Subgroups — C₈₉×#25447 #36403295

Order 22784.

892 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3155	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#3068	C₂	1	2	yes	maximal, complemented
3	C₈₉×#3067	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#472×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2957	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#2048	C₂	1	2	yes	maximal, complemented
7	C₈₉×#2008	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1978	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1953	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1914	C₂	1	2	yes	maximal, complemented
11	C₈₉×#1757	C₂	1	1	yes	maximal, characteristic, complemented
12	#575×C₈₉	C₂²	1	2	yes	—
13	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	2	yes	—
14	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	2	yes	—
15	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
16	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic
17	#568×C₈₉	C₂²	1	2	yes	—
18	C₂×#131×C₈₉	C₂²	1	1	yes	characteristic
19	#535×C₈₉	C₂²	1	2	yes	—
20	#496×C₈₉	—	2	2	no	—
21	#495×C₈₉	—	2	1	no	—
22	#493×C₈₉	—	2	1	no	—
23	#473×C₈₉	—	2	1	no	—
24	#472×C₈₉	—	2	1	no	—
25	#472×C₈₉	—	2	2	no	—
26	#472×C₈₉	—	2	2	no	—
27	#472×C₈₉	—	2	4	no	—
28	#471×C₈₉	—	2	2	no	—
29	#471×C₈₉	—	2	4	no	—
30	#444×C₈₉	—	2	2	no	—
31	#443×C₈₉	—	2	2	no	—
32	#430×C₈₉	C₂²	1	2	yes	—
33	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
35	#421×C₈₉	C₂²	1	2	yes	—
36	#418×C₈₉	C₂²	1	2	yes	—
37	#413×C₈₉	C₂²	1	2	yes	—
38	#413×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×#100×C₈₉	C₂²	1	2	yes	—
40	C₂×#100×C₈₉	C₂²	1	2	yes	—
41	#410×C₈₉	C₂²	1	1	yes	characteristic
42	#405×C₈₉	C₂²	1	1	yes	characteristic
43	#405×C₈₉	C₂²	1	1	yes	characteristic
44	#344×C₈₉	C₂²	1	1	yes	characteristic
45	#344×C₈₉	C₂²	1	1	yes	characteristic
46	#344×C₈₉	C₂²	1	2	yes	—
47	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×#41×C₈₉	C₂³	1	2	yes	—
49	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
50	#137×C₈₉	—	2	2	no	—
51	#137×C₈₉	—	2	2	no	—
52	AGL(1,ℤ/8)×C₈₉	—	2	2	no	—
53	AGL(1,ℤ/8)×C₈₉	—	2	2	no	—
54	AGL(1,ℤ/8)×C₈₉	—	2	4	no	—
55	AGL(1,ℤ/8)×C₈₉	—	2	4	no	—
56	AGL(1,ℤ/8)×C₈₉	—	4	2	no	—
57	AGL(1,ℤ/8)×C₈₉	—	4	2	no	—
58	#135×C₈₉	—	2	1	no	—
59	#135×C₈₉	—	2	1	no	—
60	#135×C₈₉	—	2	1	no	—
61	#135×C₈₉	—	2	1	no	—
62	#135×C₈₉	—	4	2	no	—
63	C₂×Q₁₆×C₈₉	D₄	1	1	yes	characteristic
64	C₂×SD₁₆×C₈₉	—	2	2	no	—
