Subgroups — C₈₉×#35463 #36413311

Order 22784.

3810 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#534×C₈₉	C₂	1	3	yes	maximal, complemented
2	C₂²×#123×C₈₉	C₂	1	3	yes	maximal, complemented
3	C₈₉×#2539	C₂	1	12	yes	maximal, complemented
4	C₈₉×#2501	C₂	1	12	yes	maximal, complemented
5	C₈₉×#2273	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₄×C₈₉	C₂²	1	3	yes	—
7	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
8	#534×C₈₉	—	2	24	no	—
9	C₂×#123×C₈₉	—	2	24	no	—
10	C₂×#123×C₈₉	C₂²	1	24	yes	complemented
11	C₂×#123×C₈₉	C₂²	1	12	yes	complemented
12	C₂×#121×C₈₉	C₂²	1	12	yes	—
13	C₂×#120×C₈₉	C₂²	1	12	yes	complemented
14	C₂²×C₄⋊C₄×C₈₉	C₂²	1	6	yes	complemented
15	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	3	yes	—
16	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	6	yes	complemented
17	#399×C₈₉	C₂²	1	24	yes	complemented
18	#392×C₈₉	C₂²	1	8	yes	complemented
19	#390×C₈₉	C₂²	1	24	yes	complemented
20	#384×C₈₉	C₂²	1	8	yes	complemented
21	C₄×C₂²⋊C₄×C₈₉	C₂²	1	12	yes	complemented
22	C₂⁵×C₈₉	C₂³	1	3	yes	—
23	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
24	C₂²×D₄×C₈₉	—	2	12	no	—
25	C₂²×D₄×C₈₉	—	2	24	no	—
26	C₂²×D₄×C₈₉	C₂³	1	12	yes	—
27	C₂²×D₄×C₈₉	C₂³	1	12	yes	—
28	C₂³×C₄×C₈₉	—	2	12	no	—
29	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
30	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
31	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
32	#123×C₈₉	—	2	96	no	—
33	#123×C₈₉	—	2	48	no	—
34	#121×C₈₉	—	2	48	no	—
35	#120×C₈₉	—	2	48	no	—
36	C₂×C₄⋊C₄×C₈₉	—	2	24	no	—
37	C₂×C₄⋊C₄×C₈₉	C₂³	1	12	yes	—
38	C₂×C₄⋊C₄×C₈₉	C₂³	1	24	yes	—
39	C₂×C₂²⋊C₄×C₈₉	—	2	24	no	—
40	C₂×C₂²⋊C₄×C₈₉	D₄	1	24	yes	complemented
41	C₂×C₂²⋊C₄×C₈₉	C₂³	1	24	yes	—
42	C₂×C₂²⋊C₄×C₈₉	C₂³	1	24	yes	—
43	C₂×C₂²⋊C₄×C₈₉	C₂³	1	12	yes	—
44	C₂×C₄²×C₈₉	C₂³	1	8	yes	—
45	#95×C₈₉	C₂³	1	8	yes	—
46	C₂⁴×C₈₉	—	2	6	no	—
47	C₂⁴×C₈₉	—	2	12	no	—
48	C₂⁴×C₈₉	—	4	4	no	—
49	C₂⁴×C₈₉	—	4	12	no	—
50	C₂⁴×C₈₉	C₂×D₄	1	12	yes	—
51	C₂⁴×C₈₉	C₂⁴	1	6	yes	—
52	C₂⁴×C₈₉	C₂⁴	1	3	yes	—
53	C₂×D₄×C₈₉	—	2	24	no	—
54	C₂×D₄×C₈₉	—	2	24	no	—
55	C₂×D₄×C₈₉	—	2	24	no	—
56	C₂×D₄×C₈₉	—	2	24	no	—
