Subgroups — C₂×#26020410 #36554828

Order 22784.

1358 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667946	C₂	1	1	yes	maximal, characteristic
2	C₂²×#15667945	C₂	1	2	yes	maximal
3	C₂²×C₈×Dic₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151467	C₂	1	2	yes	maximal
5	#26020410	C₂	1	8	yes	maximal, direct product factor
6	C₂²×#107×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂³×#10623494	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊Dic₈₉	C₂²	1	2	yes	—
9	C₂²×#13083553	—	2	4	no	—
10	C₂²×C₄×Dic₈₉	C₂²	1	1	yes	characteristic
11	C₂×#15667946	—	2	4	no	—
12	C₂×#15667946	C₂²	1	4	yes	—
13	C₂×#15667945	—	2	8	no	—
14	C₂×#15667945	C₂²	1	8	yes	—
15	C₂×C₈×Dic₈₉	C₄	1	8	yes	complemented
16	C₂×C₈×Dic₈₉	C₂²	1	4	yes	—
17	#19151467	—	2	4	no	—
18	#19151467	C₂²	1	8	yes	—
19	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
21	C₂×#107×C₈₉	—	2	4	no	—
22	C₂×#107×C₈₉	C₂²	1	4	yes	—
23	C₂³×Dic₈₉	—	2	2	no	—
24	C₂³×Dic₈₉	D₄	1	2	yes	—
25	C₂²×#10623494	C₂×C₄	1	4	yes	—
26	C₂²×#10623494	D₄	1	4	yes	—
27	C₂²×#10623494	Q₈	1	4	yes	—
28	C₂²×#10623494	C₂³	1	2	yes	—
29	C₂×C₄⋊Dic₈₉	—	2	4	no	—
30	C₂×C₄⋊Dic₈₉	—	2	4	no	—
31	C₂×C₄⋊Dic₈₉	—	2	2	no	—
32	C₂×C₄⋊Dic₈₉	C₂³	1	4	yes	—
33	C₂×#13083553	—	2	16	no	—
34	C₂×#13083553	—	4	16	no	—
35	C₂×C₄×Dic₈₉	C₂×C₄	1	4	yes	—
36	C₂×C₄×Dic₈₉	D₄	1	2	yes	—
37	C₂×C₄×Dic₈₉	Q₈	1	4	yes	—
38	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
39	#15667946	—	2	8	no	—
40	#15667945	—	2	16	no	—
41	C₈×Dic₈₉	C₂×C₄	1	16	yes	complemented
42	C₂³×C₄×C₈₉	—	2	2	no	—
43	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
45	C₂²×C₈×C₈₉	D₄	1	4	yes	—
46	C₂²×C₈×C₈₉	Q₈	1	4	yes	—
47	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
49	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
50	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
51	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
52	#107×C₈₉	—	2	8	no	—
53	C₂²×Dic₈₉	—	2	4	no	—
54	C₂²×Dic₈₉	—	4	4	no	—
55	C₂²×Dic₈₉	—	4	8	no	—
56	C₂²×Dic₈₉	C₄⋊C₄	1	8	yes	—
57	C₂²×Dic₈₉	D₈	1	8	yes	—
58	C₂²×Dic₈₉	Q₁₆	1	8	yes	—
59	C₂²×Dic₈₉	C₂×D₄	1	4	yes	—
60	C₂×#10623494	C₄⋊C₄	1	16	yes	complemented
61	C₂×#10623494	C₂²×C₄	1	4	yes	—
62	C₂×#10623494	C₂×D₄	1	4	yes	—
63	C₂×#10623494	C₂×Q₈	1	4	yes	—
64	C₄⋊Dic₈₉	—	2	4	no	—
65	C₄⋊Dic₈₉	—	2	4	no	—
66	C₄⋊Dic₈₉	—	4	4	no	—
67	#13083553	—	4	32	no	—
68	C₄×Dic₈₉	C₄⋊C₄	1	8	yes	—
69	C₄×Dic₈₉	C₂²×C₄	1	4	yes	—
70	C₄×Dic₈₉	C₂×Q₈	1	4	yes	—
71	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	—	2	4	no	—
73	C₂²×C₄×C₈₉	—	4	4	no	—
74	C₂²×C₄×C₈₉	—	4	8	no	—
