Subgroups — C₈₉×#63091 #36440939

Order 22784.

10894 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×H(7,𝔽₂)	C₂	1	2	yes	maximal, complemented
2	C₈₉×#3585	C₂	1	30	yes	maximal, complemented
3	C₈₉×#3584	C₂	1	15	yes	maximal, complemented
4	C₂×#576×C₈₉	C₂	1	15	yes	maximal, complemented
5	C₈₉×#3573	C₂	1	1	yes	maximal, characteristic, complemented
6	#585×C₈₉	—	2	12	no	—
7	#585×C₈₉	C₂²	1	1	yes	characteristic
8	#585×C₈₉	C₂²	1	30	yes	complemented
9	C₂×H(5,𝔽₂)×C₈₉	—	2	20	no	—
10	C₂×H(5,𝔽₂)×C₈₉	C₂²	1	30	yes	complemented
11	#578×C₈₉	C₂²	1	20	yes	complemented
12	#577×C₈₉	C₂²	1	120	yes	complemented
13	#576×C₈₉	C₂²	1	60	yes	complemented
14	#576×C₈₉	C₂²	1	120	yes	complemented
15	#575×C₈₉	C₂²	1	90	yes	complemented
16	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	90	yes	complemented
17	C₂×#135×C₈₉	C₂²	1	45	yes	complemented
18	C₂²×D₈×C₈₉	C₂²	1	15	yes	complemented
19	#568×C₈₉	C₂²	1	15	yes	complemented
20	C₂×#131×C₈₉	C₂²	1	15	yes	complemented
21	2₋¹⁺⁴×C₈₉	—	2	30	no	—
22	2₋¹⁺⁴×C₈₉	D₄	1	6	yes	—
23	2₋¹⁺⁴×C₈₉	C₂³	1	40	yes	complemented
24	H(5,𝔽₂)×C₈₉	—	2	120	no	—
25	H(5,𝔽₂)×C₈₉	—	2	90	no	—
26	H(5,𝔽₂)×C₈₉	D₄	1	10	yes	—
27	H(5,𝔽₂)×C₈₉	C₂³	1	120	yes	complemented
28	C₂×#41×C₈₉	—	2	120	no	—
29	C₂×#41×C₈₉	—	2	30	no	—
30	C₂×#41×C₈₉	C₂³	1	15	yes	—
31	C₂×#41×C₈₉	C₂³	1	90	yes	complemented
32	C₂²×D₄×C₈₉	—	2	90	no	—
33	C₂²×D₄×C₈₉	C₂³	1	30	yes	complemented
34	#137×C₈₉	C₂³	1	120	yes	complemented
35	AGL(1,ℤ/8)×C₈₉	C₂³	1	360	yes	complemented
36	#135×C₈₉	C₂³	1	60	yes	complemented
37	#135×C₈₉	C₂³	1	180	yes	complemented
38	C₂×Q₁₆×C₈₉	C₂³	1	15	yes	complemented
39	C₂×SD₁₆×C₈₉	C₂³	1	90	yes	complemented
40	C₂×D₈×C₈₉	C₂³	1	45	yes	complemented
41	C₂×D₈×C₈₉	C₂³	1	90	yes	complemented
42	#131×C₈₉	C₂³	1	20	yes	complemented
43	#131×C₈₉	C₂³	1	60	yes	complemented
44	C₂×M₁₆×C₈₉	C₂³	1	45	yes	complemented
45	C₂²×C₈×C₈₉	C₂³	1	15	yes	complemented
46	C₂⁴×C₈₉	—	2	30	no	—
47	#41×C₈₉	—	2	120	no	—
48	#41×C₈₉	—	2	120	no	—
49	#41×C₈₉	—	2	180	no	—
50	#41×C₈₉	C₂×D₄	1	60	yes	—
51	#41×C₈₉	C₂⁴	1	120	yes	—
52	#41×C₈₉	C₂⁴	1	20	yes	—
53	C₂×Q₈×C₈₉	—	2	40	no	—
54	C₂×Q₈×C₈₉	C₂×D₄	1	15	yes	—
55	C₂×Q₈×C₈₉	C₂⁴	1	30	yes	—
56	C₂×D₄×C₈₉	—	2	90	no	—
57	C₂×D₄×C₈₉	—	2	360	no	—
58	C₂×D₄×C₈₉	—	2	90	no	—
59	C₂×D₄×C₈₉	C₂×D₄	1	45	yes	—
