Subgroups — C₂×#26018541 #36520027

Order 22784.

966 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151651	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#13083558	C₂	1	1	yes	maximal, characteristic
3	#26018541	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151320	C₂	1	2	yes	maximal, complemented
5	C₂×#19151319	C₂	1	2	yes	maximal, complemented
6	C₂²×C₄×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#13083558	C₄	1	2	yes	—
8	#19151651	C₄	1	4	yes	—
9	#19151651	C₂²	1	4	yes	—
10	C₂×C₄⋊#10623494	C₄	1	2	yes	—
11	C₂×C₂²⋊#10623494	C₄	1	2	yes	—
12	C₄×#13083558	C₄	1	4	yes	—
13	C₄×#13083558	C₂²	1	4	yes	—
14	C₂×#15667923	C₄	1	2	yes	—
15	C₂×C₄×#10623494	C₂²	1	2	yes	—
16	C₂×C₄×#10623494	C₂²	1	2	yes	—
17	#19151320	C₂²	1	8	yes	complemented
18	#19151319	C₂²	1	8	yes	complemented
19	C₂³×C₈×C₈₉	C₄	1	2	yes	—
20	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₈₉	C₄	1	4	yes	—
22	C₂×C₄×C₈×C₈₉	C₄	1	2	yes	—
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	C₂²	1	4	yes	—
26	C₂×#13083558	—	2	8	no	—
27	C₂×#13083558	C₂×C₄	1	8	yes	—
28	C₂²×#10623494	C₂×C₄	1	4	yes	—
29	C₂²×#10623494	C₂×C₄	1	4	yes	—
30	C₄⋊#10623494	C₂×C₄	1	8	yes	—
31	C₂²⋊#10623494	C₂×C₄	1	8	yes	—
32	#15667923	C₂×C₄	1	8	yes	—
33	C₄×#10623494	C₂×C₄	1	4	yes	—
34	C₄×#10623494	C₂×C₄	1	4	yes	—
35	C₄×#10623494	C₂³	1	4	yes	—
36	C₄×#10623494	C₂³	1	4	yes	—
37	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₈×C₈₉	—	2	8	no	—
41	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₈₉	C₂×C₄	1	8	yes	—
43	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
51	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
52	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₈₉	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₈₉	C₂³	1	2	yes	—
56	C₄×C₈×C₈₉	C₂³	1	2	yes	—
57	#13083558	—	2	16	no	—
58	C₂×#10623494	—	2	8	no	—
59	C₂×#10623494	—	2	8	no	—
60	C₂×#10623494	C₂²×C₄	1	8	yes	—
61	C₂×#10623494	C₂²×C₄	1	8	yes	—
62	C₂⁴×C₈₉	C₄²	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	—	2	4	no	—
64	C₂²×C₄×C₈₉	—	2	4	no	—
65	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
66	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
68	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
69	C₂²×C₄×C₈₉	C₄²	1	2	yes	—
70	C₂²×C₄×C₈₉	C₄²	1	1	yes	characteristic
71	C₂²×C₄×C₈₉	M₁₆	1	8	yes	—
72	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₈₉	—	2	4	no	—
79	C₂×C₈×C₈₉	—	2	16	no	—
80	C₂×C₈×C₈₉	M₁₆	1	8	yes	complemented
81	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
82	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
83	C₄²×C₈₉	C₄²	1	2	yes	—
84	C₄²×C₈₉	C₄²	1	2	yes	—
85	C₄²×C₈₉	C₄²	1	2	yes	—
86	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
87	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
89	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
90	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
91	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
93	C₂³×C₈₉	—	2	2	no	—
94	C₂³×C₈₉	—	2	2	no	—
95	C₂³×C₈₉	#97	1	4	yes	—
96	C₂³×C₈₉	C₂×C₄²	1	2	yes	—
97	C₂³×C₈₉	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₈₉	—	2	2	no	—
99	C₂×C₄×C₈₉	—	2	2	no	—
100	C₂×C₄×C₈₉	—	2	4	no	—
101	C₂×C₄×C₈₉	—	2	4	no	—
102	C₂×C₄×C₈₉	—	2	4	no	—
103	C₂×C₄×C₈₉	—	2	4	no	—
104	C₂×C₄×C₈₉	#97	1	8	yes	—
105	C₂×C₄×C₈₉	#97	1	4	yes	—
106	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
107	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×C₄²	1	2	yes	—
110	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
