Subgroups — C₂³×AGL(1,ℤ/8)×C₈₉ #36440929

Order 22784.

13310 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#41×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×AGL(1,ℤ/8)×C₈₉	C₂	1	56	yes	maximal, direct product factor
4	C₂³×SD₁₆×C₈₉	C₂	1	2	yes	maximal, complemented
5	C₂³×D₈×C₈₉	C₂	1	2	yes	maximal, complemented
6	C₂³×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁶×C₈₉	—	2	1	no	—
8	C₂²×#41×C₈₉	—	2	14	no	—
9	C₂²×#41×C₈₉	C₂²	1	28	yes	complemented
10	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
11	C₂³×D₄×C₈₉	—	2	1	no	—
12	C₂³×D₄×C₈₉	—	2	14	no	—
13	C₂³×D₄×C₈₉	—	2	1	no	—
14	C₂³×D₄×C₈₉	C₂²	1	2	yes	—
15	C₂³×D₄×C₈₉	C₂²	1	28	yes	complemented
16	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
17	C₂⁴×C₄×C₈₉	—	2	1	no	—
18	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	448	yes	direct product factor
20	C₂²×SD₁₆×C₈₉	C₂²	1	56	yes	complemented
21	C₂²×D₈×C₈₉	C₂²	1	56	yes	complemented
22	C₂²×M₁₆×C₈₉	C₂²	1	28	yes	complemented
23	C₂³×C₈×C₈₉	C₂²	1	2	yes	complemented
24	C₂⁵×C₈₉	—	2	28	no	—
25	C₂⁵×C₈₉	—	2	1	no	—
26	C₂⁵×C₈₉	—	2	2	no	—
27	C₂⁵×C₈₉	—	4	16	no	—
28	C₂⁵×C₈₉	D₄	1	1	yes	characteristic
29	C₂×#41×C₈₉	—	2	168	no	—
30	C₂×#41×C₈₉	C₂³	1	112	yes	complemented
31	C₂²×Q₈×C₈₉	—	2	7	no	—
32	C₂²×Q₈×C₈₉	C₂³	1	14	yes	complemented
33	C₂²×D₄×C₈₉	—	2	28	no	—
34	C₂²×D₄×C₈₉	—	2	14	no	—
35	C₂²×D₄×C₈₉	—	2	28	no	—
36	C₂²×D₄×C₈₉	—	2	168	no	—
37	C₂²×D₄×C₈₉	—	2	7	no	—
38	C₂²×D₄×C₈₉	C₂³	1	28	yes	—
39	C₂²×D₄×C₈₉	C₂³	1	112	yes	complemented
40	C₂²×D₄×C₈₉	C₂³	1	14	yes	complemented
41	C₂³×C₄×C₈₉	—	2	1	no	—
42	C₂³×C₄×C₈₉	—	2	28	no	—
43	C₂³×C₄×C₈₉	D₄	1	14	yes	—
44	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₈₉	C₂³	1	14	yes	—
46	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
47	AGL(1,ℤ/8)×C₈₉	C₂³	1	512	yes	direct product factor
48	C₂×SD₁₆×C₈₉	C₂³	1	224	yes	complemented
49	C₂×D₈×C₈₉	C₂³	1	224	yes	complemented
50	C₂×M₁₆×C₈₉	C₂³	1	112	yes	complemented
51	C₂²×C₈×C₈₉	C₂³	1	28	yes	complemented
52	C₂⁴×C₈₉	—	2	8	no	—
53	C₂⁴×C₈₉	—	2	14	no	—
54	C₂⁴×C₈₉	—	2	28	no	—
55	C₂⁴×C₈₉	—	2	112	no	—
56	C₂⁴×C₈₉	—	4	224	no	—
57	C₂⁴×C₈₉	—	4	8	no	—
58	C₂⁴×C₈₉	—	4	16	no	—
59	C₂⁴×C₈₉	C₂×D₄	1	14	yes	—
60	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
61	#41×C₈₉	—	2	224	no	—
62	#41×C₈₉	C₂⁴	1	64	yes	complemented
63	C₂×Q₈×C₈₉	—	2	42	no	—
