Subgroups — C₈₉×#25295 #36403143

Order 22784.

886 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#460×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2582	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2548	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2065	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2055	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1968	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1957	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#1948	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1934	C₂	1	2	yes	maximal, complemented
11	C₄×#103×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	—
13	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×#124×C₈₉	—	2	1	no	—
15	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
17	#460×C₈₉	—	2	4	no	—
18	C₈×Q₈×C₈₉	—	2	4	no	—
19	C₂×#106×C₈₉	C₂²	1	2	yes	—
20	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#102×C₈₉	C₂²	1	2	yes	—
25	C₂×#102×C₈₉	C₂²	1	2	yes	—
26	C₂×#102×C₈₉	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
29	#398×C₈₉	—	2	2	no	—
30	#394×C₈₉	—	2	2	no	—
31	#391×C₈₉	—	2	1	no	—
32	#391×C₈₉	C₂²	1	2	yes	—
33	#390×C₈₉	—	2	1	no	—
34	#390×C₈₉	C₂²	1	2	yes	—
35	#390×C₈₉	C₂²	1	2	yes	—
36	#389×C₈₉	—	2	1	no	—
37	#389×C₈₉	C₂²	1	2	yes	—
38	#384×C₈₉	C₂²	1	1	yes	characteristic
39	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
40	#340×C₈₉	C₂²	1	2	yes	—
41	#340×C₈₉	C₂²	1	1	yes	characteristic
42	#340×C₈₉	C₂²	1	1	yes	characteristic
43	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
44	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₈₉	—	2	1	no	—
46	C₂²×D₄×C₈₉	—	2	1	no	—
47	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
48	C₂×SD₁₆×C₈₉	—	2	4	no	—
49	C₂×SD₁₆×C₈₉	—	2	4	no	—
50	C₂×SD₁₆×C₈₉	—	4	2	no	—
51	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
53	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
54	#127×C₈₉	—	2	2	no	—
55	#127×C₈₉	—	2	2	no	—
56	#124×C₈₉	—	4	4	no	—
57	C₄×Q₈×C₈₉	—	2	2	no	—
58	C₄×Q₈×C₈₉	—	2	2	no	—
59	C₄×Q₈×C₈₉	—	2	2	no	—
60	C₄×Q₈×C₈₉	—	2	2	no	—
61	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
62	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
63	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
64	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
65	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
67	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
70	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
75	C₂×C₄²×C₈₉	—	2	1	no	—
76	C₂×C₄²×C₈₉	—	2	1	no	—
77	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
78	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
79	#106×C₈₉	—	4	2	no	—
80	C₄⋊C₈×C₈₉	—	2	4	no	—
81	C₄⋊C₈×C₈₉	D₄	1	4	yes	—
82	#103×C₈₉	—	2	2	no	—
83	#103×C₈₉	—	2	2	no	—
84	#102×C₈₉	—	2	4	no	—
85	#102×C₈₉	—	2	4	no	—
86	#102×C₈₉	—	4	2	no	—
87	C₄×C₈×C₈₉	—	2	4	no	—
88	C₄×C₈×C₈₉	—	2	2	no	—
89	#95×C₈₉	—	2	1	no	—
90	#95×C₈₉	—	2	2	no	—
91	#95×C₈₉	—	2	2	no	—
92	#95×C₈₉	—	2	2	no	—
93	C₂⁴×C₈₉	—	2	2	no	—
94	C₂×Q₈×C₈₉	—	2	2	no	—
95	C₂×Q₈×C₈₉	—	2	2	no	—
