Subgroups — C₈₉×#35369 #36413217

Order 22784.

3270 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	C₈₉×#2480	C₂	1	24	yes	maximal, complemented
2	C₈₉×#2315	C₂	1	3	yes	maximal, complemented
3	C₂×#379×C₈₉	C₂	1	3	yes	maximal
4	C₈₉×#2273	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂⁶×C₈₉	C₂²	1	1	yes	characteristic
6	C₂×#126×C₈₉	C₂²	1	8	yes	complemented
7	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	3	yes	—
8	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	3	yes	—
9	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	12	yes	—
10	#397×C₈₉	C₂²	1	24	yes	complemented
11	#396×C₈₉	C₂²	1	8	yes	complemented
12	#388×C₈₉	C₂²	1	24	yes	complemented
13	#385×C₈₉	C₂²	1	12	yes	complemented
14	#385×C₈₉	C₂²	1	12	yes	complemented
15	#380×C₈₉	C₂²	1	12	yes	complemented
16	#379×C₈₉	C₂²	1	12	yes	—
17	#379×C₈₉	C₂²	1	12	yes	—
18	C₄×C₂²⋊C₄×C₈₉	C₂²	1	12	yes	complemented
19	C₂⁵×C₈₉	—	2	12	no	—
20	C₂⁵×C₈₉	—	4	8	no	—
21	C₂⁵×C₈₉	C₂³	1	6	yes	—
22	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
23	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
24	C₂³×C₄×C₈₉	C₂³	1	6	yes	—
25	#126×C₈₉	—	4	16	no	—
26	C₂×C₄⋊C₄×C₈₉	C₂³	1	24	yes	complemented
27	C₂×C₂²⋊C₄×C₈₉	—	2	12	no	—
28	C₂×C₂²⋊C₄×C₈₉	—	2	12	no	—
29	C₂×C₂²⋊C₄×C₈₉	—	4	24	no	—
30	C₂×C₂²⋊C₄×C₈₉	C₂³	1	6	yes	—
31	C₂×C₂²⋊C₄×C₈₉	C₂³	1	6	yes	—
32	C₂×C₂²⋊C₄×C₈₉	C₂³	1	6	yes	—
33	C₂×C₂²⋊C₄×C₈₉	C₂³	1	6	yes	—
34	C₂×C₂²⋊C₄×C₈₉	C₂³	1	24	yes	—
35	C₂×C₂²⋊C₄×C₈₉	C₂³	1	24	yes	—
36	C₂×C₄²×C₈₉	C₂³	1	8	yes	complemented
37	#95×C₈₉	C₂³	1	8	yes	complemented
38	#95×C₈₉	C₂³	1	24	yes	complemented
39	C₂⁴×C₈₉	—	2	12	no	—
40	C₂⁴×C₈₉	—	2	24	no	—
41	C₂⁴×C₈₉	—	2	12	no	—
42	C₂⁴×C₈₉	—	2	12	no	—
43	C₂⁴×C₈₉	—	4	4	no	—
44	C₂⁴×C₈₉	—	4	12	no	—
45	C₂⁴×C₈₉	—	4	96	no	—
46	C₂⁴×C₈₉	—	4	16	no	—
47	C₂⁴×C₈₉	#41	1	12	yes	—
48	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
49	C₂⁴×C₈₉	C₂⁴	1	3	yes	—
50	C₂²×C₄×C₈₉	—	2	12	no	—
51	C₂²×C₄×C₈₉	—	2	12	no	—
52	C₂²×C₄×C₈₉	—	4	12	no	—
53	C₂²×C₄×C₈₉	—	4	12	no	—
54	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
55	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
56	C₄⋊C₄×C₈₉	—	4	24	no	—
57	C₂²⋊C₄×C₈₉	—	2	12	no	—
58	C₂²⋊C₄×C₈₉	—	2	12	no	—
59	C₂²⋊C₄×C₈₉	—	2	12	no	—
60	C₂²⋊C₄×C₈₉	—	2	12	no	—
61	C₂²⋊C₄×C₈₉	—	2	48	no	—
62	C₂²⋊C₄×C₈₉	—	4	24	no	—
63	C₄²×C₈₉	—	4	8	no	—
64	C₂³×C₈₉	—	2	12	no	—
65	C₂³×C₈₉	—	2	48	no	—
66	C₂³×C₈₉	—	2	12	no	—
67	C₂³×C₈₉	—	2	12	no	—
68	C₂³×C₈₉	—	4	24	no	—
69	C₂³×C₈₉	—	4	48	no	—
70	C₂³×C₈₉	—	4	48	no	—
71	C₂³×C₈₉	—	4	24	no	—
72	C₂³×C₈₉	—	4	128	no	—
73	C₂³×C₈₉	—	4	24	no	—
74	C₂³×C₈₉	#126	1	16	yes	complemented
75	C₂³×C₈₉	C₂×#41	1	6	yes	—
