Subgroups — C₈₉×#23530 #36401378

Order 22784.

1350 conjugacy classes of subgroups.

The subgroup lattice has 626 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	C₂²×#108×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2896	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#416×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#413×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#408×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2052	C₂	1	4	yes	maximal, complemented
8	C₈₉×#2051	C₂	1	4	yes	maximal, complemented
9	C₈₉×#1742	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
11	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
12	#575×C₈₉	—	2	4	no	—
13	#575×C₈₉	—	2	4	no	—
14	C₂×#137×C₈₉	—	2	1	no	—
15	C₂×#137×C₈₉	C₂²	1	2	yes	—
16	C₂×AGL(1,ℤ/8)×C₈₉	—	2	1	no	—
17	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	2	yes	—
18	C₂×#135×C₈₉	—	2	2	no	—
19	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
22	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#108×C₈₉	—	2	4	no	—
25	C₂×#108×C₈₉	C₂²	1	4	yes	complemented
26	#419×C₈₉	—	2	1	no	—
27	#418×C₈₉	—	2	1	no	—
28	#417×C₈₉	C₂²	1	4	yes	—
29	#416×C₈₉	—	2	2	no	—
30	#416×C₈₉	—	2	2	no	—
31	#416×C₈₉	—	2	2	no	—
32	C₂×#103×C₈₉	C₂²	1	2	yes	complemented
33	C₂×#102×C₈₉	C₂²	1	2	yes	complemented
34	#413×C₈₉	—	2	4	no	—
35	C₂×#101×C₈₉	C₂²	1	2	yes	complemented
36	C₂×#100×C₈₉	C₂²	1	2	yes	complemented
37	#408×C₈₉	—	2	2	no	—
38	#408×C₈₉	C₂²	1	4	yes	—
39	C₂×C₂²⋊C₈×C₈₉	—	2	2	no	—
40	#343×C₈₉	C₂²	1	4	yes	complemented
41	C₂×#41×C₈₉	—	2	1	no	—
42	C₂×#41×C₈₉	—	2	1	no	—
43	C₂×#41×C₈₉	—	2	2	no	—
44	C₂×#41×C₈₉	—	2	1	no	—
45	C₂×#41×C₈₉	—	2	1	no	—
46	C₂×#41×C₈₉	—	2	1	no	—
47	C₂×#41×C₈₉	—	2	1	no	—
48	C₂×#41×C₈₉	—	2	2	no	—
49	C₂×#41×C₈₉	—	2	2	no	—
50	C₂×#41×C₈₉	—	2	2	no	—
51	C₂×#41×C₈₉	—	2	2	no	—
52	C₂×#41×C₈₉	—	2	2	no	—
53	C₂×#41×C₈₉	—	2	2	no	—
54	C₂×#41×C₈₉	D₄	1	4	yes	—
55	C₂×#41×C₈₉	D₄	1	2	yes	—
56	C₂×#41×C₈₉	C₂³	1	2	yes	—
57	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
59	C₂²×D₄×C₈₉	—	2	1	no	—
60	C₂²×D₄×C₈₉	—	2	1	no	—
61	C₂²×D₄×C₈₉	D₄	1	1	yes	characteristic
62	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₈₉	—	2	1	no	—
64	C₂³×C₄×C₈₉	—	2	1	no	—
65	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
66	#137×C₈₉	—	2	8	no	—
67	#137×C₈₉	—	2	4	no	—
68	#137×C₈₉	—	2	4	no	—
69	AGL(1,ℤ/8)×C₈₉	—	2	8	no	—
70	AGL(1,ℤ/8)×C₈₉	—	2	4	no	—
71	AGL(1,ℤ/8)×C₈₉	—	2	4	no	—
72	#135×C₈₉	—	2	16	no	—
