Subgroups — C₈₉×#33523 #36411371

Order 22784.

1138 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#576×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#510×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×SD₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#3059	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2189	C₂	1	4	yes	maximal, complemented
6	C₈₉×#2187	C₂	1	4	yes	maximal, complemented
7	C₈₉×#2140	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#2138	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2113	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×H(5,𝔽₂)×C₈₉	—	2	1	no	—
11	#576×C₈₉	—	2	4	no	—
12	#576×C₈₉	—	2	4	no	—
13	C₂×AGL(1,ℤ/8)×C₈₉	—	2	1	no	—
14	C₂×AGL(1,ℤ/8)×C₈₉	—	2	2	no	—
15	C₂×#135×C₈₉	C₂²	1	2	yes	—
16	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
18	C₂²×D₈×C₈₉	C₂²	1	2	yes	—
19	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
21	#536×C₈₉	—	2	1	no	—
22	#510×C₈₉	—	2	4	no	—
23	C₂×Q₃₂×C₈₉	C₂²	1	2	yes	complemented
24	C₂×SD₃₂×C₈₉	—	2	4	no	—
25	C₂×SD₃₂×C₈₉	C₂²	1	4	yes	complemented
26	C₂×SD₃₂×C₈₉	C₂²	1	2	yes	complemented
27	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic, complemented
28	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
29	#470×C₈₉	—	2	2	no	—
30	#467×C₈₉	C₂²	1	4	yes	complemented
31	#427×C₈₉	C₂²	1	1	yes	characteristic, complemented
32	#419×C₈₉	—	2	1	no	—
33	#358×C₈₉	C₂²	1	2	yes	complemented
34	#357×C₈₉	C₂²	1	2	yes	complemented
35	#357×C₈₉	C₂²	1	4	yes	complemented
36	C₂²⋊C₁₆×C₈₉	C₂²	1	4	yes	complemented
37	H(5,𝔽₂)×C₈₉	—	2	8	no	—
38	H(5,𝔽₂)×C₈₉	—	4	4	no	—
39	C₂×#41×C₈₉	—	2	2	no	—
40	C₂×#41×C₈₉	—	2	1	no	—
41	C₂×#41×C₈₉	—	4	1	no	—
42	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
43	C₂²×Q₈×C₈₉	—	2	1	no	—
44	C₂²×D₄×C₈₉	—	2	2	no	—
45	C₂²×D₄×C₈₉	—	2	1	no	—
46	C₂²×D₄×C₈₉	—	4	2	no	—
47	C₂²×D₄×C₈₉	—	4	1	no	—
48	AGL(1,ℤ/8)×C₈₉	—	2	8	no	—
49	AGL(1,ℤ/8)×C₈₉	—	2	16	no	—
50	#135×C₈₉	—	2	4	no	—
51	#135×C₈₉	—	2	4	no	—
52	#135×C₈₉	—	2	2	no	—
53	#135×C₈₉	D₄	1	4	yes	—
54	C₂×Q₁₆×C₈₉	—	2	2	no	—
55	C₂×Q₁₆×C₈₉	—	2	2	no	—
56	C₂×Q₁₆×C₈₉	—	2	1	no	—
57	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
58	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
59	C₂×SD₁₆×C₈₉	—	2	2	no	—
60	C₂×SD₁₆×C₈₉	—	2	1	no	—
61	C₂×D₈×C₈₉	—	2	4	no	—
62	C₂×D₈×C₈₉	—	2	4	no	—
63	C₂×D₈×C₈₉	—	2	2	no	—
64	C₂×D₈×C₈₉	—	2	1	no	—
65	C₂×D₈×C₈₉	—	2	2	no	—
66	C₂×D₈×C₈₉	D₄	1	4	yes	—
67	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
