Subgroups — C₈₉×#9254 #36387102

Order 22784.

534 conjugacy classes of subgroups.

The subgroup lattice has 414 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	Q₈×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#1965	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1545	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1487	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#1292	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1283	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×SD₁₆×C₈₉	C₄	1	2	yes	—
9	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
11	#460×C₈₉	—	2	1	no	—
12	#460×C₈₉	—	2	1	no	—
13	#460×C₈₉	—	2	1	no	—
14	#460×C₈₉	—	2	1	no	—
15	#446×C₈₉	—	2	1	no	—
16	#446×C₈₉	—	2	2	no	—
17	C₈×Q₈×C₈₉	—	2	1	no	—
18	#431×C₈₉	C₂²	1	1	yes	characteristic
19	#423×C₈₉	—	2	1	no	—
20	#423×C₈₉	C₂²	1	1	yes	characteristic
21	#423×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#102×C₈₉	C₄	1	2	yes	—
24	C₂×#100×C₈₉	—	2	1	no	—
25	C₂×#100×C₈₉	—	2	1	no	—
26	C₄×M₁₆×C₈₉	—	2	1	no	—
27	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
28	#331×C₈₉	—	2	2	no	—
29	Q₈⋊C₈×C₈₉	—	2	1	no	—
30	Q₈⋊C₈×C₈₉	—	2	1	no	—
31	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×D₄×C₈₉	—	2	1	no	—
33	C₂²×D₄×C₈₉	—	2	1	no	—
34	C₂²×D₄×C₈₉	C₂×C₄	1	2	yes	—
35	C₂×SD₁₆×C₈₉	—	2	2	no	—
36	C₂×SD₁₆×C₈₉	—	2	2	no	—
37	C₂×SD₁₆×C₈₉	—	4	4	no	—
38	C₂×M₁₆×C₈₉	—	2	1	no	—
39	C₂×M₁₆×C₈₉	—	2	1	no	—
40	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
41	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
43	#127×C₈₉	—	2	1	no	—
44	#127×C₈₉	—	2	1	no	—
45	#127×C₈₉	—	2	1	no	—
46	#127×C₈₉	—	2	1	no	—
47	C₄×Q₈×C₈₉	—	2	1	no	—
48	C₄×Q₈×C₈₉	—	2	1	no	—
49	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
50	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
51	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
52	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₈₉	—	2	1	no	—
56	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
57	C₄⋊C₈×C₈₉	—	2	2	no	—
58	C₄⋊C₈×C₈₉	—	2	2	no	—
59	C₄⋊C₈×C₈₉	—	2	1	no	—
60	C₄⋊C₈×C₈₉	—	2	1	no	—
61	C₄⋊C₈×C₈₉	—	2	1	no	—
62	C₄⋊C₈×C₈₉	—	2	1	no	—
63	C₄⋊C₈×C₈₉	—	2	1	no	—
64	C₄⋊C₈×C₈₉	—	2	1	no	—
65	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
66	#102×C₈₉	—	2	2	no	—
67	#102×C₈₉	—	2	2	no	—
68	#100×C₈₉	—	4	2	no	—
69	#100×C₈₉	—	4	2	no	—
70	C₂²⋊C₈×C₈₉	—	2	1	no	—
71	#97×C₈₉	—	2	2	no	—
72	#97×C₈₉	—	2	1	no	—
73	C₄×C₈×C₈₉	—	2	2	no	—
74	C₄×C₈×C₈₉	—	2	1	no	—
75	C₄×C₈×C₈₉	—	2	1	no	—
76	C₂⁴×C₈₉	—	2	2	no	—
77	C₂×Q₈×C₈₉	—	2	1	no	—
78	C₂×Q₈×C₈₉	—	2	2	no	—
79	C₂×Q₈×C₈₉	—	4	1	no	—