75	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
82	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₈₉	C₄⋊C₄	1	16	yes	complemented
84	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
86	C₂×C₈×C₈₉	C₂×Q₈	1	4	yes	—
87	C₄⋊C₄×C₈₉	—	2	4	no	—
88	C₄⋊C₄×C₈₉	—	2	4	no	—
89	C₄⋊C₄×C₈₉	—	4	4	no	—
90	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
91	C₂³×C₈₉	#127	1	2	yes	—
92	C₂³×C₈₉	C₄⋊Q₈	1	2	yes	—
93	C₂³×C₈₉	C₂×D₈	1	4	yes	—
94	C₂³×C₈₉	C₂×Q₁₆	1	4	yes	—
95	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	—	4	4	no	—
97	C₂×C₄×C₈₉	—	4	8	no	—
98	C₂×C₄×C₈₉	—	8	8	no	—
99	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	4	yes	—
102	C₂×C₄×C₈₉	#127	1	2	yes	—
103	C₂×C₄×C₈₉	C₄⋊Q₈	1	4	yes	—
104	C₂×C₄×C₈₉	C₄⋊Q₈	1	4	yes	—
105	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
106	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
112	C₂×Dic₈₉	—	4	4	no	—
113	C₂×Dic₈₉	—	4	8	no	—
114	C₂×Dic₈₉	—	8	8	no	—
115	C₂×Dic₈₉	#107	1	32	yes	complemented
116	C₂×Dic₈₉	C₂×C₄⋊C₄	1	8	yes	—
117	C₂×Dic₈₉	C₂×D₈	1	8	yes	—
118	C₂×Dic₈₉	C₂×Q₁₆	1	8	yes	—
119	C₈×C₈₉	C₂×C₄⋊C₄	1	8	yes	complemented
120	#10623494	C₂×C₄⋊C₄	1	8	yes	complemented
121	C₂²×C₈₉	#384	1	4	yes	—
122	C₂²×C₈₉	C₂×#107	1	8	yes	—
123	C₂²×C₈₉	#493	1	4	yes	—
124	C₂²×C₈₉	#494	1	4	yes	—
125	C₂²×C₈₉	#500	1	8	yes	—
126	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
128	C₂²×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
129	C₂²×C₈₉	C₂²×D₈	1	2	yes	—
130	C₂²×C₈₉	C₂²×Q₁₆	1	2	yes	—
131	C₄×C₈₉	—	8	4	no	—
132	C₄×C₈₉	#384	1	4	yes	—
133	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
134	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
135	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
136	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
137	Dic₈₉	—	8	4	no	—
138	Dic₈₉	C₂×#107	1	16	yes	complemented
139	C₂×#1845	—	89	1	no	maximal, Sylow
140	C₂×C₈₉	#1845	1	8	yes	complemented
141	C₂×C₈₉	C₂×#384	1	1	yes	characteristic
142	C₂×C₈₉	C₂²×#107	1	2	yes	—
143	C₂×C₈₉	C₂×#493	1	1	yes	characteristic
144	C₂×C₈₉	C₂×#494	1	1	yes	characteristic
145	C₂×C₈₉	C₂×#500	1	2	yes	—
146	C₂²×#107	—	89	1	no	—
147	C₂²×#107	—	89	2	no	—
148	C₂²×#107	D₈₉	1	1	yes	characteristic
149	C₂²×C₄×C₈	—	89	1	no	—
150	C₂×#384	—	89	2	no	—
151	#1845	—	89	8	no	—
152	C₈₉	C₂×#1845	1	1	yes	characteristic, complemented, Sylow
153	C₂³×C₈	—	89	1	no	—
154	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
155	C₂²×C₄⋊C₄	—	89	2	no	—
156	C₂²×C₄⋊C₄	—	178	4	no	—
157	C₂²×C₄⋊C₄	C₂×D₈₉	1	2	yes	—
158	C₂²×C₄²	—	89	1	no	—
