Subgroups — C₂×#367×D₈₉ #36707450

Order 22784.

782 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×D₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#107×D₈₉	C₂	1	2	yes	maximal
3	C₂×#19151426	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19151425	C₂	1	2	yes	maximal, complemented
5	#367×D₈₉	C₂	1	8	yes	maximal, direct product factor
6	C₂×#367×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₈₉	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄×D₈₉	—	2	2	no	—
9	C₂²×#13083549	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×D₈₉	C₄	1	8	yes	complemented
11	C₂×C₁₆×D₈₉	C₂²	1	4	yes	—
12	C₂×#15667946	C₂²	1	2	yes	—
13	C₂×#15667945	—	2	2	no	—
14	#107×D₈₉	—	2	4	no	—
15	#107×D₈₉	C₂²	1	8	yes	—
16	#19151426	C₂²	1	4	yes	complemented
17	#19151425	C₂²	1	8	yes	complemented
18	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×#107×C₈₉	C₂²	1	2	yes	—
20	#367×C₈₉	C₂²	1	4	yes	complemented
21	C₂²×C₄×D₈₉	—	4	2	no	—
22	C₂²×C₄×D₈₉	D₄	1	1	yes	characteristic
23	C₂²×#10623494	D₄	1	1	yes	characteristic
24	C₂×C₈×D₈₉	C₂×C₄	1	4	yes	—
25	C₂×C₈×D₈₉	D₄	1	2	yes	—
26	C₂×C₈×D₈₉	Q₈	1	4	yes	—
27	C₂×C₈×D₈₉	C₂³	1	2	yes	—
28	C₂×C₄⋊Dic₈₉	—	2	2	no	—
29	C₂×#13083553	—	4	2	no	—
30	C₄⋊C₄×D₈₉	—	2	8	no	—
31	C₄⋊C₄×D₈₉	—	4	4	no	—
32	C₂×#13083549	C₂×C₄	1	4	yes	complemented
33	C₂×#13083549	C₂³	1	2	yes	—
34	C₁₆×D₈₉	C₂×C₄	1	16	yes	complemented
35	#15667946	—	2	2	no	—
36	#15667946	C₂³	1	4	yes	—
37	#15667945	—	2	8	no	—
38	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
40	C₂×C₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
41	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
42	#107×C₈₉	—	2	2	no	—
43	#107×C₈₉	C₂³	1	4	yes	—
44	C₂³×D₈₉	D₈	1	1	yes	characteristic
45	C₂²×Dic₈₉	—	4	2	no	—
46	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
47	C₂×C₄×D₈₉	—	4	8	no	—
48	C₂×C₄×D₈₉	—	8	8	no	—
49	C₂×C₄×D₈₉	C₄⋊C₄	1	4	yes	—
50	C₂×C₄×D₈₉	D₈	1	2	yes	—
51	C₂×C₄×D₈₉	Q₁₆	1	4	yes	—
52	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
53	C₂×#10623494	C₄⋊C₄	1	4	yes	—
54	C₂×#10623494	C₂×D₄	1	2	yes	—
55	C₈×D₈₉	C₄⋊C₄	1	8	yes	—
56	C₈×D₈₉	C₂²×C₄	1	4	yes	—
57	C₈×D₈₉	C₂×Q₈	1	4	yes	—
58	C₄⋊Dic₈₉	—	2	4	no	—
59	C₄⋊Dic₈₉	—	4	2	no	—
60	#13083553	—	4	8	no	—
61	#13083549	C₂²×C₄	1	4	yes	complemented
62	C₂²×C₄×C₈₉	—	4	2	no	—
63	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
65	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
67	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
68	C₄⋊C₄×C₈₉	—	2	4	no	—
69	C₄⋊C₄×C₈₉	—	4	2	no	—
70	C₁₆×C₈₉	C₂²×C₄	1	4	yes	complemented
71	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
72	C₂²×D₈₉	#107	1	4	yes	—
73	C₂²×D₈₉	SD₃₂	1	8	yes	—
74	C₂²×D₈₉	C₂×D₈	1	2	yes	—
75	C₂×C₄×C₈₉	—	4	4	no	—
76	C₂×C₄×C₈₉	—	8	4	no	—
77	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
80	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
81	C₂×Dic₈₉	—	4	4	no	—
82	C₂×Dic₈₉	—	8	4	no	—
83	C₂×Dic₈₉	#107	1	4	yes	—
84	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
85	C₄×D₈₉	—	8	16	no	—
86	C₄×D₈₉	#107	1	8	yes	—
87	C₄×D₈₉	C₂×C₄⋊C₄	1	4	yes	—
88	C₄×D₈₉	C₂×Q₁₆	1	4	yes	—
89	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
90	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
91	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
92	#10623494	C₂×C₄⋊C₄	1	4	yes	—
93	C₂²×C₈₉	C₂×#107	1	2	yes	—
94	C₂²×C₈₉	C₂×SD₃₂	1	4	yes	—
95	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
96	C₂×D₈₉	#367	1	16	yes	direct product factor
97	C₂×D₈₉	C₂×#107	1	4	yes	—
98	C₂×D₈₉	C₂×SD₃₂	1	8	yes	—
99	C₄×C₈₉	—	8	4	no	—
100	C₄×C₈₉	C₂×#107	1	2	yes	—
101	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
102	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
103	Dic₈₉	—	8	4	no	—
104	Dic₈₉	C₂×#107	1	4	yes	—
105	C₂²×#367	—	89	1	no	maximal, Sylow