73	C₂×Q₁₆×C₈₉	—	2	2	no	—
74	C₂×SD₁₆×C₈₉	—	2	2	no	—
75	C₂×SD₁₆×C₈₉	—	2	2	no	—
76	C₂×D₈×C₈₉	—	2	2	no	—
77	C₂×M₁₆×C₈₉	—	2	1	no	—
78	C₂×M₁₆×C₈₉	—	2	2	no	—
79	C₂×M₁₆×C₈₉	—	2	1	no	—
80	C₂×M₁₆×C₈₉	—	2	1	no	—
81	C₂×M₁₆×C₈₉	—	2	2	no	—
82	C₂×M₁₆×C₈₉	—	2	2	no	—
83	C₂×M₁₆×C₈₉	—	2	2	no	—
84	C₂×M₁₆×C₈₉	—	2	2	no	—
85	C₂×M₁₆×C₈₉	D₄	1	4	yes	—
86	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
87	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
88	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
89	C₂²×C₈×C₈₉	—	2	2	no	—
90	C₂²×C₈×C₈₉	—	2	1	no	—
91	C₂²×C₈×C₈₉	—	2	2	no	—
92	C₂²×C₈×C₈₉	—	2	1	no	—
93	C₂²×C₈×C₈₉	—	2	1	no	—
94	C₂²×C₈×C₈₉	D₄	1	2	yes	—
95	C₂²×C₈×C₈₉	D₄	1	2	yes	—
96	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
97	#117×C₈₉	—	2	1	no	—
98	#117×C₈₉	—	2	1	no	—
99	#117×C₈₉	—	2	1	no	—
100	#117×C₈₉	—	2	1	no	—
101	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
102	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
103	C₂×C₄²×C₈₉	—	2	1	no	—
104	#108×C₈₉	—	2	8	no	—
105	#103×C₈₉	—	2	4	no	—
106	#103×C₈₉	—	2	4	no	—
107	#102×C₈₉	—	2	4	no	—
108	#102×C₈₉	—	2	4	no	—
109	#101×C₈₉	—	2	4	no	—
110	#100×C₈₉	—	2	4	no	—
111	C₂²⋊C₈×C₈₉	—	2	8	no	—
112	C₂⁴×C₈₉	—	2	1	no	—
113	C₂⁴×C₈₉	—	2	1	no	—
114	C₂⁴×C₈₉	#41	1	1	yes	characteristic
115	#41×C₈₉	—	2	4	no	—
116	#41×C₈₉	—	2	4	no	—
117	#41×C₈₉	—	2	4	no	—
118	#41×C₈₉	—	2	4	no	—
119	#41×C₈₉	—	2	4	no	—
120	#41×C₈₉	—	2	4	no	—
121	#41×C₈₉	—	2	2	no	—
122	#41×C₈₉	—	2	2	no	—
123	#41×C₈₉	—	2	2	no	—
124	#41×C₈₉	—	2	2	no	—
125	#41×C₈₉	—	4	2	no	—
126	#41×C₈₉	—	4	2	no	—
127	#41×C₈₉	—	4	2	no	—
128	#41×C₈₉	—	4	2	no	—
129	#41×C₈₉	—	4	2	no	—
130	#41×C₈₉	—	4	2	no	—
131	#41×C₈₉	—	4	2	no	—
132	#41×C₈₉	—	4	2	no	—
133	C₂×Q₈×C₈₉	—	2	1	no	—
134	C₂×Q₈×C₈₉	—	2	1	no	—
135	C₂×Q₈×C₈₉	—	2	2	no	—
136	C₂×Q₈×C₈₉	—	2	2	no	—
137	C₂×Q₈×C₈₉	—	2	1	no	—
138	C₂×Q₈×C₈₉	—	2	1	no	—
139	C₂×Q₈×C₈₉	—	2	1	no	—
140	C₂×Q₈×C₈₉	—	2	1	no	—
141	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
142	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
143	C₂×D₄×C₈₉	—	2	2	no	—
144	C₂×D₄×C₈₉	—	2	4	no	—
145	C₂×D₄×C₈₉	—	2	2	no	—
146	C₂×D₄×C₈₉	—	2	2	no	—
147	C₂×D₄×C₈₉	—	2	1	no	—
148	C₂×D₄×C₈₉	—	2	2	no	—
149	C₂×D₄×C₈₉	—	2	1	no	—
150	C₂×D₄×C₈₉	—	2	1	no	—
151	C₂×D₄×C₈₉	—	2	1	no	—
152	C₂×D₄×C₈₉	—	2	1	no	—
153	C₂×D₄×C₈₉	—	2	1	no	—
154	C₂×D₄×C₈₉	—	4	1	no	—
155	C₂×D₄×C₈₉	—	4	1	no	—
156	C₂×D₄×C₈₉	—	4	2	no	—
157	C₂×D₄×C₈₉	—	4	2	no	—
158	C₂×D₄×C₈₉	—	4	2	no	—
