Subgroups — C₂×C₈₉×#3585 #36440936

Order 22784.

11138 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#585×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×H(5,𝔽₂)×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3585	C₂	1	32	yes	maximal, direct product factor
4	C₂×#577×C₈₉	C₂	1	8	yes	maximal, complemented
5	C₂×#576×C₈₉	C₂	1	8	yes	maximal, complemented
6	C₂×#575×C₈₉	C₂	1	6	yes	maximal, complemented
7	C₂²×AGL(1,ℤ/8)×C₈₉	C₂	1	6	yes	maximal, complemented
8	C₂×#568×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	#585×C₈₉	—	2	8	no	—
10	#585×C₈₉	C₂²	1	16	yes	complemented
11	C₂×2₋¹⁺⁴×C₈₉	—	2	1	no	—
12	C₂×2₋¹⁺⁴×C₈₉	C₂²	1	4	yes	complemented
13	C₂×H(5,𝔽₂)×C₈₉	—	2	4	no	—
14	C₂×H(5,𝔽₂)×C₈₉	—	2	8	no	—
15	C₂×H(5,𝔽₂)×C₈₉	—	2	3	no	—
16	C₂×H(5,𝔽₂)×C₈₉	C₂²	1	8	yes	complemented
17	C₂×H(5,𝔽₂)×C₈₉	C₂²	1	4	yes	complemented
18	C₂×H(5,𝔽₂)×C₈₉	C₂²	1	16	yes	complemented
19	C₂²×#41×C₈₉	—	2	4	no	—
20	C₂²×#41×C₈₉	—	2	1	no	—
21	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
22	C₂²×#41×C₈₉	C₂²	1	3	yes	complemented
23	C₂²×#41×C₈₉	C₂²	1	6	yes	complemented
24	C₂³×D₄×C₈₉	—	2	3	no	—
25	C₂³×D₄×C₈₉	C₂²	1	3	yes	complemented
26	#577×C₈₉	C₂²	1	128	yes	complemented
27	#576×C₈₉	C₂²	1	128	yes	complemented
28	#575×C₈₉	C₂²	1	96	yes	complemented
29	C₂×#137×C₈₉	C₂²	1	4	yes	complemented
30	C₂×#137×C₈₉	C₂²	1	12	yes	complemented
31	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	12	yes	complemented
32	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	24	yes	complemented
33	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	12	yes	complemented
34	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	96	yes	complemented
35	C₂×#135×C₈₉	C₂²	1	24	yes	complemented
36	C₂²×SD₁₆×C₈₉	C₂²	1	12	yes	complemented
37	C₂²×D₈×C₈₉	C₂²	1	12	yes	complemented
38	#568×C₈₉	C₂²	1	16	yes	complemented
39	C₂×#131×C₈₉	C₂²	1	8	yes	complemented
40	C₂²×M₁₆×C₈₉	C₂²	1	6	yes	complemented
41	C₂⁵×C₈₉	—	2	3	no	—
42	2₋¹⁺⁴×C₈₉	—	2	16	no	—
43	2₋¹⁺⁴×C₈₉	—	2	16	no	—
44	2₋¹⁺⁴×C₈₉	C₂³	1	32	yes	complemented
45	H(5,𝔽₂)×C₈₉	—	2	32	no	—
46	H(5,𝔽₂)×C₈₉	—	2	64	no	—
47	H(5,𝔽₂)×C₈₉	—	2	16	no	—
48	H(5,𝔽₂)×C₈₉	—	2	48	no	—
49	H(5,𝔽₂)×C₈₉	C₂³	1	64	yes	complemented
50	H(5,𝔽₂)×C₈₉	C₂³	1	32	yes	complemented
51	C₂×#41×C₈₉	—	2	64	no	—
52	C₂×#41×C₈₉	—	2	8	no	—
53	C₂×#41×C₈₉	—	2	16	no	—
54	C₂×#41×C₈₉	—	2	4	no	—
55	C₂×#41×C₈₉	—	2	12	no	—
56	C₂×#41×C₈₉	—	2	6	no	—
57	C₂×#41×C₈₉	—	2	6	no	—
58	C₂×#41×C₈₉	—	2	12	no	—
59	C₂×#41×C₈₉	—	2	6	no	—
60	C₂×#41×C₈₉	—	2	24	no	—
61	C₂×#41×C₈₉	D₄	1	4	yes	—
62	C₂×#41×C₈₉	D₄	1	8	yes	—
63	C₂×#41×C₈₉	C₂³	1	12	yes	—
64	C₂×#41×C₈₉	C₂³	1	4	yes	—
65	C₂×#41×C₈₉	C₂³	1	4	yes	—
66	C₂×#41×C₈₉	C₂³	1	8	yes	—
67	C₂×#41×C₈₉	C₂³	1	24	yes	complemented