64	C₂×Q₈×C₈₉	C₂⁴	1	28	yes	complemented
65	C₂×D₄×C₈₉	—	2	112	no	—
66	C₂×D₄×C₈₉	—	2	84	no	—
67	C₂×D₄×C₈₉	—	2	112	no	—
68	C₂×D₄×C₈₉	—	2	224	no	—
69	C₂×D₄×C₈₉	—	2	42	no	—
70	C₂×D₄×C₈₉	C₂⁴	1	56	yes	—
71	C₂×D₄×C₈₉	C₂⁴	1	64	yes	complemented
72	C₂×D₄×C₈₉	C₂⁴	1	28	yes	complemented
73	C₂²×C₄×C₈₉	—	2	14	no	—
74	C₂²×C₄×C₈₉	—	2	112	no	—
75	C₂²×C₄×C₈₉	C₂×D₄	1	14	yes	—
76	C₂²×C₄×C₈₉	C₂×D₄	1	7	yes	—
77	C₂²×C₄×C₈₉	C₂×D₄	1	84	yes	—
78	C₂²×C₄×C₈₉	C₂⁴	1	7	yes	—
79	C₂²×C₄×C₈₉	C₂⁴	1	28	yes	—
80	SD₁₆×C₈₉	C₂⁴	1	128	yes	complemented
81	D₈×C₈₉	C₂⁴	1	128	yes	complemented
82	M₁₆×C₈₉	C₂⁴	1	64	yes	complemented
83	C₂×C₈×C₈₉	C₂⁴	1	56	yes	complemented
84	C₂³×C₈₉	—	2	56	no	—
85	C₂³×C₈₉	—	2	28	no	—
86	C₂³×C₈₉	—	2	56	no	—
87	C₂³×C₈₉	—	2	64	no	—
88	C₂³×C₈₉	—	4	448	no	—
89	C₂³×C₈₉	—	4	56	no	—
90	C₂³×C₈₉	—	4	112	no	—
91	C₂³×C₈₉	AGL(1,ℤ/8)	1	8	yes	direct product factor
92	C₂³×C₈₉	C₂²×D₄	1	28	yes	—
93	C₂³×C₈₉	C₂²×D₄	1	7	yes	—
94	Q₈×C₈₉	—	2	28	no	—
95	Q₈×C₈₉	C₂⁵	1	8	yes	complemented
96	D₄×C₈₉	—	2	64	no	—
97	D₄×C₈₉	—	2	56	no	—
98	D₄×C₈₉	—	2	64	no	—
99	D₄×C₈₉	—	2	28	no	—
100	D₄×C₈₉	C₂⁵	1	16	yes	—
101	D₄×C₈₉	C₂⁵	1	8	yes	complemented
102	C₂×C₄×C₈₉	—	2	28	no	—
103	C₂×C₄×C₈₉	—	2	64	no	—
104	C₂×C₄×C₈₉	C₂²×D₄	1	28	yes	—
105	C₂×C₄×C₈₉	C₂²×D₄	1	21	yes	—
106	C₂×C₄×C₈₉	C₂²×D₄	1	56	yes	—
107	C₂×C₄×C₈₉	C₂⁵	1	7	yes	—
108	C₂×C₄×C₈₉	C₂⁵	1	8	yes	—
109	C₈×C₈₉	C₂⁵	1	16	yes	complemented
110	C₂²×C₈₉	—	2	56	no	—
111	C₂²×C₈₉	—	2	8	no	—
112	C₂²×C₈₉	—	2	16	no	—
113	C₂²×C₈₉	—	4	128	no	—
114	C₂²×C₈₉	—	4	56	no	—
115	C₂²×C₈₉	—	4	112	no	—
116	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	28	yes	direct product factor
117	C₂²×C₈₉	C₂³×D₄	1	8	yes	—
118	C₂²×C₈₉	C₂³×D₄	1	7	yes	—
119	C₄×C₈₉	—	2	8	no	—
120	C₄×C₈₉	C₂³×D₄	1	8	yes	—
121	C₄×C₈₉	C₂³×D₄	1	7	yes	—
122	C₄×C₈₉	C₂⁶	1	1	yes	characteristic
123	C₂³×AGL(1,ℤ/8)	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
124	C₂×C₈₉	—	2	8	no	—
125	C₂×C₈₉	—	4	8	no	—
126	C₂×C₈₉	—	4	16	no	—
127	C₂×C₈₉	C₂²×AGL(1,ℤ/8)	1	14	yes	direct product factor
128	C₂×C₈₉	C₂⁴×D₄	1	1	yes	characteristic
129	C₂³×#41	C₂×C₈₉	1	1	yes	characteristic, complemented
130	C₂⁴×D₄	C₂×C₈₉	1	1	yes	characteristic, complemented
131	C₂²×AGL(1,ℤ/8)	C₂×C₈₉	1	56	yes	direct product factor
132	C₂³×SD₁₆	C₂×C₈₉	1	2	yes	complemented
133	C₂³×D₈	C₂×C₈₉	1	2	yes	complemented
