Subgroups — C₂×#26022049 #36707491

Order 22784.

726 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₈₉	C₂	1	1	yes	maximal, characteristic
2	#26022049	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151416	C₂	1	2	yes	maximal, complemented
4	C₂×#19151377	C₂	1	2	yes	maximal
5	C₂×C₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₈₉	C₄	1	1	yes	characteristic
8	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
9	C₂×C₄²×D₈₉	C₄	1	1	yes	characteristic
10	C₂²×#13083549	C₂²	1	2	yes	—
11	C₂×C₄×#10623494	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
14	C₄×C₈×D₈₉	C₄	1	4	yes	—
15	C₄×C₈×D₈₉	C₂²	1	4	yes	—
16	#19151416	C₂²	1	8	yes	complemented
17	#19151377	C₂²	1	8	yes	—
18	C₄×#13083549	C₂²	1	4	yes	complemented
19	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₄×C₁₆×C₈₉	C₂²	1	4	yes	complemented
22	C₂²×C₄×D₈₉	C₈	1	1	yes	characteristic
23	C₂²×C₄×D₈₉	C₈	1	1	yes	characteristic
24	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
26	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×D₈₉	—	2	4	no	—
28	C₂×C₈×D₈₉	—	2	4	no	—
29	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₈₉	C₂×C₄	1	4	yes	—
31	C₂×C₈×D₈₉	C₂³	1	2	yes	—
32	C₂×C₄×Dic₈₉	C₈	1	1	yes	characteristic
33	C₂×C₄×Dic₈₉	C₈	1	1	yes	characteristic
34	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
35	C₄²×D₈₉	C₈	1	4	yes	—
36	C₄²×D₈₉	C₂×C₄	1	4	yes	—
37	C₂×#13083549	—	2	4	no	—
38	C₂×#13083549	C₂³	1	4	yes	—
39	C₄×#10623494	C₂×C₄	1	4	yes	—
40	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
41	C₈×Dic₈₉	C₂×C₄	1	4	yes	—
42	C₈×Dic₈₉	C₂³	1	2	yes	—
43	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₆×C₈₉	—	2	4	no	—
47	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
48	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₈₉	C₂³	1	2	yes	—
50	C₂³×D₈₉	C₂×C₈	1	1	yes	characteristic
51	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
52	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
53	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
54	C₂×C₄×D₈₉	—	2	4	no	—
55	C₂×C₄×D₈₉	—	2	2	no	—
56	C₂×C₄×D₈₉	—	2	2	no	—
57	C₂×C₄×D₈₉	C₁₆	1	8	yes	complemented
58	C₂×C₄×D₈₉	C₂×C₈	1	4	yes	—
59	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
60	C₂×C₄×D₈₉	C₂×C₈	1	4	yes	—
61	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
62	C₂×#10623494	—	2	2	no	—
63	C₂×#10623494	—	2	2	no	—
64	C₂×#10623494	C₂²×C₄	1	2	yes	—
65	C₂×#10623494	C₂²×C₄	1	2	yes	—
66	C₈×D₈₉	—	2	4	no	—
67	C₈×D₈₉	—	2	8	no	—
68	C₈×D₈₉	—	2	4	no	—
69	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
70	C₄×Dic₈₉	C₂×C₈	1	4	yes	—
71	C₄×Dic₈₉	C₂×C₈	1	4	yes	—
72	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
73	#13083549	—	2	4	no	—
74	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₈₉	M₁₆	1	4	yes	—
78	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
80	C₂×C₈×C₈₉	#41	1	4	yes	—
81	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
82	C₄²×C₈₉	C₂×C₈	1	2	yes	—
83	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
84	C₁₆×C₈₉	—	2	4	no	—
85	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
86	C₂²×D₈₉	—	2	2	no	—
87	C₂²×D₈₉	—	2	2	no	—
88	C₂²×D₈₉	C₂×C₁₆	1	4	yes	—
89	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
90	C₂×C₄×C₈₉	C₂×C₁₆	1	4	yes	—
