Subgroups — C₈₉×#23892 #36401740

Order 22784.

1674 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3585	C₂	1	2	yes	maximal, complemented
2	C₈₉×#3019	C₂	1	2	yes	maximal, complemented
3	C₈₉×#3008	C₂	1	2	yes	maximal, complemented
4	C₈₉×#2018	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2017	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#2008	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1903	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1881	C₂	1	2	yes	maximal, complemented
9	C₈₉×#1794	C₂	1	1	yes	maximal, characteristic, complemented
10	#585×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×H(5,𝔽₂)×C₈₉	C₂²	1	2	yes	complemented
12	#577×C₈₉	—	2	4	no	—
13	#577×C₈₉	—	2	4	no	—
14	#576×C₈₉	—	2	4	no	—
15	#576×C₈₉	—	2	4	no	—
16	#575×C₈₉	—	2	4	no	—
17	#575×C₈₉	C₂²	1	2	yes	—
18	#575×C₈₉	C₂²	1	2	yes	complemented
19	#575×C₈₉	C₂²	1	1	yes	characteristic, complemented
20	C₂×AGL(1,ℤ/8)×C₈₉	—	2	4	no	—
21	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	2	yes	—
22	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	2	yes	complemented
23	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
24	#568×C₈₉	C₂²	1	2	yes	complemented
25	#535×C₈₉	C₂²	1	2	yes	complemented
26	#455×C₈₉	—	2	4	no	—
27	#453×C₈₉	—	2	4	no	—
28	#449×C₈₉	—	2	4	no	—
29	#447×C₈₉	—	2	4	no	—
30	#421×C₈₉	C₂²	1	2	yes	complemented
31	#421×C₈₉	C₂²	1	2	yes	complemented
32	#418×C₈₉	C₂²	1	2	yes	complemented
33	#416×C₈₉	C₂²	1	2	yes	complemented
34	C₂×#100×C₈₉	C₂²	1	2	yes	complemented
35	#410×C₈₉	C₂²	1	1	yes	characteristic, complemented
36	#408×C₈₉	C₂²	1	2	yes	complemented
37	#408×C₈₉	C₂²	1	1	yes	characteristic, complemented
38	#344×C₈₉	C₂²	1	2	yes	complemented
39	#341×C₈₉	C₂²	1	1	yes	characteristic, complemented
40	#338×C₈₉	C₂²	1	1	yes	characteristic, complemented
41	2₋¹⁺⁴×C₈₉	—	2	2	no	—
42	2₋¹⁺⁴×C₈₉	D₄	1	2	yes	complemented
43	H(5,𝔽₂)×C₈₉	—	2	4	no	—
44	H(5,𝔽₂)×C₈₉	—	2	2	no	—
45	H(5,𝔽₂)×C₈₉	—	2	4	no	—
46	H(5,𝔽₂)×C₈₉	—	2	4	no	—
47	H(5,𝔽₂)×C₈₉	—	4	1	no	—
48	H(5,𝔽₂)×C₈₉	—	4	2	no	—
49	H(5,𝔽₂)×C₈₉	D₄	1	2	yes	complemented
50	C₂×#41×C₈₉	—	2	4	no	—
51	C₂×#41×C₈₉	—	2	2	no	—
52	C₂×#41×C₈₉	—	2	2	no	—
53	C₂×#41×C₈₉	—	4	1	no	—
54	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
55	C₂×#41×C₈₉	C₂³	1	2	yes	—
56	C₂×#41×C₈₉	C₂³	1	2	yes	—
57	C₂²×D₄×C₈₉	—	2	2	no	—
58	C₂²×D₄×C₈₉	—	2	2	no	—
