Subgroups — C₈₉×#29368 #36407216

Order 22784.

822 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#2293	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2100	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2094	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2043	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#2013	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1927	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1880	C₂	1	2	yes	maximal
9	C₈₉×#1877	C₂	1	2	yes	maximal
10	C₈₉×#1865	C₂	1	2	yes	maximal, complemented
11	C₈₉×#1839	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	—
13	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#117×C₈₉	—	2	1	no	—
17	C₂×#106×C₈₉	C₂²	1	2	yes	—
18	C₂×#103×C₈₉	C₂²	1	2	yes	—
19	C₂×#103×C₈₉	C₂²	1	2	yes	—
20	C₂×#103×C₈₉	C₂²	1	2	yes	—
21	C₂×#102×C₈₉	C₂²	1	2	yes	—
22	#391×C₈₉	C₂²	1	2	yes	—
23	#390×C₈₉	C₂²	1	1	yes	characteristic
24	#390×C₈₉	C₂²	1	1	yes	characteristic
25	#389×C₈₉	—	2	1	no	—
26	#389×C₈₉	—	2	1	no	—
27	#389×C₈₉	C₂²	1	1	yes	characteristic
28	#389×C₈₉	C₂²	1	1	yes	characteristic
29	#388×C₈₉	—	2	1	no	—
30	#388×C₈₉	—	2	1	no	—
31	#384×C₈₉	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
33	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
34	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₈₉	—	2	1	no	—
36	C₄³×C₈₉	C₂²	1	1	yes	characteristic
37	#340×C₈₉	C₂²	1	2	yes	—
38	#340×C₈₉	C₂²	1	2	yes	—
39	#336×C₈₉	C₂²	1	1	yes	characteristic
40	#336×C₈₉	C₂²	1	1	yes	characteristic
41	#336×C₈₉	C₂²	1	2	yes	—
42	#336×C₈₉	C₂²	1	2	yes	—
43	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
44	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₈₉	—	2	1	no	—
46	C₂×SD₁₆×C₈₉	—	2	4	no	—
47	C₂×SD₁₆×C₈₉	—	2	4	no	—
48	C₂×SD₁₆×C₈₉	—	4	2	no	—
49	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
50	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
51	C₄⋊Q₈×C₈₉	—	2	4	no	—
52	C₄×Q₈×C₈₉	—	4	2	no	—
53	C₄×Q₈×C₈₉	—	4	2	no	—
54	C₄×D₄×C₈₉	—	2	2	no	—
55	C₄×D₄×C₈₉	—	2	2	no	—
56	#117×C₈₉	—	4	4	no	—
57	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
59	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
60	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
61	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
62	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
63	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
64	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
70	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
73	C₂×C₄²×C₈₉	—	2	1	no	—
74	C₂×C₄²×C₈₉	—	2	2	no	—
75	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
77	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
80	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
82	#106×C₈₉	—	4	2	no	—
83	#103×C₈₉	—	2	4	no	—
84	#103×C₈₉	—	2	4	no	—
85	#103×C₈₉	—	4	2	no	—
86	#102×C₈₉	—	4	2	no	—
87	#95×C₈₉	—	2	1	no	—
88	#95×C₈₉	—	2	1	no	—
89	#95×C₈₉	—	2	2	no	—
90	#95×C₈₉	—	2	2	no	—
91	#95×C₈₉	—	2	1	no	—
92	C₂⁴×C₈₉	—	2	1	no	—
93	C₂×Q₈×C₈₉	—	2	2	no	—
