Subgroups — C₈₉×#12831 #36390679

Order 22784.

396 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×M₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#365×C₈₉	C₂	1	2	yes	maximal, complemented
3	C₈₉×#1810	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1807	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1383	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
7	C₂×M₃₂×C₈₉	—	2	4	no	—
8	C₂×M₃₂×C₈₉	C₄	1	2	yes	complemented
9	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₈₉	C₄	1	2	yes	complemented
12	C₂×#108×C₈₉	C₂²	1	2	yes	—
13	C₂×#106×C₈₉	C₂²	1	2	yes	—
14	#407×C₈₉	—	2	1	no	—
15	#380×C₈₉	—	2	1	no	—
16	#365×C₈₉	—	2	4	no	—
17	#341×C₈₉	—	2	1	no	—
18	#340×C₈₉	—	2	1	no	—
19	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
20	C₂×M₁₆×C₈₉	—	2	2	no	—
21	C₂²×C₈×C₈₉	—	2	2	no	—
22	C₂²×C₈×C₈₉	—	2	2	no	—
23	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
27	C₂²×C₈×C₈₉	Q₈	1	1	yes	characteristic
28	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
30	C₂×C₂²⋊C₄×C₈₉	—	4	1	no	—
31	M₃₂×C₈₉	—	2	2	no	—
32	M₃₂×C₈₉	—	2	4	no	—
33	M₃₂×C₈₉	D₄	1	4	yes	—
34	C₂×C₁₆×C₈₉	—	2	4	no	—
35	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
36	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	—
37	#108×C₈₉	—	2	2	no	—
38	#108×C₈₉	—	2	2	no	—
39	#106×C₈₉	—	2	2	no	—
40	#106×C₈₉	—	2	2	no	—
41	C₂²⋊C₈×C₈₉	—	4	1	no	—
42	C₂²⋊C₈×C₈₉	—	4	1	no	—
43	#95×C₈₉	—	4	1	no	—
44	C₂⁴×C₈₉	SD₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₈₉	—	2	2	no	—
46	C₂²×C₄×C₈₉	—	2	2	no	—
47	C₂²×C₄×C₈₉	—	4	2	no	—
48	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	SD₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
54	M₁₆×C₈₉	—	4	2	no	—
55	M₁₆×C₈₉	—	4	2	no	—
56	C₂×C₈×C₈₉	—	2	2	no	—
57	C₂×C₈×C₈₉	—	2	1	no	—
58	C₂×C₈×C₈₉	—	2	1	no	—
59	C₂×C₈×C₈₉	—	2	1	no	—
60	C₂×C₈×C₈₉	—	2	1	no	—
61	C₂×C₈×C₈₉	—	2	2	no	—
62	C₂×C₈×C₈₉	—	2	2	no	—
63	C₂×C₈×C₈₉	—	4	2	no	—
64	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₈₉	C₄⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₈×C₈₉	#41	1	2	yes	—
70	C₄⋊C₄×C₈₉	—	4	2	no	—
71	C₄⋊C₄×C₈₉	—	4	2	no	—
72	C₂²⋊C₄×C₈₉	—	8	2	no	—
73	C₁₆×C₈₉	—	2	2	no	—
74	C₁₆×C₈₉	C₂²⋊C₄	1	4	yes	complemented
75	C₂³×C₈₉	—	2	2	no	—
76	C₂³×C₈₉	—	2	2	no	—
77	C₂³×C₈₉	—	4	1	no	—
78	C₂³×C₈₉	#106	1	1	yes	characteristic
79	C₂³×C₈₉	#106	1	1	yes	characteristic
80	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₈₉	—	2	2	no	—
82	C₂×C₄×C₈₉	—	2	1	no	—
83	C₂×C₄×C₈₉	—	2	1	no	—
84	C₂×C₄×C₈₉	—	2	1	no	—
85	C₂×C₄×C₈₉	—	2	1	no	—
86	C₂×C₄×C₈₉	—	2	2	no	—
87	C₂×C₄×C₈₉	—	2	2	no	—
88	C₂×C₄×C₈₉	—	8	2	no	—
89	C₂×C₄×C₈₉	—	8	2	no	—
90	C₂×C₄×C₈₉	—	8	2	no	—
91	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
