	1	8	yes	—
77	C₈×D₈₉	C₂²×C₄	1	4	yes	—
78	C₈×D₈₉	C₂×Q₈	1	4	yes	—
79	C₄⋊Dic₈₉	—	2	2	no	—
80	C₄⋊Dic₈₉	—	2	2	no	—
81	C₄⋊Dic₈₉	—	4	1	no	—
82	C₄⋊Dic₈₉	—	4	1	no	—
83	#13083553	—	4	4	no	—
84	#13083551	—	4	4	no	—
85	C₄×Dic₈₉	—	4	4	no	—
86	#13083549	C₂²×C₄	1	4	yes	complemented
87	C₂²×C₄×C₈₉	—	4	1	no	—
88	C₂²×C₄×C₈₉	—	4	1	no	—
89	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₈₉	—	2	2	no	—
95	C₄⋊C₄×C₈₉	—	2	2	no	—
96	C₄⋊C₄×C₈₉	—	4	1	no	—
97	C₄⋊C₄×C₈₉	—	4	1	no	—
98	C₁₆×C₈₉	C₂²×C₄	1	4	yes	complemented
99	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
100	C₂²×D₈₉	—	2	2	no	—
101	C₂²×D₈₉	—	2	2	no	—
102	C₂²×D₈₉	#107	1	4	yes	—
103	C₂²×D₈₉	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₈₉	—	4	2	no	—
105	C₂×C₄×C₈₉	—	4	2	no	—
106	C₂×C₄×C₈₉	—	8	2	no	—
107	C₂×C₄×C₈₉	—	8	2	no	—
108	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
111	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
112	C₂×Dic₈₉	—	4	2	no	—
113	C₂×Dic₈₉	—	4	2	no	—
114	C₂×Dic₈₉	—	8	2	no	—
115	C₂×Dic₈₉	—	8	2	no	—
116	C₂×Dic₈₉	#107	1	4	yes	—
117	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
118	C₄×D₈₉	—	8	8	no	—
119	C₄×D₈₉	#107	1	8	yes	—
120	C₄×D₈₉	C₂×C₄⋊C₄	1	4	yes	—
121	C₄×D₈₉	C₂×Q₁₆	1	4	yes	—
122	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
123	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
124	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
125	#10623494	C₂×C₄⋊C₄	1	4	yes	—
126	C₂²×C₈₉	C₂×#107	1	2	yes	—
127	C₂²×C₈₉	#509	1	2	yes	—
128	C₂²×C₈₉	#510	1	2	yes	—
129	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
130	C₂×D₈₉	—	2	2	no	—
131	C₂×D₈₉	—	2	2	no	—
132	C₂×D₈₉	—	2	8	no	—
133	C₂×D₈₉	C₂×#107	1	4	yes	—
134	C₄×C₈₉	—	8	2	no	—
135	C₄×C₈₉	—	8	2	no	—
136	C₄×C₈₉	C₂×#107	1	2	yes	—
137	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
139	Dic₈₉	—	8	2	no	—
140	Dic₈₉	—	8	2	no	—
141	Dic₈₉	C₂×#107	1	4	yes	—
142	C₂×#2159	—	89	1	no	maximal, Sylow
143	C₂×C₈₉	#2159	1	4	yes	complemented
144	C₂×C₈₉	C₂²×#107	1	1	yes	characteristic
145	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
146	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
147	D₈₉	—	2	4	no	—
148	C₂²×M₃₂	—	89	1	no	—
149	C₂×#427	—	89	1	no	—
150	C₂²×#107	—	89	1	no	—
151	#2159	—	89	8	no	—
152	C₂×#367	—	89	1	no	—
153	C₂×#367	—	89	1	no	—
154	C₂×#366	—	89	1	no	—
155	C₂×#366	D₈₉	1	1	yes	characteristic, complemented
156	C₈₉	C₂×#2159	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₈	—	89	1	no	—
158	C₂×#117	—	178	1	no	—
159	C₂²×C₄⋊C₄	—	178	1	no	—
160	C₂×M₃₂	—	89	8	no	—
161	C₂×M₃₂	—	89	4	no	—
162	C₂²×C₁₆	—	89	1	no	—
163	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
164	#427	—	89	4	no	—
165	#427	—	178	2	no	—
166	C₂×#107	—	89	1	no	—
167	C₂×#107	—	89	1	no	—
168	C₂×#107	—	89	4	no	—
169	C₂×#107	—	178	2	no	—
170	C₂×#107	—	178	1	no	—
171	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
172	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
173	C₂×#106	—	178	1	no	—
174	#367	—	89	4	no	—
175	#367	—	89	4	no	—
176	#366	—	89	4	no	—
177	#366	C₂×D₈₉	1	4	yes	complemented
178	C₂³×C₄	—	89	1	no	—
179	C₂³×C₄	—	356	1	no	—
180	C₂²×C₈	—	89	1	no	—
181	C₂²×C₈	—	89	4	no	—
182	C₂²×C₈	—	89	2	no	—
183	C₂²×C₈	—	89	4	no	—
184	C₂²×C₈	—	89	2	no	—
185	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
186	#117	—	356	2	no	—
187	#117	—	178	4	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	—	2	1	no	—
190	C₂×C₄⋊C₄	—	356	2	no	—
191	C₂×C₄⋊C₄	—	356	1	no	—
192	C₂×C₄⋊C₄	—	178	4	no	—
193	C₂×C₄⋊C₄	—	178	1	no	—
194	C₂×C₄⋊C₄	—	178	1	no	—
195	C₂×C₂²⋊C₄	—	356	1	no	—
196	C₂×C₄²	—	356	1	no	—
197	M₃₂	—	89	16	no	—
198	C₂×C₁₆	—	89	4	no	—
199	C₂×C₁₆	—	89	2	no	—
200	C₂×C₁₆	C₄×D₈₉	1	4	yes	complemented
201	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
202	#107	—	2	1	no	—
203	#107	—	2	1	no	—
204	#107	—	89	2	no	—
205	#107	—	89	2	no	—