65	C₂×SD₁₆×C₈₉	—	2	2	no	—
66	C₂×SD₁₆×C₈₉	D₄	1	2	yes	—
67	C₂×D₈×C₈₉	—	2	2	no	—
68	C₂×D₈×C₈₉	—	2	2	no	—
69	C₂×D₈×C₈₉	—	2	1	no	—
70	C₂×D₈×C₈₉	—	2	1	no	—
71	C₂×D₈×C₈₉	—	2	1	no	—
72	C₂×D₈×C₈₉	—	2	1	no	—
73	C₂×D₈×C₈₉	—	2	1	no	—
74	C₂×D₈×C₈₉	—	2	1	no	—
75	C₂×D₈×C₈₉	D₄	1	1	yes	characteristic
76	#131×C₈₉	—	2	4	no	—
77	#131×C₈₉	—	2	4	no	—
78	#131×C₈₉	—	2	1	no	—
79	#131×C₈₉	—	2	1	no	—
80	#131×C₈₉	—	2	1	no	—
81	#131×C₈₉	—	2	1	no	—
82	C₂×M₁₆×C₈₉	—	2	1	no	—
83	C₂×M₁₆×C₈₉	—	4	2	no	—
84	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
85	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
86	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
87	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
88	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
89	C₂²×C₈×C₈₉	—	2	1	no	—
90	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
91	#127×C₈₉	—	2	2	no	—
92	#124×C₈₉	—	2	2	no	—
93	#121×C₈₉	—	4	2	no	—
94	#120×C₈₉	—	2	4	no	—
95	#117×C₈₉	C₂³	1	1	yes	characteristic
96	#117×C₈₉	C₂³	1	1	yes	characteristic
97	#108×C₈₉	—	2	4	no	—
98	#108×C₈₉	—	2	1	no	—
99	#108×C₈₉	—	2	1	no	—
100	#108×C₈₉	D₄	1	2	yes	complemented
101	#108×C₈₉	D₄	1	2	yes	complemented
102	#104×C₈₉	—	2	4	no	—
103	#102×C₈₉	—	4	2	no	—
104	#101×C₈₉	—	2	2	no	—
105	#101×C₈₉	D₄	1	2	yes	complemented
106	#100×C₈₉	—	2	2	no	—
107	#100×C₈₉	—	2	4	no	—
108	#100×C₈₉	—	2	4	no	—
109	#100×C₈₉	D₄	1	2	yes	complemented
110	#99×C₈₉	—	2	2	no	—
111	#97×C₈₉	—	2	1	no	—
112	#97×C₈₉	—	2	1	no	—
113	C₄×C₈×C₈₉	—	2	1	no	—
114	C₄×C₈×C₈₉	—	2	1	no	—
115	C₂⁴×C₈₉	—	2	2	no	—
116	#41×C₈₉	—	2	2	no	—
117	#41×C₈₉	—	2	1	no	—
118	#41×C₈₉	—	2	2	no	—
119	#41×C₈₉	—	2	1	no	—
120	#41×C₈₉	—	2	2	no	—
121	#41×C₈₉	—	2	1	no	—
122	#41×C₈₉	—	2	2	no	—
123	#41×C₈₉	—	2	1	no	—
124	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
125	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₂×D₄×C₈₉	—	2	1	no	—
127	C₂×D₄×C₈₉	—	2	2	no	—
128	C₂×D₄×C₈₉	—	2	2	no	—
129	C₂×D₄×C₈₉	—	2	2	no	—
130	C₂×D₄×C₈₉	—	2	2	no	—
131	C₂×D₄×C₈₉	—	2	2	no	—
132	C₂×D₄×C₈₉	—	2	2	no	—
133	C₂×D₄×C₈₉	—	2	2	no	—
134	C₂×D₄×C₈₉	—	4	4	no	—
135	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
136	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₈₉	—	2	1	no	—
138	C₂²×C₄×C₈₉	—	2	2	no	—
139	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
140	Q₁₆×C₈₉	—	2	1	no	—
141	Q₁₆×C₈₉	—	2	1	no	—
142	Q₁₆×C₈₉	—	4	2	no	—
143	SD₁₆×C₈₉	—	2	2	no	—
144	SD₁₆×C₈₉	—	2	2	no	—
145	SD₁₆×C₈₉	—	4	4	no	—