96	C₂×Q₈×C₈₉	—	2	1	no	—
97	C₂×Q₈×C₈₉	—	2	1	no	—
98	C₂×Q₈×C₈₉	—	2	1	no	—
99	C₂×Q₈×C₈₉	—	2	1	no	—
100	C₂×Q₈×C₈₉	—	2	1	no	—
101	C₂×Q₈×C₈₉	SD₁₆	1	4	yes	complemented
102	C₂×Q₈×C₈₉	#41	1	2	yes	—
103	C₂×D₄×C₈₉	—	2	2	no	—
104	C₂×D₄×C₈₉	—	2	2	no	—
105	C₂×D₄×C₈₉	—	2	2	no	—
106	C₂×D₄×C₈₉	—	2	2	no	—
107	C₂×D₄×C₈₉	—	2	2	no	—
108	C₂×D₄×C₈₉	—	2	2	no	—
109	C₂×D₄×C₈₉	—	4	2	no	—
110	C₂×D₄×C₈₉	—	4	2	no	—
111	C₂×D₄×C₈₉	—	4	2	no	—
112	C₂×D₄×C₈₉	—	4	2	no	—
113	C₂×D₄×C₈₉	—	8	1	no	—
114	C₂×D₄×C₈₉	—	8	1	no	—
115	C₂²×C₄×C₈₉	—	2	1	no	—
116	C₂²×C₄×C₈₉	—	2	2	no	—
117	C₂²×C₄×C₈₉	—	2	1	no	—
118	C₂²×C₄×C₈₉	—	2	1	no	—
119	C₂²×C₄×C₈₉	—	2	1	no	—
120	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
125	SD₁₆×C₈₉	—	4	8	no	—
126	C₂×C₈×C₈₉	—	2	1	no	—
127	C₂×C₈×C₈₉	—	2	2	no	—
128	C₂×C₈×C₈₉	—	2	2	no	—
129	C₂×C₈×C₈₉	—	2	1	no	—
130	C₂×C₈×C₈₉	—	2	1	no	—
131	C₂×C₈×C₈₉	—	2	1	no	—
132	C₂×C₈×C₈₉	—	2	1	no	—
133	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
134	C₂×C₈×C₈₉	#41	1	2	yes	—
135	C₄⋊C₄×C₈₉	—	2	4	no	—
136	C₄⋊C₄×C₈₉	—	2	2	no	—
137	C₄⋊C₄×C₈₉	—	2	1	no	—
138	C₄⋊C₄×C₈₉	—	2	2	no	—
139	C₄⋊C₄×C₈₉	—	2	1	no	—
140	C₄⋊C₄×C₈₉	—	2	1	no	—
141	C₄⋊C₄×C₈₉	—	2	1	no	—
142	C₄⋊C₄×C₈₉	—	2	2	no	—
143	C₄⋊C₄×C₈₉	—	2	1	no	—
144	C₄⋊C₄×C₈₉	—	2	1	no	—
145	C₄⋊C₄×C₈₉	—	4	2	no	—
146	C₄⋊C₄×C₈₉	—	4	2	no	—
147	C₄⋊C₄×C₈₉	—	8	1	no	—
148	C₄⋊C₄×C₈₉	—	8	2	no	—
149	C₂²⋊C₄×C₈₉	—	4	4	no	—
150	C₂²⋊C₄×C₈₉	—	4	4	no	—
151	C₂²⋊C₄×C₈₉	—	8	2	no	—
152	C₂²⋊C₄×C₈₉	—	8	2	no	—
153	C₄²×C₈₉	—	2	2	no	—
154	C₄²×C₈₉	—	2	2	no	—
155	C₄²×C₈₉	—	4	2	no	—
156	C₄²×C₈₉	—	4	2	no	—
157	C₄²×C₈₉	C₂×D₄	1	2	yes	—
158	C₄²×C₈₉	#41	1	2	yes	—
159	C₂³×C₈₉	—	4	2	no	—
160	C₂³×C₈₉	—	4	2	no	—
161	C₂³×C₈₉	—	4	2	no	—
162	C₂³×C₈₉	—	4	2	no	—
163	C₂³×C₈₉	—	4	2	no	—
164	C₂³×C₈₉	—	4	2	no	—
165	C₂³×C₈₉	—	4	2	no	—
166	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
167	Q₈×C₈₉	—	2	8	no	—
168	Q₈×C₈₉	—	2	2	no	—
169	Q₈×C₈₉	—	2	2	no	—
170	Q₈×C₈₉	—	4	1	no	—
171	Q₈×C₈₉	—	4	1	no	—
172	D₄×C₈₉	—	4	2	no	—
173	D₄×C₈₉	—	4	2	no	—
174	D₄×C₈₉	—	4	2	no	—
175	D₄×C₈₉	—	4	2	no	—
176	D₄×C₈₉	—	8	4	no	—
177	D₄×C₈₉	—	8	4	no	—
178	C₂×C₄×C₈₉	—	2	2	no	—
179	C₂×C₄×C₈₉	—	2	2	no	—
180	C₂×C₄×C₈₉	—	2	1	no	—
181	C₂×C₄×C₈₉	—	2	1	no	—
182	C₂×C₄×C₈₉	—	2	2	no	—
183	C₂×C₄×C₈₉	—	2	2	no	—
184	C₂×C₄×C₈₉	—	2	2	no	—
185	C₂×C₄×C₈₉	—	2	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	1	no	—
190	C₂×C₄×C₈₉	—	4	1	no	—
191	C₂×C₄×C₈₉	—	4	2	no	—
192	C₂×C₄×C₈₉	—	4	1	no	—
193	C₂×C₄×C₈₉	—	4	1	no	—
194	C₂×C₄×C₈₉	—	4	2	no	—
195	C₂×C₄×C₈₉	—	4	1	no	—
196	C₂×C₄×C₈₉	—	4	2	no	—
197	C₂×C₄×C₈₉	—	4	1	no	—
198	C₂×C₄×C₈₉	—	4	1	no	—
199	C₂×C₄×C₈₉	—	4	1	no	—
200	C₂×C₄×C₈₉	#121	1	2	yes	—
201	C₂×C₄×C₈₉	#121	1	2	yes	—
202	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
203	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
204	C₂×C₄×C₈₉	#135	1	2	yes	—
205	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
207	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
208	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
209	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
210	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
