Subgroups — C₂×#26022461 #36709207

Order 22784.

586 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151839	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151636	C₂	1	1	yes	maximal, characteristic
3	#26022461	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151426	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151424	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151403	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#19151403	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
10	C₂×#15667991	—	2	1	no	—
11	C₂×#15667971	—	2	1	no	—
12	#19151839	—	2	2	no	—
13	#19151839	C₂²	1	4	yes	—
14	#19151636	—	2	2	no	—
15	#19151636	C₂²	1	4	yes	—
16	C₂×#15667946	C₂²	1	1	yes	characteristic
17	C₂×#15667946	C₂²	1	1	yes	characteristic
18	C₂×#15667946	C₂²	1	1	yes	characteristic
19	C₂×#15667944	—	2	1	no	—
20	C₂×#15667937	—	2	1	no	—
21	#19151426	C₂²	1	4	yes	complemented
22	#19151424	C₂²	1	4	yes	complemented
23	#19151403	C₂²	1	4	yes	complemented
24	#19151403	C₂²	1	4	yes	complemented
25	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
28	C₂²⋊C₁₆×C₈₉	C₂²	1	4	yes	—
29	C₂²×Dic₁₇₈	—	2	1	no	—
30	C₂×Dic₃₅₆	—	2	1	no	—
31	C₂×Dic₃₅₆	—	2	2	no	—
32	C₂×Dic₃₅₆	—	2	2	no	—
33	C₂×Dic₃₅₆	C₂³	1	2	yes	—
34	#15667991	—	2	4	no	—
35	#15667991	—	4	2	no	—
36	#15667971	—	2	4	no	—
37	#15667971	—	4	2	no	—
38	C₂×C₄⋊Dic₈₉	—	2	1	no	—
39	C₂×C₄⋊Dic₈₉	—	2	1	no	—
40	C₂×C₄⋊Dic₈₉	—	2	1	no	—
41	C₂×#13083553	—	4	1	no	—
42	C₂×C₂²⋊Dic₈₉	—	4	1	no	—
43	C₂×C₂²⋊Dic₈₉	—	4	1	no	—
44	C₂×C₄×Dic₈₉	—	4	1	no	—
45	#15667946	—	2	1	no	—
46	#15667946	—	2	1	no	—
47	#15667946	—	2	1	no	—
48	#15667946	C₂³	1	2	yes	—
49	#15667946	C₂³	1	2	yes	—
50	#15667946	C₂³	1	2	yes	—
51	#15667944	—	2	4	no	—
52	#15667937	—	2	4	no	—
53	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	—	2	2	no	—
55	C₂²×C₈×C₈₉	—	2	2	no	—
56	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₈₉	D₄	1	2	yes	—
58	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
59	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₂×C₁₆×C₈₉	—	2	2	no	—
61	C₂×C₁₆×C₈₉	—	2	2	no	—
62	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
63	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
64	C₂×Dic₁₇₈	—	2	2	no	—
65	C₂×Dic₁₇₈	—	4	2	no	—
66	C₂×Dic₁₇₈	—	4	2	no	—
67	C₂×Dic₁₇₈	—	4	1	no	—
68	C₂²×Dic₈₉	—	4	1	no	—
69	C₂²×Dic₈₉	—	4	1	no	—
70	C₂²×Dic₈₉	—	4	1	no	—
71	C₂²×Dic₈₉	—	4	1	no	—
72	Dic₃₅₆	—	2	2	no	—
73	Dic₃₅₆	—	2	2	no	—
74	Dic₃₅₆	—	4	2	no	—
75	C₄⋊Dic₈₉	—	2	2	no	—
76	C₄⋊Dic₈₉	—	2	2	no	—
77	C₄⋊Dic₈₉	—	2	2	no	—
78	C₄⋊Dic₈₉	—	4	1	no	—
79	C₄⋊Dic₈₉	—	4	1	no	—
80	C₄⋊Dic₈₉	—	4	1	no	—
81	#13083553	—	4	4	no	—
82	C₂²⋊Dic₈₉	—	4	4	no	—
83	C₂²⋊Dic₈₉	—	4	4	no	—
84	C₄×Dic₈₉	—	4	4	no	—
85	C₂⁴×C₈₉	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	—	2	2	no	—
87	C₂²×C₄×C₈₉	—	2	2	no	—
88	C₂²×C₄×C₈₉	D₈	1	2	yes	—
89	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	—	2	2	no	—
93	C₂×C₈×C₈₉	—	2	4	no	—
94	C₂×C₈×C₈₉	—	2	2	no	—
95	C₂×C₈×C₈₉	—	2	2	no	—
96	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₈₉	#41	1	2	yes	—
99	C₂×C₈×C₈₉	#41	1	2	yes	—
100	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
101	C₁₆×C₈₉	—	2	2	no	—
102	C₁₆×C₈₉	—	2	2	no	—
103	C₂³×C₈₉	—	2	2	no	—
104	C₂³×C₈₉	—	2	2	no	—
105	C₂³×C₈₉	—	2	2	no	—
106	C₂³×C₈₉	C₂×D₈	1	2	yes	—
