Subgroups — C₈₉×#33469 #36411317

Order 22784.

646 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3414	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2957	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2952	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×AGL(1,ℤ/16)	C₂	1	4	yes	maximal, complemented
5	C₈₉×#2180	C₂	1	4	yes	maximal, complemented
6	C₂×#365×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2148	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#360×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2110	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
11	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
12	#568×C₈₉	—	2	1	no	—
13	#520×C₈₉	—	2	1	no	—
14	#510×C₈₉	C₄	1	2	yes	complemented
15	#510×C₈₉	C₂²	1	2	yes	—
16	#509×C₈₉	C₄	1	2	yes	complemented
17	#509×C₈₉	C₂²	1	2	yes	—
18	#508×C₈₉	C₄	1	4	yes	complemented
19	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
21	#444×C₈₉	C₂²	1	4	yes	—
22	#443×C₈₉	—	2	2	no	—
23	#442×C₈₉	C₂²	1	2	yes	complemented
24	#441×C₈₉	C₂²	1	2	yes	complemented
25	C₄×SD₁₆×C₈₉	—	2	2	no	—
26	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
28	#421×C₈₉	—	2	1	no	—
29	#417×C₈₉	—	2	1	no	—
30	#405×C₈₉	C₂²	1	1	yes	characteristic
31	#365×C₈₉	C₂²	1	4	yes	complemented
32	#362×C₈₉	C₂²	1	2	yes	complemented
33	#361×C₈₉	C₂²	1	2	yes	complemented
34	#360×C₈₉	C₂²	1	4	yes	—
35	#347×C₈₉	C₂²	1	4	yes	complemented
36	C₂×#41×C₈₉	—	2	1	no	—
37	C₂×#41×C₈₉	—	2	1	no	—
38	#135×C₈₉	—	2	1	no	—
39	#135×C₈₉	—	2	1	no	—
40	#135×C₈₉	—	2	1	no	—
41	#135×C₈₉	—	2	1	no	—
42	#135×C₈₉	C₂×C₄	1	2	yes	—
43	#135×C₈₉	C₂×C₄	1	2	yes	—
44	#135×C₈₉	C₂×C₄	1	2	yes	complemented
45	#135×C₈₉	C₂³	1	2	yes	—
46	C₂×Q₁₆×C₈₉	C₂×C₄	1	1	yes	characteristic, complemented
47	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×SD₁₆×C₈₉	—	2	1	no	—
49	C₂×SD₁₆×C₈₉	—	2	1	no	—
50	C₂×D₈×C₈₉	C₂×C₄	1	1	yes	characteristic, complemented
51	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
52	#131×C₈₉	—	2	4	no	—
53	#131×C₈₉	—	4	2	no	—
54	C₂×M₁₆×C₈₉	—	2	1	no	—
55	C₂×M₁₆×C₈₉	—	4	2	no	—
56	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
58	C₄×Q₈×C₈₉	—	2	2	no	—
59	C₄×D₄×C₈₉	—	2	2	no	—
60	C₄×D₄×C₈₉	—	4	2	no	—
61	#117×C₈₉	—	4	1	no	—
62	#117×C₈₉	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
64	C₂×C₄⋊C₄×C₈₉	—	4	1	no	—
65	Q₃₂×C₈₉	C₂×C₄	1	4	yes	complemented
66	SD₃₂×C₈₉	C₂×C₄	1	4	yes	complemented