146	SD₁₆×C₈₉	—	4	4	no	—
147	SD₁₆×C₈₉	—	4	2	no	—
148	SD₁₆×C₈₉	—	4	2	no	—
149	D₈×C₈₉	—	2	1	no	—
150	D₈×C₈₉	—	2	1	no	—
151	D₈×C₈₉	—	4	4	no	—
152	D₈×C₈₉	—	4	4	no	—
153	D₈×C₈₉	—	4	2	no	—
154	D₈×C₈₉	—	4	1	no	—
155	D₈×C₈₉	—	4	1	no	—
156	D₈×C₈₉	—	4	1	no	—
157	D₈×C₈₉	—	4	1	no	—
158	M₁₆×C₈₉	—	2	1	no	—
159	M₁₆×C₈₉	—	2	2	no	—
160	M₁₆×C₈₉	—	2	1	no	—
161	M₁₆×C₈₉	—	2	1	no	—
162	M₁₆×C₈₉	—	2	1	no	—
163	M₁₆×C₈₉	—	2	2	no	—
164	M₁₆×C₈₉	—	2	2	no	—
165	M₁₆×C₈₉	—	4	4	no	—
166	M₁₆×C₈₉	—	4	1	no	—
167	M₁₆×C₈₉	—	4	1	no	—
168	M₁₆×C₈₉	C₂×D₄	1	2	yes	—
169	M₁₆×C₈₉	C₂×D₄	1	4	yes	—
170	M₁₆×C₈₉	C₂×D₄	1	2	yes	—
171	C₂×C₈×C₈₉	—	2	1	no	—
172	C₂×C₈×C₈₉	—	2	1	no	—
173	C₂×C₈×C₈₉	—	2	2	no	—
174	C₂×C₈×C₈₉	—	2	1	no	—
175	C₂×C₈×C₈₉	—	2	2	no	—
176	C₂×C₈×C₈₉	—	4	4	no	—
177	C₂×C₈×C₈₉	—	4	1	no	—
178	C₂×C₈×C₈₉	—	4	1	no	—
179	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
180	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
181	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
184	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
185	C₄⋊C₄×C₈₉	—	2	1	no	—
186	C₄⋊C₄×C₈₉	—	2	1	no	—
187	C₂²⋊C₄×C₈₉	—	2	1	no	—
188	C₂²⋊C₄×C₈₉	—	4	4	no	—
189	C₂²⋊C₄×C₈₉	#41	1	2	yes	—
190	C₄²×C₈₉	—	2	1	no	—
191	C₄²×C₈₉	—	2	1	no	—
192	C₂³×C₈₉	—	2	1	no	—
193	C₂³×C₈₉	—	2	2	no	—
194	C₂³×C₈₉	—	4	2	no	—
195	C₂³×C₈₉	—	4	2	no	—
196	C₂³×C₈₉	—	4	2	no	—
197	C₂³×C₈₉	—	16	2	no	—
198	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
199	Q₈×C₈₉	—	2	2	no	—
200	Q₈×C₈₉	—	2	1	no	—
201	Q₈×C₈₉	—	2	2	no	—
202	Q₈×C₈₉	—	2	1	no	—
203	D₄×C₈₉	—	2	2	no	—
204	D₄×C₈₉	—	2	1	no	—
205	D₄×C₈₉	—	2	2	no	—
206	D₄×C₈₉	—	2	1	no	—
207	D₄×C₈₉	—	4	2	no	—
208	D₄×C₈₉	—	4	1	no	—
209	D₄×C₈₉	—	4	1	no	—
210	D₄×C₈₉	—	4	2	no	—
211	D₄×C₈₉	—	4	2	no	—
212	D₄×C₈₉	—	4	2	no	—
213	D₄×C₈₉	—	4	2	no	—
214	D₄×C₈₉	—	4	2	no	—
215	D₄×C₈₉	—	16	4	no	—
216	C₂×C₄×C₈₉	—	2	1	no	—
217	C₂×C₄×C₈₉	—	2	2	no	—
218	C₂×C₄×C₈₉	—	2	2	no	—
219	C₂×C₄×C₈₉	—	4	1	no	—
220	C₂×C₄×C₈₉	—	4	1	no	—
221	C₂×C₄×C₈₉	—	4	2	no	—
222	C₂×C₄×C₈₉	—	4	2	no	—
223	C₂×C₄×C₈₉	—	4	1	no	—
224	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
228	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
229	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
230	C₈×C₈₉	—	2	2	no	—
231	C₈×C₈₉	—	2	4	no	—
232	C₈×C₈₉	—	2	2	no	—
233	C₈×C₈₉	—	4	1	no	—
234	C₈×C₈₉	—	4	2	no	—
235	C₈×C₈₉	#121	1	1	yes	characteristic
236	C₈×C₈₉	#121	1	1	yes	characteristic
237	C₈×C₈₉	#121	1	1	yes	characteristic
