Subgroups — C₂×D₁₆×Dic₈₉ #36707689

Order 22784.

806 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈×Dic₈₉	C₂	1	2	yes	maximal, complemented
2	D₁₆×Dic₈₉	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19151424	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19151399	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂×D₄×Dic₈₉	—	2	2	no	—
8	C₂²×#13083549	C₂²	1	1	yes	characteristic
9	D₈×Dic₈₉	—	2	4	no	—
10	D₈×Dic₈₉	C₂²	1	8	yes	complemented
11	C₂×#15667946	C₂²	1	2	yes	—
12	C₂×#15667933	—	2	2	no	—
13	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
14	#19151424	C₂²	1	4	yes	complemented
15	#19151399	C₂²	1	8	yes	complemented
16	C₁₆×Dic₈₉	C₂²	1	4	yes	complemented
17	C₂²×D₈×C₈₉	C₂²	1	2	yes	—
18	C₂×D₁₆×C₈₉	C₄	1	8	yes	complemented
19	C₂×D₁₆×C₈₉	C₂²	1	4	yes	—
20	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	C₂³×Dic₈₉	—	4	2	no	—
22	C₂²×#10623494	D₄	1	1	yes	characteristic
23	D₄×Dic₈₉	—	2	8	no	—
24	D₄×Dic₈₉	—	4	4	no	—
25	C₂×C₄⋊Dic₈₉	—	2	2	no	—
26	C₂×C₂²⋊Dic₈₉	—	4	2	no	—
27	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
28	C₂×#13083549	—	2	2	no	—
29	C₂×#13083549	C₂³	1	2	yes	—
30	#15667946	—	2	2	no	—
31	#15667946	C₂³	1	4	yes	—
32	#15667933	—	2	8	no	—
33	C₈×Dic₈₉	D₄	1	2	yes	—
34	C₈×Dic₈₉	C₂³	1	2	yes	—
35	C₂²×D₄×C₈₉	—	2	2	no	—
36	C₂×D₈×C₈₉	—	2	2	no	—
37	C₂×D₈×C₈₉	—	2	4	no	—
38	C₂×D₈×C₈₉	C₂×C₄	1	8	yes	complemented
39	C₂×D₈×C₈₉	C₂³	1	4	yes	—
40	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
41	D₁₆×C₈₉	C₂×C₄	1	16	yes	complemented
42	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
43	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
44	C₂²×Dic₈₉	—	4	2	no	—
45	C₂²×Dic₈₉	—	4	8	no	—
46	C₂²×Dic₈₉	—	8	8	no	—
47	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
48	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
49	C₂×#10623494	—	2	2	no	—
50	C₂×#10623494	C₂×D₄	1	2	yes	—
51	C₄⋊Dic₈₉	—	2	4	no	—
52	C₄⋊Dic₈₉	—	4	2	no	—
53	C₂²⋊Dic₈₉	—	4	8	no	—
54	C₄×Dic₈₉	D₈	1	2	yes	—
55	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
56	#13083549	—	2	2	no	—
57	C₂⁴×C₈₉	—	4	2	no	—
58	C₂×D₄×C₈₉	—	2	8	no	—
59	C₂×D₄×C₈₉	—	2	4	no	—
60	C₂×D₄×C₈₉	—	4	2	no	—
61	C₂×D₄×C₈₉	—	4	4	no	—
62	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
63	D₈×C₈₉	—	2	8	no	—
64	D₈×C₈₉	—	2	4	no	—
65	D₈×C₈₉	C₂²×C₄	1	8	yes	complemented
66	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₈₉	#41	1	2	yes	—
69	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
70	C₁₆×C₈₉	C₂²×C₄	1	4	yes	complemented
71	C₂³×C₈₉	—	4	4	no	—
72	C₂³×C₈₉	—	4	8	no	—
73	C₂³×C₈₉	—	8	4	no	—
74	C₂³×C₈₉	—	8	4	no	—
75	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
76	D₄×C₈₉	—	2	8	no	—
77	D₄×C₈₉	—	4	4	no	—
78	D₄×C₈₉	—	4	8	no	—
79	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
80	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₈₉	#135	1	2	yes	—
82	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