59	C₂²×D₄×C₈₉	—	4	2	no	—
60	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
61	#137×C₈₉	—	2	8	no	—
62	#137×C₈₉	—	2	2	no	—
63	#137×C₈₉	—	2	2	no	—
64	#137×C₈₉	—	2	2	no	—
65	#137×C₈₉	—	2	2	no	—
66	#137×C₈₉	D₄	1	4	yes	complemented
67	AGL(1,ℤ/8)×C₈₉	—	2	8	no	—
68	AGL(1,ℤ/8)×C₈₉	—	2	8	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	AGL(1,ℤ/8)×C₈₉	—	2	2	no	—
73	AGL(1,ℤ/8)×C₈₉	—	2	4	no	—
74	AGL(1,ℤ/8)×C₈₉	—	2	2	no	—
75	AGL(1,ℤ/8)×C₈₉	—	2	2	no	—
76	AGL(1,ℤ/8)×C₈₉	—	2	2	no	—
77	AGL(1,ℤ/8)×C₈₉	—	2	4	no	—
78	AGL(1,ℤ/8)×C₈₉	D₄	1	4	yes	complemented
79	#135×C₈₉	—	2	8	no	—
80	#135×C₈₉	—	2	8	no	—
81	#135×C₈₉	—	2	4	no	—
82	#135×C₈₉	—	2	4	no	—
83	#135×C₈₉	—	4	1	no	—
84	C₂×SD₁₆×C₈₉	—	2	8	no	—
85	C₂×SD₁₆×C₈₉	—	2	2	no	—
86	C₂×SD₁₆×C₈₉	D₄	1	4	yes	complemented
87	C₂×D₈×C₈₉	—	2	8	no	—
88	C₂×D₈×C₈₉	—	2	2	no	—
89	C₂×D₈×C₈₉	—	2	4	no	—
90	C₂×D₈×C₈₉	—	4	1	no	—
91	C₂×D₈×C₈₉	D₄	1	4	yes	complemented
92	#131×C₈₉	—	2	4	no	—
93	#131×C₈₉	—	2	4	no	—
94	C₂×M₁₆×C₈₉	—	2	4	no	—
95	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
96	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
97	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
98	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
99	#127×C₈₉	—	2	2	no	—
100	#124×C₈₉	—	2	2	no	—
101	#121×C₈₉	—	2	4	no	—
102	#120×C₈₉	—	4	2	no	—
103	#117×C₈₉	C₂³	1	2	yes	—
104	#104×C₈₉	—	2	4	no	—
105	#104×C₈₉	—	2	4	no	—
106	#103×C₈₉	—	2	2	no	—
107	#102×C₈₉	—	2	2	no	—
108	#102×C₈₉	—	2	4	no	—
109	#100×C₈₉	—	2	4	no	—
110	#99×C₈₉	—	4	1	no	—
111	C₂²⋊C₈×C₈₉	—	2	4	no	—
112	C₂²⋊C₈×C₈₉	—	4	1	no	—
113	C₂⁴×C₈₉	—	2	2	no	—
114	C₂⁴×C₈₉	—	4	2	no	—
115	#41×C₈₉	—	2	2	no	—
116	#41×C₈₉	—	2	2	no	—
117	#41×C₈₉	—	2	8	no	—
118	#41×C₈₉	—	2	4	no	—
119	#41×C₈₉	—	2	4	no	—
120	#41×C₈₉	—	2	2	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	—	2	2	no	—
126	#41×C₈₉	—	4	2	no	—
127	#41×C₈₉	—	4	2	no	—
128	#41×C₈₉	—	4	4	no	—
129	#41×C₈₉	—	4	4	no	—
130	#41×C₈₉	—	4	4	no	—
131	#41×C₈₉	—	4	2	no	—
132	#41×C₈₉	—	4	4	no	—
133	#41×C₈₉	C₂×D₄	1	4	yes	—
134	#41×C₈₉	C₂×D₄	1	4	yes	—
135	C₂×Q₈×C₈₉	—	2	4	no	—
136	C₂×Q₈×C₈₉	—	4	1	no	—
137	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
138	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
139	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
140	C₂×D₄×C₈₉	—	2	2	no	—
141	C₂×D₄×C₈₉	—	2	8	no	—
142	C₂×D₄×C₈₉	—	2	4	no	—