94	C₂×Q₈×C₈₉	—	2	2	no	—
95	C₂×Q₈×C₈₉	—	2	2	no	—
96	C₂×Q₈×C₈₉	—	2	2	no	—
97	C₂×Q₈×C₈₉	—	2	2	no	—
98	C₂×Q₈×C₈₉	#41	1	4	yes	—
99	C₂×D₄×C₈₉	—	2	2	no	—
100	C₂×D₄×C₈₉	—	2	1	no	—
101	C₂×D₄×C₈₉	—	2	2	no	—
102	C₂×D₄×C₈₉	#41	1	2	yes	—
103	C₂²×C₄×C₈₉	—	2	4	no	—
104	C₂²×C₄×C₈₉	—	2	2	no	—
105	C₂²×C₄×C₈₉	—	2	1	no	—
106	C₂²×C₄×C₈₉	—	2	1	no	—
107	C₂²×C₄×C₈₉	—	2	1	no	—
108	C₂²×C₄×C₈₉	—	2	2	no	—
109	C₂²×C₄×C₈₉	—	4	2	no	—
110	C₂²×C₄×C₈₉	—	4	2	no	—
111	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
118	SD₁₆×C₈₉	—	4	8	no	—
119	C₂×C₈×C₈₉	—	2	2	no	—
120	C₂×C₈×C₈₉	—	2	2	no	—
121	C₂×C₈×C₈₉	—	2	2	no	—
122	C₂×C₈×C₈₉	—	2	2	no	—
123	C₂×C₈×C₈₉	—	2	2	no	—
124	C₂×C₈×C₈₉	—	2	2	no	—
125	C₄⋊C₄×C₈₉	—	2	4	no	—
126	C₄⋊C₄×C₈₉	—	2	4	no	—
127	C₄⋊C₄×C₈₉	—	2	1	no	—
128	C₄⋊C₄×C₈₉	—	2	2	no	—
129	C₄⋊C₄×C₈₉	—	2	2	no	—
130	C₄⋊C₄×C₈₉	—	2	2	no	—
131	C₄⋊C₄×C₈₉	—	2	2	no	—
132	C₄⋊C₄×C₈₉	—	4	2	no	—
133	C₄⋊C₄×C₈₉	—	4	2	no	—
134	C₄⋊C₄×C₈₉	—	4	4	no	—
135	C₄⋊C₄×C₈₉	—	4	2	no	—
136	C₄⋊C₄×C₈₉	—	4	2	no	—
137	C₄⋊C₄×C₈₉	#41	1	2	yes	—
138	C₂²⋊C₄×C₈₉	—	2	4	no	—
139	C₂²⋊C₄×C₈₉	—	4	2	no	—
140	C₂²⋊C₄×C₈₉	—	4	2	no	—
141	C₂²⋊C₄×C₈₉	—	4	2	no	—
142	C₂²⋊C₄×C₈₉	—	4	2	no	—
143	C₄²×C₈₉	—	2	2	no	—
144	C₄²×C₈₉	—	2	1	no	—
145	C₄²×C₈₉	—	2	1	no	—
146	C₄²×C₈₉	—	2	2	no	—
147	C₄²×C₈₉	—	2	2	no	—
148	C₄²×C₈₉	—	4	1	no	—
149	C₄²×C₈₉	—	4	1	no	—
150	C₄²×C₈₉	—	4	4	no	—
151	C₄²×C₈₉	—	4	2	no	—
152	C₄²×C₈₉	SD₁₆	1	4	yes	complemented
153	C₂³×C₈₉	—	2	2	no	—
154	C₂³×C₈₉	—	4	1	no	—
155	C₂³×C₈₉	—	4	1	no	—
156	C₂³×C₈₉	—	4	1	no	—
157	C₂³×C₈₉	—	4	1	no	—
158	C₂³×C₈₉	—	4	1	no	—
159	C₂³×C₈₉	—	4	1	no	—
160	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
161	Q₈×C₈₉	—	2	4	no	—
162	Q₈×C₈₉	—	2	4	no	—
163	Q₈×C₈₉	—	4	2	no	—
164	Q₈×C₈₉	—	4	2	no	—
165	D₄×C₈₉	—	2	4	no	—
166	D₄×C₈₉	—	4	2	no	—
167	C₂×C₄×C₈₉	—	2	1	no	—
168	C₂×C₄×C₈₉	—	2	1	no	—
169	C₂×C₄×C₈₉	—	2	1	no	—
170	C₂×C₄×C₈₉	—	2	1	no	—
171	C₂×C₄×C₈₉	—	2	1	no	—
172	C₂×C₄×C₈₉	—	2	1	no	—
173	C₂×C₄×C₈₉	—	2	1	no	—
174	C₂×C₄×C₈₉	—	2	1	no	—
175	C₂×C₄×C₈₉	—	2	1	no	—
176	C₂×C₄×C₈₉	—	2	2	no	—
177	C₂×C₄×C₈₉	—	2	4	no	—
178	C₂×C₄×C₈₉	—	2	1	no	—
179	C₂×C₄×C₈₉	—	2	1	no	—
180	C₂×C₄×C₈₉	—	2	1	no	—
181	C₂×C₄×C₈₉	—	2	1	no	—
182	C₂×C₄×C₈₉	—	2	1	no	—
183	C₂×C₄×C₈₉	—	4	8	no	—
184	C₂×C₄×C₈₉	—	4	2	no	—
185	C₂×C₄×C₈₉	—	4	2	no	—
186	C₂×C₄×C₈₉	—	4	1	no	—
187	C₂×C₄×C₈₉	—	4	1	no	—
188	C₂×C₄×C₈₉	—	4	1	no	—
189	C₂×C₄×C₈₉	—	4	2	no	—
190	C₂×C₄×C₈₉	—	4	1	no	—
191	C₂×C₄×C₈₉	—	4	1	no	—
192	C₂×C₄×C₈₉	—	4	1	no	—
193	C₂×C₄×C₈₉	—	4	1	no	—
194	C₂×C₄×C₈₉	—	4	1	no	—
195	C₂×C₄×C₈₉	—	4	2	no	—
196	C₂×C₄×C₈₉	—	4	2	no	—
197	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
198	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
199	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
200	C₂×C₄×C₈₉	#137	1	2	yes	—
201	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
202	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
203	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
204	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
205	C₈×C₈₉	—	4	2	no	—
