Subgroups — #36605996

Order 22784.

860 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029986	C₂	1	2	yes	maximal, complemented
2	#26029968	C₂	1	1	yes	maximal, characteristic
3	C₂×#19151870	C₂	1	1	yes	maximal, characteristic, complemented
4	#26029169	C₂	1	1	yes	maximal, characteristic, complemented
5	#26021529	C₂	1	2	yes	maximal, complemented
6	#26021312	C₂	1	2	yes	maximal, complemented
7	#26021168	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021042	C₂	1	1	yes	maximal, characteristic, complemented
9	#26020805	C₂	1	2	yes	maximal, complemented
10	#26020053	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#3156	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152396	C₂²	1	2	yes	—
13	C₂×#15668043	C₂²	1	2	yes	—
14	#19152365	C₂²	1	2	yes	—
15	C₂×#15668018	C₂²	1	1	yes	characteristic
16	#19151871	—	2	4	no	—
17	#19151870	—	2	2	no	—
18	#19151870	—	2	2	no	—
19	#19151870	—	2	1	no	—
20	#19151865	—	2	1	no	—
21	#19151863	—	2	4	no	—
22	#19151858	—	2	2	no	—
23	#19151720	—	2	2	no	—
24	#19151715	—	2	2	no	—
25	#19151646	C₂²	1	2	yes	—
26	C₂×#15667948	C₂²	1	1	yes	characteristic
27	C₂×#15667948	C₂²	1	1	yes	characteristic
28	#19151615	C₂²	1	2	yes	—
29	#19151607	C₂²	1	2	yes	—
30	#19151582	C₂²	1	2	yes	—
31	#19151580	C₂²	1	1	yes	characteristic
32	C₂×#15667931	C₂²	1	2	yes	—
33	C₂×#15667931	C₂²	1	2	yes	—
34	#19151571	C₂²	1	1	yes	characteristic
35	#19151553	C₂²	1	1	yes	characteristic
36	#19151370	C₂²	1	1	yes	characteristic
37	#19151368	C₂²	1	2	yes	—
38	#19151368	C₂²	1	1	yes	characteristic
39	C₂×#137×C₈₉	C₂²	1	2	yes	—
40	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
41	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
42	#536×C₈₉	C₂²	1	2	yes	—
43	#497×C₈₉	—	2	2	no	—
44	#495×C₈₉	—	2	1	no	—
45	#494×C₈₉	—	2	1	no	—
46	#405×C₈₉	C₂²	1	1	yes	characteristic
47	C₂×#13083574	C₂³	1	1	yes	characteristic
48	#15668043	—	2	4	no	—
49	#15668043	—	2	2	no	—
50	#15668043	—	4	2	no	—
51	#15668042	—	2	2	no	—
52	#15668038	—	2	2	no	—
53	#15668037	—	2	2	no	—
54	#15668032	—	4	2	no	—
55	C₂×#13083564	—	2	2	no	—
56	C₂×#13083561	—	2	2	no	—
57	#15668019	—	2	4	no	—
58	#15668019	—	2	4	no	—
59	#15668018	—	2	1	no	—
60	#15668018	—	2	1	no	—
61	#15668018	—	2	1	no	—
62	#15668018	—	2	1	no	—
63	C₂×#13083558	—	4	2	no	—
64	C₂×#13083558	C₂³	1	1	yes	characteristic
65	C₂×#13083558	C₂³	1	2	yes	—
66	C₂×#13083558	C₂³	1	2	yes	—
67	C₂×#13083558	C₂³	1	1	yes	characteristic
68	C₂×#13083557	—	2	1	no	—
69	C₂×C₈×D₈₉	—	2	1	no	—
70	#15667971	C₂³	1	1	yes	characteristic
71	#15667948	—	2	1	no	—
72	#15667948	—	2	1	no	—
73	#15667948	D₄	1	2	yes	—