91	C₂×C₄×C₈₉	M₃₂	1	4	yes	—
92	C₂×C₄×C₈₉	#117	1	2	yes	—
93	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
95	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
96	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
98	C₂×Dic₈₉	—	2	2	no	—
99	C₂×Dic₈₉	—	2	2	no	—
100	C₂×Dic₈₉	C₂×C₁₆	1	4	yes	—
101	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
102	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
103	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
104	C₄×D₈₉	—	2	2	no	—
105	C₄×D₈₉	—	2	2	no	—
106	C₄×D₈₉	—	2	4	no	—
107	C₄×D₈₉	—	2	8	no	—
108	C₄×D₈₉	C₂×C₁₆	1	16	yes	complemented
109	C₈×C₈₉	#117	1	2	yes	—
110	C₈×C₈₉	C₂×M₁₆	1	4	yes	—
111	C₈×C₈₉	C₂×#41	1	2	yes	—
112	#10623494	—	2	2	no	—
113	#10623494	—	2	2	no	—
114	C₂²×C₈₉	#431	1	2	yes	—
115	C₂²×C₈₉	C₂²×C₁₆	1	2	yes	—
116	C₂²×C₈₉	C₂×M₃₂	1	2	yes	—
117	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
118	C₂×D₈₉	—	2	2	no	—
119	C₂×D₈₉	—	2	2	no	—
120	C₂×D₈₉	—	2	8	no	—
121	C₂×D₈₉	C₂²×C₁₆	1	4	yes	—
122	C₄×C₈₉	#431	1	2	yes	—
123	C₄×C₈₉	C₂²×C₁₆	1	4	yes	—
124	C₄×C₈₉	C₂×M₃₂	1	4	yes	—
125	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
126	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
127	Dic₈₉	—	2	2	no	—
128	Dic₈₉	—	2	2	no	—
129	Dic₈₉	C₂²×C₁₆	1	4	yes	—
130	C₂×#2162	—	89	1	no	maximal, Sylow
131	C₂×C₈₉	#2162	1	4	yes	complemented
132	C₂×C₈₉	C₂×#431	1	1	yes	characteristic
133	C₂×C₈₉	C₂³×C₁₆	1	1	yes	characteristic
134	C₂×C₈₉	C₂²×M₃₂	1	1	yes	characteristic
135	D₈₉	—	2	4	no	—
136	C₂²×C₄×C₈	—	89	1	no	—
137	#2162	—	89	8	no	—
138	C₂×C₄⋊C₁₆	—	89	2	no	—
139	C₂×C₂²⋊C₁₆	—	89	2	no	—
140	C₂×C₄×C₁₆	—	89	1	no	—
141	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
142	C₈₉	C₂×#2162	1	1	yes	characteristic, complemented, Sylow
143	C₂³×C₈	—	89	1	no	—
144	C₂³×C₈	—	89	1	no	—
145	C₂²×C₄²	—	89	1	no	—
146	C₂²×C₁₆	—	89	2	no	—
147	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
148	C₂×C₄×C₈	—	89	1	no	—
149	C₂×C₄×C₈	—	89	1	no	—
150	C₂×C₄×C₈	—	89	1	no	—
151	C₂×C₄×C₈	—	89	4	no	—
152	C₂×C₄×C₈	—	89	4	no	—
153	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
154	C₄⋊C₁₆	—	89	8	no	—
155	C₂²⋊C₁₆	—	89	8	no	—
156	C₄×C₁₆	—	89	4	no	—
157	C₄×C₁₆	C₂×D₈₉	1	4	yes	complemented
158	C₂³×C₄	—	89	1	no	—
159	C₂³×C₄	—	89	1	no	—
160	C₂³×C₄	—	89	1	no	—
161	C₂²×C₈	—	89	1	no	—
162	C₂²×C₈	—	89	1	no	—
163	C₂²×C₈	—	89	2	no	—
164	C₂²×C₈	—	89	4	no	—
165	C₂²×C₈	—	89	2	no	—
166	C₂²×C₈	—	178	4	no	—
167	C₂²×C₈	—	178	4	no	—
168	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
170	C₂×C₄²	—	89	1	no	—
171	C₂×C₄²	—	89	1	no	—
172	C₂×C₄²	—	89	1	no	—
173	C₂×C₄²	—	89	4	no	—
174	C₂×C₄²	—	89	4	no	—
175	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
176	C₂×C₁₆	—	2	4	no	—
177	C₂×C₁₆	—	89	4	no	—
178	C₂×C₁₆	—	178	4	no	—
179	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
180	C₄×C₈	—	89	2	no	—
181	C₄×C₈	—	89	4	no	—
182	C₄×C₈	—	89	2	no	—
183	C₄×C₈	—	89	4	no	—
184	C₄×C₈	C₄×D₈₉	1	2	yes	—
185	C₄×C₈	C₂²×D₈₉	1	2	yes	—
186	C₂⁴	—	89	1	no	—
187	C₂²×C₄	—	89	1	no	—
188	C₂²×C₄	—	89	4	no	—
189	C₂²×C₄	—	89	8	no	—
190	C₂²×C₄	—	89	1	no	—
191	C₂²×C₄	—	89	1	no	—
192	C₂²×C₄	—	89	2	no	—
193	C₂²×C₄	—	89	4	no	—
194	C₂²×C₄	—	89	2	no	—
195	C₂²×C₄	—	178	2	no	—
196	C₂²×C₄	—	178	2	no	—
197	C₂²×C₄	—	178	4	no	—
