Subgroups — C₈₉×#60161 #36438009

Order 22784.

2706 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#575×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#3428	C₂	1	2	yes	maximal, complemented
3	C₈₉×#2947	C₂	1	8	yes	maximal, complemented
4	C₈₉×#2946	C₂	1	8	yes	maximal, complemented
5	C₈₉×#2943	C₂	1	4	yes	maximal, complemented
6	C₄×#135×C₈₉	C₂	1	4	yes	maximal, complemented
7	C₈₉×#2911	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#416×C₈₉	C₂	1	2	yes	maximal, complemented
9	C₂×#405×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₈₉	C₂²	1	2	yes	—
11	#575×C₈₉	C₄	1	16	yes	complemented
12	C₂×#137×C₈₉	C₂²	1	4	yes	—
13	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	4	yes	—
14	C₂×#135×C₈₉	C₂²	1	4	yes	—
15	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#520×C₈₉	—	2	4	no	—
18	#520×C₈₉	C₂²	1	8	yes	—
19	#519×C₈₉	C₂²	1	4	yes	complemented
20	#518×C₈₉	—	2	4	no	—
21	#518×C₈₉	C₂²	1	4	yes	complemented
22	C₄×#41×C₈₉	—	2	4	no	—
23	C₄×#41×C₈₉	C₂²	1	8	yes	complemented
24	C₂×#117×C₈₉	—	2	4	no	—
25	C₂×#117×C₈₉	C₂²	1	2	yes	—
26	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
27	#442×C₈₉	C₂²	1	8	yes	complemented
28	#441×C₈₉	C₂²	1	8	yes	complemented
29	#440×C₈₉	C₂²	1	16	yes	complemented
30	C₄×Q₁₆×C₈₉	C₂²	1	8	yes	complemented
31	C₄×SD₁₆×C₈₉	C₂²	1	16	yes	complemented
32	C₄×D₈×C₈₉	C₂²	1	8	yes	complemented
33	#428×C₈₉	C₂²	1	8	yes	complemented
34	#427×C₈₉	C₂²	1	4	yes	complemented
35	#416×C₈₉	C₂²	1	8	yes	—
36	#416×C₈₉	C₂²	1	4	yes	complemented
37	#416×C₈₉	C₂²	1	4	yes	complemented
38	C₂×#103×C₈₉	C₂²	1	4	yes	complemented
39	C₂×#102×C₈₉	C₂²	1	4	yes	complemented
40	#405×C₈₉	C₂²	1	8	yes	complemented
41	C₂×#97×C₈₉	C₂²	1	2	yes	complemented
42	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	complemented
43	C₂×#41×C₈₉	—	2	2	no	—
44	C₂×#41×C₈₉	—	2	2	no	—
45	C₂×#41×C₈₉	—	2	8	no	—
46	C₂×#41×C₈₉	C₂×C₄	1	16	yes	complemented
47	C₂×#41×C₈₉	C₂³	1	4	yes	—
48	C₂×#41×C₈₉	C₂³	1	4	yes	—
49	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
50	C₂²×D₄×C₈₉	—	2	2	no	—
51	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
52	C₂³×C₄×C₈₉	—	2	2	no	—
53	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
54	#137×C₈₉	C₂×C₄	1	32	yes	complemented
55	AGL(1,ℤ/8)×C₈₉	C₂×C₄	1	32	yes	complemented
56	#135×C₈₉	C₂×C₄	1	32	yes	complemented
57	C₂×Q₁₆×C₈₉	C₂³	1	4	yes	—
58	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	—
59	C₂×D₈×C₈₉	C₂³	1	4	yes	—
60	C₂×M₁₆×C₈₉	C₂×C₄	1	8	yes	complemented
61	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
62	C₂²×C₈×C₈₉	—	2	4	no	—
63	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
66	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
67	C₄×Q₈×C₈₉	—	2	8	no	—
68	C₄×Q₈×C₈₉	C₂³	1	16	yes	—
69	C₄×D₄×C₈₉	—	2	16	no	—
70	C₄×D₄×C₈₉	—	2	8	no	—
71	C₄×D₄×C₈₉	—	2	16	no	—
72	C₄×D₄×C₈₉	C₂³	1	16	yes	—
73	#117×C₈₉	—	2	2	no	—
