Subgroups — C₈₉×#33385 #36411233

Order 22784.

1114 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3584	C₂	1	1	yes	maximal, characteristic
2	C₂×#504×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2916	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2143	C₂	1	6	yes	maximal, complemented
5	C₂×#359×C₈₉	C₂	1	6	yes	maximal, complemented
6	#585×C₈₉	—	2	1	no	—
7	#578×C₈₉	C₄	1	4	yes	—
8	#577×C₈₉	—	2	4	no	—
9	#576×C₈₉	C₄	1	4	yes	—
10	#575×C₈₉	—	2	3	no	—
11	C₂×#135×C₈₉	C₂²	1	6	yes	—
12	#568×C₈₉	—	2	1	no	—
13	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
14	#504×C₈₉	—	2	4	no	—
15	C₂×M₃₂×C₈₉	C₂²	1	3	yes	complemented
16	C₂²×C₁₆×C₈₉	C₂²	1	3	yes	complemented
17	#430×C₈₉	—	2	3	no	—
18	C₂×#108×C₈₉	C₂²	1	6	yes	complemented
19	#359×C₈₉	C₂²	1	4	yes	complemented
20	#359×C₈₉	C₂²	1	12	yes	complemented
21	2₋¹⁺⁴×C₈₉	—	2	4	no	—
22	2₋¹⁺⁴×C₈₉	—	4	1	no	—
23	H(5,𝔽₂)×C₈₉	—	2	4	no	—
24	H(5,𝔽₂)×C₈₉	—	4	3	no	—
25	C₂×#41×C₈₉	—	2	6	no	—
26	C₂×#41×C₈₉	—	4	4	no	—
27	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
28	#137×C₈₉	—	2	12	no	—
29	AGL(1,ℤ/8)×C₈₉	—	2	12	no	—
30	#135×C₈₉	—	2	6	no	—
31	#135×C₈₉	—	2	6	no	—
32	#135×C₈₉	C₂×C₄	1	12	yes	—
33	#135×C₈₉	C₂³	1	4	yes	—
34	C₂×Q₁₆×C₈₉	C₂×C₄	1	6	yes	—
35	C₂×SD₁₆×C₈₉	—	2	6	no	—
36	C₂×D₈×C₈₉	C₂×C₄	1	6	yes	—
37	#131×C₈₉	—	4	4	no	—
38	#131×C₈₉	C₂×C₄	1	4	yes	—
39	#131×C₈₉	D₄	1	4	yes	—
40	C₂×M₁₆×C₈₉	—	2	6	no	—
41	C₂×M₁₆×C₈₉	D₄	1	3	yes	—
42	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
43	M₃₂×C₈₉	—	2	6	no	—
44	C₂×C₁₆×C₈₉	—	2	6	no	—
45	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
46	C₂×C₁₆×C₈₉	C₂³	1	3	yes	—
47	#108×C₈₉	—	2	6	no	—
48	#108×C₈₉	C₂³	1	4	yes	—
49	#41×C₈₉	—	2	12	no	—
50	#41×C₈₉	—	2	4	no	—
51	#41×C₈₉	—	4	4	no	—
52	#41×C₈₉	—	4	12	no	—
53	#41×C₈₉	—	4	6	no	—
54	#41×C₈₉	—	4	6	no	—
55	#41×C₈₉	C₂²⋊C₄	1	4	yes	—
56	#41×C₈₉	D₈	1	4	yes	—
57	C₂×Q₈×C₈₉	—	2	6	no	—
58	C₂×Q₈×C₈₉	—	4	4	no	—
59	C₂×Q₈×C₈₉	SD₁₆	1	1	yes	characteristic
60	C₂×D₄×C₈₉	—	2	6	no	—
61	C₂×D₄×C₈₉	—	4	6	no	—
62	C₂×D₄×C₈₉	—	4	12	no	—
63	C₂×D₄×C₈₉	SD₁₆	1	3	yes	—
64	C₂²×C₄×C₈₉	—	4	6	no	—
65	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
66	Q₁₆×C₈₉	—	2	6	no	—
67	Q₁₆×C₈₉	C₂²×C₄	1	4	yes	—
68	SD₁₆×C₈₉	—	2	12	no	—
69	SD₁₆×C₈₉	—	2	4	no	—
70	D₈×C₈₉	—	2	6	no	—
71	D₈×C₈₉	C₂²×C₄	1	4	yes	—
72	M₁₆×C₈₉	—	2	4	no	—
73	M₁₆×C₈₉	—	4	6	no	—
74	M₁₆×C₈₉	C₂²⋊C₄	1	12	yes	—
75	C₂×C₈×C₈₉	—	4	6	no	—
76	C₂×C₈×C₈₉	C₂²×C₄	1	6	yes	—
77	C₂×C₈×C₈₉	C₂×D₄	1	3	yes	—
78	C₂×C₈×C₈₉	C₂×D₄	1	6	yes	—
79	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
80	C₁₆×C₈₉	—	2	1	no	—
81	C₁₆×C₈₉	—	2	3	no	—
82	C₂³×C₈₉	—	4	6	no	—
83	C₂³×C₈₉	C₂×SD₁₆	1	3	yes	—
84	Q₈×C₈₉	—	2	4	no	—
85	Q₈×C₈₉	—	4	6	no	—
86	Q₈×C₈₉	#102	1	4	yes	—
87	D₄×C₈₉	—	2	4	no	—
88	D₄×C₈₉	—	4	4	no	—
89	D₄×C₈₉	—	4	12	no	—
