Subgroups — C₈₉×#24804 #36402652

Order 22784.

858 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#473×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#3056	C₂	1	2	yes	maximal
3	C₂×#467×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2929	C₂	1	1	yes	maximal, characteristic
5	C₂²×#108×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#2064	C₂	1	2	yes	maximal, complemented
7	C₈₉×#2052	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1979	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1952	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1901	C₂	1	2	yes	maximal, complemented
11	C₈₉×#1816	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#137×C₈₉	C₂²	1	2	yes	—
13	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
16	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
17	#525×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	2	yes	—
19	#473×C₈₉	—	2	4	no	—
20	#470×C₈₉	—	2	2	no	—
21	#468×C₈₉	—	2	2	no	—
22	#467×C₈₉	—	2	1	no	—
23	#467×C₈₉	—	2	1	no	—
24	#467×C₈₉	—	2	1	no	—
25	#467×C₈₉	—	2	1	no	—
26	#467×C₈₉	—	2	2	no	—
27	#465×C₈₉	—	2	2	no	—
28	#435×C₈₉	—	2	1	no	—
29	#435×C₈₉	—	2	1	no	—
30	C₈×D₄×C₈₉	—	2	1	no	—
31	C₈×D₄×C₈₉	—	2	1	no	—
32	C₂×#108×C₈₉	—	2	4	no	—
33	C₂×#108×C₈₉	C₂²	1	2	yes	—
34	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
36	#428×C₈₉	C₂²	1	2	yes	—
37	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
38	#424×C₈₉	C₂²	1	1	yes	characteristic
39	#419×C₈₉	C₂²	1	2	yes	—
40	#417×C₈₉	C₂²	1	2	yes	—
41	C₂×#103×C₈₉	C₂²	1	2	yes	—
42	C₂×#101×C₈₉	C₂²	1	2	yes	—
43	C₂×#101×C₈₉	C₂²	1	2	yes	—
44	#408×C₈₉	C₂²	1	1	yes	characteristic
45	#407×C₈₉	C₂²	1	1	yes	characteristic
46	#405×C₈₉	C₂²	1	1	yes	characteristic
47	#343×C₈₉	C₂²	1	2	yes	—
48	#343×C₈₉	C₂²	1	2	yes	—
49	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
50	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
51	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
52	#137×C₈₉	—	2	4	no	—
53	#137×C₈₉	—	4	2	no	—
54	C₂×Q₁₆×C₈₉	—	2	2	no	—
55	C₂×Q₁₆×C₈₉	—	2	1	no	—
56	C₂×Q₁₆×C₈₉	—	2	1	no	—
57	C₂×Q₁₆×C₈₉	—	2	2	no	—
58	C₂×Q₁₆×C₈₉	—	2	2	no	—
59	C₂×SD₁₆×C₈₉	—	2	2	no	—
60	#131×C₈₉	—	2	2	no	—
61	#131×C₈₉	—	2	2	no	—
62	C₂×M₁₆×C₈₉	—	2	2	no	—
63	C₂×M₁₆×C₈₉	—	2	1	no	—
64	C₂×M₁₆×C₈₉	—	2	4	no	—
65	C₂×M₁₆×C₈₉	—	4	2	no	—
66	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
68	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
69	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
70	C₂²×C₈×C₈₉	—	2	2	no	—
71	C₂²×C₈×C₈₉	—	2	2	no	—
72	C₂²×C₈×C₈₉	—	2	2	no	—
73	C₂²×C₈×C₈₉	—	2	1	no	—
74	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