76	C₂³×C₈₉	C₂×#41	1	6	yes	—
77	C₂³×C₈₉	C₂×#41	1	6	yes	—
78	C₂³×C₈₉	H(5,𝔽₂)	1	8	yes	—
79	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
80	C₂×C₄×C₈₉	—	2	12	no	—
81	C₂×C₄×C₈₉	—	2	12	no	—
82	C₂×C₄×C₈₉	—	2	24	no	—
83	C₂×C₄×C₈₉	—	2	24	no	—
84	C₂×C₄×C₈₉	—	8	12	no	—
85	C₂×C₄×C₈₉	—	8	12	no	—
86	C₂²×C₈₉	—	2	12	no	—
87	C₂²×C₈₉	—	2	12	no	—
88	C₂²×C₈₉	—	2	12	no	—
89	C₂²×C₈₉	—	2	24	no	—
90	C₂²×C₈₉	—	4	16	no	—
91	C₂²×C₈₉	—	4	96	no	—
92	C₂²×C₈₉	—	4	4	no	—
93	C₂²×C₈₉	—	4	12	no	—
94	C₂²×C₈₉	C₂×#126	1	12	yes	—
95	C₂²×C₈₉	C₂²×#41	1	3	yes	—
96	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	4	yes	—
97	C₄×C₈₉	—	4	12	no	—
98	C₄×C₈₉	—	4	12	no	—
99	#35369	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂×C₈₉	—	2	12	no	—
101	C₂×C₈₉	—	4	8	no	—
102	C₂×C₈₉	C₂²×#126	1	1	yes	characteristic
103	C₂×C₈₉	#3490	1	6	yes	—
104	#2480	C₂×C₈₉	1	24	yes	complemented
105	#2315	C₂×C₈₉	1	3	yes	complemented
106	C₂×#379	C₂×C₈₉	1	3	yes	—
107	#2273	C₂×C₈₉	1	1	yes	characteristic, complemented
108	C₈₉	#35369	1	1	yes	characteristic, direct product factor, Sylow
109	C₂⁶	C₂²×C₈₉	1	1	yes	characteristic
110	C₂×#126	C₂²×C₈₉	1	8	yes	complemented
111	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	3	yes	—
112	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	3	yes	—
113	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	12	yes	—
114	#397	C₂²×C₈₉	1	24	yes	complemented
115	#396	C₂²×C₈₉	1	8	yes	complemented
116	#388	C₂²×C₈₉	1	24	yes	complemented
117	#385	C₂²×C₈₉	1	12	yes	complemented
118	#385	C₂²×C₈₉	1	12	yes	complemented
119	#380	C₂²×C₈₉	1	12	yes	complemented
120	#379	C₂²×C₈₉	1	12	yes	—
121	#379	C₂²×C₈₉	1	12	yes	—
122	C₄×C₂²⋊C₄	C₂²×C₈₉	1	12	yes	complemented
123	C₂⁵	—	2	12	no	—
124	C₂⁵	—	4	8	no	—
125	C₂⁵	C₂³×C₈₉	1	6	yes	—
126	C₂⁵	C₂³×C₈₉	1	1	yes	characteristic
127	C₂³×C₄	C₂³×C₈₉	1	6	yes	—
128	C₂³×C₄	C₂³×C₈₉	1	6	yes	—
129	#126	—	4	16	no	—
130	C₂×C₄⋊C₄	C₂³×C₈₉	1	24	yes	complemented
131	C₂×C₂²⋊C₄	—	2	12	no	—
132	C₂×C₂²⋊C₄	—	2	12	no	—
133	C₂×C₂²⋊C₄	—	4	24	no	—
134	C₂×C₂²⋊C₄	C₂³×C₈₉	1	24	yes	—
135	C₂×C₂²⋊C₄	C₂³×C₈₉	1	6	yes	—
136	C₂×C₂²⋊C₄	C₂³×C₈₉	1	6	yes	—
137	C₂×C₂²⋊C₄	C₂³×C₈₉	1	6	yes	—
138	C₂×C₂²⋊C₄	C₂³×C₈₉	1	6	yes	—
139	C₂×C₂²⋊C₄	C₂³×C₈₉	1	24	yes	—
140	C₂×C₄²	C₂³×C₈₉	1	8	yes	complemented
141	#95	C₂³×C₈₉	1	8	yes	complemented
142	#95	C₂³×C₈₉	1	24	yes	complemented
143	C₂⁴	—	2	12	no	—
144	C₂⁴	—	2	24	no	—
145	C₂⁴	—	2	12	no	—
146	C₂⁴	—	2	12	no	—
147	C₂⁴	—	4	4	no	—
148	C₂⁴	—	4	12	no	—
149	C₂⁴	—	4	96	no	—
150	C₂⁴	—	4	16	no	—
151	C₂⁴	#41×C₈₉	1	12	yes	—
152	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
153	C₂⁴	C₂⁴×C₈₉	1	3	yes	—
154	C₂²×C₄	—	2	12	no	—
155	C₂²×C₄	—	2	12	no	—