68	C₂×D₈×C₈₉	C₂³	1	2	yes	—
69	#131×C₈₉	—	2	2	no	—
70	#131×C₈₉	—	2	2	no	—
71	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
72	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
73	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
74	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₄⋊Q₈×C₈₉	—	4	1	no	—
76	#124×C₈₉	—	4	1	no	—
77	#123×C₈₉	—	4	2	no	—
78	C₂²⋊Q₈×C₈₉	—	2	4	no	—
79	#117×C₈₉	—	2	1	no	—
80	Q₃₂×C₈₉	—	2	4	no	—
81	SD₃₂×C₈₉	—	2	8	no	—
82	SD₃₂×C₈₉	—	2	4	no	—
83	M₃₂×C₈₉	—	2	2	no	—
84	C₂×C₁₆×C₈₉	—	2	2	no	—
85	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
86	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
87	#107×C₈₉	C₂³	1	2	yes	—
88	#106×C₈₉	—	2	1	no	—
89	#103×C₈₉	—	2	4	no	—
90	C₂⁴×C₈₉	—	4	2	no	—
91	C₂⁴×C₈₉	—	4	1	no	—
92	#41×C₈₉	—	2	2	no	—
93	#41×C₈₉	—	2	2	no	—
94	#41×C₈₉	—	2	8	no	—
95	#41×C₈₉	—	2	4	no	—
96	#41×C₈₉	—	4	4	no	—
97	#41×C₈₉	—	4	8	no	—
98	#41×C₈₉	—	4	2	no	—
99	C₂×Q₈×C₈₉	—	2	1	no	—
100	C₂×Q₈×C₈₉	—	2	2	no	—
101	C₂×Q₈×C₈₉	—	4	1	no	—
102	C₂×Q₈×C₈₉	—	4	2	no	—
103	C₂×Q₈×C₈₉	—	4	2	no	—
104	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
105	C₂×D₄×C₈₉	—	2	8	no	—
106	C₂×D₄×C₈₉	—	2	4	no	—
107	C₂×D₄×C₈₉	—	2	1	no	—
108	C₂×D₄×C₈₉	—	2	2	no	—
109	C₂×D₄×C₈₉	—	4	4	no	—
110	C₂×D₄×C₈₉	—	4	16	no	—
111	C₂×D₄×C₈₉	—	4	8	no	—
112	C₂×D₄×C₈₉	—	4	2	no	—
113	C₂×D₄×C₈₉	—	4	1	no	—
114	C₂×D₄×C₈₉	—	4	2	no	—
115	C₂×D₄×C₈₉	—	4	1	no	—
116	C₂×D₄×C₈₉	—	4	2	no	—
117	C₂×D₄×C₈₉	D₈	1	2	yes	—
118	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
119	C₂²×C₄×C₈₉	—	4	2	no	—
120	C₂²×C₄×C₈₉	—	4	1	no	—
121	C₂²×C₄×C₈₉	—	4	1	no	—
122	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
123	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
124	Q₁₆×C₈₉	—	2	1	no	—
125	Q₁₆×C₈₉	—	2	2	no	—
126	Q₁₆×C₈₉	—	2	2	no	—
127	Q₁₆×C₈₉	—	4	2	no	—
128	Q₁₆×C₈₉	C₂×D₄	1	2	yes	—
129	SD₁₆×C₈₉	—	2	8	no	—
130	SD₁₆×C₈₉	—	2	4	no	—
131	D₈×C₈₉	—	2	8	no	—
132	D₈×C₈₉	—	2	4	no	—
133	D₈×C₈₉	—	2	2	no	—
134	D₈×C₈₉	—	2	1	no	—
135	D₈×C₈₉	C₂×D₄	1	2	yes	—
136	D₈×C₈₉	C₂×D₄	1	4	yes	—
137	M₁₆×C₈₉	—	2	2	no	—
138	M₁₆×C₈₉	—	2	4	no	—
139	C₂×C₈×C₈₉	—	2	2	no	—
140	C₂×C₈×C₈₉	—	2	2	no	—
141	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
142	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
144	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
145	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
146	C₄⋊C₄×C₈₉	—	2	2	no	—
147	C₄⋊C₄×C₈₉	—	4	4	no	—
148	C₂²⋊C₄×C₈₉	—	4	1	no	—
149	C₂²⋊C₄×C₈₉	—	4	4	no	—
