Subgroups — C₂×#26019772 #36536960

Order 22784.

694 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151885	C₂	1	2	yes	maximal
2	C₂×#19151677	C₂	1	1	yes	maximal, characteristic
3	#26019772	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151349	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151347	C₂	1	2	yes	maximal, complemented
6	C₂×#500×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#13083557	—	2	1	no	—
8	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
9	#19151885	—	2	4	no	—
10	#19151885	C₂²	1	8	yes	—
11	#19151677	—	2	2	no	—
12	#19151677	C₂²	1	4	yes	—
13	C₂×#15667943	—	2	2	no	—
14	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
16	C₂×#15667932	—	2	2	no	—
17	C₂×#15667924	—	2	1	no	—
18	#19151349	C₂²	1	4	yes	complemented
19	#19151347	C₂²	1	8	yes	complemented
20	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
21	#500×C₈₉	—	2	2	no	—
22	#500×C₈₉	C₂²	1	4	yes	complemented
23	C₂×#107×C₈₉	—	2	1	no	—
24	C₂×#107×C₈₉	—	2	1	no	—
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×D₃₅₆	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₈₉	—	2	1	no	—
28	C₂×#13083557	—	2	4	no	—
29	C₂×#13083557	—	4	4	no	—
30	C₂²×#10623494	—	2	2	no	—
31	C₂²×#10623494	—	2	1	no	—
32	C₄×D₃₅₆	—	2	1	no	—
33	C₄×D₃₅₆	—	2	1	no	—
34	C₄×D₃₅₆	D₄	1	2	yes	—
35	C₄×D₃₅₆	C₂³	1	2	yes	—
36	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
37	C₂×#13083551	—	2	1	no	—
38	#15667943	—	2	8	no	—
39	C₄⋊#10623494	—	2	2	no	—
40	C₄⋊#10623494	—	2	1	no	—
41	C₄⋊#10623494	C₂³	1	4	yes	—
42	C₄⋊#10623494	C₂³	1	2	yes	—
43	#15667932	—	2	8	no	—
44	#15667924	—	2	4	no	—
45	C₂²×Q₈×C₈₉	—	2	2	no	—
46	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
48	C₄⋊Q₈×C₈₉	—	2	2	no	—
49	C₄⋊Q₈×C₈₉	—	2	2	no	—
50	C₄⋊Q₈×C₈₉	—	2	2	no	—
51	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
52	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
53	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
54	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	#107×C₈₉	—	2	4	no	—
56	#107×C₈₉	—	2	4	no	—
57	C₄×C₈×C₈₉	D₄	1	2	yes	—
58	C₄×C₈×C₈₉	C₂³	1	2	yes	—
59	C₂³×D₈₉	—	2	1	no	—
60	C₂×D₃₅₆	—	2	4	no	—
61	C₂×D₃₅₆	—	2	1	no	—
62	C₂×D₃₅₆	C₂×D₄	1	2	yes	—
63	C₂²×Dic₈₉	—	2	1	no	—
64	C₂×C₄×D₈₉	—	2	2	no	—
65	C₂×C₄×D₈₉	—	2	2	no	—
66	C₂×C₄×D₈₉	—	4	2	no	—
67	C₂×C₄×D₈₉	—	4	2	no	—
68	#13083557	—	4	8	no	—
69	C₂×#10623494	—	2	2	no	—
70	C₂×#10623494	—	2	4	no	—
71	C₂×#10623494	—	4	2	no	—
72	C₂×#10623494	—	4	4	no	—
73	C₄⋊Dic₈₉	—	2	1	no	—
74	C₄⋊Dic₈₉	C₂×D₄	1	2	yes	—
75	#13083551	—	2	2	no	—
76	#13083551	—	2	2	no	—
77	C₂×Q₈×C₈₉	—	2	4	no	—
78	C₂×Q₈×C₈₉	—	4	2	no	—
79	C₂×Q₈×C₈₉	—	8	4	no	—
80	C₂²×C₄×C₈₉	—	4	2	no	—
81	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
84	C₂×C₈×C₈₉	—	2	2	no	—
85	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
86	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
88	C₄⋊C₄×C₈₉	—	2	4	no	—
89	C₄⋊C₄×C₈₉	—	2	4	no	—
90	C₄⋊C₄×C₈₉	—	4	2	no	—
91	C₄⋊C₄×C₈₉	—	4	2	no	—
92	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₈₉	#41	1	1	yes	characteristic
95	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
96	C₂³×C₈₉	#121	1	1	yes	characteristic
97	C₂²×D₈₉	—	2	2	no	—
98	C₂²×D₈₉	—	4	2	no	—
99	C₂²×D₈₉	—	4	2	no	—
100	C₂²×D₈₉	—	4	2	no	—
