Subgroups — C₂×#26018246 #36464586

Order 22784.

428 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151380	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151356	C₂	1	1	yes	maximal, characteristic
3	#26018246	C₂	1	4	yes	maximal, direct product factor
4	C₂×#470×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13083558	—	2	1	no	—
6	C₂×#15667952	—	2	1	no	—
7	#19151380	—	2	1	no	—
8	#19151380	C₂²	1	2	yes	—
9	#19151356	—	2	1	no	—
10	#19151356	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₈₉	C₄	1	1	yes	characteristic
12	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
14	#470×C₈₉	—	2	1	no	—
15	#470×C₈₉	—	2	1	no	—
16	#470×C₈₉	C₄	1	2	yes	—
17	#470×C₈₉	C₂²	1	2	yes	—
18	C₂×#106×C₈₉	C₄	1	1	yes	characteristic
19	C₂×#103×C₈₉	—	2	1	no	—
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083558	—	2	2	no	—
22	C₂×#13083558	—	4	1	no	—
23	C₂×#13083558	—	4	1	no	—
24	C₂×#13083558	—	4	1	no	—
25	C₂×#13083558	—	4	1	no	—
26	C₂²×#10623494	—	2	2	no	—
27	#15667952	—	2	2	no	—
28	#15667952	—	4	1	no	—
29	C₂×#13083549	—	4	1	no	—
30	C₂²×Q₈×C₈₉	—	2	1	no	—
31	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
32	C₂×Q₁₆×C₈₉	—	2	1	no	—
33	C₂×Q₁₆×C₈₉	—	4	1	no	—
34	C₂×Q₁₆×C₈₉	—	4	1	no	—
35	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
36	C₂×M₁₆×C₈₉	—	4	1	no	—
37	C₂×M₁₆×C₈₉	—	4	1	no	—
38	C₂×M₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
40	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
41	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
44	C₂²⋊Q₈×C₈₉	—	2	2	no	—
45	C₂²⋊Q₈×C₈₉	—	2	2	no	—
46	C₂²⋊Q₈×C₈₉	—	4	1	no	—
47	C₂²⋊Q₈×C₈₉	—	4	1	no	—
48	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
49	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
50	C₂×C₂²⋊C₄×C₈₉	—	4	1	no	—
51	#106×C₈₉	—	2	1	no	—
52	#106×C₈₉	C₂×C₄	1	2	yes	—
53	#103×C₈₉	—	2	2	no	—
54	#103×C₈₉	—	2	2	no	—
55	#13083558	—	4	2	no	—
56	#13083558	—	4	2	no	—
57	#13083558	—	4	2	no	—
58	#13083558	—	4	2	no	—
59	C₂×#10623494	—	2	4	no	—
60	C₂×#10623494	—	4	2	no	—
61	C₂×#10623494	—	4	2	no	—
62	#13083549	—	4	2	no	—
63	C₂⁴×C₈₉	SD₁₆	1	1	yes	characteristic
64	C₂×Q₈×C₈₉	—	2	2	no	—
65	C₂×Q₈×C₈₉	—	4	1	no	—
66	C₂×Q₈×C₈₉	—	4	1	no	—
67	C₂×Q₈×C₈₉	—	4	1	no	—
68	C₂×Q₈×C₈₉	—	4	1	no	—
69	C₂×Q₈×C₈₉	—	4	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	—	2	1	no	—
74	C₂²×C₄×C₈₉	—	4	1	no	—
75	C₂²×C₄×C₈₉	—	4	1	no	—
76	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
81	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
82	Q₁₆×C₈₉	—	4	2	no	—
83	Q₁₆×C₈₉	—	4	2	no	—
84	M₁₆×C₈₉	—	4	1	no	—
85	M₁₆×C₈₉	—	4	1	no	—
86	M₁₆×C₈₉	—	4	1	no	—
87	M₁₆×C₈₉	—	4	1	no	—
88	C₂×C₈×C₈₉	—	2	1	no	—
89	C₂×C₈×C₈₉	—	2	1	no	—
90	C₂×C₈×C₈₉	—	2	1	no	—
91	C₂×C₈×C₈₉	—	2	1	no	—
92	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₄⋊C₄×C₈₉	—	2	2	no	—
97	C₄⋊C₄×C₈₉	—	4	1	no	—
98	C₄⋊C₄×C₈₉	—	4	2	no	—
99	C₄⋊C₄×C₈₉	—	4	2	no	—
100	C₂²⋊C₄×C₈₉	—	4	2	no	—
101	C₂²⋊C₄×C₈₉	—	4	2	no	—
102	C₂³×C₈₉	—	2	1	no	—
103	C₂³×C₈₉	—	2	1	no	—
104	C₂³×C₈₉	—	4	2	no	—
105	C₂³×C₈₉	#102	1	1	yes	characteristic
106	C₂³×C₈₉	#102	1	1	yes	characteristic
107	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
