Subgroups — C₈₉×#29204 #36407052

Order 22784.

770 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2647	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#2328	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2067	C₂	1	2	yes	maximal, complemented
4	C₈₉×#2065	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1961	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#1957	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1864	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1862	C₂	1	2	yes	maximal, complemented
9	C₈₉×#1847	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1835	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1768	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#107×C₈₉	C₂²	1	2	yes	—
16	C₂×#106×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
18	C₂×#102×C₈₉	C₂²	1	2	yes	—
19	C₂×#102×C₈₉	C₂²	1	2	yes	—
20	C₂×#102×C₈₉	C₂²	1	2	yes	—
21	C₂×#102×C₈₉	C₂²	1	2	yes	—
22	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	#400×C₈₉	—	2	2	no	—
25	#398×C₈₉	—	2	2	no	—
26	#390×C₈₉	C₂²	1	1	yes	characteristic
27	#390×C₈₉	C₂²	1	1	yes	characteristic
28	#389×C₈₉	C₂²	1	2	yes	—
29	#389×C₈₉	C₂²	1	2	yes	—
30	#386×C₈₉	—	2	2	no	—
31	#386×C₈₉	—	2	2	no	—
32	#384×C₈₉	C₂²	1	1	yes	characteristic
33	#382×C₈₉	C₂²	1	1	yes	characteristic
34	#382×C₈₉	C₂²	1	1	yes	characteristic
35	#376×C₈₉	C₂²	1	1	yes	characteristic
36	#340×C₈₉	C₂²	1	2	yes	—
37	#340×C₈₉	C₂²	1	2	yes	—
38	#336×C₈₉	C₂²	1	2	yes	—
39	C₂²×D₄×C₈₉	—	2	2	no	—
40	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
41	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
42	C₂³×C₄×C₈₉	—	2	1	no	—
43	C₂³×C₄×C₈₉	—	2	1	no	—
44	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
45	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
46	#127×C₈₉	—	2	2	no	—
47	#127×C₈₉	Q₈	1	4	yes	—
48	C₄×D₄×C₈₉	—	4	2	no	—
49	C₄×D₄×C₈₉	—	4	2	no	—
50	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
51	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
52	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
53	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
57	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
58	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
59	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
61	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
64	#107×C₈₉	—	4	2	no	—
65	#106×C₈₉	—	4	2	no	—
66	C₄⋊C₈×C₈₉	—	2	4	no	—
67	#102×C₈₉	—	2	4	no	—
68	#102×C₈₉	—	2	4	no	—
69	#102×C₈₉	—	4	2	no	—
70	#102×C₈₉	—	4	2	no	—
71	#97×C₈₉	—	2	2	no	—
72	C₄×C₈×C₈₉	—	2	2	no	—