92	C₂×C₄×C₈₉	#106	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂×C₄×C₈₉	#121	1	2	yes	—
95	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
96	C₂×C₄×C₈₉	#135	1	2	yes	—
97	C₈×C₈₉	—	2	1	no	—
98	C₈×C₈₉	—	2	1	no	—
99	C₈×C₈₉	—	8	2	no	—
100	C₈×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₈×C₈₉	#117	1	1	yes	characteristic
102	C₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
103	C₂²×C₈₉	—	2	1	no	—
104	C₂²×C₈₉	—	2	1	no	—
105	C₂²×C₈₉	—	2	1	no	—
106	C₂²×C₈₉	—	2	1	no	—
107	C₂²×C₈₉	—	2	1	no	—
108	C₂²×C₈₉	—	2	1	no	—
109	C₂²×C₈₉	—	2	1	no	—
110	C₂²×C₈₉	—	4	4	no	—
111	C₂²×C₈₉	#365	1	2	yes	complemented
112	C₂²×C₈₉	C₂×#106	1	1	yes	characteristic
113	C₂²×C₈₉	#465	1	2	yes	—
114	C₄×C₈₉	—	2	1	no	—
115	C₄×C₈₉	—	2	1	no	—
116	C₄×C₈₉	—	16	2	no	—
117	C₄×C₈₉	#380	1	1	yes	characteristic
118	C₄×C₈₉	#427	1	1	yes	characteristic
119	C₄×C₈₉	#465	1	2	yes	—
120	#12831	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
121	C₂×C₈₉	—	2	2	no	—
122	C₂×C₈₉	—	2	2	no	—
123	C₂×C₈₉	—	4	1	no	—
124	C₂×C₈₉	#1807	1	1	yes	characteristic
125	C₂×C₈₉	C₂×#365	1	1	yes	characteristic
126	C₂×C₈₉	#2155	1	1	yes	characteristic
127	C₂²×M₃₂	C₂×C₈₉	1	1	yes	characteristic
128	C₂×#365	C₂×C₈₉	1	2	yes	complemented
129	#1810	C₂×C₈₉	1	1	yes	characteristic
130	#1807	C₂×C₈₉	1	1	yes	characteristic
131	#1383	C₂×C₈₉	1	2	yes	complemented
132	C₈₉	#12831	1	1	yes	characteristic, direct product factor, Sylow
133	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
134	C₂×M₃₂	—	2	4	no	—
135	C₂×M₃₂	C₄×C₈₉	1	2	yes	complemented
136	C₂×M₃₂	C₂²×C₈₉	1	1	yes	characteristic
137	C₂×M₃₂	C₂²×C₈₉	1	1	yes	characteristic
138	C₂²×C₁₆	C₄×C₈₉	1	2	yes	complemented
139	C₂×#108	C₂²×C₈₉	1	2	yes	—
140	C₂×#106	C₂²×C₈₉	1	2	yes	—
141	#407	—	2	1	no	—
142	#380	—	2	1	no	—
143	#365	—	2	4	no	—
144	#341	—	2	1	no	—
145	#340	—	2	1	no	—
146	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
147	C₂×M₁₆	—	2	2	no	—
148	C₂²×C₈	—	2	2	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
151	C₂²×C₈	C₂×C₄×C₈₉	1	1	yes	characteristic
152	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
153	C₂²×C₈	Q₈×C₈₉	1	1	yes	characteristic
154	C₂²×C₈	Q₈×C₈₉	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₂²⋊C₄	—	4	1	no	—
158	M₃₂	—	2	2	no	—
159	M₃₂	—	2	4	no	—
160	M₃₂	D₄×C₈₉	1	4	yes	—
161	C₂×C₁₆	—	2	4	no	—
162	C₂×C₁₆	C₂×C₄×C₈₉	1	2	yes	—
163	C₂×C₁₆	C₂×C₄×C₈₉	1	2	yes	—
164	#108	—	2	2	no	—
165	#108	—	2	2	no	—
166	#106	—	2	2	no	—
167	#106	—	2	2	no	—
168	C₂²⋊C₈	—	4	1	no	—
169	C₂²⋊C₈	—	4	1	no	—
170	#95	—	4	1	no	—
171	C₂⁴	SD₁₆×C₈₉	1	1	yes	characteristic
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	C₄⋊C₄×C₈₉	1	1	yes	characteristic
176	C₂²×C₄	C₄⋊C₄×C₈₉	1	1	yes	characteristic
177	C₂²×C₄	SD₁₆×C₈₉	1	1	yes	characteristic