90	D₄×C₈₉	—	4	6	no	—
91	D₄×C₈₉	#102	1	12	yes	—
92	C₂×C₄×C₈₉	—	4	6	no	—
93	C₂×C₄×C₈₉	—	4	4	no	—
94	C₂×C₄×C₈₉	—	4	12	no	—
95	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	6	yes	—
96	C₂×C₄×C₈₉	C₂×D₈	1	6	yes	—
97	C₂×C₄×C₈₉	C₂×SD₁₆	1	3	yes	—
98	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
99	C₈×C₈₉	—	4	4	no	—
100	C₈×C₈₉	C₂×C₂²⋊C₄	1	6	yes	—
101	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
102	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
103	C₂²×C₈₉	—	4	3	no	—
104	C₂²×C₈₉	—	4	4	no	—
105	C₂²×C₈₉	—	16	6	no	—
106	C₂²×C₈₉	C₂×#102	1	6	yes	—
107	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
108	C₄×C₈₉	—	4	4	no	—
109	C₄×C₈₉	C₂×#102	1	6	yes	—
110	C₄×C₈₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
111	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
112	#33385	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₈₉	—	2	1	no	—
114	C₂×C₈₉	—	2	6	no	—
115	C₂×C₈₉	—	8	4	no	—
116	C₂×C₈₉	C₂²×#102	1	1	yes	characteristic
117	#3584	C₂×C₈₉	1	1	yes	characteristic
118	C₂×#504	C₂×C₈₉	1	1	yes	characteristic, complemented
119	#2916	C₂×C₈₉	1	1	yes	characteristic, complemented
120	#2143	C₂×C₈₉	1	6	yes	complemented
121	C₂×#359	C₂×C₈₉	1	6	yes	complemented
122	C₈₉	#33385	1	1	yes	characteristic, direct product factor, Sylow
123	#585	—	2	1	no	—
124	#578	C₄×C₈₉	1	4	yes	—
125	#577	—	2	4	no	—
126	#576	C₄×C₈₉	1	4	yes	—
127	#575	—	2	3	no	—
128	C₂×#135	C₂²×C₈₉	1	6	yes	—
129	#568	—	2	1	no	—
130	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
131	#504	—	2	4	no	—
132	C₂×M₃₂	C₂²×C₈₉	1	3	yes	complemented
133	C₂²×C₁₆	C₂²×C₈₉	1	3	yes	complemented
134	#430	—	2	3	no	—
135	C₂×#108	C₂²×C₈₉	1	6	yes	complemented
136	#359	C₂²×C₈₉	1	4	yes	complemented
137	#359	C₂²×C₈₉	1	12	yes	complemented
138	2₋¹⁺⁴	—	2	4	no	—
139	2₋¹⁺⁴	—	4	1	no	—
140	H(5,𝔽₂)	—	2	4	no	—
141	H(5,𝔽₂)	—	4	3	no	—
142	C₂×#41	—	2	6	no	—
143	C₂×#41	—	4	4	no	—
144	C₂×#41	D₄×C₈₉	1	1	yes	characteristic
145	#137	—	2	12	no	—
146	AGL(1,ℤ/8)	—	2	12	no	—
147	#135	—	2	6	no	—
148	#135	—	2	6	no	—
149	#135	C₂×C₄×C₈₉	1	12	yes	—
150	#135	C₂³×C₈₉	1	4	yes	—
151	C₂×Q₁₆	C₂×C₄×C₈₉	1	6	yes	—
152	C₂×SD₁₆	—	2	6	no	—
153	C₂×D₈	C₂×C₄×C₈₉	1	6	yes	—
154	#131	—	4	4	no	—
155	#131	C₂×C₄×C₈₉	1	4	yes	—
156	#131	D₄×C₈₉	1	4	yes	—
157	C₂×M₁₆	—	2	6	no	—
158	C₂×M₁₆	D₄×C₈₉	1	3	yes	—
159	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
160	M₃₂	—	2	6	no	—
161	C₂×C₁₆	—	2	6	no	—
162	C₂×C₁₆	C₂³×C₈₉	1	1	yes	characteristic
163	C₂×C₁₆	C₂³×C₈₉	1	3	yes	—
164	#108	—	2	6	no	—
165	#108	C₂³×C₈₉	1	4	yes	—
166	#41	—	2	12	no	—
167	#41	—	2	4	no	—
168	#41	—	4	4	no	—
169	#41	—	4	12	no	—
170	#41	—	4	6	no	—
171	#41	—	4	6	no	—
172	#41	C₂²⋊C₄×C₈₉	1	4	yes	—
173	#41	D₈×C₈₉	1	4	yes	—
174	C₂×Q₈	—	2	6	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	SD₁₆×C₈₉	1	1	yes	characteristic