198	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
199	C₂²×C₄	C₈×D₈₉	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
201	C₂×C₈	—	89	2	no	—
202	C₂×C₈	—	89	2	no	—
203	C₂×C₈	—	178	2	no	—
204	C₂×C₈	—	178	2	no	—
205	C₂×C₈	—	178	4	no	—
206	C₂×C₈	—	178	8	no	—
207	C₂×C₈	—	178	4	no	—
208	C₂×C₈	#13083557	1	4	yes	—
209	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
210	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
211	C₂×C₈	#13083574	1	4	yes	—
212	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
213	C₄²	—	89	2	no	—
214	C₄²	—	89	4	no	—
215	C₄²	—	89	4	no	—
216	C₄²	—	89	2	no	—
217	C₄²	C₈×D₈₉	1	2	yes	—
218	C₄²	C₂×C₄×D₈₉	1	2	yes	—
219	C₁₆	—	2	4	no	—
220	C₁₆	—	178	4	no	—
221	C₂³	—	89	4	no	—
222	C₂³	—	89	2	no	—
223	C₂³	—	178	2	no	—
224	C₂³	—	178	2	no	—
225	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
226	C₂×C₄	—	89	4	no	—
227	C₂×C₄	—	89	2	no	—
228	C₂×C₄	—	89	16	no	—
229	C₂×C₄	—	89	2	no	—
230	C₂×C₄	—	89	2	no	—
231	C₂×C₄	—	178	2	no	—
232	C₂×C₄	—	178	2	no	—
233	C₂×C₄	—	178	4	no	—
234	C₂×C₄	—	178	8	no	—
235	C₂×C₄	—	178	2	no	—
236	C₂×C₄	—	178	2	no	—
237	C₂×C₄	C₁₆×D₈₉	1	4	yes	—
238	C₂×C₄	#15667951	1	4	yes	—
239	C₂×C₄	#15667969	1	2	yes	—
240	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
241	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
242	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
243	C₂×C₄	C₂×#13083557	1	2	yes	—
244	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
245	C₈	—	178	2	no	—
246	C₈	—	178	2	no	—
247	C₈	#15667969	1	2	yes	—
248	C₈	C₂×#13083557	1	4	yes	—
249	C₈	C₂×#13083574	1	2	yes	—
250	C₂²	—	89	4	no	—
251	C₂²	—	178	2	no	—
252	C₂²	—	178	8	no	—
253	C₂²	—	178	2	no	—
254	C₂²	#19151649	1	2	yes	—
255	C₂²	C₂×C₁₆×D₈₉	1	2	yes	—
256	C₂²	C₂×#15667951	1	2	yes	—
257	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
258	C₄	—	89	4	no	—
259	C₄	—	178	2	no	—
260	C₄	—	178	2	no	—
261	C₄	#19151649	1	2	yes	—
262	C₄	C₂×C₁₆×D₈₉	1	4	yes	—
263	C₄	C₂×#15667951	1	4	yes	—
264	C₄	C₂×#15667969	1	1	yes	characteristic
265	C₄	C₂²×#13083557	1	1	yes	characteristic
266	C₂	—	178	4	no	—
267	C₂	#26022049	1	4	yes	direct product factor
268	C₂	C₂×#19151649	1	1	yes	characteristic
269	C₂	C₂²×C₁₆×D₈₉	1	1	yes	characteristic
270	C₂	C₂²×#15667951	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₈₉
commutator	C₂×C₈₉	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle	C₂³×C₈₉	C₂²×C₈
prime radical	C₂³×D₈₉	C₂×C₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	C₂×#2162
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #26022049	C₂	direct (N × H)
3. C₂×#19151416	C₂	semidirect
5. C₂×C₄×#13083549	C₂	semidirect
6. C₂×C₄×C₁₆×C₈₉	C₂	semidirect
16. #19151416	C₂²	semidirect
18. C₄×#13083549	C₂²	semidirect
21. C₄×C₁₆×C₈₉	C₂²	semidirect
57. C₂×C₄×D₈₉	C₁₆	semidirect
108. C₄×D₈₉	C₂×C₁₆	semidirect
131. C₂×C₈₉	#2162	semidirect
141. C₂×C₄×C₁₆	D₈₉	semidirect
142. C₈₉	C₂×#2162	semidirect
157. C₄×C₁₆	C₂×D₈₉	semidirect
267. C₂	#26022049	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×C₄×C₈×D₈₉	2
#26022049	2
C₂×#19151416	2
C₂×#19151377	2
C₂×C₄×#13083549	2
C₂×C₄×C₁₆×C₈₉	2
C₂×#2162	89

Subgroups — C2×#26022049 #36707491

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022049 #36707491