159	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
160	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₈₉	—	2	1	no	—
162	C₂²×C₄×C₈₉	—	2	1	no	—
163	C₂²×C₄×C₈₉	—	2	2	no	—
164	C₂²×C₄×C₈₉	—	2	4	no	—
165	C₂²×C₄×C₈₉	—	2	2	no	—
166	C₂²×C₄×C₈₉	—	2	2	no	—
167	C₂²×C₄×C₈₉	—	4	1	no	—
168	C₂²×C₄×C₈₉	—	4	2	no	—
169	C₂²×C₄×C₈₉	—	4	2	no	—
170	C₂²×C₄×C₈₉	—	4	1	no	—
171	C₂²×C₄×C₈₉	—	4	2	no	—
172	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₈₉	#41	1	2	yes	—
179	C₂²×C₄×C₈₉	#41	1	2	yes	—
180	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
181	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
182	Q₁₆×C₈₉	—	2	8	no	—
183	SD₁₆×C₈₉	—	2	8	no	—
184	SD₁₆×C₈₉	—	2	8	no	—
185	D₈×C₈₉	—	2	8	no	—
186	M₁₆×C₈₉	—	2	4	no	—
187	M₁₆×C₈₉	—	2	2	no	—
188	M₁₆×C₈₉	—	2	2	no	—
189	M₁₆×C₈₉	—	2	2	no	—
190	M₁₆×C₈₉	—	2	2	no	—
191	M₁₆×C₈₉	—	2	2	no	—
192	M₁₆×C₈₉	—	4	2	no	—
193	M₁₆×C₈₉	—	4	2	no	—
194	M₁₆×C₈₉	C₂×D₄	1	4	yes	—
195	C₂×C₈×C₈₉	—	2	4	no	—
196	C₂×C₈×C₈₉	—	2	4	no	—
197	C₂×C₈×C₈₉	—	2	4	no	—
198	C₂×C₈×C₈₉	—	2	2	no	—
199	C₂×C₈×C₈₉	—	2	2	no	—
200	C₂×C₈×C₈₉	—	2	2	no	—
201	C₂×C₈×C₈₉	—	2	2	no	—
202	C₂×C₈×C₈₉	—	2	2	no	—
203	C₂×C₈×C₈₉	—	4	2	no	—
204	C₂×C₈×C₈₉	—	4	2	no	—
205	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
206	C₄⋊C₄×C₈₉	—	2	2	no	—
207	C₄⋊C₄×C₈₉	—	2	2	no	—
208	C₂²⋊C₄×C₈₉	—	4	1	no	—
209	C₂²⋊C₄×C₈₉	—	4	1	no	—
210	C₄²×C₈₉	—	4	1	no	—
211	C₄²×C₈₉	—	4	1	no	—
212	C₂³×C₈₉	—	2	2	no	—
213	C₂³×C₈₉	—	2	2	no	—
214	C₂³×C₈₉	—	4	2	no	—
215	C₂³×C₈₉	—	4	4	no	—
216	C₂³×C₈₉	—	4	1	no	—
217	C₂³×C₈₉	—	4	1	no	—
218	C₂³×C₈₉	—	4	2	no	—
219	C₂³×C₈₉	—	8	2	no	—
220	C₂³×C₈₉	#121	1	1	yes	characteristic
221	C₂³×C₈₉	#121	1	1	yes	characteristic
222	C₂³×C₈₉	#123	1	2	yes	—
223	C₂³×C₈₉	#124	1	2	yes	—
224	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
225	Q₈×C₈₉	—	2	2	no	—
226	Q₈×C₈₉	—	2	2	no	—
227	Q₈×C₈₉	—	2	2	no	—
228	Q₈×C₈₉	—	2	2	no	—
229	Q₈×C₈₉	—	4	1	no	—
230	Q₈×C₈₉	—	4	1	no	—
231	Q₈×C₈₉	—	4	2	no	—
232	D₄×C₈₉	—	2	2	no	—
233	D₄×C₈₉	—	2	2	no	—
234	D₄×C₈₉	—	2	2	no	—
235	D₄×C₈₉	—	2	2	no	—
236	D₄×C₈₉	—	4	2	no	—
237	D₄×C₈₉	—	4	2	no	—
238	D₄×C₈₉	—	4	4	no	—
239	D₄×C₈₉	—	4	4	no	—
240	D₄×C₈₉	—	4	4	no	—
241	D₄×C₈₉	—	4	1	no	—
242	D₄×C₈₉	—	4	1	no	—
243	D₄×C₈₉	—	4	2	no	—
244	C₂×C₄×C₈₉	—	2	2	no	—
245	C₂×C₄×C₈₉	—	2	2	no	—
246	C₂×C₄×C₈₉	—	4	1	no	—
247	C₂×C₄×C₈₉	—	4	1	no	—
248	C₂×C₄×C₈₉	—	4	2	no	—
249	C₂×C₄×C₈₉	—	4	2	no	—
250	C₂×C₄×C₈₉	—	4	2	no	—
251	C₂×C₄×C₈₉	—	4	4	no	—
252	C₂×C₄×C₈₉	—	4	4	no	—
