Subgroups — C₂×#26021847 #36707061

Order 22784.

854 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	C₂²×#15667951	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151552	C₂	1	1	yes	maximal, characteristic
4	#26021847	C₂	1	8	yes	maximal, direct product factor
5	C₂×#19151374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151373	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#13083557	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
11	C₂×#15667969	C₄	1	1	yes	characteristic
12	C₂×#15667951	C₄	1	8	yes	complemented
13	C₂×#15667951	C₂²	1	4	yes	—
14	C₂²×#13083549	C₂²	1	1	yes	characteristic
15	C₂²×#13083549	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₈₉	C₄	1	8	yes	complemented
17	C₂×C₁₆×D₈₉	C₂²	1	4	yes	—
18	#19151552	C₄	1	4	yes	—
19	#19151552	C₂²	1	4	yes	—
20	C₂×#15667923	C₄	1	1	yes	characteristic
21	C₂×#15667922	C₄	1	1	yes	characteristic
22	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
23	#19151374	C₂²	1	4	yes	complemented
24	#19151373	C₂²	1	4	yes	complemented
25	C₁₆×Dic₈₉	C₂²	1	4	yes	complemented
26	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₄×C₁₆×C₈₉	C₂²	1	4	yes	complemented
30	C₂²×C₄×D₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂×#13083557	C₈	1	8	yes	—
32	C₂×#13083557	C₂×C₄	1	4	yes	—
33	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
34	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₈₉	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₈₉	C₂×C₄	1	4	yes	—
37	C₂×C₈×D₈₉	C₂×C₄	1	4	yes	—
38	C₂×C₈×D₈₉	C₂³	1	2	yes	—
39	#15667969	C₈	1	4	yes	—
40	#15667969	C₂×C₄	1	4	yes	—
41	C₂×#13083553	C₈	1	2	yes	—
42	C₂×#13083551	C₈	1	2	yes	—
43	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
44	#15667951	C₂×C₄	1	16	yes	complemented
45	C₂×#13083549	C₂×C₄	1	4	yes	—
46	C₂×#13083549	C₂×C₄	1	4	yes	—
47	C₂×#13083549	C₂³	1	2	yes	—
48	C₂×#13083549	C₂³	1	2	yes	—
49	C₁₆×D₈₉	C₂×C₄	1	16	yes	complemented
50	#15667923	C₂×C₄	1	4	yes	—
51	#15667922	C₂×C₄	1	4	yes	—
52	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
53	C₈×Dic₈₉	C₂³	1	2	yes	—
54	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
57	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
58	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	—
59	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
60	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
61	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₈₉	C₂³	1	2	yes	—
63	C₂³×D₈₉	C₂×C₈	1	1	yes	characteristic
64	C₂²×Dic₈₉	C₂×C₈	1	2	yes	—
65	C₂²×Dic₈₉	C₂×C₈	1	1	yes	characteristic
66	C₂×C₄×D₈₉	C₄²	1	4	yes	—
67	C₂×C₄×D₈₉	C₂×C₈	1	2	yes	—
68	C₂×C₄×D₈₉	C₂×C₈	1	4	yes	—
69	C₂×C₄×D₈₉	C₂²×C₄	1	2	yes	—
70	#13083557	C₂×C₈	1	16	yes	—
71	C₂×#10623494	C₄²	1	4	yes	—
72	C₂×#10623494	C₂×C₈	1	4	yes	—
73	C₂×#10623494	C₂²×C₄	1	2	yes	—
74	C₂×#10623494	C₂²×C₄	1	2	yes	—
75	C₈×D₈₉	C₄²	1	8	yes	—
76	C₈×D₈₉	C₂²×C₄	1	4	yes	—
77	C₈×D₈₉	C₂²×C₄	1	4	yes	—
78	#13083553	C₂×C₈	1	8	yes	—
79	#13083551	C₂×C₈	1	8	yes	—
80	C₄×Dic₈₉	C₂×C₈	1	2	yes	—
81	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
82	#13083549	C₂²×C₄	1	4	yes	—
83	#13083549	C₂²×C₄	1	4	yes	—
84	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
85	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₈₉	C₂×C₈	1	4	yes	—
87	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
92	C₄²×C₈₉	C₂×C₈	1	2	yes	—
93	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
94	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
95	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
96	C₂³×C₈₉	C₂²×C₈	1	1	yes	characteristic
97	C₂²×D₈₉	—	2	4	no	—
98	C₂²×D₈₉	C₄×C₈	1	4	yes	—
99	C₂²×D₈₉	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₈₉	—	2	4	no	—
101	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₈₉	C₂²×C₈	1	4	yes	—
103	C₂×C₄×C₈₉	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
106	C₂×Dic₈₉	—	2	4	no	—
107	C₂×Dic₈₉	C₄×C₈	1	4	yes	—
108	C₂×Dic₈₉	C₂²×C₈	1	4	yes	—
109	C₂×Dic₈₉	C₂²×C₈	1	2	yes	—
110	C₄×D₈₉	C₄×C₈	1	8	yes	—
111	C₄×D₈₉	C₂×C₄²	1	4	yes	—
112	C₄×D₈₉	C₂²×C₈	1	4	yes	—
113	C₈×C₈₉	C₂×C₄²	1	2	yes	—
114	C₈×C₈₉	C₂²×C₈	1	4	yes	—
115	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
116	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
117	#10623494	C₂×C₄²	1	4	yes	—
118	#10623494	C₂²×C₈	1	4	yes	—
119	C₂²×C₈₉	C₂×C₄×C₈	1	2	yes	—
120	C₂²×C₈₉	#504	1	4	yes	—
121	C₂²×C₈₉	C₂³×C₈	1	1	yes	characteristic
122	C₂×D₈₉	—	2	8	no	—
123	C₂×D₈₉	—	2	4	no	—