67	SD₃₂×C₈₉	C₂×C₄	1	4	yes	complemented
68	D₁₆×C₈₉	C₂×C₄	1	4	yes	complemented
69	M₃₂×C₈₉	—	2	1	no	—
70	M₃₂×C₈₉	C₂×C₄	1	2	yes	complemented
71	M₃₂×C₈₉	C₂³	1	2	yes	—
72	M₃₂×C₈₉	C₂³	1	2	yes	—
73	C₂×C₁₆×C₈₉	—	2	1	no	—
74	C₂×C₁₆×C₈₉	C₂×C₄	1	2	yes	complemented
75	#108×C₈₉	—	2	1	no	—
76	#108×C₈₉	C₂³	1	2	yes	—
77	#106×C₈₉	—	2	1	no	—
78	#106×C₈₉	C₂³	1	2	yes	—
79	#104×C₈₉	—	2	4	no	—
80	#103×C₈₉	—	2	2	no	—
81	#102×C₈₉	—	2	2	no	—
82	#97×C₈₉	C₂³	1	2	yes	—
83	C₄×C₈×C₈₉	D₄	1	2	yes	—
84	#41×C₈₉	—	2	2	no	—
85	#41×C₈₉	—	2	2	no	—
86	#41×C₈₉	—	2	2	no	—
87	#41×C₈₉	—	2	2	no	—
88	#41×C₈₉	—	4	1	no	—
89	#41×C₈₉	—	4	1	no	—
90	#41×C₈₉	—	4	1	no	—
91	#41×C₈₉	—	4	1	no	—
92	C₂×Q₈×C₈₉	—	2	1	no	—
93	C₂×Q₈×C₈₉	—	2	1	no	—
94	C₂×D₄×C₈₉	—	2	1	no	—
95	C₂×D₄×C₈₉	—	2	1	no	—
96	C₂×D₄×C₈₉	—	4	1	no	—
97	C₂×D₄×C₈₉	—	4	1	no	—
98	C₂²×C₄×C₈₉	—	4	2	no	—
99	C₂²×C₄×C₈₉	—	4	1	no	—
100	C₂²×C₄×C₈₉	—	4	1	no	—
101	C₂²×C₄×C₈₉	—	4	1	no	—
102	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
103	Q₁₆×C₈₉	—	2	1	no	—
104	Q₁₆×C₈₉	—	2	1	no	—
105	Q₁₆×C₈₉	C₂²×C₄	1	2	yes	—
106	Q₁₆×C₈₉	C₂²×C₄	1	2	yes	—
107	SD₁₆×C₈₉	—	2	2	no	—
108	SD₁₆×C₈₉	—	2	2	no	—
109	SD₁₆×C₈₉	—	2	2	no	—
110	SD₁₆×C₈₉	—	2	2	no	—
111	D₈×C₈₉	—	2	1	no	—
112	D₈×C₈₉	—	2	1	no	—
113	D₈×C₈₉	C₂²×C₄	1	2	yes	—
114	D₈×C₈₉	C₂²×C₄	1	2	yes	—
115	M₁₆×C₈₉	—	2	1	no	—
116	M₁₆×C₈₉	—	2	1	no	—
117	M₁₆×C₈₉	—	2	2	no	—
118	M₁₆×C₈₉	—	4	4	no	—
119	M₁₆×C₈₉	—	4	1	no	—
120	C₂×C₈×C₈₉	—	4	4	no	—
121	C₂×C₈×C₈₉	—	4	1	no	—
122	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
125	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
127	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
128	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
129	C₄⋊C₄×C₈₉	—	2	1	no	—
130	C₄⋊C₄×C₈₉	—	2	2	no	—
131	C₄⋊C₄×C₈₉	—	4	2	no	—
132	C₄⋊C₄×C₈₉	—	4	1	no	—
133	C₂²⋊C₄×C₈₉	—	2	1	no	—
134	C₂²⋊C₄×C₈₉	—	4	2	no	—
135	C₄²×C₈₉	—	4	2	no	—
136	C₄²×C₈₉	C₂×D₄	1	2	yes	—
137	C₁₆×C₈₉	—	2	2	no	—
138	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
139	C₂³×C₈₉	—	4	1	no	—
140	C₂³×C₈₉	—	4	1	no	—
141	C₂³×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
142	Q₈×C₈₉	—	2	2	no	—
143	Q₈×C₈₉	—	2	2	no	—
144	Q₈×C₈₉	—	4	1	no	—
145	Q₈×C₈₉	—	4	1	no	—
146	D₄×C₈₉	—	2	2	no	—
147	D₄×C₈₉	—	2	2	no	—
148	D₄×C₈₉	—	4	2	no	—
149	D₄×C₈₉	—	4	2	no	—
150	D₄×C₈₉	—	4	2	no	—
151	D₄×C₈₉	—	4	2	no	—
152	D₄×C₈₉	—	4	1	no	—
153	D₄×C₈₉	—	4	1	no	—
154	C₂×C₄×C₈₉	—	2	2	no	—