74	#117×C₈₉	—	2	2	no	—
75	#117×C₈₉	—	2	8	no	—
76	#117×C₈₉	—	2	8	no	—
77	#117×C₈₉	—	2	8	no	—
78	#117×C₈₉	D₄	1	4	yes	—
79	#117×C₈₉	C₂³	1	4	yes	—
80	#117×C₈₉	C₂³	1	4	yes	—
81	#117×C₈₉	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
83	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
84	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
85	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
87	C₂×C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
88	C₂×C₄²×C₈₉	—	2	2	no	—
89	C₂×C₄²×C₈₉	—	2	8	no	—
90	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
91	#107×C₈₉	C₂³	1	8	yes	—
92	#106×C₈₉	C₂³	1	8	yes	—
93	#103×C₈₉	C₂³	1	8	yes	—
94	#103×C₈₉	C₂³	1	8	yes	—
95	#102×C₈₉	C₂³	1	8	yes	—
96	#102×C₈₉	C₂³	1	8	yes	—
97	#97×C₈₉	C₂³	1	8	yes	—
98	C₄×C₈×C₈₉	C₂³	1	8	yes	—
99	C₂⁴×C₈₉	—	2	2	no	—
100	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
101	#41×C₈₉	—	2	16	no	—
102	#41×C₈₉	—	2	16	no	—
103	#41×C₈₉	—	2	8	no	—
104	#41×C₈₉	—	2	8	no	—
105	#41×C₈₉	C₂²×C₄	1	16	yes	—
106	#41×C₈₉	C₂²×C₄	1	16	yes	—
107	C₂×Q₈×C₈₉	—	2	2	no	—
108	C₂×Q₈×C₈₉	—	2	4	no	—
109	C₂×Q₈×C₈₉	C₂²×C₄	1	8	yes	—
110	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
111	C₂×D₄×C₈₉	—	2	16	no	—
112	C₂×D₄×C₈₉	—	2	4	no	—
113	C₂×D₄×C₈₉	—	2	4	no	—
114	C₂×D₄×C₈₉	—	2	2	no	—
115	C₂×D₄×C₈₉	—	2	4	no	—
116	C₂×D₄×C₈₉	C₂²×C₄	1	8	yes	—
117	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
118	C₂²×C₄×C₈₉	—	2	2	no	—
119	C₂²×C₄×C₈₉	—	2	16	no	—
120	C₂²×C₄×C₈₉	—	2	4	no	—
121	C₂²×C₄×C₈₉	—	2	4	no	—
122	C₂²×C₄×C₈₉	—	2	4	no	—
123	C₂²×C₄×C₈₉	—	2	16	no	—
124	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
125	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
126	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₈₉	#41	1	4	yes	—
129	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
130	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
131	Q₁₆×C₈₉	C₂²×C₄	1	16	yes	—
132	SD₁₆×C₈₉	C₂²×C₄	1	32	yes	—
133	D₈×C₈₉	C₂²×C₄	1	16	yes	—
134	M₁₆×C₈₉	C₂²×C₄	1	16	yes	—
135	C₂×C₈×C₈₉	—	2	8	no	—
136	C₂×C₈×C₈₉	—	2	2	no	—
137	C₂×C₈×C₈₉	—	2	2	no	—
138	C₂×C₈×C₈₉	C₂²×C₄	1	8	yes	—
139	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
140	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
141	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
142	C₄⋊C₄×C₈₉	—	2	4	no	—
143	C₄⋊C₄×C₈₉	—	2	16	no	—
144	C₄⋊C₄×C₈₉	C₂×D₄	1	8	yes	—
145	C₄⋊C₄×C₈₉	C₂⁴	1	8	yes	—
146	C₂²⋊C₄×C₈₉	—	2	4	no	—
147	C₂²⋊C₄×C₈₉	—	2	4	no	—
148	C₂²⋊C₄×C₈₉	—	2	16	no	—
149	C₂²⋊C₄×C₈₉	C₂×D₄	1	8	yes	—
150	C₄²×C₈₉	—	2	16	no	—
151	C₄²×C₈₉	—	2	4	no	—
152	C₄²×C₈₉	—	2	4	no	—
153	C₄²×C₈₉	C₂×D₄	1	4	yes	—
154	C₄²×C₈₉	C₂⁴	1	4	yes	—
155	C₂³×C₈₉	—	2	4	no	—
156	C₂³×C₈₉	—	2	8	no	—
157	C₂³×C₈₉	—	4	2	no	—
158	C₂³×C₈₉	—	8	4	no	—
159	C₂³×C₈₉	C₄×D₄	1	4	yes	—
160	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
161	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