107	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	—	2	2	no	—
109	C₂×C₄×C₈₉	—	2	2	no	—
110	C₂×C₄×C₈₉	—	2	2	no	—
111	C₂×C₄×C₈₉	—	2	4	no	—
112	C₂×C₄×C₈₉	#123	1	2	yes	—
113	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
114	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
115	C₂×C₄×C₈₉	#137	1	2	yes	—
116	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
117	C₂×Dic₈₉	—	4	2	no	—
118	C₂×Dic₈₉	—	4	2	no	—
119	C₂×Dic₈₉	—	4	2	no	—
120	C₂×Dic₈₉	—	4	2	no	—
121	C₂×Dic₈₉	—	8	2	no	—
122	C₂×Dic₈₉	—	8	2	no	—
123	C₂×Dic₈₉	—	8	2	no	—
124	C₂×Dic₈₉	—	8	2	no	—
125	Dic₁₇₈	—	4	2	no	—
126	Dic₁₇₈	—	4	2	no	—
127	Dic₁₇₈	—	8	2	no	—
128	C₈×C₈₉	—	2	2	no	—
129	C₈×C₈₉	#123	1	2	yes	—
130	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
131	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
132	C₂²×C₈₉	—	2	2	no	—
133	C₂²×C₈₉	—	2	2	no	—
134	C₂²×C₈₉	—	2	2	no	—
135	C₂²×C₈₉	—	4	4	no	—
136	C₂²×C₈₉	#480	1	2	yes	—
137	C₂²×C₈₉	#508	1	2	yes	—
138	C₂²×C₈₉	#510	1	2	yes	—
139	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
140	C₄×C₈₉	—	2	2	no	—
141	C₄×C₈₉	#480	1	2	yes	—
142	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
143	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
144	Dic₈₉	—	8	2	no	—
145	Dic₈₉	—	8	2	no	—
146	Dic₈₉	—	8	2	no	—
147	Dic₈₉	—	8	2	no	—
148	C₂×#2237	—	89	1	no	maximal, Sylow
149	C₂×C₈₉	—	4	2	no	—
150	C₂×C₈₉	#2237	1	4	yes	complemented
151	C₂×C₈₉	C₂×#480	1	1	yes	characteristic
152	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
153	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
154	C₂×#467	—	89	1	no	—
155	C₂×#427	—	89	1	no	—
156	#2237	—	89	8	no	—
157	C₂×#367	—	89	1	no	—
158	C₂×#366	—	89	1	no	—
159	C₂×#358	—	89	1	no	—
160	C₂×#358	—	89	1	no	—
161	C₂×C₂²⋊C₁₆	D₈₉	1	1	yes	characteristic
162	C₈₉	C₂×#2237	1	1	yes	characteristic, complemented, Sylow
163	C₂²×Q₁₆	—	89	1	no	—
164	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
165	C₂×C₂²⋊Q₈	—	178	1	no	—
166	C₂×#117	—	178	1	no	—
167	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
168	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
169	#467	—	89	4	no	—
170	#467	—	178	2	no	—
171	#427	—	89	4	no	—
172	#427	—	178	2	no	—
173	C₂×#107	—	89	1	no	—
174	C₂×#107	—	89	1	no	—
175	C₂×#107	—	89	1	no	—
176	C₂×#106	—	178	1	no	—
177	C₂×#103	—	178	1	no	—
178	#367	—	89	4	no	—
179	#366	—	89	4	no	—
180	#358	—	89	4	no	—
181	#358	—	89	4	no	—
182	C₂²⋊C₁₆	C₂×D₈₉	1	4	yes	—
183	C₂²×Q₈	—	178	1	no	—
184	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
185	C₂×Q₁₆	—	89	2	no	—
186	C₂×Q₁₆	—	178	1	no	—
187	C₂×Q₁₆	—	178	2	no	—
188	C₂×Q₁₆	—	178	2	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
192	C₂²×C₈	D₃₅₆	1	2	yes	—
193	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
194	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
195	C₂²⋊Q₈	—	356	2	no	—
196	C₂²⋊Q₈	—	178	4	no	—
197	#117	—	356	2	no	—
198	#117	—	178	4	no	—
199	C₂×C₄⋊C₄	—	356	1	no	—
200	C₂×C₄⋊C₄	—	178	1	no	—
201	C₂×C₄⋊C₄	—	178	1	no	—
202	C₂×C₄⋊C₄	—	178	1	no	—
203	C₂×C₂²⋊C₄	—	356	1	no	—
204	C₂×C₂²⋊C₄	—	356	1	no	—
205	C₂×C₄²	—	356	1	no	—
206	C₂×C₁₆	—	2	2	no	—
207	C₂×C₁₆	—	2	2	no	—
208	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
209	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
210	#107	—	89	2	no	—
211	#107	—	89	2	no	—
212	#107	—	89	2	no	—