143	C₂×D₄×C₈₉	—	2	1	no	—
144	C₂×D₄×C₈₉	—	2	2	no	—
145	C₂×D₄×C₈₉	—	2	2	no	—
146	C₂×D₄×C₈₉	—	2	2	no	—
147	C₂×D₄×C₈₉	—	2	2	no	—
148	C₂×D₄×C₈₉	—	2	2	no	—
149	C₂×D₄×C₈₉	—	2	2	no	—
150	C₂×D₄×C₈₉	—	4	4	no	—
151	C₂×D₄×C₈₉	—	4	2	no	—
152	C₂×D₄×C₈₉	—	4	2	no	—
153	C₂×D₄×C₈₉	—	4	2	no	—
154	C₂×D₄×C₈₉	—	4	4	no	—
155	C₂×D₄×C₈₉	—	4	1	no	—
156	C₂×D₄×C₈₉	—	4	8	no	—
157	C₂×D₄×C₈₉	—	4	4	no	—
158	C₂×D₄×C₈₉	—	4	8	no	—
159	C₂×D₄×C₈₉	—	4	4	no	—
160	C₂×D₄×C₈₉	—	4	4	no	—
161	C₂×D₄×C₈₉	C₂×D₄	1	4	yes	—
162	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
163	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
164	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₈₉	—	2	2	no	—
166	C₂²×C₄×C₈₉	—	2	2	no	—
167	C₂²×C₄×C₈₉	—	2	1	no	—
168	C₂²×C₄×C₈₉	—	4	2	no	—
169	C₂²×C₄×C₈₉	—	4	2	no	—
170	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
171	Q₁₆×C₈₉	—	2	4	no	—
172	Q₁₆×C₈₉	—	2	4	no	—
173	Q₁₆×C₈₉	—	4	1	no	—
174	SD₁₆×C₈₉	—	2	8	no	—
175	SD₁₆×C₈₉	—	2	4	no	—
176	SD₁₆×C₈₉	—	2	4	no	—
177	SD₁₆×C₈₉	—	2	4	no	—
178	SD₁₆×C₈₉	—	2	8	no	—
179	SD₁₆×C₈₉	—	2	4	no	—
180	SD₁₆×C₈₉	—	4	2	no	—
181	D₈×C₈₉	—	2	8	no	—
182	D₈×C₈₉	—	2	4	no	—
183	D₈×C₈₉	—	2	4	no	—
184	D₈×C₈₉	—	2	8	no	—
185	D₈×C₈₉	—	4	4	no	—
186	D₈×C₈₉	—	4	1	no	—
187	M₁₆×C₈₉	—	2	8	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	C₂×D₄	1	4	yes	—
193	C₂×C₈×C₈₉	—	2	8	no	—
194	C₂×C₈×C₈₉	—	2	2	no	—
195	C₂×C₈×C₈₉	—	2	2	no	—
196	C₂×C₈×C₈₉	—	2	1	no	—
197	C₂×C₈×C₈₉	—	2	1	no	—
198	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
199	C₄⋊C₄×C₈₉	—	2	2	no	—
200	C₂²⋊C₄×C₈₉	—	2	2	no	—
201	C₂²⋊C₄×C₈₉	—	4	4	no	—
202	C₄²×C₈₉	—	2	2	no	—
203	C₂³×C₈₉	—	2	2	no	—
204	C₂³×C₈₉	—	2	2	no	—
205	C₂³×C₈₉	—	2	1	no	—
206	C₂³×C₈₉	—	2	4	no	—
207	C₂³×C₈₉	—	4	2	no	—
208	C₂³×C₈₉	—	4	8	no	—
209	C₂³×C₈₉	—	4	2	no	—
210	C₂³×C₈₉	—	4	2	no	—
211	C₂³×C₈₉	—	16	2	no	—
212	C₂³×C₈₉	—	16	4	no	—
213	C₂³×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
214	Q₈×C₈₉	—	2	2	no	—
215	Q₈×C₈₉	—	2	2	no	—
216	Q₈×C₈₉	—	2	2	no	—
217	Q₈×C₈₉	—	2	2	no	—
218	Q₈×C₈₉	—	4	2	no	—
219	Q₈×C₈₉	#120	1	4	yes	—
220	D₄×C₈₉	—	2	4	no	—
221	D₄×C₈₉	—	2	4	no	—
222	D₄×C₈₉	—	2	2	no	—
223	D₄×C₈₉	—	2	2	no	—
224	D₄×C₈₉	—	2	4	no	—
225	D₄×C₈₉	—	2	2	no	—
226	D₄×C₈₉	—	2	4	no	—
227	D₄×C₈₉	—	2	2	no	—
228	D₄×C₈₉	—	4	4	no	—
229	D₄×C₈₉	—	4	8	no	—
230	D₄×C₈₉	—	4	2	no	—
231	D₄×C₈₉	—	4	8	no	—
232	D₄×C₈₉	—	4	2	no	—
233	D₄×C₈₉	—	4	2	no	—
