Subgroups — C₂×#26019641 #36536685

Order 22784.

790 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152003	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151803	C₂	1	2	yes	maximal
3	#26019641	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151351	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151343	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#13083562	—	2	2	no	—
8	C₂×#15668012	C₂²	1	2	yes	—
9	#19152003	—	2	2	no	—
10	#19152003	C₂²	1	4	yes	complemented
11	#19151803	—	2	4	no	—
12	#19151803	C₂²	1	8	yes	—
13	C₂×#15667946	—	2	1	no	—
14	C₂×#15667946	—	2	1	no	—
15	C₂×C₄⋊#10623494	C₂²	1	2	yes	—
16	C₂×#15667935	—	2	2	no	—
17	#19151351	C₂²	1	4	yes	complemented
18	#19151343	C₂²	1	8	yes	complemented
19	C₂³×C₈×C₈₉	—	2	1	no	—
20	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
21	C₈×D₄×C₈₉	—	2	2	no	—
22	C₈×D₄×C₈₉	C₂²	1	4	yes	—
23	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₈×C₈₉	—	2	1	no	—
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂²×Dic₁₇₈	—	2	2	no	—
27	C₂×#13083562	—	2	8	no	—
28	C₂×#13083562	—	4	8	no	—
29	C₂²×#10623494	—	2	2	no	—
30	#15668012	—	2	2	no	—
31	#15668012	—	2	2	no	—
32	#15668012	—	2	2	no	—
33	#15668012	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₈₉	—	2	2	no	—
35	C₂×C₄⋊Dic₈₉	—	2	2	no	—
36	#15667946	—	2	4	no	—
37	#15667946	—	2	4	no	—
38	C₄⋊#10623494	—	2	2	no	—
39	C₄⋊#10623494	C₂³	1	4	yes	—
40	#15667935	—	2	8	no	—
41	C₂²×D₄×C₈₉	D₄	1	1	yes	characteristic
42	C₂³×C₄×C₈₉	—	2	1	no	—
43	C₂²×C₈×C₈₉	—	2	1	no	—
44	C₂²×C₈×C₈₉	—	2	4	no	—
45	C₂²×C₈×C₈₉	—	4	4	no	—
46	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
48	C₄×D₄×C₈₉	—	2	1	no	—
49	C₄×D₄×C₈₉	—	2	1	no	—
50	C₄×D₄×C₈₉	D₄	1	2	yes	—
51	C₄×D₄×C₈₉	C₂³	1	2	yes	—
52	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
53	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
54	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₈₉	—	2	1	no	—
56	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
57	C₂²⋊C₈×C₈₉	—	2	4	no	—
58	C₄×C₈×C₈₉	D₄	1	2	yes	—
59	C₄×C₈×C₈₉	C₂³	1	2	yes	—
60	C₂×Dic₁₇₈	—	2	4	no	—
61	C₂×Dic₁₇₈	—	4	4	no	—
62	C₂×Dic₁₇₈	—	4	4	no	—
63	C₂×Dic₁₇₈	—	4	2	no	—
64	C₂²×Dic₈₉	—	4	2	no	—
65	#13083562	—	4	16	no	—
66	C₂×#10623494	—	2	4	no	—
67	C₂×#10623494	—	4	4	no	—
68	C₄⋊Dic₈₉	—	2	4	no	—
69	C₄⋊Dic₈₉	—	2	4	no	—
70	C₄⋊Dic₈₉	—	4	2	no	—
71	C₄⋊Dic₈₉	—	4	2	no	—
72	C₂⁴×C₈₉	—	2	1	no	—
73	C₂×D₄×C₈₉	—	2	1	no	—
74	C₂×D₄×C₈₉	—	2	2	no	—
75	C₂×D₄×C₈₉	D₈	1	4	yes	—
76	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₈₉	—	2	1	no	—
78	C₂²×C₄×C₈₉	—	2	2	no	—
79	C₂²×C₄×C₈₉	—	2	2	no	—
80	C₂²×C₄×C₈₉	—	4	2	no	—
81	C₂²×C₄×C₈₉	—	4	2	no	—
82	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
85	C₂×C₈×C₈₉	—	2	2	no	—
86	C₂×C₈×C₈₉	—	2	2	no	—
87	C₂×C₈×C₈₉	—	4	2	no	—
88	C₂×C₈×C₈₉	—	4	8	no	—
89	C₂×C₈×C₈₉	SD₁₆	1	4	yes	complemented
90	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
92	C₄⋊C₄×C₈₉	—	2	1	no	—
93	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
94	C₂²⋊C₄×C₈₉	—	2	2	no	—
95	C₂²⋊C₄×C₈₉	—	2	2	no	—
96	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₈₉	#41	1	1	yes	characteristic
99	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
100	C₂³×C₈₉	—	2	2	no	—
101	C₂³×C₈₉	—	2	4	no	—
102	C₂³×C₈₉	—	4	2	no	—
103	C₂³×C₈₉	—	4	2	no	—
104	C₂³×C₈₉	#121	1	1	yes	characteristic
105	D₄×C₈₉	—	2	2	no	—
106	D₄×C₈₉	—	2	4	no	—
107	D₄×C₈₉	C₂×D₈	1	4	yes	—
108	C₂×C₄×C₈₉	—	2	2	no	—
109	C₂×C₄×C₈₉	—	2	2	no	—
110	C₂×C₄×C₈₉	—	4	2	no	—
