Subgroups — C₈₉×#23378 #36401226

Order 22784.

642 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#126×C₈₉	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×#1851	C₂	1	3	yes	maximal
6	C₈₉×#1842	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1839	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
9	C₂×#126×C₈₉	C₄	1	1	yes	characteristic
10	C₂×C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
11	C₂×C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
12	C₂×C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
13	C₂×C₂²⋊C₈×C₈₉	C₂²	1	3	yes	—
14	C₂×#97×C₈₉	C₂²	1	3	yes	—
15	#388×C₈₉	C₄	1	3	yes	—
16	#387×C₈₉	C₄	1	3	yes	—
17	#383×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	C₄³×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	C₂×M₁₆×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	#126×C₈₉	—	2	4	no	—
32	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	3	yes	—
33	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	3	yes	—
34	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	3	yes	—
35	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	3	yes	—
36	C₂×C₄²×C₈₉	C₂×C₄	1	3	yes	—
37	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
39	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
40	C₄⋊C₈×C₈₉	—	4	3	no	—
41	C₄⋊C₈×C₈₉	—	4	3	no	—
42	C₄⋊C₈×C₈₉	—	4	3	no	—
43	C₂²⋊C₈×C₈₉	—	4	3	no	—
44	#97×C₈₉	—	4	3	no	—
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	#95×C₈₉	C₂×C₄	1	3	yes	—
49	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	—	2	3	no	—
51	C₂²×C₄×C₈₉	—	2	3	no	—
52	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
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	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
58	M₁₆×C₈₉	—	4	4	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	3	no	—
64	C₂×C₈×C₈₉	—	2	3	no	—
65	C₂×C₈×C₈₉	—	2	6	no	—
66	C₄⋊C₄×C₈₉	—	2	6	no	—
67	C₄⋊C₄×C₈₉	—	4	3	no	—
68	C₂²⋊C₄×C₈₉	—	2	6	no	—
69	C₂²⋊C₄×C₈₉	—	4	3	no	—
70	C₄²×C₈₉	—	2	3	no	—
71	C₄²×C₈₉	—	2	3	no	—
72	C₄²×C₈₉	—	2	3	no	—
73	C₄²×C₈₉	—	2	3	no	—
74	C₄²×C₈₉	—	4	3	no	—
75	C₄²×C₈₉	M₁₆	1	4	yes	complemented
76	C₂³×C₈₉	—	2	3	no	—
77	C₂³×C₈₉	—	2	1	no	—
78	C₂³×C₈₉	—	2	3	no	—
79	C₂³×C₈₉	C₂³×C₄	1	1	yes	characteristic
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	—	2	3	no	—
91	C₂×C₄×C₈₉	—	2	3	no	—
92	C₂×C₄×C₈₉	—	4	1	no	—
93	C₂×C₄×C₈₉	—	4	3	no	—
94	C₂×C₄×C₈₉	C₂×M₁₆	1	6	yes	—
95	C₂×C₄×C₈₉	H(5,𝔽₂)	1	3	yes	—
96	C₂×C₄×C₈₉	2₋¹⁺⁴	1	3	yes	—
97	C₈×C₈₉	—	4	3	no	—
98	C₈×C₈₉	—	4	1	no	—
99	C₈×C₈₉	—	4	1	no	—
100	C₈×C₈₉	—	4	3	no	—
101	C₂²×C₈₉	—	4	1	no	—
102	C₂²×C₈₉	—	4	3	no	—
103	C₂²×C₈₉	—	4	3	no	—
104	C₂²×C₈₉	#520	1	3	yes	—
105	C₂²×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
106	C₂²×C₈₉	#568	1	3	yes	—
107	C₄×C₈₉	—	2	6	no	—
108	C₄×C₈₉	—	4	1	no	—
109	C₄×C₈₉	—	4	3	no	—
110	C₄×C₈₉	—	4	3	no	—
111	C₄×C₈₉	—	4	3	no	—
112	C₄×C₈₉	#560	1	1	yes	characteristic
113	C₄×C₈₉	#563	1	3	yes	—
114	#23378	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂×C₈₉	—	8	1	no	—
116	C₂×C₈₉	#2383	1	1	yes	characteristic
117	C₂×C₈₉	#2989	1	3	yes	—
118	C₂×C₈₉	#2995	1	3	yes	—
119	C₄×#126	C₂×C₈₉	1	1	yes	characteristic
120	#1940	C₂×C₈₉	1	3	yes	complemented
121	#1924	C₂×C₈₉	1	3	yes	—
122	#1917	C₂×C₈₉	1	3	yes	complemented
123	#1851	C₂×C₈₉	1	3	yes	—
124	#1842	C₂×C₈₉	1	1	yes	characteristic, complemented
125	#1839	C₂×C₈₉	1	1	yes	characteristic, complemented
126	C₈₉	#23378	1	1	yes	characteristic, direct product factor, Sylow
127	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
128	C₂×#126	C₄×C₈₉	1	1	yes	characteristic