68	C₂×#41×C₈₉	C₂³	1	12	yes	complemented
69	C₂×#41×C₈₉	C₂³	1	48	yes	complemented
70	C₂²×Q₈×C₈₉	—	2	4	no	—
71	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
72	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₂²×Q₈×C₈₉	C₂³	1	6	yes	complemented
74	C₂²×D₄×C₈₉	—	2	3	no	—
75	C₂²×D₄×C₈₉	—	2	6	no	—
76	C₂²×D₄×C₈₉	—	2	24	no	—
77	C₂²×D₄×C₈₉	—	2	48	no	—
78	C₂²×D₄×C₈₉	—	2	12	no	—
79	C₂²×D₄×C₈₉	—	2	3	no	—
80	C₂²×D₄×C₈₉	—	2	6	no	—
81	C₂²×D₄×C₈₉	—	2	12	no	—
82	C₂²×D₄×C₈₉	D₄	1	3	yes	—
83	C₂²×D₄×C₈₉	C₂³	1	12	yes	—
84	C₂²×D₄×C₈₉	C₂³	1	3	yes	—
85	C₂²×D₄×C₈₉	C₂³	1	24	yes	complemented
86	C₂²×D₄×C₈₉	C₂³	1	6	yes	complemented
87	C₂³×C₄×C₈₉	—	2	3	no	—
88	C₂³×C₄×C₈₉	—	2	6	no	—
89	C₂³×C₄×C₈₉	C₂³	1	3	yes	—
90	#137×C₈₉	C₂³	1	32	yes	complemented
91	#137×C₈₉	C₂³	1	96	yes	complemented
92	AGL(1,ℤ/8)×C₈₉	C₂³	1	96	yes	complemented
93	AGL(1,ℤ/8)×C₈₉	C₂³	1	192	yes	complemented
94	AGL(1,ℤ/8)×C₈₉	C₂³	1	96	yes	complemented
95	#135×C₈₉	C₂³	1	192	yes	complemented
96	C₂×Q₁₆×C₈₉	C₂³	1	8	yes	complemented
97	C₂×SD₁₆×C₈₉	C₂³	1	24	yes	complemented
98	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	complemented
99	C₂×SD₁₆×C₈₉	C₂³	1	96	yes	complemented
100	C₂×D₈×C₈₉	C₂³	1	24	yes	complemented
101	C₂×D₈×C₈₉	C₂³	1	96	yes	complemented
102	#131×C₈₉	C₂³	1	64	yes	complemented
103	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
104	C₂×M₁₆×C₈₉	C₂³	1	12	yes	complemented
105	C₂×M₁₆×C₈₉	C₂³	1	48	yes	complemented
106	C₂²×C₈×C₈₉	C₂³	1	12	yes	complemented
107	C₂⁴×C₈₉	—	2	3	no	—
108	C₂⁴×C₈₉	—	2	6	no	—
109	C₂⁴×C₈₉	—	2	12	no	—
110	C₂⁴×C₈₉	—	2	24	no	—
111	C₂⁴×C₈₉	—	8	12	no	—
112	C₂⁴×C₈₉	C₂×D₄	1	3	yes	—
113	#41×C₈₉	—	2	64	no	—
114	#41×C₈₉	—	2	32	no	—
115	#41×C₈₉	—	2	96	no	—
116	#41×C₈₉	—	2	8	no	—
117	#41×C₈₉	—	2	48	no	—
118	#41×C₈₉	—	2	24	no	—
119	#41×C₈₉	—	2	48	no	—
120	#41×C₈₉	—	2	48	no	—
121	#41×C₈₉	—	2	24	no	—
122	#41×C₈₉	C₂×D₄	1	32	yes	—
123	#41×C₈₉	C₂×D₄	1	16	yes	—
124	#41×C₈₉	C₂⁴	1	48	yes	—
125	#41×C₈₉	C₂⁴	1	16	yes	—
126	#41×C₈₉	C₂⁴	1	16	yes	—
127	#41×C₈₉	C₂⁴	1	48	yes	complemented
128	C₂×Q₈×C₈₉	—	2	32	no	—
129	C₂×Q₈×C₈₉	—	2	6	no	—
130	C₂×Q₈×C₈₉	—	2	2	no	—
131	C₂×Q₈×C₈₉	—	2	1	no	—
132	C₂×Q₈×C₈₉	—	2	12	no	—
133	C₂×Q₈×C₈₉	C₂×D₄	1	4	yes	—
134	C₂×Q₈×C₈₉	C₂×D₄	1	8	yes	—
135	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
136	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
137	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
138	C₂×Q₈×C₈₉	C₂⁴	1	24	yes	complemented
139	C₂×D₄×C₈₉	—	2	12	no	—
140	C₂×D₄×C₈₉	—	2	24	no	—
141	C₂×D₄×C₈₉	—	2	4	no	—
142	C₂×D₄×C₈₉	—	2	12	no	—
143	C₂×D₄×C₈₉	—	2	48	no	—
144	C₂×D₄×C₈₉	—	2	192	no	—
145	C₂×D₄×C₈₉	—	2	96	no	—
146	C₂×D₄×C₈₉	—	2	6	no	—
147	C₂×D₄×C₈₉	—	2	3	no	—
148	C₂×D₄×C₈₉	—	2	12	no	—
