Subgroups — #568×D₈₉ #26034959

Order 11392.

1460 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₈₉	C₂	1	1	yes	maximal, characteristic
2	#19152365	C₂	1	1	yes	maximal, characteristic, complemented
3	#19152364	C₂	1	6	yes	maximal, complemented
4	#19152363	C₂	1	8	yes	maximal, complemented
5	#131×D₈₉	C₂	1	8	yes	maximal, complemented
6	C₂×M₁₆×D₈₉	C₂	1	6	yes	maximal, complemented
7	#568×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13083575	C₄	1	1	yes	characteristic
9	C₂×#13083574	C₂²	1	3	yes	—
10	C₂×#13083573	C₄	1	3	yes	—
11	#41×D₈₉	C₄	1	4	yes	—
12	#41×D₈₉	C₄	1	4	yes	—
13	#41×D₈₉	C₂²	1	4	yes	—
14	#41×D₈₉	C₂²	1	4	yes	—
15	C₂×Q₈×D₈₉	C₄	1	1	yes	characteristic
16	C₂×D₄×D₈₉	C₄	1	3	yes	—
17	C₂²×C₄×D₈₉	C₂²	1	3	yes	—
18	#15668020	C₂²	1	24	yes	complemented
19	#15668019	C₂²	1	8	yes	complemented
20	#15668018	C₂²	1	24	yes	complemented
21	C₂×#13083558	C₂²	1	6	yes	complemented
22	C₂×#13083557	C₂²	1	12	yes	complemented
23	M₁₆×D₈₉	C₂²	1	4	yes	—
24	M₁₆×D₈₉	C₂²	1	12	yes	complemented
25	M₁₆×D₈₉	C₂²	1	12	yes	complemented
26	M₁₆×D₈₉	C₂²	1	12	yes	complemented
27	C₂×C₈×D₈₉	C₂²	1	12	yes	complemented
28	C₂×#41×C₈₉	C₂²	1	1	yes	characteristic
29	#131×C₈₉	C₂²	1	8	yes	complemented
30	C₂×M₁₆×C₈₉	C₂²	1	6	yes	complemented
31	C₂³×D₈₉	C₂×C₄	1	3	yes	—
32	#13083575	C₂×C₄	1	4	yes	—
33	#13083575	C₂×C₄	1	4	yes	—
34	#13083574	C₂×C₄	1	12	yes	—
35	#13083574	C₂³	1	12	yes	—
36	#13083573	C₂×C₄	1	12	yes	—
37	#13083573	C₂×C₄	1	12	yes	—
38	C₂×Dic₁₇₈	C₂×C₄	1	3	yes	—
39	Q₈×D₈₉	C₂×C₄	1	4	yes	—
40	Q₈×D₈₉	C₂×C₄	1	4	yes	—
41	C₂×D₃₅₆	C₂×C₄	1	3	yes	—
42	C₂×C₂²⋊D₈₉	C₂×C₄	1	6	yes	—
43	D₄×D₈₉	C₂×C₄	1	12	yes	—
44	D₄×D₈₉	C₂×C₄	1	12	yes	—
45	C₂²×Dic₈₉	C₂×C₄	1	3	yes	—
46	C₂×C₄×D₈₉	C₂×C₄	1	3	yes	—
47	C₂×C₄×D₈₉	C₂×C₄	1	6	yes	—
48	C₂×C₄×D₈₉	C₂×C₄	1	6	yes	—
49	C₂×C₄×D₈₉	C₂×C₄	1	3	yes	—
50	C₂×C₄×D₈₉	C₂³	1	6	yes	—
51	C₂×C₄×D₈₉	C₂³	1	1	yes	characteristic
52	C₂×C₄×D₈₉	C₂³	1	3	yes	—
53	C₂×C₄×D₈₉	C₂³	1	6	yes	—
54	#13083558	C₂³	1	4	yes	—
55	#13083558	C₂³	1	12	yes	complemented
56	#13083557	C₂³	1	8	yes	—
57	#13083557	C₂³	1	24	yes	complemented
58	C₂×#10623494	C₂³	1	12	yes	complemented
59	C₈×D₈₉	C₂³	1	8	yes	—
60	C₈×D₈₉	C₂³	1	24	yes	complemented
61	#41×C₈₉	C₂×C₄	1	4	yes	—
62	#41×C₈₉	C₂³	1	4	yes	—
63	C₂×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
64	C₂×D₄×C₈₉	C₂×C₄	1	3	yes	—
65	C₂²×C₄×C₈₉	C₂³	1	3	yes	—
66	M₁₆×C₈₉	C₂³	1	4	yes	—
67	M₁₆×C₈₉	C₂³	1	12	yes	complemented
68	C₂×C₈×C₈₉	C₂³	1	12	yes	complemented
69	C₂³×C₈₉	C₂²×C₄	1	3	yes	—
70	C₂²×D₈₉	—	4	6	no	—