73	#95×C₈₉	—	2	2	no	—
74	#95×C₈₉	—	2	2	no	—
75	#95×C₈₉	—	2	2	no	—
76	#95×C₈₉	—	2	2	no	—
77	#95×C₈₉	D₄	1	2	yes	—
78	#95×C₈₉	D₄	1	2	yes	—
79	C₂⁴×C₈₉	—	2	1	no	—
80	C₂⁴×C₈₉	—	2	1	no	—
81	C₂×D₄×C₈₉	—	2	8	no	—
82	C₂×D₄×C₈₉	—	2	2	no	—
83	C₂×D₄×C₈₉	—	2	2	no	—
84	C₂×D₄×C₈₉	—	2	1	no	—
85	C₂×D₄×C₈₉	—	2	1	no	—
86	C₂×D₄×C₈₉	—	2	2	no	—
87	C₂×D₄×C₈₉	—	2	2	no	—
88	C₂×D₄×C₈₉	—	8	2	no	—
89	C₂×D₄×C₈₉	—	8	2	no	—
90	C₂×D₄×C₈₉	C₂×Q₈	1	2	yes	—
91	C₂×D₄×C₈₉	C₂×Q₈	1	2	yes	—
92	C₂²×C₄×C₈₉	—	2	2	no	—
93	C₂²×C₄×C₈₉	—	2	1	no	—
94	C₂²×C₄×C₈₉	—	2	1	no	—
95	C₂²×C₄×C₈₉	—	2	2	no	—
96	C₂²×C₄×C₈₉	—	4	2	no	—
97	C₂²×C₄×C₈₉	—	4	2	no	—
98	C₂²×C₄×C₈₉	—	8	2	no	—
99	C₂²×C₄×C₈₉	—	8	2	no	—
100	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
105	C₂×C₈×C₈₉	—	2	2	no	—
106	C₂×C₈×C₈₉	—	2	2	no	—
107	C₂×C₈×C₈₉	—	2	2	no	—
108	C₂×C₈×C₈₉	—	2	2	no	—
109	C₂×C₈×C₈₉	—	2	2	no	—
110	C₂×C₈×C₈₉	#41	1	4	yes	—
111	C₄⋊C₄×C₈₉	—	2	2	no	—
112	C₄⋊C₄×C₈₉	—	2	2	no	—
113	C₄⋊C₄×C₈₉	—	2	2	no	—
114	C₄⋊C₄×C₈₉	—	2	2	no	—
115	C₄⋊C₄×C₈₉	—	2	2	no	—
116	C₄⋊C₄×C₈₉	—	2	2	no	—
117	C₄⋊C₄×C₈₉	—	8	2	no	—
118	C₄⋊C₄×C₈₉	—	8	2	no	—
119	C₄⋊C₄×C₈₉	—	8	2	no	—
120	C₂²⋊C₄×C₈₉	—	4	2	no	—
121	C₂²⋊C₄×C₈₉	—	4	2	no	—
122	C₂²⋊C₄×C₈₉	—	8	2	no	—
123	C₂²⋊C₄×C₈₉	—	8	2	no	—
124	C₄²×C₈₉	—	2	2	no	—
125	C₄²×C₈₉	—	2	2	no	—
126	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
127	C₄²×C₈₉	#41	1	2	yes	—
128	C₂³×C₈₉	—	2	2	no	—
129	C₂³×C₈₉	—	2	2	no	—
130	C₂³×C₈₉	—	4	1	no	—
131	C₂³×C₈₉	—	4	1	no	—
132	C₂³×C₈₉	—	4	1	no	—
133	C₂³×C₈₉	—	4	1	no	—
134	C₂³×C₈₉	—	4	2	no	—
135	C₂³×C₈₉	—	4	2	no	—
136	C₂³×C₈₉	—	4	2	no	—
137	C₂³×C₈₉	—	4	2	no	—
138	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
139	D₄×C₈₉	—	2	4	no	—
140	D₄×C₈₉	—	2	4	no	—
141	D₄×C₈₉	—	4	2	no	—
142	D₄×C₈₉	—	4	2	no	—
143	D₄×C₈₉	—	8	8	no	—
144	C₂×C₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	—	2	2	no	—
146	C₂×C₄×C₈₉	—	2	2	no	—
147	C₂×C₄×C₈₉	—	2	1	no	—
148	C₂×C₄×C₈₉	—	2	1	no	—
149	C₂×C₄×C₈₉	—	2	2	no	—
150	C₂×C₄×C₈₉	—	2	2	no	—
151	C₂×C₄×C₈₉	—	2	2	no	—
152	C₂×C₄×C₈₉	—	2	2	no	—
153	C₂×C₄×C₈₉	—	4	2	no	—
154	C₂×C₄×C₈₉	—	4	1	no	—
155	C₂×C₄×C₈₉	—	4	1	no	—
156	C₂×C₄×C₈₉	—	4	2	no	—
157	C₂×C₄×C₈₉	—	4	2	no	—
158	C₂×C₄×C₈₉	—	4	2	no	—
159	C₂×C₄×C₈₉	—	4	2	no	—
160	C₂×C₄×C₈₉	—	4	2	no	—
161	C₂×C₄×C₈₉	—	4	2	no	—
162	C₂×C₄×C₈₉	—	4	2	no	—
163	C₂×C₄×C₈₉	—	8	4	no	—
164	C₂×C₄×C₈₉	—	8	4	no	—
165	C₂×C₄×C₈₉	C₂²⋊Q₈	1	4	yes	—
166	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
167	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
168	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
169	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