80	C₂×Q₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×Q₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×D₄×C₈₉	—	2	2	no	—
83	C₂×D₄×C₈₉	—	2	2	no	—
84	C₂×D₄×C₈₉	—	4	2	no	—
85	C₂×D₄×C₈₉	—	4	2	no	—
86	C₂×D₄×C₈₉	—	4	1	no	—
87	C₂×D₄×C₈₉	—	4	1	no	—
88	C₂×D₄×C₈₉	—	4	1	no	—
89	C₂×D₄×C₈₉	—	4	1	no	—
90	C₂×D₄×C₈₉	—	4	1	no	—
91	C₂×D₄×C₈₉	—	4	1	no	—
92	C₂×D₄×C₈₉	—	4	1	no	—
93	C₂×D₄×C₈₉	—	4	1	no	—
94	C₂×D₄×C₈₉	—	8	2	no	—
95	C₂²×C₄×C₈₉	—	2	1	no	—
96	C₂²×C₄×C₈₉	—	2	1	no	—
97	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
100	SD₁₆×C₈₉	—	4	4	no	—
101	SD₁₆×C₈₉	—	4	4	no	—
102	M₁₆×C₈₉	—	2	1	no	—
103	M₁₆×C₈₉	—	2	1	no	—
104	M₁₆×C₈₉	—	2	1	no	—
105	M₁₆×C₈₉	—	4	1	no	—
106	M₁₆×C₈₉	—	4	1	no	—
107	M₁₆×C₈₉	—	4	1	no	—
108	M₁₆×C₈₉	—	4	1	no	—
109	M₁₆×C₈₉	SD₁₆	1	2	yes	complemented
110	C₂×C₈×C₈₉	—	2	1	no	—
111	C₂×C₈×C₈₉	—	2	2	no	—
112	C₂×C₈×C₈₉	—	2	1	no	—
113	C₂×C₈×C₈₉	—	2	2	no	—
114	C₂×C₈×C₈₉	—	4	2	no	—
115	C₂×C₈×C₈₉	—	4	1	no	—
116	C₂×C₈×C₈₉	—	4	1	no	—
117	C₄⋊C₄×C₈₉	—	2	1	no	—
118	C₄⋊C₄×C₈₉	—	2	1	no	—
119	C₄⋊C₄×C₈₉	—	2	1	no	—
120	C₄⋊C₄×C₈₉	—	2	1	no	—
121	C₄⋊C₄×C₈₉	—	2	1	no	—
122	C₄⋊C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
123	C₄⋊C₄×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
124	C₄²×C₈₉	—	2	1	no	—
125	C₄²×C₈₉	—	2	1	no	—
126	C₄²×C₈₉	—	2	1	no	—
127	C₄²×C₈₉	—	2	1	no	—
128	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
129	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₈₉	#41	1	1	yes	characteristic
132	C₂³×C₈₉	—	4	2	no	—
133	C₂³×C₈₉	—	4	2	no	—
134	C₂³×C₈₉	—	4	2	no	—
135	C₂³×C₈₉	—	8	2	no	—
136	C₂³×C₈₉	—	8	2	no	—
137	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
138	Q₈×C₈₉	—	2	1	no	—
139	Q₈×C₈₉	—	2	1	no	—
140	Q₈×C₈₉	—	2	1	no	—
141	Q₈×C₈₉	—	4	2	no	—
142	Q₈×C₈₉	#100	1	2	yes	complemented
143	D₄×C₈₉	—	4	2	no	—
144	D₄×C₈₉	—	4	2	no	—
145	D₄×C₈₉	—	8	2	no	—
146	D₄×C₈₉	—	8	2	no	—
147	D₄×C₈₉	—	8	2	no	—
148	D₄×C₈₉	—	8	1	no	—
149	D₄×C₈₉	—	8	1	no	—
150	D₄×C₈₉	—	8	1	no	—
151	D₄×C₈₉	—	8	1	no	—
152	C₂×C₄×C₈₉	—	2	1	no	—
153	C₂×C₄×C₈₉	—	2	1	no	—
154	C₂×C₄×C₈₉	—	2	1	no	—
155	C₂×C₄×C₈₉	—	2	1	no	—
156	C₂×C₄×C₈₉	—	2	1	no	—
157	C₂×C₄×C₈₉	—	2	1	no	—
158	C₂×C₄×C₈₉	—	4	1	no	—
159	C₂×C₄×C₈₉	—	4	2	no	—
160	C₂×C₄×C₈₉	—	8	1	no	—
161	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
163	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
164	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
165	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
166	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
167	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
168	C₂×C₄×C₈₉	#135	1	1	yes	characteristic
169	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