74	#15667948	D₄	1	2	yes	—
75	#15667947	—	2	4	no	—
76	#15667939	—	2	4	no	—
77	#15667936	—	4	2	no	—
78	#15667931	—	2	4	no	—
79	#15667931	—	2	4	no	—
80	#15667931	—	2	2	no	—
81	#15667930	D₄	1	2	yes	—
82	#15667929	—	2	2	no	—
83	#15667928	D₄	1	2	yes	—
84	#15667926	—	2	2	no	—
85	#15667922	—	2	1	no	—
86	C₈×Dic₈₉	—	2	1	no	—
87	C₂×#41×C₈₉	C₂³	1	2	yes	—
88	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
89	#137×C₈₉	—	2	4	no	—
90	#137×C₈₉	—	4	2	no	—
91	#135×C₈₉	—	2	1	no	—
92	#135×C₈₉	—	2	1	no	—
93	#135×C₈₉	—	2	1	no	—
94	#135×C₈₉	—	2	1	no	—
95	C₂×Q₁₆×C₈₉	—	2	2	no	—
96	C₂×Q₁₆×C₈₉	—	2	1	no	—
97	C₂×Q₁₆×C₈₉	—	2	1	no	—
98	C₂×Q₁₆×C₈₉	—	2	1	no	—
99	C₂×Q₁₆×C₈₉	—	2	1	no	—
100	C₂×Q₁₆×C₈₉	—	2	1	no	—
101	C₂×Q₁₆×C₈₉	—	2	1	no	—
102	C₂×Q₁₆×C₈₉	D₄	1	1	yes	characteristic
103	C₂×SD₁₆×C₈₉	—	2	2	no	—
104	C₂×SD₁₆×C₈₉	D₄	1	2	yes	—
105	C₂×D₈×C₈₉	D₄	1	1	yes	characteristic
106	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
107	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
108	C₄⋊Q₈×C₈₉	—	2	2	no	—
109	#124×C₈₉	—	2	2	no	—
110	#123×C₈₉	—	2	4	no	—
111	C₂²⋊Q₈×C₈₉	—	4	2	no	—
112	#117×C₈₉	C₂³	1	1	yes	characteristic
113	#97×C₈₉	—	2	1	no	—
114	C₄×C₈×C₈₉	—	2	1	no	—
115	#13083574	—	2	1	no	—
116	#13083574	—	2	1	no	—
117	#13083574	—	2	1	no	—
118	#13083574	—	2	1	no	—
119	C₂×Dic₁₇₈	C₂×D₄	1	1	yes	characteristic
120	C₂×D₃₅₆	C₂×D₄	1	1	yes	characteristic
121	C₂×C₂²⋊D₈₉	—	2	1	no	—
122	C₂×C₄×D₈₉	—	2	1	no	—
123	#13083564	—	4	4	no	—
124	#13083564	—	4	2	no	—
125	#13083562	—	4	2	no	—
126	#13083561	—	4	4	no	—
127	#13083561	—	4	2	no	—
128	#13083559	—	4	2	no	—
129	#13083558	—	2	1	no	—
130	#13083558	—	2	2	no	—
131	#13083558	—	2	1	no	—
132	#13083558	—	2	1	no	—
133	#13083558	—	2	2	no	—
134	#13083558	—	2	2	no	—
135	#13083558	—	4	4	no	—
136	#13083558	C₂×D₄	1	2	yes	—
137	#13083558	C₂×D₄	1	4	yes	—
138	#13083557	—	4	1	no	—
139	#13083557	—	4	1	no	—
140	C₂×#10623494	—	2	1	no	—
141	C₂×#10623494	—	2	2	no	—
142	C₂×#10623494	—	2	1	no	—
143	C₂×#10623494	—	2	2	no	—
144	C₂×#10623494	—	4	4	no	—
145	C₈×D₈₉	—	4	1	no	—
146	C₈×D₈₉	—	4	1	no	—
147	C₄⋊Dic₈₉	—	2	1	no	—
148	C₂²⋊Dic₈₉	—	2	1	no	—
149	C₄×Dic₈₉	—	2	1	no	—
150	#41×C₈₉	—	2	2	no	—
151	#41×C₈₉	—	2	2	no	—
152	#41×C₈₉	—	2	2	no	—
153	#41×C₈₉	—	2	2	no	—
154	C₂×Q₈×C₈₉	—	2	2	no	—
155	C₂×Q₈×C₈₉	—	2	2	no	—
156	C₂×Q₈×C₈₉	—	2	2	no	—
157	C₂×Q₈×C₈₉	—	2	2	no	—
158	C₂×Q₈×C₈₉	—	2	2	no	—
159	C₂×Q₈×C₈₉	—	2	2	no	—
160	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
161	C₂×D₄×C₈₉	—	2	2	no	—
162	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₈₉	—	2	2	no	—
164	C₂²×C₄×C₈₉	—	2	2	no	—