238	C₈×C₈₉	#121	1	1	yes	characteristic
239	C₂²×C₈₉	—	4	1	no	—
240	C₂²×C₈₉	—	4	1	no	—
241	C₂²×C₈₉	—	4	2	no	—
242	C₂²×C₈₉	—	8	2	no	—
243	C₂²×C₈₉	—	8	1	no	—
244	C₂²×C₈₉	—	8	4	no	—
245	C₂²×C₈₉	—	8	2	no	—
246	C₂²×C₈₉	—	16	2	no	—
247	C₂²×C₈₉	—	16	2	no	—
248	C₂²×C₈₉	#532	1	1	yes	characteristic
249	C₂²×C₈₉	#546	1	1	yes	characteristic
250	C₂²×C₈₉	#550	1	1	yes	characteristic
251	C₄×C₈₉	—	4	2	no	—
252	C₄×C₈₉	—	4	1	no	—
253	C₄×C₈₉	—	4	2	no	—
254	C₄×C₈₉	—	8	1	no	—
255	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
256	C₄×C₈₉	#540	1	1	yes	characteristic
257	C₄×C₈₉	D₄²	1	1	yes	characteristic
258	C₄×C₈₉	D₄²	1	1	yes	characteristic
259	#25447	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
260	C₂×C₈₉	—	2	1	no	—
261	C₂×C₈₉	—	2	1	no	—
262	C₂×C₈₉	—	2	1	no	—
263	C₂×C₈₉	—	8	2	no	—
264	C₂×C₈₉	—	8	1	no	—
265	C₂×C₈₉	—	16	2	no	—
266	C₂×C₈₉	#2428	1	1	yes	characteristic
267	#3155	C₂×C₈₉	1	1	yes	characteristic, complemented
268	#3068	C₂×C₈₉	1	2	yes	complemented
269	#3067	C₂×C₈₉	1	1	yes	characteristic, complemented
270	C₂×#472	C₂×C₈₉	1	1	yes	characteristic, complemented
271	#2957	C₂×C₈₉	1	1	yes	characteristic, complemented
272	#2048	C₂×C₈₉	1	2	yes	complemented
273	#2008	C₂×C₈₉	1	2	yes	complemented
274	#1978	C₂×C₈₉	1	1	yes	characteristic, complemented
275	#1953	C₂×C₈₉	1	1	yes	characteristic, complemented
276	#1914	C₂×C₈₉	1	2	yes	complemented
277	#1757	C₂×C₈₉	1	1	yes	characteristic, complemented
278	C₈₉	#25447	1	1	yes	characteristic, direct product factor, Sylow
279	#575	C₂²×C₈₉	1	2	yes	—
280	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	2	yes	—
281	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	2	yes	—
282	C₂×#135	C₂²×C₈₉	1	1	yes	characteristic
283	C₂²×D₈	C₂²×C₈₉	1	1	yes	characteristic
284	#568	C₂²×C₈₉	1	2	yes	—
285	C₂×#131	C₂²×C₈₉	1	1	yes	characteristic
286	#535	C₂²×C₈₉	1	2	yes	—
287	#496	—	2	2	no	—
288	#495	—	2	1	no	—
289	#493	—	2	1	no	—
290	#473	—	2	1	no	—
291	#472	—	2	1	no	—
292	#472	—	2	2	no	—
293	#472	—	2	2	no	—
294	#472	—	2	4	no	—
295	#471	—	2	2	no	—
296	#471	—	2	4	no	—
297	#444	—	2	2	no	—
298	#443	—	2	2	no	—
299	#430	C₂²×C₈₉	1	2	yes	—
300	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
301	C₂×#108	C₂²×C₈₉	1	1	yes	characteristic
302	#421	C₂²×C₈₉	1	2	yes	—
303	#418	C₂²×C₈₉	1	2	yes	—
304	#413	C₂²×C₈₉	1	2	yes	—
305	#413	C₂²×C₈₉	1	1	yes	characteristic
306	C₂×#100	C₂²×C₈₉	1	2	yes	—
307	C₂×#100	C₂²×C₈₉	1	2	yes	—
308	#410	C₂²×C₈₉	1	1	yes	characteristic
