Subgroups — C₈₉×#11629 #36389477

Order 22784.

430 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2162	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2162	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×D₄⋊C₁₆	C₂	1	4	yes	maximal, complemented
5	C₂³×C₈×C₈₉	—	2	1	no	—
6	C₂×C₄×D₄×C₈₉	C₄	1	1	yes	characteristic
7	C₈×D₄×C₈₉	—	2	1	no	—
8	C₈×D₄×C₈₉	—	2	1	no	—
9	C₈×D₄×C₈₉	C₄	1	2	yes	—
10	C₈×D₄×C₈₉	C₂²	1	2	yes	—
11	C₂×C₄⋊C₈×C₈₉	C₄	1	1	yes	characteristic
12	C₂×C₂²⋊C₈×C₈₉	—	2	1	no	—
13	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₁₆×C₈₉	—	2	1	no	—
15	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	complemented
16	C₂²⋊C₁₆×C₈₉	—	2	1	no	—
17	C₂²⋊C₁₆×C₈₉	—	2	1	no	—
18	C₄×C₁₆×C₈₉	—	2	1	no	—
19	C₄×C₁₆×C₈₉	C₂²	1	2	yes	complemented
20	C₂²×D₄×C₈₉	C₈	1	1	yes	characteristic
21	C₂³×C₄×C₈₉	—	2	1	no	—
22	C₂²×C₈×C₈₉	—	2	1	no	—
23	C₂²×C₈×C₈₉	—	2	2	no	—
24	C₂²×C₈×C₈₉	—	2	2	no	—
25	C₂²×C₈×C₈₉	—	4	4	no	—
26	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
28	C₄×D₄×C₈₉	—	2	1	no	—
29	C₄×D₄×C₈₉	—	2	1	no	—
30	C₄×D₄×C₈₉	C₈	1	2	yes	—
31	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₈₉	C₈	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
34	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₆×C₈₉	—	2	2	no	—
36	C₂×C₁₆×C₈₉	—	2	2	no	—
37	C₄⋊C₈×C₈₉	—	2	1	no	—
38	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
39	C₂²⋊C₈×C₈₉	—	2	2	no	—
40	C₂²⋊C₈×C₈₉	—	2	2	no	—
41	C₄×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
43	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
44	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂⁴×C₈₉	—	2	1	no	—
46	C₂×D₄×C₈₉	—	2	1	no	—
47	C₂×D₄×C₈₉	—	2	2	no	—
48	C₂×D₄×C₈₉	C₁₆	1	4	yes	complemented
49	C₂×D₄×C₈₉	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₈₉	—	2	1	no	—
51	C₂²×C₄×C₈₉	—	2	2	no	—
52	C₂²×C₄×C₈₉	—	2	2	no	—
53	C₂²×C₄×C₈₉	—	4	2	no	—
54	C₂²×C₄×C₈₉	—	4	2	no	—
55	C₂²×C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	M₁₆	1	1	yes	characteristic
58	C₂×C₈×C₈₉	—	2	2	no	—
59	C₂×C₈×C₈₉	—	2	2	no	—
60	C₂×C₈×C₈₉	—	4	1	no	—
61	C₂×C₈×C₈₉	—	4	4	no	—
62	C₂×C₈×C₈₉	—	4	2	no	—
63	C₂×C₈×C₈₉	—	4	4	no	—
64	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₈₉	—	2	1	no	—
69	C₄⋊C₄×C₈₉	C₂×C₈	1	2	yes	—
70	C₂²⋊C₄×C₈₉	—	2	2	no	—
71	C₂²⋊C₄×C₈₉	—	2	2	no	—
72	C₄²×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₈₉	C₂×C₈	1	1	yes	characteristic
74	C₄²×C₈₉	M₁₆	1	1	yes	characteristic
75	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
76	C₁₆×C₈₉	—	4	2	no	—
77	C₁₆×C₈₉	—	4	2	no	—
78	C₂³×C₈₉	—	2	4	no	—
79	C₂³×C₈₉	—	2	2	no	—
80	C₂³×C₈₉	—	4	2	no	—
81	C₂³×C₈₉	—	4	2	no	—
82	C₂³×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
83	D₄×C₈₉	—	2	4	no	—
84	D₄×C₈₉	—	2	2	no	—
85	D₄×C₈₉	C₂×C₁₆	1	4	yes	—
86	C₂×C₄×C₈₉	—	2	1	no	—
87	C₂×C₄×C₈₉	—	2	1	no	—
88	C₂×C₄×C₈₉	—	2	2	no	—
89	C₂×C₄×C₈₉	—	4	1	no	—
90	C₂×C₄×C₈₉	—	4	4	no	—
91	C₂×C₄×C₈₉	—	4	2	no	—
92	C₂×C₄×C₈₉	—	4	2	no	—
93	C₂×C₄×C₈₉	—	4	2	no	—