253	C₂×C₄×C₈₉	—	4	4	no	—
254	C₂×C₄×C₈₉	—	4	2	no	—
255	C₂×C₄×C₈₉	—	8	2	no	—
256	C₂×C₄×C₈₉	#120	1	2	yes	—
257	C₂×C₄×C₈₉	#120	1	2	yes	—
258	C₂×C₄×C₈₉	#121	1	2	yes	—
259	C₂×C₄×C₈₉	#121	1	2	yes	—
260	C₂×C₄×C₈₉	#121	1	2	yes	—
261	C₂×C₄×C₈₉	#121	1	2	yes	—
262	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
263	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
264	C₂×C₄×C₈₉	#123	1	2	yes	—
265	C₂×C₄×C₈₉	#123	1	2	yes	—
266	C₂×C₄×C₈₉	#123	1	2	yes	—
267	C₂×C₄×C₈₉	#124	1	2	yes	—
268	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
269	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
273	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
274	C₈×C₈₉	—	2	4	no	—
275	C₈×C₈₉	—	2	4	no	—
276	C₈×C₈₉	—	4	2	no	—
277	C₈×C₈₉	—	4	2	no	—
278	C₂²×C₈₉	—	2	2	no	—
279	C₂²×C₈₉	—	2	2	no	—
280	C₂²×C₈₉	—	2	2	no	—
281	C₂²×C₈₉	—	4	2	no	—
282	C₂²×C₈₉	—	4	2	no	—
283	C₂²×C₈₉	—	4	2	no	—
284	C₂²×C₈₉	—	4	2	no	—
285	C₂²×C₈₉	—	4	4	no	—
286	C₂²×C₈₉	—	8	4	no	—
287	C₂²×C₈₉	—	8	2	no	—
288	C₂²×C₈₉	—	8	2	no	—
289	C₂²×C₈₉	—	8	2	no	—
290	C₂²×C₈₉	#394	1	2	yes	—
291	C₂²×C₈₉	#394	1	2	yes	—
292	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
293	C₂²×C₈₉	C₂×#123	1	1	yes	characteristic
294	C₂²×C₈₉	C₂×#124	1	1	yes	characteristic
295	C₄×C₈₉	—	4	2	no	—
296	C₄×C₈₉	—	4	2	no	—
297	C₄×C₈₉	—	8	2	no	—
298	C₄×C₈₉	#394	1	2	yes	—
299	C₄×C₈₉	#394	1	2	yes	—
300	C₄×C₈₉	C₂×#120	1	1	yes	characteristic
301	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
302	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
303	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
304	#23530	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
305	C₂×C₈₉	—	2	1	no	—
306	C₂×C₈₉	—	2	1	no	—
307	C₂×C₈₉	—	2	2	no	—
308	C₂×C₈₉	—	2	2	no	—
309	C₂×C₈₉	—	8	2	no	—
310	C₂×C₈₉	—	8	2	no	—
311	C₂×C₈₉	C₂×#394	1	1	yes	characteristic
312	C₂×C₈₉	#3067	1	2	yes	—
313	C₂×#575	C₂×C₈₉	1	1	yes	characteristic
314	C₂²×#108	C₂×C₈₉	1	1	yes	characteristic, complemented
315	#2896	C₂×C₈₉	1	1	yes	characteristic, complemented
316	C₂×#416	C₂×C₈₉	1	1	yes	characteristic, complemented
317	C₂×#413	C₂×C₈₉	1	1	yes	characteristic, complemented
318	C₂×#408	C₂×C₈₉	1	1	yes	characteristic, complemented
319	#2052	C₂×C₈₉	1	4	yes	complemented
320	#2051	C₂×C₈₉	1	4	yes	complemented
321	#1742	C₂×C₈₉	1	1	yes	characteristic, complemented
322	C₈₉	#23530	1	1	yes	characteristic, direct product factor, Sylow
323	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
