Subgroups — C₂×#26022489 #36709735

Order 22784.

634 conjugacy classes of subgroups.

The subgroup lattice has 268 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×#19151982	C₂	1	1	yes	maximal, characteristic
3	#26022489	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151403	C₂	1	2	yes	maximal, complemented
5	C₂×#19151400	C₂	1	2	yes	maximal
6	C₂×#346×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₃₅₆	C₂²	1	1	yes	characteristic
8	C₂²×#13083563	—	2	1	no	—
9	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
10	C₂×#15668012	—	2	1	no	—
11	C₂×#15668000	—	2	1	no	—
12	#19152003	—	2	2	no	—
13	#19152003	C₂²	1	4	yes	complemented
14	#19151982	—	2	2	no	—
15	#19151982	C₂²	1	4	yes	—
16	C₂×#15667946	—	2	1	no	—
17	C₂×#15667946	C₂²	1	2	yes	—
18	#19151403	C₂²	1	8	yes	complemented
19	#19151400	C₂²	1	8	yes	—
20	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	#346×C₈₉	C₂²	1	4	yes	complemented
23	C₂²×Dic₁₇₈	—	2	1	no	—
24	C₂²×Dic₁₇₈	—	4	1	no	—
25	C₂²×D₃₅₆	—	2	1	no	—
26	C₂×Dic₃₅₆	—	2	1	no	—
27	C₂×Dic₃₅₆	—	2	4	no	—
28	C₂×Dic₃₅₆	C₂³	1	2	yes	—
29	C₂×#13083563	—	2	4	no	—
30	C₂×#13083563	—	4	4	no	—
31	C₂×D₇₁₂	—	2	1	no	—
32	C₂×D₇₁₂	—	2	4	no	—
33	C₂×D₇₁₂	C₂³	1	2	yes	—
34	#15668012	—	2	4	no	—
35	#15668012	—	4	2	no	—
36	#15668000	—	2	4	no	—
37	#15668000	—	4	2	no	—
38	C₂×C₄⋊Dic₈₉	—	2	2	no	—
39	C₂×C₄⋊Dic₈₉	—	4	1	no	—
40	C₂×#13083551	—	4	1	no	—
41	C₂×#13083551	—	4	1	no	—
42	#15667946	—	2	4	no	—
43	#15667946	—	2	2	no	—
44	#15667946	C₂³	1	4	yes	—
45	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
48	C₂×C₁₆×C₈₉	—	2	4	no	—
49	C₂×C₁₆×C₈₉	C₂³	1	4	yes	—
50	C₄×C₈×C₈₉	D₄	1	2	yes	—
51	C₄×C₈×C₈₉	C₂³	1	2	yes	—
52	C₂³×D₈₉	—	4	1	no	—
53	C₂×Dic₁₇₈	—	2	2	no	—
54	C₂×Dic₁₇₈	—	4	2	no	—
55	C₂×Dic₁₇₈	—	4	2	no	—
56	C₂×Dic₁₇₈	—	4	1	no	—
57	C₂×Dic₁₇₈	—	4	4	no	—
58	C₂×Dic₁₇₈	—	8	4	no	—
59	C₂×D₃₅₆	—	2	2	no	—
60	C₂×D₃₅₆	—	4	2	no	—
61	C₂×D₃₅₆	—	4	2	no	—
62	C₂×D₃₅₆	—	4	1	no	—
63	C₂²×Dic₈₉	—	4	1	no	—
64	C₂²×Dic₈₉	—	4	2	no	—
65	Dic₃₅₆	—	2	4	no	—
66	Dic₃₅₆	—	4	2	no	—
67	#13083563	—	4	8	no	—
68	D₇₁₂	—	2	4	no	—
69	D₇₁₂	—	4	2	no	—
70	C₄⋊Dic₈₉	—	2	4	no	—
71	C₄⋊Dic₈₉	—	4	4	no	—
72	C₄⋊Dic₈₉	—	4	2	no	—
73	#13083551	—	4	4	no	—
74	#13083551	—	4	4	no	—
75	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₈₉	SD₁₆	1	4	yes	—
79	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₈₉	#41	1	4	yes	—
82	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
83	C₄²×C₈₉	D₈	1	2	yes	—
84	C₄²×C₈₉	C₂×D₄	1	2	yes	—
85	C₁₆×C₈₉	—	2	4	no	—
86	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
87	C₂²×D₈₉	—	4	2	no	—
88	C₂²×D₈₉	—	8	2	no	—
89	C₂²×D₈₉	—	8	2	no	—
90	C₂²×D₈₉	—	8	2	no	—
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	2	2	no	—
93	C₂×C₄×C₈₉	#124	1	2	yes	—
94	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
98	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
99	C₂×Dic₈₉	—	4	2	no	—