129	C₂×C₄⋊C₈	C₂²×C₈₉	1	3	yes	—
130	C₂×C₄⋊C₈	C₂²×C₈₉	1	3	yes	—
131	C₂×C₄⋊C₈	C₂²×C₈₉	1	3	yes	—
132	C₂×C₂²⋊C₈	C₂²×C₈₉	1	3	yes	—
133	C₂×#97	C₂²×C₈₉	1	3	yes	—
134	#388	C₄×C₈₉	1	3	yes	—
135	#387	C₄×C₈₉	1	3	yes	—
136	#383	C₄×C₈₉	1	1	yes	characteristic
137	C₄×C₄⋊C₄	C₂²×C₈₉	1	3	yes	—
138	C₄×C₂²⋊C₄	C₂²×C₈₉	1	3	yes	—
139	C₄³	C₂²×C₈₉	1	1	yes	characteristic
140	#336	C₂²×C₈₉	1	3	yes	—
141	#336	C₂²×C₈₉	1	3	yes	—
142	#336	C₂²×C₈₉	1	3	yes	—
143	#336	C₂²×C₈₉	1	3	yes	—
144	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
145	C₂×M₁₆	—	4	3	no	—
146	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
147	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
149	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
150	#126	—	2	4	no	—
151	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	3	yes	—
152	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	3	yes	—
153	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	3	yes	—
154	C₂×C₂²⋊C₄	C₂×C₄×C₈₉	1	3	yes	—
155	C₂×C₄²	C₂×C₄×C₈₉	1	3	yes	—
156	C₂×C₄²	C₂×C₄×C₈₉	1	1	yes	characteristic
157	C₂×C₄²	C₂³×C₈₉	1	3	yes	—
158	C₂×C₄²	C₂³×C₈₉	1	3	yes	—
159	C₄⋊C₈	—	4	3	no	—
160	C₄⋊C₈	—	4	3	no	—
161	C₄⋊C₈	—	4	3	no	—
162	C₂²⋊C₈	—	4	3	no	—
163	#97	—	4	3	no	—
164	#95	C₂×C₄×C₈₉	1	3	yes	—
165	#95	C₂×C₄×C₈₉	1	3	yes	—
166	#95	C₂×C₄×C₈₉	1	3	yes	—
167	#95	C₂×C₄×C₈₉	1	3	yes	—
168	C₂⁴	C₂²×C₄×C₈₉	1	1	yes	characteristic
169	C₂²×C₄	—	2	3	no	—
170	C₂²×C₄	—	2	3	no	—
171	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
172	C₂²×C₄	C₂²×C₄×C₈₉	1	3	yes	—
173	C₂²×C₄	C₂²×C₄×C₈₉	1	3	yes	—
174	C₂²×C₄	C₂²×C₄×C₈₉	1	3	yes	—
175	C₂²×C₄	C₂²×C₄×C₈₉	1	3	yes	—
176	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
177	M₁₆	—	4	4	no	—
178	C₂×C₈	—	2	3	no	—
179	C₂×C₈	—	2	3	no	—
180	C₂×C₈	—	2	3	no	—
181	C₂×C₈	—	2	3	no	—
182	C₂×C₈	—	2	3	no	—
183	C₂×C₈	—	2	3	no	—
184	C₂×C₈	—	2	6	no	—
185	C₄⋊C₄	—	2	6	no	—
186	C₄⋊C₄	—	4	3	no	—
187	C₂²⋊C₄	—	2	6	no	—
188	C₂²⋊C₄	—	4	3	no	—
189	C₄²	—	2	3	no	—
190	C₄²	—	2	3	no	—
191	C₄²	—	2	3	no	—
192	C₄²	—	2	3	no	—
193	C₄²	—	4	3	no	—
194	C₄²	M₁₆×C₈₉	1	4	yes	complemented
195	C₂³	—	2	3	no	—
196	C₂³	—	2	1	no	—
197	C₂³	—	2	3	no	—
198	C₂³	C₂³×C₄×C₈₉	1	1	yes	characteristic
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	—	2	3	no	—
207	C₂×C₄	—	2	3	no	—
208	C₂×C₄	—	2	3	no	—
209	C₂×C₄	—	2	3	no	—
210	C₂×C₄	—	2	3	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	4	3	no	—
213	C₂×C₄	C₂×M₁₆×C₈₉	1	6	yes	—
214	C₂×C₄	H(5,𝔽₂)×C₈₉	1	3	yes	—
215	C₂×C₄	2₋¹⁺⁴×C₈₉	1	3	yes	—
216	C₈	—	4	3	no	—
217	C₈	—	4	1	no	—
218	C₈	—	4	1	no	—
219	C₈	—	4	3	no	—
220	C₂²	—	4	1	no	—
221	C₂²	—	4	3	no	—
222	C₂²	—	4	3	no	—
223	C₂²	#520×C₈₉	1	3	yes	—
224	C₂²	C₂²×M₁₆×C₈₉	1	1	yes	characteristic
225	C₂²	#568×C₈₉	1	3	yes	—
226	C₄	—	2	6	no	—
227	C₄	—	4	1	no	—
228	C₄	—	4	3	no	—
229	C₄	—	4	3	no	—
230	C₄	—	4	3	no	—
231	C₄	#560×C₈₉	1	1	yes	characteristic
232	C₄	#563×C₈₉	1	3	yes	—
233	C₂	—	8	1	no	—
234	C₂	C₈₉×#2383	1	1	yes	characteristic
235	C₂	C₈₉×#2989	1	3	yes	—
236	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×#1842	C₂	semidirect
7. C₈₉×#1839	C₂	semidirect
75. C₄²×C₈₉	M₁₆	semidirect
114. #23378	C₈₉	direct (N × H)
120. #1940	C₂×C₈₉	semidirect
122. #1917	C₂×C₈₉	semidirect
124. #1842	C₂×C₈₉	semidirect
125. #1839	C₂×C₈₉	semidirect
126. C₈₉	#23378	direct (N × H)
194. C₄²	M₁₆×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×#126×C₈₉	2
C₈₉×#1940	2
C₈₉×#1924	2
C₈₉×#1917	2
C₈₉×#1851	2
C₈₉×#1842	2
C₈₉×#1839	2
#23378	89

Subgroups — C89×#23378 #36401226

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#23378 #36401226