Subgroups — C₂×#26020399 #36554691

Order 22784.

730 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
2	#26020399	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151368	C₂	1	2	yes	maximal
4	C₂×#19151357	C₂	1	2	yes	maximal
5	C₂×#19151356	C₂	1	1	yes	maximal, characteristic
6	C₂×H(3,ℤ/4)×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#13083558	C₂²	1	2	yes	—
8	C₂²×#13083558	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
10	M₁₆×Dic₈₉	C₄	1	4	yes	complemented
11	M₁₆×Dic₈₉	C₂²	1	4	yes	—
12	C₂×#15667922	C₄	1	2	yes	complemented
13	C₂×C₈×Dic₈₉	C₄	1	2	yes	complemented
14	#19151368	C₂²	1	8	yes	—
15	#19151357	C₂²	1	8	yes	—
16	#19151356	C₂²	1	4	yes	—
17	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#117×C₈₉	C₂²	1	2	yes	—
19	H(3,ℤ/4)×C₈₉	C₂²	1	4	yes	—
20	C₂³×Dic₈₉	—	2	1	no	—
21	C₂×#13083558	—	2	2	no	—
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₂×#13083558	—	2	2	no	—
26	C₂×#13083558	—	2	2	no	—
27	C₂×#13083558	—	2	2	no	—
28	C₂×#13083558	C₂×C₄	1	2	yes	—
29	C₂×#13083558	C₂³	1	4	yes	—
30	C₂×#13083558	C₂³	1	2	yes	—
31	C₂²×#10623494	—	2	2	no	—
32	C₂²×#10623494	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₈₉	—	2	1	no	—
34	C₂×C₄×Dic₈₉	—	2	1	no	—
35	C₂×C₄×Dic₈₉	—	2	1	no	—
36	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄×Dic₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₈₉	C₂×C₄	1	2	yes	—
39	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
40	#15667922	C₂×C₄	1	8	yes	complemented
41	C₈×Dic₈₉	C₂×C₄	1	8	yes	complemented
42	C₂³×C₄×C₈₉	C₂³	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	2	yes	—
48	#117×C₈₉	—	2	2	no	—
49	#117×C₈₉	—	2	2	no	—
50	#117×C₈₉	—	2	2	no	—
51	#117×C₈₉	C₂³	1	4	yes	—
52	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
54	C₂×C₄²×C₈₉	—	2	2	no	—
55	C₂²×Dic₈₉	—	2	1	no	—
56	C₂²×Dic₈₉	—	2	2	no	—
57	C₂²×Dic₈₉	—	2	1	no	—
58	C₂²×Dic₈₉	—	2	2	no	—
59	C₂²×Dic₈₉	—	4	4	no	—
60	#13083558	—	2	4	no	—
61	#13083558	—	2	4	no	—
62	#13083558	—	2	4	no	—
63	#13083558	—	2	4	no	—
64	#13083558	—	4	2	no	—
65	#13083558	—	4	2	no	—
66	C₂×#10623494	—	2	2	no	—
67	C₂×#10623494	—	2	2	no	—
68	C₂×#10623494	—	2	4	no	—
69	C₂×#10623494	—	4	2	no	—
70	C₂×#10623494	—	4	2	no	—
71	C₂×#10623494	C₂²×C₄	1	4	yes	—
72	C₄×Dic₈₉	—	2	1	no	—
73	C₄×Dic₈₉	—	2	2	no	—
74	C₄×Dic₈₉	—	2	1	no	—
75	C₄×Dic₈₉	—	2	2	no	—
76	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
77	C₄×Dic₈₉	C₂²×C₄	1	2	yes	—
78	C₂⁴×C₈₉	#41	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	—	4	2	no	—
80	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	#41	1	2	yes	—
84	C₂²×C₄×C₈₉	#41	1	2	yes	—