309	#405	C₂²×C₈₉	1	1	yes	characteristic
310	#405	C₂²×C₈₉	1	1	yes	characteristic
311	#344	C₂²×C₈₉	1	1	yes	characteristic
312	#344	C₂²×C₈₉	1	1	yes	characteristic
313	#344	C₂²×C₈₉	1	2	yes	—
314	C₂×#41	C₂³×C₈₉	1	2	yes	—
315	C₂×#41	C₂³×C₈₉	1	1	yes	characteristic
316	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
317	#137	—	2	2	no	—
318	#137	—	2	2	no	—
319	AGL(1,ℤ/8)	—	2	2	no	—
320	AGL(1,ℤ/8)	—	2	2	no	—
321	AGL(1,ℤ/8)	—	2	4	no	—
322	AGL(1,ℤ/8)	—	2	4	no	—
323	AGL(1,ℤ/8)	—	4	2	no	—
324	AGL(1,ℤ/8)	—	4	2	no	—
325	#135	—	2	1	no	—
326	#135	—	2	1	no	—
327	#135	—	2	1	no	—
328	#135	—	2	1	no	—
329	#135	—	4	2	no	—
330	C₂×Q₁₆	D₄×C₈₉	1	1	yes	characteristic
331	C₂×SD₁₆	—	2	2	no	—
332	C₂×SD₁₆	—	2	2	no	—
333	C₂×SD₁₆	D₄×C₈₉	1	2	yes	—
334	C₂×D₈	—	2	2	no	—
335	C₂×D₈	—	2	2	no	—
336	C₂×D₈	—	2	1	no	—
337	C₂×D₈	—	2	1	no	—
338	C₂×D₈	—	2	1	no	—
339	C₂×D₈	—	2	1	no	—
340	C₂×D₈	—	2	1	no	—
341	C₂×D₈	—	2	1	no	—
342	C₂×D₈	D₄×C₈₉	1	1	yes	characteristic
343	#131	—	2	4	no	—
344	#131	—	2	4	no	—
345	#131	—	2	1	no	—
346	#131	—	2	1	no	—
347	#131	—	2	1	no	—
348	#131	—	2	1	no	—
349	C₂×M₁₆	—	2	1	no	—
350	C₂×M₁₆	—	4	2	no	—
351	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
352	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
353	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
354	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
355	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
356	C₂²×C₈	—	2	1	no	—
357	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
358	#127	—	2	2	no	—
359	#124	—	2	2	no	—
360	#121	—	4	2	no	—
361	#120	—	2	4	no	—
362	#117	C₂³×C₈₉	1	1	yes	characteristic
363	#117	C₂³×C₈₉	1	1	yes	characteristic
364	#108	—	2	4	no	—
365	#108	—	2	1	no	—
366	#108	—	2	1	no	—
367	#108	D₄×C₈₉	1	2	yes	complemented
368	#108	D₄×C₈₉	1	2	yes	complemented
369	#104	—	2	4	no	—
370	#102	—	4	2	no	—
371	#101	—	2	2	no	—
372	#101	D₄×C₈₉	1	2	yes	complemented
373	#100	—	2	2	no	—
374	#100	—	2	4	no	—
375	#100	—	2	4	no	—
376	#100	D₄×C₈₉	1	2	yes	complemented
377	#99	—	2	2	no	—
378	#97	—	2	1	no	—
379	#97	—	2	1	no	—
380	C₄×C₈	—	2	1	no	—
381	C₄×C₈	—	2	1	no	—
382	C₂⁴	—	2	2	no	—
383	#41	—	2	2	no	—
384	#41	—	2	1	no	—
385	#41	—	2	2	no	—
386	#41	—	2	1	no	—
387	#41	—	2	2	no	—
388	#41	—	2	1	no	—
389	#41	—	2	2	no	—
390	#41	—	2	1	no	—
391	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
392	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
393	C₂×D₄	—	2	1	no	—
394	C₂×D₄	—	2	2	no	—