149	C₂×D₄×C₈₉	—	2	24	no	—
150	C₂×D₄×C₈₉	—	2	24	no	—
151	C₂×D₄×C₈₉	—	2	6	no	—
152	C₂×D₄×C₈₉	—	2	48	no	—
153	C₂×D₄×C₈₉	—	2	12	no	—
154	C₂×D₄×C₈₉	C₂×D₄	1	12	yes	—
155	C₂×D₄×C₈₉	C₂×D₄	1	24	yes	—
156	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
157	C₂×D₄×C₈₉	C₂⁴	1	6	yes	—
158	C₂×D₄×C₈₉	C₂⁴	1	48	yes	—
159	C₂×D₄×C₈₉	C₂⁴	1	12	yes	—
160	C₂×D₄×C₈₉	C₂⁴	1	24	yes	complemented
161	C₂²×C₄×C₈₉	—	2	1	no	—
162	C₂²×C₄×C₈₉	—	2	6	no	—
163	C₂²×C₄×C₈₉	—	2	4	no	—
164	C₂²×C₄×C₈₉	—	2	12	no	—
165	C₂²×C₄×C₈₉	—	2	24	no	—
166	C₂²×C₄×C₈₉	—	2	12	no	—
167	C₂²×C₄×C₈₉	—	2	48	no	—
168	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
169	C₂²×C₄×C₈₉	C₂×D₄	1	6	yes	—
170	C₂²×C₄×C₈₉	C₂×D₄	1	12	yes	—
171	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
172	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
173	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
174	Q₁₆×C₈₉	C₂⁴	1	32	yes	complemented
175	SD₁₆×C₈₉	C₂⁴	1	96	yes	complemented
176	SD₁₆×C₈₉	C₂⁴	1	32	yes	complemented
177	D₈×C₈₉	C₂⁴	1	96	yes	complemented
178	M₁₆×C₈₉	C₂⁴	1	16	yes	—
179	M₁₆×C₈₉	C₂⁴	1	48	yes	complemented
180	C₂×C₈×C₈₉	C₂⁴	1	8	yes	—
181	C₂×C₈×C₈₉	C₂⁴	1	48	yes	complemented
182	C₂³×C₈₉	—	2	12	no	—
183	C₂³×C₈₉	—	2	4	no	—
184	C₂³×C₈₉	—	2	6	no	—
185	C₂³×C₈₉	—	2	24	no	—
186	C₂³×C₈₉	—	2	48	no	—
187	C₂³×C₈₉	—	4	6	no	—
188	C₂³×C₈₉	—	4	12	no	—
189	C₂³×C₈₉	—	8	12	no	—
190	C₂³×C₈₉	—	8	24	no	—
191	C₂³×C₈₉	—	8	48	no	—
192	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
193	C₂³×C₈₉	C₂²×D₄	1	6	yes	—
194	C₂³×C₈₉	C₂²×D₄	1	12	yes	—
195	Q₈×C₈₉	—	2	4	no	—
196	Q₈×C₈₉	—	2	24	no	—
197	Q₈×C₈₉	—	2	4	no	—
198	Q₈×C₈₉	—	2	2	no	—
199	Q₈×C₈₉	C₂²×D₄	1	16	yes	—
200	Q₈×C₈₉	C₂⁵	1	8	yes	—
201	Q₈×C₈₉	C₂⁵	1	4	yes	—
202	D₄×C₈₉	—	2	48	no	—
203	D₄×C₈₉	—	2	16	no	—
204	D₄×C₈₉	—	2	48	no	—
205	D₄×C₈₉	—	2	12	no	—
206	D₄×C₈₉	—	2	48	no	—
207	D₄×C₈₉	—	2	6	no	—
208	D₄×C₈₉	—	2	4	no	—
209	D₄×C₈₉	—	2	24	no	—
210	D₄×C₈₉	C₂²×D₄	1	48	yes	—
211	D₄×C₈₉	C₂⁵	1	8	yes	—
212	D₄×C₈₉	C₂⁵	1	12	yes	—
213	C₂×C₄×C₈₉	—	2	4	no	—
214	C₂×C₄×C₈₉	—	2	24	no	—
215	C₂×C₄×C₈₉	—	2	2	no	—
216	C₂×C₄×C₈₉	—	2	4	no	—
217	C₂×C₄×C₈₉	—	2	16	no	—
218	C₂×C₄×C₈₉	—	2	48	no	—
219	C₂×C₄×C₈₉	—	2	48	no	—
220	C₂×C₄×C₈₉	C₂²×D₄	1	12	yes	—
221	C₂×C₄×C₈₉	C₂²×D₄	1	24	yes	—
222	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
223	C₂×C₄×C₈₉	C₂²×D₄	1	12	yes	—
224	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₈₉	C₂²×D₄	1	6	yes	—
226	C₂×C₄×C₈₉	C₂⁵	1	12	yes	—
227	C₂×C₄×C₈₉	C₂⁵	1	2	yes	—
228	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
229	C₈×C₈₉	C₂⁵	1	16	yes	—
230	C₂²×C₈₉	—	2	2	no	—
231	C₂²×C₈₉	—	2	12	no	—
232	C₂²×C₈₉	—	2	8	no	—
233	C₂²×C₈₉	—	2	12	no	—
234	C₂²×C₈₉	—	4	12	no	—
235	C₂²×C₈₉	—	4	24	no	—
236	C₂²×C₈₉	—	4	12	no	—