170	C₈×C₈₉	—	2	4	no	—
171	C₈×C₈₉	—	4	2	no	—
172	C₂²×C₈₉	—	4	2	no	—
173	C₂²×C₈₉	—	4	2	no	—
174	C₂²×C₈₉	—	4	4	no	—
175	C₂²×C₈₉	—	4	4	no	—
176	C₂²×C₈₉	—	8	2	no	—
177	C₂²×C₈₉	—	8	2	no	—
178	C₂²×C₈₉	—	8	1	no	—
179	C₂²×C₈₉	—	8	1	no	—
180	C₂²×C₈₉	—	8	1	no	—
181	C₂²×C₈₉	—	8	1	no	—
182	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
183	C₂²×C₈₉	#532	1	1	yes	characteristic
184	C₂²×C₈₉	#535	1	1	yes	characteristic
185	C₂²×C₈₉	#576	1	2	yes	—
186	C₂²×C₈₉	#577	1	2	yes	—
187	C₄×C₈₉	—	2	4	no	—
188	C₄×C₈₉	—	4	2	no	—
189	C₄×C₈₉	—	4	1	no	—
190	C₄×C₈₉	—	4	1	no	—
191	C₄×C₈₉	—	8	2	no	—
192	C₄×C₈₉	—	8	2	no	—
193	C₄×C₈₉	—	8	1	no	—
194	C₄×C₈₉	—	8	1	no	—
195	C₄×C₈₉	#553	1	2	yes	—
196	C₄×C₈₉	#554	1	2	yes	—
197	#29204	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂×C₈₉	—	8	2	no	—
199	C₂×C₈₉	—	8	2	no	—
200	C₂×C₈₉	#2566	1	1	yes	characteristic
201	C₂×C₈₉	#3081	1	2	yes	—
202	C₂×C₈₉	#3179	1	2	yes	—
203	C₂×C₈₉	#3220	1	1	yes	characteristic
204	C₂×C₈₉	#3222	1	1	yes	characteristic
205	#2647	C₂×C₈₉	1	1	yes	characteristic, complemented
206	#2328	C₂×C₈₉	1	1	yes	characteristic
207	#2067	C₂×C₈₉	1	2	yes	complemented
208	#2065	C₂×C₈₉	1	2	yes	complemented
209	#1961	C₂×C₈₉	1	1	yes	characteristic
210	#1957	C₂×C₈₉	1	1	yes	characteristic
211	#1864	C₂×C₈₉	1	2	yes	complemented
212	#1862	C₂×C₈₉	1	2	yes	complemented
213	#1847	C₂×C₈₉	1	1	yes	characteristic, complemented
214	#1835	C₂×C₈₉	1	1	yes	characteristic, complemented
215	#1768	C₂×C₈₉	1	1	yes	characteristic, complemented
216	C₈₉	#29204	1	1	yes	characteristic, direct product factor, Sylow
217	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
218	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
219	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
220	C₂×#107	C₂²×C₈₉	1	2	yes	—
221	C₂×#106	C₂²×C₈₉	1	2	yes	—
222	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
223	C₂×#102	C₂²×C₈₉	1	2	yes	—
224	C₂×#102	C₂²×C₈₉	1	2	yes	—
225	C₂×#102	C₂²×C₈₉	1	2	yes	—
226	C₂×#102	C₂²×C₈₉	1	2	yes	—
227	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
228	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
229	#400	—	2	2	no	—
230	#398	—	2	2	no	—
231	#390	C₂²×C₈₉	1	1	yes	characteristic
232	#390	C₂²×C₈₉	1	1	yes	characteristic
233	#389	C₂²×C₈₉	1	2	yes	—
234	#389	C₂²×C₈₉	1	2	yes	—
235	#386	—	2	2	no	—
236	#386	—	2	2	no	—
237	#384	C₂²×C₈₉	1	1	yes	characteristic
238	#382	C₂²×C₈₉	1	1	yes	characteristic
239	#382	C₂²×C₈₉	1	1	yes	characteristic
240	#376	C₂²×C₈₉	1	1	yes	characteristic
241	#340	C₂²×C₈₉	1	2	yes	—
242	#340	C₂²×C₈₉	1	2	yes	—