71	C₂²×D₈₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×D₈₉	C₂²×C₄	1	6	yes	—
73	C₂²×D₈₉	C₂²×C₄	1	6	yes	—
74	C₂²×D₈₉	C₂²×C₄	1	3	yes	—
75	Q₈×C₈₉	C₂²×C₄	1	4	yes	—
76	D₄×C₈₉	C₂²×C₄	1	12	yes	—
77	C₂×C₄×C₈₉	C₂²×C₄	1	3	yes	—
78	C₂×C₄×C₈₉	C₂²×C₄	1	6	yes	—
79	C₂×C₄×C₈₉	C₂⁴	1	6	yes	—
80	C₂×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
81	C₂²⋊D₈₉	C₂²×C₄	1	24	yes	—
82	C₂×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×Dic₈₉	C₂²×C₄	1	3	yes	—
84	C₂×Dic₈₉	C₂²×C₄	1	6	yes	—
85	C₂×Dic₈₉	C₂²×C₄	1	6	yes	—
86	D₃₅₆	C₂²×C₄	1	12	yes	—
87	C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
88	C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
89	C₄×D₈₉	C₂²×C₄	1	6	yes	—
90	C₄×D₈₉	C₂²×C₄	1	6	yes	—
91	C₄×D₈₉	C₂²×C₄	1	6	yes	—
92	C₄×D₈₉	C₂²×C₄	1	6	yes	—
93	C₄×D₈₉	C₂²×C₄	1	6	yes	—
94	C₄×D₈₉	C₂⁴	1	1	yes	characteristic
95	C₄×D₈₉	C₂⁴	1	6	yes	—
96	C₄×D₈₉	C₂⁴	1	1	yes	characteristic
97	Dic₁₇₈	C₂²×C₄	1	12	yes	—
98	C₈×C₈₉	C₂⁴	1	8	yes	—
99	#10623494	C₂⁴	1	8	yes	—
100	C₂²×C₈₉	—	4	3	no	—
101	C₂²×C₈₉	C₂³×C₄	1	1	yes	characteristic
102	C₂²×C₈₉	C₂³×C₄	1	6	yes	—
103	C₂×D₈₉	—	2	2	no	—
104	C₂×D₈₉	—	2	12	no	—
105	C₂×D₈₉	—	4	6	no	—
106	C₂×D₈₉	—	4	3	no	—
107	C₂×D₈₉	C₂³×C₄	1	1	yes	characteristic
108	C₂×D₈₉	C₂³×C₄	1	6	yes	—
109	C₂×D₈₉	C₂³×C₄	1	1	yes	characteristic
110	C₄×C₈₉	C₂³×C₄	1	6	yes	—
111	C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
112	C₄×C₈₉	C₂⁵	1	1	yes	characteristic
113	Dic₈₉	C₂³×C₄	1	6	yes	—
114	Dic₈₉	C₂³×C₄	1	1	yes	characteristic
115	Dic₈₉	C₂³×C₄	1	1	yes	characteristic
116	C₂×C₈₉	—	2	1	no	—
117	C₂×C₈₉	—	2	6	no	—
118	C₂×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
119	D₈₉	—	2	1	no	—
120	D₈₉	—	2	6	no	—
121	D₈₉	#568	1	2	yes	direct product factor
122	C₂×#568	—	89	1	no	maximal, Sylow
123	C₈₉	C₂×#568	1	1	yes	characteristic, complemented, Sylow
124	C₂²×#41	—	89	1	no	—
125	#568	—	89	1	no	—
126	#568	—	89	8	no	—
127	#568	—	89	6	no	—
128	#568	D₈₉	1	1	yes	characteristic, direct product factor
129	C₂×#131	—	89	8	no	—
130	C₂²×M₁₆	—	89	6	no	—
131	C₂×#41	—	89	4	no	—
132	C₂×#41	—	89	4	no	—
133	C₂×#41	—	89	4	no	—
134	C₂×#41	—	89	4	no	—
135	C₂×#41	—	89	1	no	—
136	C₂×#41	—	89	3	no	—
137	C₂×#41	—	89	3	no	—
138	C₂×#41	C₂×D₈₉	1	1	yes	characteristic
139	C₂²×Q₈	—	89	1	no	—
140	C₂²×D₄	—	89	3	no	—
141	C₂³×C₄	—	89	3	no	—
142	#131	—	89	8	no	—
143	#131	—	89	24	no	—
144	#131	—	89	24	no	—
145	#131	C₂×D₈₉	1	8	yes	complemented
146	C₂×M₁₆	—	89	6	no	—
147	C₂×M₁₆	—	89	4	no	—
148	C₂×M₁₆	—	89	12	no	—
149	C₂×M₁₆	—	89	12	no	—
150	C₂×M₁₆	—	89	12	no	—
151	C₂×M₁₆	—	89	12	no	—
