Subgroups — C₂×#26022065 #36707552

Order 22784.

538 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151556	C₂	1	1	yes	maximal, characteristic
2	#26022065	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151418	C₂	1	2	yes	maximal, complemented
4	C₂×#19151380	C₂	1	1	yes	maximal, characteristic
5	C₂×#19151375	C₂	1	2	yes	maximal, complemented
6	C₂×#349×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×#10623494	C₄	1	1	yes	characteristic
8	C₂×#15667971	C₄	1	1	yes	characteristic
9	C₂×#15667952	C₂²	1	2	yes	—
10	#19151556	C₄	1	4	yes	—
11	#19151556	C₂²	1	4	yes	—
12	C₂×#15667922	C₄	1	1	yes	characteristic
13	C₂×#15667922	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₈₉	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
16	#19151418	C₂²	1	8	yes	complemented
17	#19151380	C₂²	1	4	yes	—
18	#19151375	C₂²	1	8	yes	complemented
19	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
21	#349×C₈₉	C₂²	1	4	yes	—
22	C₂²×#10623494	—	2	2	no	—
23	C₂²×#10623494	—	2	2	no	—
24	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
25	C₂²×#10623494	C₂×C₄	1	4	yes	—
26	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
27	#15667971	—	2	2	no	—
28	#15667971	C₂×C₄	1	4	yes	—
29	C₂×C₄⋊Dic₈₉	—	2	1	no	—
30	C₂×C₂²⋊Dic₈₉	—	2	1	no	—
31	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
33	#15667952	—	2	2	no	—
34	#15667952	C₂³	1	4	yes	—
35	C₂×#13083549	—	2	2	no	—
36	#15667922	C₂×C₄	1	4	yes	—
37	#15667922	C₂×C₄	1	2	yes	—
38	#15667922	C₂³	1	2	yes	—
39	C₈×Dic₈₉	C₂×C₄	1	4	yes	—
40	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
41	C₈×Dic₈₉	C₂³	1	2	yes	—
42	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂×M₁₆×C₈₉	—	2	2	no	—
44	C₂×M₁₆×C₈₉	—	2	2	no	—
45	C₂×M₁₆×C₈₉	C₂×C₄	1	2	yes	—
46	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
47	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
48	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
49	M₃₂×C₈₉	—	2	2	no	—
50	M₃₂×C₈₉	C₂³	1	4	yes	—
51	C₂×C₁₆×C₈₉	—	2	2	no	—
52	C₂²×Dic₈₉	—	2	1	no	—
53	C₂²×Dic₈₉	—	2	1	no	—
54	C₂×#10623494	—	2	1	no	—
55	C₂×#10623494	—	2	1	no	—
56	C₂×#10623494	—	2	8	no	—
57	C₂×#10623494	M₁₆	1	2	yes	—
58	C₂×#10623494	M₁₆	1	2	yes	—
59	C₂×#10623494	C₂²×C₄	1	2	yes	—
60	C₂×#10623494	C₂²×C₄	1	2	yes	—
61	C₄⋊Dic₈₉	—	2	4	no	—
62	C₂²⋊Dic₈₉	—	2	4	no	—
63	C₄×Dic₈₉	—	2	1	no	—
64	C₄×Dic₈₉	—	2	1	no	—
65	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
66	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
67	#13083549	—	2	4	no	—
68	C₂⁴×C₈₉	M₁₆	1	1	yes	characteristic
69	C₂²×C₄×C₈₉	—	2	1	no	—
70	C₂²×C₄×C₈₉	—	2	1	no	—
71	C₂²×C₄×C₈₉	—	2	1	no	—
72	C₂²×C₄×C₈₉	—	2	1	no	—
73	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
74	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	M₁₆×C₈₉	—	2	4	no	—
78	M₁₆×C₈₉	—	2	4	no	—
79	C₂×C₈×C₈₉	—	2	1	no	—
80	C₂×C₈×C₈₉	—	2	1	no	—
81	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
82	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₈₉	#41	1	2	yes	—