85	C₂²×C₄×C₈₉	#41	1	2	yes	—
86	C₂²×C₄×C₈₉	#41	1	2	yes	—
87	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
89	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
90	M₁₆×C₈₉	—	2	4	no	—
91	M₁₆×C₈₉	—	2	4	no	—
92	C₂×C₈×C₈₉	—	2	2	no	—
93	C₂×C₈×C₈₉	—	2	2	no	—
94	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
95	C₄⋊C₄×C₈₉	—	2	4	no	—
96	C₄⋊C₄×C₈₉	—	4	2	no	—
97	C₂²⋊C₄×C₈₉	—	2	4	no	—
98	C₂²⋊C₄×C₈₉	—	4	2	no	—
99	C₄²×C₈₉	—	2	4	no	—
100	C₄²×C₈₉	—	4	2	no	—
101	C₂³×C₈₉	—	4	2	no	—
102	C₂³×C₈₉	#117	1	1	yes	characteristic
103	C₂³×C₈₉	#117	1	1	yes	characteristic
104	C₂³×C₈₉	#126	1	2	yes	—
105	C₂³×C₈₉	#126	1	2	yes	—
106	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₈₉	—	4	4	no	—
108	C₂×C₄×C₈₉	—	8	4	no	—
109	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
111	C₂×C₄×C₈₉	#117	1	2	yes	—
112	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
113	C₂×C₄×C₈₉	#117	1	2	yes	—
114	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
115	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#126	1	2	yes	—
118	C₂×C₄×C₈₉	#126	1	2	yes	—
119	C₂×C₄×C₈₉	#126	1	2	yes	—
120	C₂×C₄×C₈₉	#126	1	2	yes	—
121	C₂×C₄×C₈₉	#126	1	2	yes	—
122	C₂×C₄×C₈₉	#126	1	2	yes	—
123	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
125	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
126	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
127	C₂×Dic₈₉	—	2	4	no	—
128	C₂×Dic₈₉	—	2	2	no	—
129	C₂×Dic₈₉	—	2	2	no	—
130	C₂×Dic₈₉	—	4	4	no	—
131	C₂×Dic₈₉	—	4	4	no	—
132	C₂×Dic₈₉	—	4	2	no	—
133	C₈×C₈₉	—	2	4	no	—
134	#10623494	—	2	4	no	—
135	#10623494	—	4	4	no	—
136	C₂²×C₈₉	—	2	2	no	—
137	C₂²×C₈₉	—	2	4	no	—
138	C₂²×C₈₉	—	4	1	no	—
139	C₂²×C₈₉	—	4	1	no	—
140	C₂²×C₈₉	—	4	2	no	—
141	C₂²×C₈₉	#383	1	1	yes	characteristic
142	C₂²×C₈₉	#383	1	2	yes	—
143	C₂²×C₈₉	#383	1	1	yes	characteristic
144	C₂²×C₈₉	C₂×#117	1	1	yes	characteristic
145	C₂²×C₈₉	C₂×#126	1	2	yes	—
146	C₄×C₈₉	—	8	4	no	—
147	C₄×C₈₉	#383	1	1	yes	characteristic
148	C₄×C₈₉	#383	1	2	yes	—
149	C₄×C₈₉	#383	1	1	yes	characteristic
150	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
151	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
152	C₄×C₈₉	C₂×#126	1	2	yes	—
153	Dic₈₉	—	2	4	no	—
154	Dic₈₉	—	4	2	no	—
155	C₂×#1841	—	89	1	no	maximal, Sylow
156	C₂×C₈₉	—	2	1	no	—
157	C₂×C₈₉	—	2	2	no	—
158	C₂×C₈₉	—	2	1	no	—
159	C₂×C₈₉	—	2	2	no	—
160	C₂×C₈₉	#1841	1	2	yes	complemented
161	C₂×C₈₉	C₂×#383	1	1	yes	characteristic
162	C₂×C₄×M₁₆	—	89	1	no	—
163	#1841	—	89	8	no	—
164	C₂×#344	—	89	2	no	—
165	C₂×#339	—	89	2	no	—
166	C₂×#338	—	89	1	no	—
167	C₂×H(3,ℤ/4)	D₈₉	1	1	yes	characteristic
