Subgroups — C₂×C₈₉×#3042 #36438265

Order 22784.

2882 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×AGL(1,ℤ/8)×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂³×SD₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#536×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#518×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#3042	C₂	1	16	yes	maximal, direct product factor
6	C₂×#463×C₈₉	C₂	1	4	yes	maximal, complemented
7	C₂×#461×C₈₉	C₂	1	4	yes	maximal, complemented
8	C₂×#424×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#419×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#416×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
12	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×AGL(1,ℤ/8)×C₈₉	—	2	8	no	—
15	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	8	yes	—
16	C₂²×SD₁₆×C₈₉	—	2	8	no	—
17	C₂²×SD₁₆×C₈₉	C₂²	1	4	yes	—
18	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	—
19	C₂²×SD₁₆×C₈₉	C₂²	1	8	yes	complemented
20	C₂²×D₈×C₈₉	C₂²	1	2	yes	—
21	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
22	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	#536×C₈₉	—	2	4	no	—
24	#536×C₈₉	C₂²	1	8	yes	complemented
25	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	complemented
26	C₂×#124×C₈₉	C₂²	1	2	yes	complemented
27	C₂×#123×C₈₉	—	2	2	no	—
28	C₂×C₂²⋊Q₈×C₈₉	—	2	2	no	—
29	#518×C₈₉	—	2	4	no	—
30	#518×C₈₉	C₂²	1	8	yes	complemented
31	C₂×C₄×D₄×C₈₉	—	2	2	no	—
32	C₂×C₄×D₄×C₈₉	C₂²	1	4	yes	—
33	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
35	C₂²×C₂²⋊C₄×C₈₉	—	2	2	no	—
36	#463×C₈₉	C₂²	1	32	yes	complemented
37	#461×C₈₉	C₂²	1	32	yes	complemented
38	#424×C₈₉	C₂²	1	8	yes	complemented
39	C₂×C₄⋊C₈×C₈₉	C₂²	1	4	yes	complemented
40	#419×C₈₉	C₂²	1	8	yes	complemented
41	#416×C₈₉	C₂²	1	8	yes	complemented
42	C₂×#103×C₈₉	C₂²	1	2	yes	complemented
43	C₂×#103×C₈₉	C₂²	1	4	yes	complemented
44	C₂×#102×C₈₉	C₂²	1	2	yes	complemented
45	C₂⁵×C₈₉	—	2	1	no	—
46	C₂×#41×C₈₉	—	2	2	no	—
47	C₂×#41×C₈₉	—	2	4	no	—
48	C₂×#41×C₈₉	—	2	4	no	—
49	C₂×#41×C₈₉	C₂³	1	4	yes	—
50	C₂²×Q₈×C₈₉	—	2	1	no	—
51	C₂²×Q₈×C₈₉	—	2	2	no	—
52	C₂²×Q₈×C₈₉	—	2	4	no	—
53	C₂²×Q₈×C₈₉	—	2	4	no	—
54	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
55	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
56	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
57	C₂²×D₄×C₈₉	—	2	1	no	—
58	C₂²×D₄×C₈₉	—	2	1	no	—
59	C₂²×D₄×C₈₉	—	2	2	no	—
60	C₂²×D₄×C₈₉	—	2	4	no	—
61	C₂²×D₄×C₈₉	D₄	1	8	yes	—
62	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
64	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
65	C₂³×C₄×C₈₉	—	2	1	no	—
66	C₂³×C₄×C₈₉	—	2	1	no	—
67	C₂³×C₄×C₈₉	—	2	4	no	—
68	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
69	AGL(1,ℤ/8)×C₈₉	—	2	32	no	—
70	C₂×SD₁₆×C₈₉	—	2	16	no	—
71	C₂×SD₁₆×C₈₉	—	2	32	no	—
