Subgroups — C₂³×C₄×D₄×C₈₉ #36440490

Order 22784.

19998 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₄×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂⁵×C₄×C₈₉	C₂	1	2	yes	maximal, complemented
3	C₂²×C₄×D₄×C₈₉	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₂²⋊C₄×C₈₉	C₂	1	2	yes	maximal, complemented
6	C₂³×C₄²×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁶×C₈₉	C₂²	1	2	yes	—
8	C₂³×D₄×C₈₉	C₄	1	32	yes	direct product factor
9	C₂³×D₄×C₈₉	C₂²	1	28	yes	—
10	C₂⁴×C₄×C₈₉	—	2	32	no	—
11	C₂⁴×C₄×C₈₉	C₂²	1	2	yes	—
12	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₈₉	C₂²	1	56	yes	complemented
16	C₂×C₄×D₄×C₈₉	C₂²	1	448	yes	direct product factor
17	C₂²×C₄⋊C₄×C₈₉	C₂²	1	28	yes	complemented
18	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	56	yes	complemented
19	C₂²×C₄²×C₈₉	C₂²	1	28	yes	complemented
20	C₂⁵×C₈₉	—	2	16	no	—
21	C₂⁵×C₈₉	—	2	16	no	—
22	C₂⁵×C₈₉	C₂×C₄	1	32	yes	complemented
23	C₂⁵×C₈₉	C₂³	1	28	yes	—
24	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
25	C₂²×D₄×C₈₉	C₂×C₄	1	448	yes	direct product factor
26	C₂²×D₄×C₈₉	C₂³	1	112	yes	—
27	C₂³×C₄×C₈₉	—	2	16	no	—
28	C₂³×C₄×C₈₉	—	2	8	no	—
29	C₂³×C₄×C₈₉	—	2	448	no	—
30	C₂³×C₄×C₈₉	C₂×C₄	1	16	yes	complemented
31	C₂³×C₄×C₈₉	D₄	1	16	yes	direct product factor
32	C₂³×C₄×C₈₉	C₂³	1	14	yes	—
33	C₂³×C₄×C₈₉	C₂³	1	28	yes	—
34	C₂³×C₄×C₈₉	C₂³	1	14	yes	—
35	C₂³×C₄×C₈₉	C₂³	1	14	yes	—
36	C₂³×C₄×C₈₉	C₂³	1	224	yes	complemented
37	C₄×D₄×C₈₉	C₂³	1	512	yes	direct product factor
38	C₂×C₄⋊C₄×C₈₉	C₂³	1	112	yes	complemented
39	C₂×C₂²⋊C₄×C₈₉	C₂³	1	224	yes	complemented
40	C₂×C₄²×C₈₉	C₂³	1	112	yes	complemented
41	C₂⁴×C₈₉	—	2	256	no	—
42	C₂⁴×C₈₉	—	2	112	no	—
43	C₂⁴×C₈₉	—	2	112	no	—
44	C₂⁴×C₈₉	C₂²×C₄	1	8	yes	—
45	C₂⁴×C₈₉	C₂²×C₄	1	224	yes	complemented
46	C₂⁴×C₈₉	C₂×D₄	1	8	yes	—
47	C₂⁴×C₈₉	#41	1	8	yes	—
48	C₂⁴×C₈₉	C₂⁴	1	56	yes	—
49	C₂⁴×C₈₉	C₂⁴	1	7	yes	—
50	C₂×D₄×C₈₉	C₂²×C₄	1	896	yes	direct product factor
51	C₂×D₄×C₈₉	C₂⁴	1	64	yes	—
52	C₂²×C₄×C₈₉	—	2	112	no	—
53	C₂²×C₄×C₈₉	—	2	56	no	—
54	C₂²×C₄×C₈₉	—	2	896	no	—
55	C₂²×C₄×C₈₉	C₂²×C₄	1	112	yes	complemented