162	Q₈×C₈₉	—	2	8	no	—
163	Q₈×C₈₉	—	2	4	no	—
164	Q₈×C₈₉	C₂³×C₄	1	8	yes	—
165	D₄×C₈₉	—	2	16	no	—
166	D₄×C₈₉	—	2	16	no	—
167	D₄×C₈₉	—	2	8	no	—
168	D₄×C₈₉	—	2	4	no	—
169	D₄×C₈₉	C₂³×C₄	1	8	yes	—
170	C₂×C₄×C₈₉	—	2	8	no	—
171	C₂×C₄×C₈₉	—	2	4	no	—
172	C₂×C₄×C₈₉	—	2	16	no	—
173	C₂×C₄×C₈₉	—	2	16	no	—
174	C₂×C₄×C₈₉	—	2	8	no	—
175	C₂×C₄×C₈₉	—	4	4	no	—
176	C₂×C₄×C₈₉	—	8	16	no	—
177	C₂×C₄×C₈₉	—	8	4	no	—
178	C₂×C₄×C₈₉	C₄×D₄	1	8	yes	—
179	C₂×C₄×C₈₉	C₄×D₄	1	4	yes	—
180	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
181	C₂×C₄×C₈₉	C₂³×C₄	1	4	yes	—
182	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
183	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
184	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
186	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
187	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
188	C₈×C₈₉	—	2	2	no	—
189	C₈×C₈₉	—	2	2	no	—
190	C₈×C₈₉	C₂³×C₄	1	8	yes	—
191	C₂²×C₈₉	—	2	2	no	—
192	C₂²×C₈₉	—	2	4	no	—
193	C₂²×C₈₉	—	2	8	no	—
194	C₂²×C₈₉	—	4	4	no	—
195	C₂²×C₈₉	—	4	8	no	—
196	C₂²×C₈₉	—	8	8	no	—
197	C₂²×C₈₉	C₂×C₄×D₄	1	4	yes	—
198	C₂²×C₈₉	C₂×C₄×D₄	1	2	yes	—
199	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
200	C₄×C₈₉	—	2	8	no	—
201	C₄×C₈₉	—	2	8	no	—
202	C₄×C₈₉	—	4	8	no	—
203	C₄×C₈₉	C₂×C₄×D₄	1	2	yes	—
204	C₄×C₈₉	C₂×C₄×D₄	1	4	yes	—
205	C₄×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
206	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
207	#60161	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
208	C₂×C₈₉	—	2	4	no	—
209	C₂×C₈₉	—	2	2	no	—
210	C₂×C₈₉	—	4	8	no	—
211	C₂×C₈₉	C₂²×C₄×D₄	1	1	yes	characteristic
212	C₂×C₈₉	#3584	1	2	yes	—
213	C₂×#575	C₂×C₈₉	1	1	yes	characteristic
214	#3428	C₂×C₈₉	1	2	yes	complemented
215	#2947	C₂×C₈₉	1	8	yes	complemented
216	#2946	C₂×C₈₉	1	8	yes	complemented
217	#2943	C₂×C₈₉	1	4	yes	complemented
218	C₄×#135	C₂×C₈₉	1	4	yes	complemented
219	#2911	C₂×C₈₉	1	1	yes	characteristic, complemented
220	C₂×#416	C₂×C₈₉	1	2	yes	complemented
221	C₂×#405	C₂×C₈₉	1	1	yes	characteristic, complemented
222	C₈₉	#60161	1	1	yes	characteristic, direct product factor, Sylow
223	C₂²×#41	C₂²×C₈₉	1	2	yes	—
224	#575	C₄×C₈₉	1	16	yes	complemented
225	C₂×#137	C₂²×C₈₉	1	4	yes	—
226	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	4	yes	—
227	C₂×#135	C₂²×C₈₉	1	4	yes	—
228	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
229	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
230	#520	—	2	4	no	—
231	#520	C₂²×C₈₉	1	8	yes	—
232	#519	C₂²×C₈₉	1	4	yes	complemented
233	#518	—	2	4	no	—
234	#518	C₂²×C₈₉	1	4	yes	complemented
235	C₄×#41	—	2	4	no	—
236	C₄×#41	C₂²×C₈₉	1	8	yes	complemented
237	C₂×#117	—	2	4	no	—
238	C₂×#117	C₂²×C₈₉	1	2	yes	—
239	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
240	#442	C₂²×C₈₉	1	8	yes	complemented
241	#441	C₂²×C₈₉	1	8	yes	complemented
242	#440	C₂²×C₈₉	1	16	yes	complemented