72	C₂×SD₁₆×C₈₉	—	2	8	no	—
73	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	—
74	C₂×SD₁₆×C₈₉	C₂³	1	16	yes	—
75	C₂×D₈×C₈₉	—	2	8	no	—
76	C₂×D₈×C₈₉	C₂³	1	8	yes	—
77	C₂×M₁₆×C₈₉	—	2	4	no	—
78	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
79	C₂²×C₈×C₈₉	—	2	4	no	—
80	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
81	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
82	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
83	C₄⋊Q₈×C₈₉	—	2	4	no	—
84	C₄⋊Q₈×C₈₉	C₂³	1	8	yes	complemented
85	#124×C₈₉	—	2	4	no	—
86	#124×C₈₉	C₂³	1	8	yes	complemented
87	#123×C₈₉	—	2	16	no	—
88	C₂²⋊Q₈×C₈₉	—	2	16	no	—
89	C₄×D₄×C₈₉	—	2	16	no	—
90	C₄×D₄×C₈₉	—	2	8	no	—
91	C₄×D₄×C₈₉	C₂³	1	16	yes	—
92	#117×C₈₉	—	2	2	no	—
93	#117×C₈₉	D₄	1	4	yes	—
94	#117×C₈₉	C₂³	1	4	yes	—
95	#117×C₈₉	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
97	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
98	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
99	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
100	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
101	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
102	C₂×C₂²⋊C₄×C₈₉	—	2	16	no	—
103	C₂×C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
104	C₂×C₄²×C₈₉	—	2	1	no	—
105	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
106	C₄⋊C₈×C₈₉	C₂³	1	16	yes	complemented
107	#103×C₈₉	C₂³	1	8	yes	complemented
108	#103×C₈₉	C₂³	1	16	yes	complemented
109	#102×C₈₉	C₂³	1	8	yes	complemented
110	C₂⁴×C₈₉	—	2	2	no	—
111	C₂⁴×C₈₉	—	2	8	no	—
112	C₂⁴×C₈₉	—	2	4	no	—
113	C₂⁴×C₈₉	—	2	1	no	—
114	C₂⁴×C₈₉	—	8	4	no	—
115	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
116	#41×C₈₉	—	2	8	no	—
117	#41×C₈₉	—	2	8	no	—
118	#41×C₈₉	—	4	8	no	—
119	C₂×Q₈×C₈₉	—	2	4	no	—
120	C₂×Q₈×C₈₉	—	2	2	no	—
121	C₂×Q₈×C₈₉	—	2	1	no	—
122	C₂×Q₈×C₈₉	—	2	4	no	—
123	C₂×Q₈×C₈₉	—	2	8	no	—
124	C₂×Q₈×C₈₉	—	2	2	no	—
125	C₂×Q₈×C₈₉	—	2	4	no	—
126	C₂×Q₈×C₈₉	—	2	8	no	—
127	C₂×Q₈×C₈₉	—	2	2	no	—
128	C₂×Q₈×C₈₉	—	4	4	no	—
129	C₂×Q₈×C₈₉	—	4	8	no	—
130	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
131	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
132	C₂×D₄×C₈₉	—	2	4	no	—
133	C₂×D₄×C₈₉	—	2	8	no	—
134	C₂×D₄×C₈₉	—	2	16	no	—
135	C₂×D₄×C₈₉	—	2	4	no	—
136	C₂×D₄×C₈₉	—	2	2	no	—
137	C₂×D₄×C₈₉	—	2	2	no	—
138	C₂×D₄×C₈₉	—	2	1	no	—
139	C₂×D₄×C₈₉	—	2	4	no	—
140	C₂×D₄×C₈₉	—	2	8	no	—
141	C₂×D₄×C₈₉	—	2	2	no	—
142	C₂×D₄×C₈₉	—	4	4	no	—
143	C₂×D₄×C₈₉	C₂×D₄	1	8	yes	—
144	C₂×D₄×C₈₉	C₂×D₄	1	16	yes	—
145	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
146	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
147	C₂²×C₄×C₈₉	—	2	4	no	—
148	C₂²×C₄×C₈₉	—	2	2	no	—
149	C₂²×C₄×C₈₉	—	2	2	no	—
150	C₂²×C₄×C₈₉	—	2	1	no	—
151	C₂²×C₄×C₈₉	—	2	4	no	—
152	C₂²×C₄×C₈₉	—	2	16	no	—
153	C₂²×C₄×C₈₉	—	4	4	no	—
154	C₂²×C₄×C₈₉	—	4	4	no	—