177	C₂×D₄	—	2	6	no	—
178	C₂×D₄	—	4	6	no	—
179	C₂×D₄	—	4	12	no	—
180	C₂×D₄	SD₁₆×C₈₉	1	3	yes	—
181	C₂²×C₄	—	4	6	no	—
182	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
183	Q₁₆	—	2	6	no	—
184	Q₁₆	C₂²×C₄×C₈₉	1	4	yes	—
185	SD₁₆	—	2	12	no	—
186	SD₁₆	—	2	4	no	—
187	D₈	—	2	6	no	—
188	D₈	C₂²×C₄×C₈₉	1	4	yes	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	4	6	no	—
191	M₁₆	C₂²⋊C₄×C₈₉	1	12	yes	—
192	C₂×C₈	—	4	6	no	—
193	C₂×C₈	C₂²×C₄×C₈₉	1	6	yes	—
194	C₂×C₈	C₂×D₄×C₈₉	1	3	yes	—
195	C₂×C₈	C₂×D₄×C₈₉	1	6	yes	—
196	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
197	C₁₆	—	2	1	no	—
198	C₁₆	—	2	3	no	—
199	C₂³	—	4	6	no	—
200	C₂³	C₂×SD₁₆×C₈₉	1	3	yes	—
201	Q₈	—	2	4	no	—
202	Q₈	—	4	6	no	—
203	Q₈	#102×C₈₉	1	4	yes	—
204	D₄	—	2	4	no	—
205	D₄	—	4	12	no	—
206	D₄	—	4	4	no	—
207	D₄	—	4	6	no	—
208	D₄	#102×C₈₉	1	12	yes	—
209	C₂×C₄	—	4	6	no	—
210	C₂×C₄	—	4	12	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	6	yes	—
213	C₂×C₄	C₂×D₈×C₈₉	1	6	yes	—
214	C₂×C₄	C₂×SD₁₆×C₈₉	1	3	yes	—
215	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
216	C₈	—	4	4	no	—
217	C₈	C₂×C₂²⋊C₄×C₈₉	1	6	yes	—
218	C₈	C₂³×C₄×C₈₉	1	1	yes	characteristic
219	C₈	C₂²×D₄×C₈₉	1	1	yes	characteristic
220	C₂²	—	4	3	no	—
221	C₂²	—	4	4	no	—
222	C₂²	—	16	6	no	—
223	C₂²	C₂×#102×C₈₉	1	6	yes	—
224	C₂²	C₂²×SD₁₆×C₈₉	1	1	yes	characteristic
225	C₄	—	4	4	no	—
226	C₄	C₂×#102×C₈₉	1	6	yes	—
227	C₄	C₂²×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
228	C₄	C₂²×D₈×C₈₉	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	6	no	—
231	C₂	—	8	4	no	—
232	C₂	C₂²×#102×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈₉	C₂²×D₈
commutator	C₈	C₂³×C₄×C₈₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₈₉
socle	C₂×C₈₉	C₂²×#102
prime radical	C₈₉×#3584	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈₉ ◁ C₂×#41×C₈₉ ◁ C₂×#131×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#504×C₈₉	C₂	semidirect
3. C₈₉×#2916	C₂	semidirect
4. C₈₉×#2143	C₂	semidirect
5. C₂×#359×C₈₉	C₂	semidirect
15. C₂×M₃₂×C₈₉	C₂²	semidirect
16. C₂²×C₁₆×C₈₉	C₂²	semidirect
18. C₂×#108×C₈₉	C₂²	semidirect
19. #359×C₈₉	C₂²	semidirect
20. #359×C₈₉	C₂²	semidirect
112. #33385	C₈₉	direct (N × H)
118. C₂×#504	C₂×C₈₉	semidirect
119. #2916	C₂×C₈₉	semidirect
120. #2143	C₂×C₈₉	semidirect
121. C₂×#359	C₂×C₈₉	semidirect
122. C₈₉	#33385	direct (N × H)
132. C₂×M₃₂	C₂²×C₈₉	semidirect
133. C₂²×C₁₆	C₂²×C₈₉	semidirect
135. C₂×#108	C₂²×C₈₉	semidirect
136. #359	C₂²×C₈₉	semidirect
137. #359	C₂²×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3584	2
C₂×#504×C₈₉	2
C₈₉×#2916	2
C₈₉×#2143	2
C₂×#359×C₈₉	2
#33385	89

Subgroups — C89×#33385 #36411233

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#33385 #36411233