57	C₂×D₄×C₈₉	—	4	24	no	—
58	C₂×D₄×C₈₉	—	4	48	no	—
59	C₂²×C₄×C₈₉	—	2	12	no	—
60	C₂²×C₄×C₈₉	—	2	6	no	—
61	C₂²×C₄×C₈₉	—	2	24	no	—
62	C₂²×C₄×C₈₉	—	2	12	no	—
63	C₂²×C₄×C₈₉	—	4	12	no	—
64	C₂²×C₄×C₈₉	—	4	16	no	—
65	C₂²×C₄×C₈₉	C₂×D₄	1	24	yes	—
66	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
67	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
68	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
69	C₄⋊C₄×C₈₉	—	2	48	no	—
70	C₄⋊C₄×C₈₉	—	2	24	no	—
71	C₂²⋊C₄×C₈₉	—	2	48	no	—
72	C₂²⋊C₄×C₈₉	—	2	48	no	—
73	C₂²⋊C₄×C₈₉	C₂×D₄	1	48	yes	—
74	C₄²×C₈₉	—	4	8	no	—
75	C₂³×C₈₉	—	2	12	no	—
76	C₂³×C₈₉	—	2	6	no	—
77	C₂³×C₈₉	—	2	12	no	—
78	C₂³×C₈₉	—	2	24	no	—
79	C₂³×C₈₉	—	4	24	no	—
80	C₂³×C₈₉	—	4	24	no	—
81	C₂³×C₈₉	—	4	12	no	—
82	C₂³×C₈₉	—	8	24	no	—
83	C₂³×C₈₉	#127	1	16	yes	—
84	C₂³×C₈₉	C₂²×D₄	1	12	yes	—
85	C₂³×C₈₉	C₂²×D₄	1	6	yes	—
86	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
87	D₄×C₈₉	—	4	48	no	—
88	D₄×C₈₉	—	8	24	no	—
89	C₂×C₄×C₈₉	—	2	24	no	—
90	C₂×C₄×C₈₉	—	2	12	no	—
91	C₂×C₄×C₈₉	—	2	12	no	—
92	C₂×C₄×C₈₉	—	2	6	no	—
93	C₂×C₄×C₈₉	—	4	24	no	—
94	C₂×C₄×C₈₉	—	4	24	no	—
95	C₂×C₄×C₈₉	—	4	12	no	—
96	C₂×C₄×C₈₉	C₂²×D₄	1	24	yes	—
97	C₂²×C₈₉	—	2	12	no	—
98	C₂²×C₈₉	—	2	24	no	—
99	C₂²×C₈₉	—	4	16	no	—
100	C₂²×C₈₉	—	4	6	no	—
101	C₂²×C₈₉	—	4	24	no	—
102	C₂²×C₈₉	—	4	12	no	—
103	C₂²×C₈₉	—	8	24	no	—
104	C₂²×C₈₉	C₂×#127	1	12	yes	—
105	C₂²×C₈₉	C₂³×D₄	1	3	yes	—
106	C₂²×C₈₉	#585	1	4	yes	—
107	C₄×C₈₉	—	2	24	no	—
108	C₄×C₈₉	—	4	4	no	—
109	C₄×C₈₉	—	4	6	no	—
110	#35463	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
111	C₂×C₈₉	—	2	12	no	—
112	C₂×C₈₉	—	8	6	no	—
113	C₂×C₈₉	C₂²×#127	1	1	yes	characteristic
114	C₂×C₈₉	#3485	1	6	yes	—
115	C₂×#534	C₂×C₈₉	1	3	yes	complemented
116	C₂²×#123	C₂×C₈₉	1	3	yes	complemented
117	#2539	C₂×C₈₉	1	12	yes	complemented
118	#2501	C₂×C₈₉	1	12	yes	complemented
119	#2273	C₂×C₈₉	1	1	yes	characteristic, complemented
120	C₈₉	#35463	1	1	yes	characteristic, direct product factor, Sylow
121	C₂³×D₄	C₂²×C₈₉	1	3	yes	—
122	C₂⁴×C₄	C₂²×C₈₉	1	1	yes	characteristic
123	#534	—	2	24	no	—
124	C₂×#123	—	2	24	no	—