75	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
76	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
77	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
80	#123×C₈₉	—	2	1	no	—
81	C₂²⋊Q₈×C₈₉	—	2	2	no	—
82	C₂²⋊Q₈×C₈₉	—	2	2	no	—
83	C₂²⋊Q₈×C₈₉	—	2	2	no	—
84	C₂²⋊Q₈×C₈₉	—	2	2	no	—
85	C₂²⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
86	C₂²⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
87	#121×C₈₉	D₄	1	1	yes	characteristic
88	#121×C₈₉	D₄	1	1	yes	characteristic
89	#120×C₈₉	—	2	1	no	—
90	C₄×D₄×C₈₉	—	2	1	no	—
91	C₄×D₄×C₈₉	—	2	1	no	—
92	#117×C₈₉	C₂³	1	1	yes	characteristic
93	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
94	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
95	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
96	#108×C₈₉	—	2	4	no	—
97	#108×C₈₉	—	2	2	no	—
98	#108×C₈₉	D₄	1	4	yes	complemented
99	#107×C₈₉	—	2	1	no	—
100	#107×C₈₉	—	2	1	no	—
101	#107×C₈₉	—	2	2	no	—
102	#106×C₈₉	—	2	2	no	—
103	C₄⋊C₈×C₈₉	—	2	1	no	—
104	C₄⋊C₈×C₈₉	—	2	1	no	—
105	#103×C₈₉	—	2	2	no	—
106	#103×C₈₉	—	2	2	no	—
107	#103×C₈₉	—	2	2	no	—
108	#103×C₈₉	—	2	2	no	—
109	#103×C₈₉	—	2	2	no	—
110	#102×C₈₉	—	2	2	no	—
111	#101×C₈₉	—	2	4	no	—
112	#101×C₈₉	—	4	2	no	—
113	C₂²⋊C₈×C₈₉	—	2	1	no	—
114	C₂²⋊C₈×C₈₉	—	2	1	no	—
115	C₂²⋊C₈×C₈₉	—	2	1	no	—
116	C₂²⋊C₈×C₈₉	—	2	1	no	—
117	#97×C₈₉	—	2	1	no	—
118	C₄×C₈×C₈₉	—	2	1	no	—
119	C₂⁴×C₈₉	#41	1	1	yes	characteristic
120	#41×C₈₉	—	2	1	no	—
121	#41×C₈₉	—	2	1	no	—
122	#41×C₈₉	—	2	1	no	—
123	#41×C₈₉	—	2	1	no	—
124	C₂×Q₈×C₈₉	—	2	2	no	—
125	C₂×Q₈×C₈₉	—	2	2	no	—
126	C₂×Q₈×C₈₉	—	2	2	no	—
127	C₂×Q₈×C₈₉	—	2	2	no	—
128	C₂×Q₈×C₈₉	—	2	2	no	—
129	C₂×Q₈×C₈₉	—	2	2	no	—
130	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₂×D₄×C₈₉	—	2	1	no	—
132	C₂×D₄×C₈₉	—	2	1	no	—
133	C₂×D₄×C₈₉	—	2	1	no	—
134	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₈₉	—	2	1	no	—
136	C₂²×C₄×C₈₉	—	2	1	no	—
137	C₂²×C₄×C₈₉	—	2	1	no	—
138	C₂²×C₄×C₈₉	—	2	1	no	—
139	C₂²×C₄×C₈₉	—	2	1	no	—
140	C₂²×C₄×C₈₉	—	2	2	no	—
141	C₂²×C₄×C₈₉	—	2	2	no	—
142	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
147	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
148	Q₁₆×C₈₉	—	4	4	no	—
149	Q₁₆×C₈₉	—	4	2	no	—
150	Q₁₆×C₈₉	—	4	2	no	—
151	SD₁₆×C₈₉	—	4	4	no	—
152	M₁₆×C₈₉	—	2	2	no	—
153	M₁₆×C₈₉	—	2	1	no	—
154	M₁₆×C₈₉	—	2	2	no	—
155	M₁₆×C₈₉	—	2	1	no	—
156	M₁₆×C₈₉	—	2	1	no	—
157	M₁₆×C₈₉	—	2	2	no	—
158	M₁₆×C₈₉	—	2	1	no	—
159	M₁₆×C₈₉	—	4	4	no	—
160	M₁₆×C₈₉	—	4	2	no	—
161	M₁₆×C₈₉	C₂×D₄	1	4	yes	—
162	C₂×C₈×C₈₉	—	2	2	no	—
163	C₂×C₈×C₈₉	—	2	1	no	—
164	C₂×C₈×C₈₉	—	2	1	no	—
165	C₂×C₈×C₈₉	—	2	2	no	—