234	D₄×C₈₉	—	4	2	no	—
235	D₄×C₈₉	—	4	2	no	—
236	D₄×C₈₉	—	4	2	no	—
237	D₄×C₈₉	—	16	4	no	—
238	D₄×C₈₉	#120	1	4	yes	—
239	C₂×C₄×C₈₉	—	2	2	no	—
240	C₂×C₄×C₈₉	—	2	4	no	—
241	C₂×C₄×C₈₉	—	2	4	no	—
242	C₂×C₄×C₈₉	—	2	1	no	—
243	C₂×C₄×C₈₉	—	2	2	no	—
244	C₂×C₄×C₈₉	—	4	2	no	—
245	C₂×C₄×C₈₉	—	4	4	no	—
246	C₂×C₄×C₈₉	—	4	8	no	—
247	C₂×C₄×C₈₉	—	4	2	no	—
248	C₂×C₄×C₈₉	—	4	1	no	—
249	C₂×C₄×C₈₉	—	4	2	no	—
250	C₂×C₄×C₈₉	—	4	1	no	—
251	C₂×C₄×C₈₉	—	4	2	no	—
252	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
253	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
254	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
255	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
256	C₈×C₈₉	—	2	4	no	—
257	C₈×C₈₉	—	2	4	no	—
258	C₈×C₈₉	—	4	1	no	—
259	C₈×C₈₉	—	4	1	no	—
260	C₂²×C₈₉	—	2	4	no	—
261	C₂²×C₈₉	—	4	1	no	—
262	C₂²×C₈₉	—	4	4	no	—
263	C₂²×C₈₉	—	4	1	no	—
264	C₂²×C₈₉	—	4	2	no	—
265	C₂²×C₈₉	—	4	4	no	—
266	C₂²×C₈₉	—	8	2	no	—
267	C₂²×C₈₉	—	8	4	no	—
268	C₂²×C₈₉	—	8	4	no	—
269	C₂²×C₈₉	—	8	1	no	—
270	C₂²×C₈₉	—	16	2	no	—
271	C₂²×C₈₉	—	16	2	no	—
272	C₂²×C₈₉	—	16	8	no	—
273	C₂²×C₈₉	#534	1	1	yes	characteristic
274	C₂²×C₈₉	D₄²	1	2	yes	—
275	C₄×C₈₉	—	2	4	no	—
276	C₄×C₈₉	—	4	2	no	—
277	C₄×C₈₉	—	4	1	no	—
278	C₄×C₈₉	—	8	2	no	—
279	C₄×C₈₉	C₂×#120	1	2	yes	—
280	C₄×C₈₉	D₄²	1	1	yes	characteristic
281	C₄×C₈₉	D₄²	1	1	yes	characteristic
282	#23892	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
283	C₂×C₈₉	—	2	2	no	—
284	C₂×C₈₉	—	2	1	no	—
285	C₂×C₈₉	—	4	4	no	—
286	C₂×C₈₉	—	8	2	no	—
287	C₂×C₈₉	—	8	4	no	—
288	C₂×C₈₉	—	8	1	no	—
289	C₂×C₈₉	#2403	1	1	yes	characteristic
290	#3585	C₂×C₈₉	1	2	yes	complemented
291	#3019	C₂×C₈₉	1	2	yes	complemented
292	#3008	C₂×C₈₉	1	2	yes	complemented
293	#2018	C₂×C₈₉	1	1	yes	characteristic, complemented
294	#2017	C₂×C₈₉	1	1	yes	characteristic, complemented
295	#2008	C₂×C₈₉	1	2	yes	complemented
296	#1903	C₂×C₈₉	1	2	yes	complemented
297	#1881	C₂×C₈₉	1	2	yes	complemented
298	#1794	C₂×C₈₉	1	1	yes	characteristic, complemented
299	C₈₉	#23892	1	1	yes	characteristic, direct product factor, Sylow
300	#585	C₂²×C₈₉	1	1	yes	characteristic
301	C₂×H(5,𝔽₂)	C₂²×C₈₉	1	2	yes	complemented
302	#577	—	2	4	no	—
303	#577	—	2	4	no	—
304	#576	—	2	4	no	—
305	#576	—	2	4	no	—
306	#575	—	2	4	no	—
307	#575	C₂²×C₈₉	1	2	yes	—
308	#575	C₂²×C₈₉	1	2	yes	complemented
309	#575	C₂²×C₈₉	1	1	yes	characteristic, complemented