125	C₂×#123	C₂²×C₈₉	1	24	yes	complemented
126	C₂×#123	C₂²×C₈₉	1	12	yes	complemented
127	C₂×#121	C₂²×C₈₉	1	12	yes	—
128	C₂×#120	C₂²×C₈₉	1	12	yes	complemented
129	C₂²×C₄⋊C₄	C₂²×C₈₉	1	6	yes	complemented
130	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	3	yes	—
131	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	6	yes	complemented
132	#399	C₂²×C₈₉	1	24	yes	complemented
133	#392	C₂²×C₈₉	1	8	yes	complemented
134	#390	C₂²×C₈₉	1	24	yes	complemented
135	#384	C₂²×C₈₉	1	8	yes	complemented
136	C₄×C₂²⋊C₄	C₂²×C₈₉	1	12	yes	complemented
137	C₂⁵	C₂³×C₈₉	1	3	yes	—
138	C₂⁵	C₂³×C₈₉	1	1	yes	characteristic
139	C₂²×D₄	—	2	12	no	—
140	C₂²×D₄	—	2	24	no	—
141	C₂²×D₄	C₂³×C₈₉	1	12	yes	—
142	C₂²×D₄	C₂³×C₈₉	1	12	yes	—
143	C₂³×C₄	—	2	12	no	—
144	C₂³×C₄	C₂³×C₈₉	1	6	yes	—
145	C₂³×C₄	C₂³×C₈₉	1	3	yes	—
146	C₂³×C₄	C₂³×C₈₉	1	6	yes	—
147	#123	—	2	96	no	—
148	#123	—	2	48	no	—
149	#121	—	2	48	no	—
150	#120	—	2	48	no	—
151	C₂×C₄⋊C₄	—	2	24	no	—
152	C₂×C₄⋊C₄	C₂³×C₈₉	1	24	yes	—
153	C₂×C₄⋊C₄	C₂³×C₈₉	1	12	yes	—
154	C₂×C₂²⋊C₄	—	2	24	no	—
155	C₂×C₂²⋊C₄	D₄×C₈₉	1	24	yes	complemented
156	C₂×C₂²⋊C₄	C₂³×C₈₉	1	24	yes	—
157	C₂×C₂²⋊C₄	C₂³×C₈₉	1	24	yes	—
158	C₂×C₂²⋊C₄	C₂³×C₈₉	1	12	yes	—
159	C₂×C₄²	C₂³×C₈₉	1	8	yes	—
160	#95	C₂³×C₈₉	1	8	yes	—
161	C₂⁴	—	2	6	no	—
162	C₂⁴	—	2	12	no	—
163	C₂⁴	—	4	4	no	—
164	C₂⁴	—	4	12	no	—
165	C₂⁴	C₂×D₄×C₈₉	1	12	yes	—
166	C₂⁴	C₂⁴×C₈₉	1	6	yes	—
167	C₂⁴	C₂⁴×C₈₉	1	3	yes	—
168	C₂×D₄	—	2	24	no	—
169	C₂×D₄	—	2	24	no	—
170	C₂×D₄	—	2	24	no	—
171	C₂×D₄	—	2	24	no	—
172	C₂×D₄	—	4	24	no	—
173	C₂×D₄	—	4	48	no	—
174	C₂²×C₄	—	2	12	no	—
175	C₂²×C₄	—	2	6	no	—
176	C₂²×C₄	—	2	24	no	—
177	C₂²×C₄	—	2	12	no	—
178	C₂²×C₄	—	4	12	no	—
179	C₂²×C₄	—	4	16	no	—
180	C₂²×C₄	C₂×D₄×C₈₉	1	24	yes	—
181	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
182	C₂²×C₄	C₂⁴×C₈₉	1	6	yes	—
183	C₂²×C₄	C₂⁴×C₈₉	1	12	yes	—
184	C₄⋊C₄	—	2	48	no	—
185	C₄⋊C₄	—	2	24	no	—
186	C₂²⋊C₄	—	2	48	no	—
187	C₂²⋊C₄	—	2	48	no	—
188	C₂²⋊C₄	C₂×D₄×C₈₉	1	48	yes	—
189	C₄²	—	4	8	no	—
190	C₂³	—	2	12	no	—
191	C₂³	—	2	6	no	—
192	C₂³	—	2	12	no	—
193	C₂³	—	2	24	no	—
194	C₂³	—	4	24	no	—
195	C₂³	—	4	24	no	—
196	C₂³	—	4	12	no	—