152	C₂×M₁₆	C₂×D₈₉	1	6	yes	complemented
153	C₂²×C₈	—	89	12	no	—
154	C₂⁴	—	89	3	no	—
155	#41	—	89	4	no	—
156	#41	—	89	4	no	—
157	#41	—	89	12	no	—
158	#41	—	89	12	no	—
159	#41	—	89	12	no	—
160	#41	—	89	12	no	—
161	#41	C₄×D₈₉	1	4	yes	—
162	#41	C₂²×D₈₉	1	4	yes	—
163	C₂×Q₈	—	89	4	no	—
164	C₂×Q₈	—	89	4	no	—
165	C₂×Q₈	—	89	3	no	—
166	C₂×Q₈	C₄×D₈₉	1	1	yes	characteristic
167	C₂×D₄	—	89	6	no	—
168	C₂×D₄	—	89	12	no	—
169	C₂×D₄	—	89	12	no	—
170	C₂×D₄	—	89	3	no	—
171	C₂×D₄	C₄×D₈₉	1	3	yes	—
172	C₂²×C₄	—	89	3	no	—
173	C₂²×C₄	—	89	3	no	—
174	C₂²×C₄	—	89	3	no	—
175	C₂²×C₄	—	89	6	no	—
176	C₂²×C₄	—	89	6	no	—
177	C₂²×C₄	—	89	6	no	—
178	C₂²×C₄	—	89	1	no	—
179	C₂²×C₄	—	89	3	no	—
180	C₂²×C₄	—	89	6	no	—
181	C₂²×C₄	C₂²×D₈₉	1	3	yes	—
182	M₁₆	—	89	4	no	—
183	M₁₆	—	89	12	no	—
184	M₁₆	—	89	8	no	—
185	M₁₆	—	89	24	no	—
186	M₁₆	C₂²×D₈₉	1	4	yes	—
187	M₁₆	C₂²×D₈₉	1	12	yes	complemented
188	C₂×C₈	—	89	12	no	—
189	C₂×C₈	—	89	8	no	—
190	C₂×C₈	—	89	24	no	—
191	C₂×C₈	C₂²×D₈₉	1	12	yes	complemented
192	C₂³	—	89	6	no	—
193	C₂³	—	89	3	no	—
194	C₂³	—	89	1	no	—
195	C₂³	—	89	6	no	—
196	C₂³	—	356	6	no	—
197	C₂³	C₂×C₄×D₈₉	1	3	yes	—
198	Q₈	—	89	12	no	—
199	Q₈	C₂×C₄×D₈₉	1	4	yes	—
200	D₄	—	89	24	no	—
201	D₄	—	89	12	no	—
202	D₄	C₂×C₄×D₈₉	1	12	yes	—
203	C₂×C₄	—	89	1	no	—
204	C₂×C₄	—	89	3	no	—
205	C₂×C₄	—	89	6	no	—
206	C₂×C₄	—	89	6	no	—
207	C₂×C₄	—	89	1	no	—
208	C₂×C₄	—	89	1	no	—
209	C₂×C₄	—	89	6	no	—
210	C₂×C₄	—	89	6	no	—
211	C₂×C₄	—	89	6	no	—
212	C₂×C₄	—	89	6	no	—
213	C₂×C₄	—	89	6	no	—
214	C₂×C₄	—	89	1	no	—
215	C₂×C₄	—	89	6	no	—
216	C₂×C₄	—	89	1	no	—
217	C₂×C₄	C₂×C₄×D₈₉	1	3	yes	—
218	C₂×C₄	C₂×C₄×D₈₉	1	6	yes	—
219	C₂×C₄	C₂³×D₈₉	1	6	yes	—
220	C₂×C₄	C₂³×D₈₉	1	1	yes	characteristic
221	C₈	—	89	8	no	—
222	C₈	C₂³×D₈₉	1	8	yes	—
223	C₂²	—	4	3	no	—
224	C₂²	—	89	1	no	—
225	C₂²	—	89	1	no	—
226	C₂²	—	89	6	no	—
227	C₂²	—	356	3	no	—
228	C₂²	—	356	6	no	—
229	C₂²	—	178	2	no	—
230	C₂²	—	178	12	no	—
231	C₂²	C₂²×C₄×D₈₉	1	1	yes	characteristic
232	C₂²	C₂²×C₄×D₈₉	1	6	yes	—
233	C₄	—	89	1	no	—
234	C₄	—	89	1	no	—
235	C₄	—	89	6	no	—
236	C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
237	C₄	C₂²×C₄×D₈₉	1	6	yes	—
238	C₄	C₂⁴×D₈₉	1	1	yes	characteristic
239	C₂	—	2	1	no	—
240	C₂	—	2	6	no	—
241	C₂	—	89	2	no	—
242	C₂	—	178	1	no	—
243	C₂	—	178	6	no	—
244	C₂	C₂³×C₄×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂⁴×D₈₉
commutator, socle	C₂×C₈₉	C₂⁴×C₄
Fitting	#568×C₈₉	C₂
prime radical	C₂×#41×D₈₉	C₂
cosocle	C₄×C₈₉	C₂⁵
hypercenter	#568	D₈₉
hypocenter	C₈₉	C₂×#568
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ #568