168	C₈₉	C₂×#1841	1	1	yes	characteristic, complemented, Sylow
169	C₂²×M₁₆	—	89	1	no	—
170	C₂²×M₁₆	—	89	2	no	—
171	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
172	C₂×#117	C₂×D₈₉	1	2	yes	—
173	C₂²×C₄²	—	89	1	no	—
174	C₄×M₁₆	—	89	4	no	—
175	C₄×M₁₆	—	89	4	no	—
176	C₂×#97	—	89	2	no	—
177	C₂×C₄×C₈	—	89	2	no	—
178	#344	—	89	8	no	—
179	#339	—	89	8	no	—
180	#338	—	89	4	no	—
181	H(3,ℤ/4)	C₂×D₈₉	1	4	yes	—
182	C₂³×C₄	—	178	1	no	—
183	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	89	2	no	—
187	C₂×M₁₆	—	89	2	no	—
188	C₂×M₁₆	—	89	4	no	—
189	C₂×M₁₆	—	178	2	no	—
190	C₂×M₁₆	—	178	2	no	—
191	C₂×M₁₆	—	178	2	no	—
192	C₂×M₁₆	—	178	2	no	—
193	C₂×M₁₆	—	178	2	no	—
194	C₂×M₁₆	—	178	2	no	—
195	C₂×M₁₆	—	178	2	no	—
196	C₂×M₁₆	C₄×D₈₉	1	2	yes	—
197	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
198	C₂²×C₈	—	89	2	no	—
199	C₂²×C₈	—	178	2	no	—
200	C₂²×C₈	C₄×D₈₉	1	2	yes	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	#117	C₂²×D₈₉	1	4	yes	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₂²⋊C₄	—	2	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	89	1	no	—
209	C₂×C₄²	—	89	1	no	—
210	C₂×C₄²	—	89	2	no	—
211	C₂×C₄²	—	89	2	no	—
212	C₂×C₄²	—	178	1	no	—
213	C₂×C₄²	—	178	1	no	—
214	C₂×C₄²	—	178	1	no	—
215	#97	—	89	8	no	—
216	C₄×C₈	—	89	8	no	—
217	C₂⁴	#13083573	1	1	yes	characteristic
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	—	356	4	no	—
220	C₂²×C₄	—	178	1	no	—
221	C₂²×C₄	—	178	2	no	—
222	C₂²×C₄	—	178	2	no	—
223	C₂²×C₄	—	178	1	no	—
224	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
225	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
226	C₂²×C₄	#13083573	1	2	yes	—
227	C₂²×C₄	#13083573	1	2	yes	—
228	C₂²×C₄	#13083573	1	1	yes	characteristic
229	C₂²×C₄	#13083574	1	2	yes	—
230	C₂²×C₄	#13083574	1	2	yes	—
231	C₂²×C₄	#13083575	1	1	yes	characteristic
232	C₂²×C₄	#13083575	1	1	yes	characteristic
233	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	356	2	no	—
237	M₁₆	—	356	2	no	—
238	M₁₆	—	178	4	no	—
239	M₁₆	—	178	4	no	—
240	M₁₆	—	178	4	no	—
241	M₁₆	—	178	4	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	89	4	no	—
245	C₂×C₈	—	356	2	no	—
246	C₂×C₈	—	356	2	no	—
247	C₂×C₈	—	178	4	no	—
248	C₂×C₈	—	178	2	no	—
249	C₂×C₈	—	178	2	no	—
250	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	2	4	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₄²	—	2	4	no	—
256	C₄²	—	4	2	no	—
257	C₄²	—	89	2	no	—
258	C₄²	—	89	2	no	—
259	C₄²	—	178	1	no	—
260	C₄²	—	178	2	no	—
261	C₄²	—	178	1	no	—
262	C₄²	—	178	2	no	—
263	C₂³	—	4	2	no	—
264	C₂³	#15667970	1	1	yes	characteristic
265	C₂³	#15667970	1	1	yes	characteristic