310	C₂×AGL(1,ℤ/8)	—	2	4	no	—
311	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	2	yes	—
312	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	2	yes	complemented
313	C₂²×D₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
314	#568	C₂²×C₈₉	1	2	yes	complemented
315	#535	C₂²×C₈₉	1	2	yes	complemented
316	#455	—	2	4	no	—
317	#453	—	2	4	no	—
318	#449	—	2	4	no	—
319	#447	—	2	4	no	—
320	#421	C₂²×C₈₉	1	2	yes	complemented
321	#421	C₂²×C₈₉	1	2	yes	complemented
322	#418	C₂²×C₈₉	1	2	yes	complemented
323	#416	C₂²×C₈₉	1	2	yes	complemented
324	C₂×#100	C₂²×C₈₉	1	2	yes	complemented
325	#410	C₂²×C₈₉	1	1	yes	characteristic, complemented
326	#408	C₂²×C₈₉	1	2	yes	complemented
327	#408	C₂²×C₈₉	1	1	yes	characteristic, complemented
328	#344	C₂²×C₈₉	1	2	yes	complemented
329	#341	C₂²×C₈₉	1	1	yes	characteristic, complemented
330	#338	C₂²×C₈₉	1	1	yes	characteristic, complemented
331	2₋¹⁺⁴	—	2	2	no	—
332	2₋¹⁺⁴	D₄×C₈₉	1	2	yes	complemented
333	H(5,𝔽₂)	—	2	4	no	—
334	H(5,𝔽₂)	—	2	2	no	—
335	H(5,𝔽₂)	—	2	4	no	—
336	H(5,𝔽₂)	—	2	4	no	—
337	H(5,𝔽₂)	—	4	1	no	—
338	H(5,𝔽₂)	—	4	2	no	—
339	H(5,𝔽₂)	D₄×C₈₉	1	2	yes	complemented
340	C₂×#41	—	2	4	no	—
341	C₂×#41	—	2	2	no	—
342	C₂×#41	—	2	2	no	—
343	C₂×#41	—	4	1	no	—
344	C₂×#41	C₂³×C₈₉	1	2	yes	—
345	C₂×#41	C₂³×C₈₉	1	1	yes	characteristic
346	C₂×#41	C₂³×C₈₉	1	2	yes	—
347	C₂²×D₄	—	2	2	no	—
348	C₂²×D₄	—	2	2	no	—
349	C₂²×D₄	—	4	2	no	—
350	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
351	#137	—	2	8	no	—
352	#137	—	2	2	no	—
353	#137	—	2	2	no	—
354	#137	—	2	2	no	—
355	#137	—	2	2	no	—
356	#137	D₄×C₈₉	1	4	yes	complemented
357	AGL(1,ℤ/8)	—	2	8	no	—
358	AGL(1,ℤ/8)	—	2	8	no	—
359	AGL(1,ℤ/8)	—	2	8	no	—
360	AGL(1,ℤ/8)	—	2	4	no	—
361	AGL(1,ℤ/8)	—	2	4	no	—
362	AGL(1,ℤ/8)	—	2	2	no	—
363	AGL(1,ℤ/8)	—	2	4	no	—
364	AGL(1,ℤ/8)	—	2	2	no	—
365	AGL(1,ℤ/8)	—	2	2	no	—
366	AGL(1,ℤ/8)	—	2	2	no	—
367	AGL(1,ℤ/8)	—	2	4	no	—
368	AGL(1,ℤ/8)	D₄×C₈₉	1	4	yes	complemented
369	#135	—	2	8	no	—
370	#135	—	2	8	no	—
371	#135	—	2	4	no	—
372	#135	—	2	4	no	—
373	#135	—	4	1	no	—
374	C₂×SD₁₆	—	2	8	no	—
375	C₂×SD₁₆	—	2	2	no	—
376	C₂×SD₁₆	D₄×C₈₉	1	4	yes	complemented
377	C₂×D₈	—	2	8	no	—
378	C₂×D₈	—	2	2	no	—
379	C₂×D₈	—	2	4	no	—
380	C₂×D₈	—	4	1	no	—
381	C₂×D₈	D₄×C₈₉	1	4	yes	complemented
382	#131	—	2	4	no	—
383	#131	—	2	4	no	—
384	C₂×M₁₆	—	2	4	no	—