243	#336	C₂²×C₈₉	1	2	yes	—
244	C₂²×D₄	—	2	2	no	—
245	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
246	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
247	C₂³×C₄	—	2	1	no	—
248	C₂³×C₄	—	2	1	no	—
249	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
250	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
251	#127	—	2	2	no	—
252	#127	Q₈×C₈₉	1	4	yes	—
253	C₄×D₄	—	4	2	no	—
254	C₄×D₄	—	4	2	no	—
255	C₂×C₄⋊C₄	—	2	2	no	—
256	C₂×C₄⋊C₄	—	2	2	no	—
257	C₂×C₄⋊C₄	—	2	2	no	—
258	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
259	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
260	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
261	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
262	C₂×C₂²⋊C₄	—	2	1	no	—
263	C₂×C₂²⋊C₄	—	2	1	no	—
264	C₂×C₂²⋊C₄	—	2	2	no	—
265	C₂×C₂²⋊C₄	—	2	2	no	—
266	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
267	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
268	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
269	#107	—	4	2	no	—
270	#106	—	4	2	no	—
271	C₄⋊C₈	—	2	4	no	—
272	#102	—	2	4	no	—
273	#102	—	2	4	no	—
274	#102	—	4	2	no	—
275	#102	—	4	2	no	—
276	#97	—	2	2	no	—
277	C₄×C₈	—	2	2	no	—
278	#95	—	2	2	no	—
279	#95	—	2	2	no	—
280	#95	—	2	2	no	—
281	#95	—	2	2	no	—
282	#95	D₄×C₈₉	1	2	yes	—
283	#95	D₄×C₈₉	1	2	yes	—
284	C₂⁴	—	2	1	no	—
285	C₂⁴	—	2	1	no	—
286	C₂×D₄	—	2	8	no	—
287	C₂×D₄	—	2	2	no	—
288	C₂×D₄	—	2	2	no	—
289	C₂×D₄	—	2	1	no	—
290	C₂×D₄	—	2	1	no	—
291	C₂×D₄	—	2	2	no	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	—	8	2	no	—
294	C₂×D₄	—	8	2	no	—
295	C₂×D₄	C₂×Q₈×C₈₉	1	2	yes	—
296	C₂×D₄	C₂×Q₈×C₈₉	1	2	yes	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	1	no	—
299	C₂²×C₄	—	2	1	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	4	2	no	—
302	C₂²×C₄	—	4	2	no	—
303	C₂²×C₄	—	8	2	no	—
304	C₂²×C₄	—	8	2	no	—
305	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
306	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
307	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
309	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
310	C₂×C₈	—	2	2	no	—
311	C₂×C₈	—	2	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	#41×C₈₉	1	4	yes	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	2	2	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	8	2	no	—
323	C₄⋊C₄	—	8	2	no	—
324	C₄⋊C₄	—	8	2	no	—
325	C₂²⋊C₄	—	4	2	no	—
326	C₂²⋊C₄	—	4	2	no	—
327	C₂²⋊C₄	—	8	2	no	—
328	C₂²⋊C₄	—	8	2	no	—
329	C₄²	—	2	2	no	—
330	C₄²	—	2	2	no	—
331	C₄²	C₂×Q₈×C₈₉	1	2	yes	—
332	C₄²	#41×C₈₉	1	2	yes	—