108	Q₈×C₈₉	—	4	2	no	—
109	Q₈×C₈₉	—	4	2	no	—
110	Q₈×C₈₉	—	8	1	no	—
111	Q₈×C₈₉	—	8	1	no	—
112	C₂×C₄×C₈₉	—	2	1	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	—	8	2	no	—
125	C₂×C₄×C₈₉	—	8	1	no	—
126	C₂×C₄×C₈₉	—	8	1	no	—
127	C₂×C₄×C₈₉	#99	1	1	yes	characteristic
128	C₂×C₄×C₈₉	#99	1	1	yes	characteristic
129	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
130	C₂×C₄×C₈₉	#102	1	1	yes	characteristic
131	C₂×C₄×C₈₉	#104	1	1	yes	characteristic
132	C₂×C₄×C₈₉	#104	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
135	C₈×C₈₉	—	2	1	no	—
136	C₈×C₈₉	—	2	1	no	—
137	C₈×C₈₉	—	2	1	no	—
138	C₈×C₈₉	—	2	1	no	—
139	#10623494	—	4	4	no	—
140	C₂²×C₈₉	—	2	2	no	—
141	C₂²×C₈₉	—	2	1	no	—
142	C₂²×C₈₉	—	2	1	no	—
143	C₂²×C₈₉	—	4	1	no	—
144	C₂²×C₈₉	—	4	1	no	—
145	C₂²×C₈₉	—	4	2	no	—
146	C₂²×C₈₉	—	8	1	no	—
147	C₂²×C₈₉	#327	1	1	yes	characteristic
148	C₂²×C₈₉	#327	1	1	yes	characteristic
149	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	—	2	1	no	—
152	C₄×C₈₉	—	8	2	no	—
153	C₄×C₈₉	—	8	2	no	—
154	C₄×C₈₉	#327	1	1	yes	characteristic
155	C₄×C₈₉	#327	1	1	yes	characteristic
156	C₄×C₈₉	C₂×#99	1	1	yes	characteristic
157	C₄×C₈₉	C₂×#104	1	1	yes	characteristic
158	C₂×#1342	—	89	1	no	maximal, Sylow
159	C₂×C₈₉	—	2	1	no	—
160	C₂×C₈₉	—	2	1	no	—
161	C₂×C₈₉	—	4	1	no	—
162	C₂×C₈₉	—	4	1	no	—
163	C₂×C₈₉	#1342	1	2	yes	complemented
164	C₂×C₈₉	C₂×#327	1	1	yes	characteristic
165	C₂×#470	D₈₉	1	1	yes	characteristic
166	C₂×#349	—	89	1	no	—
167	C₂×#338	—	89	1	no	—
168	#1342	—	89	4	no	—
169	C₈₉	C₂×#1342	1	1	yes	characteristic, complemented, Sylow
170	C₂²×Q₁₆	Dic₈₉	1	1	yes	characteristic
171	C₂²×M₁₆	—	178	1	no	—
172	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
173	C₂×C₂²⋊Q₈	—	2	1	no	—
174	C₂×M₃₂	—	178	1	no	—
175	#470	—	2	1	no	—
176	#470	—	2	1	no	—
177	#470	Dic₈₉	1	2	yes	—
178	#470	C₂×D₈₉	1	2	yes	—
179	C₂×#106	Dic₈₉	1	1	yes	characteristic
180	C₂×#103	—	2	1	no	—
181	#349	—	89	2	no	—
182	#349	—	178	1	no	—
183	#338	—	89	2	no	—
184	#338	—	178	1	no	—
185	C₂²×Q₈	—	2	1	no	—
186	C₂³×C₄	C₂²⋊D₈₉	1	1	yes	characteristic
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	—	4	1	no	—
189	C₂×Q₁₆	—	4	1	no	—
190	C₂×Q₁₆	C₂×Dic₈₉	1	2	yes	—
191	C₂×M₁₆	—	4	1	no	—
192	C₂×M₁₆	—	4	1	no	—
193	C₂×M₁₆	—	356	1	no	—
194	C₂×M₁₆	—	356	1	no	—
195	C₂×M₁₆	—	356	1	no	—
196	C₂×M₁₆	—	356	1	no	—
197	C₂×M₁₆	—	178	2	no	—
198	C₂×M₁₆	—	178	2	no	—
199	C₂×M₁₆	C₂×Dic₈₉	1	1	yes	characteristic
200	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
201	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
202	C₂×M₁₆	C₂²×D₈₉	1	1	yes	characteristic
203	C₂²×C₈	—	178	2	no	—
204	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
205	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
206	C₂²⋊Q₈	—	2	2	no	—
207	C₂²⋊Q₈	—	2	2	no	—
208	C₂²⋊Q₈	—	4	1	no	—
209	C₂²⋊Q₈	—	4	1	no	—
210	C₂×C₄⋊C₄	—	2	1	no	—
211	C₂×C₄⋊C₄	—	4	1	no	—
212	C₂×C₂²⋊C₄	—	4	1	no	—
213	M₃₂	—	356	1	no	—
214	M₃₂	—	178	2	no	—
215	C₂×C₁₆	—	356	1	no	—
216	#106	—	2	1	no	—
217	#106	C₂×Dic₈₉	1	2	yes	—
218	#103	—	2	2	no	—
219	#103	—	2	2	no	—
220	C₂⁴	#13083562	1	1	yes	characteristic
221	C₂×Q₈	—	2	2	no	—
222	C₂×Q₈	—	4	1	no	—
223	C₂×Q₈	—	4	1	no	—
224	C₂×Q₈	—	4	1	no	—