385	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
386	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
387	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
388	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
389	#127	—	2	2	no	—
390	#124	—	2	2	no	—
391	#121	—	2	4	no	—
392	#120	—	4	2	no	—
393	#117	C₂³×C₈₉	1	2	yes	—
394	#104	—	2	4	no	—
395	#104	—	2	4	no	—
396	#103	—	2	2	no	—
397	#102	—	2	2	no	—
398	#102	—	2	4	no	—
399	#100	—	2	4	no	—
400	#99	—	4	1	no	—
401	C₂²⋊C₈	—	2	4	no	—
402	C₂²⋊C₈	—	4	1	no	—
403	C₂⁴	—	2	2	no	—
404	C₂⁴	—	4	2	no	—
405	#41	—	2	2	no	—
406	#41	—	2	2	no	—
407	#41	—	2	8	no	—
408	#41	—	2	4	no	—
409	#41	—	2	4	no	—
410	#41	—	2	2	no	—
411	#41	—	2	2	no	—
412	#41	—	2	2	no	—
413	#41	—	2	2	no	—
414	#41	—	2	2	no	—
415	#41	—	2	2	no	—
416	#41	—	4	2	no	—
417	#41	—	4	2	no	—
418	#41	—	4	4	no	—
419	#41	—	4	4	no	—
420	#41	—	4	4	no	—
421	#41	—	4	2	no	—
422	#41	—	4	4	no	—
423	#41	C₂×D₄×C₈₉	1	4	yes	—
424	#41	C₂×D₄×C₈₉	1	4	yes	—
425	C₂×Q₈	—	2	4	no	—
426	C₂×Q₈	—	4	1	no	—
427	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
428	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
429	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
430	C₂×D₄	—	2	2	no	—
431	C₂×D₄	—	2	8	no	—
432	C₂×D₄	—	2	4	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	2	no	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	2	2	no	—
439	C₂×D₄	—	2	2	no	—
440	C₂×D₄	—	4	4	no	—
441	C₂×D₄	—	4	2	no	—
442	C₂×D₄	—	4	2	no	—
443	C₂×D₄	—	4	2	no	—
444	C₂×D₄	—	4	4	no	—
445	C₂×D₄	—	4	1	no	—
446	C₂×D₄	—	4	8	no	—
447	C₂×D₄	—	4	4	no	—
448	C₂×D₄	—	4	8	no	—
449	C₂×D₄	—	4	4	no	—
450	C₂×D₄	—	4	4	no	—
451	C₂×D₄	C₂×D₄×C₈₉	1	4	yes	—
452	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
453	C₂×D₄	C₂×D₄×C₈₉	1	1	yes	characteristic
454	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
455	C₂²×C₄	—	2	2	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	4	2	no	—
459	C₂²×C₄	—	4	2	no	—
460	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
461	Q₁₆	—	2	4	no	—
462	Q₁₆	—	2	4	no	—
463	Q₁₆	—	4	1	no	—
464	SD₁₆	—	2	8	no	—
465	SD₁₆	—	2	4	no	—
466	SD₁₆	—	2	4	no	—
467	SD₁₆	—	2	4	no	—
468	SD₁₆	—	2	8	no	—
469	SD₁₆	—	2	4	no	—
470	SD₁₆	—	4	2	no	—
471	D₈	—	2	8	no	—
472	D₈	—	2	4	no	—
473	D₈	—	2	4	no	—
474	D₈	—	2	8	no	—
475	D₈	—	4	4	no	—
476	D₈	—	4	1	no	—
477	M₁₆	—	2	8	no	—
478	M₁₆	—	2	2	no	—
479	M₁₆	—	2	2	no	—