206	C₈×C₈₉	—	4	2	no	—
207	C₂²×C₈₉	—	4	2	no	—
208	C₂²×C₈₉	—	4	2	no	—
209	C₂²×C₈₉	—	4	2	no	—
210	C₂²×C₈₉	—	8	1	no	—
211	C₂²×C₈₉	—	8	1	no	—
212	C₂²×C₈₉	—	8	1	no	—
213	C₂²×C₈₉	—	8	1	no	—
214	C₂²×C₈₉	#525	1	1	yes	characteristic
215	C₂²×C₈₉	#525	1	1	yes	characteristic
216	C₂²×C₈₉	#535	1	1	yes	characteristic
217	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
218	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
219	C₂²×C₈₉	#575	1	1	yes	characteristic
220	C₂²×C₈₉	#575	1	1	yes	characteristic
221	C₄×C₈₉	—	2	2	no	—
222	C₄×C₈₉	—	2	2	no	—
223	C₄×C₈₉	—	2	2	no	—
224	C₄×C₈₉	—	2	2	no	—
225	C₄×C₈₉	—	4	2	no	—
226	C₄×C₈₉	—	4	4	no	—
227	C₄×C₈₉	—	4	1	no	—
228	C₄×C₈₉	—	4	1	no	—
229	C₄×C₈₉	—	8	1	no	—
230	C₄×C₈₉	—	8	1	no	—
231	C₄×C₈₉	—	8	2	no	—
232	C₄×C₈₉	#551	1	2	yes	—
233	C₄×C₈₉	#553	1	2	yes	—
234	#29368	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
235	C₂×C₈₉	—	8	2	no	—
236	C₂×C₈₉	#2598	1	1	yes	characteristic
237	C₂×C₈₉	#3101	1	1	yes	characteristic
238	C₂×C₈₉	#3103	1	1	yes	characteristic
239	C₂×C₈₉	#3116	1	1	yes	characteristic
240	C₂×C₈₉	#3117	1	1	yes	characteristic
241	C₂×C₈₉	#3206	1	1	yes	characteristic
242	C₂×C₈₉	#3219	1	1	yes	characteristic
243	C₄×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
244	#2293	C₂×C₈₉	1	1	yes	characteristic
245	#2100	C₂×C₈₉	1	1	yes	characteristic, complemented
246	#2094	C₂×C₈₉	1	1	yes	characteristic
247	#2043	C₂×C₈₉	1	1	yes	characteristic, complemented
248	#2013	C₂×C₈₉	1	1	yes	characteristic
249	#1927	C₂×C₈₉	1	2	yes	complemented
250	#1880	C₂×C₈₉	1	2	yes	—
251	#1877	C₂×C₈₉	1	2	yes	—
252	#1865	C₂×C₈₉	1	2	yes	complemented
253	#1839	C₂×C₈₉	1	1	yes	characteristic, complemented
254	C₈₉	#29368	1	1	yes	characteristic, direct product factor, Sylow
255	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	—
256	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
257	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
258	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
259	C₂×#117	—	2	1	no	—
260	C₂×#106	C₂²×C₈₉	1	2	yes	—
261	C₂×#103	C₂²×C₈₉	1	2	yes	—
262	C₂×#103	C₂²×C₈₉	1	2	yes	—
263	C₂×#103	C₂²×C₈₉	1	2	yes	—
264	C₂×#102	C₂²×C₈₉	1	2	yes	—
265	#391	C₂²×C₈₉	1	2	yes	—
266	#390	C₂²×C₈₉	1	1	yes	characteristic
267	#390	C₂²×C₈₉	1	1	yes	characteristic
268	#389	—	2	1	no	—
269	#389	—	2	1	no	—
270	#389	C₂²×C₈₉	1	1	yes	characteristic
271	#389	C₂²×C₈₉	1	1	yes	characteristic
272	#388	—	2	1	no	—
273	#388	—	2	1	no	—
274	#384	C₂²×C₈₉	1	1	yes	characteristic
275	C₄×C₄⋊C₄	—	2	1	no	—
276	C₄×C₄⋊C₄	—	2	1	no	—
277	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
278	C₄×C₂²⋊C₄	—	2	1	no	—
279	C₄³	C₂²×C₈₉	1	1	yes	characteristic
280	#340	C₂²×C₈₉	1	2	yes	—
281	#340	C₂²×C₈₉	1	2	yes	—
282	#336	C₂²×C₈₉	1	1	yes	characteristic
283	#336	C₂²×C₈₉	1	1	yes	characteristic
284	#336	C₂²×C₈₉	1	2	yes	—
285	#336	C₂²×C₈₉	1	2	yes	—
286	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
287	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
288	C₂³×C₄	—	2	1	no	—
289	C₂×SD₁₆	—	2	4	no	—
