Subgroups — C₂×#26022338 #36708846

Order 22784.

806 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151834	C₂	1	2	yes	maximal, complemented
3	#26022338	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151424	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151400	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₇₁₂	C₂²	1	2	yes	—
8	C₂×#15667986	—	2	2	no	—
9	C₂×D₁₄₂₄	—	2	4	no	—
10	C₂×D₁₄₂₄	C₂²	1	4	yes	complemented
11	#19151834	—	2	4	no	—
12	#19151834	C₂²	1	8	yes	complemented
13	C₂×#15667946	C₂²	1	2	yes	—
14	C₂×#15667934	—	2	2	no	—
15	#19151424	C₂²	1	4	yes	complemented
16	#19151400	C₂²	1	8	yes	complemented
17	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₈₉	—	2	4	no	—
19	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₈₉	C₂²	1	4	yes	complemented
22	C₂²×D₃₅₆	—	2	2	no	—
23	C₂²×C₂²⋊D₈₉	—	4	2	no	—
24	C₂×D₇₁₂	—	2	4	no	—
25	C₂×D₇₁₂	—	2	2	no	—
26	C₂×D₇₁₂	—	2	4	no	—
27	C₂×D₇₁₂	C₂³	1	4	yes	—
28	#15667986	—	2	8	no	—
29	#15667986	—	4	4	no	—
30	C₂×C₄⋊Dic₈₉	—	2	2	no	—
31	C₂×#13083551	—	4	2	no	—
32	D₁₄₂₄	—	2	8	no	—
33	#15667946	—	2	2	no	—
34	#15667946	C₂³	1	4	yes	—
35	#15667934	—	2	8	no	—
36	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₉	—	2	2	no	—
38	C₂²×C₈×C₈₉	—	2	2	no	—
39	C₂²×C₈×C₈₉	D₄	1	2	yes	—
40	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
42	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
43	C₂×C₁₆×C₈₉	—	2	2	no	—
44	C₂×C₁₆×C₈₉	—	2	8	no	—
45	C₂×C₁₆×C₈₉	D₄	1	4	yes	complemented
46	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
47	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
48	C₂³×D₈₉	—	4	2	no	—
49	C₂×D₃₅₆	—	2	4	no	—
50	C₂×D₃₅₆	—	4	4	no	—
51	C₂×D₃₅₆	—	4	4	no	—
52	C₂×D₃₅₆	—	4	2	no	—
53	C₂×C₂²⋊D₈₉	—	4	8	no	—
54	C₂×C₂²⋊D₈₉	—	8	8	no	—
55	C₂²×Dic₈₉	—	4	2	no	—
56	D₇₁₂	—	2	4	no	—
57	D₇₁₂	—	2	4	no	—
58	D₇₁₂	—	4	4	no	—
59	C₄⋊Dic₈₉	—	2	4	no	—
60	C₄⋊Dic₈₉	—	4	2	no	—
61	#13083551	—	4	8	no	—
62	C₂⁴×C₈₉	D₈	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	—	2	2	no	—
64	C₂²×C₄×C₈₉	—	2	2	no	—
65	C₂²×C₄×C₈₉	D₈	1	2	yes	—
66	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
67	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₈₉	—	2	2	no	—
70	C₂×C₈×C₈₉	—	2	4	no	—
71	C₂×C₈×C₈₉	—	2	2	no	—
72	C₂×C₈×C₈₉	—	2	2	no	—
73	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₈₉	#41	1	2	yes	—
77	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₁₆×C₈₉	—	2	2	no	—
79	C₁₆×C₈₉	C₂×D₄	1	4	yes	complemented
80	C₂³×C₈₉	—	2	2	no	—
81	C₂³×C₈₉	—	2	2	no	—
82	C₂³×C₈₉	D₁₆	1	4	yes	complemented
83	C₂³×C₈₉	C₂×D₈	1	2	yes	—
84	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
85	C₂²×D₈₉	—	4	4	no	—
86	C₂²×D₈₉	—	8	4	no	—
87	C₂²×D₈₉	—	8	4	no	—
88	C₂²×D₈₉	—	8	4	no	—
89	C₂×C₄×C₈₉	—	2	2	no	—
90	C₂×C₄×C₈₉	—	2	2	no	—
91	C₂×C₄×C₈₉	—	2	2	no	—
92	C₂×C₄×C₈₉	—	2	4	no	—
93	C₂×C₄×C₈₉	#121	1	2	yes	—
94	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
95	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₈₉	#135	1	2	yes	—
97	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