237	C₂²×C₈₉	—	4	8	no	—
238	C₂²×C₈₉	—	8	24	no	—
239	C₂²×C₈₉	—	8	48	no	—
240	C₂²×C₈₉	C₂³×D₄	1	2	yes	—
241	C₂²×C₈₉	C₂³×D₄	1	12	yes	—
242	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
243	C₄×C₈₉	—	2	4	no	—
244	C₄×C₈₉	—	2	8	no	—
245	C₄×C₈₉	C₂³×D₄	1	2	yes	—
246	C₄×C₈₉	C₂³×D₄	1	12	yes	—
247	C₄×C₈₉	C₂³×D₄	1	1	yes	characteristic
248	C₄×C₈₉	C₂⁶	1	1	yes	characteristic
249	C₂×#3585	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
250	C₂×C₈₉	—	2	2	no	—
251	C₂×C₈₉	—	2	12	no	—
252	C₂×C₈₉	—	4	12	no	—
253	C₂×C₈₉	—	4	8	no	—
254	C₂×C₈₉	#3585	1	2	yes	direct product factor
255	C₂×C₈₉	C₂⁴×D₄	1	1	yes	characteristic
256	C₂×#585	C₂×C₈₉	1	1	yes	characteristic, complemented
257	C₂²×H(5,𝔽₂)	C₂×C₈₉	1	1	yes	characteristic, complemented
258	#3585	C₂×C₈₉	1	32	yes	direct product factor
259	C₂×#577	C₂×C₈₉	1	8	yes	complemented
260	C₂×#576	C₂×C₈₉	1	8	yes	complemented
261	C₂×#575	C₂×C₈₉	1	6	yes	complemented
262	C₂²×AGL(1,ℤ/8)	C₂×C₈₉	1	6	yes	complemented
263	C₂×#568	C₂×C₈₉	1	1	yes	characteristic, complemented
264	C₈₉	C₂×#3585	1	1	yes	characteristic, direct product factor, Sylow
265	#585	—	2	8	no	—
266	#585	C₂²×C₈₉	1	16	yes	complemented
267	C₂×2₋¹⁺⁴	—	2	1	no	—
268	C₂×2₋¹⁺⁴	C₂²×C₈₉	1	4	yes	complemented
269	C₂×H(5,𝔽₂)	—	2	4	no	—
270	C₂×H(5,𝔽₂)	—	2	8	no	—
271	C₂×H(5,𝔽₂)	—	2	3	no	—
272	C₂×H(5,𝔽₂)	C₂²×C₈₉	1	8	yes	complemented
273	C₂×H(5,𝔽₂)	C₂²×C₈₉	1	16	yes	complemented
274	C₂×H(5,𝔽₂)	C₂²×C₈₉	1	4	yes	complemented
275	C₂²×#41	—	2	4	no	—
276	C₂²×#41	—	2	1	no	—
277	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
278	C₂²×#41	C₂²×C₈₉	1	3	yes	complemented
279	C₂²×#41	C₂²×C₈₉	1	6	yes	complemented
280	C₂³×D₄	—	2	3	no	—
281	C₂³×D₄	C₂²×C₈₉	1	3	yes	complemented
282	#577	C₂²×C₈₉	1	128	yes	complemented
283	#576	C₂²×C₈₉	1	128	yes	complemented
284	#575	C₂²×C₈₉	1	96	yes	complemented
285	C₂×#137	C₂²×C₈₉	1	4	yes	complemented
286	C₂×#137	C₂²×C₈₉	1	12	yes	complemented
287	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	12	yes	complemented
288	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	24	yes	complemented
289	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	12	yes	complemented
290	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	96	yes	complemented
291	C₂×#135	C₂²×C₈₉	1	24	yes	complemented
292	C₂²×SD₁₆	C₂²×C₈₉	1	12	yes	complemented
293	C₂²×D₈	C₂²×C₈₉	1	12	yes	complemented
294	#568	C₂²×C₈₉	1	16	yes	complemented
295	C₂×#131	C₂²×C₈₉	1	8	yes	complemented
296	C₂²×M₁₆	C₂²×C₈₉	1	6	yes	complemented
297	C₂⁵	—	2	3	no	—
298	2₋¹⁺⁴	—	2	16	no	—
299	2₋¹⁺⁴	—	2	16	no	—
300	2₋¹⁺⁴	C₂³×C₈₉	1	32	yes	complemented
301	H(5,𝔽₂)	—	2	32	no	—
302	H(5,𝔽₂)	—	2	64	no	—
303	H(5,𝔽₂)	—	2	16	no	—
304	H(5,𝔽₂)	—	2	48	no	—