211	C₈×C₈₉	—	2	4	no	—
212	C₈×C₈₉	—	2	2	no	—
213	C₈×C₈₉	—	4	1	no	—
214	C₂²×C₈₉	—	8	2	no	—
215	C₂²×C₈₉	—	8	2	no	—
216	C₂²×C₈₉	—	8	2	no	—
217	C₂²×C₈₉	—	8	2	no	—
218	C₂²×C₈₉	—	8	2	no	—
219	C₂²×C₈₉	—	8	2	no	—
220	C₂²×C₈₉	—	8	2	no	—
221	C₂²×C₈₉	C₂×#121	1	1	yes	characteristic
222	C₂²×C₈₉	#548	1	1	yes	characteristic
223	C₂²×C₈₉	#550	1	1	yes	characteristic
224	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
225	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
226	C₂²×C₈₉	#576	1	1	yes	characteristic
227	C₂²×C₈₉	#577	1	1	yes	characteristic
228	C₄×C₈₉	—	2	4	no	—
229	C₄×C₈₉	—	2	2	no	—
230	C₄×C₈₉	—	4	2	no	—
231	C₄×C₈₉	—	4	2	no	—
232	C₄×C₈₉	—	8	1	no	—
233	C₄×C₈₉	—	8	1	no	—
234	C₄×C₈₉	—	8	2	no	—
235	C₄×C₈₉	—	8	1	no	—
236	C₄×C₈₉	#465	1	4	yes	—
237	C₄×C₈₉	#529	1	1	yes	characteristic
238	C₄×C₈₉	#542	1	1	yes	characteristic
239	C₄×C₈₉	#546	1	1	yes	characteristic
240	C₄×C₈₉	#548	1	1	yes	characteristic
241	#25295	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
242	C₂×C₈₉	—	16	2	no	—
243	C₂×C₈₉	#2426	1	1	yes	characteristic
244	C₂×C₈₉	C₂×#465	1	1	yes	characteristic
245	C₂×C₈₉	#3062	1	1	yes	characteristic
246	C₂×C₈₉	#3362	1	1	yes	characteristic
247	C₂×C₈₉	#3365	1	1	yes	characteristic
248	C₂×C₈₉	#3392	1	1	yes	characteristic
249	C₂×C₈₉	#3396	1	1	yes	characteristic
250	C₂×#460	C₂×C₈₉	1	1	yes	characteristic
251	C₂×C₈×Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
252	#2582	C₂×C₈₉	1	1	yes	characteristic, complemented
253	#2548	C₂×C₈₉	1	1	yes	characteristic
254	#2065	C₂×C₈₉	1	2	yes	complemented
255	#2055	C₂×C₈₉	1	2	yes	complemented
256	#1968	C₂×C₈₉	1	2	yes	complemented
257	#1957	C₂×C₈₉	1	1	yes	characteristic
258	#1948	C₂×C₈₉	1	1	yes	characteristic, complemented
259	#1934	C₂×C₈₉	1	2	yes	complemented
260	C₄×#103	C₂×C₈₉	1	1	yes	characteristic, complemented
261	C₈₉	#25295	1	1	yes	characteristic, direct product factor, Sylow
262	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	—
263	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
264	C₂×#124	—	2	1	no	—
265	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
266	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
267	#460	—	2	4	no	—
268	C₈×Q₈	—	2	4	no	—
269	C₂×#106	C₂²×C₈₉	1	2	yes	—
270	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
271	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
272	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
273	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
274	C₂×#102	C₂²×C₈₉	1	2	yes	—
275	C₂×#102	C₂²×C₈₉	1	2	yes	—
276	C₂×#102	C₂²×C₈₉	1	2	yes	—
277	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	—
278	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
279	#398	—	2	2	no	—
280	#394	—	2	2	no	—
281	#391	—	2	1	no	—
282	#391	C₂²×C₈₉	1	2	yes	—
283	#390	—	2	1	no	—
284	#390	C₂²×C₈₉	1	2	yes	—
285	#390	C₂²×C₈₉	1	2	yes	—
286	#389	—	2	1	no	—
287	#389	C₂²×C₈₉	1	2	yes	—
288	#384	C₂²×C₈₉	1	1	yes	characteristic
289	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
290	#340	C₂²×C₈₉	1	2	yes	—
291	#340	C₂²×C₈₉	1	1	yes	characteristic