134	C₂³×M₁₆	C₂×C₈₉	1	1	yes	characteristic, complemented
135	C₈₉	C₂³×AGL(1,ℤ/8)	1	1	yes	characteristic, direct product factor, Sylow
136	C₂⁶	—	2	1	no	—
137	C₂²×#41	—	2	14	no	—
138	C₂²×#41	C₂²×C₈₉	1	28	yes	complemented
139	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
140	C₂³×D₄	—	2	1	no	—
141	C₂³×D₄	—	2	1	no	—
142	C₂³×D₄	—	2	14	no	—
143	C₂³×D₄	C₂²×C₈₉	1	2	yes	—
144	C₂³×D₄	C₂²×C₈₉	1	28	yes	complemented
145	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic, complemented
146	C₂⁴×C₄	—	2	1	no	—
147	C₂⁴×C₄	C₂²×C₈₉	1	1	yes	characteristic
148	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	448	yes	direct product factor
149	C₂²×SD₁₆	C₂²×C₈₉	1	56	yes	complemented
150	C₂²×D₈	C₂²×C₈₉	1	56	yes	complemented
151	C₂²×M₁₆	C₂²×C₈₉	1	28	yes	complemented
152	C₂³×C₈	C₂²×C₈₉	1	2	yes	complemented
153	C₂⁵	—	2	1	no	—
154	C₂⁵	—	2	2	no	—
155	C₂⁵	—	2	28	no	—
156	C₂⁵	—	4	16	no	—
157	C₂⁵	D₄×C₈₉	1	1	yes	characteristic
158	C₂×#41	—	2	168	no	—
159	C₂×#41	C₂³×C₈₉	1	112	yes	complemented
160	C₂²×Q₈	—	2	7	no	—
161	C₂²×Q₈	C₂³×C₈₉	1	14	yes	complemented
162	C₂²×D₄	—	2	28	no	—
163	C₂²×D₄	—	2	14	no	—
164	C₂²×D₄	—	2	28	no	—
165	C₂²×D₄	—	2	168	no	—
166	C₂²×D₄	—	2	7	no	—
167	C₂²×D₄	C₂³×C₈₉	1	28	yes	—
168	C₂²×D₄	C₂³×C₈₉	1	112	yes	complemented
169	C₂²×D₄	C₂³×C₈₉	1	14	yes	complemented
170	C₂³×C₄	—	2	1	no	—
171	C₂³×C₄	—	2	28	no	—
172	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
173	C₂³×C₄	D₄×C₈₉	1	14	yes	—
174	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
175	C₂³×C₄	C₂³×C₈₉	1	14	yes	—
176	AGL(1,ℤ/8)	C₂³×C₈₉	1	512	yes	direct product factor
177	C₂×SD₁₆	C₂³×C₈₉	1	224	yes	complemented
178	C₂×D₈	C₂³×C₈₉	1	224	yes	complemented
179	C₂×M₁₆	C₂³×C₈₉	1	112	yes	complemented
180	C₂²×C₈	C₂³×C₈₉	1	28	yes	complemented
181	C₂⁴	—	2	8	no	—
182	C₂⁴	—	2	14	no	—
183	C₂⁴	—	2	28	no	—
184	C₂⁴	—	2	112	no	—
185	C₂⁴	—	4	224	no	—
186	C₂⁴	—	4	16	no	—
187	C₂⁴	—	4	8	no	—
188	C₂⁴	C₂×D₄×C₈₉	1	14	yes	—
189	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
190	#41	—	2	224	no	—
191	#41	C₂⁴×C₈₉	1	64	yes	complemented
192	C₂×Q₈	—	2	42	no	—
193	C₂×Q₈	C₂⁴×C₈₉	1	28	yes	complemented
194	C₂×D₄	—	2	112	no	—
195	C₂×D₄	—	2	84	no	—
196	C₂×D₄	—	2	112	no	—
197	C₂×D₄	—	2	224	no	—
198	C₂×D₄	—	2	42	no	—
199	C₂×D₄	C₂⁴×C₈₉	1	56	yes	—
200	C₂×D₄	C₂⁴×C₈₉	1	64	yes	complemented