150	C₄²×C₈₉	—	4	1	no	—
151	C₁₆×C₈₉	—	2	2	no	—
152	C₁₆×C₈₉	—	2	2	no	—
153	C₂³×C₈₉	—	4	8	no	—
154	C₂³×C₈₉	—	4	2	no	—
155	C₂³×C₈₉	—	4	1	no	—
156	C₂³×C₈₉	—	4	4	no	—
157	C₂³×C₈₉	—	16	2	no	—
158	C₂³×C₈₉	—	16	4	no	—
159	C₂³×C₈₉	C₂×D₈	1	2	yes	—
160	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
161	Q₈×C₈₉	—	2	2	no	—
162	Q₈×C₈₉	—	2	2	no	—
163	Q₈×C₈₉	—	4	1	no	—
164	Q₈×C₈₉	—	4	2	no	—
165	Q₈×C₈₉	—	4	2	no	—
166	Q₈×C₈₉	—	8	2	no	—
167	D₄×C₈₉	—	2	4	no	—
168	D₄×C₈₉	—	2	2	no	—
169	D₄×C₈₉	—	2	4	no	—
170	D₄×C₈₉	—	2	2	no	—
171	D₄×C₈₉	—	4	8	no	—
172	D₄×C₈₉	—	4	8	no	—
173	D₄×C₈₉	—	4	4	no	—
174	D₄×C₈₉	—	4	1	no	—
175	D₄×C₈₉	—	4	2	no	—
176	D₄×C₈₉	—	4	4	no	—
177	C₂×C₄×C₈₉	—	2	2	no	—
178	C₂×C₄×C₈₉	—	2	2	no	—
179	C₂×C₄×C₈₉	—	4	8	no	—
180	C₂×C₄×C₈₉	—	4	1	no	—
181	C₂×C₄×C₈₉	—	4	4	no	—
182	C₂×C₄×C₈₉	—	4	2	no	—
183	C₂×C₄×C₈₉	—	4	2	no	—
184	C₂×C₄×C₈₉	—	8	2	no	—
185	C₂×C₄×C₈₉	#120	1	2	yes	—
186	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
187	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
188	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
189	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
190	C₈×C₈₉	—	2	2	no	—
191	C₈×C₈₉	#120	1	2	yes	—
192	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
193	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
194	C₂²×C₈₉	—	2	2	no	—
195	C₂²×C₈₉	—	2	4	no	—
196	C₂²×C₈₉	—	2	2	no	—
197	C₂²×C₈₉	—	4	4	no	—
198	C₂²×C₈₉	—	4	2	no	—
199	C₂²×C₈₉	—	8	4	no	—
200	C₂²×C₈₉	—	8	2	no	—
201	C₂²×C₈₉	—	16	4	no	—
202	C₂²×C₈₉	—	16	8	no	—
203	C₂²×C₈₉	#447	1	2	yes	—
204	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
205	C₄×C₈₉	—	2	2	no	—
206	C₄×C₈₉	—	4	2	no	—
207	C₄×C₈₉	—	8	2	no	—
208	C₄×C₈₉	—	8	2	no	—
209	C₄×C₈₉	#447	1	2	yes	—
210	C₄×C₈₉	C₂×#120	1	1	yes	characteristic
211	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
212	#33523	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
213	C₂×C₈₉	—	2	2	no	—
214	C₂×C₈₉	—	4	2	no	—
215	C₂×C₈₉	—	8	4	no	—
216	C₂×C₈₉	—	8	2	no	—
217	C₂×C₈₉	C₂×#447	1	1	yes	characteristic
218	C₂×C₈₉	#3416	1	2	yes	—
219	C₂×#576	C₂×C₈₉	1	1	yes	characteristic
220	C₂×#510	C₂×C₈₉	1	1	yes	characteristic, complemented
221	C₂²×SD₃₂	C₂×C₈₉	1	1	yes	characteristic, complemented
222	#3059	C₂×C₈₉	1	1	yes	characteristic, complemented
223	#2189	C₂×C₈₉	1	4	yes	complemented
224	#2187	C₂×C₈₉	1	4	yes	complemented
225	#2140	C₂×C₈₉	1	1	yes	characteristic, complemented
226	#2138	C₂×C₈₉	1	1	yes	characteristic, complemented