305	H(5,𝔽₂)	C₂³×C₈₉	1	64	yes	complemented
306	H(5,𝔽₂)	C₂³×C₈₉	1	32	yes	complemented
307	C₂×#41	—	2	64	no	—
308	C₂×#41	—	2	8	no	—
309	C₂×#41	—	2	16	no	—
310	C₂×#41	—	2	4	no	—
311	C₂×#41	—	2	12	no	—
312	C₂×#41	—	2	6	no	—
313	C₂×#41	—	2	6	no	—
314	C₂×#41	—	2	12	no	—
315	C₂×#41	—	2	6	no	—
316	C₂×#41	—	2	24	no	—
317	C₂×#41	D₄×C₈₉	1	4	yes	—
318	C₂×#41	D₄×C₈₉	1	8	yes	—
319	C₂×#41	C₂³×C₈₉	1	12	yes	—
320	C₂×#41	C₂³×C₈₉	1	4	yes	—
321	C₂×#41	C₂³×C₈₉	1	4	yes	—
322	C₂×#41	C₂³×C₈₉	1	8	yes	—
323	C₂×#41	C₂³×C₈₉	1	24	yes	complemented
324	C₂×#41	C₂³×C₈₉	1	12	yes	complemented
325	C₂×#41	C₂³×C₈₉	1	48	yes	complemented
326	C₂²×Q₈	—	2	4	no	—
327	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
328	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
329	C₂²×Q₈	C₂³×C₈₉	1	6	yes	complemented
330	C₂²×D₄	—	2	3	no	—
331	C₂²×D₄	—	2	6	no	—
332	C₂²×D₄	—	2	24	no	—
333	C₂²×D₄	—	2	48	no	—
334	C₂²×D₄	—	2	12	no	—
335	C₂²×D₄	—	2	3	no	—
336	C₂²×D₄	—	2	6	no	—
337	C₂²×D₄	—	2	12	no	—
338	C₂²×D₄	D₄×C₈₉	1	3	yes	—
339	C₂²×D₄	C₂³×C₈₉	1	12	yes	—
340	C₂²×D₄	C₂³×C₈₉	1	3	yes	—
341	C₂²×D₄	C₂³×C₈₉	1	24	yes	complemented
342	C₂²×D₄	C₂³×C₈₉	1	6	yes	complemented
343	C₂³×C₄	—	2	3	no	—
344	C₂³×C₄	—	2	6	no	—
345	C₂³×C₄	C₂³×C₈₉	1	3	yes	—
346	#137	C₂³×C₈₉	1	32	yes	complemented
347	#137	C₂³×C₈₉	1	96	yes	complemented
348	AGL(1,ℤ/8)	C₂³×C₈₉	1	96	yes	complemented
349	AGL(1,ℤ/8)	C₂³×C₈₉	1	192	yes	complemented
350	AGL(1,ℤ/8)	C₂³×C₈₉	1	96	yes	complemented
351	#135	C₂³×C₈₉	1	192	yes	complemented
352	C₂×Q₁₆	C₂³×C₈₉	1	8	yes	complemented
353	C₂×SD₁₆	C₂³×C₈₉	1	24	yes	complemented
354	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	complemented
355	C₂×SD₁₆	C₂³×C₈₉	1	96	yes	complemented
356	C₂×D₈	C₂³×C₈₉	1	24	yes	complemented
357	C₂×D₈	C₂³×C₈₉	1	96	yes	complemented
358	#131	C₂³×C₈₉	1	64	yes	complemented
359	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
360	C₂×M₁₆	C₂³×C₈₉	1	12	yes	complemented
361	C₂×M₁₆	C₂³×C₈₉	1	48	yes	complemented
362	C₂²×C₈	C₂³×C₈₉	1	12	yes	complemented
363	C₂⁴	—	2	3	no	—
364	C₂⁴	—	2	6	no	—
365	C₂⁴	—	2	12	no	—
366	C₂⁴	—	2	24	no	—
367	C₂⁴	—	8	12	no	—
368	C₂⁴	C₂×D₄×C₈₉	1	3	yes	—
369	#41	—	2	64	no	—
370	#41	—	2	32	no	—
371	#41	—	2	96	no	—
372	#41	—	2	8	no	—
373	#41	—	2	48	no	—
374	#41	—	2	24	no	—
375	#41	—	2	48	no	—
376	#41	—	2	48	no	—
377	#41	—	2	24	no	—
378	#41	C₂×D₄×C₈₉	1	32	yes	—
379	#41	C₂×D₄×C₈₉	1	16	yes	—
380	#41	C₂⁴×C₈₉	1	48	yes	—
381	#41	C₂⁴×C₈₉	1	16	yes	—
382	#41	C₂⁴×C₈₉	1	16	yes	—
383	#41	C₂⁴×C₈₉	1	48	yes	complemented
384	C₂×Q₈	—	2	32	no	—
385	C₂×Q₈	—	2	6	no	—
386	C₂×Q₈	—	2	1	no	—
387	C₂×Q₈	—	2	2	no	—
388	C₂×Q₈	—	2	12	no	—
389	C₂×Q₈	C₂×D₄×C₈₉	1	4	yes	—
390	C₂×Q₈	C₂×D₄×C₈₉	1	8	yes	—
391	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