290	C₂×SD₁₆	—	2	4	no	—
291	C₂×SD₁₆	—	4	2	no	—
292	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
293	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
294	C₄⋊Q₈	—	2	4	no	—
295	C₄×Q₈	—	4	2	no	—
296	C₄×Q₈	—	4	2	no	—
297	C₄×D₄	—	2	2	no	—
298	C₄×D₄	—	2	2	no	—
299	#117	—	4	4	no	—
300	C₂×C₄⋊C₄	—	2	1	no	—
301	C₂×C₄⋊C₄	—	2	1	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	2	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄⋊C₄	—	2	1	no	—
307	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
308	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
309	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
310	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
311	C₂×C₂²⋊C₄	—	2	1	no	—
312	C₂×C₂²⋊C₄	—	2	1	no	—
313	C₂×C₂²⋊C₄	—	2	1	no	—
314	C₂×C₂²⋊C₄	—	2	1	no	—
315	C₂×C₂²⋊C₄	—	2	1	no	—
316	C₂×C₄²	—	2	1	no	—
317	C₂×C₄²	—	2	2	no	—
318	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
319	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
320	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
321	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
322	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
323	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
324	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
325	#106	—	4	2	no	—
326	#103	—	2	4	no	—
327	#103	—	2	4	no	—
328	#103	—	4	2	no	—
329	#102	—	4	2	no	—
330	#95	—	2	1	no	—
331	#95	—	2	1	no	—
332	#95	—	2	2	no	—
333	#95	—	2	2	no	—
334	#95	—	2	1	no	—
335	C₂⁴	—	2	1	no	—
336	C₂×Q₈	—	2	2	no	—
337	C₂×Q₈	—	2	2	no	—
338	C₂×Q₈	—	2	2	no	—
339	C₂×Q₈	—	2	2	no	—
340	C₂×Q₈	—	2	2	no	—
341	C₂×Q₈	#41×C₈₉	1	4	yes	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	2	1	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	#41×C₈₉	1	2	yes	—
346	C₂²×C₄	—	2	4	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	1	no	—
349	C₂²×C₄	—	2	1	no	—
350	C₂²×C₄	—	2	1	no	—
351	C₂²×C₄	—	2	2	no	—
352	C₂²×C₄	—	4	2	no	—
353	C₂²×C₄	—	4	2	no	—
354	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
356	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
357	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
358	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
359	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
360	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
361	SD₁₆	—	4	8	no	—
362	C₂×C₈	—	2	2	no	—
363	C₂×C₈	—	2	2	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	2	no	—
368	C₄⋊C₄	—	2	4	no	—
369	C₄⋊C₄	—	2	4	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	2	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	—	2	2	no	—
374	C₄⋊C₄	—	2	2	no	—
375	C₄⋊C₄	—	4	4	no	—
376	C₄⋊C₄	—	4	2	no	—
377	C₄⋊C₄	—	4	2	no	—
378	C₄⋊C₄	—	4	2	no	—
379	C₄⋊C₄	—	4	2	no	—
380	C₄⋊C₄	#41×C₈₉	1	2	yes	—
381	C₂²⋊C₄	—	2	4	no	—
382	C₂²⋊C₄	—	4	2	no	—
383	C₂²⋊C₄	—	4	2	no	—
384	C₂²⋊C₄	—	4	2	no	—
385	C₂²⋊C₄	—	4	2	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	1	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	2	no	—
390	C₄²	—	2	2	no	—