101	Q₈×C₈₉	—	8	4	no	—
102	Q₈×C₈₉	—	8	4	no	—
103	C₂×C₄×C₈₉	—	2	2	no	—
104	C₂×C₄×C₈₉	—	4	4	no	—
105	C₂×C₄×C₈₉	—	8	4	no	—
106	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
108	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
110	C₂×C₄×C₈₉	#137	1	2	yes	—
111	C₂×C₄×C₈₉	#137	1	4	yes	—
112	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
115	C₂×Dic₈₉	—	2	2	no	—
116	C₂×Dic₈₉	—	4	2	no	—
117	D₃₅₆	—	2	4	no	—
118	D₃₅₆	—	4	2	no	—
119	C₄×D₈₉	—	4	2	no	—
120	C₄×D₈₉	—	4	2	no	—
121	C₄×D₈₉	—	4	4	no	—
122	C₈×C₈₉	—	2	2	no	—
123	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
124	#10623494	—	4	2	no	—
125	#10623494	—	4	4	no	—
126	C₂²×C₈₉	#461	1	2	yes	—
127	C₂²×C₈₉	#470	1	2	yes	—
128	C₂²×C₈₉	#472	1	2	yes	—
129	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
130	C₂×D₈₉	—	4	2	no	—
131	C₂×D₈₉	—	4	2	no	—
132	C₂×D₈₉	—	4	2	no	—
133	C₂×D₈₉	—	8	4	no	—
134	C₄×C₈₉	—	2	1	no	—
135	C₄×C₈₉	—	2	1	no	—
136	C₄×C₈₉	—	8	4	no	—
137	C₄×C₈₉	#461	1	2	yes	—
138	C₄×C₈₉	#470	1	2	yes	—
139	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
140	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
141	C₄×C₈₉	C₂×#137	1	2	yes	—
142	Dic₈₉	—	4	2	no	—
143	C₂×#1695	—	89	1	no	maximal, Sylow
144	C₂×C₈₉	#1695	1	4	yes	complemented
145	C₂×C₈₉	C₂×#461	1	1	yes	characteristic
146	C₂×C₈₉	C₂×#470	1	1	yes	characteristic
147	C₂×C₈₉	C₂×#472	1	1	yes	characteristic
148	D₈₉	—	8	2	no	—
149	C₂×#500	D₈₉	1	1	yes	characteristic, complemented
150	C₂×#476	—	89	2	no	—
151	C₂×#436	—	89	1	no	—
152	#1695	—	89	8	no	—
153	C₂×#334	—	89	1	no	—
154	C₂×#333	—	89	2	no	—
155	C₈₉	C₂×#1695	1	1	yes	characteristic, complemented, Sylow
156	C₂²×M₁₆	—	178	1	no	—
157	C₂×C₄⋊Q₈	C₂×D₈₉	1	2	yes	—
158	C₂×C₄×D₄	—	89	1	no	—
159	#500	—	2	2	no	—
160	#500	C₂×D₈₉	1	4	yes	complemented
161	#476	—	89	8	no	—
162	#476	—	178	4	no	—
163	#436	—	89	4	no	—
164	#436	—	178	2	no	—
165	C₂×#107	—	2	1	no	—
166	C₂×#107	—	2	1	no	—
167	C₂×#106	—	178	2	no	—
168	C₂×C₄⋊C₈	—	89	1	no	—
169	C₂×C₄⋊C₈	—	89	2	no	—
170	C₂×#102	—	178	2	no	—
171	C₂×C₂²⋊C₈	—	178	1	no	—
172	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
173	#334	—	89	4	no	—
174	#333	—	89	8	no	—
175	C₂²×Q₈	—	2	2	no	—
176	C₂²×D₄	—	89	1	no	—
177	C₂³×C₄	—	178	1	no	—
178	C₂×M₁₆	—	356	4	no	—
179	C₂×M₁₆	—	178	4	no	—
180	C₂²×C₈	—	178	2	no	—
181	C₂²×C₈	—	178	1	no	—
182	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
183	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
184	C₄⋊Q₈	—	2	2	no	—
185	C₄⋊Q₈	—	2	2	no	—
186	C₄⋊Q₈	—	2	2	no	—
187	C₄⋊Q₈	C₂²×D₈₉	1	4	yes	—
188	C₄×D₄	—	89	2	no	—
189	C₄×D₄	—	89	2	no	—
190	C₄×D₄	—	178	1	no	—
191	C₄×D₄	—	178	1	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	89	1	no	—
195	C₂×C₂²⋊C₄	—	178	1	no	—
196	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
197	#107	—	2	4	no	—
198	#107	—	2	4	no	—
199	#106	—	178	8	no	—
200	C₄⋊C₈	—	89	2	no	—
201	C₄⋊C₈	—	89	4	no	—
202	C₄⋊C₈	—	178	2	no	—
203	C₄⋊C₈	—	178	1	no	—
204	#102	—	178	8	no	—
205	C₂²⋊C₈	—	178	4	no	—
206	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
207	C₄×C₈	C₂²×D₈₉	1	2	yes	—
208	C₂⁴	—	178	1	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	—	8	4	no	—
212	C₂×D₄	—	89	2	no	—
213	C₂×D₄	—	178	1	no	—
214	C₂×D₄	—	178	4	no	—
215	C₂²×C₄	—	4	2	no	—