156	C₂²×C₄	—	4	12	no	—
157	C₂²×C₄	—	4	12	no	—
158	C₂²×C₄	C₂⁴×C₈₉	1	12	yes	—
159	C₂²×C₄	C₂⁴×C₈₉	1	12	yes	—
160	C₄⋊C₄	—	4	24	no	—
161	C₂²⋊C₄	—	2	12	no	—
162	C₂²⋊C₄	—	2	12	no	—
163	C₂²⋊C₄	—	2	12	no	—
164	C₂²⋊C₄	—	2	12	no	—
165	C₂²⋊C₄	—	2	48	no	—
166	C₂²⋊C₄	—	4	24	no	—
167	C₄²	—	4	8	no	—
168	C₂³	—	2	12	no	—
169	C₂³	—	2	48	no	—
170	C₂³	—	2	12	no	—
171	C₂³	—	2	12	no	—
172	C₂³	—	4	24	no	—
173	C₂³	—	4	48	no	—
174	C₂³	—	4	48	no	—
175	C₂³	—	4	24	no	—
176	C₂³	—	4	128	no	—
177	C₂³	—	4	24	no	—
178	C₂³	#126×C₈₉	1	16	yes	complemented
179	C₂³	C₂×#41×C₈₉	1	6	yes	—
180	C₂³	C₂×#41×C₈₉	1	6	yes	—
181	C₂³	C₂×#41×C₈₉	1	6	yes	—
182	C₂³	H(5,𝔽₂)×C₈₉	1	8	yes	—
183	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
184	C₂×C₄	—	2	12	no	—
185	C₂×C₄	—	2	12	no	—
186	C₂×C₄	—	2	24	no	—
187	C₂×C₄	—	2	24	no	—
188	C₂×C₄	—	8	12	no	—
189	C₂×C₄	—	8	12	no	—
190	C₂²	—	2	12	no	—
191	C₂²	—	2	12	no	—
192	C₂²	—	2	12	no	—
193	C₂²	—	2	24	no	—
194	C₂²	—	4	16	no	—
195	C₂²	—	4	96	no	—
196	C₂²	—	4	4	no	—
197	C₂²	—	4	12	no	—
198	C₂²	C₂×#126×C₈₉	1	12	yes	—
199	C₂²	C₂²×#41×C₈₉	1	3	yes	—
200	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	4	yes	—
201	C₄	—	4	12	no	—
202	C₄	—	4	12	no	—
203	C₂	—	2	12	no	—
204	C₂	—	4	8	no	—
205	C₂	C₂²×#126×C₈₉	1	1	yes	characteristic
206	C₂	C₈₉×#3490	1	6	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂⁶×C₈₉	C₂²
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. C₈₉×#2480	C₂	semidirect
2. C₈₉×#2315	C₂	semidirect
4. C₈₉×#2273	C₂	semidirect
6. C₂×#126×C₈₉	C₂²	semidirect
10. #397×C₈₉	C₂²	semidirect
11. #396×C₈₉	C₂²	semidirect
12. #388×C₈₉	C₂²	semidirect
13. #385×C₈₉	C₂²	semidirect
14. #385×C₈₉	C₂²	semidirect
15. #380×C₈₉	C₂²	semidirect
18. C₄×C₂²⋊C₄×C₈₉	C₂²	semidirect
26. C₂×C₄⋊C₄×C₈₉	C₂³	semidirect
36. C₂×C₄²×C₈₉	C₂³	semidirect
37. #95×C₈₉	C₂³	semidirect
38. #95×C₈₉	C₂³	semidirect
74. C₂³×C₈₉	#126	semidirect
99. #35369	C₈₉	direct (N × H)
104. #2480	C₂×C₈₉	semidirect
105. #2315	C₂×C₈₉	semidirect
107. #2273	C₂×C₈₉	semidirect
108. C₈₉	#35369	direct (N × H)
110. C₂×#126	C₂²×C₈₉	semidirect
114. #397	C₂²×C₈₉	semidirect
115. #396	C₂²×C₈₉	semidirect
116. #388	C₂²×C₈₉	semidirect
117. #385	C₂²×C₈₉	semidirect
118. #385	C₂²×C₈₉	semidirect
119. #380	C₂²×C₈₉	semidirect
122. C₄×C₂²⋊C₄	C₂²×C₈₉	semidirect
130. C₂×C₄⋊C₄	C₂³×C₈₉	semidirect
140. C₂×C₄²	C₂³×C₈₉	semidirect
141. #95	C₂³×C₈₉	semidirect
142. #95	C₂³×C₈₉	semidirect
178. C₂³	#126×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#2480	2
C₈₉×#2315	2
C₂×#379×C₈₉	2
C₈₉×#2273	2
#35369	89

Subgroups — C89×#35369 #36413217

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35369 #36413217