106	C₂×C₈₉	C₂×#367	1	4	yes	complemented
107	C₂×C₈₉	C₂²×#107	1	1	yes	characteristic
108	C₂×C₈₉	C₂²×SD₃₂	1	2	yes	—
109	D₈₉	C₂×#367	1	8	yes	direct product factor
110	C₂³×C₁₆	—	89	1	no	—
111	C₂²×#107	—	89	2	no	—
112	C₂×#367	—	89	1	no	—
113	C₂×#367	—	89	2	no	—
114	C₂×#367	—	89	8	no	—
115	C₂×#367	D₈₉	1	1	yes	characteristic, direct product factor
116	C₈₉	C₂²×#367	1	1	yes	characteristic, complemented, Sylow
117	C₂³×C₈	—	89	1	no	—
118	C₂²×C₄⋊C₄	—	178	2	no	—
119	C₂²×C₁₆	—	89	1	no	—
120	C₂²×C₁₆	—	89	8	no	—
121	C₂²×C₁₆	—	89	4	no	—
122	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
123	C₂×#107	—	89	2	no	—
124	C₂×#107	—	89	8	no	—
125	C₂×#107	—	178	4	no	—
126	C₂×#107	—	178	2	no	—
127	C₂×#107	C₂×D₈₉	1	2	yes	—
128	#367	—	89	4	no	—
129	#367	—	89	8	no	—
130	#367	C₂×D₈₉	1	4	yes	direct product factor
131	C₂³×C₄	—	89	1	no	—
132	C₂³×C₄	—	356	2	no	—
133	C₂²×C₈	—	89	1	no	—
134	C₂²×C₈	—	89	4	no	—
135	C₂²×C₈	—	89	2	no	—
136	C₂²×C₈	—	89	4	no	—
137	C₂²×C₈	—	89	2	no	—
138	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	356	4	no	—
141	C₂×C₄⋊C₄	—	356	2	no	—
142	C₂×C₄⋊C₄	—	178	8	no	—
143	C₂×C₄⋊C₄	—	178	2	no	—
144	C₂×C₁₆	—	89	4	no	—
145	C₂×C₁₆	—	89	2	no	—
146	C₂×C₁₆	—	89	16	no	—
147	C₂×C₁₆	C₄×D₈₉	1	4	yes	complemented
148	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
149	#107	—	2	2	no	—
150	#107	—	89	4	no	—
151	#107	—	178	8	no	—
152	#107	—	178	2	no	—
153	#107	C₂²×D₈₉	1	4	yes	—
154	C₂⁴	—	89	1	no	—
155	C₂²×C₄	—	4	2	no	—
156	C₂²×C₄	—	89	1	no	—
157	C₂²×C₄	—	89	2	no	—
158	C₂²×C₄	—	89	4	no	—
159	C₂²×C₄	—	89	2	no	—
160	C₂²×C₄	—	89	4	no	—
161	C₂²×C₄	—	356	8	no	—
162	C₂²×C₄	—	356	2	no	—
163	C₂²×C₄	—	712	8	no	—
164	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
165	C₂×C₈	—	89	2	no	—
166	C₂×C₈	—	89	4	no	—
167	C₂×C₈	—	89	4	no	—
168	C₂×C₈	—	89	4	no	—
169	C₂×C₈	—	89	8	no	—
170	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
171	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
172	C₂×C₈	Q₈×D₈₉	1	2	yes	—
173	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
174	C₄⋊C₄	—	2	4	no	—
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	356	8	no	—
177	C₄⋊C₄	—	356	2	no	—
178	C₄⋊C₄	—	178	4	no	—
179	C₁₆	—	89	4	no	—
180	C₁₆	C₂×C₄×D₈₉	1	4	yes	complemented
181	C₂³	—	89	4	no	—
182	C₂³	—	89	8	no	—
183	C₂³	—	89	2	no	—
184	C₂³	D₈×D₈₉	1	1	yes	characteristic
185	C₂×C₄	—	4	4	no	—
186	C₂×C₄	—	8	4	no	—
187	C₂×C₄	—	89	4	no	—
188	C₂×C₄	—	89	2	no	—
189	C₂×C₄	—	89	8	no	—
190	C₂×C₄	—	89	4	no	—
191	C₂×C₄	—	89	4	no	—
192	C₂×C₄	—	356	4	no	—
193	C₂×C₄	—	712	16	no	—
194	C₂×C₄	—	712	4	no	—
195	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
196	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
197	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
198	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
199	C₈	—	89	4	no	—
200	C₈	C₄⋊C₄×D₈₉	1	2	yes	—
201	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
202	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
203	C₂²	—	89	4	no	—
204	C₂²	—	89	16	no	—
205	C₂²	—	89	8	no	—
206	C₂²	#107×D₈₉	1	2	yes	—
207	C₂²	SD₃₂×D₈₉	1	4	yes	—
208	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
209	C₄	—	8	4	no	—
210	C₄	—	89	4	no	—
211	C₄	—	712	4	no	—
212	C₄	#107×D₈₉	1	2	yes	—
213	C₄	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
214	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
215	C₂	—	89	8	no	—
216	C₂	#367×D₈₉	1	4	yes	direct product factor
217	C₂	C₂×#107×D₈₉	1	1	yes	characteristic
218	C₂	C₂×SD₃₂×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂³	D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×#367×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂³×D₈₉	D₈
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	C₂×#367	D₈₉
hypocenter	C₈₉	C₂²×#367
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