324	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
325	#575	—	2	4	no	—
326	#575	—	2	4	no	—
327	C₂×#137	—	2	1	no	—
328	C₂×#137	C₂²×C₈₉	1	2	yes	—
329	C₂×AGL(1,ℤ/8)	—	2	1	no	—
330	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	2	yes	—
331	C₂×#135	—	2	2	no	—
332	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
333	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
334	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
335	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
336	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
337	C₂×#108	—	2	4	no	—
338	C₂×#108	C₂²×C₈₉	1	4	yes	complemented
339	#419	—	2	1	no	—
340	#418	—	2	1	no	—
341	#417	C₂²×C₈₉	1	4	yes	—
342	#416	—	2	2	no	—
343	#416	—	2	2	no	—
344	#416	—	2	2	no	—
345	C₂×#103	C₂²×C₈₉	1	2	yes	complemented
346	C₂×#102	C₂²×C₈₉	1	2	yes	complemented
347	#413	—	2	4	no	—
348	C₂×#101	C₂²×C₈₉	1	2	yes	complemented
349	C₂×#100	C₂²×C₈₉	1	2	yes	complemented
350	#408	—	2	2	no	—
351	#408	C₂²×C₈₉	1	4	yes	—
352	C₂×C₂²⋊C₈	—	2	2	no	—
353	#343	C₂²×C₈₉	1	4	yes	complemented
354	C₂×#41	—	2	1	no	—
355	C₂×#41	—	2	1	no	—
356	C₂×#41	—	2	2	no	—
357	C₂×#41	—	2	1	no	—
358	C₂×#41	—	2	1	no	—
359	C₂×#41	—	2	2	no	—
360	C₂×#41	—	2	2	no	—
361	C₂×#41	—	2	2	no	—
362	C₂×#41	—	2	1	no	—
363	C₂×#41	—	2	1	no	—
364	C₂×#41	—	2	2	no	—
365	C₂×#41	—	2	2	no	—
366	C₂×#41	—	2	2	no	—
367	C₂×#41	D₄×C₈₉	1	2	yes	—
368	C₂×#41	D₄×C₈₉	1	4	yes	—
369	C₂×#41	C₂³×C₈₉	1	2	yes	—
370	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
371	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
372	C₂²×D₄	—	2	1	no	—
373	C₂²×D₄	—	2	1	no	—
374	C₂²×D₄	D₄×C₈₉	1	1	yes	characteristic
375	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
376	C₂³×C₄	—	2	1	no	—
377	C₂³×C₄	—	2	1	no	—
378	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
379	#137	—	2	8	no	—
380	#137	—	2	4	no	—
381	#137	—	2	4	no	—
382	AGL(1,ℤ/8)	—	2	8	no	—
383	AGL(1,ℤ/8)	—	2	4	no	—
384	AGL(1,ℤ/8)	—	2	4	no	—
385	#135	—	2	16	no	—
386	C₂×Q₁₆	—	2	2	no	—
387	C₂×SD₁₆	—	2	2	no	—
388	C₂×SD₁₆	—	2	2	no	—
389	C₂×D₈	—	2	2	no	—
390	C₂×M₁₆	—	2	1	no	—
391	C₂×M₁₆	—	2	2	no	—
392	C₂×M₁₆	—	2	1	no	—
393	C₂×M₁₆	—	2	1	no	—
394	C₂×M₁₆	—	2	2	no	—
395	C₂×M₁₆	—	2	2	no	—
396	C₂×M₁₆	—	2	2	no	—
397	C₂×M₁₆	—	2	2	no	—
398	C₂×M₁₆	D₄×C₈₉	1	4	yes	—
399	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
400	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
401	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