165	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
166	Q₁₆×C₈₉	—	2	1	no	—
167	Q₁₆×C₈₉	—	2	1	no	—
168	Q₁₆×C₈₉	—	4	4	no	—
169	Q₁₆×C₈₉	—	4	1	no	—
170	Q₁₆×C₈₉	—	4	1	no	—
171	Q₁₆×C₈₉	—	4	1	no	—
172	Q₁₆×C₈₉	—	4	1	no	—
173	SD₁₆×C₈₉	—	2	2	no	—
174	SD₁₆×C₈₉	—	2	2	no	—
175	SD₁₆×C₈₉	—	4	4	no	—
176	D₈×C₈₉	—	2	1	no	—
177	D₈×C₈₉	—	2	1	no	—
178	M₁₆×C₈₉	—	2	1	no	—
179	M₁₆×C₈₉	C₂×D₄	1	2	yes	—
180	C₂×C₈×C₈₉	—	2	1	no	—
181	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
182	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
183	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
184	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
185	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
186	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
187	C₄⋊C₄×C₈₉	—	2	1	no	—
188	C₄⋊C₄×C₈₉	—	4	4	no	—
189	C₂²⋊C₄×C₈₉	—	4	4	no	—
190	C₂²⋊C₄×C₈₉	#41	1	2	yes	—
191	C₄²×C₈₉	—	2	1	no	—
192	C₂³×C₈₉	—	2	2	no	—
193	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
194	C₂²×D₈₉	—	2	1	no	—
195	Q₈×C₈₉	—	2	2	no	—
196	Q₈×C₈₉	—	2	2	no	—
197	Q₈×C₈₉	—	4	2	no	—
198	Q₈×C₈₉	—	4	2	no	—
199	Q₈×C₈₉	—	4	2	no	—
200	Q₈×C₈₉	—	4	2	no	—
201	D₄×C₈₉	—	2	2	no	—
202	D₄×C₈₉	—	2	2	no	—
203	D₄×C₈₉	—	4	2	no	—
204	D₄×C₈₉	—	4	2	no	—
205	C₂×C₄×C₈₉	—	2	2	no	—
206	C₂×C₄×C₈₉	—	2	2	no	—
207	C₂×C₄×C₈₉	—	4	2	no	—
208	C₂×C₄×C₈₉	—	4	2	no	—
209	C₂×C₄×C₈₉	—	4	2	no	—
210	C₂×C₄×C₈₉	—	4	2	no	—
211	C₂×C₄×C₈₉	—	4	2	no	—
212	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
217	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
218	C₂²⋊D₈₉	—	4	1	no	—
219	C₂²⋊D₈₉	—	4	1	no	—
220	C₂×Dic₈₉	—	2	1	no	—
221	C₂×Dic₈₉	—	4	1	no	—
222	D₃₅₆	—	2	1	no	—
223	D₃₅₆	—	2	1	no	—
224	C₄×D₈₉	—	4	1	no	—
225	C₄×D₈₉	—	4	1	no	—
226	Dic₁₇₈	—	2	1	no	—
227	Dic₁₇₈	—	2	1	no	—
228	C₈×C₈₉	—	2	2	no	—
229	C₈×C₈₉	#121	1	1	yes	characteristic
230	C₈×C₈₉	#121	1	1	yes	characteristic
231	C₈×C₈₉	#121	1	1	yes	characteristic
232	C₈×C₈₉	#121	1	1	yes	characteristic
233	#10623494	—	2	2	no	—
234	#10623494	—	2	4	no	—
235	#10623494	—	4	1	no	—
236	#10623494	—	4	2	no	—
237	C₂²×C₈₉	—	4	1	no	—
238	C₂²×C₈₉	—	4	2	no	—
239	C₂²×C₈₉	—	8	2	no	—
240	C₂²×C₈₉	#532	1	1	yes	characteristic
241	C₂²×C₈₉	#546	1	1	yes	characteristic
242	C₂²×C₈₉	#550	1	1	yes	characteristic
243	C₂×D₈₉	—	4	1	no	—
244	C₂×D₈₉	—	8	1	no	—
245	C₄×C₈₉	—	4	2	no	—
246	C₄×C₈₉	—	4	2	no	—
247	C₄×C₈₉	—	8	2	no	—
248	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
249	C₄×C₈₉	#540	1	1	yes	characteristic
250	C₄×C₈₉	D₄²	1	1	yes	characteristic
251	C₄×C₈₉	D₄²	1	1	yes	characteristic
252	Dic₈₉	—	4	1	no	—
253	Dic₈₉	—	8	1	no	—
254	#25448	—	89	1	no	maximal, Sylow