98	C₂²⋊D₈₉	—	8	16	no	—
99	C₂×Dic₈₉	—	4	4	no	—
100	C₂×Dic₈₉	—	8	4	no	—
101	D₃₅₆	—	4	4	no	—
102	D₃₅₆	—	4	4	no	—
103	D₃₅₆	—	8	4	no	—
104	C₈×C₈₉	—	2	2	no	—
105	C₈×C₈₉	#121	1	2	yes	—
106	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
108	C₂²×C₈₉	—	2	2	no	—
109	C₂²×C₈₉	—	2	8	no	—
110	C₂²×C₈₉	—	2	2	no	—
111	C₂²×C₈₉	#466	1	2	yes	—
112	C₂²×C₈₉	C₂×D₁₆	1	2	yes	—
113	C₂²×C₈₉	C₂×D₁₆	1	4	yes	complemented
114	C₂²×C₈₉	#508	1	2	yes	—
115	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
116	C₂×D₈₉	—	8	4	no	—
117	C₂×D₈₉	—	8	4	no	—
118	C₂×D₈₉	—	8	4	no	—
119	C₂×D₈₉	—	16	8	no	—
120	C₄×C₈₉	—	2	2	no	—
121	C₄×C₈₉	#466	1	2	yes	—
122	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
123	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
124	Dic₈₉	—	8	4	no	—
125	C₂×#2215	—	89	1	no	maximal, Sylow
126	C₂×C₈₉	—	2	4	no	—
127	C₂×C₈₉	#2215	1	4	yes	complemented
128	C₂×C₈₉	C₂×#466	1	1	yes	characteristic
129	C₂×C₈₉	C₂²×D₁₆	1	1	yes	characteristic
130	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
131	D₈₉	—	16	4	no	—
132	C₂²×D₁₆	—	89	1	no	—
133	C₂³×C₁₆	D₈₉	1	1	yes	characteristic, complemented
134	C₂×#466	—	89	2	no	—
135	#2215	—	89	8	no	—
136	C₂×#366	—	89	1	no	—
137	C₂×#357	—	89	2	no	—
138	C₈₉	C₂×#2215	1	1	yes	characteristic, complemented, Sylow
139	C₂²×D₈	—	89	2	no	—
140	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
141	C₂×#121	—	178	2	no	—
142	C₂×D₁₆	—	89	4	no	—
143	C₂×D₁₆	—	178	4	no	—
144	C₂²×C₁₆	—	2	4	no	—
145	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
146	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
147	C₂²×C₁₆	C₂×D₈₉	1	4	yes	complemented
148	#466	—	89	8	no	—
149	#466	—	178	4	no	—
150	C₂×#107	—	89	2	no	—
151	C₂×#102	—	178	2	no	—
152	#366	—	89	4	no	—
153	#357	—	89	8	no	—
154	C₂²×D₄	—	356	2	no	—
155	C₂²×D₄	—	178	2	no	—
156	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
157	C₂×D₈	—	89	4	no	—
158	C₂×D₈	—	178	4	no	—
159	C₂×D₈	—	178	2	no	—
160	C₂×D₈	—	178	4	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
164	C₂²×C₈	D₃₅₆	1	2	yes	—
165	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
166	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
167	#121	—	356	4	no	—
168	#121	—	178	8	no	—
169	C₂×C₄⋊C₄	—	178	2	no	—
170	C₂×C₂²⋊C₄	—	356	2	no	—
171	D₁₆	—	178	8	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	2	8	no	—
174	C₂×C₁₆	C₂²⋊D₈₉	1	4	yes	complemented
175	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
176	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
177	#107	—	89	4	no	—
178	#107	—	178	2	no	—
179	#102	—	178	8	no	—
180	C₂⁴	—	356	2	no	—
181	C₂⁴	D₇₁₂	1	1	yes	characteristic
182	C₂×D₄	—	356	8	no	—
183	C₂×D₄	—	356	4	no	—
184	C₂×D₄	—	356	2	no	—
185	C₂×D₄	—	356	4	no	—
186	C₂×D₄	—	178	4	no	—
187	C₂×D₄	—	712	8	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	356	2	no	—
191	C₂²×C₄	D₇₁₂	1	2	yes	—
192	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
194	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
195	D₈	—	356	4	no	—
196	D₈	—	178	4	no	—