392	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
393	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
394	C₂×Q₈	C₂⁴×C₈₉	1	24	yes	complemented
395	C₂×D₄	—	2	12	no	—
396	C₂×D₄	—	2	24	no	—
397	C₂×D₄	—	2	4	no	—
398	C₂×D₄	—	2	12	no	—
399	C₂×D₄	—	2	48	no	—
400	C₂×D₄	—	2	192	no	—
401	C₂×D₄	—	2	96	no	—
402	C₂×D₄	—	2	3	no	—
403	C₂×D₄	—	2	12	no	—
404	C₂×D₄	—	2	6	no	—
405	C₂×D₄	—	2	24	no	—
406	C₂×D₄	—	2	24	no	—
407	C₂×D₄	—	2	6	no	—
408	C₂×D₄	—	2	48	no	—
409	C₂×D₄	—	2	12	no	—
410	C₂×D₄	C₂×D₄×C₈₉	1	12	yes	—
411	C₂×D₄	C₂×D₄×C₈₉	1	24	yes	—
412	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
413	C₂×D₄	C₂⁴×C₈₉	1	6	yes	—
414	C₂×D₄	C₂⁴×C₈₉	1	48	yes	—
415	C₂×D₄	C₂⁴×C₈₉	1	12	yes	—
416	C₂×D₄	C₂⁴×C₈₉	1	24	yes	complemented
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	6	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	12	no	—
421	C₂²×C₄	—	2	24	no	—
422	C₂²×C₄	—	2	12	no	—
423	C₂²×C₄	—	2	48	no	—
424	C₂²×C₄	C₂×D₄×C₈₉	1	3	yes	—
425	C₂²×C₄	C₂×D₄×C₈₉	1	6	yes	—
426	C₂²×C₄	C₂×D₄×C₈₉	1	12	yes	—
427	C₂²×C₄	C₂⁴×C₈₉	1	6	yes	—
428	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
429	C₂²×C₄	C₂⁴×C₈₉	1	12	yes	—
430	Q₁₆	C₂⁴×C₈₉	1	32	yes	complemented
431	SD₁₆	C₂⁴×C₈₉	1	96	yes	complemented
432	SD₁₆	C₂⁴×C₈₉	1	32	yes	complemented
433	D₈	C₂⁴×C₈₉	1	96	yes	complemented
434	M₁₆	C₂⁴×C₈₉	1	16	yes	—
435	M₁₆	C₂⁴×C₈₉	1	48	yes	complemented
436	C₂×C₈	C₂⁴×C₈₉	1	8	yes	—
437	C₂×C₈	C₂⁴×C₈₉	1	48	yes	complemented
438	C₂³	—	2	4	no	—
439	C₂³	—	2	12	no	—
440	C₂³	—	2	6	no	—
441	C₂³	—	2	24	no	—
442	C₂³	—	2	48	no	—
443	C₂³	—	4	6	no	—
444	C₂³	—	4	12	no	—
445	C₂³	—	8	12	no	—
446	C₂³	—	8	24	no	—
447	C₂³	—	8	48	no	—
448	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
449	C₂³	C₂²×D₄×C₈₉	1	6	yes	—
450	C₂³	C₂²×D₄×C₈₉	1	12	yes	—
451	Q₈	—	2	4	no	—
452	Q₈	—	2	24	no	—
453	Q₈	—	2	4	no	—
454	Q₈	—	2	2	no	—
455	Q₈	C₂²×D₄×C₈₉	1	16	yes	—
456	Q₈	C₂⁵×C₈₉	1	8	yes	—
457	Q₈	C₂⁵×C₈₉	1	4	yes	—
458	D₄	—	2	48	no	—
459	D₄	—	2	16	no	—
460	D₄	—	2	48	no	—
461	D₄	—	2	12	no	—
462	D₄	—	2	48	no	—
463	D₄	—	2	6	no	—
464	D₄	—	2	4	no	—
465	D₄	—	2	24	no	—
466	D₄	C₂²×D₄×C₈₉	1	48	yes	—
467	D₄	C₂⁵×C₈₉	1	8	yes	—
468	D₄	C₂⁵×C₈₉	1	12	yes	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	24	no	—
473	C₂×C₄	—	2	16	no	—
474	C₂×C₄	—	2	48	no	—
475	C₂×C₄	—	2	48	no	—
476	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
477	C₂×C₄	C₂²×D₄×C₈₉	1	6	yes	—
478	C₂×C₄	C₂²×D₄×C₈₉	1	12	yes	—
479	C₂×C₄	C₂²×D₄×C₈₉	1	24	yes	—
480	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
481	C₂×C₄	C₂²×D₄×C₈₉	1	12	yes	—
482	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
483	C₂×C₄	C₂⁵×C₈₉	1	12	yes	—
484	C₂×C₄	C₂⁵×C₈₉	1	2	yes	—
485	C₈	C₂⁵×C₈₉	1	16	yes	—
486	C₂²	—	2	2	no	—