170	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
171	C₈×C₈₉	—	2	2	no	—
172	C₈×C₈₉	—	4	2	no	—
173	C₈×C₈₉	—	4	1	no	—
174	C₈×C₈₉	—	4	2	no	—
175	C₈×C₈₉	—	8	1	no	—
176	C₈×C₈₉	—	8	1	no	—
177	C₂²×C₈₉	—	2	1	no	—
178	C₂²×C₈₉	—	2	1	no	—
179	C₂²×C₈₉	—	8	2	no	—
180	C₂²×C₈₉	—	8	2	no	—
181	C₂²×C₈₉	—	8	2	no	—
182	C₂²×C₈₉	—	16	2	no	—
183	C₂²×C₈₉	—	16	2	no	—
184	C₂²×C₈₉	#386	1	1	yes	characteristic
185	C₂²×C₈₉	C₄×SD₁₆	1	1	yes	characteristic
186	C₂²×C₈₉	#440	1	1	yes	characteristic
187	C₄×C₈₉	—	2	1	no	—
188	C₄×C₈₉	—	2	2	no	—
189	C₄×C₈₉	—	2	1	no	—
190	C₄×C₈₉	—	2	1	no	—
191	C₄×C₈₉	—	4	1	no	—
192	C₄×C₈₉	—	4	1	no	—
193	C₄×C₈₉	—	4	1	no	—
194	C₄×C₈₉	C₂×#100	1	1	yes	characteristic
195	C₄×C₈₉	#413	1	1	yes	characteristic
196	C₄×C₈₉	#448	1	1	yes	characteristic
197	C₄×C₈₉	#449	1	1	yes	characteristic
198	C₄×C₈₉	#460	1	1	yes	characteristic
199	C₄×C₈₉	#464	1	1	yes	characteristic
200	#9254	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂×C₈₉	—	2	1	no	—
202	C₂×C₈₉	—	2	1	no	—
203	C₂×C₈₉	—	16	2	no	—
204	C₂×C₈₉	#1880	1	1	yes	characteristic
205	C₂×C₈₉	#1900	1	1	yes	characteristic
206	C₂×C₈₉	#1903	1	1	yes	characteristic
207	C₂×#460	C₂×C₈₉	1	1	yes	characteristic
208	Q₈×M₁₆	C₂×C₈₉	1	1	yes	characteristic, complemented
209	#1965	C₂×C₈₉	1	1	yes	characteristic
210	#1545	C₂×C₈₉	1	1	yes	characteristic
211	#1487	C₂×C₈₉	1	1	yes	characteristic, complemented
212	#1292	C₂×C₈₉	1	1	yes	characteristic, complemented
213	#1283	C₂×C₈₉	1	1	yes	characteristic, complemented
214	C₈₉	#9254	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²×SD₁₆	C₄×C₈₉	1	2	yes	—
216	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
217	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
218	#460	—	2	1	no	—
219	#460	—	2	1	no	—
220	#460	—	2	1	no	—
221	#460	—	2	1	no	—
222	#446	—	2	1	no	—
223	#446	—	2	2	no	—
224	C₈×Q₈	—	2	1	no	—
225	#431	C₂²×C₈₉	1	1	yes	characteristic
226	#423	—	2	1	no	—
227	#423	C₂²×C₈₉	1	1	yes	characteristic
228	#423	C₂²×C₈₉	1	1	yes	characteristic
229	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
230	C₂×#102	C₄×C₈₉	1	2	yes	—
231	C₂×#100	—	2	1	no	—
232	C₂×#100	—	2	1	no	—
233	C₄×M₁₆	—	2	1	no	—
234	C₄×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
235	#331	—	2	2	no	—
236	Q₈⋊C₈	—	2	1	no	—
237	Q₈⋊C₈	—	2	1	no	—
238	C₂²×Q₈	C₂×C₄×C₈₉	1	1	yes	characteristic
239	C₂²×D₄	—	2	1	no	—
240	C₂²×D₄	—	2	1	no	—
241	C₂²×D₄	C₂×C₄×C₈₉	1	2	yes	—
242	C₂×SD₁₆	—	2	2	no	—
243	C₂×SD₁₆	—	2	2	no	—
244	C₂×SD₁₆	—	4	4	no	—
245	C₂×M₁₆	—	2	1	no	—
246	C₂×M₁₆	—	2	1	no	—
247	C₂×M₁₆	D₄×C₈₉	1	1	yes	characteristic
248	C₂×M₁₆	D₄×C₈₉	1	1	yes	characteristic