155	C₂×C₄×C₈₉	—	4	1	no	—
156	C₂×C₄×C₈₉	—	4	1	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	—	16	2	no	—
163	C₂×C₄×C₈₉	—	16	1	no	—
164	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
165	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
166	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
170	C₈×C₈₉	—	2	2	no	—
171	C₈×C₈₉	—	4	2	no	—
172	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
173	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
174	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
175	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
176	C₂²×C₈₉	—	4	1	no	—
177	C₂²×C₈₉	—	4	2	no	—
178	C₂²×C₈₉	—	4	2	no	—
179	C₂²×C₈₉	—	8	1	no	—
180	C₂²×C₈₉	—	8	1	no	—
181	C₂²×C₈₉	—	16	1	no	—
182	C₂²×C₈₉	#442	1	1	yes	characteristic
183	C₂²×C₈₉	#442	1	1	yes	characteristic
184	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
185	C₄×C₈₉	—	4	2	no	—
186	C₄×C₈₉	—	4	2	no	—
187	C₄×C₈₉	—	4	2	no	—
188	C₄×C₈₉	—	8	2	no	—
189	C₄×C₈₉	#442	1	1	yes	characteristic
190	C₄×C₈₉	#442	1	1	yes	characteristic
191	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
192	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
193	#33469	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂×C₈₉	—	2	1	no	—
195	C₂×C₈₉	—	2	1	no	—
196	C₂×C₈₉	—	2	1	no	—
197	C₂×C₈₉	—	8	2	no	—
198	C₂×C₈₉	—	8	2	no	—
199	C₂×C₈₉	C₂×#442	1	1	yes	characteristic
200	#3414	C₂×C₈₉	1	1	yes	characteristic
201	#2957	C₂×C₈₉	1	1	yes	characteristic, complemented
202	#2952	C₂×C₈₉	1	1	yes	characteristic, complemented
203	AGL(1,ℤ/16)	C₂×C₈₉	1	4	yes	complemented
204	#2180	C₂×C₈₉	1	4	yes	complemented
205	C₂×#365	C₂×C₈₉	1	1	yes	characteristic, complemented
206	#2148	C₂×C₈₉	1	1	yes	characteristic, complemented
207	C₂×#360	C₂×C₈₉	1	1	yes	characteristic, complemented
208	#2110	C₂×C₈₉	1	1	yes	characteristic, complemented
209	C₈₉	#33469	1	1	yes	characteristic, direct product factor, Sylow
210	C₂×#135	C₂²×C₈₉	1	1	yes	characteristic
211	C₂×#135	C₂²×C₈₉	1	1	yes	characteristic
212	#568	—	2	1	no	—
213	#520	—	2	1	no	—
214	#510	C₄×C₈₉	1	2	yes	complemented
215	#510	C₂²×C₈₉	1	2	yes	—
216	#509	C₄×C₈₉	1	2	yes	complemented
217	#509	C₂²×C₈₉	1	2	yes	—
218	#508	C₄×C₈₉	1	4	yes	complemented
219	C₂×M₃₂	C₂²×C₈₉	1	1	yes	characteristic
220	C₂×M₃₂	C₂²×C₈₉	1	1	yes	characteristic
221	#444	C₂²×C₈₉	1	4	yes	—
222	#443	—	2	2	no	—
223	#442	C₂²×C₈₉	1	2	yes	complemented