155	C₂²×C₄×C₈₉	—	8	8	no	—
156	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
157	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
158	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₈₉	#41	1	4	yes	—
161	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
163	SD₁₆×C₈₉	—	2	32	no	—
164	SD₁₆×C₈₉	—	2	16	no	—
165	D₈×C₈₉	—	2	16	no	—
166	M₁₆×C₈₉	—	2	8	no	—
167	C₂×C₈×C₈₉	—	2	4	no	—
168	C₂×C₈×C₈₉	—	2	8	no	—
169	C₂×C₈×C₈₉	—	2	4	no	—
170	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
171	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
172	C₄⋊C₄×C₈₉	—	2	16	no	—
173	C₄⋊C₄×C₈₉	—	2	2	no	—
174	C₄⋊C₄×C₈₉	C₂×D₄	1	8	yes	—
175	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
176	C₂²⋊C₄×C₈₉	—	2	4	no	—
177	C₂²⋊C₄×C₈₉	—	2	16	no	—
178	C₂²⋊C₄×C₈₉	—	2	16	no	—
179	C₂²⋊C₄×C₈₉	C₂×D₄	1	8	yes	—
180	C₄²×C₈₉	—	2	4	no	—
181	C₄²×C₈₉	C₂×D₄	1	4	yes	—
182	C₄²×C₈₉	C₂⁴	1	4	yes	—
183	C₂³×C₈₉	—	2	4	no	—
184	C₂³×C₈₉	—	2	8	no	—
185	C₂³×C₈₉	—	2	4	no	—
186	C₂³×C₈₉	—	2	16	no	—
187	C₂³×C₈₉	—	2	2	no	—
188	C₂³×C₈₉	—	4	8	no	—
189	C₂³×C₈₉	—	4	2	no	—
190	C₂³×C₈₉	—	4	4	no	—
191	C₂³×C₈₉	—	8	16	no	—
192	C₂³×C₈₉	—	8	8	no	—
193	C₂³×C₈₉	#121	1	4	yes	—
194	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
195	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
196	Q₈×C₈₉	—	2	4	no	—
197	Q₈×C₈₉	—	2	2	no	—
198	Q₈×C₈₉	—	2	4	no	—
199	Q₈×C₈₉	—	2	2	no	—
200	Q₈×C₈₉	—	4	8	no	—
201	Q₈×C₈₉	—	4	4	no	—
202	Q₈×C₈₉	—	4	2	no	—
203	D₄×C₈₉	—	2	16	no	—
204	D₄×C₈₉	—	2	8	no	—
205	D₄×C₈₉	—	2	2	no	—
206	D₄×C₈₉	—	2	4	no	—
207	D₄×C₈₉	—	2	2	no	—
208	D₄×C₈₉	—	4	8	no	—
209	D₄×C₈₉	—	4	2	no	—
210	D₄×C₈₉	C₂²×D₄	1	8	yes	—
211	C₂×C₄×C₈₉	—	2	8	no	—
212	C₂×C₄×C₈₉	—	2	4	no	—
213	C₂×C₄×C₈₉	—	2	4	no	—
214	C₂×C₄×C₈₉	—	2	2	no	—
215	C₂×C₄×C₈₉	—	4	2	no	—
216	C₂×C₄×C₈₉	—	4	8	no	—
217	C₂×C₄×C₈₉	—	4	4	no	—
218	C₂×C₄×C₈₉	—	4	4	no	—
219	C₂×C₄×C₈₉	—	4	8	no	—
220	C₂×C₄×C₈₉	—	8	16	no	—
221	C₂×C₄×C₈₉	#121	1	4	yes	—
222	C₂×C₄×C₈₉	#121	1	8	yes	—
223	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
224	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
227	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
228	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
229	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
230	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
231	C₈×C₈₉	—	2	4	no	—
232	C₈×C₈₉	—	2	4	no	—
233	C₂²×C₈₉	—	2	8	no	—
234	C₂²×C₈₉	—	2	4	no	—
235	C₂²×C₈₉	—	2	8	no	—
236	C₂²×C₈₉	—	4	16	no	—
237	C₂²×C₈₉	—	4	8	no	—
238	C₂²×C₈₉	—	4	2	no	—
239	C₂²×C₈₉	—	4	4	no	—
240	C₂²×C₈₉	—	8	16	no	—
241	C₂²×C₈₉	—	8	4	no	—
242	C₂²×C₈₉	C₂×#121	1	4	yes	—
243	C₂²×C₈₉	C₂×#121	1	2	yes	—
244	C₂²×C₈₉	#577	1	4	yes	—
245	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
246	C₄×C₈₉	—	2	8	no	—
247	C₄×C₈₉	—	4	2	no	—
248	C₄×C₈₉	—	4	4	no	—
249	C₄×C₈₉	—	4	4	no	—