402	C₂²×C₈	—	2	2	no	—
403	C₂²×C₈	—	2	1	no	—
404	C₂²×C₈	—	2	2	no	—
405	C₂²×C₈	—	2	1	no	—
406	C₂²×C₈	—	2	1	no	—
407	C₂²×C₈	D₄×C₈₉	1	2	yes	—
408	C₂²×C₈	D₄×C₈₉	1	2	yes	—
409	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
410	#117	—	2	1	no	—
411	#117	—	2	1	no	—
412	#117	—	2	1	no	—
413	#117	—	2	1	no	—
414	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
415	C₂×C₂²⋊C₄	—	2	1	no	—
416	C₂×C₄²	—	2	1	no	—
417	#108	—	2	8	no	—
418	#103	—	2	4	no	—
419	#103	—	2	4	no	—
420	#102	—	2	4	no	—
421	#102	—	2	4	no	—
422	#101	—	2	4	no	—
423	#100	—	2	4	no	—
424	C₂²⋊C₈	—	2	8	no	—
425	C₂⁴	—	2	1	no	—
426	C₂⁴	—	2	1	no	—
427	C₂⁴	#41×C₈₉	1	1	yes	characteristic
428	#41	—	2	4	no	—
429	#41	—	2	4	no	—
430	#41	—	2	4	no	—
431	#41	—	2	4	no	—
432	#41	—	2	4	no	—
433	#41	—	2	4	no	—
434	#41	—	2	2	no	—
435	#41	—	2	2	no	—
436	#41	—	2	2	no	—
437	#41	—	2	2	no	—
438	#41	—	4	2	no	—
439	#41	—	4	2	no	—
440	#41	—	4	2	no	—
441	#41	—	4	2	no	—
442	#41	—	4	2	no	—
443	#41	—	4	2	no	—
444	#41	—	4	2	no	—
445	#41	—	4	2	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	2	1	no	—
448	C₂×Q₈	—	2	2	no	—
449	C₂×Q₈	—	2	2	no	—
450	C₂×Q₈	—	2	1	no	—
451	C₂×Q₈	—	2	1	no	—
452	C₂×Q₈	—	2	1	no	—
453	C₂×Q₈	—	2	1	no	—
454	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
455	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
456	C₂×D₄	—	2	2	no	—
457	C₂×D₄	—	2	4	no	—
458	C₂×D₄	—	2	2	no	—
459	C₂×D₄	—	2	2	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	—	2	2	no	—
462	C₂×D₄	—	2	1	no	—
463	C₂×D₄	—	2	1	no	—
464	C₂×D₄	—	2	1	no	—
465	C₂×D₄	—	2	1	no	—
466	C₂×D₄	—	2	1	no	—
467	C₂×D₄	—	4	1	no	—
468	C₂×D₄	—	4	1	no	—
469	C₂×D₄	—	4	2	no	—
470	C₂×D₄	—	4	2	no	—
471	C₂×D₄	—	4	2	no	—
472	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
473	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	2	2	no	—
477	C₂²×C₄	—	2	4	no	—
478	C₂²×C₄	—	2	2	no	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	4	2	no	—
481	C₂²×C₄	—	4	2	no	—
482	C₂²×C₄	—	4	1	no	—
483	C₂²×C₄	—	4	1	no	—
484	C₂²×C₄	—	4	2	no	—
485	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
486	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
487	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
488	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
489	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
490	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
491	C₂²×C₄	#41×C₈₉	1	2	yes	—