Complements

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

N	H ≅ G/N	type
3. C₂×#19151426	C₂	semidirect
4. C₂×#19151425	C₂	semidirect
5. #367×D₈₉	C₂	direct (N × H)
6. C₂×#367×C₈₉	C₂	semidirect
10. C₂×C₁₆×D₈₉	C₄	semidirect
16. #19151426	C₂²	semidirect
17. #19151425	C₂²	semidirect
20. #367×C₈₉	C₂²	semidirect
32. C₂×#13083549	C₂×C₄	semidirect
34. C₁₆×D₈₉	C₂×C₄	semidirect
40. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
61. #13083549	C₂²×C₄	semidirect
70. C₁₆×C₈₉	C₂²×C₄	semidirect
96. C₂×D₈₉	#367	direct (N × H)
106. C₂×C₈₉	C₂×#367	semidirect
109. D₈₉	C₂×#367	direct (N × H)
115. C₂×#367	D₈₉	direct (N × H)
116. C₈₉	C₂²×#367	semidirect
130. #367	C₂×D₈₉	direct (N × H)
147. C₂×C₁₆	C₄×D₈₉	semidirect
180. C₁₆	C₂×C₄×D₈₉	semidirect
216. C₂	#367×D₈₉	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×C₁₆×D₈₉	2
C₂×#107×D₈₉	2
C₂×#19151426	2
C₂×#19151425	2
#367×D₈₉	2
C₂×#367×C₈₉	2
C₂²×#367	89

Subgroups — C2×#367×D89 #36707450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#367×D₈₉ #36707450