166	C₂×C₈×C₈₉	—	2	2	no	—
167	C₂×C₈×C₈₉	—	2	2	no	—
168	C₂×C₈×C₈₉	—	2	2	no	—
169	C₂×C₈×C₈₉	—	2	1	no	—
170	C₂×C₈×C₈₉	—	2	1	no	—
171	C₂×C₈×C₈₉	—	2	2	no	—
172	C₂×C₈×C₈₉	—	4	4	no	—
173	C₂×C₈×C₈₉	—	4	2	no	—
174	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
175	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
176	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
177	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
178	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
179	C₂×C₈×C₈₉	#41	1	2	yes	—
180	C₄⋊C₄×C₈₉	—	2	1	no	—
181	C₄⋊C₄×C₈₉	—	2	1	no	—
182	C₄⋊C₄×C₈₉	—	2	2	no	—
183	C₄⋊C₄×C₈₉	—	2	2	no	—
184	C₄⋊C₄×C₈₉	—	4	2	no	—
185	C₄⋊C₄×C₈₉	—	4	2	no	—
186	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
187	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
188	C₂²⋊C₄×C₈₉	—	2	1	no	—
189	C₂²⋊C₄×C₈₉	—	2	1	no	—
190	C₂²⋊C₄×C₈₉	—	2	1	no	—
191	C₂²⋊C₄×C₈₉	—	2	1	no	—
192	C₂²⋊C₄×C₈₉	—	2	1	no	—
193	C₂²⋊C₄×C₈₉	—	4	2	no	—
194	C₂²⋊C₄×C₈₉	—	4	2	no	—
195	C₄²×C₈₉	—	2	1	no	—
196	C₂³×C₈₉	—	2	1	no	—
197	C₂³×C₈₉	—	2	1	no	—
198	C₂³×C₈₉	—	2	2	no	—
199	C₂³×C₈₉	—	2	1	no	—
200	C₂³×C₈₉	—	4	1	no	—
201	C₂³×C₈₉	#121	1	1	yes	characteristic
202	C₂³×C₈₉	#121	1	1	yes	characteristic
203	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
204	Q₈×C₈₉	—	2	1	no	—
205	Q₈×C₈₉	—	2	1	no	—
206	Q₈×C₈₉	—	4	2	no	—
207	Q₈×C₈₉	—	4	2	no	—
208	Q₈×C₈₉	—	4	2	no	—
209	Q₈×C₈₉	—	4	2	no	—
210	D₄×C₈₉	—	2	1	no	—
211	D₄×C₈₉	—	2	1	no	—
212	D₄×C₈₉	—	4	2	no	—
213	D₄×C₈₉	—	4	1	no	—
214	D₄×C₈₉	—	4	1	no	—
215	D₄×C₈₉	—	4	2	no	—
216	C₂×C₄×C₈₉	—	2	1	no	—
217	C₂×C₄×C₈₉	—	2	1	no	—
218	C₂×C₄×C₈₉	—	2	1	no	—
219	C₂×C₄×C₈₉	—	2	1	no	—
220	C₂×C₄×C₈₉	—	2	1	no	—
221	C₂×C₄×C₈₉	—	2	1	no	—
222	C₂×C₄×C₈₉	—	2	1	no	—
223	C₂×C₄×C₈₉	—	2	1	no	—
224	C₂×C₄×C₈₉	—	2	1	no	—
225	C₂×C₄×C₈₉	—	2	1	no	—
226	C₂×C₄×C₈₉	—	2	1	no	—
227	C₂×C₄×C₈₉	—	2	1	no	—
228	C₂×C₄×C₈₉	—	2	1	no	—
229	C₂×C₄×C₈₉	—	4	1	no	—
230	C₂×C₄×C₈₉	—	4	1	no	—
231	C₂×C₄×C₈₉	—	4	2	no	—
232	C₂×C₄×C₈₉	—	4	2	no	—
233	C₂×C₄×C₈₉	—	4	2	no	—
234	C₂×C₄×C₈₉	—	4	2	no	—
235	C₂×C₄×C₈₉	—	8	2	no	—
236	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
237	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
238	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
242	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
243	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
244	C₈×C₈₉	—	2	1	no	—
245	C₈×C₈₉	—	2	1	no	—
246	C₈×C₈₉	—	2	4	no	—
247	C₈×C₈₉	—	4	1	no	—
248	C₈×C₈₉	—	4	1	no	—
249	C₈×C₈₉	—	4	2	no	—
250	C₈×C₈₉	#121	1	2	yes	—
251	C₈×C₈₉	#121	1	2	yes	—
252	C₂²×C₈₉	—	2	1	no	—
253	C₂²×C₈₉	—	2	1	no	—
254	C₂²×C₈₉	—	2	1	no	—