178	C₂²×C₄	SD₁₆×C₈₉	1	1	yes	characteristic
179	C₂²×C₄	SD₁₆×C₈₉	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
181	M₁₆	—	4	2	no	—
182	M₁₆	—	4	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	1	no	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	C₄⋊C₄×C₈₉	1	1	yes	characteristic
192	C₂×C₈	C₄⋊C₄×C₈₉	1	1	yes	characteristic
193	C₂×C₈	C₂²×C₄×C₈₉	1	1	yes	characteristic
194	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
195	C₂×C₈	C₂×Q₈×C₈₉	1	1	yes	characteristic
196	C₂×C₈	#41×C₈₉	1	2	yes	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₂²⋊C₄	—	8	2	no	—
200	C₁₆	—	2	2	no	—
201	C₁₆	C₂²⋊C₄×C₈₉	1	4	yes	complemented
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	4	1	no	—
205	C₂³	#106×C₈₉	1	1	yes	characteristic
206	C₂³	#106×C₈₉	1	1	yes	characteristic
207	C₂³	C₂×SD₁₆×C₈₉	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	#106×C₈₉	1	1	yes	characteristic
219	C₂×C₄	#106×C₈₉	1	1	yes	characteristic
220	C₂×C₄	C₂×C₄⋊C₄×C₈₉	1	1	yes	characteristic
221	C₂×C₄	#121×C₈₉	1	2	yes	—
222	C₂×C₄	C₂×SD₁₆×C₈₉	1	1	yes	characteristic
223	C₂×C₄	#135×C₈₉	1	2	yes	—
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	8	2	no	—
227	C₈	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
228	C₈	#117×C₈₉	1	1	yes	characteristic
229	C₈	C₂²⋊Q₈×C₈₉	1	2	yes	—
230	C₂²	—	2	1	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	4	4	no	—
238	C₂²	#365×C₈₉	1	2	yes	complemented
239	C₂²	C₂×#106×C₈₉	1	1	yes	characteristic
240	C₂²	#465×C₈₉	1	2	yes	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	16	2	no	—
244	C₄	#380×C₈₉	1	1	yes	characteristic
245	C₄	#427×C₈₉	1	1	yes	characteristic
246	C₄	#465×C₈₉	1	2	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	2	2	no	—
249	C₂	—	4	1	no	—
250	C₂	C₈₉×#1807	1	1	yes	characteristic
251	C₂	C₂×#365×C₈₉	1	1	yes	characteristic
252	C₂	C₈₉×#2155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₈₉
Frattini, cosocle	C₂²×C₈	C₂³×C₈₉
prime radical	C₂⁴×C₈₉	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂³×C₄×C₈₉ ◁ C₂³×C₈×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#365×C₈₉	C₂	semidirect
5. C₈₉×#1383	C₂	semidirect
8. C₂×M₃₂×C₈₉	C₄	semidirect
11. C₂²×C₁₆×C₈₉	C₄	semidirect
74. C₁₆×C₈₉	C₂²⋊C₄	semidirect
111. C₂²×C₈₉	#365	semidirect
120. #12831	C₈₉	direct (N × H)
128. C₂×#365	C₂×C₈₉	semidirect
131. #1383	C₂×C₈₉	semidirect
132. C₈₉	#12831	direct (N × H)
135. C₂×M₃₂	C₄×C₈₉	semidirect
138. C₂²×C₁₆	C₄×C₈₉	semidirect
201. C₁₆	C₂²⋊C₄×C₈₉	semidirect
238. C₂²	#365×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂²×M₃₂×C₈₉	2
C₂×#365×C₈₉	2
C₈₉×#1810	2
C₈₉×#1807	2
C₈₉×#1383	2
#12831	89

Subgroups — C89×#12831 #36390679

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#12831 #36390679