Subgroups — C₂×#26022095 #36707663

Order 22784.

654 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152014	C₂	1	2	yes	maximal, complemented
2	C₂×#19151716	C₂	1	1	yes	maximal, characteristic
3	#26022095	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151419	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151381	C₂	1	2	yes	maximal, complemented
6	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#15668034	C₄	1	1	yes	characteristic
8	C₂×#15668018	C₂²	1	2	yes	—
9	#19152014	—	2	4	no	—
10	#19152014	C₂²	1	8	yes	complemented
11	C₂×#15667952	C₂²	1	2	yes	—
12	C₂×#15667951	—	2	2	no	—
13	C₂×C₁₆×D₈₉	—	2	2	no	—
14	#19151716	C₄	1	4	yes	—
15	#19151716	C₂²	1	4	yes	—
16	C₂×#15667948	C₄	1	1	yes	characteristic
17	C₂×#15667940	C₄	1	2	yes	—
18	#19151419	C₂²	1	4	yes	complemented
19	#19151381	C₂²	1	8	yes	complemented
20	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	#346×C₈₉	C₂²	1	4	yes	complemented
24	C₂×#13083574	C₂×C₄	1	2	yes	—
25	#15668034	C₈	1	4	yes	—
26	#15668034	C₂×C₄	1	4	yes	—
27	C₂×Dic₃₅₆	C₈	1	1	yes	characteristic
28	C₂×#13083563	C₈	1	2	yes	—
29	C₂×D₇₁₂	C₈	1	1	yes	characteristic
30	#15668018	—	2	2	no	—
31	#15668018	—	2	2	no	—
32	#15668018	C₂×C₄	1	4	yes	—
33	#15668018	C₂³	1	4	yes	—
34	C₂×#13083558	C₂×C₄	1	2	yes	—
35	C₂×#13083557	—	2	2	no	—
36	C₂×C₈×D₈₉	—	2	2	no	—
37	#15667952	—	2	2	no	—
38	#15667952	C₂³	1	4	yes	—
39	#15667951	—	2	8	no	—
40	C₂×#13083549	—	2	2	no	—
41	C₁₆×D₈₉	—	2	8	no	—
42	#15667948	C₂×C₄	1	4	yes	—
43	#15667940	C₂×C₄	1	8	yes	—
44	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₆×C₈₉	—	2	1	no	—
48	C₂×C₁₆×C₈₉	—	2	1	no	—
49	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
50	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
51	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
52	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
53	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₈₉	C₂³	1	2	yes	—
55	#13083574	—	2	2	no	—
56	#13083574	—	2	2	no	—
57	#13083574	C₂×C₈	1	4	yes	—
58	#13083574	C₂²×C₄	1	4	yes	—
59	C₂×Dic₁₇₈	C₂×C₈	1	2	yes	—
60	C₂×D₃₅₆	C₂×C₈	1	2	yes	—
61	C₂×C₂²⋊D₈₉	—	2	2	no	—
62	C₂×C₄×D₈₉	—	2	2	no	—
63	Dic₃₅₆	C₂×C₈	1	4	yes	—
64	#13083563	C₂×C₈	1	8	yes	—
65	D₇₁₂	C₂×C₈	1	4	yes	—
66	#13083558	—	2	2	no	—
67	#13083558	C₂²×C₄	1	4	yes	—
68	#13083557	—	2	4	no	—
69	#13083557	—	2	4	no	—
70	C₂×#10623494	—	2	2	no	—
71	C₈×D₈₉	—	2	4	no	—
72	C₈×D₈₉	—	2	4	no	—
73	#13083549	—	2	4	no	—
74	C₂²×C₄×C₈₉	—	2	1	no	—
75	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
77	C₂×C₈×C₈₉	C₂×C₈	1	1	yes	characteristic
78	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
84	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
86	C₄²×C₈₉	—	2	1	no	—
87	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
88	C₁₆×C₈₉	—	2	2	no	—
89	C₁₆×C₈₉	C₂×D₄	1	4	yes	—
90	C₂³×C₈₉	#131	1	1	yes	characteristic
91	C₂²×D₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	2	2	no	—
93	C₂×C₄×C₈₉	—	4	2	no	—
94	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₈₉	#131	1	1	yes	characteristic
99	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
100	C₂²⋊D₈₉	—	2	4	no	—