266	C₂³	#15668005	1	2	yes	—
267	C₂³	#15668005	1	2	yes	—
268	C₂³	C₂×#13083573	1	1	yes	characteristic
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	8	4	no	—
271	C₂×C₄	—	356	2	no	—
272	C₂×C₄	—	356	4	no	—
273	C₂×C₄	—	356	4	no	—
274	C₂×C₄	—	178	2	no	—
275	C₂×C₄	—	178	4	no	—
276	C₂×C₄	—	178	2	no	—
277	C₂×C₄	#15667968	1	1	yes	characteristic
278	C₂×C₄	#15667968	1	1	yes	characteristic
279	C₂×C₄	#15667968	1	2	yes	—
280	C₂×C₄	#15667969	1	1	yes	characteristic
281	C₂×C₄	#15667969	1	2	yes	—
282	C₂×C₄	#15667969	1	1	yes	characteristic
283	C₂×C₄	#15667970	1	1	yes	characteristic
284	C₂×C₄	#15667970	1	1	yes	characteristic
285	C₂×C₄	#15668005	1	2	yes	—
286	C₂×C₄	#15668005	1	2	yes	—
287	C₂×C₄	#15668007	1	2	yes	—
288	C₂×C₄	#15668007	1	2	yes	—
289	C₂×C₄	#15668007	1	2	yes	—
290	C₂×C₄	#15668007	1	2	yes	—
291	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
292	C₂×C₄	C₂×#13083573	1	2	yes	—
293	C₂×C₄	C₂×#13083574	1	2	yes	—
294	C₂×C₄	C₂×#13083575	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	356	4	no	—
297	C₈	—	178	4	no	—
298	C₂²	—	2	4	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	#19151465	1	1	yes	characteristic
304	C₂²	#19151465	1	2	yes	—
305	C₂²	#19151465	1	1	yes	characteristic
306	C₂²	C₂×#15667970	1	1	yes	characteristic
307	C₂²	C₂×#15668005	1	2	yes	—
308	C₄	—	8	4	no	—
309	C₄	—	356	2	no	—
310	C₄	—	178	4	no	—
311	C₄	#19151465	1	1	yes	characteristic
312	C₄	#19151465	1	2	yes	—
313	C₄	#19151465	1	1	yes	characteristic
314	C₄	C₂×#15667968	1	1	yes	characteristic
315	C₄	C₂×#15667969	1	1	yes	characteristic
316	C₄	C₂×#15668007	1	2	yes	—
317	C₂	—	2	2	no	—
318	C₂	—	2	1	no	—
319	C₂	—	2	2	no	—
320	C₂	—	2	1	no	—
321	C₂	#26020399	1	2	yes	direct product factor
322	C₂	C₂×#19151465	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667969
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×H(3,ℤ/4)×C₈₉	C₂
socle	C₂²×C₈₉	#383
prime radical	C₂⁴×C₈₉	#41
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×H(3,ℤ/4)	D₈₉
hypocenter	C₈₉	C₂×#1841
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
2. #26020399	C₂	direct (N × H)
10. M₁₆×Dic₈₉	C₄	semidirect
12. C₂×#15667922	C₄	semidirect
13. C₂×C₈×Dic₈₉	C₄	semidirect
40. #15667922	C₂×C₄	semidirect
41. C₈×Dic₈₉	C₂×C₄	semidirect
160. C₂×C₈₉	#1841	semidirect
168. C₈₉	C₂×#1841	semidirect
321. C₂	#26020399	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×M₁₆×Dic₈₉	2
#26020399	2
C₂×#19151368	2
C₂×#19151357	2
C₂×#19151356	2
C₂×H(3,ℤ/4)×C₈₉	2
C₂×#1841	89

Subgroups — C2×#26020399 #36554691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26020399 #36554691