84	C₂×C₈×C₈₉	#41	1	2	yes	—
85	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
86	C₁₆×C₈₉	—	2	4	no	—
87	C₂³×C₈₉	—	2	1	no	—
88	C₂³×C₈₉	—	2	1	no	—
89	C₂³×C₈₉	—	4	2	no	—
90	C₂³×C₈₉	C₂×M₁₆	1	1	yes	characteristic
91	C₂³×C₈₉	C₂×M₁₆	1	2	yes	—
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	2	no	—
95	C₂×C₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	—	2	2	no	—
98	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
99	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
105	C₂×Dic₈₉	—	2	2	no	—
106	C₂×Dic₈₉	—	2	2	no	—
107	C₂×Dic₈₉	—	4	2	no	—
108	C₂×Dic₈₉	—	4	2	no	—
109	C₈×C₈₉	—	2	2	no	—
110	C₈×C₈₉	#117	1	2	yes	—
111	C₈×C₈₉	C₂×#41	1	2	yes	—
112	#10623494	—	2	2	no	—
113	#10623494	C₂×M₁₆	1	4	yes	—
114	C₂²×C₈₉	—	2	2	no	—
115	C₂²×C₈₉	—	2	2	no	—
116	C₂²×C₈₉	—	4	2	no	—
117	C₂²×C₈₉	—	4	2	no	—
118	C₂²×C₈₉	—	4	2	no	—
119	C₂²×C₈₉	#432	1	1	yes	characteristic
120	C₂²×C₈₉	#432	1	1	yes	characteristic
121	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
122	C₄×C₈₉	—	2	2	no	—
123	C₄×C₈₉	#432	1	1	yes	characteristic
124	C₄×C₈₉	#432	1	1	yes	characteristic
125	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
126	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
127	Dic₈₉	—	4	2	no	—
128	Dic₈₉	—	4	2	no	—
129	C₂×#2166	—	89	1	no	maximal, Sylow
130	C₂×C₈₉	—	2	1	no	—
131	C₂×C₈₉	—	2	1	no	—
132	C₂×C₈₉	—	4	2	no	—
133	C₂×C₈₉	#2166	1	2	yes	complemented
134	C₂×C₈₉	C₂×#432	1	1	yes	characteristic
135	C₂×#405	—	89	1	no	—
136	#2166	—	89	8	no	—
137	C₂×#364	—	89	2	no	—
138	C₂×#349	—	89	1	no	—
139	C₂×#349	D₈₉	1	1	yes	characteristic
140	C₂×#347	—	89	2	no	—
141	C₈₉	C₂×#2166	1	1	yes	characteristic, complemented, Sylow
142	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
143	C₂³×C₈	—	89	1	no	—
144	C₂×#117	—	89	1	no	—
145	C₂×M₃₂	—	89	2	no	—
146	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
147	#405	—	89	4	no	—
148	#405	—	89	4	no	—
149	C₂×#97	—	89	1	no	—
150	C₂×#97	—	89	1	no	—
151	C₂×C₄×C₈	—	89	1	no	—
152	C₂×C₄×C₈	—	89	1	no	—
153	#364	—	89	8	no	—
154	#349	—	89	4	no	—
155	#349	C₂×D₈₉	1	4	yes	—
156	#347	—	89	8	no	—
157	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	C₄×D₈₉	1	2	yes	—
161	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
162	C₂²×C₈	—	89	1	no	—
163	C₂²×C₈	—	89	1	no	—
164	C₂²×C₈	—	89	4	no	—
165	C₂²×C₈	—	178	2	no	—
166	C₂²×C₈	—	178	2	no	—
167	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
169	#117	—	89	4	no	—
170	#117	—	178	2	no	—
171	C₂×C₄⋊C₄	—	178	1	no	—
172	C₂×C₂²⋊C₄	—	178	1	no	—
173	C₂×C₄²	—	89	1	no	—
174	C₂×C₄²	—	89	1	no	—
175	M₃₂	—	2	2	no	—
176	M₃₂	—	89	4	no	—
177	M₃₂	—	178	2	no	—
178	M₃₂	C₂²×D₈₉	1	4	yes	—
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	—	178	2	no	—
181	#97	—	89	4	no	—
182	#97	—	89	2	no	—
183	#97	—	89	2	no	—
184	C₄×C₈	—	89	4	no	—
185	C₄×C₈	—	89	2	no	—
186	C₄×C₈	—	89	2	no	—
187	C₂⁴	#13083557	1	1	yes	characteristic
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	178	1	no	—