216	C₂²×C₄	—	356	2	no	—
217	C₂²×C₄	—	356	2	no	—
218	C₂²×C₄	—	178	1	no	—
219	C₂²×C₄	—	178	2	no	—
220	C₂²×C₄	—	178	2	no	—
221	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
223	C₂²×C₄	#13083573	1	1	yes	characteristic
224	M₁₆	—	356	8	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	356	2	no	—
227	C₂×C₈	—	356	4	no	—
228	C₂×C₈	—	178	4	no	—
229	C₂×C₈	—	178	2	no	—
230	C₂×C₈	#13083561	1	4	yes	—
231	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
232	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
233	C₄⋊C₄	—	2	4	no	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	4	2	no	—
236	C₄⋊C₄	—	4	2	no	—
237	C₄⋊C₄	—	89	2	no	—
238	C₄⋊C₄	—	178	1	no	—
239	C₂²⋊C₄	—	178	2	no	—
240	C₂²⋊C₄	—	178	2	no	—
241	C₄²	D₄×D₈₉	1	1	yes	characteristic
242	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
243	C₄²	#13083573	1	1	yes	characteristic
244	C₄²	C₂³×D₈₉	1	1	yes	characteristic
245	C₂³	—	356	2	no	—
246	C₂³	—	356	2	no	—
247	C₂³	—	356	2	no	—
248	C₂³	—	178	2	no	—
249	C₂³	#15667983	1	1	yes	characteristic
250	Q₈	—	8	4	no	—
251	Q₈	—	8	4	no	—
252	D₄	—	356	2	no	—
253	D₄	—	178	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	8	4	no	—
257	C₂×C₄	—	356	2	no	—
258	C₂×C₄	—	356	2	no	—
259	C₂×C₄	—	356	4	no	—
260	C₂×C₄	—	356	2	no	—
261	C₂×C₄	—	178	2	no	—
262	C₂×C₄	#15667983	1	1	yes	characteristic
263	C₂×C₄	#15667983	1	1	yes	characteristic
264	C₂×C₄	#15667983	1	1	yes	characteristic
265	C₂×C₄	C₂×#13083561	1	2	yes	—
266	C₂×C₄	#15668043	1	2	yes	—
267	C₂×C₄	#15668044	1	4	yes	—
268	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
270	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	356	4	no	—
273	C₈	—	356	2	no	—
274	C₈	C₂×#13083561	1	4	yes	—
275	C₂²	—	356	2	no	—
276	C₂²	—	356	2	no	—
277	C₂²	—	356	2	no	—
278	C₂²	—	712	4	no	—
279	C₂²	#19151800	1	2	yes	—
280	C₂²	#19151853	1	2	yes	—
281	C₂²	#19151865	1	2	yes	—
282	C₂²	C₂×#15667983	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	8	4	no	—
286	C₄	—	356	2	no	—
287	C₄	#19151800	1	2	yes	—
288	C₄	#19151853	1	2	yes	—
289	C₄	C₂²×#13083561	1	1	yes	characteristic
290	C₄	C₂×#15668043	1	1	yes	characteristic
291	C₄	C₂×#15668044	1	2	yes	—
292	C₂	—	712	2	no	—
293	C₂	#26019772	1	4	yes	direct product factor
294	C₂	C₂×#19151800	1	1	yes	characteristic
295	C₂	C₂×#19151853	1	1	yes	characteristic
296	C₂	C₂×#19151865	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667983
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×#500×C₈₉	C₂
socle	C₂³×C₈₉	#121
prime radical	C₂²×D₃₅₆	D₄
hypercenter	C₂×#500	D₈₉
hypocenter	C₈₉	C₂×#1695
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
3. #26019772	C₂	direct (N × H)
4. C₂×#19151349	C₂	semidirect
5. C₂×#19151347	C₂	semidirect
6. C₂×#500×C₈₉	C₂	semidirect
18. #19151349	C₂²	semidirect
19. #19151347	C₂²	semidirect
22. #500×C₈₉	C₂²	semidirect
144. C₂×C₈₉	#1695	semidirect
149. C₂×#500	D₈₉	semidirect
155. C₈₉	C₂×#1695	semidirect
160. #500	C₂×D₈₉	semidirect
293. C₂	#26019772	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×#19151885	2
C₂×#19151677	2
#26019772	2
C₂×#19151349	2
C₂×#19151347	2
C₂×#500×C₈₉	2
C₂×#1695	89

Subgroups — C2×#26019772 #36536960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019772 #36536960