101	C₂²⋊D₈₉	—	2	4	no	—
102	C₂×Dic₈₉	—	2	2	no	—
103	D₃₅₆	—	2	2	no	—
104	D₃₅₆	C₂²×C₈	1	4	yes	—
105	C₄×D₈₉	—	2	4	no	—
106	C₄×D₈₉	—	2	4	no	—
107	Dic₁₇₈	—	2	2	no	—
108	Dic₁₇₈	C₂²×C₈	1	4	yes	—
109	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
110	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
111	C₈×C₈₉	C₂²×C₈	1	2	yes	—
112	C₈×C₈₉	C₂²×C₈	1	2	yes	—
113	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
114	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
115	#10623494	—	2	4	no	—
116	C₂²×C₈₉	—	2	2	no	—
117	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
118	C₂²×C₈₉	C₈×D₄	1	1	yes	characteristic
119	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
120	C₂×D₈₉	—	2	4	no	—
121	C₂×D₈₉	—	4	4	no	—
122	C₄×C₈₉	—	4	2	no	—
123	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
124	C₄×C₈₉	C₈×D₄	1	1	yes	characteristic
125	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
126	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
127	Dic₈₉	—	2	4	no	—
128	C₂×#2171	—	89	1	no	maximal, Sylow
129	C₂×C₈₉	—	2	1	no	—
130	C₂×C₈₉	—	2	1	no	—
131	C₂×C₈₉	#2171	1	2	yes	complemented
132	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
133	D₈₉	—	4	4	no	—
134	C₂×#504	—	89	2	no	—
135	C₂×#443	—	89	1	no	—
136	#2171	—	89	8	no	—
137	C₂×#364	—	89	1	no	—
138	C₂×#350	—	89	2	no	—
139	C₂×#346	D₈₉	1	1	yes	characteristic, complemented
140	C₈₉	C₂×#2171	1	1	yes	characteristic, complemented, Sylow
141	C₂×#135	—	89	1	no	—
142	C₂×#131	—	89	2	no	—
143	#504	—	89	8	no	—
144	#504	—	178	4	no	—
145	C₂×M₃₂	—	89	2	no	—
146	C₂×M₃₂	—	178	2	no	—
147	C₂²×C₁₆	—	178	2	no	—
148	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
149	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
150	#443	—	89	4	no	—
151	#443	—	89	4	no	—
152	C₂×#108	—	89	1	no	—
153	C₂×#104	—	89	2	no	—
154	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
155	#364	—	89	4	no	—
156	#350	—	89	8	no	—
157	#346	C₂×D₈₉	1	4	yes	complemented
158	C₂×#41	—	89	2	no	—
159	#135	—	89	4	no	—
160	#135	—	89	4	no	—
161	C₂×Q₁₆	—	89	1	no	—
162	C₂×SD₁₆	—	89	2	no	—
163	C₂×D₈	—	89	1	no	—
164	#131	—	89	4	no	—
165	#131	—	89	4	no	—
166	#131	—	178	2	no	—
167	#131	—	178	2	no	—
168	C₂×M₁₆	—	89	2	no	—
169	C₂×M₁₆	—	178	2	no	—
170	C₂²×C₈	—	178	2	no	—
171	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
173	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
174	M₃₂	—	89	4	no	—
175	M₃₂	—	178	2	no	—
176	M₃₂	—	178	8	no	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	—	2	1	no	—
179	C₂×C₁₆	—	178	2	no	—
180	C₂×C₁₆	—	178	8	no	—
181	C₂×C₁₆	D₃₅₆	1	2	yes	—
182	C₂×C₁₆	D₃₅₆	1	2	yes	—
183	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
184	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
185	#108	—	89	4	no	—
186	#104	—	89	8	no	—
187	C₄×C₈	C₄×D₈₉	1	2	yes	—
188	C₄×C₈	C₂²×D₈₉	1	2	yes	—
189	#41	—	89	4	no	—
190	#41	—	89	4	no	—
191	#41	—	178	2	no	—
192	#41	—	178	2	no	—
193	C₂×Q₈	—	89	2	no	—
194	C₂×D₄	—	89	2	no	—
195	C₂×D₄	—	178	2	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	178	2	no	—
198	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