159	C₂×#107	—	2	4	no	—
160	C₂×#107	—	89	4	no	—
161	C₂×#107	—	89	8	no	—
162	C₂×#107	—	178	8	no	—
163	C₂×#107	—	178	4	no	—
164	C₂×#107	C₂×D₈₉	1	4	yes	—
165	C₂×C₄×C₈	—	89	8	no	—
166	C₂×C₄×C₈	—	89	4	no	—
167	#384	—	89	8	no	—
168	#384	—	178	4	no	—
169	C₂³×C₄	—	2	2	no	—
170	C₂³×C₄	—	89	2	no	—
171	C₂³×C₄	—	178	2	no	—
172	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
173	C₂²×C₈	—	89	4	no	—
174	C₂²×C₈	—	89	4	no	—
175	C₂²×C₈	—	89	4	no	—
176	C₂²×C₈	—	89	2	no	—
177	C₂²×C₈	Dic₁₇₈	1	4	yes	—
178	C₂²×C₈	C₄×D₈₉	1	4	yes	—
179	C₂²×C₈	C₂²⋊D₈₉	1	4	yes	—
180	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	—	2	4	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	89	4	no	—
185	C₂×C₄⋊C₄	—	356	16	no	—
186	C₂×C₄⋊C₄	—	178	16	no	—
187	C₂×C₄⋊C₄	—	178	2	no	—
188	C₂×C₄⋊C₄	—	178	4	no	—
189	C₂×C₄⋊C₄	—	178	4	no	—
190	C₂×C₄⋊C₄	C₂²×D₈₉	1	4	yes	—
191	C₂×C₄²	—	89	4	no	—
192	C₂×C₄²	—	89	2	no	—
193	C₂×C₄²	—	89	4	no	—
194	C₂×C₄²	—	89	2	no	—
195	#107	—	2	8	no	—
196	#107	—	178	16	no	—
197	#107	—	178	8	no	—
198	C₄×C₈	—	89	16	no	—
199	C₂⁴	D₄×D₈₉	1	1	yes	characteristic
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	4	8	no	—
203	C₂²×C₄	—	89	8	no	—
204	C₂²×C₄	—	89	8	no	—
205	C₂²×C₄	—	89	4	no	—
206	C₂²×C₄	—	89	8	no	—
207	C₂²×C₄	—	356	4	no	—
208	C₂²×C₄	—	356	8	no	—
209	C₂²×C₄	—	178	4	no	—
210	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
211	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
212	C₂²×C₄	D₄×D₈₉	1	2	yes	—
213	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
214	C₂²×C₄	Q₈×D₈₉	1	2	yes	—
215	C₂²×C₄	Q₈×D₈₉	1	2	yes	—
216	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
217	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
218	C₂×C₈	—	89	4	no	—
219	C₂×C₈	—	89	4	no	—
220	C₂×C₈	—	89	4	no	—
221	C₂×C₈	—	89	16	no	—
222	C₂×C₈	#13083553	1	16	yes	complemented
223	C₂×C₈	C₂×C₄×D₈₉	1	4	yes	—
224	C₂×C₈	C₂×C₂²⋊D₈₉	1	4	yes	—
225	C₂×C₈	C₂×Dic₁₇₈	1	4	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	356	32	no	—
230	C₄⋊C₄	—	356	4	no	—
231	C₄⋊C₄	—	178	4	no	—
232	C₄⋊C₄	—	178	4	no	—
233	C₄²	—	89	4	no	—
234	C₄²	—	89	4	no	—
235	C₄²	—	89	8	no	—
236	C₂³	C₄⋊C₄×D₈₉	1	2	yes	—
237	C₂³	#15668008	1	2	yes	—
238	C₂³	#15668011	1	2	yes	—
239	C₂³	D₈×D₈₉	1	4	yes	—
240	C₂³	Q₁₆×D₈₉	1	4	yes	—
241	C₂³	C₂×D₄×D₈₉	1	1	yes	characteristic
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	8	no	—
244	C₂×C₄	—	8	8	no	—
245	C₂×C₄	—	89	32	no	—
246	C₂×C₄	—	89	8	no	—
247	C₂×C₄	—	89	8	no	—
248	C₂×C₄	—	89	8	no	—
249	C₂×C₄	—	356	4	no	—
250	C₂×C₄	—	356	8	no	—