243	C₄×Q₁₆	C₂²×C₈₉	1	8	yes	complemented
244	C₄×SD₁₆	C₂²×C₈₉	1	16	yes	complemented
245	C₄×D₈	C₂²×C₈₉	1	8	yes	complemented
246	#428	C₂²×C₈₉	1	8	yes	complemented
247	#427	C₂²×C₈₉	1	4	yes	complemented
248	#416	C₂²×C₈₉	1	8	yes	—
249	#416	C₂²×C₈₉	1	4	yes	complemented
250	#416	C₂²×C₈₉	1	4	yes	complemented
251	C₂×#103	C₂²×C₈₉	1	4	yes	complemented
252	C₂×#102	C₂²×C₈₉	1	4	yes	complemented
253	#405	C₂²×C₈₉	1	8	yes	complemented
254	C₂×#97	C₂²×C₈₉	1	2	yes	complemented
255	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	complemented
256	C₂×#41	—	2	2	no	—
257	C₂×#41	—	2	2	no	—
258	C₂×#41	—	2	8	no	—
259	C₂×#41	C₂×C₄×C₈₉	1	16	yes	complemented
260	C₂×#41	C₂³×C₈₉	1	4	yes	—
261	C₂×#41	C₂³×C₈₉	1	4	yes	—
262	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
263	C₂²×D₄	—	2	2	no	—
264	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
265	C₂³×C₄	—	2	2	no	—
266	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
267	#137	C₂×C₄×C₈₉	1	32	yes	complemented
268	AGL(1,ℤ/8)	C₂×C₄×C₈₉	1	32	yes	complemented
269	#135	C₂×C₄×C₈₉	1	32	yes	complemented
270	C₂×Q₁₆	C₂³×C₈₉	1	4	yes	—
271	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	—
272	C₂×D₈	C₂³×C₈₉	1	4	yes	—
273	C₂×M₁₆	C₂×C₄×C₈₉	1	8	yes	complemented
274	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
275	C₂²×C₈	—	2	4	no	—
276	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
277	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
278	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
279	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
280	C₄×Q₈	—	2	8	no	—
281	C₄×Q₈	C₂³×C₈₉	1	16	yes	—
282	C₄×D₄	—	2	16	no	—
283	C₄×D₄	—	2	8	no	—
284	C₄×D₄	—	2	16	no	—
285	C₄×D₄	C₂³×C₈₉	1	16	yes	—
286	#117	—	2	2	no	—
287	#117	—	2	2	no	—
288	#117	—	2	8	no	—
289	#117	—	2	8	no	—
290	#117	—	2	8	no	—
291	#117	D₄×C₈₉	1	4	yes	—
292	#117	C₂³×C₈₉	1	4	yes	—
293	#117	C₂³×C₈₉	1	4	yes	—
294	#117	C₂³×C₈₉	1	4	yes	—
295	C₂×C₄⋊C₄	—	2	8	no	—
296	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
297	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
298	C₂×C₂²⋊C₄	—	2	2	no	—
299	C₂×C₂²⋊C₄	—	2	8	no	—
300	C₂×C₂²⋊C₄	D₄×C₈₉	1	2	yes	—
301	C₂×C₄²	—	2	2	no	—
302	C₂×C₄²	—	2	8	no	—
303	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
304	#107	C₂³×C₈₉	1	8	yes	—
305	#106	C₂³×C₈₉	1	8	yes	—
306	#103	C₂³×C₈₉	1	8	yes	—
307	#103	C₂³×C₈₉	1	8	yes	—
308	#102	C₂³×C₈₉	1	8	yes	—
309	#102	C₂³×C₈₉	1	8	yes	—
310	#97	C₂³×C₈₉	1	8	yes	—
311	C₄×C₈	C₂³×C₈₉	1	8	yes	—
312	C₂⁴	—	2	2	no	—
313	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
314	#41	—	2	16	no	—
315	#41	—	2	16	no	—
316	#41	—	2	8	no	—
317	#41	—	2	8	no	—
318	#41	C₂²×C₄×C₈₉	1	16	yes	—
319	#41	C₂²×C₄×C₈₉	1	16	yes	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	2	4	no	—
322	C₂×Q₈	C₂²×C₄×C₈₉	1	8	yes	—
323	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
324	C₂×D₄	—	2	16	no	—