395	C₂×D₄	—	2	2	no	—
396	C₂×D₄	—	2	2	no	—
397	C₂×D₄	—	2	2	no	—
398	C₂×D₄	—	2	2	no	—
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	—	4	4	no	—
402	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
403	C₂×D₄	C₂×D₄×C₈₉	1	1	yes	characteristic
404	C₂²×C₄	—	2	1	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
407	Q₁₆	—	2	1	no	—
408	Q₁₆	—	2	1	no	—
409	Q₁₆	—	4	2	no	—
410	SD₁₆	—	2	2	no	—
411	SD₁₆	—	2	2	no	—
412	SD₁₆	—	4	4	no	—
413	SD₁₆	—	4	4	no	—
414	SD₁₆	—	4	2	no	—
415	SD₁₆	—	4	2	no	—
416	D₈	—	2	1	no	—
417	D₈	—	2	1	no	—
418	D₈	—	4	4	no	—
419	D₈	—	4	4	no	—
420	D₈	—	4	2	no	—
421	D₈	—	4	1	no	—
422	D₈	—	4	1	no	—
423	D₈	—	4	1	no	—
424	D₈	—	4	1	no	—
425	M₁₆	—	2	1	no	—
426	M₁₆	—	2	2	no	—
427	M₁₆	—	2	1	no	—
428	M₁₆	—	2	1	no	—
429	M₁₆	—	2	1	no	—
430	M₁₆	—	2	2	no	—
431	M₁₆	—	2	2	no	—
432	M₁₆	—	4	4	no	—
433	M₁₆	—	4	1	no	—
434	M₁₆	—	4	1	no	—
435	M₁₆	C₂×D₄×C₈₉	1	2	yes	—
436	M₁₆	C₂×D₄×C₈₉	1	4	yes	—
437	M₁₆	C₂×D₄×C₈₉	1	2	yes	—
438	C₂×C₈	—	2	1	no	—
439	C₂×C₈	—	2	1	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	—	4	4	no	—
444	C₂×C₈	—	4	1	no	—
445	C₂×C₈	—	4	1	no	—
446	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
447	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
448	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
449	C₂×C₈	C₂×D₄×C₈₉	1	1	yes	characteristic
450	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
451	C₂×C₈	#41×C₈₉	1	1	yes	characteristic
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₂²⋊C₄	—	2	1	no	—
455	C₂²⋊C₄	—	4	4	no	—
456	C₂²⋊C₄	#41×C₈₉	1	2	yes	—
457	C₄²	—	2	1	no	—
458	C₄²	—	2	1	no	—
459	C₂³	—	2	1	no	—
460	C₂³	—	2	2	no	—
461	C₂³	—	4	2	no	—
462	C₂³	—	4	2	no	—
463	C₂³	—	4	2	no	—
464	C₂³	—	16	2	no	—
465	C₂³	C₂×#41×C₈₉	1	1	yes	characteristic
466	Q₈	—	2	2	no	—
467	Q₈	—	2	1	no	—
468	Q₈	—	2	2	no	—
469	Q₈	—	2	1	no	—
470	D₄	—	2	2	no	—
471	D₄	—	2	1	no	—
472	D₄	—	2	2	no	—
473	D₄	—	2	1	no	—
474	D₄	—	4	2	no	—
475	D₄	—	4	1	no	—
476	D₄	—	4	1	no	—
477	D₄	—	4	2	no	—
478	D₄	—	4	2	no	—
479	D₄	—	4	2	no	—
480	D₄	—	4	2	no	—
481	D₄	—	4	2	no	—
482	D₄	—	16	4	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	2	2	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	2	no	—
490	C₂×C₄	—	4	1	no	—
491	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
492	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
493	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
494	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
