Subgroups — C₈₉×#23380 #36401228

Order 22784.

612 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2401	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#1940	C₂	1	3	yes	maximal, complemented
3	C₈₉×#1924	C₂	1	3	yes	maximal
4	C₈₉×#1917	C₂	1	3	yes	maximal, complemented
5	C₈₉×#1850	C₂	1	3	yes	maximal
6	C₈₉×#1843	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1840	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
10	C₂×C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
11	C₂×C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
12	C₂×C₂²⋊C₈×C₈₉	C₂²	1	3	yes	—
13	C₂×#97×C₈₉	C₂²	1	3	yes	—
14	#401×C₈₉	C₄	1	1	yes	characteristic
15	#400×C₈₉	C₄	1	3	yes	—
16	#397×C₈₉	C₄	1	3	yes	—
17	#396×C₈₉	C₄	1	1	yes	characteristic
18	C₄×C₄⋊C₄×C₈₉	C₂²	1	3	yes	—
19	C₄×C₂²⋊C₄×C₈₉	C₂²	1	3	yes	—
20	#376×C₈₉	C₂²	1	1	yes	characteristic
21	#336×C₈₉	C₂²	1	3	yes	—
22	#336×C₈₉	C₂²	1	3	yes	—
23	#336×C₈₉	C₂²	1	3	yes	—
24	#336×C₈₉	C₂²	1	3	yes	—
25	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
26	C₂²×C₈×C₈₉	—	4	3	no	—
27	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
28	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
29	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
30	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	3	yes	—
32	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	3	yes	—
33	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	3	yes	—
34	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	3	yes	—
35	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
36	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
37	C₄⋊C₈×C₈₉	—	4	3	no	—
38	C₄⋊C₈×C₈₉	—	4	3	no	—
39	C₄⋊C₈×C₈₉	—	4	3	no	—
40	C₂²⋊C₈×C₈₉	—	4	3	no	—
41	#97×C₈₉	—	4	3	no	—
42	#95×C₈₉	C₂×C₄	1	1	yes	characteristic
43	#95×C₈₉	C₂×C₄	1	3	yes	—
44	#95×C₈₉	C₂×C₄	1	3	yes	—
45	#95×C₈₉	C₂×C₄	1	3	yes	—
46	#95×C₈₉	C₂×C₄	1	3	yes	—
47	#95×C₈₉	C₂×C₄	1	3	yes	—
48	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₈₉	—	2	3	no	—
50	C₂²×C₄×C₈₉	—	2	3	no	—
51	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
55	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
57	C₂×C₈×C₈₉	—	2	3	no	—
58	C₂×C₈×C₈₉	—	2	3	no	—
59	C₂×C₈×C₈₉	—	2	3	no	—
60	C₂×C₈×C₈₉	—	2	3	no	—
61	C₂×C₈×C₈₉	—	2	3	no	—
62	C₂×C₈×C₈₉	—	2	3	no	—
63	C₂×C₈×C₈₉	—	2	6	no	—
64	C₂×C₈×C₈₉	—	4	4	no	—
65	C₄⋊C₄×C₈₉	—	4	3	no	—
66	C₄⋊C₄×C₈₉	—	4	3	no	—
67	C₂²⋊C₄×C₈₉	—	2	6	no	—
68	C₂²⋊C₄×C₈₉	—	4	3	no	—
69	C₄²×C₈₉	—	2	3	no	—
70	C₄²×C₈₉	—	2	3	no	—
71	C₄²×C₈₉	—	2	3	no	—
72	C₄²×C₈₉	—	2	3	no	—
73	C₂³×C₈₉	—	2	3	no	—
74	C₂³×C₈₉	—	2	3	no	—
75	C₂³×C₈₉	#131	1	2	yes	—