227	#2113	C₂×C₈₉	1	1	yes	characteristic, complemented
228	C₈₉	#33523	1	1	yes	characteristic, direct product factor, Sylow
229	C₂×H(5,𝔽₂)	—	2	1	no	—
230	#576	—	2	4	no	—
231	#576	—	2	4	no	—
232	C₂×AGL(1,ℤ/8)	—	2	1	no	—
233	C₂×AGL(1,ℤ/8)	—	2	2	no	—
234	C₂×#135	C₂²×C₈₉	1	2	yes	—
235	C₂×#135	C₂²×C₈₉	1	1	yes	characteristic
236	C₂²×Q₁₆	C₂²×C₈₉	1	1	yes	characteristic, complemented
237	C₂²×D₈	C₂²×C₈₉	1	2	yes	—
238	C₂²×D₈	C₂²×C₈₉	1	1	yes	characteristic
239	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
240	#536	—	2	1	no	—
241	#510	—	2	4	no	—
242	C₂×Q₃₂	C₂²×C₈₉	1	2	yes	complemented
243	C₂×SD₃₂	—	2	4	no	—
244	C₂×SD₃₂	C₂²×C₈₉	1	4	yes	complemented
245	C₂×SD₃₂	C₂²×C₈₉	1	2	yes	complemented
246	C₂×M₃₂	C₂²×C₈₉	1	1	yes	characteristic, complemented
247	C₂²×C₁₆	C₂²×C₈₉	1	1	yes	characteristic, complemented
248	#470	—	2	2	no	—
249	#467	C₂²×C₈₉	1	4	yes	complemented
250	#427	C₂²×C₈₉	1	1	yes	characteristic, complemented
251	#419	—	2	1	no	—
252	#358	C₂²×C₈₉	1	2	yes	complemented
253	#357	C₂²×C₈₉	1	2	yes	complemented
254	#357	C₂²×C₈₉	1	4	yes	complemented
255	C₂²⋊C₁₆	C₂²×C₈₉	1	4	yes	complemented
256	H(5,𝔽₂)	—	2	8	no	—
257	H(5,𝔽₂)	—	4	4	no	—
258	C₂×#41	—	2	2	no	—
259	C₂×#41	—	2	1	no	—
260	C₂×#41	—	4	1	no	—
261	C₂×#41	D₄×C₈₉	1	1	yes	characteristic
262	C₂²×Q₈	—	2	1	no	—
263	C₂²×D₄	—	2	2	no	—
264	C₂²×D₄	—	2	1	no	—
265	C₂²×D₄	—	4	2	no	—
266	C₂²×D₄	—	4	1	no	—
267	AGL(1,ℤ/8)	—	2	8	no	—
268	AGL(1,ℤ/8)	—	2	16	no	—
269	#135	—	2	4	no	—
270	#135	—	2	4	no	—
271	#135	—	2	2	no	—
272	#135	D₄×C₈₉	1	4	yes	—
273	C₂×Q₁₆	—	2	2	no	—
274	C₂×Q₁₆	—	2	2	no	—
275	C₂×Q₁₆	—	2	1	no	—
276	C₂×Q₁₆	C₂³×C₈₉	1	2	yes	—
277	C₂×Q₁₆	C₂³×C₈₉	1	1	yes	characteristic
278	C₂×SD₁₆	—	2	2	no	—
279	C₂×SD₁₆	—	2	1	no	—
280	C₂×D₈	—	2	4	no	—
281	C₂×D₈	—	2	4	no	—
282	C₂×D₈	—	2	2	no	—
283	C₂×D₈	—	2	1	no	—
284	C₂×D₈	—	2	2	no	—
285	C₂×D₈	D₄×C₈₉	1	4	yes	—
286	C₂×D₈	C₂³×C₈₉	1	1	yes	characteristic
287	C₂×D₈	C₂³×C₈₉	1	2	yes	—
288	#131	—	2	2	no	—
289	#131	—	2	2	no	—
290	C₂×M₁₆	D₄×C₈₉	1	1	yes	characteristic
291	C₂×M₁₆	D₄×C₈₉	1	2	yes	—
292	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
293	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
294	C₄⋊Q₈	—	4	1	no	—
295	#124	—	4	1	no	—
296	#123	—	4	2	no	—
297	C₂²⋊Q₈	—	2	4	no	—
298	#117	—	2	1	no	—
299	Q₃₂	—	2	4	no	—
300	SD₃₂	—	2	8	no	—
301	SD₃₂	—	2	4	no	—
302	M₃₂	—	2	2	no	—
303	C₂×C₁₆	—	2	2	no	—
304	C₂×C₁₆	C₂³×C₈₉	1	2	yes	—
305	C₂×C₁₆	C₂³×C₈₉	1	2	yes	—
306	#107	C₂³×C₈₉	1	2	yes	—
307	#106	—	2	1	no	—