255	C₂×C₈₉	—	2	1	no	—
256	C₂×C₈₉	—	2	1	no	—
257	C₂×C₈₉	—	2	1	no	—
258	C₂×C₈₉	—	8	2	no	—
259	C₂×C₈₉	#2428	1	1	yes	characteristic
260	D₈₉	—	8	1	no	—
261	#3156	D₈₉	1	1	yes	characteristic, complemented
262	#3069	—	89	2	no	—
263	#3067	—	89	1	no	—
264	C₂×#473	—	89	1	no	—
265	#2957	—	89	1	no	—
266	#2049	—	89	2	no	—
267	#2009	—	89	2	no	—
268	#1979	—	89	1	no	—
269	#1953	—	89	1	no	—
270	#1915	—	89	2	no	—
271	#1757	—	89	1	no	—
272	C₈₉	#25448	1	1	yes	characteristic, complemented, Sylow
273	#575	—	89	2	no	—
274	C₂×#137	—	89	2	no	—
275	C₂×#137	C₂×D₈₉	1	2	yes	—
276	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
277	C₂²×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
278	#568	—	89	2	no	—
279	C₂×#131	—	89	1	no	—
280	#536	C₂×D₈₉	1	2	yes	—
281	#497	—	2	2	no	—
282	#495	—	2	1	no	—
283	#494	—	2	1	no	—
284	#473	—	178	4	no	—
285	#473	—	178	2	no	—
286	#473	—	178	2	no	—
287	#473	—	178	1	no	—
288	#472	—	178	1	no	—
289	#471	—	178	4	no	—
290	#471	—	178	2	no	—
291	#444	—	178	2	no	—
292	#443	—	178	2	no	—
293	#430	—	89	2	no	—
294	C₂×#108	—	89	1	no	—
295	C₂×#108	—	89	1	no	—
296	#421	—	89	2	no	—
297	#419	—	89	2	no	—
298	#413	—	89	1	no	—
299	#413	—	89	2	no	—
300	C₂×#101	—	89	2	no	—
301	C₂×#101	—	89	2	no	—
302	#410	—	89	1	no	—
303	#405	—	89	1	no	—
304	#405	C₂×D₈₉	1	1	yes	characteristic
305	#344	—	89	1	no	—
306	#344	—	89	1	no	—
307	#344	—	89	2	no	—
308	C₂×#41	—	89	1	no	—
309	C₂×#41	C₂²×D₈₉	1	2	yes	—
310	C₂²×Q₈	C₂²×D₈₉	1	2	yes	—
311	#137	—	2	4	no	—
312	#137	—	4	2	no	—
313	#137	—	356	2	no	—
314	#137	—	178	4	no	—
315	#137	—	178	2	no	—
316	#137	—	178	2	no	—
317	AGL(1,ℤ/8)	—	178	2	no	—
318	AGL(1,ℤ/8)	—	178	2	no	—
319	#135	—	2	1	no	—
320	#135	—	2	1	no	—
321	#135	—	2	1	no	—
322	#135	—	2	1	no	—
323	#135	—	356	2	no	—
324	C₂×Q₁₆	—	2	2	no	—
325	C₂×Q₁₆	—	2	1	no	—
326	C₂×Q₁₆	—	2	1	no	—
327	C₂×Q₁₆	—	2	1	no	—
328	C₂×Q₁₆	—	2	1	no	—
329	C₂×Q₁₆	—	2	1	no	—
330	C₂×Q₁₆	—	2	1	no	—
331	C₂×Q₁₆	—	178	2	no	—
332	C₂×Q₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
333	C₂×SD₁₆	—	2	2	no	—
334	C₂×SD₁₆	—	178	2	no	—
335	C₂×SD₁₆	C₂²⋊D₈₉	1	2	yes	—
336	C₂×D₈	C₂²⋊D₈₉	1	1	yes	characteristic
337	#131	—	178	4	no	—
338	#131	—	178	4	no	—
339	#131	—	178	1	no	—
340	#131	—	178	1	no	—
341	#131	—	178	1	no	—
342	#131	—	178	1	no	—
343	C₂×M₁₆	—	89	1	no	—
344	C₂×M₁₆	—	89	2	no	—
345	C₂×M₁₆	—	89	1	no	—
346	C₂×M₁₆	—	89	2	no	—
347	C₂×M₁₆	—	356	2	no	—
348	C₂×M₁₆	—	178	1	no	—
349	C₂×M₁₆	C₂²×D₈₉	1	1	yes	characteristic
350	C₂²×C₈	—	178	1	no	—
351	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
352	C₄⋊Q₈	—	2	2	no	—
353	#124	—	2	2	no	—
354	#123	—	2	4	no	—