94	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂²⋊C₈	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂×C₁₆	1	2	yes	—
98	C₂×C₄×C₈₉	M₃₂	1	2	yes	—
99	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
102	C₈×C₈₉	—	2	1	no	—
103	C₈×C₈₉	—	2	1	no	—
104	C₈×C₈₉	—	2	1	no	—
105	C₈×C₈₉	—	4	2	no	—
106	C₈×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
107	C₂²×C₈₉	—	2	1	no	—
108	C₂²×C₈₉	—	2	1	no	—
109	C₂²×C₈₉	—	2	4	no	—
110	C₂²×C₈₉	—	4	8	no	—
111	C₂²×C₈₉	—	4	2	no	—
112	C₂²×C₈₉	—	4	2	no	—
113	C₂²×C₈₉	C₂²⋊C₁₆	1	2	yes	—
114	C₂²×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
115	C₄×C₈₉	—	2	1	no	—
116	C₄×C₈₉	—	2	1	no	—
117	C₄×C₈₉	—	2	1	no	—
118	C₄×C₈₉	—	4	2	no	—
119	C₄×C₈₉	C₂²⋊C₁₆	1	2	yes	—
120	C₄×C₈₉	#418	1	1	yes	characteristic
121	C₄×C₈₉	#421	1	1	yes	characteristic
122	C₄×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
123	C₄×C₈₉	C₂×M₃₂	1	1	yes	characteristic
124	#11629	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
125	C₂×C₈₉	—	2	2	no	—
126	C₂×C₈₉	—	4	4	no	—
127	C₂×C₈₉	#1478	1	1	yes	characteristic
128	C₂×C₈₉	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
129	C₂×C₈₉	#2117	1	1	yes	characteristic
130	C₂×C₈×D₄	C₂×C₈₉	1	1	yes	characteristic
131	#2162	C₂×C₈₉	1	1	yes	characteristic, complemented
132	#2162	C₂×C₈₉	1	1	yes	characteristic, complemented
133	D₄⋊C₁₆	C₂×C₈₉	1	4	yes	complemented
134	C₈₉	#11629	1	1	yes	characteristic, direct product factor, Sylow
135	C₂³×C₈	—	2	1	no	—
136	C₂×C₄×D₄	C₄×C₈₉	1	1	yes	characteristic
137	C₈×D₄	—	2	1	no	—
138	C₈×D₄	—	2	1	no	—
139	C₈×D₄	C₄×C₈₉	1	2	yes	—
140	C₈×D₄	C₂²×C₈₉	1	2	yes	—
141	C₂×C₄⋊C₈	C₄×C₈₉	1	1	yes	characteristic
142	C₂×C₂²⋊C₈	—	2	1	no	—
143	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
144	C₄⋊C₁₆	—	2	1	no	—
145	C₄⋊C₁₆	C₂²×C₈₉	1	2	yes	complemented
146	C₂²⋊C₁₆	—	2	1	no	—
147	C₂²⋊C₁₆	—	2	1	no	—
148	C₄×C₁₆	—	2	1	no	—
149	C₄×C₁₆	C₂²×C₈₉	1	2	yes	complemented
150	C₂²×D₄	C₈×C₈₉	1	1	yes	characteristic
151	C₂³×C₄	—	2	1	no	—
152	C₂²×C₈	—	2	1	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	4	4	no	—
156	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
157	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
158	C₄×D₄	—	2	1	no	—
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	C₈×C₈₉	1	2	yes	—
161	C₄×D₄	C₂×C₄×C₈₉	1	2	yes	—
162	C₂×C₄⋊C₄	C₈×C₈₉	1	1	yes	characteristic
163	C₂×C₂²⋊C₄	—	2	1	no	—
164	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	C₄⋊C₈	—	2	1	no	—
168	C₄⋊C₈	C₂×C₄×C₈₉	1	2	yes	—
169	C₂²⋊C₈	—	2	2	no	—
170	C₂²⋊C₈	—	2	2	no	—
171	C₄×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
172	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
173	C₄×C₈	D₄×C₈₉	1	1	yes	characteristic
174	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
175	C₂⁴	—	2	1	no	—
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	—	2	2	no	—
178	C₂×D₄	C₁₆×C₈₉	1	4	yes	complemented
179	C₂×D₄	C₂×C₈×C₈₉	1	2	yes	—
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