56	C₂²×C₄×C₈₉	C₂×D₄	1	112	yes	direct product factor
57	C₂²×C₄×C₈₉	C₂⁴	1	28	yes	—
58	C₂²×C₄×C₈₉	C₂⁴	1	56	yes	—
59	C₂²×C₄×C₈₉	C₂⁴	1	28	yes	—
60	C₂²×C₄×C₈₉	C₂⁴	1	28	yes	—
61	C₂²×C₄×C₈₉	C₂⁴	1	128	yes	complemented
62	C₄⋊C₄×C₈₉	C₂⁴	1	64	yes	complemented
63	C₂²⋊C₄×C₈₉	C₂⁴	1	128	yes	complemented
64	C₄²×C₈₉	C₂⁴	1	64	yes	complemented
65	C₂³×C₈₉	—	2	896	no	—
66	C₂³×C₈₉	—	2	112	no	—
67	C₂³×C₈₉	—	2	112	no	—
68	C₂³×C₈₉	C₄×D₄	1	64	yes	direct product factor
69	C₂³×C₈₉	C₂³×C₄	1	28	yes	—
70	C₂³×C₈₉	C₂³×C₄	1	224	yes	complemented
71	C₂³×C₈₉	C₂²×D₄	1	28	yes	—
72	C₂³×C₈₉	C₂×#41	1	28	yes	—
73	C₂³×C₈₉	C₂⁵	1	16	yes	—
74	C₂³×C₈₉	C₂⁵	1	7	yes	—
75	D₄×C₈₉	C₂³×C₄	1	256	yes	direct product factor
76	C₂×C₄×C₈₉	—	2	112	no	—
77	C₂×C₄×C₈₉	—	2	56	no	—
78	C₂×C₄×C₈₉	—	2	256	no	—
79	C₂×C₄×C₈₉	C₂³×C₄	1	112	yes	complemented
80	C₂×C₄×C₈₉	C₂²×D₄	1	112	yes	direct product factor
81	C₂×C₄×C₈₉	C₂⁵	1	8	yes	—
82	C₂×C₄×C₈₉	C₂⁵	1	16	yes	—
83	C₂×C₄×C₈₉	C₂⁵	1	8	yes	—
84	C₂×C₄×C₈₉	C₂⁵	1	8	yes	—
85	C₂²×C₈₉	—	2	16	no	—
86	C₂²×C₈₉	—	2	16	no	—
87	C₂²×C₈₉	—	2	448	no	—
88	C₂²×C₈₉	C₂×C₄×D₄	1	112	yes	direct product factor
89	C₂²×C₈₉	C₂⁴×C₄	1	14	yes	—
90	C₂²×C₈₉	C₂⁴×C₄	1	32	yes	complemented
91	C₂²×C₈₉	C₂³×D₄	1	14	yes	—
92	C₂²×C₈₉	C₂²×#41	1	14	yes	—
93	C₂²×C₈₉	C₂⁶	1	1	yes	characteristic
94	C₄×C₈₉	—	2	16	no	—
95	C₄×C₈₉	—	2	8	no	—
96	C₄×C₈₉	C₂⁴×C₄	1	16	yes	complemented
97	C₄×C₈₉	C₂³×D₄	1	16	yes	direct product factor
98	C₂³×C₄×D₄	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
99	C₂×C₈₉	—	2	32	no	—
100	C₂×C₈₉	C₂²×C₄×D₄	1	28	yes	direct product factor
101	C₂×C₈₉	C₂⁵×C₄	1	1	yes	characteristic
102	C₂×C₈₉	C₂⁴×D₄	1	1	yes	characteristic
103	C₂×C₈₉	C₂³×#41	1	1	yes	characteristic
104	C₂⁴×D₄	C₂×C₈₉	1	1	yes	characteristic
105	C₂⁵×C₄	C₂×C₈₉	1	2	yes	complemented
106	C₂²×C₄×D₄	C₂×C₈₉	1	56	yes	direct product factor
107	C₂³×C₄⋊C₄	C₂×C₈₉	1	1	yes	characteristic, complemented
108	C₂³×C₂²⋊C₄	C₂×C₈₉	1	2	yes	complemented
109	C₂³×C₄²	C₂×C₈₉	1	1	yes	characteristic, complemented
110	C₈₉	C₂³×C₄×D₄	1	1	yes	characteristic, direct product factor, Sylow
111	C₂⁶	C₂²×C₈₉	1	2	yes	—
112	C₂³×D₄	C₄×C₈₉	1	32	yes	direct product factor