255	C₂²×C₈₉	—	2	1	no	—
256	C₂²×C₈₉	—	2	1	no	—
257	C₂²×C₈₉	—	2	1	no	—
258	C₂²×C₈₉	—	2	1	no	—
259	C₂²×C₈₉	—	4	2	no	—
260	C₂²×C₈₉	—	4	2	no	—
261	C₂²×C₈₉	—	4	1	no	—
262	C₂²×C₈₉	—	4	1	no	—
263	C₂²×C₈₉	—	4	1	no	—
264	C₂²×C₈₉	#473	1	2	yes	complemented
265	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
266	C₂²×C₈₉	#548	1	1	yes	characteristic
267	C₂²×C₈₉	#553	1	1	yes	characteristic
268	C₄×C₈₉	—	2	1	no	—
269	C₄×C₈₉	—	2	1	no	—
270	C₄×C₈₉	—	4	1	no	—
271	C₄×C₈₉	—	4	1	no	—
272	C₄×C₈₉	—	4	1	no	—
273	C₄×C₈₉	—	8	2	no	—
274	C₄×C₈₉	—	8	2	no	—
275	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
276	C₄×C₈₉	#537	1	1	yes	characteristic
277	C₄×C₈₉	D₄²	1	1	yes	characteristic
278	C₄×C₈₉	#547	1	1	yes	characteristic
279	#24804	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
280	C₂×C₈₉	—	2	1	no	—
281	C₂×C₈₉	—	2	1	no	—
282	C₂×C₈₉	—	2	2	no	—
283	C₂×C₈₉	—	4	1	no	—
284	C₂×C₈₉	—	8	1	no	—
285	C₂×C₈₉	#2416	1	1	yes	characteristic
286	C₂×C₈₉	C₂×#473	1	1	yes	characteristic
287	C₂×C₈₉	#3067	1	1	yes	characteristic
288	C₂×#473	C₂×C₈₉	1	1	yes	characteristic, complemented
289	#3056	C₂×C₈₉	1	2	yes	—
290	C₂×#467	C₂×C₈₉	1	1	yes	characteristic, complemented
291	#2929	C₂×C₈₉	1	1	yes	characteristic
292	C₂²×#108	C₂×C₈₉	1	1	yes	characteristic, complemented
293	#2064	C₂×C₈₉	1	2	yes	complemented
294	#2052	C₂×C₈₉	1	2	yes	complemented
295	#1979	C₂×C₈₉	1	1	yes	characteristic, complemented
296	#1952	C₂×C₈₉	1	1	yes	characteristic, complemented
297	#1901	C₂×C₈₉	1	2	yes	complemented
298	#1816	C₂×C₈₉	1	1	yes	characteristic, complemented
299	C₈₉	#24804	1	1	yes	characteristic, direct product factor, Sylow
300	C₂×#137	C₂²×C₈₉	1	2	yes	—
301	C₂²×Q₁₆	C₂²×C₈₉	1	1	yes	characteristic
302	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
303	C₂²×M₁₆	C₂²×C₈₉	1	2	yes	—
304	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
305	#525	C₂²×C₈₉	1	1	yes	characteristic
306	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	2	yes	—
307	#473	—	2	4	no	—
308	#470	—	2	2	no	—
309	#468	—	2	2	no	—
310	#467	—	2	1	no	—
311	#467	—	2	1	no	—
312	#467	—	2	1	no	—
313	#467	—	2	1	no	—
314	#467	—	2	2	no	—
315	#465	—	2	2	no	—
316	#435	—	2	1	no	—
317	#435	—	2	1	no	—
318	C₈×D₄	—	2	1	no	—
319	C₈×D₄	—	2	1	no	—
320	C₂×#108	—	2	4	no	—
321	C₂×#108	C₂²×C₈₉	1	2	yes	—
322	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
323	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
324	#428	C₂²×C₈₉	1	2	yes	—
325	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
326	#424	C₂²×C₈₉	1	1	yes	characteristic
327	#419	C₂²×C₈₉	1	2	yes	—
328	#417	C₂²×C₈₉	1	2	yes	—
329	C₂×#103	C₂²×C₈₉	1	2	yes	—