197	C₂³	—	8	24	no	—
198	C₂³	#127×C₈₉	1	16	yes	—
199	C₂³	C₂²×D₄×C₈₉	1	12	yes	—
200	C₂³	C₂²×D₄×C₈₉	1	6	yes	—
201	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
202	D₄	—	4	48	no	—
203	D₄	—	8	24	no	—
204	C₂×C₄	—	2	24	no	—
205	C₂×C₄	—	2	12	no	—
206	C₂×C₄	—	2	12	no	—
207	C₂×C₄	—	2	6	no	—
208	C₂×C₄	—	4	24	no	—
209	C₂×C₄	—	4	24	no	—
210	C₂×C₄	—	4	12	no	—
211	C₂×C₄	C₂²×D₄×C₈₉	1	24	yes	—
212	C₂²	—	2	12	no	—
213	C₂²	—	2	24	no	—
214	C₂²	—	4	16	no	—
215	C₂²	—	4	6	no	—
216	C₂²	—	4	24	no	—
217	C₂²	—	4	12	no	—
218	C₂²	—	8	24	no	—
219	C₂²	C₂×#127×C₈₉	1	12	yes	—
220	C₂²	C₂³×D₄×C₈₉	1	3	yes	—
221	C₂²	#585×C₈₉	1	4	yes	—
222	C₄	—	2	24	no	—
223	C₄	—	4	4	no	—
224	C₄	—	4	6	no	—
225	C₂	—	2	12	no	—
226	C₂	—	8	6	no	—
227	C₂	C₂²×#127×C₈₉	1	1	yes	characteristic
228	C₂	C₈₉×#3485	1	6	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#534×C₈₉	C₂	semidirect
2. C₂²×#123×C₈₉	C₂	semidirect
3. C₈₉×#2539	C₂	semidirect
4. C₈₉×#2501	C₂	semidirect
5. C₈₉×#2273	C₂	semidirect
10. C₂×#123×C₈₉	C₂²	semidirect
11. C₂×#123×C₈₉	C₂²	semidirect
13. C₂×#120×C₈₉	C₂²	semidirect
14. C₂²×C₄⋊C₄×C₈₉	C₂²	semidirect
16. C₂²×C₂²⋊C₄×C₈₉	C₂²	semidirect
17. #399×C₈₉	C₂²	semidirect
18. #392×C₈₉	C₂²	semidirect
19. #390×C₈₉	C₂²	semidirect
20. #384×C₈₉	C₂²	semidirect
21. C₄×C₂²⋊C₄×C₈₉	C₂²	semidirect
40. C₂×C₂²⋊C₄×C₈₉	D₄	semidirect
110. #35463	C₈₉	direct (N × H)
115. C₂×#534	C₂×C₈₉	semidirect
116. C₂²×#123	C₂×C₈₉	semidirect
117. #2539	C₂×C₈₉	semidirect
118. #2501	C₂×C₈₉	semidirect
119. #2273	C₂×C₈₉	semidirect
120. C₈₉	#35463	direct (N × H)
125. C₂×#123	C₂²×C₈₉	semidirect
126. C₂×#123	C₂²×C₈₉	semidirect
128. C₂×#120	C₂²×C₈₉	semidirect
129. C₂²×C₄⋊C₄	C₂²×C₈₉	semidirect
131. C₂²×C₂²⋊C₄	C₂²×C₈₉	semidirect
132. #399	C₂²×C₈₉	semidirect
133. #392	C₂²×C₈₉	semidirect
134. #390	C₂²×C₈₉	semidirect
135. #384	C₂²×C₈₉	semidirect
136. C₄×C₂²⋊C₄	C₂²×C₈₉	semidirect
155. C₂×C₂²⋊C₄	D₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#534×C₈₉	2
C₂²×#123×C₈₉	2
C₈₉×#2539	2
C₈₉×#2501	2
C₈₉×#2273	2
#35463	89

Subgroups — C89×#35463 #36413311

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35463 #36413311