292	#340	C₂²×C₈₉	1	1	yes	characteristic
293	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
294	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
295	C₂²×D₄	—	2	1	no	—
296	C₂²×D₄	—	2	1	no	—
297	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
298	C₂×SD₁₆	—	2	4	no	—
299	C₂×SD₁₆	—	2	4	no	—
300	C₂×SD₁₆	—	4	2	no	—
301	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
302	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
303	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
304	#127	—	2	2	no	—
305	#127	—	2	2	no	—
306	#124	—	4	4	no	—
307	C₄×Q₈	—	2	2	no	—
308	C₄×Q₈	—	2	2	no	—
309	C₄×Q₈	—	2	2	no	—
310	C₄×Q₈	—	2	2	no	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	—	2	1	no	—
313	C₂×C₄⋊C₄	—	2	1	no	—
314	C₂×C₄⋊C₄	—	2	2	no	—
315	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
316	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
317	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
318	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
319	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
320	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
321	C₂×C₂²⋊C₄	—	2	2	no	—
322	C₂×C₂²⋊C₄	—	2	2	no	—
323	C₂×C₂²⋊C₄	—	2	2	no	—
324	C₂×C₂²⋊C₄	—	2	2	no	—
325	C₂×C₄²	—	2	1	no	—
326	C₂×C₄²	—	2	1	no	—
327	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
328	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
329	#106	—	4	2	no	—
330	C₄⋊C₈	—	2	4	no	—
331	C₄⋊C₈	D₄×C₈₉	1	4	yes	—
332	#103	—	2	2	no	—
333	#103	—	2	2	no	—
334	#102	—	2	4	no	—
335	#102	—	2	4	no	—
336	#102	—	4	2	no	—
337	C₄×C₈	—	2	4	no	—
338	C₄×C₈	—	2	2	no	—
339	#95	—	2	1	no	—
340	#95	—	2	2	no	—
341	#95	—	2	2	no	—
342	#95	—	2	2	no	—
343	C₂⁴	—	2	2	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	—	2	1	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	2	1	no	—
350	C₂×Q₈	—	2	1	no	—
351	C₂×Q₈	SD₁₆×C₈₉	1	4	yes	complemented
352	C₂×Q₈	#41×C₈₉	1	2	yes	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	—	2	2	no	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	4	2	no	—
360	C₂×D₄	—	4	2	no	—
361	C₂×D₄	—	4	2	no	—
362	C₂×D₄	—	4	2	no	—
363	C₂×D₄	—	8	1	no	—
364	C₂×D₄	—	8	1	no	—
365	C₂²×C₄	—	2	1	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
371	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
373	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
374	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
375	SD₁₆	—	4	8	no	—
376	C₂×C₈	—	2	1	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	C₂×D₄×C₈₉	1	4	yes	—
384	C₂×C₈	#41×C₈₉	1	2	yes	—
385	C₄⋊C₄	—	2	4	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	4	2	no	—
396	C₄⋊C₄	—	4	2	no	—
397	C₄⋊C₄	—	8	1	no	—
398	C₄⋊C₄	—	8	2	no	—
399	C₂²⋊C₄	—	4	4	no	—
400	C₂²⋊C₄	—	4	4	no	—
401	C₂²⋊C₄	—	8	2	no	—
402	C₂²⋊C₄	—	8	2	no	—
403	C₄²	—	2	2	no	—
404	C₄²	—	2	2	no	—
405	C₄²	—	4	2	no	—
406	C₄²	—	4	2	no	—
407	C₄²	C₂×D₄×C₈₉	1	2	yes	—
408	C₄²	#41×C₈₉	1	2	yes	—
409	C₂³	—	4	2	no	—
410	C₂³	—	4	2	no	—
411	C₂³	—	4	2	no	—
412	C₂³	—	4	2	no	—
413	C₂³	—	4	2	no	—
414	C₂³	—	4	2	no	—
415	C₂³	—	4	2	no	—
416	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