249	C₂²×C₈	C₂×C₄×C₈₉	1	2	yes	—
250	#127	—	2	1	no	—
251	#127	—	2	1	no	—
252	#127	—	2	1	no	—
253	#127	—	2	1	no	—
254	C₄×Q₈	—	2	1	no	—
255	C₄×Q₈	—	2	1	no	—
256	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
257	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
258	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
259	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
260	C₂×C₄⋊C₄	—	2	1	no	—
261	C₂×C₄⋊C₄	C₂×C₄×C₈₉	1	1	yes	characteristic
262	C₂×C₄²	—	2	1	no	—
263	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
264	C₄⋊C₈	—	2	2	no	—
265	C₄⋊C₈	—	2	2	no	—
266	C₄⋊C₈	—	2	1	no	—
267	C₄⋊C₈	—	2	1	no	—
268	C₄⋊C₈	—	2	1	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	2	1	no	—
271	C₄⋊C₈	—	2	1	no	—
272	C₄⋊C₈	D₄×C₈₉	1	2	yes	—
273	#102	—	2	2	no	—
274	#102	—	2	2	no	—
275	#100	—	4	2	no	—
276	#100	—	4	2	no	—
277	C₂²⋊C₈	—	2	1	no	—
278	#97	—	2	2	no	—
279	#97	—	2	1	no	—
280	C₄×C₈	—	2	2	no	—
281	C₄×C₈	—	2	1	no	—
282	C₄×C₈	—	2	1	no	—
283	C₂⁴	—	2	2	no	—
284	C₂×Q₈	—	2	1	no	—
285	C₂×Q₈	—	2	2	no	—
286	C₂×Q₈	—	4	1	no	—
287	C₂×Q₈	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
288	C₂×Q₈	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
289	C₂×D₄	—	2	2	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	—	4	2	no	—
292	C₂×D₄	—	4	2	no	—
293	C₂×D₄	—	4	1	no	—
294	C₂×D₄	—	4	1	no	—
295	C₂×D₄	—	4	1	no	—
296	C₂×D₄	—	4	1	no	—
297	C₂×D₄	—	4	1	no	—
298	C₂×D₄	—	4	1	no	—
299	C₂×D₄	—	4	1	no	—
300	C₂×D₄	—	4	1	no	—
301	C₂×D₄	—	8	2	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
305	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
306	C₂²×C₄	#41×C₈₉	1	1	yes	characteristic
307	SD₁₆	—	4	4	no	—
308	SD₁₆	—	4	4	no	—
309	M₁₆	—	2	1	no	—
310	M₁₆	—	2	1	no	—
311	M₁₆	—	2	1	no	—
312	M₁₆	—	4	1	no	—
313	M₁₆	—	4	1	no	—
314	M₁₆	—	4	1	no	—
315	M₁₆	—	4	1	no	—
316	M₁₆	SD₁₆×C₈₉	1	2	yes	complemented
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	4	2	no	—
322	C₂×C₈	—	4	1	no	—
323	C₂×C₈	—	4	1	no	—
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	2	1	no	—
327	C₄⋊C₄	—	2	1	no	—
328	C₄⋊C₄	—	2	1	no	—
329	C₄⋊C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
330	C₄⋊C₄	C₂²⋊C₄×C₈₉	1	1	yes	characteristic
331	C₄²	—	2	1	no	—
332	C₄²	—	2	1	no	—
333	C₄²	—	2	1	no	—
334	C₄²	—	2	1	no	—
335	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
336	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
337	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
338	C₄²	#41×C₈₉	1	1	yes	characteristic
339	C₂³	—	4	2	no	—
340	C₂³	—	4	2	no	—
341	C₂³	—	4	2	no	—
342	C₂³	—	8	2	no	—
343	C₂³	—	8	2	no	—
344	C₂³	C₄×D₄×C₈₉	1	1	yes	characteristic
345	Q₈	—	2	1	no	—