225	C₂×Q₈	—	4	1	no	—
226	C₂×Q₈	—	4	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	4	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
234	C₂²×C₄	C₂²⋊Dic₈₉	1	1	yes	characteristic
235	C₂²×C₄	#13083559	1	1	yes	characteristic
236	C₂²×C₄	#13083559	1	1	yes	characteristic
237	C₂²×C₄	#13083562	1	1	yes	characteristic
238	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
239	Q₁₆	—	4	2	no	—
240	Q₁₆	—	4	2	no	—
241	M₁₆	—	4	1	no	—
242	M₁₆	—	4	1	no	—
243	M₁₆	—	4	1	no	—
244	M₁₆	—	4	1	no	—
245	M₁₆	—	356	2	no	—
246	M₁₆	—	356	2	no	—
247	M₁₆	—	356	2	no	—
248	M₁₆	—	356	2	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	356	2	no	—
254	C₂×C₈	—	356	2	no	—
255	C₂×C₈	—	178	4	no	—
256	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
257	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
258	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
259	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₁₆	—	356	2	no	—
267	C₂³	—	2	1	no	—
268	C₂³	—	2	1	no	—
269	C₂³	—	4	2	no	—
270	C₂³	#15667933	1	1	yes	characteristic
271	C₂³	#15667933	1	1	yes	characteristic
272	C₂³	C₂×#13083562	1	1	yes	characteristic
273	Q₈	—	4	2	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	8	1	no	—
276	Q₈	—	8	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	8	2	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	#15667927	1	1	yes	characteristic
293	C₂×C₄	#15667927	1	1	yes	characteristic
294	C₂×C₄	#15667933	1	1	yes	characteristic
295	C₂×C₄	#15667933	1	1	yes	characteristic
296	C₂×C₄	#15667939	1	1	yes	characteristic
297	C₂×C₄	#15667939	1	1	yes	characteristic
298	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
299	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
300	C₈	—	2	1	no	—
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	356	4	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	4	2	no	—
311	C₂²	—	8	1	no	—
312	C₂²	#19151332	1	1	yes	characteristic
313	C₂²	#19151332	1	1	yes	characteristic
314	C₂²	C₂×#15667933	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	2	no	—
319	C₄	#19151332	1	1	yes	characteristic
320	C₄	#19151332	1	1	yes	characteristic
321	C₄	C₂×#15667927	1	1	yes	characteristic
322	C₄	C₂×#15667939	1	1	yes	characteristic
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	4	1	no	—
326	C₂	—	4	1	no	—
327	C₂	#26018246	1	2	yes	direct product factor
328	C₂	C₂×#19151332	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151332
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂×M₁₆	C₂²×D₈₉
Fitting	C₂×#470×C₈₉	C₂
socle	C₂²×C₈₉	#327
prime radical	C₂⁴×C₈₉	SD₁₆
cosocle	C₂×M₁₆×C₈₉	C₂³
hypercenter	C₂×#470	D₈₉
hypocenter	C₈₉	C₂×#1342
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ C₂×#470

Complements

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

N	H ≅ G/N	type
3. #26018246	C₂	direct (N × H)
163. C₂×C₈₉	#1342	semidirect
169. C₈₉	C₂×#1342	semidirect
327. C₂	#26018246	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×#19151380	2
C₂×#19151356	2
#26018246	2
C₂×#470×C₈₉	2
C₂×#1342	89

Subgroups — C2×#26018246 #36464586

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26018246 #36464586