325	C₂×D₄	—	2	4	no	—
326	C₂×D₄	—	2	4	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	4	no	—
329	C₂×D₄	C₂²×C₄×C₈₉	1	8	yes	—
330	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
331	C₂²×C₄	—	2	2	no	—
332	C₂²×C₄	—	2	16	no	—
333	C₂²×C₄	—	2	4	no	—
334	C₂²×C₄	—	2	4	no	—
335	C₂²×C₄	—	2	4	no	—
336	C₂²×C₄	—	2	16	no	—
337	C₂²×C₄	C₂²×C₄×C₈₉	1	4	yes	—
338	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
339	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
340	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
341	C₂²×C₄	#41×C₈₉	1	4	yes	—
342	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
343	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
344	Q₁₆	C₂²×C₄×C₈₉	1	16	yes	—
345	SD₁₆	C₂²×C₄×C₈₉	1	32	yes	—
346	D₈	C₂²×C₄×C₈₉	1	16	yes	—
347	M₁₆	C₂²×C₄×C₈₉	1	16	yes	—
348	C₂×C₈	—	2	8	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	C₂²×C₄×C₈₉	1	8	yes	—
352	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
353	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
354	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
355	C₄⋊C₄	—	2	4	no	—
356	C₄⋊C₄	—	2	16	no	—
357	C₄⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
358	C₄⋊C₄	C₂⁴×C₈₉	1	8	yes	—
359	C₂²⋊C₄	—	2	4	no	—
360	C₂²⋊C₄	—	2	4	no	—
361	C₂²⋊C₄	—	2	16	no	—
362	C₂²⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
363	C₄²	—	2	4	no	—
364	C₄²	—	2	16	no	—
365	C₄²	—	2	4	no	—
366	C₄²	C₂×D₄×C₈₉	1	4	yes	—
367	C₄²	C₂⁴×C₈₉	1	4	yes	—
368	C₂³	—	2	4	no	—
369	C₂³	—	2	8	no	—
370	C₂³	—	4	2	no	—
371	C₂³	—	8	4	no	—
372	C₂³	C₄×D₄×C₈₉	1	4	yes	—
373	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
374	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
375	Q₈	—	2	8	no	—
376	Q₈	—	2	4	no	—
377	Q₈	C₂³×C₄×C₈₉	1	8	yes	—
378	D₄	—	2	16	no	—
379	D₄	—	2	16	no	—
380	D₄	—	2	8	no	—
381	D₄	—	2	4	no	—
382	D₄	C₂³×C₄×C₈₉	1	8	yes	—
383	C₂×C₄	—	2	8	no	—
384	C₂×C₄	—	2	4	no	—
385	C₂×C₄	—	2	16	no	—
386	C₂×C₄	—	2	16	no	—
387	C₂×C₄	—	2	8	no	—
388	C₂×C₄	—	4	4	no	—
389	C₂×C₄	—	8	16	no	—
390	C₂×C₄	—	8	4	no	—
391	C₂×C₄	C₄×D₄×C₈₉	1	8	yes	—
392	C₂×C₄	C₄×D₄×C₈₉	1	4	yes	—
393	C₂×C₄	C₂³×C₄×C₈₉	1	2	yes	—
394	C₂×C₄	C₂³×C₄×C₈₉	1	4	yes	—
395	C₂×C₄	C₂²×D₄×C₈₉	1	8	yes	—
396	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
397	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
398	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
399	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
400	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
401	C₈	—	2	2	no	—
402	C₈	—	2	2	no	—
403	C₈	C₂³×C₄×C₈₉	1	8	yes	—
404	C₂²	—	2	2	no	—
405	C₂²	—	2	4	no	—
406	C₂²	—	2	8	no	—
407	C₂²	—	4	4	no	—
408	C₂²	—	4	8	no	—
409	C₂²	—	8	8	no	—
410	C₂²	C₂×C₄×D₄×C₈₉	1	4	yes	—
411	C₂²	C₂×C₄×D₄×C₈₉	1	2	yes	—
412	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
413	C₄	—	2	8	no	—
414	C₄	—	2	8	no	—
415	C₄	—	4	8	no	—