60	C₂×D₄×C₈₉	C₂⁴	1	90	yes	—
61	C₂²×C₄×C₈₉	—	2	90	no	—
62	C₂²×C₄×C₈₉	C₂⁴	1	15	yes	—
63	Q₁₆×C₈₉	C₂⁴	1	6	yes	—
64	Q₁₆×C₈₉	C₂⁴	1	30	yes	—
65	SD₁₆×C₈₉	C₂⁴	1	120	yes	—
66	D₈×C₈₉	C₂⁴	1	10	yes	—
67	D₈×C₈₉	C₂⁴	1	90	yes	—
68	M₁₆×C₈₉	C₂⁴	1	60	yes	—
69	C₂×C₈×C₈₉	C₂⁴	1	15	yes	—
70	C₂×C₈×C₈₉	C₂⁴	1	45	yes	—
71	C₂³×C₈₉	—	2	90	no	—
72	C₂³×C₈₉	—	16	30	no	—
73	C₂³×C₈₉	C₂²×D₄	1	15	yes	—
74	Q₈×C₈₉	—	2	30	no	—
75	Q₈×C₈₉	C₂²×D₄	1	20	yes	—
76	Q₈×C₈₉	C₂⁵	1	12	yes	—
77	D₄×C₈₉	—	2	120	no	—
78	D₄×C₈₉	—	2	90	no	—
79	D₄×C₈₉	C₂²×D₄	1	60	yes	—
80	D₄×C₈₉	C₂⁵	1	20	yes	—
81	C₂×C₄×C₈₉	—	2	30	no	—
82	C₂×C₄×C₈₉	—	2	120	no	—
83	C₂×C₄×C₈₉	C₂²×D₄	1	45	yes	—
84	C₂×C₄×C₈₉	C₂⁵	1	15	yes	—
85	C₈×C₈₉	C₂⁵	1	1	yes	characteristic
86	C₈×C₈₉	C₂⁵	1	15	yes	—
87	C₂²×C₈₉	—	2	20	no	—
88	C₂²×C₈₉	—	4	15	no	—
89	C₂²×C₈₉	—	8	90	no	—
90	C₂²×C₈₉	C₂³×D₄	1	15	yes	—
91	C₄×C₈₉	—	2	12	no	—
92	C₄×C₈₉	C₂³×D₄	1	15	yes	—
93	C₄×C₈₉	C₂⁶	1	1	yes	characteristic
94	#63091	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂×C₈₉	—	2	15	no	—
96	C₂×C₈₉	—	4	20	no	—
97	C₂×C₈₉	C₂⁴×D₄	1	1	yes	characteristic
98	H(7,𝔽₂)	C₂×C₈₉	1	2	yes	complemented
99	#3585	C₂×C₈₉	1	30	yes	complemented
100	#3584	C₂×C₈₉	1	15	yes	complemented
101	C₂×#576	C₂×C₈₉	1	15	yes	complemented
102	#3573	C₂×C₈₉	1	1	yes	characteristic, complemented
103	C₈₉	#63091	1	1	yes	characteristic, direct product factor, Sylow
104	#585	—	2	12	no	—
105	#585	C₂²×C₈₉	1	1	yes	characteristic
106	#585	C₂²×C₈₉	1	30	yes	complemented
107	C₂×H(5,𝔽₂)	—	2	20	no	—
108	C₂×H(5,𝔽₂)	C₂²×C₈₉	1	30	yes	complemented
109	#578	C₂²×C₈₉	1	20	yes	complemented
110	#577	C₂²×C₈₉	1	120	yes	complemented
111	#576	C₂²×C₈₉	1	60	yes	complemented
112	#576	C₂²×C₈₉	1	120	yes	complemented
113	#575	C₂²×C₈₉	1	90	yes	complemented
114	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	90	yes	complemented
115	C₂×#135	C₂²×C₈₉	1	45	yes	complemented
116	C₂²×D₈	C₂²×C₈₉	1	15	yes	complemented
117	#568	C₂²×C₈₉	1	15	yes	complemented
118	C₂×#131	C₂²×C₈₉	1	15	yes	complemented
119	2₋¹⁺⁴	—	2	30	no	—
120	2₋¹⁺⁴	D₄×C₈₉	1	6	yes	—
121	2₋¹⁺⁴	C₂³×C₈₉	1	40	yes	complemented
122	H(5,𝔽₂)	—	2	120	no	—
123	H(5,𝔽₂)	—	2	90	no	—
124	H(5,𝔽₂)	D₄×C₈₉	1	10	yes	—
125	H(5,𝔽₂)	C₂³×C₈₉	1	120	yes	complemented
126	C₂×#41	—	2	120	no	—
127	C₂×#41	—	2	30	no	—