100	C₂×Dic₈₉	—	4	4	no	—
101	C₂×Dic₈₉	—	8	2	no	—
102	C₂×Dic₈₉	—	8	4	no	—
103	D₃₅₆	—	4	2	no	—
104	D₃₅₆	—	4	2	no	—
105	D₃₅₆	—	8	2	no	—
106	Dic₁₇₈	—	4	2	no	—
107	Dic₁₇₈	—	4	2	no	—
108	Dic₁₇₈	—	8	8	no	—
109	Dic₁₇₈	—	8	2	no	—
110	C₈×C₈₉	#124	1	2	yes	—
111	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
112	C₈×C₈₉	C₂×#41	1	2	yes	—
113	C₂²×C₈₉	#486	1	2	yes	—
114	C₂²×C₈₉	#509	1	2	yes	—
115	C₂²×C₈₉	#510	1	2	yes	—
116	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
117	C₂×D₈₉	—	8	2	no	—
118	C₂×D₈₉	—	8	2	no	—
119	C₂×D₈₉	—	8	2	no	—
120	C₂×D₈₉	—	16	4	no	—
121	C₄×C₈₉	—	2	2	no	—
122	C₄×C₈₉	—	2	2	no	—
123	C₄×C₈₉	#486	1	2	yes	—
124	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
125	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
126	Dic₈₉	—	8	2	no	—
127	Dic₈₉	—	8	4	no	—
128	C₂×#2243	—	89	1	no	maximal, Sylow
129	C₂×C₈₉	#2243	1	4	yes	complemented
130	C₂×C₈₉	C₂×#486	1	1	yes	characteristic
131	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
132	C₂×C₈₉	C₂×#510	1	1	yes	characteristic
133	D₈₉	—	16	2	no	—
134	C₂×#500	—	89	1	no	—
135	C₂×#495	—	89	1	no	—
136	#2243	—	89	8	no	—
137	C₂×#358	—	89	2	no	—
138	C₂×#357	—	89	2	no	—
139	C₂×#346	D₈₉	1	1	yes	characteristic, complemented
140	C₈₉	C₂×#2243	1	1	yes	characteristic, complemented, Sylow
141	C₂²×Q₁₆	—	89	1	no	—
142	C₂²×SD₁₆	—	178	1	no	—
143	C₂²×D₈	—	89	1	no	—
144	C₂×C₄⋊Q₈	—	178	1	no	—
145	C₂×#124	—	178	1	no	—
146	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
147	#500	—	89	4	no	—
148	#500	—	178	2	no	—
149	#495	—	89	4	no	—
150	#495	—	178	2	no	—
151	C₂×#107	—	89	2	no	—
152	C₂×#107	—	178	1	no	—
153	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
154	#358	—	89	8	no	—
155	#357	—	89	8	no	—
156	#346	C₂×D₈₉	1	4	yes	complemented
157	C₂²×Q₈	—	356	1	no	—
158	C₂²×Q₈	—	178	1	no	—
159	C₂²×D₄	—	178	1	no	—
160	C₂×Q₁₆	—	89	2	no	—
161	C₂×Q₁₆	—	178	1	no	—
162	C₂×Q₁₆	—	178	4	no	—
163	C₂×SD₁₆	—	356	4	no	—
164	C₂×SD₁₆	—	178	4	no	—
165	C₂×D₈	—	89	2	no	—
166	C₂×D₈	—	178	1	no	—
167	C₂×D₈	—	178	4	no	—
168	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
169	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
170	C₄⋊Q₈	—	356	2	no	—
171	C₄⋊Q₈	—	178	4	no	—
172	#124	—	356	2	no	—
173	#124	—	178	4	no	—
174	C₂×C₄⋊C₄	—	356	1	no	—
175	C₂×C₄⋊C₄	—	178	2	no	—
176	C₂×C₂²⋊C₄	—	356	1	no	—
177	C₂×C₂²⋊C₄	—	356	1	no	—
178	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
179	C₂×C₁₆	—	2	4	no	—
180	C₂×C₁₆	C₂²×D₈₉	1	4	yes	—
181	#107	—	89	4	no	—
182	#107	—	178	2	no	—
183	#107	—	178	4	no	—
184	C₄×C₈	D₃₅₆	1	2	yes	—
185	C₄×C₈	C₂²×D₈₉	1	2	yes	—
186	C₂⁴	—	356	1	no	—
187	C₂×Q₈	—	356	1	no	—
188	C₂×Q₈	—	356	4	no	—
189	C₂×Q₈	—	356	2	no	—
190	C₂×Q₈	—	356	2	no	—
191	C₂×Q₈	—	178	2	no	—
192	C₂×Q₈	—	712	4	no	—
193	C₂×D₄	—	356	2	no	—
194	C₂×D₄	—	356	1	no	—
195	C₂×D₄	—	356	2	no	—
196	C₂×D₄	—	178	2	no	—
197	C₂²×C₄	—	356	1	no	—