251	C₂×C₄	—	712	8	no	—
252	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
253	C₂×C₄	C₄⋊C₄×D₈₉	1	4	yes	—
254	C₂×C₄	C₂×#13083553	1	4	yes	—
255	C₂×C₄	#15668008	1	2	yes	—
256	C₂×C₄	#15668010	1	4	yes	—
257	C₂×C₄	#15668011	1	4	yes	—
258	C₂×C₄	#15668011	1	2	yes	—
259	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
260	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
261	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
262	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
263	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
264	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
265	C₈	—	89	8	no	—
266	C₈	C₂×#13083553	1	8	yes	complemented
267	C₂²	#19151467	1	4	yes	—
268	C₂²	#107×D₈₉	1	8	yes	—
269	C₂²	#19151973	1	4	yes	—
270	C₂²	#19151976	1	4	yes	—
271	C₂²	#19152002	1	8	yes	—
272	C₂²	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
273	C₂²	C₂×#15668008	1	1	yes	characteristic
274	C₂²	C₂×#15668011	1	1	yes	characteristic
275	C₂²	C₂×D₈×D₈₉	1	2	yes	—
276	C₂²	C₂×Q₁₆×D₈₉	1	2	yes	—
277	C₄	—	8	4	no	—
278	C₄	—	89	16	no	—
279	C₄	—	712	4	no	—
280	C₄	#19151467	1	4	yes	—
281	C₄	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
282	C₄	C₂²×#13083553	1	1	yes	characteristic
283	C₄	C₂×#15668010	1	1	yes	characteristic
284	C₄	C₂×#15668011	1	1	yes	characteristic
285	C₂	#26020410	1	8	yes	direct product factor
286	C₂	C₂×#19151467	1	1	yes	characteristic
287	C₂	C₂×#107×D₈₉	1	2	yes	—
288	C₂	C₂×#19151973	1	1	yes	characteristic
289	C₂	C₂×#19151976	1	1	yes	characteristic
290	C₂	C₂×#19152002	1	2	yes	—

	subgroup	G/H
center	C₂⁴	D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂²×#107×C₈₉	C₂
socle, prime radical	C₂⁴×C₈₉	C₂×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂²×#107	D₈₉
hypocenter	C₈₉	C₂×#1845
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
5. #26020410	C₂	direct (N × H)
15. C₂×C₈×Dic₈₉	C₄	semidirect
41. C₈×Dic₈₉	C₂×C₄	semidirect
60. C₂×#10623494	C₄⋊C₄	semidirect
83. C₂×C₈×C₈₉	C₄⋊C₄	semidirect
115. C₂×Dic₈₉	#107	semidirect
119. C₈×C₈₉	C₂×C₄⋊C₄	semidirect
120. #10623494	C₂×C₄⋊C₄	semidirect
138. Dic₈₉	C₂×#107	semidirect
140. C₂×C₈₉	#1845	semidirect
152. C₈₉	C₂×#1845	semidirect
222. C₂×C₈	#13083553	semidirect
266. C₈	C₂×#13083553	semidirect
285. C₂	#26020410	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#15667946	2
C₂²×#15667945	2
C₂²×C₈×Dic₈₉	2
C₂×#19151467	2
#26020410	2
C₂²×#107×C₈₉	2
C₂×#1845	89

Subgroups — C2×#26020410 #36554828

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26020410 #36554828