417	Q₈	—	2	8	no	—
418	Q₈	—	2	2	no	—
419	Q₈	—	2	2	no	—
420	Q₈	—	4	1	no	—
421	Q₈	—	4	1	no	—
422	D₄	—	4	2	no	—
423	D₄	—	4	2	no	—
424	D₄	—	4	2	no	—
425	D₄	—	4	2	no	—
426	D₄	—	8	4	no	—
427	D₄	—	8	4	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	4	1	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	1	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	2	no	—
448	C₂×C₄	—	4	1	no	—
449	C₂×C₄	—	4	1	no	—
450	C₂×C₄	#121×C₈₉	1	2	yes	—
451	C₂×C₄	#121×C₈₉	1	2	yes	—
452	C₂×C₄	C₂×SD₁₆×C₈₉	1	4	yes	—
453	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
454	C₂×C₄	#135×C₈₉	1	2	yes	—
455	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
456	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
457	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
458	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
459	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
460	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
461	C₈	—	2	4	no	—
462	C₈	—	2	2	no	—
463	C₈	—	4	1	no	—
464	C₂²	—	8	2	no	—
465	C₂²	—	8	2	no	—
466	C₂²	—	8	2	no	—
467	C₂²	—	8	2	no	—
468	C₂²	—	8	2	no	—
469	C₂²	—	8	2	no	—
470	C₂²	—	8	2	no	—
471	C₂²	C₂×#121×C₈₉	1	1	yes	characteristic
472	C₂²	#548×C₈₉	1	1	yes	characteristic
473	C₂²	#550×C₈₉	1	1	yes	characteristic
474	C₂²	C₂²×SD₁₆×C₈₉	1	1	yes	characteristic
475	C₂²	C₂×#135×C₈₉	1	1	yes	characteristic
476	C₂²	#576×C₈₉	1	1	yes	characteristic
477	C₂²	#577×C₈₉	1	1	yes	characteristic
478	C₄	—	2	2	no	—
479	C₄	—	2	4	no	—
480	C₄	—	4	2	no	—
481	C₄	—	4	2	no	—
482	C₄	—	8	1	no	—
483	C₄	—	8	1	no	—
484	C₄	—	8	1	no	—
485	C₄	—	8	2	no	—
486	C₄	#465×C₈₉	1	4	yes	—
487	C₄	#529×C₈₉	1	1	yes	characteristic
488	C₄	#542×C₈₉	1	1	yes	characteristic
489	C₄	#546×C₈₉	1	1	yes	characteristic
490	C₄	#548×C₈₉	1	1	yes	characteristic
491	C₂	—	16	2	no	—
492	C₂	C₈₉×#2426	1	1	yes	characteristic
493	C₂	C₂×#465×C₈₉	1	1	yes	characteristic
494	C₂	C₈₉×#3062	1	1	yes	characteristic
495	C₂	C₈₉×#3362	1	1	yes	characteristic
496	C₂	C₈₉×#3365	1	1	yes	characteristic
497	C₂	C₈₉×#3392	1	1	yes	characteristic
498	C₂	C₈₉×#3396	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 ◁ C₂×C₄×Q₈×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Q₈×C₈₉	C₂	semidirect
3. C₈₉×#2582	C₂	semidirect
5. C₈₉×#2065	C₂	semidirect
6. C₈₉×#2055	C₂	semidirect
7. C₈₉×#1968	C₂	semidirect
9. C₈₉×#1948	C₂	semidirect
10. C₈₉×#1934	C₂	semidirect
11. C₄×#103×C₈₉	C₂	semidirect
101. C₂×Q₈×C₈₉	SD₁₆	semidirect
241. #25295	C₈₉	direct (N × H)
251. C₂×C₈×Q₈	C₂×C₈₉	semidirect
252. #2582	C₂×C₈₉	semidirect
254. #2065	C₂×C₈₉	semidirect
255. #2055	C₂×C₈₉	semidirect
256. #1968	C₂×C₈₉	semidirect
258. #1948	C₂×C₈₉	semidirect
259. #1934	C₂×C₈₉	semidirect
260. C₄×#103	C₂×C₈₉	semidirect
261. C₈₉	#25295	direct (N × H)
351. C₂×Q₈	SD₁₆×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#460×C₈₉	2
C₂×C₈×Q₈×C₈₉	2
C₈₉×#2582	2
C₈₉×#2548	2
C₈₉×#2065	2
C₈₉×#2055	2
C₈₉×#1968	2
C₈₉×#1957	2
C₈₉×#1948	2
C₈₉×#1934	2
C₄×#103×C₈₉	2
#25295	89

Subgroups — C89×#25295 #36403143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#25295 #36403143