186	C₂²×C₄	C₂×C₈×C₈₉	1	1	yes	characteristic
187	C₂²×C₄	M₁₆×C₈₉	1	1	yes	characteristic
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	4	1	no	—
191	C₂×C₈	—	4	4	no	—
192	C₂×C₈	—	4	2	no	—
193	C₂×C₈	—	4	4	no	—
194	C₂×C₈	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
195	C₂×C₈	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
196	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
197	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	C₂×C₈×C₈₉	1	2	yes	—
200	C₂²⋊C₄	—	2	2	no	—
201	C₂²⋊C₄	—	2	2	no	—
202	C₄²	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
203	C₄²	C₂×C₈×C₈₉	1	1	yes	characteristic
204	C₄²	M₁₆×C₈₉	1	1	yes	characteristic
205	C₄²	C₂²×C₄×C₈₉	1	1	yes	characteristic
206	C₁₆	—	4	2	no	—
207	C₁₆	—	4	2	no	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	4	no	—
210	C₂³	—	4	2	no	—
211	C₂³	—	4	2	no	—
212	C₂³	C₂²⋊C₈×C₈₉	1	1	yes	characteristic
213	D₄	—	2	2	no	—
214	D₄	—	2	4	no	—
215	D₄	C₂×C₁₆×C₈₉	1	4	yes	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	C₂²⋊C₈×C₈₉	1	1	yes	characteristic
225	C₂×C₄	C₂²⋊C₈×C₈₉	1	1	yes	characteristic
226	C₂×C₄	C₂²⋊C₈×C₈₉	1	1	yes	characteristic
227	C₂×C₄	C₂×C₁₆×C₈₉	1	2	yes	—
228	C₂×C₄	M₃₂×C₈₉	1	2	yes	—
229	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
230	C₂×C₄	C₂²×C₈×C₈₉	1	1	yes	characteristic
231	C₂×C₄	C₂×M₁₆×C₈₉	1	1	yes	characteristic
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	—	2	1	no	—
235	C₈	—	4	2	no	—
236	C₈	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	4	8	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	C₂²⋊C₁₆×C₈₉	1	2	yes	—
244	C₂²	C₂×C₂²⋊C₈×C₈₉	1	1	yes	characteristic
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	2	no	—
249	C₄	C₂²⋊C₁₆×C₈₉	1	2	yes	—
250	C₄	#418×C₈₉	1	1	yes	characteristic
251	C₄	#421×C₈₉	1	1	yes	characteristic
252	C₄	C₂²×C₁₆×C₈₉	1	1	yes	characteristic
253	C₄	C₂×M₃₂×C₈₉	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	4	4	no	—
256	C₂	C₈₉×#1478	1	1	yes	characteristic
257	C₂	C₂×C₂²⋊C₁₆×C₈₉	1	1	yes	characteristic
258	C₂	C₈₉×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈×C₈₉	C₂×D₄
commutator	C₄	C₂²×C₁₆×C₈₉
Frattini, cosocle	C₄×C₈	C₂³×C₈₉
socle	C₂²×C₈₉	C₂×C₂²⋊C₈
prime radical	C₂²×D₄×C₈₉	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₈₉ ◁ C₂×C₄×C₈×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2162	C₂	semidirect
3. C₈₉×#2162	C₂	semidirect
4. C₈₉×D₄⋊C₁₆	C₂	semidirect
15. C₄⋊C₁₆×C₈₉	C₂²	semidirect
19. C₄×C₁₆×C₈₉	C₂²	semidirect
48. C₂×D₄×C₈₉	C₁₆	semidirect
124. #11629	C₈₉	direct (N × H)
131. #2162	C₂×C₈₉	semidirect
132. #2162	C₂×C₈₉	semidirect
133. D₄⋊C₁₆	C₂×C₈₉	semidirect
134. C₈₉	#11629	direct (N × H)
145. C₄⋊C₁₆	C₂²×C₈₉	semidirect
149. C₄×C₁₆	C₂²×C₈₉	semidirect
178. C₂×D₄	C₁₆×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₈₉	2
C₈₉×#2162	2
C₈₉×D₄⋊C₁₆	2
#11629	89

Subgroups — C89×#11629 #36389477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#11629 #36389477