201	C₂×D₄	C₂⁴×C₈₉	1	28	yes	complemented
202	C₂²×C₄	—	2	14	no	—
203	C₂²×C₄	—	2	112	no	—
204	C₂²×C₄	C₂×D₄×C₈₉	1	14	yes	—
205	C₂²×C₄	C₂×D₄×C₈₉	1	7	yes	—
206	C₂²×C₄	C₂×D₄×C₈₉	1	84	yes	—
207	C₂²×C₄	C₂⁴×C₈₉	1	7	yes	—
208	C₂²×C₄	C₂⁴×C₈₉	1	28	yes	—
209	SD₁₆	C₂⁴×C₈₉	1	128	yes	complemented
210	D₈	C₂⁴×C₈₉	1	128	yes	complemented
211	M₁₆	C₂⁴×C₈₉	1	64	yes	complemented
212	C₂×C₈	C₂⁴×C₈₉	1	56	yes	complemented
213	C₂³	—	2	56	no	—
214	C₂³	—	2	28	no	—
215	C₂³	—	2	56	no	—
216	C₂³	—	2	64	no	—
217	C₂³	—	4	448	no	—
218	C₂³	—	4	56	no	—
219	C₂³	—	4	112	no	—
220	C₂³	AGL(1,ℤ/8)×C₈₉	1	8	yes	direct product factor
221	C₂³	C₂²×D₄×C₈₉	1	28	yes	—
222	C₂³	C₂²×D₄×C₈₉	1	7	yes	—
223	Q₈	—	2	28	no	—
224	Q₈	C₂⁵×C₈₉	1	8	yes	complemented
225	D₄	—	2	64	no	—
226	D₄	—	2	56	no	—
227	D₄	—	2	64	no	—
228	D₄	—	2	28	no	—
229	D₄	C₂⁵×C₈₉	1	16	yes	—
230	D₄	C₂⁵×C₈₉	1	8	yes	complemented
231	C₂×C₄	—	2	28	no	—
232	C₂×C₄	—	2	64	no	—
233	C₂×C₄	C₂²×D₄×C₈₉	1	28	yes	—
234	C₂×C₄	C₂²×D₄×C₈₉	1	21	yes	—
235	C₂×C₄	C₂²×D₄×C₈₉	1	56	yes	—
236	C₂×C₄	C₂⁵×C₈₉	1	7	yes	—
237	C₂×C₄	C₂⁵×C₈₉	1	8	yes	—
238	C₈	C₂⁵×C₈₉	1	16	yes	complemented
239	C₂²	—	2	56	no	—
240	C₂²	—	2	8	no	—
241	C₂²	—	2	16	no	—
242	C₂²	—	4	128	no	—
243	C₂²	—	4	112	no	—
244	C₂²	—	4	56	no	—
245	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	28	yes	direct product factor
246	C₂²	C₂³×D₄×C₈₉	1	8	yes	—
247	C₂²	C₂³×D₄×C₈₉	1	7	yes	—
248	C₄	—	2	8	no	—
249	C₄	C₂³×D₄×C₈₉	1	8	yes	—
250	C₄	C₂³×D₄×C₈₉	1	7	yes	—
251	C₄	C₂⁶×C₈₉	1	1	yes	characteristic
252	C₂	—	2	8	no	—
253	C₂	—	4	8	no	—
254	C₂	—	4	16	no	—
255	C₂	C₂²×AGL(1,ℤ/8)×C₈₉	1	14	yes	direct product factor
256	C₂	C₂⁴×D₄×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₈₉	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₈₉
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₈₉ ◁ C₂⁴×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×#41×C₈₉	C₂	semidirect
2. C₂⁴×D₄×C₈₉	C₂	semidirect
3. C₂²×AGL(1,ℤ/8)×C₈₉	C₂	direct (N × H)
4. C₂³×SD₁₆×C₈₉	C₂	semidirect
5. C₂³×D₈×C₈₉	C₂	semidirect
6. C₂³×M₁₆×C₈₉	C₂	semidirect
9. C₂²×#41×C₈₉	C₂²	semidirect
10. C₂³×Q₈×C₈₉	C₂²	semidirect
15. C₂³×D₄×C₈₉	C₂²	semidirect
16. C₂³×D₄×C₈₉	C₂²	semidirect
19. C₂×AGL(1,ℤ/8)×C₈₉	C₂²	direct (N × H)
20. C₂²×SD₁₆×C₈₉	C₂²	semidirect
21. C₂²×D₈×C₈₉	C₂²	semidirect
22. C₂²×M₁₆×C₈₉	C₂²	semidirect
23. C₂³×C₈×C₈₉	C₂²	semidirect