199	Q₁₆	—	89	4	no	—
200	SD₁₆	—	89	8	no	—
201	D₈	—	89	4	no	—
202	M₁₆	—	89	4	no	—
203	M₁₆	—	178	2	no	—
204	M₁₆	—	178	4	no	—
205	M₁₆	—	178	4	no	—
206	C₂×C₈	—	178	2	no	—
207	C₂×C₈	—	178	4	no	—
208	C₂×C₈	—	178	4	no	—
209	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
210	C₂×C₈	C₈×D₈₉	1	1	yes	characteristic
211	C₂×C₈	C₈×D₈₉	1	2	yes	—
212	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
213	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
214	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
215	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
216	C₂×C₈	#13083574	1	1	yes	characteristic
217	C₂×C₈	#13083574	1	1	yes	characteristic
218	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
219	C₄²	—	2	1	no	—
220	C₄²	C₂×C₄×D₈₉	1	2	yes	—
221	C₁₆	—	2	2	no	—
222	C₁₆	—	178	4	no	—
223	C₁₆	C₂×D₃₅₆	1	4	yes	—
224	C₂³	—	178	2	no	—
225	C₂³	#15668018	1	1	yes	characteristic
226	Q₈	—	89	4	no	—
227	Q₈	—	178	2	no	—
228	D₄	—	89	4	no	—
229	D₄	—	178	4	no	—
230	D₄	—	178	4	no	—
231	D₄	—	178	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	178	2	no	—
235	C₂×C₄	—	178	4	no	—
236	C₂×C₄	—	178	4	no	—
237	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
238	C₂×C₄	C₄×D₃₅₆	1	1	yes	characteristic
239	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
240	C₂×C₄	C₂×C₈×D₈₉	1	1	yes	characteristic
241	C₂×C₄	#15668018	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
243	C₈	—	178	4	no	—
244	C₈	C₄×D₃₅₆	1	1	yes	characteristic
245	C₈	C₄×D₃₅₆	1	1	yes	characteristic
246	C₈	C₂×C₈×D₈₉	1	2	yes	—
247	C₈	C₂×C₈×D₈₉	1	2	yes	—
248	C₈	C₂²×D₃₅₆	1	1	yes	characteristic
249	C₈	C₂×#13083574	1	1	yes	characteristic
250	C₂²	—	2	2	no	—
251	C₂²	—	356	4	no	—
252	C₂²	—	178	4	no	—
253	C₂²	C₈×D₃₅₆	1	1	yes	characteristic
254	C₂²	C₈×D₃₅₆	1	1	yes	characteristic
255	C₂²	C₂×#15668018	1	1	yes	characteristic
256	C₄	—	4	2	no	—
257	C₄	—	178	4	no	—
258	C₄	C₈×D₃₅₆	1	1	yes	characteristic
259	C₄	C₈×D₃₅₆	1	1	yes	characteristic
260	C₄	C₂×C₄×D₃₅₆	1	1	yes	characteristic
261	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	356	4	no	—
265	C₂	#26022095	1	2	yes	direct product factor
266	C₂	C₂×C₈×D₃₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₃₅₆
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#346×C₈₉	C₂
socle	C₂²×C₈₉	C₈×D₄
prime radical	C₂×#15668034	C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	C₂×#2171
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
1. C₂×#19152014	C₂	semidirect
3. #26022095	C₂	direct (N × H)
4. C₂×#19151419	C₂	semidirect
5. C₂×#19151381	C₂	semidirect
6. C₂×#346×C₈₉	C₂	semidirect
10. #19152014	C₂²	semidirect
18. #19151419	C₂²	semidirect
19. #19151381	C₂²	semidirect
23. #346×C₈₉	C₂²	semidirect
131. C₂×C₈₉	#2171	semidirect
139. C₂×#346	D₈₉	semidirect
140. C₈₉	C₂×#2171	semidirect
157. #346	C₂×D₈₉	semidirect
265. C₂	#26022095	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×#19152014	2
C₂×#19151716	2
#26022095	2
C₂×#19151419	2
C₂×#19151381	2
C₂×#346×C₈₉	2
C₂×#2171	89

Subgroups — C2×#26022095 #36707663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022095 #36707663