197	D₈	—	178	4	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
203	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
204	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
205	C₂×C₈	#13083574	1	2	yes	—
206	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
207	C₄⋊C₄	—	356	2	no	—
208	C₄⋊C₄	—	178	4	no	—
209	C₂²⋊C₄	—	356	8	no	—
210	C₁₆	—	2	2	no	—
211	C₁₆	C₂×C₂²⋊D₈₉	1	4	yes	complemented
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	356	4	no	—
215	C₂³	—	712	4	no	—
216	C₂³	—	712	4	no	—
217	C₂³	—	712	4	no	—
218	C₂³	D₁₄₂₄	1	4	yes	complemented
219	C₂³	C₂×D₇₁₂	1	2	yes	—
220	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
221	D₄	—	356	4	no	—
222	D₄	—	356	4	no	—
223	D₄	—	712	16	no	—
224	D₄	—	712	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	356	4	no	—
230	C₂×C₄	—	712	4	no	—
231	C₂×C₄	#15667986	1	2	yes	—
232	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
233	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
234	C₂×C₄	#15668034	1	2	yes	—
235	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	#15667986	1	2	yes	—
238	C₈	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
239	C₈	C₂×#13083574	1	1	yes	characteristic
240	C₂²	—	2	8	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	1424	8	no	—
244	C₂²	—	712	4	no	—
245	C₂²	—	712	4	no	—
246	C₂²	—	712	4	no	—
247	C₂²	#19151834	1	2	yes	—
248	C₂²	C₂×D₁₄₂₄	1	2	yes	—
249	C₂²	C₂×D₁₄₂₄	1	4	yes	complemented
250	C₂²	#19152030	1	2	yes	—
251	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	712	4	no	—
254	C₄	#19151834	1	2	yes	—
255	C₄	C₂×#15667986	1	1	yes	characteristic
256	C₄	C₂×#15668034	1	1	yes	characteristic
257	C₂	—	2	4	no	—
258	C₂	—	1424	4	no	—
259	C₂	#26022338	1	4	yes	direct product factor
260	C₂	C₂×#19151834	1	1	yes	characteristic
261	C₂	C₂²×D₁₄₂₄	1	1	yes	characteristic
262	C₂	C₂×#19152030	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂³×C₁₆×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
hypercenter	C₂³×C₁₆	D₈₉
hypocenter	C₈₉	C₂×#2215
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₄₂₄	C₂	semidirect
2. C₂×#19151834	C₂	semidirect
3. #26022338	C₂	direct (N × H)
4. C₂×#19151424	C₂	semidirect
5. C₂×#19151400	C₂	semidirect
6. C₂³×C₁₆×C₈₉	C₂	semidirect
10. C₂×D₁₄₂₄	C₂²	semidirect
12. #19151834	C₂²	semidirect
15. #19151424	C₂²	semidirect
16. #19151400	C₂²	semidirect
21. C₂²×C₁₆×C₈₉	C₂²	semidirect
45. C₂×C₁₆×C₈₉	D₄	semidirect
79. C₁₆×C₈₉	C₂×D₄	semidirect
82. C₂³×C₈₉	D₁₆	semidirect
113. C₂²×C₈₉	C₂×D₁₆	semidirect
127. C₂×C₈₉	#2215	semidirect
133. C₂³×C₁₆	D₈₉	semidirect
138. C₈₉	C₂×#2215	semidirect
147. C₂²×C₁₆	C₂×D₈₉	semidirect
174. C₂×C₁₆	C₂²⋊D₈₉	semidirect
211. C₁₆	C₂×C₂²⋊D₈₉	semidirect
218. C₂³	D₁₄₂₄	semidirect
249. C₂²	C₂×D₁₄₂₄	semidirect
259. C₂	#26022338	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×D₁₄₂₄	2
C₂×#19151834	2
#26022338	2
C₂×#19151424	2
C₂×#19151400	2
C₂³×C₁₆×C₈₉	2
C₂×#2215	89

Subgroups — C2×#26022338 #36708846

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022338 #36708846