355	C₂²⋊Q₈	—	4	2	no	—
356	#117	—	89	1	no	—
357	#117	C₂²×D₈₉	1	1	yes	characteristic
358	#108	—	89	2	no	—
359	#108	—	89	2	no	—
360	#108	—	178	4	no	—
361	#108	—	178	1	no	—
362	#108	—	178	1	no	—
363	#104	—	178	4	no	—
364	#103	—	356	2	no	—
365	#101	—	89	2	no	—
366	#101	—	178	4	no	—
367	#101	—	178	4	no	—
368	#101	—	178	2	no	—
369	#100	—	89	2	no	—
370	#100	—	178	2	no	—
371	#99	—	178	2	no	—
372	#97	—	2	1	no	—
373	#97	—	178	1	no	—
374	C₄×C₈	—	2	1	no	—
375	C₄×C₈	—	178	1	no	—
376	#41	—	2	2	no	—
377	#41	—	2	2	no	—
378	#41	—	2	2	no	—
379	#41	—	2	2	no	—
380	#41	—	178	1	no	—
381	#41	—	178	1	no	—
382	#41	—	178	1	no	—
383	#41	—	178	1	no	—
384	C₂×Q₈	—	2	2	no	—
385	C₂×Q₈	—	2	2	no	—
386	C₂×Q₈	—	2	2	no	—
387	C₂×Q₈	—	2	2	no	—
388	C₂×Q₈	—	2	2	no	—
389	C₂×Q₈	—	2	2	no	—
390	C₂×Q₈	—	89	1	no	—
391	C₂×Q₈	C₂×C₂²⋊D₈₉	1	2	yes	—
392	C₂×D₄	—	2	2	no	—
393	C₂×D₄	—	89	1	no	—
394	C₂×D₄	—	178	1	no	—
395	C₂×D₄	C₂×C₂²⋊D₈₉	1	2	yes	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	178	1	no	—
399	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
400	Q₁₆	—	2	1	no	—
401	Q₁₆	—	2	1	no	—
402	Q₁₆	—	4	4	no	—
403	Q₁₆	—	4	1	no	—
404	Q₁₆	—	4	1	no	—
405	Q₁₆	—	4	1	no	—
406	Q₁₆	—	4	1	no	—
407	Q₁₆	—	356	2	no	—
408	Q₁₆	—	356	4	no	—
409	SD₁₆	—	2	2	no	—
410	SD₁₆	—	2	2	no	—
411	SD₁₆	—	4	4	no	—
412	SD₁₆	—	356	2	no	—
413	SD₁₆	—	356	2	no	—
414	SD₁₆	—	356	4	no	—
415	D₈	—	2	1	no	—
416	D₈	—	2	1	no	—
417	D₈	—	356	2	no	—
418	M₁₆	—	2	1	no	—
419	M₁₆	—	89	2	no	—
420	M₁₆	—	89	4	no	—
421	M₁₆	—	356	1	no	—
422	M₁₆	—	356	1	no	—
423	M₁₆	—	356	4	no	—
424	M₁₆	—	178	1	no	—
425	M₁₆	—	178	2	no	—
426	M₁₆	—	178	1	no	—
427	M₁₆	—	178	2	no	—
428	M₁₆	—	178	1	no	—
429	M₁₆	—	178	2	no	—
430	M₁₆	D₄×D₈₉	1	2	yes	—
431	C₂×C₈	—	2	1	no	—
432	C₂×C₈	—	356	1	no	—
433	C₂×C₈	—	356	1	no	—
434	C₂×C₈	—	356	4	no	—
435	C₂×C₈	—	178	1	no	—
436	C₂×C₈	—	178	2	no	—
437	C₂×C₈	—	178	1	no	—
438	C₂×C₈	—	178	2	no	—
439	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
440	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
441	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
442	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
443	C₂×C₈	#13083573	1	1	yes	characteristic
444	C₂×C₈	#13083573	1	1	yes	characteristic
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	4	4	no	—
447	C₄⋊C₄	—	178	1	no	—
448	C₂²⋊C₄	—	4	4	no	—
449	C₂²⋊C₄	—	178	1	no	—
450	C₂²⋊C₄	#13083573	1	2	yes	—
451	C₄²	—	2	1	no	—
452	C₄²	—	178	1	no	—
453	C₂³	—	2	2	no	—
454	C₂³	—	178	1	no	—
455	C₂³	C₂×#13083573	1	1	yes	characteristic
456	Q₈	—	2	2	no	—
457	Q₈	—	2	2	no	—
458	Q₈	—	4	2	no	—
459	Q₈	—	4	2	no	—