492	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
493	C₂²×C₄	#41×C₈₉	1	2	yes	—
494	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
495	Q₁₆	—	2	8	no	—
496	SD₁₆	—	2	8	no	—
497	SD₁₆	—	2	8	no	—
498	D₈	—	2	8	no	—
499	M₁₆	—	2	4	no	—
500	M₁₆	—	2	2	no	—
501	M₁₆	—	2	2	no	—
502	M₁₆	—	2	2	no	—
503	M₁₆	—	2	2	no	—
504	M₁₆	—	2	2	no	—
505	M₁₆	—	4	2	no	—
506	M₁₆	—	4	2	no	—
507	M₁₆	C₂×D₄×C₈₉	1	4	yes	—
508	C₂×C₈	—	2	4	no	—
509	C₂×C₈	—	2	4	no	—
510	C₂×C₈	—	2	4	no	—
511	C₂×C₈	—	2	2	no	—
512	C₂×C₈	—	2	2	no	—
513	C₂×C₈	—	2	2	no	—
514	C₂×C₈	—	2	2	no	—
515	C₂×C₈	—	2	2	no	—
516	C₂×C₈	—	4	2	no	—
517	C₂×C₈	—	4	2	no	—
518	C₂×C₈	C₂×D₄×C₈₉	1	4	yes	—
519	C₄⋊C₄	—	2	2	no	—
520	C₄⋊C₄	—	2	2	no	—
521	C₂²⋊C₄	—	4	1	no	—
522	C₂²⋊C₄	—	4	1	no	—
523	C₄²	—	4	1	no	—
524	C₄²	—	4	1	no	—
525	C₂³	—	2	2	no	—
526	C₂³	—	2	2	no	—
527	C₂³	—	4	2	no	—
528	C₂³	—	4	4	no	—
529	C₂³	—	4	2	no	—
530	C₂³	—	4	1	no	—
531	C₂³	—	4	1	no	—
532	C₂³	—	8	2	no	—
533	C₂³	#121×C₈₉	1	1	yes	characteristic
534	C₂³	#121×C₈₉	1	1	yes	characteristic
535	C₂³	#123×C₈₉	1	2	yes	—
536	C₂³	#124×C₈₉	1	2	yes	—
537	C₂³	C₂×#41×C₈₉	1	1	yes	characteristic
538	Q₈	—	2	2	no	—
539	Q₈	—	2	2	no	—
540	Q₈	—	2	2	no	—
541	Q₈	—	2	2	no	—
542	Q₈	—	4	2	no	—
543	Q₈	—	4	1	no	—
544	Q₈	—	4	1	no	—
545	D₄	—	2	2	no	—
546	D₄	—	2	2	no	—
547	D₄	—	2	2	no	—
548	D₄	—	2	2	no	—
549	D₄	—	4	4	no	—
550	D₄	—	4	4	no	—
551	D₄	—	4	4	no	—
552	D₄	—	4	2	no	—
553	D₄	—	4	2	no	—
554	D₄	—	4	2	no	—
555	D₄	—	4	1	no	—
556	D₄	—	4	1	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	4	2	no	—
560	C₂×C₄	—	4	1	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	4	4	no	—
563	C₂×C₄	—	4	4	no	—
564	C₂×C₄	—	4	4	no	—
565	C₂×C₄	—	4	2	no	—
566	C₂×C₄	—	4	2	no	—
567	C₂×C₄	—	4	2	no	—
568	C₂×C₄	—	8	2	no	—
569	C₂×C₄	#120×C₈₉	1	2	yes	—
570	C₂×C₄	#120×C₈₉	1	2	yes	—
571	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
572	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
573	C₂×C₄	#121×C₈₉	1	2	yes	—
574	C₂×C₄	#121×C₈₉	1	2	yes	—
575	C₂×C₄	#121×C₈₉	1	2	yes	—
576	C₂×C₄	#121×C₈₉	1	2	yes	—
577	C₂×C₄	#123×C₈₉	1	2	yes	—
578	C₂×C₄	#123×C₈₉	1	2	yes	—
579	C₂×C₄	#123×C₈₉	1	2	yes	—
580	C₂×C₄	#124×C₈₉	1	2	yes	—
581	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
582	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
583	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
584	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
585	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