193	C₂²×C₄	—	178	1	no	—
194	C₂²×C₄	#13083557	1	1	yes	characteristic
195	C₂²×C₄	#13083557	1	2	yes	—
196	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
197	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
198	M₁₆	—	2	4	no	—
199	M₁₆	—	2	4	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	89	2	no	—
203	C₂×C₈	—	89	2	no	—
204	C₂×C₈	—	89	2	no	—
205	C₂×C₈	—	89	2	no	—
206	C₂×C₈	—	178	8	no	—
207	C₂×C₈	—	178	1	no	—
208	C₂×C₈	—	178	1	no	—
209	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
210	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
211	C₂×C₈	#13083573	1	2	yes	—
212	C₂×C₈	#13083573	1	2	yes	—
213	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
214	C₄⋊C₄	—	178	4	no	—
215	C₂²⋊C₄	—	178	4	no	—
216	C₄²	—	89	2	no	—
217	C₄²	—	89	2	no	—
218	C₄²	—	178	1	no	—
219	C₄²	—	178	1	no	—
220	C₁₆	—	2	4	no	—
221	C₁₆	—	178	4	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	2	1	no	—
224	C₂³	—	4	2	no	—
225	C₂³	C₂×#13083557	1	1	yes	characteristic
226	C₂³	C₂×#13083557	1	2	yes	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	356	2	no	—
234	C₂×C₄	—	356	2	no	—
235	C₂×C₄	—	178	2	no	—
236	C₂×C₄	—	178	2	no	—
237	C₂×C₄	#15667968	1	1	yes	characteristic
238	C₂×C₄	#15667968	1	1	yes	characteristic
239	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
240	C₂×C₄	M₁₆×D₈₉	1	1	yes	characteristic
241	C₂×C₄	C₂×#13083557	1	2	yes	—
242	C₂×C₄	C₂×#13083557	1	1	yes	characteristic
243	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	89	4	no	—
246	C₈	—	178	2	no	—
247	C₈	#15667968	1	2	yes	—
248	C₈	C₂×#13083573	1	2	yes	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	2	no	—
253	C₂²	—	4	2	no	—
254	C₂²	#19151652	1	1	yes	characteristic
255	C₂²	#19151652	1	1	yes	characteristic
256	C₂²	C₂²×#13083557	1	1	yes	characteristic
257	C₄	—	2	2	no	—
258	C₄	—	356	2	no	—
259	C₄	—	356	2	no	—
260	C₄	#19151652	1	1	yes	characteristic
261	C₄	#19151652	1	1	yes	characteristic
262	C₄	C₂×#15667968	1	1	yes	characteristic
263	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	4	2	no	—
267	C₂	#26022065	1	2	yes	direct product factor
268	C₂	C₂×#19151652	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667968
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#349×C₈₉	C₂
socle	C₂²×C₈₉	#432
prime radical	C₂⁴×C₈₉	M₁₆
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#349	D₈₉
hypocenter	C₈₉	C₂×#2166
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
2. #26022065	C₂	direct (N × H)
3. C₂×#19151418	C₂	semidirect
5. C₂×#19151375	C₂	semidirect
16. #19151418	C₂²	semidirect
18. #19151375	C₂²	semidirect
133. C₂×C₈₉	#2166	semidirect
141. C₈₉	C₂×#2166	semidirect
267. C₂	#26022065	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×#19151556	2
#26022065	2
C₂×#19151418	2
C₂×#19151380	2
C₂×#19151375	2
C₂×#349×C₈₉	2
C₂×#2166	89

Subgroups — C2×#26022065 #36707552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022065 #36707552