391	C₄²	—	4	1	no	—
392	C₄²	—	4	1	no	—
393	C₄²	—	4	4	no	—
394	C₄²	—	4	2	no	—
395	C₄²	SD₁₆×C₈₉	1	4	yes	complemented
396	C₂³	—	2	2	no	—
397	C₂³	—	4	1	no	—
398	C₂³	—	4	1	no	—
399	C₂³	—	4	1	no	—
400	C₂³	—	4	1	no	—
401	C₂³	—	4	1	no	—
402	C₂³	—	4	1	no	—
403	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
404	Q₈	—	2	4	no	—
405	Q₈	—	2	4	no	—
406	Q₈	—	4	2	no	—
407	Q₈	—	4	2	no	—
408	D₄	—	2	4	no	—
409	D₄	—	4	2	no	—
410	C₂×C₄	—	2	1	no	—
411	C₂×C₄	—	2	1	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	1	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	4	8	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	1	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
441	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
442	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
443	C₂×C₄	#137×C₈₉	1	2	yes	—
444	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
445	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
446	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
447	C₂×C₄	H(5,𝔽₂)×C₈₉	1	2	yes	—
448	C₈	—	4	2	no	—
449	C₈	—	4	2	no	—
450	C₂²	—	4	2	no	—
451	C₂²	—	4	2	no	—
452	C₂²	—	4	2	no	—
453	C₂²	—	8	1	no	—
454	C₂²	—	8	1	no	—
455	C₂²	—	8	1	no	—
456	C₂²	—	8	1	no	—
457	C₂²	#525×C₈₉	1	1	yes	characteristic
458	C₂²	#525×C₈₉	1	1	yes	characteristic
459	C₂²	#535×C₈₉	1	1	yes	characteristic
460	C₂²	C₂²×SD₁₆×C₈₉	1	1	yes	characteristic
461	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
462	C₂²	#575×C₈₉	1	1	yes	characteristic
463	C₂²	#575×C₈₉	1	1	yes	characteristic
464	C₄	—	2	2	no	—
465	C₄	—	2	2	no	—
466	C₄	—	2	2	no	—
467	C₄	—	2	2	no	—
468	C₄	—	4	2	no	—
469	C₄	—	4	4	no	—
470	C₄	—	4	1	no	—
471	C₄	—	4	1	no	—
472	C₄	—	8	1	no	—
473	C₄	—	8	1	no	—
474	C₄	—	8	2	no	—
475	C₄	#551×C₈₉	1	2	yes	—
476	C₄	#553×C₈₉	1	2	yes	—
477	C₂	—	8	2	no	—
478	C₂	C₈₉×#2598	1	1	yes	characteristic
479	C₂	C₈₉×#3101	1	1	yes	characteristic
480	C₂	C₈₉×#3103	1	1	yes	characteristic
481	C₂	C₈₉×#3116	1	1	yes	characteristic
482	C₂	C₈₉×#3117	1	1	yes	characteristic
483	C₂	C₈₉×#3206	1	1	yes	characteristic
484	C₂	C₈₉×#3219	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂²×D₄×C₈₉	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₄³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄×C₄⋊Q₈×C₈₉	C₂	semidirect
3. C₈₉×#2100	C₂	semidirect
5. C₈₉×#2043	C₂	semidirect
7. C₈₉×#1927	C₂	semidirect
10. C₈₉×#1865	C₂	semidirect
11. C₈₉×#1839	C₂	semidirect
152. C₄²×C₈₉	SD₁₆	semidirect
234. #29368	C₈₉	direct (N × H)
243. C₄×C₄⋊Q₈	C₂×C₈₉	semidirect
245. #2100	C₂×C₈₉	semidirect
247. #2043	C₂×C₈₉	semidirect
249. #1927	C₂×C₈₉	semidirect
252. #1865	C₂×C₈₉	semidirect
253. #1839	C₂×C₈₉	semidirect
254. C₈₉	#29368	direct (N × H)
395. C₄²	SD₁₆×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₈₉	2
C₈₉×#2293	2
C₈₉×#2100	2
C₈₉×#2094	2
C₈₉×#2043	2
C₈₉×#2013	2
C₈₉×#1927	2
C₈₉×#1880	2
C₈₉×#1877	2
C₈₉×#1865	2
C₈₉×#1839	2
#29368	89

Subgroups — C89×#29368 #36407216

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#29368 #36407216