124	C₂×D₈₉	C₂×C₄×C₈	1	4	yes	—
125	C₄×C₈₉	—	2	4	no	—
126	C₄×C₈₉	C₂×C₄×C₈	1	2	yes	—
127	C₄×C₈₉	C₂²×C₄²	1	1	yes	characteristic
128	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
129	Dic₈₉	—	2	4	no	—
130	Dic₈₉	C₂×C₄×C₈	1	4	yes	—
131	C₂×#2108	—	89	1	no	maximal, Sylow
132	C₂×C₈₉	#2108	1	4	yes	complemented
133	C₂×C₈₉	C₂²×C₄×C₈	1	1	yes	characteristic
134	C₂×C₈₉	C₂×#504	1	2	yes	—
135	D₈₉	—	2	4	no	—
136	C₂²×M₃₂	—	89	1	no	—
137	C₂³×C₁₆	—	89	1	no	—
138	C₂×#405	—	89	1	no	—
139	#2108	—	89	8	no	—
140	C₂×#346	—	89	1	no	—
141	C₂×#346	—	89	1	no	—
142	C₂×C₄×C₁₆	—	89	1	no	—
143	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
144	C₈₉	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
145	C₂²×M₁₆	—	89	1	no	—
146	C₂³×C₈	—	89	1	no	—
147	C₂×#117	—	89	1	no	—
148	C₂×M₃₂	—	89	8	no	—
149	C₂×M₃₂	—	89	4	no	—
150	C₂²×C₁₆	—	89	1	no	—
151	C₂²×C₁₆	—	89	1	no	—
152	C₂²×C₁₆	—	89	8	no	—
153	C₂²×C₁₆	—	89	4	no	—
154	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
155	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
156	#405	—	89	4	no	—
157	#405	—	89	4	no	—
158	C₂×#97	—	89	1	no	—
159	C₂×#97	—	89	1	no	—
160	C₂×C₄×C₈	—	89	1	no	—
161	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
162	#346	—	89	4	no	—
163	#346	—	89	4	no	—
164	C₄×C₁₆	—	89	4	no	—
165	C₄×C₁₆	C₂×D₈₉	1	4	yes	complemented
166	C₂³×C₄	—	89	1	no	—
167	C₂×M₁₆	—	89	8	no	—
168	C₂×M₁₆	—	89	4	no	—
169	C₂²×C₈	—	89	1	no	—
170	C₂²×C₈	—	89	1	no	—
171	C₂²×C₈	—	89	2	no	—
172	C₂²×C₈	—	89	4	no	—
173	C₂²×C₈	—	89	4	no	—
174	C₂²×C₈	—	89	2	no	—
175	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
177	#117	—	89	4	no	—
178	#117	—	89	4	no	—
179	C₂×C₄⋊C₄	—	89	2	no	—
180	C₂×C₂²⋊C₄	—	89	2	no	—
181	C₂×C₄²	—	89	1	no	—
182	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
183	M₃₂	—	89	16	no	—
184	C₂×C₁₆	—	89	4	no	—
185	C₂×C₁₆	—	89	4	no	—
186	C₂×C₁₆	—	89	2	no	—
187	C₂×C₁₆	—	89	2	no	—
188	C₂×C₁₆	—	89	16	no	—
189	C₂×C₁₆	C₄×D₈₉	1	4	yes	—
190	C₂×C₁₆	C₄×D₈₉	1	4	yes	—
191	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
192	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
193	#97	—	89	4	no	—
194	#97	—	89	4	no	—
195	C₄×C₈	—	89	2	no	—
196	C₄×C₈	—	89	2	no	—
197	C₄×C₈	C₄×D₈₉	1	2	yes	—
198	C₄×C₈	C₂²×D₈₉	1	2	yes	—
199	C₂⁴	—	89	1	no	—
200	C₂²×C₄	—	89	1	no	—
201	C₂²×C₄	—	89	2	no	—
202	C₂²×C₄	—	89	4	no	—
203	C₂²×C₄	—	89	2	no	—
204	C₂²×C₄	—	89	4	no	—
205	C₂²×C₄	—	89	2	no	—
206	C₂²×C₄	C₈×D₈₉	1	2	yes	—
207	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
208	M₁₆	—	89	16	no	—
209	C₂×C₈	—	89	4	no	—
210	C₂×C₈	—	89	2	no	—
211	C₂×C₈	—	89	2	no	—
212	C₂×C₈	—	89	4	no	—
213	C₂×C₈	—	89	4	no	—
214	C₂×C₈	—	89	4	no	—
215	C₂×C₈	—	89	8	no	—
216	C₂×C₈	C₈×D₈₉	1	4	yes	—
217	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
218	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
219	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
220	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
221	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
222	C₄⋊C₄	—	89	8	no	—
223	C₂²⋊C₄	—	89	8	no	—
224	C₄²	—	89	2	no	—
225	C₄²	—	89	2	no	—
226	C₄²	C₈×D₈₉	1	2	yes	—
227	C₄²	C₂×C₄×D₈₉	1	2	yes	—
228	C₁₆	—	89	4	no	—
229	C₁₆	—	89	4	no	—
230	C₁₆	C₂×C₄×D₈₉	1	4	yes	—
231	C₁₆	C₂×C₄×D₈₉	1	4	yes	—
232	C₂³	—	89	4	no	—
233	C₂³	—	89	2	no	—
234	C₂³	—	178	4	no	—
235	C₂³	C₂×C₈×D₈₉	1	1	yes	characteristic
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	89	4	no	—
238	C₂×C₄	—	89	2	no	—
239	C₂×C₄	—	89	4	no	—
240	C₂×C₄	—	89	8	no	—
241	C₂×C₄	—	89	4	no	—
242	C₂×C₄	—	89	4	no	—
243	C₂×C₄	—	178	4	no	—
244	C₂×C₄	C₄²×D₈₉	1	2	yes	—
245	C₂×C₄	C₂×C₈×D₈₉	1	4	yes	—
246	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
247	C₂×C₄	C₂×C₈×D₈₉	1	2	yes	—
248	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
249	C₈	—	89	4	no	—
250	C₈	—	89	4	no	—
251	C₈	C₄²×D₈₉	1	2	yes	—
252	C₈	C₂×C₈×D₈₉	1	4	yes	—
253	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
254	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
255	C₂²	—	89	4	no	—
256	C₂²	—	178	4	no	—
257	C₂²	—	178	8	no	—
258	C₂²	C₄×C₈×D₈₉	1	2	yes	—
259	C₂²	#19152014	1	4	yes	—
260	C₂²	C₂²×C₈×D₈₉	1	1	yes	characteristic
261	C₄	—	2	4	no	—
262	C₄	—	89	4	no	—
263	C₄	—	178	4	no	—
264	C₄	C₄×C₈×D₈₉	1	2	yes	—
265	C₄	C₂×C₄²×D₈₉	1	1	yes	characteristic
266	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
267	C₂	—	178	4	no	—
268	C₂	#26021847	1	4	yes	direct product factor
269	C₂	C₂×C₄×C₈×D₈₉	1	1	yes	characteristic
270	C₂	C₂×#19152014	1	2	yes	—