460	Q₈	—	4	2	no	—
461	Q₈	—	4	2	no	—
462	Q₈	—	178	1	no	—
463	Q₈	—	178	1	no	—
464	D₄	—	2	2	no	—
465	D₄	—	2	2	no	—
466	D₄	—	4	2	no	—
467	D₄	—	4	2	no	—
468	D₄	—	356	1	no	—
469	D₄	—	356	1	no	—
470	D₄	—	178	1	no	—
471	D₄	—	178	1	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	2	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	356	1	no	—
480	C₂×C₄	—	356	1	no	—
481	C₂×C₄	—	356	1	no	—
482	C₂×C₄	—	178	1	no	—
483	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
484	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
485	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
486	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
487	C₂×C₄	#15668057	1	1	yes	characteristic
488	C₂×C₄	#15668057	1	1	yes	characteristic
489	C₈	—	2	2	no	—
490	C₈	—	356	2	no	—
491	C₈	—	356	1	no	—
492	C₈	—	178	2	no	—
493	C₈	—	178	4	no	—
494	C₈	#15667984	1	1	yes	characteristic
495	C₈	#15667984	1	1	yes	characteristic
496	C₈	#15667984	1	1	yes	characteristic
497	C₈	#15667984	1	1	yes	characteristic
498	C₂²	—	4	1	no	—
499	C₂²	—	4	2	no	—
500	C₂²	—	8	2	no	—
501	C₂²	—	356	1	no	—
502	C₂²	—	712	1	no	—
503	C₂²	#19152146	1	1	yes	characteristic
504	C₂²	#19152242	1	1	yes	characteristic
505	C₂²	#19152268	1	1	yes	characteristic
506	C₄	—	4	2	no	—
507	C₄	—	4	2	no	—
508	C₄	—	8	2	no	—
509	C₄	—	356	1	no	—
510	C₄	—	712	1	no	—
511	C₄	C₂×#15667984	1	1	yes	characteristic
512	C₄	#19152202	1	1	yes	characteristic
513	C₄	D₄×C₂²⋊D₈₉	1	1	yes	characteristic
514	C₄	#19152233	1	1	yes	characteristic
515	C₂	—	2	1	no	—
516	C₂	—	2	1	no	—
517	C₂	—	2	1	no	—
518	C₂	—	8	2	no	—
519	C₂	—	712	1	no	—
520	C₂	#26023843	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26023843
commutator, cosocle	C₂²×C₄×C₈₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₈₉×#3156	C₂
socle	C₂×C₈₉	#2428
prime radical	#26029968	C₂
hypercenter	#3156	D₈₉
hypocenter	C₈₉	#25448
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3156

Complements

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

N	H ≅ G/N	type
1. #26029986	C₂	semidirect
3. C₂×#19151870	C₂	semidirect
4. #26029169	C₂	semidirect
5. #26021529	C₂	semidirect
6. #26021312	C₂	semidirect
7. #26021168	C₂	semidirect
8. #26021042	C₂	semidirect
9. #26020805	C₂	semidirect
10. #26020053	C₂	semidirect
11. C₈₉×#3156	C₂	semidirect
261. #3156	D₈₉	semidirect
272. C₈₉	#25448	semidirect

Maximal subgroups

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

subgroup	index
#26029986	2
#26029968	2
C₂×#19151870	2
#26029169	2
#26021529	2
#26021312	2
#26021168	2
#26021042	2
#26020805	2
#26020053	2
C₈₉×#3156	2
#25448	89