480	M₁₆	—	2	2	no	—
481	M₁₆	—	2	2	no	—
482	M₁₆	C₂×D₄×C₈₉	1	4	yes	—
483	C₂×C₈	—	2	8	no	—
484	C₂×C₈	—	2	2	no	—
485	C₂×C₈	—	2	2	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	2	1	no	—
488	C₂×C₈	C₂×D₄×C₈₉	1	4	yes	—
489	C₄⋊C₄	—	2	2	no	—
490	C₂²⋊C₄	—	2	2	no	—
491	C₂²⋊C₄	—	4	4	no	—
492	C₄²	—	2	2	no	—
493	C₂³	—	2	2	no	—
494	C₂³	—	2	2	no	—
495	C₂³	—	2	1	no	—
496	C₂³	—	2	4	no	—
497	C₂³	—	4	2	no	—
498	C₂³	—	4	8	no	—
499	C₂³	—	4	2	no	—
500	C₂³	—	4	2	no	—
501	C₂³	—	16	2	no	—
502	C₂³	—	16	4	no	—
503	C₂³	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
504	Q₈	—	2	2	no	—
505	Q₈	—	2	2	no	—
506	Q₈	—	2	2	no	—
507	Q₈	—	2	2	no	—
508	Q₈	—	4	2	no	—
509	Q₈	#120×C₈₉	1	4	yes	—
510	D₄	—	2	4	no	—
511	D₄	—	2	4	no	—
512	D₄	—	2	2	no	—
513	D₄	—	2	2	no	—
514	D₄	—	2	4	no	—
515	D₄	—	2	2	no	—
516	D₄	—	2	4	no	—
517	D₄	—	2	2	no	—
518	D₄	—	4	4	no	—
519	D₄	—	4	8	no	—
520	D₄	—	4	2	no	—
521	D₄	—	4	8	no	—
522	D₄	—	4	2	no	—
523	D₄	—	4	2	no	—
524	D₄	—	4	2	no	—
525	D₄	—	4	2	no	—
526	D₄	—	4	2	no	—
527	D₄	—	16	4	no	—
528	D₄	#120×C₈₉	1	4	yes	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	4	no	—
532	C₂×C₄	—	2	1	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	4	2	no	—
535	C₂×C₄	—	4	4	no	—
536	C₂×C₄	—	4	8	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	4	1	no	—
539	C₂×C₄	—	4	2	no	—
540	C₂×C₄	—	4	1	no	—
541	C₂×C₄	—	4	2	no	—
542	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
543	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
544	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
545	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
546	C₈	—	2	4	no	—
547	C₈	—	2	4	no	—
548	C₈	—	4	1	no	—
549	C₈	—	4	1	no	—
550	C₂²	—	2	4	no	—
551	C₂²	—	4	1	no	—
552	C₂²	—	4	4	no	—
553	C₂²	—	4	1	no	—
554	C₂²	—	4	2	no	—
555	C₂²	—	4	4	no	—
556	C₂²	—	8	2	no	—
557	C₂²	—	8	4	no	—
558	C₂²	—	8	1	no	—
559	C₂²	—	8	4	no	—
560	C₂²	—	16	2	no	—
561	C₂²	—	16	2	no	—
562	C₂²	—	16	8	no	—
563	C₂²	#534×C₈₉	1	1	yes	characteristic
564	C₂²	D₄²×C₈₉	1	2	yes	—
565	C₄	—	2	4	no	—
566	C₄	—	4	2	no	—
567	C₄	—	4	1	no	—
568	C₄	—	8	2	no	—
569	C₄	C₂×#120×C₈₉	1	2	yes	—
570	C₄	D₄²×C₈₉	1	1	yes	characteristic
571	C₄	D₄²×C₈₉	1	1	yes	characteristic
572	C₂	—	2	2	no	—
573	C₂	—	2	1	no	—
574	C₂	—	4	4	no	—
575	C₂	—	8	2	no	—
576	C₂	—	8	4	no	—
577	C₂	—	8	1	no	—
578	C₂	C₈₉×#2403	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₈₉	#2403