113	C₂³×D₄	C₂²×C₈₉	1	28	yes	—
114	C₂⁴×C₄	—	2	32	no	—
115	C₂⁴×C₄	C₂²×C₈₉	1	2	yes	—
116	C₂⁴×C₄	C₂²×C₈₉	1	1	yes	characteristic
117	C₂⁴×C₄	C₂²×C₈₉	1	1	yes	characteristic
118	C₂⁴×C₄	C₂²×C₈₉	1	1	yes	characteristic
119	C₂⁴×C₄	C₂²×C₈₉	1	56	yes	complemented
120	C₂×C₄×D₄	C₂²×C₈₉	1	448	yes	direct product factor
121	C₂²×C₄⋊C₄	C₂²×C₈₉	1	28	yes	complemented
122	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	56	yes	complemented
123	C₂²×C₄²	C₂²×C₈₉	1	28	yes	complemented
124	C₂⁵	—	2	16	no	—
125	C₂⁵	—	2	16	no	—
126	C₂⁵	C₂×C₄×C₈₉	1	32	yes	complemented
127	C₂⁵	C₂³×C₈₉	1	28	yes	—
128	C₂⁵	C₂³×C₈₉	1	1	yes	characteristic
129	C₂²×D₄	C₂×C₄×C₈₉	1	448	yes	direct product factor
130	C₂²×D₄	C₂³×C₈₉	1	112	yes	—
131	C₂³×C₄	—	2	16	no	—
132	C₂³×C₄	—	2	8	no	—
133	C₂³×C₄	—	2	448	no	—
134	C₂³×C₄	C₂×C₄×C₈₉	1	16	yes	complemented
135	C₂³×C₄	D₄×C₈₉	1	16	yes	direct product factor
136	C₂³×C₄	C₂³×C₈₉	1	14	yes	—
137	C₂³×C₄	C₂³×C₈₉	1	28	yes	—
138	C₂³×C₄	C₂³×C₈₉	1	14	yes	—
139	C₂³×C₄	C₂³×C₈₉	1	14	yes	—
140	C₂³×C₄	C₂³×C₈₉	1	224	yes	complemented
141	C₄×D₄	C₂³×C₈₉	1	512	yes	direct product factor
142	C₂×C₄⋊C₄	C₂³×C₈₉	1	112	yes	complemented
143	C₂×C₂²⋊C₄	C₂³×C₈₉	1	224	yes	complemented
144	C₂×C₄²	C₂³×C₈₉	1	112	yes	complemented
145	C₂⁴	—	2	256	no	—
146	C₂⁴	—	2	112	no	—
147	C₂⁴	—	2	112	no	—
148	C₂⁴	C₂²×C₄×C₈₉	1	8	yes	—
149	C₂⁴	C₂²×C₄×C₈₉	1	224	yes	complemented
150	C₂⁴	C₂×D₄×C₈₉	1	8	yes	—
151	C₂⁴	#41×C₈₉	1	8	yes	—
152	C₂⁴	C₂⁴×C₈₉	1	56	yes	—
153	C₂⁴	C₂⁴×C₈₉	1	7	yes	—
154	C₂×D₄	C₂²×C₄×C₈₉	1	896	yes	direct product factor
155	C₂×D₄	C₂⁴×C₈₉	1	64	yes	—
156	C₂²×C₄	—	2	112	no	—
157	C₂²×C₄	—	2	56	no	—
158	C₂²×C₄	—	2	896	no	—
159	C₂²×C₄	C₂²×C₄×C₈₉	1	112	yes	complemented
160	C₂²×C₄	C₂×D₄×C₈₉	1	112	yes	direct product factor
161	C₂²×C₄	C₂⁴×C₈₉	1	28	yes	—
162	C₂²×C₄	C₂⁴×C₈₉	1	56	yes	—
163	C₂²×C₄	C₂⁴×C₈₉	1	28	yes	—
164	C₂²×C₄	C₂⁴×C₈₉	1	28	yes	—
165	C₂²×C₄	C₂⁴×C₈₉	1	128	yes	complemented
166	C₄⋊C₄	C₂⁴×C₈₉	1	64	yes	complemented
167	C₂²⋊C₄	C₂⁴×C₈₉	1	128	yes	complemented
168	C₄²	C₂⁴×C₈₉	1	64	yes	complemented
169	C₂³	—	2	896	no	—
170	C₂³	—	2	112	no	—
171	C₂³	—	2	112	no	—
172	C₂³	C₄×D₄×C₈₉	1	64	yes	direct product factor