213	#107	—	178	1	no	—
214	#107	—	178	1	no	—
215	#107	—	178	1	no	—
216	#106	—	178	4	no	—
217	#103	—	178	4	no	—
218	C₂⁴	D₇₁₂	1	1	yes	characteristic
219	C₂×Q₈	—	356	1	no	—
220	C₂×Q₈	—	356	2	no	—
221	C₂×Q₈	—	356	2	no	—
222	C₂×Q₈	—	178	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	356	1	no	—
226	C₂²×C₄	—	356	1	no	—
227	C₂²×C₄	—	356	1	no	—
228	C₂²×C₄	—	356	1	no	—
229	C₂²×C₄	D₇₁₂	1	2	yes	—
230	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
232	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
233	Q₁₆	—	356	2	no	—
234	Q₁₆	—	178	2	no	—
235	Q₁₆	—	178	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
241	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
242	C₂×C₈	#13083573	1	2	yes	—
243	C₂×C₈	#13083573	1	2	yes	—
244	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
245	C₄⋊C₄	—	356	4	no	—
246	C₄⋊C₄	—	356	1	no	—
247	C₄⋊C₄	—	356	1	no	—
248	C₄⋊C₄	—	356	1	no	—
249	C₄⋊C₄	—	178	2	no	—
250	C₄⋊C₄	—	178	2	no	—
251	C₄⋊C₄	—	178	2	no	—
252	C₂²⋊C₄	—	356	4	no	—
253	C₂²⋊C₄	—	356	4	no	—
254	C₄²	—	356	4	no	—
255	C₁₆	—	2	2	no	—
256	C₁₆	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂×D₇₁₂	1	2	yes	—
261	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
262	Q₈	—	356	2	no	—
263	Q₈	—	356	2	no	—
264	Q₈	—	712	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	356	2	no	—
270	C₂×C₄	—	356	2	no	—
271	C₂×C₄	—	356	2	no	—
272	C₂×C₄	—	356	2	no	—
273	C₂×C₄	—	712	2	no	—
274	C₂×C₄	—	712	2	no	—
275	C₂×C₄	—	712	2	no	—
276	C₂×C₄	—	712	2	no	—
277	C₂×C₄	#15667996	1	2	yes	—
278	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
279	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
280	C₂×C₄	#15668045	1	2	yes	—
281	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	#15667996	1	2	yes	—
284	C₈	C₂×#13083573	1	1	yes	characteristic
285	C₈	C₂×#13083573	1	1	yes	characteristic
286	C₂²	—	2	2	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	4	no	—
290	C₂²	#19151912	1	2	yes	—
291	C₂²	#19152030	1	2	yes	—
292	C₂²	#19152041	1	2	yes	—
293	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	712	2	no	—
296	C₄	—	712	2	no	—
297	C₄	—	712	2	no	—
298	C₄	—	712	2	no	—
299	C₄	#19151912	1	2	yes	—
300	C₄	C₂×#15667996	1	1	yes	characteristic
301	C₄	C₂×#15668045	1	1	yes	characteristic
302	C₂	—	4	2	no	—
303	C₂	#26022461	1	4	yes	direct product factor
304	C₂	C₂×#19151912	1	1	yes	characteristic
305	C₂	C₂×#19152030	1	1	yes	characteristic
306	C₂	C₂×#19152041	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×C₂²⋊C₁₆×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂⁴×C₈₉	D₈
hypercenter	C₂×C₂²⋊C₁₆	D₈₉
hypocenter	C₈₉	C₂×#2237
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. #26022461	C₂	direct (N × H)
4. C₂×#19151426	C₂	semidirect
5. C₂×#19151424	C₂	semidirect
6. C₂×#19151403	C₂	semidirect
7. C₂×#19151403	C₂	semidirect
21. #19151426	C₂²	semidirect
22. #19151424	C₂²	semidirect
23. #19151403	C₂²	semidirect
24. #19151403	C₂²	semidirect
150. C₂×C₈₉	#2237	semidirect
162. C₈₉	C₂×#2237	semidirect
303. C₂	#26022461	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×#19151839	2
C₂×#19151636	2
#26022461	2
C₂×#19151426	2
C₂×#19151424	2
C₂×#19151403	2
C₂×C₂²⋊C₁₆×C₈₉	2
C₂×#2237	89

Subgroups — C2×#26022461 #36709207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022461 #36709207