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#3585	C₂	semidirect
2. C₈₉×#3019	C₂	semidirect
3. C₈₉×#3008	C₂	semidirect
4. C₈₉×#2018	C₂	semidirect
5. C₈₉×#2017	C₂	semidirect
6. C₈₉×#2008	C₂	semidirect
7. C₈₉×#1903	C₂	semidirect
8. C₈₉×#1881	C₂	semidirect
9. C₈₉×#1794	C₂	semidirect
11. C₂×H(5,𝔽₂)×C₈₉	C₂²	semidirect
18. #575×C₈₉	C₂²	semidirect
19. #575×C₈₉	C₂²	semidirect
22. C₂×AGL(1,ℤ/8)×C₈₉	C₂²	semidirect
23. C₂²×D₈×C₈₉	C₂²	semidirect
24. #568×C₈₉	C₂²	semidirect
25. #535×C₈₉	C₂²	semidirect
30. #421×C₈₉	C₂²	semidirect
31. #421×C₈₉	C₂²	semidirect
32. #418×C₈₉	C₂²	semidirect
33. #416×C₈₉	C₂²	semidirect
34. C₂×#100×C₈₉	C₂²	semidirect
35. #410×C₈₉	C₂²	semidirect
36. #408×C₈₉	C₂²	semidirect
37. #408×C₈₉	C₂²	semidirect
38. #344×C₈₉	C₂²	semidirect
39. #341×C₈₉	C₂²	semidirect
40. #338×C₈₉	C₂²	semidirect
42. 2₋¹⁺⁴×C₈₉	D₄	semidirect
49. H(5,𝔽₂)×C₈₉	D₄	semidirect
66. #137×C₈₉	D₄	semidirect
78. AGL(1,ℤ/8)×C₈₉	D₄	semidirect
86. C₂×SD₁₆×C₈₉	D₄	semidirect
91. C₂×D₈×C₈₉	D₄	semidirect
282. #23892	C₈₉	direct (N × H)
290. #3585	C₂×C₈₉	semidirect
291. #3019	C₂×C₈₉	semidirect
292. #3008	C₂×C₈₉	semidirect
293. #2018	C₂×C₈₉	semidirect
294. #2017	C₂×C₈₉	semidirect
295. #2008	C₂×C₈₉	semidirect
296. #1903	C₂×C₈₉	semidirect
297. #1881	C₂×C₈₉	semidirect
298. #1794	C₂×C₈₉	semidirect
299. C₈₉	#23892	direct (N × H)
301. C₂×H(5,𝔽₂)	C₂²×C₈₉	semidirect
308. #575	C₂²×C₈₉	semidirect
309. #575	C₂²×C₈₉	semidirect
312. C₂×AGL(1,ℤ/8)	C₂²×C₈₉	semidirect
313. C₂²×D₈	C₂²×C₈₉	semidirect
314. #568	C₂²×C₈₉	semidirect
315. #535	C₂²×C₈₉	semidirect
320. #421	C₂²×C₈₉	semidirect
321. #421	C₂²×C₈₉	semidirect
322. #418	C₂²×C₈₉	semidirect
323. #416	C₂²×C₈₉	semidirect
324. C₂×#100	C₂²×C₈₉	semidirect
325. #410	C₂²×C₈₉	semidirect
326. #408	C₂²×C₈₉	semidirect
327. #408	C₂²×C₈₉	semidirect
328. #344	C₂²×C₈₉	semidirect
329. #341	C₂²×C₈₉	semidirect
330. #338	C₂²×C₈₉	semidirect
332. 2₋¹⁺⁴	D₄×C₈₉	semidirect
339. H(5,𝔽₂)	D₄×C₈₉	semidirect
356. #137	D₄×C₈₉	semidirect
368. AGL(1,ℤ/8)	D₄×C₈₉	semidirect
376. C₂×SD₁₆	D₄×C₈₉	semidirect
381. C₂×D₈	D₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3585	2
C₈₉×#3019	2
C₈₉×#3008	2
C₈₉×#2018	2
C₈₉×#2017	2
C₈₉×#2008	2
C₈₉×#1903	2
C₈₉×#1881	2
C₈₉×#1794	2
#23892	89

Subgroups — C89×#23892 #36401740

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#23892 #36401740