224	#441	C₂²×C₈₉	1	2	yes	complemented
225	C₄×SD₁₆	—	2	2	no	—
226	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
227	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
228	#421	—	2	1	no	—
229	#417	—	2	1	no	—
230	#405	C₂²×C₈₉	1	1	yes	characteristic
231	#365	C₂²×C₈₉	1	4	yes	complemented
232	#362	C₂²×C₈₉	1	2	yes	complemented
233	#361	C₂²×C₈₉	1	2	yes	complemented
234	#360	C₂²×C₈₉	1	4	yes	—
235	#347	C₂²×C₈₉	1	4	yes	complemented
236	C₂×#41	—	2	1	no	—
237	C₂×#41	—	2	1	no	—
238	#135	—	2	1	no	—
239	#135	—	2	1	no	—
240	#135	—	2	1	no	—
241	#135	—	2	1	no	—
242	#135	C₂×C₄×C₈₉	1	2	yes	—
243	#135	C₂×C₄×C₈₉	1	2	yes	—
244	#135	C₂×C₄×C₈₉	1	2	yes	complemented
245	#135	C₂³×C₈₉	1	2	yes	—
246	C₂×Q₁₆	C₂×C₄×C₈₉	1	1	yes	characteristic, complemented
247	C₂×Q₁₆	C₂³×C₈₉	1	1	yes	characteristic
248	C₂×SD₁₆	—	2	1	no	—
249	C₂×SD₁₆	—	2	1	no	—
250	C₂×D₈	C₂×C₄×C₈₉	1	1	yes	characteristic, complemented
251	C₂×D₈	C₂³×C₈₉	1	1	yes	characteristic
252	#131	—	2	4	no	—
253	#131	—	4	2	no	—
254	C₂×M₁₆	—	2	1	no	—
255	C₂×M₁₆	—	4	2	no	—
256	C₂×M₁₆	D₄×C₈₉	1	1	yes	characteristic
257	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
258	C₄×Q₈	—	2	2	no	—
259	C₄×D₄	—	2	2	no	—
260	C₄×D₄	—	4	2	no	—
261	#117	—	4	1	no	—
262	#117	D₄×C₈₉	1	1	yes	characteristic
263	C₂×C₄⋊C₄	—	2	1	no	—
264	C₂×C₄⋊C₄	—	4	1	no	—
265	Q₃₂	C₂×C₄×C₈₉	1	4	yes	complemented
266	SD₃₂	C₂×C₄×C₈₉	1	4	yes	complemented
267	SD₃₂	C₂×C₄×C₈₉	1	4	yes	complemented
268	D₁₆	C₂×C₄×C₈₉	1	4	yes	complemented
269	M₃₂	—	2	1	no	—
270	M₃₂	C₂×C₄×C₈₉	1	2	yes	complemented
271	M₃₂	C₂³×C₈₉	1	2	yes	—
272	M₃₂	C₂³×C₈₉	1	2	yes	—
273	C₂×C₁₆	—	2	1	no	—
274	C₂×C₁₆	C₂×C₄×C₈₉	1	2	yes	complemented
275	#108	—	2	1	no	—
276	#108	C₂³×C₈₉	1	2	yes	—
277	#106	—	2	1	no	—
278	#106	C₂³×C₈₉	1	2	yes	—
279	#104	—	2	4	no	—
280	#103	—	2	2	no	—
281	#102	—	2	2	no	—
282	#97	C₂³×C₈₉	1	2	yes	—
283	C₄×C₈	D₄×C₈₉	1	2	yes	—
284	#41	—	2	2	no	—
285	#41	—	2	2	no	—
286	#41	—	2	2	no	—
287	#41	—	2	2	no	—
288	#41	—	4	1	no	—
289	#41	—	4	1	no	—
290	#41	—	4	1	no	—
291	#41	—	4	1	no	—
292	C₂×Q₈	—	2	1	no	—
293	C₂×Q₈	—	2	1	no	—
294	C₂×D₄	—	2	1	no	—
295	C₂×D₄	—	2	1	no	—
296	C₂×D₄	—	4	1	no	—
297	C₂×D₄	—	4	1	no	—
298	C₂²×C₄	—	4	2	no	—
299	C₂²×C₄	—	4	1	no	—
300	C₂²×C₄	—	4	1	no	—
301	C₂²×C₄	—	4	1	no	—
302	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
303	Q₁₆	—	2	1	no	—
304	Q₁₆	—	2	1	no	—
305	Q₁₆	C₂²×C₄×C₈₉	1	2	yes	—
306	Q₁₆	C₂²×C₄×C₈₉	1	2	yes	—