250	C₄×C₈₉	C₂×#121	1	4	yes	—
251	C₄×C₈₉	C₂×#121	1	2	yes	—
252	C₄×C₈₉	C₂³×D₄	1	1	yes	characteristic
253	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
254	C₂×#3042	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
255	C₂×C₈₉	—	2	4	no	—
256	C₂×C₈₉	—	4	8	no	—
257	C₂×C₈₉	—	8	2	no	—
258	C₂×C₈₉	#3042	1	4	yes	direct product factor
259	C₂×C₈₉	C₂²×#121	1	1	yes	characteristic
260	C₂×C₈₉	C₂×#577	1	2	yes	—
261	C₂²×AGL(1,ℤ/8)	C₂×C₈₉	1	1	yes	characteristic
262	C₂³×SD₁₆	C₂×C₈₉	1	1	yes	characteristic, complemented
263	C₂×#536	C₂×C₈₉	1	1	yes	characteristic, complemented
264	C₂×#518	C₂×C₈₉	1	1	yes	characteristic, complemented
265	#3042	C₂×C₈₉	1	16	yes	direct product factor
266	C₂×#463	C₂×C₈₉	1	4	yes	complemented
267	C₂×#461	C₂×C₈₉	1	4	yes	complemented
268	C₂×#424	C₂×C₈₉	1	1	yes	characteristic, complemented
269	C₂×#419	C₂×C₈₉	1	1	yes	characteristic, complemented
270	C₂×#416	C₂×C₈₉	1	1	yes	characteristic, complemented
271	C₈₉	C₂×#3042	1	1	yes	characteristic, direct product factor, Sylow
272	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
273	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic
274	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
275	C₂×AGL(1,ℤ/8)	—	2	8	no	—
276	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	8	yes	—
277	C₂²×SD₁₆	—	2	8	no	—
278	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	—
279	C₂²×SD₁₆	C₂²×C₈₉	1	4	yes	—
280	C₂²×SD₁₆	C₂²×C₈₉	1	8	yes	complemented
281	C₂²×D₈	C₂²×C₈₉	1	2	yes	—
282	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
283	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
284	#536	—	2	4	no	—
285	#536	C₂²×C₈₉	1	8	yes	complemented
286	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	complemented
287	C₂×#124	C₂²×C₈₉	1	2	yes	complemented
288	C₂×#123	—	2	2	no	—
289	C₂×C₂²⋊Q₈	—	2	2	no	—
290	#518	—	2	4	no	—
291	#518	C₂²×C₈₉	1	8	yes	complemented
292	C₂×C₄×D₄	—	2	2	no	—
293	C₂×C₄×D₄	C₂²×C₈₉	1	4	yes	—
294	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
295	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
296	C₂²×C₂²⋊C₄	—	2	2	no	—
297	#463	C₂²×C₈₉	1	32	yes	complemented
298	#461	C₂²×C₈₉	1	32	yes	complemented
299	#424	C₂²×C₈₉	1	8	yes	complemented
300	C₂×C₄⋊C₈	C₂²×C₈₉	1	4	yes	complemented
301	#419	C₂²×C₈₉	1	8	yes	complemented
302	#416	C₂²×C₈₉	1	8	yes	complemented
303	C₂×#103	C₂²×C₈₉	1	2	yes	complemented
304	C₂×#103	C₂²×C₈₉	1	4	yes	complemented
305	C₂×#102	C₂²×C₈₉	1	2	yes	complemented
306	C₂⁵	—	2	1	no	—
307	C₂×#41	—	2	2	no	—
308	C₂×#41	—	2	4	no	—
309	C₂×#41	—	2	4	no	—
310	C₂×#41	C₂³×C₈₉	1	4	yes	—
311	C₂²×Q₈	—	2	1	no	—
312	C₂²×Q₈	—	2	2	no	—
313	C₂²×Q₈	—	2	4	no	—
314	C₂²×Q₈	—	2	4	no	—
315	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
316	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
317	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