333	C₂³	—	2	2	no	—
334	C₂³	—	2	2	no	—
335	C₂³	—	4	1	no	—
336	C₂³	—	4	1	no	—
337	C₂³	—	4	1	no	—
338	C₂³	—	4	1	no	—
339	C₂³	—	4	2	no	—
340	C₂³	—	4	2	no	—
341	C₂³	—	4	2	no	—
342	C₂³	—	4	2	no	—
343	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
344	D₄	—	2	4	no	—
345	D₄	—	2	4	no	—
346	D₄	—	4	2	no	—
347	D₄	—	4	2	no	—
348	D₄	—	8	8	no	—
349	C₂×C₄	—	2	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	2	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	8	4	no	—
369	C₂×C₄	—	8	4	no	—
370	C₂×C₄	C₂²⋊Q₈×C₈₉	1	4	yes	—
371	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
372	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
373	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
374	C₂×C₄	H(5,𝔽₂)×C₈₉	1	2	yes	—
375	C₈	—	2	4	no	—
376	C₈	—	4	2	no	—
377	C₂²	—	4	2	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	4	4	no	—
380	C₂²	—	4	4	no	—
381	C₂²	—	8	2	no	—
382	C₂²	—	8	2	no	—
383	C₂²	—	8	1	no	—
384	C₂²	—	8	1	no	—
385	C₂²	—	8	1	no	—
386	C₂²	—	8	1	no	—
387	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
388	C₂²	#532×C₈₉	1	1	yes	characteristic
389	C₂²	#535×C₈₉	1	1	yes	characteristic
390	C₂²	#576×C₈₉	1	2	yes	—
391	C₂²	#577×C₈₉	1	2	yes	—
392	C₄	—	2	4	no	—
393	C₄	—	4	2	no	—
394	C₄	—	4	1	no	—
395	C₄	—	4	1	no	—
396	C₄	—	8	2	no	—
397	C₄	—	8	1	no	—
398	C₄	—	8	2	no	—
399	C₄	—	8	1	no	—
400	C₄	#553×C₈₉	1	2	yes	—
401	C₄	#554×C₈₉	1	2	yes	—
402	C₂	—	8	2	no	—
403	C₂	—	8	2	no	—
404	C₂	C₈₉×#2566	1	1	yes	characteristic
405	C₂	C₈₉×#3081	1	2	yes	—
406	C₂	C₈₉×#3179	1	2	yes	—
407	C₂	C₈₉×#3220	1	1	yes	characteristic
408	C₂	C₈₉×#3222	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂×#127×C₈₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ #376×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#2647	C₂	semidirect
3. C₈₉×#2067	C₂	semidirect
4. C₈₉×#2065	C₂	semidirect
7. C₈₉×#1864	C₂	semidirect
8. C₈₉×#1862	C₂	semidirect
9. C₈₉×#1847	C₂	semidirect
10. C₈₉×#1835	C₂	semidirect
11. C₈₉×#1768	C₂	semidirect
197. #29204	C₈₉	direct (N × H)
205. #2647	C₂×C₈₉	semidirect
207. #2067	C₂×C₈₉	semidirect
208. #2065	C₂×C₈₉	semidirect
211. #1864	C₂×C₈₉	semidirect
212. #1862	C₂×C₈₉	semidirect
213. #1847	C₂×C₈₉	semidirect
214. #1835	C₂×C₈₉	semidirect
215. #1768	C₂×C₈₉	semidirect
216. C₈₉	#29204	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈₉×#2647	2
C₈₉×#2328	2
C₈₉×#2067	2
C₈₉×#2065	2
C₈₉×#1961	2
C₈₉×#1957	2
C₈₉×#1864	2
C₈₉×#1862	2
C₈₉×#1847	2
C₈₉×#1835	2
C₈₉×#1768	2
#29204	89

Subgroups — C89×#29204 #36407052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#29204 #36407052