487	C₂²	—	2	12	no	—
488	C₂²	—	2	8	no	—
489	C₂²	—	2	12	no	—
490	C₂²	—	4	12	no	—
491	C₂²	—	4	24	no	—
492	C₂²	—	4	12	no	—
493	C₂²	—	4	8	no	—
494	C₂²	—	8	48	no	—
495	C₂²	—	8	24	no	—
496	C₂²	C₂³×D₄×C₈₉	1	2	yes	—
497	C₂²	C₂³×D₄×C₈₉	1	12	yes	—
498	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
499	C₄	—	2	4	no	—
500	C₄	—	2	8	no	—
501	C₄	C₂³×D₄×C₈₉	1	2	yes	—
502	C₄	C₂³×D₄×C₈₉	1	12	yes	—
503	C₄	C₂³×D₄×C₈₉	1	1	yes	characteristic
504	C₄	C₂⁶×C₈₉	1	1	yes	characteristic
505	C₂	—	2	2	no	—
506	C₂	—	2	12	no	—
507	C₂	—	4	12	no	—
508	C₂	—	4	8	no	—
509	C₂	C₈₉×#3585	1	2	yes	direct product factor
510	C₂	C₂⁴×D₄×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂³×D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₈₉
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂²×#41×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#585×C₈₉	C₂	semidirect
2. C₂²×H(5,𝔽₂)×C₈₉	C₂	semidirect
3. C₈₉×#3585	C₂	direct (N × H)
4. C₂×#577×C₈₉	C₂	semidirect
5. C₂×#576×C₈₉	C₂	semidirect
6. C₂×#575×C₈₉	C₂	semidirect
7. C₂²×AGL(1,ℤ/8)×C₈₉	C₂	semidirect
8. C₂×#568×C₈₉	C₂	semidirect
10. #585×C₈₉	C₂²	semidirect
12. C₂×2₋¹⁺⁴×C₈₉	C₂²	semidirect
16. C₂×H(5,𝔽₂)×C₈₉	C₂²	semidirect
17. C₂×H(5,𝔽₂)×C₈₉	C₂²	semidirect
18. C₂×H(5,𝔽₂)×C₈₉	C₂²	semidirect
22. C₂²×#41×C₈₉	C₂²	semidirect
23. C₂²×#41×C₈₉	C₂²	semidirect
25. C₂³×D₄×C₈₉	C₂²	semidirect
26. #577×C₈₉	C₂²	semidirect
27. #576×C₈₉	C₂²	semidirect
28. #575×C₈₉	C₂²	semidirect
29. C₂×#137×C₈₉	C₂²	semidirect
30. C₂×#137×C₈₉	C₂²	semidirect
31. C₂×AGL(1,ℤ/8)×C₈₉	C₂²	semidirect
32. C₂×AGL(1,ℤ/8)×C₈₉	C₂²	semidirect
33. C₂×AGL(1,ℤ/8)×C₈₉	C₂²	semidirect
34. C₂×AGL(1,ℤ/8)×C₈₉	C₂²	semidirect
35. C₂×#135×C₈₉	C₂²	semidirect
36. C₂²×SD₁₆×C₈₉	C₂²	semidirect
37. C₂²×D₈×C₈₉	C₂²	semidirect
38. #568×C₈₉	C₂²	semidirect
39. C₂×#131×C₈₉	C₂²	semidirect
40. C₂²×M₁₆×C₈₉	C₂²	semidirect
44. 2₋¹⁺⁴×C₈₉	C₂³	semidirect
49. H(5,𝔽₂)×C₈₉	C₂³	semidirect
50. H(5,𝔽₂)×C₈₉	C₂³	semidirect
67. C₂×#41×C₈₉	C₂³	semidirect
68. C₂×#41×C₈₉	C₂³	semidirect
69. C₂×#41×C₈₉	C₂³	semidirect
73. C₂²×Q₈×C₈₉	C₂³	semidirect
85. C₂²×D₄×C₈₉	C₂³	semidirect
86. C₂²×D₄×C₈₉	C₂³	semidirect
90. #137×C₈₉	C₂³	semidirect
91. #137×C₈₉	C₂³	semidirect
92. AGL(1,ℤ/8)×C₈₉	C₂³	semidirect
93. AGL(1,ℤ/8)×C₈₉	C₂³	semidirect
94. AGL(1,ℤ/8)×C₈₉	C₂³	semidirect
95. #135×C₈₉	C₂³	semidirect
96. C₂×Q₁₆×C₈₉	C₂³	semidirect
97. C₂×SD₁₆×C₈₉	C₂³	semidirect
98. C₂×SD₁₆×C₈₉	C₂³	semidirect
99. C₂×SD₁₆×C₈₉	C₂³	semidirect
100. C₂×D₈×C₈₉	C₂³	semidirect
101. C₂×D₈×C₈₉	C₂³	semidirect
102. #131×C₈₉	C₂³	semidirect
104. C₂×M₁₆×C₈₉	C₂³	semidirect
105. C₂×M₁₆×C₈₉	C₂³	semidirect
106. C₂²×C₈×C₈₉	C₂³	semidirect
127. #41×C₈₉	C₂⁴	semidirect
138. C₂×Q₈×C₈₉	C₂⁴	semidirect
160. C₂×D₄×C₈₉	C₂⁴	semidirect