113	C₂×C₄×C₈₉	C₂×M₁₆	1	8	yes	—
114	C₂×C₄×C₈₉	C₂×M₁₆	1	4	yes	—
115	C₂×C₄×C₈₉	#131	1	4	yes	—
116	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
119	C₈×C₈₉	—	2	4	no	—
120	C₈×C₈₉	C₂×M₁₆	1	8	yes	complemented
121	#10623494	—	2	8	no	—
122	#10623494	—	2	8	no	—
123	C₂²×C₈₉	—	2	2	no	—
124	C₂²×C₈₉	—	2	2	no	—
125	C₂²×C₈₉	—	2	8	no	—
126	C₂²×C₈₉	C₂×#97	1	4	yes	—
127	C₂²×C₈₉	C₂×#97	1	2	yes	—
128	C₂²×C₈₉	C₄×M₁₆	1	2	yes	—
129	C₂²×C₈₉	#405	1	2	yes	—
130	C₂²×C₈₉	C₂²×C₄²	1	1	yes	characteristic
131	C₄×C₈₉	—	2	2	no	—
132	C₄×C₈₉	—	2	2	no	—
133	C₄×C₈₉	C₂×#97	1	2	yes	—
134	C₄×C₈₉	C₂×#97	1	4	yes	—
135	C₄×C₈₉	C₄×M₁₆	1	2	yes	—
136	C₄×C₈₉	#405	1	2	yes	—
137	C₄×C₈₉	C₂²×M₁₆	1	2	yes	—
138	C₄×C₈₉	C₂²×M₁₆	1	2	yes	—
139	C₄×C₈₉	C₂×#131	1	2	yes	—
140	C₂×#1451	—	89	1	no	maximal, Sylow
141	C₂×C₈₉	—	2	4	no	—
142	C₂×C₈₉	#1451	1	4	yes	complemented
143	C₂×C₈₉	C₂²×#97	1	1	yes	characteristic
144	C₂×C₈₉	C₂×C₄×M₁₆	1	1	yes	characteristic
145	C₂×C₈₉	C₂×#405	1	1	yes	characteristic
146	C₂×#431	—	89	1	no	—
147	C₂×C₄×M₁₆	—	89	1	no	—
148	C₂²×C₄×C₈	D₈₉	1	1	yes	characteristic
149	#1451	—	89	8	no	—
150	C₂×#322	—	89	2	no	—
151	C₂×#322	—	89	2	no	—
152	C₈₉	C₂×#1451	1	1	yes	characteristic, complemented, Sylow
153	C₂²×M₁₆	—	89	2	no	—
154	C₂³×C₈	Dic₈₉	1	2	yes	—
155	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
156	#431	—	89	4	no	—
157	#431	—	89	4	no	—
158	C₂×C₄⋊C₈	—	89	2	no	—
159	C₂×C₂²⋊C₈	—	89	2	no	—
160	C₄×M₁₆	—	89	4	no	—
161	C₄×M₁₆	—	89	4	no	—
162	C₂×#97	—	89	2	no	—
163	C₂×C₄×C₈	—	89	2	no	—
164	C₂×C₄×C₈	—	89	2	no	—
165	C₂×C₄×C₈	Dic₈₉	1	4	yes	—
166	C₂×C₄×C₈	Dic₈₉	1	2	yes	—
167	C₂×C₄×C₈	C₂×D₈₉	1	4	yes	—
168	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
169	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
170	#322	—	89	8	no	—
171	#322	—	89	8	no	—
172	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
173	C₂³×C₄	C₄×D₈₉	1	1	yes	characteristic
174	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
175	C₂×M₁₆	—	89	8	no	—
176	C₂×M₁₆	—	178	8	no	—
177	C₂²×C₈	—	2	8	no	—
178	C₂²×C₈	—	89	4	no	—
179	C₂²×C₈	—	89	4	no	—
180	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
181	C₂²×C₈	C₂×Dic₈₉	1	8	yes	—
182	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
183	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
184	C₂×C₄²	C₄×D₈₉	1	2	yes	—
185	C₂×C₄²	C₄×D₈₉	1	1	yes	characteristic
186	C₂×C₄²	C₄×D₈₉	1	2	yes	—
187	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
188	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
189	C₂×C₄²	C₂²×D₈₉	1	2	yes	—
190	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
191	C₄⋊C₈	—	89	8	no	—
192	C₂²⋊C₈	—	89	8	no	—
193	#97	—	89	8	no	—
194	C₄×C₈	—	89	4	no	—
195	C₄×C₈	—	89	4	no	—
196	C₄×C₈	—	89	4	no	—
197	C₄×C₈	—	89	4	no	—
198	C₄×C₈	C₂×Dic₈₉	1	8	yes	—
199	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
200	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
201	C₄×C₈	C₂²×D₈₉	1	2	yes	—
202	C₄×C₈	C₂²×D₈₉	1	2	yes	—
203	C₂⁴	C₄×Dic₈₉	1	1	yes	characteristic
204	C₂²×C₄	—	2	4	no	—
205	C₂²×C₄	—	2	4	no	—
206	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
207	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
208	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
209	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
210	C₂²×C₄	C₄×Dic₈₉	1	2	yes	—
211	C₂²×C₄	C₄×Dic₈₉	1	1	yes	characteristic
212	C₂²×C₄	#13083557	1	8	yes	—
213	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
214	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