83	C₂×Dic₈₉	—	2	2	no	—
84	C₂×Dic₈₉	—	4	4	no	—
85	C₂×Dic₈₉	—	8	4	no	—
86	C₂×Dic₈₉	—	8	16	no	—
87	C₂×Dic₈₉	D₁₆	1	4	yes	direct product factor
88	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
89	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
90	C₈×C₈₉	C₄×D₄	1	2	yes	—
91	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
92	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
93	#10623494	—	2	2	no	—
94	C₂²×C₈₉	—	4	8	no	—
95	C₂²×C₈₉	—	8	4	no	—
96	C₂²×C₈₉	—	8	4	no	—
97	C₂²×C₈₉	—	8	16	no	—
98	C₂²×C₈₉	C₄×D₈	1	2	yes	—
99	C₂²×C₈₉	C₂×D₁₆	1	2	yes	—
100	C₂²×C₈₉	#508	1	2	yes	—
101	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
102	C₄×C₈₉	C₄×D₈	1	2	yes	—
103	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
104	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
105	Dic₈₉	—	2	2	no	—
106	Dic₈₉	—	8	4	no	—
107	Dic₈₉	C₂×D₁₆	1	4	yes	direct product factor
108	C₂×C₄×D₁₆	—	89	1	no	maximal, Sylow
109	C₂×C₈₉	—	8	8	no	—
110	C₂×C₈₉	C₄×D₁₆	1	4	yes	complemented
111	C₂×C₈₉	C₂×C₄×D₈	1	1	yes	characteristic
112	C₂×C₈₉	C₂²×D₁₆	1	1	yes	characteristic
113	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
114	C₂²×D₁₆	D₈₉	1	1	yes	characteristic
115	C₂×C₄×D₈	—	89	2	no	—
116	C₄×D₁₆	—	89	8	no	—
117	C₂×#366	—	89	1	no	—
118	C₂×#357	—	89	2	no	—
119	C₂×C₄×C₁₆	—	89	1	no	—
120	C₈₉	C₂×C₄×D₁₆	1	1	yes	characteristic, complemented, Sylow
121	C₂²×D₈	C₂×D₈₉	1	2	yes	—
122	C₂×C₄×D₄	—	178	2	no	—
123	C₂×D₁₆	Dic₈₉	1	8	yes	direct product factor
124	C₂×D₁₆	C₂×D₈₉	1	4	yes	—
125	C₂²×C₁₆	—	89	1	no	—
126	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
127	C₄×D₈	—	89	8	no	—
128	C₄×D₈	—	178	4	no	—
129	C₂×#107	—	89	2	no	—
130	C₂×#102	—	178	2	no	—
131	C₂×C₄×C₈	—	89	1	no	—
132	#366	—	89	4	no	—
133	#357	—	89	8	no	—
134	C₄×C₁₆	—	89	4	no	—
135	C₂²×D₄	—	2	2	no	—
136	C₂³×C₄	—	356	2	no	—
137	C₂×D₈	—	2	2	no	—
138	C₂×D₈	—	2	4	no	—
139	C₂×D₈	C₂×Dic₈₉	1	8	yes	complemented
140	C₂×D₈	C₂²×D₈₉	1	4	yes	—
141	C₂²×C₈	—	89	1	no	—
142	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
143	C₄×D₄	—	356	4	no	—
144	C₄×D₄	—	178	8	no	—
145	C₂×C₄⋊C₄	—	178	2	no	—
146	C₂×C₂²⋊C₄	—	356	2	no	—
147	C₂×C₄²	—	89	1	no	—
148	D₁₆	C₂×Dic₈₉	1	16	yes	direct product factor
149	C₂×C₁₆	—	89	2	no	—
150	C₂×C₁₆	—	178	2	no	—
151	C₂×C₁₆	C₂×Dic₈₉	1	4	yes	complemented
152	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
153	#107	—	89	4	no	—
154	#107	—	178	2	no	—
155	#102	—	178	8	no	—
156	C₄×C₈	—	89	2	no	—
157	C₄×C₈	—	89	2	no	—
158	C₂⁴	—	4	2	no	—
159	C₂×D₄	—	2	8	no	—
160	C₂×D₄	—	2	4	no	—
161	C₂×D₄	—	4	2	no	—
162	C₂×D₄	—	4	4	no	—
163	C₂²×C₄	—	89	1	no	—
164	C₂²×C₄	—	89	1	no	—
165	C₂²×C₄	—	356	8	no	—
166	C₂²×C₄	—	356	2	no	—
167	C₂²×C₄	—	712	8	no	—
168	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
169	D₈	—	2	8	no	—
170	D₈	—	2	4	no	—
171	D₈	C₂²×Dic₈₉	1	8	yes	complemented
172	C₂×C₈	—	89	2	no	—
173	C₂×C₈	—	178	2	no	—
174	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