318	C₂²×D₄	—	2	1	no	—
319	C₂²×D₄	—	2	1	no	—
320	C₂²×D₄	—	2	2	no	—
321	C₂²×D₄	—	2	4	no	—
322	C₂²×D₄	D₄×C₈₉	1	8	yes	—
323	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
324	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
325	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
326	C₂³×C₄	—	2	1	no	—
327	C₂³×C₄	—	2	1	no	—
328	C₂³×C₄	—	2	4	no	—
329	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
330	AGL(1,ℤ/8)	—	2	32	no	—
331	C₂×SD₁₆	—	2	16	no	—
332	C₂×SD₁₆	—	2	32	no	—
333	C₂×SD₁₆	—	2	8	no	—
334	C₂×SD₁₆	C₂³×C₈₉	1	16	yes	—
335	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	—
336	C₂×D₈	—	2	8	no	—
337	C₂×D₈	C₂³×C₈₉	1	8	yes	—
338	C₂×M₁₆	—	2	4	no	—
339	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
340	C₂²×C₈	—	2	4	no	—
341	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
342	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
343	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
344	C₄⋊Q₈	—	2	4	no	—
345	C₄⋊Q₈	C₂³×C₈₉	1	8	yes	complemented
346	#124	—	2	4	no	—
347	#124	C₂³×C₈₉	1	8	yes	complemented
348	#123	—	2	16	no	—
349	C₂²⋊Q₈	—	2	16	no	—
350	C₄×D₄	—	2	16	no	—
351	C₄×D₄	—	2	8	no	—
352	C₄×D₄	C₂³×C₈₉	1	16	yes	—
353	#117	—	2	2	no	—
354	#117	D₄×C₈₉	1	4	yes	—
355	#117	C₂³×C₈₉	1	4	yes	—
356	#117	C₂³×C₈₉	1	4	yes	—
357	C₂×C₄⋊C₄	—	2	4	no	—
358	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
359	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
360	C₂×C₂²⋊C₄	—	2	1	no	—
361	C₂×C₂²⋊C₄	—	2	4	no	—
362	C₂×C₂²⋊C₄	—	2	4	no	—
363	C₂×C₂²⋊C₄	—	2	16	no	—
364	C₂×C₂²⋊C₄	D₄×C₈₉	1	2	yes	—
365	C₂×C₄²	—	2	1	no	—
366	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
367	C₄⋊C₈	C₂³×C₈₉	1	16	yes	complemented
368	#103	C₂³×C₈₉	1	8	yes	complemented
369	#103	C₂³×C₈₉	1	16	yes	complemented
370	#102	C₂³×C₈₉	1	8	yes	complemented
371	C₂⁴	—	2	2	no	—
372	C₂⁴	—	2	8	no	—
373	C₂⁴	—	2	4	no	—
374	C₂⁴	—	2	1	no	—
375	C₂⁴	—	8	4	no	—
376	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
377	#41	—	2	8	no	—
378	#41	—	2	8	no	—
379	#41	—	4	8	no	—
380	C₂×Q₈	—	2	4	no	—
381	C₂×Q₈	—	2	2	no	—
382	C₂×Q₈	—	2	1	no	—
383	C₂×Q₈	—	2	4	no	—
384	C₂×Q₈	—	2	8	no	—
385	C₂×Q₈	—	2	2	no	—
386	C₂×Q₈	—	2	4	no	—
387	C₂×Q₈	—	2	8	no	—
388	C₂×Q₈	—	2	2	no	—
389	C₂×Q₈	—	4	4	no	—
390	C₂×Q₈	—	4	8	no	—
391	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
392	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
393	C₂×D₄	—	2	4	no	—
394	C₂×D₄	—	2	8	no	—
395	C₂×D₄	—	2	16	no	—
396	C₂×D₄	—	2	4	no	—
397	C₂×D₄	—	2	2	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	1	no	—
400	C₂×D₄	—	2	4	no	—
401	C₂×D₄	—	2	8	no	—
402	C₂×D₄	—	2	2	no	—
403	C₂×D₄	—	4	4	no	—
404	C₂×D₄	C₂×D₄×C₈₉	1	8	yes	—
405	C₂×D₄	C₂×D₄×C₈₉	1	16	yes	—
406	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
407	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	16	no	—