	subgroup	G/H
center	C₂²×C₁₆	C₂×D₈₉
commutator	C₂×C₈₉	C₂²×C₄×C₈
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×#2108
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
4. #26021847	C₂	direct (N × H)
5. C₂×#19151374	C₂	semidirect
6. C₂×#19151373	C₂	semidirect
7. C₂×C₁₆×Dic₈₉	C₂	semidirect
8. C₂×C₄×C₁₆×C₈₉	C₂	semidirect
12. C₂×#15667951	C₄	semidirect
16. C₂×C₁₆×D₈₉	C₄	semidirect
23. #19151374	C₂²	semidirect
24. #19151373	C₂²	semidirect
25. C₁₆×Dic₈₉	C₂²	semidirect
29. C₄×C₁₆×C₈₉	C₂²	semidirect
44. #15667951	C₂×C₄	semidirect
49. C₁₆×D₈₉	C₂×C₄	semidirect
132. C₂×C₈₉	#2108	semidirect
143. C₂×C₄×C₁₆	D₈₉	semidirect
144. C₈₉	C₂×#2108	semidirect
165. C₄×C₁₆	C₂×D₈₉	semidirect
268. C₂	#26021847	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
C₂²×C₁₆×D₈₉	2
C₂×#19151552	2
#26021847	2
C₂×#19151374	2
C₂×#19151373	2
C₂×C₁₆×Dic₈₉	2
C₂×C₄×C₁₆×C₈₉	2
C₂×#2108	89

Subgroups — C2×#26021847 #36707061

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26021847 #36707061