128	C₂×#41	C₂³×C₈₉	1	15	yes	—
129	C₂×#41	C₂³×C₈₉	1	90	yes	complemented
130	C₂²×D₄	—	2	90	no	—
131	C₂²×D₄	C₂³×C₈₉	1	30	yes	complemented
132	#137	C₂³×C₈₉	1	120	yes	complemented
133	AGL(1,ℤ/8)	C₂³×C₈₉	1	360	yes	complemented
134	#135	C₂³×C₈₉	1	60	yes	complemented
135	#135	C₂³×C₈₉	1	180	yes	complemented
136	C₂×Q₁₆	C₂³×C₈₉	1	15	yes	complemented
137	C₂×SD₁₆	C₂³×C₈₉	1	90	yes	complemented
138	C₂×D₈	C₂³×C₈₉	1	45	yes	complemented
139	C₂×D₈	C₂³×C₈₉	1	90	yes	complemented
140	#131	C₂³×C₈₉	1	20	yes	complemented
141	#131	C₂³×C₈₉	1	60	yes	complemented
142	C₂×M₁₆	C₂³×C₈₉	1	45	yes	complemented
143	C₂²×C₈	C₂³×C₈₉	1	15	yes	complemented
144	C₂⁴	—	2	30	no	—
145	#41	—	2	120	no	—
146	#41	—	2	120	no	—
147	#41	—	2	180	no	—
148	#41	C₂×D₄×C₈₉	1	60	yes	—
149	#41	C₂⁴×C₈₉	1	120	yes	—
150	#41	C₂⁴×C₈₉	1	20	yes	—
151	C₂×Q₈	—	2	40	no	—
152	C₂×Q₈	C₂×D₄×C₈₉	1	15	yes	—
153	C₂×Q₈	C₂⁴×C₈₉	1	30	yes	—
154	C₂×D₄	—	2	90	no	—
155	C₂×D₄	—	2	360	no	—
156	C₂×D₄	—	2	90	no	—
157	C₂×D₄	C₂×D₄×C₈₉	1	45	yes	—
158	C₂×D₄	C₂⁴×C₈₉	1	90	yes	—
159	C₂²×C₄	—	2	90	no	—
160	C₂²×C₄	C₂⁴×C₈₉	1	15	yes	—
161	Q₁₆	C₂⁴×C₈₉	1	6	yes	—
162	Q₁₆	C₂⁴×C₈₉	1	30	yes	—
163	SD₁₆	C₂⁴×C₈₉	1	120	yes	—
164	D₈	C₂⁴×C₈₉	1	10	yes	—
165	D₈	C₂⁴×C₈₉	1	90	yes	—
166	M₁₆	C₂⁴×C₈₉	1	60	yes	—
167	C₂×C₈	C₂⁴×C₈₉	1	15	yes	—
168	C₂×C₈	C₂⁴×C₈₉	1	45	yes	—
169	C₂³	—	2	90	no	—
170	C₂³	—	16	30	no	—
171	C₂³	C₂²×D₄×C₈₉	1	15	yes	—
172	Q₈	—	2	30	no	—
173	Q₈	C₂²×D₄×C₈₉	1	20	yes	—
174	Q₈	C₂⁵×C₈₉	1	12	yes	—
175	D₄	—	2	120	no	—
176	D₄	—	2	90	no	—
177	D₄	C₂²×D₄×C₈₉	1	60	yes	—
178	D₄	C₂⁵×C₈₉	1	20	yes	—
179	C₂×C₄	—	2	30	no	—
180	C₂×C₄	—	2	120	no	—
181	C₂×C₄	C₂²×D₄×C₈₉	1	45	yes	—
182	C₂×C₄	C₂⁵×C₈₉	1	15	yes	—
183	C₈	C₂⁵×C₈₉	1	1	yes	characteristic
184	C₈	C₂⁵×C₈₉	1	15	yes	—
185	C₂²	—	2	20	no	—
186	C₂²	—	4	15	no	—
187	C₂²	—	8	90	no	—
188	C₂²	C₂³×D₄×C₈₉	1	15	yes	—
189	C₄	—	2	12	no	—
190	C₄	C₂³×D₄×C₈₉	1	15	yes	—
191	C₄	C₂⁶×C₈₉	1	1	yes	characteristic
192	C₂	—	2	15	no	—
193	C₂	—	4	20	no	—
194	C₂	C₂⁴×D₄×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₈₉	C₂⁴×D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₈₉
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₉ ◁ #585×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×H(7,𝔽₂)	C₂	semidirect
2. C₈₉×#3585	C₂	semidirect