414	C₂²×C₄	—	4	4	no	—
415	C₂²×C₄	—	4	4	no	—
416	C₂²×C₄	—	8	8	no	—
417	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
418	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
419	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
420	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
421	C₂²×C₄	#41×C₈₉	1	4	yes	—
422	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
423	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
424	SD₁₆	—	2	32	no	—
425	SD₁₆	—	2	16	no	—
426	D₈	—	2	16	no	—
427	M₁₆	—	2	8	no	—
428	C₂×C₈	—	2	4	no	—
429	C₂×C₈	—	2	8	no	—
430	C₂×C₈	—	2	4	no	—
431	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
432	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
433	C₄⋊C₄	—	2	16	no	—
434	C₄⋊C₄	—	2	2	no	—
435	C₄⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
436	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
437	C₂²⋊C₄	—	2	4	no	—
438	C₂²⋊C₄	—	2	16	no	—
439	C₂²⋊C₄	—	2	16	no	—
440	C₂²⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
441	C₄²	—	2	4	no	—
442	C₄²	C₂×D₄×C₈₉	1	4	yes	—
443	C₄²	C₂⁴×C₈₉	1	4	yes	—
444	C₂³	—	2	4	no	—
445	C₂³	—	2	8	no	—
446	C₂³	—	2	4	no	—
447	C₂³	—	2	16	no	—
448	C₂³	—	2	2	no	—
449	C₂³	—	4	8	no	—
450	C₂³	—	4	2	no	—
451	C₂³	—	4	4	no	—
452	C₂³	—	8	8	no	—
453	C₂³	—	8	16	no	—
454	C₂³	#121×C₈₉	1	4	yes	—
455	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
456	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
457	Q₈	—	2	4	no	—
458	Q₈	—	2	2	no	—
459	Q₈	—	2	4	no	—
460	Q₈	—	2	2	no	—
461	Q₈	—	4	8	no	—
462	Q₈	—	4	4	no	—
463	Q₈	—	4	2	no	—
464	D₄	—	2	16	no	—
465	D₄	—	2	8	no	—
466	D₄	—	2	2	no	—
467	D₄	—	2	4	no	—
468	D₄	—	2	2	no	—
469	D₄	—	4	8	no	—
470	D₄	—	4	2	no	—
471	D₄	C₂²×D₄×C₈₉	1	8	yes	—
472	C₂×C₄	—	2	8	no	—
473	C₂×C₄	—	2	4	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	8	no	—
478	C₂×C₄	—	4	4	no	—
479	C₂×C₄	—	4	4	no	—
480	C₂×C₄	—	4	8	no	—
481	C₂×C₄	—	8	16	no	—
482	C₂×C₄	#121×C₈₉	1	4	yes	—
483	C₂×C₄	#121×C₈₉	1	8	yes	—
484	C₂×C₄	C₂²×D₄×C₈₉	1	8	yes	—
485	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
486	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
487	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
488	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
489	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
490	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
491	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
492	C₈	—	2	4	no	—
493	C₈	—	2	4	no	—
494	C₂²	—	2	4	no	—
495	C₂²	—	2	8	no	—
496	C₂²	—	2	8	no	—
497	C₂²	—	4	8	no	—
498	C₂²	—	4	16	no	—
499	C₂²	—	4	4	no	—
500	C₂²	—	4	2	no	—
501	C₂²	—	8	16	no	—
502	C₂²	—	8	4	no	—
503	C₂²	C₂×#121×C₈₉	1	4	yes	—
504	C₂²	C₂×#121×C₈₉	1	2	yes	—
505	C₂²	#577×C₈₉	1	4	yes	—
506	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
507	C₄	—	2	8	no	—