307	SD₁₆	—	2	2	no	—
308	SD₁₆	—	2	2	no	—
309	SD₁₆	—	2	2	no	—
310	SD₁₆	—	2	2	no	—
311	D₈	—	2	1	no	—
312	D₈	—	2	1	no	—
313	D₈	C₂²×C₄×C₈₉	1	2	yes	—
314	D₈	C₂²×C₄×C₈₉	1	2	yes	—
315	M₁₆	—	2	1	no	—
316	M₁₆	—	2	1	no	—
317	M₁₆	—	2	2	no	—
318	M₁₆	—	4	4	no	—
319	M₁₆	—	4	1	no	—
320	C₂×C₈	—	4	4	no	—
321	C₂×C₈	—	4	1	no	—
322	C₂×C₈	C₂²×C₄×C₈₉	1	1	yes	characteristic
323	C₂×C₈	C₂²×C₄×C₈₉	1	1	yes	characteristic
324	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
325	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
326	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
327	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
328	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	4	2	no	—
332	C₄⋊C₄	—	4	1	no	—
333	C₂²⋊C₄	—	2	1	no	—
334	C₂²⋊C₄	—	4	2	no	—
335	C₄²	—	4	2	no	—
336	C₄²	C₂×D₄×C₈₉	1	2	yes	—
337	C₁₆	—	2	2	no	—
338	C₁₆	C₂²×C₄×C₈₉	1	4	yes	—
339	C₂³	—	4	1	no	—
340	C₂³	—	4	1	no	—
341	C₂³	AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
342	Q₈	—	2	2	no	—
343	Q₈	—	2	2	no	—
344	Q₈	—	4	1	no	—
345	Q₈	—	4	1	no	—
346	D₄	—	2	2	no	—
347	D₄	—	2	2	no	—
348	D₄	—	4	2	no	—
349	D₄	—	4	2	no	—
350	D₄	—	4	2	no	—
351	D₄	—	4	2	no	—
352	D₄	—	4	1	no	—
353	D₄	—	4	1	no	—
354	C₂×C₄	—	2	2	no	—
355	C₂×C₄	—	4	1	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	4	2	no	—
359	C₂×C₄	—	4	2	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	16	2	no	—
363	C₂×C₄	—	16	1	no	—
364	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
365	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
366	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
367	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
368	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
369	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
370	C₈	—	2	2	no	—
371	C₈	—	4	2	no	—
372	C₈	C₄×D₄×C₈₉	1	1	yes	characteristic
373	C₈	C₄×D₄×C₈₉	1	1	yes	characteristic
374	C₈	C₂³×C₄×C₈₉	1	1	yes	characteristic
375	C₈	C₂×#41×C₈₉	1	1	yes	characteristic
376	C₂²	—	4	1	no	—
377	C₂²	—	4	2	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	8	1	no	—
380	C₂²	—	8	1	no	—
381	C₂²	—	16	1	no	—
382	C₂²	#442×C₈₉	1	1	yes	characteristic
383	C₂²	#442×C₈₉	1	1	yes	characteristic
384	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
385	C₄	—	4	2	no	—
386	C₄	—	4	2	no	—
387	C₄	—	4	2	no	—
388	C₄	—	8	2	no	—
389	C₄	#442×C₈₉	1	1	yes	characteristic
390	C₄	#442×C₈₉	1	1	yes	characteristic