174. Q₁₆×C₈₉	C₂⁴	semidirect
175. SD₁₆×C₈₉	C₂⁴	semidirect
176. SD₁₆×C₈₉	C₂⁴	semidirect
177. D₈×C₈₉	C₂⁴	semidirect
179. M₁₆×C₈₉	C₂⁴	semidirect
181. C₂×C₈×C₈₉	C₂⁴	semidirect
249. C₂×#3585	C₈₉	direct (N × H)
254. C₂×C₈₉	#3585	direct (N × H)
256. C₂×#585	C₂×C₈₉	semidirect
257. C₂²×H(5,𝔽₂)	C₂×C₈₉	semidirect
258. #3585	C₂×C₈₉	direct (N × H)
259. C₂×#577	C₂×C₈₉	semidirect
260. C₂×#576	C₂×C₈₉	semidirect
261. C₂×#575	C₂×C₈₉	semidirect
262. C₂²×AGL(1,ℤ/8)	C₂×C₈₉	semidirect
263. C₂×#568	C₂×C₈₉	semidirect
264. C₈₉	C₂×#3585	direct (N × H)
266. #585	C₂²×C₈₉	semidirect
268. C₂×2₋¹⁺⁴	C₂²×C₈₉	semidirect
272. C₂×H(5,𝔽₂)	C₂²×C₈₉	semidirect
273. C₂×H(5,𝔽₂)	C₂²×C₈₉	semidirect
274. C₂×H(5,𝔽₂)	C₂²×C₈₉	semidirect
278. C₂²×#41	C₂²×C₈₉	semidirect
279. C₂²×#41	C₂²×C₈₉	semidirect
281. C₂³×D₄	C₂²×C₈₉	semidirect
282. #577	C₂²×C₈₉	semidirect
283. #576	C₂²×C₈₉	semidirect
284. #575	C₂²×C₈₉	semidirect
285. C₂×#137	C₂²×C₈₉	semidirect
286. C₂×#137	C₂²×C₈₉	semidirect
287. C₂×AGL(1,ℤ/8)	C₂²×C₈₉	semidirect
288. C₂×AGL(1,ℤ/8)	C₂²×C₈₉	semidirect
289. C₂×AGL(1,ℤ/8)	C₂²×C₈₉	semidirect
290. C₂×AGL(1,ℤ/8)	C₂²×C₈₉	semidirect
291. C₂×#135	C₂²×C₈₉	semidirect
292. C₂²×SD₁₆	C₂²×C₈₉	semidirect
293. C₂²×D₈	C₂²×C₈₉	semidirect
294. #568	C₂²×C₈₉	semidirect
295. C₂×#131	C₂²×C₈₉	semidirect
296. C₂²×M₁₆	C₂²×C₈₉	semidirect
300. 2₋¹⁺⁴	C₂³×C₈₉	semidirect
305. H(5,𝔽₂)	C₂³×C₈₉	semidirect
306. H(5,𝔽₂)	C₂³×C₈₉	semidirect
323. C₂×#41	C₂³×C₈₉	semidirect
324. C₂×#41	C₂³×C₈₉	semidirect
325. C₂×#41	C₂³×C₈₉	semidirect
329. C₂²×Q₈	C₂³×C₈₉	semidirect
341. C₂²×D₄	C₂³×C₈₉	semidirect
342. C₂²×D₄	C₂³×C₈₉	semidirect
346. #137	C₂³×C₈₉	semidirect
347. #137	C₂³×C₈₉	semidirect
348. AGL(1,ℤ/8)	C₂³×C₈₉	semidirect
349. AGL(1,ℤ/8)	C₂³×C₈₉	semidirect
350. AGL(1,ℤ/8)	C₂³×C₈₉	semidirect
351. #135	C₂³×C₈₉	semidirect
352. C₂×Q₁₆	C₂³×C₈₉	semidirect
353. C₂×SD₁₆	C₂³×C₈₉	semidirect
354. C₂×SD₁₆	C₂³×C₈₉	semidirect
355. C₂×SD₁₆	C₂³×C₈₉	semidirect
356. C₂×D₈	C₂³×C₈₉	semidirect
357. C₂×D₈	C₂³×C₈₉	semidirect
358. #131	C₂³×C₈₉	semidirect
360. C₂×M₁₆	C₂³×C₈₉	semidirect
361. C₂×M₁₆	C₂³×C₈₉	semidirect
362. C₂²×C₈	C₂³×C₈₉	semidirect
383. #41	C₂⁴×C₈₉	semidirect
394. C₂×Q₈	C₂⁴×C₈₉	semidirect
416. C₂×D₄	C₂⁴×C₈₉	semidirect
430. Q₁₆	C₂⁴×C₈₉	semidirect
431. SD₁₆	C₂⁴×C₈₉	semidirect
432. SD₁₆	C₂⁴×C₈₉	semidirect
433. D₈	C₂⁴×C₈₉	semidirect
435. M₁₆	C₂⁴×C₈₉	semidirect
437. C₂×C₈	C₂⁴×C₈₉	semidirect
509. C₂	C₈₉×#3585	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#585×C₈₉	2
C₂²×H(5,𝔽₂)×C₈₉	2
C₈₉×#3585	2
C₂×#577×C₈₉	2
C₂×#576×C₈₉	2
C₂×#575×C₈₉	2
C₂²×AGL(1,ℤ/8)×C₈₉	2
C₂×#568×C₈₉	2
C₂×#3585	89

Subgroups — C2×C89×#3585 #36440936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3585 #36440936