495	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
496	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
497	C₈	—	2	2	no	—
498	C₈	—	2	4	no	—
499	C₈	—	2	2	no	—
500	C₈	—	4	1	no	—
501	C₈	—	4	2	no	—
502	C₈	#121×C₈₉	1	1	yes	characteristic
503	C₈	#121×C₈₉	1	1	yes	characteristic
504	C₈	#121×C₈₉	1	1	yes	characteristic
505	C₈	#121×C₈₉	1	1	yes	characteristic
506	C₂²	—	4	1	no	—
507	C₂²	—	4	1	no	—
508	C₂²	—	4	2	no	—
509	C₂²	—	8	2	no	—
510	C₂²	—	8	1	no	—
511	C₂²	—	8	4	no	—
512	C₂²	—	8	2	no	—
513	C₂²	—	16	2	no	—
514	C₂²	—	16	2	no	—
515	C₂²	#532×C₈₉	1	1	yes	characteristic
516	C₂²	#546×C₈₉	1	1	yes	characteristic
517	C₂²	#550×C₈₉	1	1	yes	characteristic
518	C₄	—	4	2	no	—
519	C₄	—	4	1	no	—
520	C₄	—	4	2	no	—
521	C₄	—	8	1	no	—
522	C₄	C₂×#121×C₈₉	1	1	yes	characteristic
523	C₄	#540×C₈₉	1	1	yes	characteristic
524	C₄	D₄²×C₈₉	1	1	yes	characteristic
525	C₄	D₄²×C₈₉	1	1	yes	characteristic
526	C₂	—	2	1	no	—
527	C₂	—	2	1	no	—
528	C₂	—	2	1	no	—
529	C₂	—	8	2	no	—
530	C₂	—	8	1	no	—
531	C₂	—	16	2	no	—
532	C₂	C₈₉×#2428	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₈₉	#2428
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈₉ ◁ C₂²×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#3155	C₂	semidirect
2. C₈₉×#3068	C₂	semidirect
3. C₈₉×#3067	C₂	semidirect
4. C₂×#472×C₈₉	C₂	semidirect
5. C₈₉×#2957	C₂	semidirect
6. C₈₉×#2048	C₂	semidirect
7. C₈₉×#2008	C₂	semidirect
8. C₈₉×#1978	C₂	semidirect
9. C₈₉×#1953	C₂	semidirect
10. C₈₉×#1914	C₂	semidirect
11. C₈₉×#1757	C₂	semidirect
100. #108×C₈₉	D₄	semidirect
101. #108×C₈₉	D₄	semidirect
105. #101×C₈₉	D₄	semidirect
109. #100×C₈₉	D₄	semidirect
259. #25447	C₈₉	direct (N × H)
267. #3155	C₂×C₈₉	semidirect
268. #3068	C₂×C₈₉	semidirect
269. #3067	C₂×C₈₉	semidirect
270. C₂×#472	C₂×C₈₉	semidirect
271. #2957	C₂×C₈₉	semidirect
272. #2048	C₂×C₈₉	semidirect
273. #2008	C₂×C₈₉	semidirect
274. #1978	C₂×C₈₉	semidirect
275. #1953	C₂×C₈₉	semidirect
276. #1914	C₂×C₈₉	semidirect
277. #1757	C₂×C₈₉	semidirect
278. C₈₉	#25447	direct (N × H)
367. #108	D₄×C₈₉	semidirect
368. #108	D₄×C₈₉	semidirect
372. #101	D₄×C₈₉	semidirect
376. #100	D₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3155	2
C₈₉×#3068	2
C₈₉×#3067	2
C₂×#472×C₈₉	2
C₈₉×#2957	2
C₈₉×#2048	2
C₈₉×#2008	2
C₈₉×#1978	2
C₈₉×#1953	2
C₈₉×#1914	2
C₈₉×#1757	2
#25447	89

Subgroups — C89×#25447 #36403295

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#25447 #36403295