Subgroups — C₂×#26022355 #36708874

Order 22784.

786 conjugacy classes of subgroups.

The subgroup lattice has 256 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	#26022355	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19151426	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19151400	C₂	1	2	yes	maximal, complemented
6	C₂²×M₃₂×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	#19151426	C₂²	1	4	yes	complemented
16	#19151400	C₂²	1	8	yes	complemented
17	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×M₃₂×C₈₉	—	2	4	no	—
19	C₂×M₃₂×C₈₉	C₂²	1	4	yes	complemented
20	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
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	M₃₂×C₈₉	—	2	8	no	—
44	C₂×C₁₆×C₈₉	—	2	2	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	4	no	—
82	C₂³×C₈₉	C₂×D₈	1	2	yes	—
83	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
84	C₂²×D₈₉	—	4	4	no	—
85	C₂²×D₈₉	—	8	4	no	—
86	C₂²×D₈₉	—	8	4	no	—
87	C₂²×D₈₉	—	8	4	no	—
88	C₂×C₄×C₈₉	—	2	2	no	—
89	C₂×C₄×C₈₉	—	2	2	no	—
90	C₂×C₄×C₈₉	—	2	2	no	—
91	C₂×C₄×C₈₉	—	2	4	no	—
92	C₂×C₄×C₈₉	#121	1	2	yes	—
93	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
94	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₈₉	#135	1	2	yes	—
96	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
97	C₂²⋊D₈₉	—	8	16	no	—
98	C₂×Dic₈₉	—	4	4	no	—
99	C₂×Dic₈₉	—	8	4	no	—
100	D₃₅₆	—	4	4	no	—
101	D₃₅₆	—	4	4	no	—
102	D₃₅₆	—	8	4	no	—
103	C₈×C₈₉	—	2	2	no	—
104	C₈×C₈₉	#121	1	2	yes	—
105	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
106	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
107	C₂²×C₈₉	—	2	4	no	—
108	C₂²×C₈₉	—	2	2	no	—
109	C₂²×C₈₉	—	4	4	no	—
110	C₂²×C₈₉	#466	1	2	yes	—
111	C₂²×C₈₉	#509	1	4	yes	—
112	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
113	C₂×D₈₉	—	8	4	no	—
114	C₂×D₈₉	—	8	4	no	—
115	C₂×D₈₉	—	8	4	no	—
116	C₂×D₈₉	—	16	8	no	—
117	C₄×C₈₉	—	2	2	no	—
118	C₄×C₈₉	#466	1	2	yes	—
119	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
120	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
121	Dic₈₉	—	8	4	no	—
122	C₂×#2218	—	89	1	no	maximal, Sylow
123	C₂×C₈₉	—	4	2	no	—
124	C₂×C₈₉	#2218	1	4	yes	complemented
125	C₂×C₈₉	C₂×#466	1	1	yes	characteristic
126	C₂×C₈₉	C₂×#509	1	2	yes	—
127	D₈₉	—	16	4	no	—
128	C₂²×D₁₆	—	89	1	no	—
129	C₂²×M₃₂	D₈₉	1	1	yes	characteristic, complemented
130	C₂×#466	—	89	2	no	—
131	#2218	—	89	8	no	—
132	C₂×#367	—	89	1	no	—
133	C₂×#357	—	89	2	no	—
134	C₈₉	C₂×#2218	1	1	yes	characteristic, complemented, Sylow
135	C₂²×D₈	—	89	2	no	—
136	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
137	C₂×#121	—	178	2	no	—
138	C₂×D₁₆	—	89	4	no	—
139	C₂×D₁₆	—	178	4	no	—
140	C₂×M₃₂	—	2	4	no	—
141	C₂×M₃₂	C₂×D₈₉	1	4	yes	complemented
142	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
143	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
144	#466	—	89	8	no	—
145	#466	—	178	4	no	—
146	C₂×#107	—	89	2	no	—
147	C₂×#102	—	178	2	no	—
148	#367	—	89	4	no	—
149	#357	—	89	8	no	—
150	C₂²×D₄	—	356	2	no	—
151	C₂²×D₄	—	178	2	no	—
152	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
153	C₂×D₈	—	89	4	no	—
154	C₂×D₈	—	178	4	no	—
155	C₂×D₈	—	178	2	no	—
156	C₂×D₈	—	178	4	no	—
157	C₂²×C₈	—	2	2	no	—
158	C₂²×C₈	—	2	2	no	—
159	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
160	C₂²×C₈	D₃₅₆	1	2	yes	—
161	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
162	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
163	#121	—	356	4	no	—
164	#121	—	178	8	no	—
165	C₂×C₄⋊C₄	—	178	2	no	—
166	C₂×C₂²⋊C₄	—	356	2	no	—
167	D₁₆	—	178	8	no	—
168	M₃₂	—	2	8	no	—
169	C₂×C₁₆	—	2	2	no	—