330	C₂×#101	C₂²×C₈₉	1	2	yes	—
331	C₂×#101	C₂²×C₈₉	1	2	yes	—
332	#408	C₂²×C₈₉	1	1	yes	characteristic
333	#407	C₂²×C₈₉	1	1	yes	characteristic
334	#405	C₂²×C₈₉	1	1	yes	characteristic
335	#343	C₂²×C₈₉	1	2	yes	—
336	#343	C₂²×C₈₉	1	2	yes	—
337	C₂×#41	C₂³×C₈₉	1	1	yes	characteristic
338	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
339	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
340	#137	—	2	4	no	—
341	#137	—	4	2	no	—
342	C₂×Q₁₆	—	2	2	no	—
343	C₂×Q₁₆	—	2	1	no	—
344	C₂×Q₁₆	—	2	1	no	—
345	C₂×Q₁₆	—	2	2	no	—
346	C₂×Q₁₆	—	2	2	no	—
347	C₂×SD₁₆	—	2	2	no	—
348	#131	—	2	2	no	—
349	#131	—	2	2	no	—
350	C₂×M₁₆	—	2	2	no	—
351	C₂×M₁₆	—	2	1	no	—
352	C₂×M₁₆	—	2	4	no	—
353	C₂×M₁₆	—	4	2	no	—
354	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
355	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
356	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
357	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
358	C₂²×C₈	—	2	2	no	—
359	C₂²×C₈	—	2	2	no	—
360	C₂²×C₈	—	2	2	no	—
361	C₂²×C₈	—	2	1	no	—
362	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
363	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
364	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
365	C₂²×C₈	D₄×C₈₉	1	1	yes	characteristic
366	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
367	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
368	#123	—	2	1	no	—
369	C₂²⋊Q₈	—	2	2	no	—
370	C₂²⋊Q₈	—	2	2	no	—
371	C₂²⋊Q₈	—	2	2	no	—
372	C₂²⋊Q₈	—	2	2	no	—
373	C₂²⋊Q₈	D₄×C₈₉	1	1	yes	characteristic
374	C₂²⋊Q₈	D₄×C₈₉	1	1	yes	characteristic
375	#121	D₄×C₈₉	1	1	yes	characteristic
376	#121	D₄×C₈₉	1	1	yes	characteristic
377	#120	—	2	1	no	—
378	C₄×D₄	—	2	1	no	—
379	C₄×D₄	—	2	1	no	—
380	#117	C₂³×C₈₉	1	1	yes	characteristic
381	C₂×C₄⋊C₄	—	2	2	no	—
382	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
383	C₂×C₂²⋊C₄	—	2	2	no	—
384	#108	—	2	4	no	—
385	#108	—	2	2	no	—
386	#108	D₄×C₈₉	1	4	yes	complemented
387	#107	—	2	1	no	—
388	#107	—	2	1	no	—
389	#107	—	2	2	no	—
390	#106	—	2	2	no	—
391	C₄⋊C₈	—	2	1	no	—
392	C₄⋊C₈	—	2	1	no	—
393	#103	—	2	2	no	—
394	#103	—	2	2	no	—
395	#103	—	2	2	no	—
396	#103	—	2	2	no	—
397	#103	—	2	2	no	—
398	#102	—	2	2	no	—
399	#101	—	2	4	no	—
400	#101	—	4	2	no	—
401	C₂²⋊C₈	—	2	1	no	—
402	C₂²⋊C₈	—	2	1	no	—
403	C₂²⋊C₈	—	2	1	no	—
404	C₂²⋊C₈	—	2	1	no	—
405	#97	—	2	1	no	—
406	C₄×C₈	—	2	1	no	—
407	C₂⁴	#41×C₈₉	1	1	yes	characteristic
408	#41	—	2	1	no	—
409	#41	—	2	1	no	—
410	#41	—	2	1	no	—
411	#41	—	2	1	no	—
412	C₂×Q₈	—	2	2	no	—
413	C₂×Q₈	—	2	2	no	—
414	C₂×Q₈	—	2	2	no	—
415	C₂×Q₈	—	2	2	no	—
416	C₂×Q₈	—	2	2	no	—
417	C₂×Q₈	—	2	2	no	—