76	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
77	C₂×C₄×C₈₉	—	2	3	no	—
78	C₂×C₄×C₈₉	—	2	3	no	—
79	C₂×C₄×C₈₉	—	2	3	no	—
80	C₂×C₄×C₈₉	—	2	3	no	—
81	C₂×C₄×C₈₉	—	2	3	no	—
82	C₂×C₄×C₈₉	—	2	3	no	—
83	C₂×C₄×C₈₉	—	2	3	no	—
84	C₂×C₄×C₈₉	—	2	3	no	—
85	C₂×C₄×C₈₉	—	2	3	no	—
86	C₂×C₄×C₈₉	—	2	3	no	—
87	C₂×C₄×C₈₉	—	2	3	no	—
88	C₂×C₄×C₈₉	—	2	3	no	—
89	C₂×C₄×C₈₉	—	2	3	no	—
90	C₂×C₄×C₈₉	—	4	1	no	—
91	C₂×C₄×C₈₉	—	4	3	no	—
92	C₂×C₄×C₈₉	H(5,𝔽₂)	1	3	yes	—
93	C₂×C₄×C₈₉	2₋¹⁺⁴	1	3	yes	—
94	C₈×C₈₉	—	4	3	no	—
95	C₈×C₈₉	—	4	1	no	—
96	C₈×C₈₉	—	4	1	no	—
97	C₈×C₈₉	—	4	3	no	—
98	C₂²×C₈₉	—	2	6	no	—
99	C₂²×C₈₉	—	4	1	no	—
100	C₂²×C₈₉	—	4	3	no	—
101	C₂²×C₈₉	#520	1	3	yes	—
102	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
103	C₂²×C₈₉	#568	1	3	yes	—
104	C₄×C₈₉	—	4	1	no	—
105	C₄×C₈₉	—	4	3	no	—
106	C₄×C₈₉	—	4	3	no	—
107	C₄×C₈₉	—	4	3	no	—
108	C₄×C₈₉	—	4	3	no	—
109	C₄×C₈₉	#560	1	1	yes	characteristic
110	C₄×C₈₉	#563	1	3	yes	—
111	#23380	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
112	C₂×C₈₉	—	4	2	no	—
113	C₂×C₈₉	#2383	1	1	yes	characteristic
114	C₂×C₈₉	#2992	1	3	yes	—
115	C₂×C₈₉	#2995	1	3	yes	—
116	#2401	C₂×C₈₉	1	1	yes	characteristic
117	#1940	C₂×C₈₉	1	3	yes	complemented
118	#1924	C₂×C₈₉	1	3	yes	—
119	#1917	C₂×C₈₉	1	3	yes	complemented
120	#1850	C₂×C₈₉	1	3	yes	—
121	#1843	C₂×C₈₉	1	1	yes	characteristic, complemented
122	#1840	C₂×C₈₉	1	1	yes	characteristic, complemented
123	C₈₉	#23380	1	1	yes	characteristic, direct product factor, Sylow
124	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
125	C₂×C₄⋊C₈	C₂²×C₈₉	1	3	yes	—
126	C₂×C₄⋊C₈	C₂²×C₈₉	1	3	yes	—
127	C₂×C₄⋊C₈	C₂²×C₈₉	1	3	yes	—
128	C₂×C₂²⋊C₈	C₂²×C₈₉	1	3	yes	—
129	C₂×#97	C₂²×C₈₉	1	3	yes	—
130	#401	C₄×C₈₉	1	1	yes	characteristic
131	#400	C₄×C₈₉	1	3	yes	—
132	#397	C₄×C₈₉	1	3	yes	—
133	#396	C₄×C₈₉	1	1	yes	characteristic
134	C₄×C₄⋊C₄	C₂²×C₈₉	1	3	yes	—
135	C₄×C₂²⋊C₄	C₂²×C₈₉	1	3	yes	—
136	#376	C₂²×C₈₉	1	1	yes	characteristic
137	#336	C₂²×C₈₉	1	3	yes	—
138	#336	C₂²×C₈₉	1	3	yes	—
139	#336	C₂²×C₈₉	1	3	yes	—
140	#336	C₂²×C₈₉	1	3	yes	—
141	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
142	C₂²×C₈	—	4	3	no	—
143	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
144	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
145	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
146	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
147	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	3	yes	—
148	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	3	yes	—
149	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	3	yes	—
150	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	3	yes	—