170	C₂×C₁₆	C₂²⋊D₈₉	1	4	yes	complemented
171	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
172	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
173	#107	—	89	4	no	—
174	#107	—	178	2	no	—
175	#102	—	178	8	no	—
176	C₂⁴	—	356	2	no	—
177	C₂⁴	D₇₁₂	1	1	yes	characteristic
178	C₂×D₄	—	356	8	no	—
179	C₂×D₄	—	356	4	no	—
180	C₂×D₄	—	356	2	no	—
181	C₂×D₄	—	356	4	no	—
182	C₂×D₄	—	178	4	no	—
183	C₂×D₄	—	712	8	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	356	2	no	—
187	C₂²×C₄	D₇₁₂	1	2	yes	—
188	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
189	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
190	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
191	D₈	—	356	4	no	—
192	D₈	—	178	4	no	—
193	D₈	—	178	4	no	—
194	C₂×C₈	—	2	2	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	2	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
199	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
200	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
201	C₂×C₈	#13083574	1	2	yes	—
202	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
203	C₄⋊C₄	—	356	2	no	—
204	C₄⋊C₄	—	178	4	no	—
205	C₂²⋊C₄	—	356	8	no	—
206	C₁₆	—	2	2	no	—
207	C₁₆	C₂×C₂²⋊D₈₉	1	4	yes	complemented
208	C₂³	—	2	4	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	356	4	no	—
211	C₂³	—	712	4	no	—
212	C₂³	—	712	4	no	—
213	C₂³	—	712	4	no	—
214	C₂³	C₂×D₇₁₂	1	2	yes	—
215	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
216	D₄	—	356	4	no	—
217	D₄	—	356	4	no	—
218	D₄	—	712	16	no	—
219	D₄	—	712	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	356	4	no	—
225	C₂×C₄	—	712	4	no	—
226	C₂×C₄	#15667986	1	2	yes	—
227	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
228	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
229	C₂×C₄	#15668034	1	2	yes	—
230	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	#15667986	1	2	yes	—
233	C₈	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
234	C₈	C₂×#13083574	1	1	yes	characteristic
235	C₂²	—	2	4	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	4	no	—
238	C₂²	—	1424	8	no	—
239	C₂²	—	712	4	no	—
240	C₂²	—	712	4	no	—
241	C₂²	—	712	4	no	—
242	C₂²	#19151834	1	2	yes	—
243	C₂²	#19152036	1	4	yes	—
244	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
245	C₄	—	2	2	no	—
246	C₄	—	712	4	no	—
247	C₄	#19151834	1	2	yes	—
248	C₄	C₂×#15667986	1	1	yes	characteristic
249	C₄	C₂×#15668034	1	1	yes	characteristic
250	C₂	—	4	2	no	—
251	C₂	—	1424	4	no	—
252	C₂	#26022355	1	4	yes	direct product factor
253	C₂	C₂×#19151834	1	1	yes	characteristic
254	C₂	C₂×#19152036	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂²×M₃₂×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
hypercenter	C₂²×M₃₂	D₈₉
hypocenter	C₈₉	C₂×#2218
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

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. #26022355	C₂	direct (N × H)
4. C₂×#19151426	C₂	semidirect
5. C₂×#19151400	C₂	semidirect
6. C₂²×M₃₂×C₈₉	C₂	semidirect
10. C₂×D₁₄₂₄	C₂²	semidirect
12. #19151834	C₂²	semidirect
15. #19151426	C₂²	semidirect
16. #19151400	C₂²	semidirect
19. C₂×M₃₂×C₈₉	C₂²	semidirect
45. C₂×C₁₆×C₈₉	D₄	semidirect
79. C₁₆×C₈₉	C₂×D₄	semidirect
124. C₂×C₈₉	#2218	semidirect
129. C₂²×M₃₂	D₈₉	semidirect
134. C₈₉	C₂×#2218	semidirect
141. C₂×M₃₂	C₂×D₈₉	semidirect
170. C₂×C₁₆	C₂²⋊D₈₉	semidirect
207. C₁₆	C₂×C₂²⋊D₈₉	semidirect
252. C₂	#26022355	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
#26022355	2
C₂×#19151426	2
C₂×#19151400	2
C₂²×M₃₂×C₈₉	2
C₂×#2218	89

Subgroups — C2×#26022355 #36708874

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26022355 #36708874