508	C₄	—	4	4	no	—
509	C₄	—	4	2	no	—
510	C₄	—	4	4	no	—
511	C₄	C₂×#121×C₈₉	1	4	yes	—
512	C₄	C₂×#121×C₈₉	1	2	yes	—
513	C₄	C₂³×D₄×C₈₉	1	1	yes	characteristic
514	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
515	C₂	—	2	4	no	—
516	C₂	—	4	8	no	—
517	C₂	—	8	2	no	—
518	C₂	C₈₉×#3042	1	4	yes	direct product factor
519	C₂	C₂²×#121×C₈₉	1	1	yes	characteristic
520	C₂	C₂×#577×C₈₉	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
prime radical	C₂²×AGL(1,ℤ/8)×C₈₉	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×#117×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂³×SD₁₆×C₈₉	C₂	semidirect
3. C₂×#536×C₈₉	C₂	semidirect
4. C₂×#518×C₈₉	C₂	semidirect
5. C₈₉×#3042	C₂	direct (N × H)
6. C₂×#463×C₈₉	C₂	semidirect
7. C₂×#461×C₈₉	C₂	semidirect
8. C₂×#424×C₈₉	C₂	semidirect
9. C₂×#419×C₈₉	C₂	semidirect
10. C₂×#416×C₈₉	C₂	semidirect
19. C₂²×SD₁₆×C₈₉	C₂²	semidirect
24. #536×C₈₉	C₂²	semidirect
25. C₂×C₄⋊Q₈×C₈₉	C₂²	semidirect
26. C₂×#124×C₈₉	C₂²	semidirect
30. #518×C₈₉	C₂²	semidirect
36. #463×C₈₉	C₂²	semidirect
37. #461×C₈₉	C₂²	semidirect
38. #424×C₈₉	C₂²	semidirect
39. C₂×C₄⋊C₈×C₈₉	C₂²	semidirect
40. #419×C₈₉	C₂²	semidirect
41. #416×C₈₉	C₂²	semidirect
42. C₂×#103×C₈₉	C₂²	semidirect
43. C₂×#103×C₈₉	C₂²	semidirect
44. C₂×#102×C₈₉	C₂²	semidirect
84. C₄⋊Q₈×C₈₉	C₂³	semidirect
86. #124×C₈₉	C₂³	semidirect
106. C₄⋊C₈×C₈₉	C₂³	semidirect
107. #103×C₈₉	C₂³	semidirect
108. #103×C₈₉	C₂³	semidirect
109. #102×C₈₉	C₂³	semidirect
254. C₂×#3042	C₈₉	direct (N × H)
258. C₂×C₈₉	#3042	direct (N × H)
262. C₂³×SD₁₆	C₂×C₈₉	semidirect
263. C₂×#536	C₂×C₈₉	semidirect
264. C₂×#518	C₂×C₈₉	semidirect
265. #3042	C₂×C₈₉	direct (N × H)
266. C₂×#463	C₂×C₈₉	semidirect
267. C₂×#461	C₂×C₈₉	semidirect
268. C₂×#424	C₂×C₈₉	semidirect
269. C₂×#419	C₂×C₈₉	semidirect
270. C₂×#416	C₂×C₈₉	semidirect
271. C₈₉	C₂×#3042	direct (N × H)
280. C₂²×SD₁₆	C₂²×C₈₉	semidirect
285. #536	C₂²×C₈₉	semidirect
286. C₂×C₄⋊Q₈	C₂²×C₈₉	semidirect
287. C₂×#124	C₂²×C₈₉	semidirect
291. #518	C₂²×C₈₉	semidirect
297. #463	C₂²×C₈₉	semidirect
298. #461	C₂²×C₈₉	semidirect
299. #424	C₂²×C₈₉	semidirect
300. C₂×C₄⋊C₈	C₂²×C₈₉	semidirect
301. #419	C₂²×C₈₉	semidirect
302. #416	C₂²×C₈₉	semidirect
303. C₂×#103	C₂²×C₈₉	semidirect
304. C₂×#103	C₂²×C₈₉	semidirect
305. C₂×#102	C₂²×C₈₉	semidirect
345. C₄⋊Q₈	C₂³×C₈₉	semidirect
347. #124	C₂³×C₈₉	semidirect
367. C₄⋊C₈	C₂³×C₈₉	semidirect
368. #103	C₂³×C₈₉	semidirect
369. #103	C₂³×C₈₉	semidirect
370. #102	C₂³×C₈₉	semidirect
518. C₂	C₈₉×#3042	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×AGL(1,ℤ/8)×C₈₉	2
C₂³×SD₁₆×C₈₉	2
C₂×#536×C₈₉	2
C₂×#518×C₈₉	2
C₈₉×#3042	2
C₂×#463×C₈₉	2
C₂×#461×C₈₉	2
C₂×#424×C₈₉	2
C₂×#419×C₈₉	2
C₂×#416×C₈₉	2
C₂×#3042	89

Subgroups — C2×C89×#3042 #36438265

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3042 #36438265