346	Q₈	—	2	1	no	—
347	Q₈	—	2	1	no	—
348	Q₈	—	4	2	no	—
349	Q₈	#100×C₈₉	1	2	yes	complemented
350	D₄	—	4	2	no	—
351	D₄	—	4	2	no	—
352	D₄	—	8	2	no	—
353	D₄	—	8	2	no	—
354	D₄	—	8	2	no	—
355	D₄	—	8	1	no	—
356	D₄	—	8	1	no	—
357	D₄	—	8	1	no	—
358	D₄	—	8	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	4	1	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	8	1	no	—
368	C₂×C₄	C₂×C₂²⋊C₄×C₈₉	1	1	yes	characteristic
369	C₂×C₄	C₄×D₄×C₈₉	1	1	yes	characteristic
370	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
371	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
372	C₂×C₄	#121×C₈₉	1	1	yes	characteristic
373	C₂×C₄	#123×C₈₉	1	1	yes	characteristic
374	C₂×C₄	C₂×SD₁₆×C₈₉	1	1	yes	characteristic
375	C₂×C₄	#135×C₈₉	1	1	yes	characteristic
376	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
377	C₂×C₄	#137×C₈₉	1	1	yes	characteristic
378	C₈	—	2	2	no	—
379	C₈	—	4	2	no	—
380	C₈	—	4	1	no	—
381	C₈	—	4	2	no	—
382	C₈	—	8	1	no	—
383	C₈	—	8	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	1	no	—
386	C₂²	—	8	2	no	—
387	C₂²	—	8	2	no	—
388	C₂²	—	8	2	no	—
389	C₂²	—	16	2	no	—
390	C₂²	—	16	2	no	—
391	C₂²	#386×C₈₉	1	1	yes	characteristic
392	C₂²	C₄×SD₁₆×C₈₉	1	1	yes	characteristic
393	C₂²	#440×C₈₉	1	1	yes	characteristic
394	C₄	—	2	1	no	—
395	C₄	—	2	2	no	—
396	C₄	—	2	1	no	—
397	C₄	—	2	1	no	—
398	C₄	—	4	1	no	—
399	C₄	—	4	1	no	—
400	C₄	—	4	1	no	—
401	C₄	C₂×#100×C₈₉	1	1	yes	characteristic
402	C₄	#413×C₈₉	1	1	yes	characteristic
403	C₄	#448×C₈₉	1	1	yes	characteristic
404	C₄	#449×C₈₉	1	1	yes	characteristic
405	C₄	#460×C₈₉	1	1	yes	characteristic
406	C₄	#464×C₈₉	1	1	yes	characteristic
407	C₂	—	2	1	no	—
408	C₂	—	2	1	no	—
409	C₂	—	16	2	no	—
410	C₂	C₈₉×#1880	1	1	yes	characteristic
411	C₂	C₈₉×#1900	1	1	yes	characteristic
412	C₂	C₈₉×#1903	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	#386
commutator	C₂²×C₄	C₂²×C₄×C₈₉
Frattini, cosocle	C₂×C₄²	C₂³×C₈₉
prime radical	C₂×#127×C₈₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. Q₈×M₁₆×C₈₉	C₂	semidirect
5. C₈₉×#1487	C₂	semidirect
6. C₈₉×#1292	C₂	semidirect
7. C₈₉×#1283	C₂	semidirect
109. M₁₆×C₈₉	SD₁₆	semidirect
142. Q₈×C₈₉	#100	semidirect
200. #9254	C₈₉	direct (N × H)
208. Q₈×M₁₆	C₂×C₈₉	semidirect
211. #1487	C₂×C₈₉	semidirect
212. #1292	C₂×C₈₉	semidirect
213. #1283	C₂×C₈₉	semidirect
214. C₈₉	#9254	direct (N × H)
316. M₁₆	SD₁₆×C₈₉	semidirect
349. Q₈	#100×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
Q₈×M₁₆×C₈₉	2
C₈₉×#1965	2
C₈₉×#1545	2
C₈₉×#1487	2
C₈₉×#1292	2
C₈₉×#1283	2
#9254	89

Subgroups — C89×#9254 #36387102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#9254 #36387102