215	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
217	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
218	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
219	M₁₆	—	178	16	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	16	no	—
222	C₂×C₈	—	89	8	no	—
223	C₂×C₈	—	89	8	no	—
224	C₂×C₈	—	178	8	no	—
225	C₂×C₈	—	178	8	no	—
226	C₂×C₈	#13083558	1	8	yes	complemented
227	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
228	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
229	C₄²	C₄×Dic₈₉	1	2	yes	—
230	C₄²	C₄×Dic₈₉	1	2	yes	—
231	C₄²	C₄×Dic₈₉	1	2	yes	—
232	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
233	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
234	C₄²	C₂×C₄×D₈₉	1	2	yes	—
235	C₄²	C₂×C₄×D₈₉	1	2	yes	—
236	C₄²	C₂²×Dic₈₉	1	2	yes	—
237	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
238	C₄²	C₂³×D₈₉	1	1	yes	characteristic
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	#15667922	1	4	yes	—
242	C₂³	C₂×C₄×Dic₈₉	1	2	yes	—
243	C₂³	C₂×C₄×Dic₈₉	1	1	yes	characteristic
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	#15667922	1	8	yes	—
251	C₂×C₄	#15667922	1	4	yes	—
252	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
253	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
254	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄×Dic₈₉	1	2	yes	—
256	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
257	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
258	C₂×C₄	C₂×#13083557	1	4	yes	—
259	C₂×C₄	C₂×#13083557	1	8	yes	—
260	C₂×C₄	C₂×#13083558	1	4	yes	—
261	C₂×C₄	#15668018	1	4	yes	—
262	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
263	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
264	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
265	C₈	—	2	4	no	—
266	C₈	—	178	8	no	—
267	C₈	—	178	8	no	—
268	C₈	C₂×#13083558	1	8	yes	complemented
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	8	no	—
272	C₂²	C₂×#15667922	1	4	yes	—
273	C₂²	C₂×#15667922	1	2	yes	—
274	C₂²	C₄×#13083558	1	2	yes	—
275	C₂²	#19151553	1	2	yes	—
276	C₂²	C₂²×C₄×Dic₈₉	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	C₂×#15667922	1	2	yes	—
280	C₄	C₂×#15667922	1	4	yes	—
281	C₄	C₄×#13083558	1	2	yes	—
282	C₄	#19151553	1	2	yes	—
283	C₄	C₂²×#13083557	1	2	yes	—
284	C₄	C₂²×#13083558	1	2	yes	—
285	C₄	C₂×#15668018	1	2	yes	—
286	C₂	—	2	4	no	—
287	C₂	#26018541	1	4	yes	direct product factor
288	C₂	C₂²×#15667922	1	1	yes	characteristic
289	C₂	C₂×C₄×#13083558	1	1	yes	characteristic
290	C₂	C₂×#19151553	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂²×D₈₉
commutator	C₂²×C₈₉	C₂²×C₄²
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂²×C₄×C₈×C₈₉	C₂
socle	C₂³×C₈₉	C₂×C₄²
prime radical	C₂⁴×C₈₉	C₄²
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	C₂²×C₄×C₈	D₈₉
hypocenter	C₈₉	C₂×#1451
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #26018541	C₂	direct (N × H)
4. C₂×#19151320	C₂	semidirect
5. C₂×#19151319	C₂	semidirect
17. #19151320	C₂²	semidirect
18. #19151319	C₂²	semidirect
80. C₂×C₈×C₈₉	M₁₆	semidirect
120. C₈×C₈₉	C₂×M₁₆	semidirect
142. C₂×C₈₉	#1451	semidirect
152. C₈₉	C₂×#1451	semidirect
226. C₂×C₈	#13083558	semidirect
268. C₈	C₂×#13083558	semidirect
287. C₂	#26018541	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×#19151651	2
C₂×C₄×#13083558	2
#26018541	2
C₂×#19151320	2
C₂×#19151319	2
C₂²×C₄×C₈×C₈₉	2
C₂×#1451	89

Subgroups — C2×#26018541 #36520027

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018541 #36520027