111	C₂×C₄×C₈₉	—	4	2	no	—
112	C₂×C₄×C₈₉	—	4	2	no	—
113	C₂×C₄×C₈₉	—	4	4	no	—
114	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
115	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
119	C₂×C₄×C₈₉	#135	1	2	yes	—
120	C₂×C₄×C₈₉	#137	1	2	yes	—
121	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
124	C₂×Dic₈₉	—	4	4	no	—
125	C₂×Dic₈₉	—	8	4	no	—
126	Dic₁₇₈	—	4	4	no	—
127	Dic₁₇₈	—	4	4	no	—
128	Dic₁₇₈	—	8	4	no	—
129	C₈×C₈₉	—	2	2	no	—
130	C₈×C₈₉	—	4	2	no	—
131	C₈×C₈₉	C₂×SD₁₆	1	4	yes	complemented
132	#10623494	—	4	4	no	—
133	C₂²×C₈₉	—	2	4	no	—
134	C₂²×C₈₉	—	4	8	no	—
135	C₂²×C₈₉	—	4	2	no	—
136	C₂²×C₈₉	—	4	2	no	—
137	C₂²×C₈₉	#461	1	2	yes	—
138	C₂²×C₈₉	#466	1	2	yes	—
139	C₂²×C₈₉	#473	1	2	yes	—
140	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
141	C₄×C₈₉	—	2	1	no	—
142	C₄×C₈₉	—	2	1	no	—
143	C₄×C₈₉	—	4	2	no	—
144	C₄×C₈₉	#461	1	2	yes	—
145	C₄×C₈₉	#466	1	2	yes	—
146	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
147	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
148	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
149	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
150	Dic₈₉	—	8	4	no	—
151	C₂×#1673	—	89	1	no	maximal, Sylow
152	C₂×C₈₉	—	4	4	no	—
153	C₂×C₈₉	#1673	1	4	yes	complemented
154	C₂×C₈₉	C₂×#461	1	1	yes	characteristic
155	C₂×C₈₉	C₂×#466	1	1	yes	characteristic
156	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
157	C₂×#500	—	89	1	no	—
158	C₂×#461	—	89	2	no	—
159	C₂×C₈×D₄	D₈₉	1	1	yes	characteristic
160	#1673	—	89	8	no	—
161	C₂×#335	—	89	1	no	—
162	C₂×#332	—	89	2	no	—
163	C₈₉	C₂×#1673	1	1	yes	characteristic, complemented, Sylow
164	C₂²×SD₁₆	—	178	2	no	—
165	C₂³×C₈	—	2	1	no	—
166	C₂×C₄⋊Q₈	—	89	2	no	—
167	C₂×C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
168	#500	—	89	4	no	—
169	#500	—	178	2	no	—
170	#461	—	89	8	no	—
171	#461	—	178	4	no	—
172	C₈×D₄	—	2	2	no	—
173	C₈×D₄	C₂×D₈₉	1	4	yes	—
174	C₂×#107	—	178	1	no	—
175	C₂×#107	—	178	1	no	—
176	C₂×C₄⋊C₈	—	89	2	no	—
177	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
178	C₂×#103	—	178	2	no	—
179	C₂×C₂²⋊C₈	—	2	1	no	—
180	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
181	#335	—	89	4	no	—
182	#332	—	89	8	no	—
183	C₂²×Q₈	—	178	2	no	—
184	C₂²×D₄	D₃₅₆	1	1	yes	characteristic
185	C₂³×C₄	—	2	1	no	—
186	C₂×SD₁₆	—	356	8	no	—
187	C₂×SD₁₆	—	178	8	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	4	no	—
190	C₂²×C₈	—	4	4	no	—
191	C₂²×C₈	—	178	2	no	—
192	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
193	C₂²×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
194	C₄⋊Q₈	—	89	4	no	—
195	C₄⋊Q₈	—	178	2	no	—
196	C₄⋊Q₈	—	178	2	no	—
197	C₄⋊Q₈	—	178	2	no	—
198	C₄×D₄	—	2	1	no	—
199	C₄×D₄	—	2	1	no	—
200	C₄×D₄	D₃₅₆	1	2	yes	—
201	C₄×D₄	C₂²×D₈₉	1	2	yes	—
202	C₂×C₄⋊C₄	—	178	2	no	—
203	C₂×C₄⋊C₄	—	178	2	no	—
204	C₂×C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
205	C₂×C₂²⋊C₄	—	2	1	no	—
206	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
207	#107	—	178	4	no	—
208	#107	—	178	4	no	—
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	89	4	no	—
211	C₄⋊C₈	—	178	2	no	—
212	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
213	#103	—	178	8	no	—
214	C₂²⋊C₈	—	2	4	no	—
215	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
216	C₄×C₈	C₂²×D₈₉	1	2	yes	—
217	C₂⁴	—	2	1	no	—
218	C₂×Q₈	—	356	2	no	—
219	C₂×Q₈	—	356	4	no	—
220	C₂×Q₈	—	356	4	no	—
221	C₂×Q₈	—	178	4	no	—