198	C₂²×C₄	—	356	2	no	—
199	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
200	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
202	Q₁₆	—	356	2	no	—
203	Q₁₆	—	178	4	no	—
204	SD₁₆	—	356	8	no	—
205	D₈	—	356	2	no	—
206	D₈	—	178	4	no	—
207	C₂×C₈	#13083563	1	4	yes	—
208	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
209	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
210	C₂×C₈	#13083574	1	4	yes	—
211	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
212	C₄⋊C₄	—	356	4	no	—
213	C₄⋊C₄	—	356	2	no	—
214	C₄⋊C₄	—	178	4	no	—
215	C₂²⋊C₄	—	356	4	no	—
216	C₂²⋊C₄	—	356	4	no	—
217	C₄²	D₇₁₂	1	2	yes	—
218	C₄²	C₂×D₃₅₆	1	2	yes	—
219	C₁₆	—	2	4	no	—
220	C₂³	—	356	2	no	—
221	C₂³	—	712	2	no	—
222	C₂³	—	712	2	no	—
223	C₂³	—	712	2	no	—
224	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
225	Q₈	—	356	2	no	—
226	Q₈	—	356	2	no	—
227	Q₈	—	712	8	no	—
228	Q₈	—	712	2	no	—
229	D₄	—	356	2	no	—
230	D₄	—	356	2	no	—
231	D₄	—	712	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	356	2	no	—
235	C₂×C₄	—	356	4	no	—
236	C₂×C₄	—	712	2	no	—
237	C₂×C₄	—	712	4	no	—
238	C₂×C₄	#15668000	1	2	yes	—
239	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
240	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
241	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
242	C₂×C₄	C₂×#13083563	1	2	yes	—
243	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
244	C₈	#15668000	1	2	yes	—
245	C₈	C₂×#13083563	1	4	yes	—
246	C₈	C₂×#13083574	1	2	yes	—
247	C₂²	—	1424	4	no	—
248	C₂²	—	712	2	no	—
249	C₂²	—	712	2	no	—
250	C₂²	—	712	2	no	—
251	C₂²	#19151945	1	2	yes	—
252	C₂²	#19152036	1	2	yes	—
253	C₂²	#19152041	1	2	yes	—
254	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	712	2	no	—
258	C₄	—	712	4	no	—
259	C₄	#19151945	1	2	yes	—
260	C₄	C₂×#15668000	1	1	yes	characteristic
261	C₄	C₂²×#13083563	1	1	yes	characteristic
262	C₂	—	1424	2	no	—
263	C₂	#26022489	1	4	yes	direct product factor
264	C₂	C₂×#19151945	1	1	yes	characteristic
265	C₂	C₂×#19152036	1	1	yes	characteristic
266	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×#346×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂²×D₇₁₂	C₂²
hypercenter	C₂×#346	D₈₉
hypocenter	C₈₉	C₂×#2243
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
1. C₂×#19152003	C₂	semidirect
3. #26022489	C₂	direct (N × H)
4. C₂×#19151403	C₂	semidirect
6. C₂×#346×C₈₉	C₂	semidirect
13. #19152003	C₂²	semidirect
18. #19151403	C₂²	semidirect
22. #346×C₈₉	C₂²	semidirect
129. C₂×C₈₉	#2243	semidirect
139. C₂×#346	D₈₉	semidirect
140. C₈₉	C₂×#2243	semidirect
156. #346	C₂×D₈₉	semidirect
263. C₂	#26022489	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×#19151982	2
#26022489	2
C₂×#19151403	2
C₂×#19151400	2
C₂×#346×C₈₉	2
C₂×#2243	89

Subgroups — C2×#26022489 #36709735

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022489 #36709735