151	C₂×C₄²	C₂³×C₈₉	1	3	yes	—
152	C₂×C₄²	C₂³×C₈₉	1	3	yes	—
153	C₄⋊C₈	—	4	3	no	—
154	C₄⋊C₈	—	4	3	no	—
155	C₄⋊C₈	—	4	3	no	—
156	C₂²⋊C₈	—	4	3	no	—
157	#97	—	4	3	no	—
158	#95	C₂×C₄×C₈₉	1	1	yes	characteristic
159	#95	C₂×C₄×C₈₉	1	3	yes	—
160	#95	C₂×C₄×C₈₉	1	3	yes	—
161	#95	C₂×C₄×C₈₉	1	3	yes	—
162	#95	C₂×C₄×C₈₉	1	3	yes	—
163	#95	C₂×C₄×C₈₉	1	3	yes	—
164	C₂⁴	C₂²×C₄×C₈₉	1	1	yes	characteristic
165	C₂²×C₄	—	2	3	no	—
166	C₂²×C₄	—	2	3	no	—
167	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
168	C₂²×C₄	C₂²×C₄×C₈₉	1	3	yes	—
169	C₂²×C₄	C₂²×C₄×C₈₉	1	3	yes	—
170	C₂²×C₄	C₂²×C₄×C₈₉	1	3	yes	—
171	C₂²×C₄	C₂²×C₄×C₈₉	1	3	yes	—
172	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
173	C₂×C₈	—	2	3	no	—
174	C₂×C₈	—	2	3	no	—
175	C₂×C₈	—	2	3	no	—
176	C₂×C₈	—	2	3	no	—
177	C₂×C₈	—	2	3	no	—
178	C₂×C₈	—	2	3	no	—
179	C₂×C₈	—	2	6	no	—
180	C₂×C₈	—	4	4	no	—
181	C₄⋊C₄	—	4	3	no	—
182	C₄⋊C₄	—	4	3	no	—
183	C₂²⋊C₄	—	2	6	no	—
184	C₂²⋊C₄	—	4	3	no	—
185	C₄²	—	2	3	no	—
186	C₄²	—	2	3	no	—
187	C₄²	—	2	3	no	—
188	C₄²	—	2	3	no	—
189	C₂³	—	2	3	no	—
190	C₂³	—	2	3	no	—
191	C₂³	#131×C₈₉	1	2	yes	—
192	C₂³	C₂³×C₄×C₈₉	1	1	yes	characteristic
193	C₂×C₄	—	2	3	no	—
194	C₂×C₄	—	2	3	no	—
195	C₂×C₄	—	2	3	no	—
196	C₂×C₄	—	2	3	no	—
197	C₂×C₄	—	2	3	no	—
198	C₂×C₄	—	2	3	no	—
199	C₂×C₄	—	2	3	no	—
200	C₂×C₄	—	2	3	no	—
201	C₂×C₄	—	2	3	no	—
202	C₂×C₄	—	2	3	no	—
203	C₂×C₄	—	2	3	no	—
204	C₂×C₄	—	2	3	no	—
205	C₂×C₄	—	2	3	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	3	no	—
208	C₂×C₄	H(5,𝔽₂)×C₈₉	1	3	yes	—
209	C₂×C₄	2₋¹⁺⁴×C₈₉	1	3	yes	—
210	C₈	—	4	3	no	—
211	C₈	—	4	1	no	—
212	C₈	—	4	1	no	—
213	C₈	—	4	3	no	—
214	C₂²	—	2	6	no	—
215	C₂²	—	4	1	no	—
216	C₂²	—	4	3	no	—
217	C₂²	#520×C₈₉	1	3	yes	—
218	C₂²	C₂×#131×C₈₉	1	1	yes	characteristic
219	C₂²	#568×C₈₉	1	3	yes	—
220	C₄	—	4	1	no	—
221	C₄	—	4	3	no	—
222	C₄	—	4	3	no	—
223	C₄	—	4	3	no	—
224	C₄	—	4	3	no	—
225	C₄	#560×C₈₉	1	1	yes	characteristic
226	C₄	#563×C₈₉	1	3	yes	—
227	C₂	—	4	2	no	—
228	C₂	C₈₉×#2383	1	1	yes	characteristic
229	C₂	C₈₉×#2992	1	3	yes	—
230	C₂	C₈₉×#2995	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₈₉×#1940	C₂	semidirect
4. C₈₉×#1917	C₂	semidirect
6. C₈₉×#1843	C₂	semidirect
7. C₈₉×#1840	C₂	semidirect
111. #23380	C₈₉	direct (N × H)
117. #1940	C₂×C₈₉	semidirect
119. #1917	C₂×C₈₉	semidirect
121. #1843	C₂×C₈₉	semidirect
122. #1840	C₂×C₈₉	semidirect
123. C₈₉	#23380	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈₉×#2401	2
C₈₉×#1940	2
C₈₉×#1924	2
C₈₉×#1917	2
C₈₉×#1850	2
C₈₉×#1843	2
C₈₉×#1840	2
#23380	89

Subgroups — C89×#23380 #36401228

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#23380 #36401228