418	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	C₂×D₄×C₈₉	1	1	yes	characteristic
423	C₂²×C₄	—	2	1	no	—
424	C₂²×C₄	—	2	1	no	—
425	C₂²×C₄	—	2	1	no	—
426	C₂²×C₄	—	2	1	no	—
427	C₂²×C₄	—	2	1	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
431	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
433	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
434	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
435	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
436	Q₁₆	—	4	4	no	—
437	Q₁₆	—	4	2	no	—
438	Q₁₆	—	4	2	no	—
439	SD₁₆	—	4	4	no	—
440	M₁₆	—	2	2	no	—
441	M₁₆	—	2	1	no	—
442	M₁₆	—	2	2	no	—
443	M₁₆	—	2	1	no	—
444	M₁₆	—	2	1	no	—
445	M₁₆	—	2	2	no	—
446	M₁₆	—	2	1	no	—
447	M₁₆	—	4	4	no	—
448	M₁₆	—	4	2	no	—
449	M₁₆	C₂×D₄×C₈₉	1	4	yes	—
450	C₂×C₈	—	2	2	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	2	1	no	—
453	C₂×C₈	—	2	2	no	—
454	C₂×C₈	—	2	2	no	—
455	C₂×C₈	—	2	2	no	—
456	C₂×C₈	—	2	2	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	2	no	—
460	C₂×C₈	—	4	4	no	—
461	C₂×C₈	—	4	2	no	—
462	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
463	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
464	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
465	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
466	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
467	C₂×C₈	#41×C₈₉	1	2	yes	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	2	no	—
472	C₄⋊C₄	—	4	2	no	—
473	C₄⋊C₄	—	4	2	no	—
474	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
475	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
476	C₂²⋊C₄	—	2	1	no	—
477	C₂²⋊C₄	—	2	1	no	—
478	C₂²⋊C₄	—	2	1	no	—
479	C₂²⋊C₄	—	2	1	no	—
480	C₂²⋊C₄	—	2	1	no	—
481	C₂²⋊C₄	—	4	2	no	—
482	C₂²⋊C₄	—	4	2	no	—
483	C₄²	—	2	1	no	—
484	C₂³	—	2	2	no	—
485	C₂³	—	2	1	no	—
486	C₂³	—	2	1	no	—
487	C₂³	—	2	1	no	—
488	C₂³	—	4	1	no	—
489	C₂³	#121×C₈₉	1	1	yes	characteristic
490	C₂³	#121×C₈₉	1	1	yes	characteristic
491	C₂³	C₂×#41×C₈₉	1	1	yes	characteristic
492	Q₈	—	2	1	no	—
493	Q₈	—	2	1	no	—
494	Q₈	—	4	2	no	—
495	Q₈	—	4	2	no	—
496	Q₈	—	4	2	no	—
497	Q₈	—	4	2	no	—
498	D₄	—	2	1	no	—
499	D₄	—	2	1	no	—
500	D₄	—	4	2	no	—
501	D₄	—	4	2	no	—
502	D₄	—	4	1	no	—
503	D₄	—	4	1	no	—
504	C₂×C₄	—	2	1	no	—
505	C₂×C₄	—	2	1	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	1	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	4	1	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	2	no	—
520	C₂×C₄	—	4	2	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	4	2	no	—
523	C₂×C₄	—	8	2	no	—
524	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