391	C₄	C₂×C₄×D₄×C₈₉	1	1	yes	characteristic
392	C₄	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
393	C₂	—	2	1	no	—
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	8	2	no	—
397	C₂	—	8	2	no	—
398	C₂	C₂×#442×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₈₉	C₂×#442
commutator	C₈	C₂³×C₄×C₈₉
Frattini, cosocle	C₂×C₈	C₂⁴×C₈₉
prime radical	C₈₉×#3414	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ #405×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2957	C₂	semidirect
3. C₈₉×#2952	C₂	semidirect
4. C₈₉×AGL(1,ℤ/16)	C₂	semidirect
5. C₈₉×#2180	C₂	semidirect
6. C₂×#365×C₈₉	C₂	semidirect
7. C₈₉×#2148	C₂	semidirect
8. C₂×#360×C₈₉	C₂	semidirect
9. C₈₉×#2110	C₂	semidirect
14. #510×C₈₉	C₄	semidirect
16. #509×C₈₉	C₄	semidirect
18. #508×C₈₉	C₄	semidirect
23. #442×C₈₉	C₂²	semidirect
24. #441×C₈₉	C₂²	semidirect
31. #365×C₈₉	C₂²	semidirect
32. #362×C₈₉	C₂²	semidirect
33. #361×C₈₉	C₂²	semidirect
35. #347×C₈₉	C₂²	semidirect
44. #135×C₈₉	C₂×C₄	semidirect
46. C₂×Q₁₆×C₈₉	C₂×C₄	semidirect
50. C₂×D₈×C₈₉	C₂×C₄	semidirect
65. Q₃₂×C₈₉	C₂×C₄	semidirect
66. SD₃₂×C₈₉	C₂×C₄	semidirect
67. SD₃₂×C₈₉	C₂×C₄	semidirect
68. D₁₆×C₈₉	C₂×C₄	semidirect
70. M₃₂×C₈₉	C₂×C₄	semidirect
74. C₂×C₁₆×C₈₉	C₂×C₄	semidirect
193. #33469	C₈₉	direct (N × H)
201. #2957	C₂×C₈₉	semidirect
202. #2952	C₂×C₈₉	semidirect
203. AGL(1,ℤ/16)	C₂×C₈₉	semidirect
204. #2180	C₂×C₈₉	semidirect
205. C₂×#365	C₂×C₈₉	semidirect
206. #2148	C₂×C₈₉	semidirect
207. C₂×#360	C₂×C₈₉	semidirect
208. #2110	C₂×C₈₉	semidirect
209. C₈₉	#33469	direct (N × H)
214. #510	C₄×C₈₉	semidirect
216. #509	C₄×C₈₉	semidirect
218. #508	C₄×C₈₉	semidirect
223. #442	C₂²×C₈₉	semidirect
224. #441	C₂²×C₈₉	semidirect
231. #365	C₂²×C₈₉	semidirect
232. #362	C₂²×C₈₉	semidirect
233. #361	C₂²×C₈₉	semidirect
235. #347	C₂²×C₈₉	semidirect
244. #135	C₂×C₄×C₈₉	semidirect
246. C₂×Q₁₆	C₂×C₄×C₈₉	semidirect
250. C₂×D₈	C₂×C₄×C₈₉	semidirect
265. Q₃₂	C₂×C₄×C₈₉	semidirect
266. SD₃₂	C₂×C₄×C₈₉	semidirect
267. SD₃₂	C₂×C₄×C₈₉	semidirect
268. D₁₆	C₂×C₄×C₈₉	semidirect
270. M₃₂	C₂×C₄×C₈₉	semidirect
274. C₂×C₁₆	C₂×C₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3414	2
C₈₉×#2957	2
C₈₉×#2952	2
C₈₉×AGL(1,ℤ/16)	2
C₈₉×#2180	2
C₂×#365×C₈₉	2
C₈₉×#2148	2
C₂×#360×C₈₉	2
C₈₉×#2110	2
#33469	89

Subgroups — C89×#33469 #36411317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#33469 #36411317