173	C₂³	C₂³×C₄×C₈₉	1	28	yes	—
174	C₂³	C₂³×C₄×C₈₉	1	224	yes	complemented
175	C₂³	C₂²×D₄×C₈₉	1	28	yes	—
176	C₂³	C₂×#41×C₈₉	1	28	yes	—
177	C₂³	C₂⁵×C₈₉	1	16	yes	—
178	C₂³	C₂⁵×C₈₉	1	7	yes	—
179	D₄	C₂³×C₄×C₈₉	1	256	yes	direct product factor
180	C₂×C₄	—	2	112	no	—
181	C₂×C₄	—	2	56	no	—
182	C₂×C₄	—	2	256	no	—
183	C₂×C₄	C₂³×C₄×C₈₉	1	112	yes	complemented
184	C₂×C₄	C₂²×D₄×C₈₉	1	112	yes	direct product factor
185	C₂×C₄	C₂⁵×C₈₉	1	8	yes	—
186	C₂×C₄	C₂⁵×C₈₉	1	16	yes	—
187	C₂×C₄	C₂⁵×C₈₉	1	8	yes	—
188	C₂×C₄	C₂⁵×C₈₉	1	8	yes	—
189	C₂²	—	2	16	no	—
190	C₂²	—	2	16	no	—
191	C₂²	—	2	448	no	—
192	C₂²	C₂×C₄×D₄×C₈₉	1	112	yes	direct product factor
193	C₂²	C₂⁴×C₄×C₈₉	1	14	yes	—
194	C₂²	C₂⁴×C₄×C₈₉	1	32	yes	complemented
195	C₂²	C₂³×D₄×C₈₉	1	14	yes	—
196	C₂²	C₂²×#41×C₈₉	1	14	yes	—
197	C₂²	C₂⁶×C₈₉	1	1	yes	characteristic
198	C₄	—	2	16	no	—
199	C₄	—	2	8	no	—
200	C₄	C₂⁴×C₄×C₈₉	1	16	yes	complemented
201	C₄	C₂³×D₄×C₈₉	1	16	yes	direct product factor
202	C₂	—	2	32	no	—
203	C₂	C₂²×C₄×D₄×C₈₉	1	28	yes	direct product factor
204	C₂	C₂⁵×C₄×C₈₉	1	1	yes	characteristic
205	C₂	C₂⁴×D₄×C₈₉	1	1	yes	characteristic
206	C₂	C₂³×#41×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄×C₈₉	C₂²
commutator	C₂	C₂⁵×C₄×C₈₉
Frattini, cosocle	C₂²	C₂⁶×C₈₉
socle	C₂⁵×C₈₉	C₂³
prime radical	C₂⁴×D₄×C₈₉	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁵×C₄×C₈₉	C₂	semidirect
3. C₂²×C₄×D₄×C₈₉	C₂	direct (N × H)
4. C₂³×C₄⋊C₄×C₈₉	C₂	semidirect
5. C₂³×C₂²⋊C₄×C₈₉	C₂	semidirect
6. C₂³×C₄²×C₈₉	C₂	semidirect
8. C₂³×D₄×C₈₉	C₄	direct (N × H)
15. C₂⁴×C₄×C₈₉	C₂²	semidirect
16. C₂×C₄×D₄×C₈₉	C₂²	direct (N × H)
17. C₂²×C₄⋊C₄×C₈₉	C₂²	semidirect
18. C₂²×C₂²⋊C₄×C₈₉	C₂²	semidirect
19. C₂²×C₄²×C₈₉	C₂²	semidirect
22. C₂⁵×C₈₉	C₂×C₄	semidirect
25. C₂²×D₄×C₈₉	C₂×C₄	direct (N × H)
30. C₂³×C₄×C₈₉	C₂×C₄	semidirect
31. C₂³×C₄×C₈₉	D₄	direct (N × H)
36. C₂³×C₄×C₈₉	C₂³	semidirect
37. C₄×D₄×C₈₉	C₂³	direct (N × H)
38. C₂×C₄⋊C₄×C₈₉	C₂³	semidirect
39. C₂×C₂²⋊C₄×C₈₉	C₂³	semidirect
40. C₂×C₄²×C₈₉	C₂³	semidirect
45. C₂⁴×C₈₉	C₂²×C₄	semidirect
50. C₂×D₄×C₈₉	C₂²×C₄	direct (N × H)
55. C₂²×C₄×C₈₉	C₂²×C₄	semidirect