586	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
587	C₈	—	2	4	no	—
588	C₈	—	2	4	no	—
589	C₈	—	4	2	no	—
590	C₈	—	4	2	no	—
591	C₂²	—	2	2	no	—
592	C₂²	—	2	2	no	—
593	C₂²	—	2	2	no	—
594	C₂²	—	4	2	no	—
595	C₂²	—	4	2	no	—
596	C₂²	—	4	4	no	—
597	C₂²	—	4	2	no	—
598	C₂²	—	4	2	no	—
599	C₂²	—	8	2	no	—
600	C₂²	—	8	4	no	—
601	C₂²	—	8	2	no	—
602	C₂²	—	8	2	no	—
603	C₂²	#394×C₈₉	1	2	yes	—
604	C₂²	#394×C₈₉	1	2	yes	—
605	C₂²	C₂×#121×C₈₉	1	1	yes	characteristic
606	C₂²	C₂×#123×C₈₉	1	1	yes	characteristic
607	C₂²	C₂×#124×C₈₉	1	1	yes	characteristic
608	C₄	—	4	2	no	—
609	C₄	—	4	2	no	—
610	C₄	—	8	2	no	—
611	C₄	#394×C₈₉	1	2	yes	—
612	C₄	#394×C₈₉	1	2	yes	—
613	C₄	C₂×#120×C₈₉	1	1	yes	characteristic
614	C₄	C₂×#121×C₈₉	1	1	yes	characteristic
615	C₄	C₂×#121×C₈₉	1	1	yes	characteristic
616	C₄	C₂×#123×C₈₉	1	1	yes	characteristic
617	C₂	—	2	2	no	—
618	C₂	—	2	1	no	—
619	C₂	—	2	1	no	—
620	C₂	—	2	2	no	—
621	C₂	—	8	2	no	—
622	C₂	—	8	2	no	—
623	C₂	C₂×#394×C₈₉	1	1	yes	characteristic
624	C₂	C₈₉×#3067	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈₉	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
socle	C₂²×C₈₉	C₂×#121
prime radical	C₂×#575×C₈₉	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈₉ ◁ C₂³×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×#108×C₈₉	C₂	semidirect
3. C₈₉×#2896	C₂	semidirect
4. C₂×#416×C₈₉	C₂	semidirect
5. C₂×#413×C₈₉	C₂	semidirect
6. C₂×#408×C₈₉	C₂	semidirect
7. C₈₉×#2052	C₂	semidirect
8. C₈₉×#2051	C₂	semidirect
9. C₈₉×#1742	C₂	semidirect
25. C₂×#108×C₈₉	C₂²	semidirect
32. C₂×#103×C₈₉	C₂²	semidirect
33. C₂×#102×C₈₉	C₂²	semidirect
35. C₂×#101×C₈₉	C₂²	semidirect
36. C₂×#100×C₈₉	C₂²	semidirect
40. #343×C₈₉	C₂²	semidirect
304. #23530	C₈₉	direct (N × H)
314. C₂²×#108	C₂×C₈₉	semidirect
315. #2896	C₂×C₈₉	semidirect
316. C₂×#416	C₂×C₈₉	semidirect
317. C₂×#413	C₂×C₈₉	semidirect
318. C₂×#408	C₂×C₈₉	semidirect
319. #2052	C₂×C₈₉	semidirect
320. #2051	C₂×C₈₉	semidirect
321. #1742	C₂×C₈₉	semidirect
322. C₈₉	#23530	direct (N × H)
338. C₂×#108	C₂²×C₈₉	semidirect
345. C₂×#103	C₂²×C₈₉	semidirect
346. C₂×#102	C₂²×C₈₉	semidirect
348. C₂×#101	C₂²×C₈₉	semidirect
349. C₂×#100	C₂²×C₈₉	semidirect
353. #343	C₂²×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
C₂²×#108×C₈₉	2
C₈₉×#2896	2
C₂×#416×C₈₉	2
C₂×#413×C₈₉	2
C₂×#408×C₈₉	2
C₈₉×#2052	2
C₈₉×#2051	2
C₈₉×#1742	2
#23530	89

Subgroups — C89×#23530 #36401378

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#23530 #36401378