30. C₂×#41×C₈₉	C₂³	semidirect
32. C₂²×Q₈×C₈₉	C₂³	semidirect
39. C₂²×D₄×C₈₉	C₂³	semidirect
40. C₂²×D₄×C₈₉	C₂³	semidirect
47. AGL(1,ℤ/8)×C₈₉	C₂³	direct (N × H)
48. C₂×SD₁₆×C₈₉	C₂³	semidirect
49. C₂×D₈×C₈₉	C₂³	semidirect
50. C₂×M₁₆×C₈₉	C₂³	semidirect
51. C₂²×C₈×C₈₉	C₂³	semidirect
62. #41×C₈₉	C₂⁴	semidirect
64. C₂×Q₈×C₈₉	C₂⁴	semidirect
71. C₂×D₄×C₈₉	C₂⁴	semidirect
72. C₂×D₄×C₈₉	C₂⁴	semidirect
80. SD₁₆×C₈₉	C₂⁴	semidirect
81. D₈×C₈₉	C₂⁴	semidirect
82. M₁₆×C₈₉	C₂⁴	semidirect
83. C₂×C₈×C₈₉	C₂⁴	semidirect
91. C₂³×C₈₉	AGL(1,ℤ/8)	direct (N × H)
95. Q₈×C₈₉	C₂⁵	semidirect
101. D₄×C₈₉	C₂⁵	semidirect
109. C₈×C₈₉	C₂⁵	semidirect
116. C₂²×C₈₉	C₂×AGL(1,ℤ/8)	direct (N × H)
123. C₂³×AGL(1,ℤ/8)	C₈₉	direct (N × H)
127. C₂×C₈₉	C₂²×AGL(1,ℤ/8)	direct (N × H)
129. C₂³×#41	C₂×C₈₉	semidirect
130. C₂⁴×D₄	C₂×C₈₉	semidirect
131. C₂²×AGL(1,ℤ/8)	C₂×C₈₉	direct (N × H)
132. C₂³×SD₁₆	C₂×C₈₉	semidirect
133. C₂³×D₈	C₂×C₈₉	semidirect
134. C₂³×M₁₆	C₂×C₈₉	semidirect
135. C₈₉	C₂³×AGL(1,ℤ/8)	direct (N × H)
138. C₂²×#41	C₂²×C₈₉	semidirect
139. C₂³×Q₈	C₂²×C₈₉	semidirect
144. C₂³×D₄	C₂²×C₈₉	semidirect
145. C₂³×D₄	C₂²×C₈₉	semidirect
148. C₂×AGL(1,ℤ/8)	C₂²×C₈₉	direct (N × H)
149. C₂²×SD₁₆	C₂²×C₈₉	semidirect
150. C₂²×D₈	C₂²×C₈₉	semidirect
151. C₂²×M₁₆	C₂²×C₈₉	semidirect
152. C₂³×C₈	C₂²×C₈₉	semidirect
159. C₂×#41	C₂³×C₈₉	semidirect
161. C₂²×Q₈	C₂³×C₈₉	semidirect
168. C₂²×D₄	C₂³×C₈₉	semidirect
169. C₂²×D₄	C₂³×C₈₉	semidirect
176. AGL(1,ℤ/8)	C₂³×C₈₉	direct (N × H)
177. C₂×SD₁₆	C₂³×C₈₉	semidirect
178. C₂×D₈	C₂³×C₈₉	semidirect
179. C₂×M₁₆	C₂³×C₈₉	semidirect
180. C₂²×C₈	C₂³×C₈₉	semidirect
191. #41	C₂⁴×C₈₉	semidirect
193. C₂×Q₈	C₂⁴×C₈₉	semidirect
200. C₂×D₄	C₂⁴×C₈₉	semidirect
201. C₂×D₄	C₂⁴×C₈₉	semidirect
209. SD₁₆	C₂⁴×C₈₉	semidirect
210. D₈	C₂⁴×C₈₉	semidirect
211. M₁₆	C₂⁴×C₈₉	semidirect
212. C₂×C₈	C₂⁴×C₈₉	semidirect
220. C₂³	AGL(1,ℤ/8)×C₈₉	direct (N × H)
224. Q₈	C₂⁵×C₈₉	semidirect
230. D₄	C₂⁵×C₈₉	semidirect
238. C₈	C₂⁵×C₈₉	semidirect
245. C₂²	C₂×AGL(1,ℤ/8)×C₈₉	direct (N × H)
255. C₂	C₂²×AGL(1,ℤ/8)×C₈₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#41×C₈₉	2
C₂⁴×D₄×C₈₉	2
C₂²×AGL(1,ℤ/8)×C₈₉	2
C₂³×SD₁₆×C₈₉	2
C₂³×D₈×C₈₉	2
C₂³×M₁₆×C₈₉	2
C₂³×AGL(1,ℤ/8)	89

Subgroups — C23×AGL(1,ℤ/8)×C89 #36440929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×AGL(1,ℤ/8)×C₈₉ #36440929