175	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
176	C₂×C₈	#13083573	1	2	yes	—
177	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
178	C₄⋊C₄	—	356	2	no	—
179	C₄⋊C₄	—	178	4	no	—
180	C₂²⋊C₄	—	356	8	no	—
181	C₄²	—	89	2	no	—
182	C₄²	—	89	2	no	—
183	C₁₆	—	178	2	no	—
184	C₁₆	C₂²×Dic₈₉	1	4	yes	complemented
185	C₂³	—	4	8	no	—
186	C₂³	—	4	4	no	—
187	C₂³	—	8	4	no	—
188	C₂³	—	8	4	no	—
189	C₂³	D₈×D₈₉	1	1	yes	characteristic
190	D₄	—	2	8	no	—
191	D₄	—	4	8	no	—
192	D₄	—	4	4	no	—
193	C₂×C₄	—	89	4	no	—
194	C₂×C₄	—	89	2	no	—
195	C₂×C₄	—	89	2	no	—
196	C₂×C₄	—	356	4	no	—
197	C₂×C₄	—	178	2	no	—
198	C₂×C₄	—	712	16	no	—
199	C₂×C₄	—	712	4	no	—
200	C₂×C₄	D₄×Dic₈₉	1	2	yes	—
201	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
202	C₂×C₄	#15668031	1	2	yes	—
203	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
204	C₈	—	178	2	no	—
205	C₈	D₄×Dic₈₉	1	2	yes	—
206	C₈	C₂³×Dic₈₉	1	1	yes	characteristic
207	C₈	C₂×#13083573	1	1	yes	characteristic
208	C₂²	—	4	8	no	—
209	C₂²	—	8	4	no	—
210	C₂²	—	8	16	no	—
211	C₂²	—	8	4	no	—
212	C₂²	D₈×Dic₈₉	1	2	yes	—
213	C₂²	D₁₆×D₈₉	1	2	yes	—
214	C₂²	#19152027	1	2	yes	—
215	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
216	C₄	—	89	4	no	—
217	C₄	—	178	2	no	—
218	C₄	—	712	4	no	—
219	C₄	D₈×Dic₈₉	1	2	yes	—
220	C₄	C₂×D₄×Dic₈₉	1	1	yes	characteristic
221	C₄	C₂×#15668031	1	1	yes	characteristic
222	C₂	—	8	8	no	—
223	C₂	D₁₆×Dic₈₉	1	4	yes	direct product factor
224	C₂	C₂×D₈×Dic₈₉	1	1	yes	characteristic
225	C₂	C₂×D₁₆×D₈₉	1	1	yes	characteristic
226	C₂	C₂×#19152027	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting, prime radical	C₂²×D₁₆×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂²×D₁₆	D₈₉
hypocenter	C₈₉	C₂×C₄×D₁₆
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×D₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×D₈×Dic₈₉	C₂	semidirect
2. D₁₆×Dic₈₉	C₂	direct (N × H)
3. C₂×#19151424	C₂	semidirect
4. C₂×#19151399	C₂	semidirect
5. C₂×C₁₆×Dic₈₉	C₂	semidirect
10. D₈×Dic₈₉	C₂²	semidirect
14. #19151424	C₂²	semidirect
15. #19151399	C₂²	semidirect
16. C₁₆×Dic₈₉	C₂²	semidirect
18. C₂×D₁₆×C₈₉	C₄	semidirect
38. C₂×D₈×C₈₉	C₂×C₄	semidirect
41. D₁₆×C₈₉	C₂×C₄	semidirect
42. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
65. D₈×C₈₉	C₂²×C₄	semidirect
70. C₁₆×C₈₉	C₂²×C₄	semidirect
87. C₂×Dic₈₉	D₁₆	direct (N × H)
107. Dic₈₉	C₂×D₁₆	direct (N × H)
110. C₂×C₈₉	C₄×D₁₆	semidirect
120. C₈₉	C₂×C₄×D₁₆	semidirect
123. C₂×D₁₆	Dic₈₉	direct (N × H)
139. C₂×D₈	C₂×Dic₈₉	semidirect
148. D₁₆	C₂×Dic₈₉	direct (N × H)
151. C₂×C₁₆	C₂×Dic₈₉	semidirect
171. D₈	C₂²×Dic₈₉	semidirect
184. C₁₆	C₂²×Dic₈₉	semidirect
223. C₂	D₁₆×Dic₈₉	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×D₈×Dic₈₉	2
D₁₆×Dic₈₉	2
C₂×#19151424	2
C₂×#19151399	2
C₂×C₁₆×Dic₈₉	2
C₂²×D₁₆×C₈₉	2
C₂×C₄×D₁₆	89

Subgroups — C2×D16×Dic89 #36707689

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₁₆×Dic₈₉ #36707689