206	#107	—	178	4	no	—
207	#107	—	178	1	no	—
208	#107	—	178	1	no	—
209	#107	C₂²×D₈₉	1	2	yes	—
210	#107	C₂²×D₈₉	1	2	yes	—
211	#106	—	178	4	no	—
212	C₂⁴	—	89	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	89	1	no	—
216	C₂²×C₄	—	89	2	no	—
217	C₂²×C₄	—	89	4	no	—
218	C₂²×C₄	—	89	2	no	—
219	C₂²×C₄	—	89	4	no	—
220	C₂²×C₄	—	356	4	no	—
221	C₂²×C₄	—	356	1	no	—
222	C₂²×C₄	—	356	1	no	—
223	C₂²×C₄	—	712	4	no	—
224	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
225	C₂×C₈	—	89	2	no	—
226	C₂×C₈	—	89	4	no	—
227	C₂×C₈	—	89	4	no	—
228	C₂×C₈	—	89	4	no	—
229	C₂×C₈	—	89	8	no	—
230	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
231	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
232	C₂×C₈	Q₈×D₈₉	1	2	yes	—
233	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
234	C₄⋊C₄	—	2	2	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	4	1	no	—
237	C₄⋊C₄	—	4	1	no	—
238	C₄⋊C₄	—	356	4	no	—
239	C₄⋊C₄	—	356	1	no	—
240	C₄⋊C₄	—	356	1	no	—
241	C₄⋊C₄	—	178	2	no	—
242	C₄⋊C₄	—	178	2	no	—
243	C₂²⋊C₄	—	356	4	no	—
244	C₄²	—	356	4	no	—
245	C₁₆	—	89	4	no	—
246	C₁₆	C₂×C₄×D₈₉	1	4	yes	complemented
247	C₂³	—	89	4	no	—
248	C₂³	—	89	2	no	—
249	C₂³	—	178	2	no	—
250	C₂³	—	178	2	no	—
251	C₂³	D₈×D₈₉	1	1	yes	characteristic
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	8	2	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	—	89	4	no	—
257	C₂×C₄	—	89	2	no	—
258	C₂×C₄	—	89	8	no	—
259	C₂×C₄	—	89	4	no	—
260	C₂×C₄	—	89	4	no	—
261	C₂×C₄	—	356	2	no	—
262	C₂×C₄	—	356	2	no	—
263	C₂×C₄	—	712	8	no	—
264	C₂×C₄	—	712	2	no	—
265	C₂×C₄	—	712	2	no	—
266	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
267	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
268	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
269	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
270	C₈	—	89	4	no	—
271	C₈	C₄⋊C₄×D₈₉	1	2	yes	—
272	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
273	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
274	C₂²	—	89	4	no	—
275	C₂²	—	178	8	no	—
276	C₂²	—	178	2	no	—
277	C₂²	—	178	2	no	—
278	C₂²	#107×D₈₉	1	2	yes	—
279	C₂²	#19152032	1	2	yes	—
280	C₂²	#19152040	1	2	yes	—
281	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	89	4	no	—
285	C₄	—	712	2	no	—
286	C₄	—	712	2	no	—
287	C₄	#107×D₈₉	1	2	yes	—
288	C₄	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
289	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
290	C₂	—	178	4	no	—
291	C₂	#26022037	1	4	yes	direct product factor
292	C₂	C₂×#107×D₈₉	1	1	yes	characteristic
293	C₂	C₂×#19152032	1	1	yes	characteristic
294	C₂	C₂×#19152040	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#366×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂³×D₈₉	D₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#366	D₈₉
hypocenter	C₈₉	C₂×#2159
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
4. #26022037	C₂	direct (N × H)
5. C₂×#19151426	C₂	semidirect
6. C₂×#19151425	C₂	semidirect
7. C₂×#19151423	C₂	semidirect
8. C₂×#366×C₈₉	C₂	semidirect
12. C₂×#15667951	C₄	semidirect
23. #19151426	C₂²	semidirect
24. #19151425	C₂²	semidirect
25. #19151423	C₂²	semidirect
29. #366×C₈₉	C₂²	semidirect
46. #15667951	C₂×C₄	semidirect
47. C₂×#13083549	C₂×C₄	semidirect
58. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
86. #13083549	C₂²×C₄	semidirect
98. C₁₆×C₈₉	C₂²×C₄	semidirect
143. C₂×C₈₉	#2159	semidirect
155. C₂×#366	D₈₉	semidirect
156. C₈₉	C₂×#2159	semidirect
177. #366	C₂×D₈₉	semidirect
200. C₂×C₁₆	C₄×D₈₉	semidirect
246. C₁₆	C₂×C₄×D₈₉	semidirect
291. C₂	#26022037	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#15667951	2
C₂×#19151635	2
C₂×#107×D₈₉	2
#26022037	2
C₂×#19151426	2
C₂×#19151425	2
C₂×#19151423	2
C₂×#366×C₈₉	2
C₂×#2159	89

Subgroups — C2×#26022037 #36707459

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022037 #36707459