222	C₂×D₄	—	2	1	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	D₇₁₂	1	4	yes	—
225	C₂×D₄	C₂×D₃₅₆	1	2	yes	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	—	356	2	no	—
232	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
234	C₂²×C₄	#13083574	1	1	yes	characteristic
235	SD₁₆	—	356	16	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	8	no	—
240	C₂×C₈	—	356	4	no	—
241	C₂×C₈	—	178	4	no	—
242	C₂×C₈	#13083562	1	4	yes	complemented
243	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
244	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	356	2	no	—
247	C₄⋊C₄	—	356	2	no	—
248	C₄⋊C₄	—	178	4	no	—
249	C₄⋊C₄	—	178	4	no	—
250	C₄⋊C₄	C₂×D₃₅₆	1	2	yes	—
251	C₂²⋊C₄	—	2	2	no	—
252	C₂²⋊C₄	—	2	2	no	—
253	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
254	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
255	C₄²	#13083574	1	1	yes	characteristic
256	C₄²	C₂³×D₈₉	1	1	yes	characteristic
257	C₂³	—	2	4	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	4	2	no	—
260	C₂³	—	4	2	no	—
261	C₂³	#15667986	1	1	yes	characteristic
262	Q₈	—	356	4	no	—
263	Q₈	—	356	4	no	—
264	Q₈	—	712	4	no	—
265	D₄	—	2	4	no	—
266	D₄	—	2	2	no	—
267	D₄	C₂×D₇₁₂	1	4	yes	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	356	4	no	—
275	C₂×C₄	—	712	4	no	—
276	C₂×C₄	#15667986	1	1	yes	characteristic
277	C₂×C₄	#15667986	1	1	yes	characteristic
278	C₂×C₄	#15667986	1	1	yes	characteristic
279	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
280	C₂×C₄	C₂×#13083562	1	2	yes	—
281	C₂×C₄	#15668034	1	2	yes	—
282	C₂×C₄	#15668042	1	2	yes	—
283	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
284	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
285	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
286	C₈	—	2	2	no	—
287	C₈	—	4	2	no	—
288	C₈	—	356	4	no	—
289	C₈	C₂×#13083562	1	4	yes	complemented
290	C₂²	—	2	4	no	—
291	C₂²	—	4	8	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	#19151803	1	2	yes	—
295	C₂²	#19151834	1	2	yes	—
296	C₂²	#19151873	1	2	yes	—
297	C₂²	C₂×#15667986	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	2	no	—
301	C₄	—	712	4	no	—
302	C₄	#19151803	1	2	yes	—
303	C₄	#19151834	1	2	yes	—
304	C₄	C₂²×D₇₁₂	1	1	yes	characteristic
305	C₄	C₂²×#13083562	1	1	yes	characteristic
306	C₄	C₂×#15668034	1	1	yes	characteristic
307	C₄	C₂×#15668042	1	1	yes	characteristic
308	C₂	—	4	4	no	—
309	C₂	#26019641	1	4	yes	direct product factor
310	C₂	C₂×#19151803	1	1	yes	characteristic
311	C₂	C₂×#19151834	1	1	yes	characteristic
312	C₂	C₂×#19151873	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15667986
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×C₈×D₄×C₈₉	C₂
socle	C₂³×C₈₉	#121
prime radical	C₂²×D₄×C₈₉	D₄
hypercenter	C₂×C₈×D₄	D₈₉
hypocenter	C₈₉	C₂×#1673
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#19152003	C₂	semidirect
3. #26019641	C₂	direct (N × H)
4. C₂×#19151351	C₂	semidirect
5. C₂×#19151343	C₂	semidirect
10. #19152003	C₂²	semidirect
17. #19151351	C₂²	semidirect
18. #19151343	C₂²	semidirect
89. C₂×C₈×C₈₉	SD₁₆	semidirect
131. C₈×C₈₉	C₂×SD₁₆	semidirect
153. C₂×C₈₉	#1673	semidirect
163. C₈₉	C₂×#1673	semidirect
242. C₂×C₈	#13083562	semidirect
289. C₈	C₂×#13083562	semidirect
309. C₂	#26019641	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×#19152003	2
C₂×#19151803	2
#26019641	2
C₂×#19151351	2
C₂×#19151343	2
C₂×C₈×D₄×C₈₉	2
C₂×#1673	89

Subgroups — C2×#26019641 #36536685

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26019641 #36536685