Complements

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

N	H ≅ G/N	type
2. #19152365	C₂	semidirect
3. #19152364	C₂	semidirect
4. #19152363	C₂	semidirect
5. #131×D₈₉	C₂	semidirect
6. C₂×M₁₆×D₈₉	C₂	semidirect
7. #568×C₈₉	C₂	semidirect
18. #15668020	C₂²	semidirect
19. #15668019	C₂²	semidirect
20. #15668018	C₂²	semidirect
21. C₂×#13083558	C₂²	semidirect
22. C₂×#13083557	C₂²	semidirect
24. M₁₆×D₈₉	C₂²	semidirect
25. M₁₆×D₈₉	C₂²	semidirect
26. M₁₆×D₈₉	C₂²	semidirect
27. C₂×C₈×D₈₉	C₂²	semidirect
29. #131×C₈₉	C₂²	semidirect
30. C₂×M₁₆×C₈₉	C₂²	semidirect
55. #13083558	C₂³	semidirect
57. #13083557	C₂³	semidirect
58. C₂×#10623494	C₂³	semidirect
60. C₈×D₈₉	C₂³	semidirect
67. M₁₆×C₈₉	C₂³	semidirect
68. C₂×C₈×C₈₉	C₂³	semidirect
121. D₈₉	#568	direct (N × H)
123. C₈₉	C₂×#568	semidirect
128. #568	D₈₉	direct (N × H)
145. #131	C₂×D₈₉	semidirect
152. C₂×M₁₆	C₂×D₈₉	semidirect
187. M₁₆	C₂²×D₈₉	semidirect
191. C₂×C₈	C₂²×D₈₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#41×D₈₉	2
#19152365	2
#19152364	2
#19152363	2
#131×D₈₉	2
C₂×M₁₆×D₈₉	2
#568×C₈₉	2
C₂×#568	89

Subgroups — #568×D89 #26034959

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #568×D₈₉ #26034959