308	#103	—	2	4	no	—
309	C₂⁴	—	4	2	no	—
310	C₂⁴	—	4	1	no	—
311	#41	—	2	2	no	—
312	#41	—	2	2	no	—
313	#41	—	2	8	no	—
314	#41	—	2	4	no	—
315	#41	—	4	4	no	—
316	#41	—	4	8	no	—
317	#41	—	4	2	no	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	—	2	2	no	—
320	C₂×Q₈	—	4	1	no	—
321	C₂×Q₈	—	4	2	no	—
322	C₂×Q₈	—	4	2	no	—
323	C₂×Q₈	D₈×C₈₉	1	1	yes	characteristic
324	C₂×D₄	—	2	8	no	—
325	C₂×D₄	—	2	4	no	—
326	C₂×D₄	—	2	1	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	4	4	no	—
329	C₂×D₄	—	4	16	no	—
330	C₂×D₄	—	4	8	no	—
331	C₂×D₄	—	4	2	no	—
332	C₂×D₄	—	4	1	no	—
333	C₂×D₄	—	4	2	no	—
334	C₂×D₄	—	4	1	no	—
335	C₂×D₄	—	4	2	no	—
336	C₂×D₄	D₈×C₈₉	1	2	yes	—
337	C₂×D₄	D₈×C₈₉	1	1	yes	characteristic
338	C₂²×C₄	—	4	2	no	—
339	C₂²×C₄	—	4	1	no	—
340	C₂²×C₄	—	4	1	no	—
341	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
342	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
343	Q₁₆	—	2	1	no	—
344	Q₁₆	—	2	2	no	—
345	Q₁₆	—	2	2	no	—
346	Q₁₆	—	4	2	no	—
347	Q₁₆	C₂×D₄×C₈₉	1	2	yes	—
348	SD₁₆	—	2	8	no	—
349	SD₁₆	—	2	4	no	—
350	D₈	—	2	8	no	—
351	D₈	—	2	4	no	—
352	D₈	—	2	2	no	—
353	D₈	—	2	1	no	—
354	D₈	C₂×D₄×C₈₉	1	2	yes	—
355	D₈	C₂×D₄×C₈₉	1	4	yes	—
356	M₁₆	—	2	2	no	—
357	M₁₆	—	2	4	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
361	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
362	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
363	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
364	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	4	4	no	—
367	C₂²⋊C₄	—	4	1	no	—
368	C₂²⋊C₄	—	4	4	no	—
369	C₄²	—	4	1	no	—
370	C₁₆	—	2	2	no	—
371	C₁₆	—	2	2	no	—
372	C₂³	—	4	8	no	—
373	C₂³	—	4	2	no	—
374	C₂³	—	4	1	no	—
375	C₂³	—	4	4	no	—
376	C₂³	—	16	2	no	—
377	C₂³	—	16	4	no	—
378	C₂³	C₂×D₈×C₈₉	1	2	yes	—
379	C₂³	C₂×D₈×C₈₉	1	1	yes	characteristic
380	Q₈	—	2	2	no	—
381	Q₈	—	2	2	no	—
382	Q₈	—	4	1	no	—
383	Q₈	—	4	2	no	—
384	Q₈	—	4	2	no	—
385	Q₈	—	8	2	no	—
386	D₄	—	2	4	no	—
387	D₄	—	2	2	no	—
388	D₄	—	2	2	no	—
389	D₄	—	2	4	no	—
390	D₄	—	4	8	no	—
391	D₄	—	4	8	no	—
392	D₄	—	4	4	no	—
393	D₄	—	4	4	no	—
394	D₄	—	4	1	no	—
395	D₄	—	4	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	4	8	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	4	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	8	2	no	—
404	C₂×C₄	#120×C₈₉	1	2	yes	—
405	C₂×C₄	C₂×D₈×C₈₉	1	2	yes	—
406	C₂×C₄	C₂×D₈×C₈₉	1	1	yes	characteristic
407	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
408	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