3. C₈₉×#3584	C₂	semidirect
4. C₂×#576×C₈₉	C₂	semidirect
5. C₈₉×#3573	C₂	semidirect
8. #585×C₈₉	C₂²	semidirect
10. C₂×H(5,𝔽₂)×C₈₉	C₂²	semidirect
11. #578×C₈₉	C₂²	semidirect
12. #577×C₈₉	C₂²	semidirect
13. #576×C₈₉	C₂²	semidirect
14. #576×C₈₉	C₂²	semidirect
15. #575×C₈₉	C₂²	semidirect
16. C₂×AGL(1,ℤ/8)×C₈₉	C₂²	semidirect
17. C₂×#135×C₈₉	C₂²	semidirect
18. C₂²×D₈×C₈₉	C₂²	semidirect
19. #568×C₈₉	C₂²	semidirect
20. C₂×#131×C₈₉	C₂²	semidirect
23. 2₋¹⁺⁴×C₈₉	C₂³	semidirect
27. H(5,𝔽₂)×C₈₉	C₂³	semidirect
31. C₂×#41×C₈₉	C₂³	semidirect
33. C₂²×D₄×C₈₉	C₂³	semidirect
34. #137×C₈₉	C₂³	semidirect
35. AGL(1,ℤ/8)×C₈₉	C₂³	semidirect
36. #135×C₈₉	C₂³	semidirect
37. #135×C₈₉	C₂³	semidirect
38. C₂×Q₁₆×C₈₉	C₂³	semidirect
39. C₂×SD₁₆×C₈₉	C₂³	semidirect
40. C₂×D₈×C₈₉	C₂³	semidirect
41. C₂×D₈×C₈₉	C₂³	semidirect
42. #131×C₈₉	C₂³	semidirect
43. #131×C₈₉	C₂³	semidirect
44. C₂×M₁₆×C₈₉	C₂³	semidirect
45. C₂²×C₈×C₈₉	C₂³	semidirect
94. #63091	C₈₉	direct (N × H)
98. H(7,𝔽₂)	C₂×C₈₉	semidirect
99. #3585	C₂×C₈₉	semidirect
100. #3584	C₂×C₈₉	semidirect
101. C₂×#576	C₂×C₈₉	semidirect
102. #3573	C₂×C₈₉	semidirect
103. C₈₉	#63091	direct (N × H)
106. #585	C₂²×C₈₉	semidirect
108. C₂×H(5,𝔽₂)	C₂²×C₈₉	semidirect
109. #578	C₂²×C₈₉	semidirect
110. #577	C₂²×C₈₉	semidirect
111. #576	C₂²×C₈₉	semidirect
112. #576	C₂²×C₈₉	semidirect
113. #575	C₂²×C₈₉	semidirect
114. C₂×AGL(1,ℤ/8)	C₂²×C₈₉	semidirect
115. C₂×#135	C₂²×C₈₉	semidirect
116. C₂²×D₈	C₂²×C₈₉	semidirect
117. #568	C₂²×C₈₉	semidirect
118. C₂×#131	C₂²×C₈₉	semidirect
121. 2₋¹⁺⁴	C₂³×C₈₉	semidirect
125. H(5,𝔽₂)	C₂³×C₈₉	semidirect
129. C₂×#41	C₂³×C₈₉	semidirect
131. C₂²×D₄	C₂³×C₈₉	semidirect
132. #137	C₂³×C₈₉	semidirect
133. AGL(1,ℤ/8)	C₂³×C₈₉	semidirect
134. #135	C₂³×C₈₉	semidirect
135. #135	C₂³×C₈₉	semidirect
136. C₂×Q₁₆	C₂³×C₈₉	semidirect
137. C₂×SD₁₆	C₂³×C₈₉	semidirect
138. C₂×D₈	C₂³×C₈₉	semidirect
139. C₂×D₈	C₂³×C₈₉	semidirect
140. #131	C₂³×C₈₉	semidirect
141. #131	C₂³×C₈₉	semidirect
142. C₂×M₁₆	C₂³×C₈₉	semidirect
143. C₂²×C₈	C₂³×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×H(7,𝔽₂)	2
C₈₉×#3585	2
C₈₉×#3584	2
C₂×#576×C₈₉	2
C₈₉×#3573	2
#63091	89

Subgroups — C89×#63091 #36440939

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#63091 #36440939