56. C₂²×C₄×C₈₉	C₂×D₄	direct (N × H)
61. C₂²×C₄×C₈₉	C₂⁴	semidirect
62. C₄⋊C₄×C₈₉	C₂⁴	semidirect
63. C₂²⋊C₄×C₈₉	C₂⁴	semidirect
64. C₄²×C₈₉	C₂⁴	semidirect
68. C₂³×C₈₉	C₄×D₄	direct (N × H)
70. C₂³×C₈₉	C₂³×C₄	semidirect
75. D₄×C₈₉	C₂³×C₄	direct (N × H)
79. C₂×C₄×C₈₉	C₂³×C₄	semidirect
80. C₂×C₄×C₈₉	C₂²×D₄	direct (N × H)
88. C₂²×C₈₉	C₂×C₄×D₄	direct (N × H)
90. C₂²×C₈₉	C₂⁴×C₄	semidirect
96. C₄×C₈₉	C₂⁴×C₄	semidirect
97. C₄×C₈₉	C₂³×D₄	direct (N × H)
98. C₂³×C₄×D₄	C₈₉	direct (N × H)
100. C₂×C₈₉	C₂²×C₄×D₄	direct (N × H)
105. C₂⁵×C₄	C₂×C₈₉	semidirect
106. C₂²×C₄×D₄	C₂×C₈₉	direct (N × H)
107. C₂³×C₄⋊C₄	C₂×C₈₉	semidirect
108. C₂³×C₂²⋊C₄	C₂×C₈₉	semidirect
109. C₂³×C₄²	C₂×C₈₉	semidirect
110. C₈₉	C₂³×C₄×D₄	direct (N × H)
112. C₂³×D₄	C₄×C₈₉	direct (N × H)
119. C₂⁴×C₄	C₂²×C₈₉	semidirect
120. C₂×C₄×D₄	C₂²×C₈₉	direct (N × H)
121. C₂²×C₄⋊C₄	C₂²×C₈₉	semidirect
122. C₂²×C₂²⋊C₄	C₂²×C₈₉	semidirect
123. C₂²×C₄²	C₂²×C₈₉	semidirect
126. C₂⁵	C₂×C₄×C₈₉	semidirect
129. C₂²×D₄	C₂×C₄×C₈₉	direct (N × H)
134. C₂³×C₄	C₂×C₄×C₈₉	semidirect
135. C₂³×C₄	D₄×C₈₉	direct (N × H)
140. C₂³×C₄	C₂³×C₈₉	semidirect
141. C₄×D₄	C₂³×C₈₉	direct (N × H)
142. C₂×C₄⋊C₄	C₂³×C₈₉	semidirect
143. C₂×C₂²⋊C₄	C₂³×C₈₉	semidirect
144. C₂×C₄²	C₂³×C₈₉	semidirect
149. C₂⁴	C₂²×C₄×C₈₉	semidirect
154. C₂×D₄	C₂²×C₄×C₈₉	direct (N × H)
159. C₂²×C₄	C₂²×C₄×C₈₉	semidirect
160. C₂²×C₄	C₂×D₄×C₈₉	direct (N × H)
165. C₂²×C₄	C₂⁴×C₈₉	semidirect
166. C₄⋊C₄	C₂⁴×C₈₉	semidirect
167. C₂²⋊C₄	C₂⁴×C₈₉	semidirect
168. C₄²	C₂⁴×C₈₉	semidirect
172. C₂³	C₄×D₄×C₈₉	direct (N × H)
174. C₂³	C₂³×C₄×C₈₉	semidirect
179. D₄	C₂³×C₄×C₈₉	direct (N × H)
183. C₂×C₄	C₂³×C₄×C₈₉	semidirect
184. C₂×C₄	C₂²×D₄×C₈₉	direct (N × H)
192. C₂²	C₂×C₄×D₄×C₈₉	direct (N × H)
194. C₂²	C₂⁴×C₄×C₈₉	semidirect
200. C₄	C₂⁴×C₄×C₈₉	semidirect
201. C₄	C₂³×D₄×C₈₉	direct (N × H)
203. C₂	C₂²×C₄×D₄×C₈₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₄×C₈₉	2
C₂⁵×C₄×C₈₉	2
C₂²×C₄×D₄×C₈₉	2
C₂³×C₄⋊C₄×C₈₉	2
C₂³×C₂²⋊C₄×C₈₉	2
C₂³×C₄²×C₈₉	2
C₂³×C₄×D₄	89

Subgroups — C23×C4×D4×C89 #36440490

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×D₄×C₈₉ #36440490