409	C₈	—	2	2	no	—
410	C₈	#120×C₈₉	1	2	yes	—
411	C₈	C₂²×D₄×C₈₉	1	1	yes	characteristic
412	C₈	C₂²×D₄×C₈₉	1	1	yes	characteristic
413	C₂²	—	2	2	no	—
414	C₂²	—	2	4	no	—
415	C₂²	—	2	2	no	—
416	C₂²	—	4	4	no	—
417	C₂²	—	4	2	no	—
418	C₂²	—	8	4	no	—
419	C₂²	—	8	2	no	—
420	C₂²	—	16	4	no	—
421	C₂²	—	16	8	no	—
422	C₂²	#447×C₈₉	1	2	yes	—
423	C₂²	C₂²×D₈×C₈₉	1	1	yes	characteristic
424	C₄	—	2	2	no	—
425	C₄	—	4	2	no	—
426	C₄	—	8	2	no	—
427	C₄	—	8	2	no	—
428	C₄	#447×C₈₉	1	2	yes	—
429	C₄	C₂×#120×C₈₉	1	1	yes	characteristic
430	C₄	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
431	C₂	—	2	2	no	—
432	C₂	—	4	2	no	—
433	C₂	—	8	4	no	—
434	C₂	—	8	2	no	—
435	C₂	C₂×#447×C₈₉	1	1	yes	characteristic
436	C₂	C₈₉×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₈₉
prime radical	C₂×#576×C₈₉	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×#41×C₈₉ ◁ C₂×#131×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#510×C₈₉	C₂	semidirect
3. C₂²×SD₃₂×C₈₉	C₂	semidirect
4. C₈₉×#3059	C₂	semidirect
5. C₈₉×#2189	C₂	semidirect
6. C₈₉×#2187	C₂	semidirect
7. C₈₉×#2140	C₂	semidirect
8. C₈₉×#2138	C₂	semidirect
9. C₈₉×#2113	C₂	semidirect
17. C₂²×Q₁₆×C₈₉	C₂²	semidirect
23. C₂×Q₃₂×C₈₉	C₂²	semidirect
25. C₂×SD₃₂×C₈₉	C₂²	semidirect
26. C₂×SD₃₂×C₈₉	C₂²	semidirect
27. C₂×M₃₂×C₈₉	C₂²	semidirect
28. C₂²×C₁₆×C₈₉	C₂²	semidirect
30. #467×C₈₉	C₂²	semidirect
31. #427×C₈₉	C₂²	semidirect
33. #358×C₈₉	C₂²	semidirect
34. #357×C₈₉	C₂²	semidirect
35. #357×C₈₉	C₂²	semidirect
36. C₂²⋊C₁₆×C₈₉	C₂²	semidirect
212. #33523	C₈₉	direct (N × H)
220. C₂×#510	C₂×C₈₉	semidirect
221. C₂²×SD₃₂	C₂×C₈₉	semidirect
222. #3059	C₂×C₈₉	semidirect
223. #2189	C₂×C₈₉	semidirect
224. #2187	C₂×C₈₉	semidirect
225. #2140	C₂×C₈₉	semidirect
226. #2138	C₂×C₈₉	semidirect
227. #2113	C₂×C₈₉	semidirect
228. C₈₉	#33523	direct (N × H)
236. C₂²×Q₁₆	C₂²×C₈₉	semidirect
242. C₂×Q₃₂	C₂²×C₈₉	semidirect
244. C₂×SD₃₂	C₂²×C₈₉	semidirect
245. C₂×SD₃₂	C₂²×C₈₉	semidirect
246. C₂×M₃₂	C₂²×C₈₉	semidirect
247. C₂²×C₁₆	C₂²×C₈₉	semidirect
249. #467	C₂²×C₈₉	semidirect
250. #427	C₂²×C₈₉	semidirect
252. #358	C₂²×C₈₉	semidirect
253. #357	C₂²×C₈₉	semidirect
254. #357	C₂²×C₈₉	semidirect
255. C₂²⋊C₁₆	C₂²×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#576×C₈₉	2
C₂×#510×C₈₉	2
C₂²×SD₃₂×C₈₉	2
C₈₉×#3059	2
C₈₉×#2189	2
C₈₉×#2187	2
C₈₉×#2140	2
C₈₉×#2138	2
C₈₉×#2113	2
#33523	89

Subgroups — C89×#33523 #36411371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#33523 #36411371