416	C₄	C₂×C₄×D₄×C₈₉	1	2	yes	—
417	C₄	C₂×C₄×D₄×C₈₉	1	4	yes	—
418	C₄	C₂⁴×C₄×C₈₉	1	1	yes	characteristic
419	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
420	C₂	—	2	4	no	—
421	C₂	—	2	2	no	—
422	C₂	—	4	8	no	—
423	C₂	C₂²×C₄×D₄×C₈₉	1	1	yes	characteristic
424	C₂	C₈₉×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈₉	C₂²×D₄
commutator	C₄	C₂⁴×C₄×C₈₉
Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
socle	C₂²×C₈₉	C₂³×D₄
prime radical	C₂×#575×C₈₉	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈₉ ◁ C₂×#117×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#3428	C₂	semidirect
3. C₈₉×#2947	C₂	semidirect
4. C₈₉×#2946	C₂	semidirect
5. C₈₉×#2943	C₂	semidirect
6. C₄×#135×C₈₉	C₂	semidirect
7. C₈₉×#2911	C₂	semidirect
8. C₂×#416×C₈₉	C₂	semidirect
9. C₂×#405×C₈₉	C₂	semidirect
11. #575×C₈₉	C₄	semidirect
19. #519×C₈₉	C₂²	semidirect
21. #518×C₈₉	C₂²	semidirect
23. C₄×#41×C₈₉	C₂²	semidirect
27. #442×C₈₉	C₂²	semidirect
28. #441×C₈₉	C₂²	semidirect
29. #440×C₈₉	C₂²	semidirect
30. C₄×Q₁₆×C₈₉	C₂²	semidirect
31. C₄×SD₁₆×C₈₉	C₂²	semidirect
32. C₄×D₈×C₈₉	C₂²	semidirect
33. #428×C₈₉	C₂²	semidirect
34. #427×C₈₉	C₂²	semidirect
36. #416×C₈₉	C₂²	semidirect
37. #416×C₈₉	C₂²	semidirect
38. C₂×#103×C₈₉	C₂²	semidirect
39. C₂×#102×C₈₉	C₂²	semidirect
40. #405×C₈₉	C₂²	semidirect
41. C₂×#97×C₈₉	C₂²	semidirect
42. C₂×C₄×C₈×C₈₉	C₂²	semidirect
46. C₂×#41×C₈₉	C₂×C₄	semidirect
54. #137×C₈₉	C₂×C₄	semidirect
55. AGL(1,ℤ/8)×C₈₉	C₂×C₄	semidirect
56. #135×C₈₉	C₂×C₄	semidirect
60. C₂×M₁₆×C₈₉	C₂×C₄	semidirect
207. #60161	C₈₉	direct (N × H)
214. #3428	C₂×C₈₉	semidirect
215. #2947	C₂×C₈₉	semidirect
216. #2946	C₂×C₈₉	semidirect
217. #2943	C₂×C₈₉	semidirect
218. C₄×#135	C₂×C₈₉	semidirect
219. #2911	C₂×C₈₉	semidirect
220. C₂×#416	C₂×C₈₉	semidirect
221. C₂×#405	C₂×C₈₉	semidirect
222. C₈₉	#60161	direct (N × H)
224. #575	C₄×C₈₉	semidirect
232. #519	C₂²×C₈₉	semidirect
234. #518	C₂²×C₈₉	semidirect
236. C₄×#41	C₂²×C₈₉	semidirect
240. #442	C₂²×C₈₉	semidirect
241. #441	C₂²×C₈₉	semidirect
242. #440	C₂²×C₈₉	semidirect
243. C₄×Q₁₆	C₂²×C₈₉	semidirect
244. C₄×SD₁₆	C₂²×C₈₉	semidirect
245. C₄×D₈	C₂²×C₈₉	semidirect
246. #428	C₂²×C₈₉	semidirect
247. #427	C₂²×C₈₉	semidirect
249. #416	C₂²×C₈₉	semidirect
250. #416	C₂²×C₈₉	semidirect
251. C₂×#103	C₂²×C₈₉	semidirect
252. C₂×#102	C₂²×C₈₉	semidirect
253. #405	C₂²×C₈₉	semidirect
254. C₂×#97	C₂²×C₈₉	semidirect
255. C₂×C₄×C₈	C₂²×C₈₉	semidirect
259. C₂×#41	C₂×C₄×C₈₉	semidirect
267. #137	C₂×C₄×C₈₉	semidirect
268. AGL(1,ℤ/8)	C₂×C₄×C₈₉	semidirect
269. #135	C₂×C₄×C₈₉	semidirect
273. C₂×M₁₆	C₂×C₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#575×C₈₉	2
C₈₉×#3428	2
C₈₉×#2947	2
C₈₉×#2946	2
C₈₉×#2943	2
C₄×#135×C₈₉	2
C₈₉×#2911	2
C₂×#416×C₈₉	2
C₂×#405×C₈₉	2
#60161	89

Subgroups — C89×#60161 #36438009

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60161 #36438009