525	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
526	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
527	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
528	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
529	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
530	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
531	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
532	C₈	—	2	1	no	—
533	C₈	—	2	1	no	—
534	C₈	—	2	4	no	—
535	C₈	—	4	1	no	—
536	C₈	—	4	1	no	—
537	C₈	—	4	2	no	—
538	C₈	#121×C₈₉	1	2	yes	—
539	C₈	#121×C₈₉	1	2	yes	—
540	C₂²	—	2	1	no	—
541	C₂²	—	2	1	no	—
542	C₂²	—	2	1	no	—
543	C₂²	—	2	1	no	—
544	C₂²	—	2	1	no	—
545	C₂²	—	2	1	no	—
546	C₂²	—	2	1	no	—
547	C₂²	—	4	2	no	—
548	C₂²	—	4	2	no	—
549	C₂²	—	4	1	no	—
550	C₂²	—	4	1	no	—
551	C₂²	—	4	1	no	—
552	C₂²	#473×C₈₉	1	2	yes	complemented
553	C₂²	C₂×#121×C₈₉	1	1	yes	characteristic
554	C₂²	#548×C₈₉	1	1	yes	characteristic
555	C₂²	#553×C₈₉	1	1	yes	characteristic
556	C₄	—	2	1	no	—
557	C₄	—	2	1	no	—
558	C₄	—	4	1	no	—
559	C₄	—	4	1	no	—
560	C₄	—	4	1	no	—
561	C₄	—	8	2	no	—
562	C₄	—	8	2	no	—
563	C₄	C₂×#121×C₈₉	1	1	yes	characteristic
564	C₄	#537×C₈₉	1	1	yes	characteristic
565	C₄	D₄²×C₈₉	1	1	yes	characteristic
566	C₄	#547×C₈₉	1	1	yes	characteristic
567	C₂	—	2	2	no	—
568	C₂	—	2	1	no	—
569	C₂	—	2	1	no	—
570	C₂	—	4	1	no	—
571	C₂	—	8	1	no	—
572	C₂	C₈₉×#2416	1	1	yes	characteristic
573	C₂	C₂×#473×C₈₉	1	1	yes	characteristic
574	C₂	C₈₉×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	#525×C₈₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂³×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#473×C₈₉	C₂	semidirect
3. C₂×#467×C₈₉	C₂	semidirect
5. C₂²×#108×C₈₉	C₂	semidirect
6. C₈₉×#2064	C₂	semidirect
7. C₈₉×#2052	C₂	semidirect
8. C₈₉×#1979	C₂	semidirect
9. C₈₉×#1952	C₂	semidirect
10. C₈₉×#1901	C₂	semidirect
11. C₈₉×#1816	C₂	semidirect
98. #108×C₈₉	D₄	semidirect
264. C₂²×C₈₉	#473	semidirect
279. #24804	C₈₉	direct (N × H)
288. C₂×#473	C₂×C₈₉	semidirect
290. C₂×#467	C₂×C₈₉	semidirect
292. C₂²×#108	C₂×C₈₉	semidirect
293. #2064	C₂×C₈₉	semidirect
294. #2052	C₂×C₈₉	semidirect
295. #1979	C₂×C₈₉	semidirect
296. #1952	C₂×C₈₉	semidirect
297. #1901	C₂×C₈₉	semidirect
298. #1816	C₂×C₈₉	semidirect
299. C₈₉	#24804	direct (N × H)
386. #108	D₄×C₈₉	semidirect
552. C₂²	#473×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#473×C₈₉	2
C₈₉×#3056	2
C₂×#467×C₈₉	2
C₈₉×#2929	2
C₂²×#108×C₈₉	2
C₈₉×#2064	2
C₈₉×#2052	2
C₈₉×#1979	2
C₈₉×#1952	2
C₈₉×#1901	2
C₈₉×#1816	2
#24804	89

Subgroups — C89×#24804 #36402652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#24804 #36402652