Subgroups — #36707666

Order 22784.

646 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#504×D₈₉	C₂	1	2	yes	maximal, complemented
2	#26029150	C₂	1	1	yes	maximal, characteristic
3	#26022095	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022092	C₂	1	2	yes	maximal, complemented
5	#26022089	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022088	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022086	C₂	1	2	yes	maximal, complemented
8	#26022016	C₂	1	1	yes	maximal, characteristic, complemented
9	#26021878	C₂	1	2	yes	maximal, complemented
10	#26021850	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2171	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152395	C₄	1	1	yes	characteristic
13	#131×D₈₉	C₂²	1	2	yes	—
14	#19152016	—	2	2	no	—
15	#19152016	C₂²	1	2	yes	complemented
16	#19152015	C₂²	1	2	yes	complemented
17	#19152014	—	2	2	no	—
18	#19152014	C₂²	1	2	yes	complemented
19	C₂×#15667951	C₂²	1	1	yes	characteristic
20	M₃₂×D₈₉	—	2	2	no	—
21	M₃₂×D₈₉	C₂²	1	2	yes	—
22	C₂×C₁₆×D₈₉	—	2	2	no	—
23	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
24	#19151720	C₄	1	1	yes	characteristic
25	#19151719	C₄	1	1	yes	characteristic
26	#19151717	C₄	1	2	yes	—
27	#19151716	C₂²	1	1	yes	characteristic
28	#19151713	C₂²	1	2	yes	—
29	#19151645	C₄	1	1	yes	characteristic
30	#19151613	C₄	1	2	yes	—
31	#19151552	C₂²	1	1	yes	characteristic
32	#19151419	C₂²	1	1	yes	characteristic, complemented
33	#19151418	C₂²	1	2	yes	complemented
34	#19151383	C₂²	1	2	yes	complemented
35	#19151382	C₂²	1	2	yes	complemented
36	#19151381	C₂²	1	2	yes	complemented
37	#19151373	C₂²	1	1	yes	characteristic, complemented
38	C₄×#13083549	C₂²	1	1	yes	characteristic, complemented
39	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic, complemented
40	#504×C₈₉	C₂²	1	2	yes	complemented
41	#443×C₈₉	C₂²	1	1	yes	characteristic
42	#364×C₈₉	C₂²	1	1	yes	characteristic, complemented
43	#350×C₈₉	C₂²	1	2	yes	complemented
44	#346×C₈₉	C₂²	1	1	yes	characteristic, complemented
45	#41×D₈₉	C₂×C₄	1	2	yes	—
46	#15668044	C₈	1	2	yes	—
47	#15668041	C₈	1	2	yes	—
48	#15668039	C₈	1	2	yes	—
49	#15668036	C₈	1	2	yes	—
50	#15668034	C₂×C₄	1	1	yes	characteristic
51	#15668032	C₂×C₄	1	2	yes	—
52	#15668020	—	2	2	no	—
53	#15668020	C₂×C₄	1	2	yes	—
54	#15668019	C₂×C₄	1	2	yes	—
55	#15668018	—	2	2	no	—
56	#15668018	C₂³	1	2	yes	—
57	C₂×#13083557	C₂×C₄	1	1	yes	characteristic
58	M₁₆×D₈₉	—	2	2	no	—
59	M₁₆×D₈₉	C₂×C₄	1	2	yes	—
60	C₂×C₈×D₈₉	—	2	2	no	—
61	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
62	#15667969	C₂×C₄	1	1	yes	characteristic
63	#15667952	—	2	2	no	—
64	#15667952	C₂³	1	2	yes	—
65	#15667951	—	2	2	no	—
66	#15667951	—	2	2	no	—
67	#15667951	—	2	2	no	—
68	#15667951	—	2	2	no	—
69	C₂×#13083549	—	2	2	no	—
70	C₂×#13083549	C₂³	1	1	yes	characteristic
71	C₂×#13083549	C₂³	1	1	yes	characteristic
72	C₁₆×D₈₉	—	2	2	no	—
73	C₁₆×D₈₉	—	2	2	no	—
74	C₁₆×D₈₉	—	2	2	no	—
75	C₁₆×D₈₉	D₄	1	2	yes	—
76	C₁₆×D₈₉	D₄	1	2	yes	—
77	#15667948	C₂×C₄	1	1	yes	characteristic
78	#15667947	C₂×C₄	1	2	yes	—
79	#15667940	C₂×C₄	1	2	yes	—
80	#15667939	C₂×C₄	1	2	yes	—
81	#15667938	C₂×C₄	1	2	yes	—
82	#15667923	C₂×C₄	1	1	yes	characteristic
83	#15667922	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
85	#135×C₈₉	C₂×C₄	1	1	yes	characteristic
86	#131×C₈₉	C₂³	1	2	yes	—
87	M₃₂×C₈₉	—	2	2	no	—
88	M₃₂×C₈₉	C₂³	1	2	yes	—
89	C₂×C₁₆×C₈₉	—	2	2	no	—
90	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
91	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
92	#108×C₈₉	C₂×C₄	1	1	yes	characteristic
93	#104×C₈₉	C₂×C₄	1	2	yes	—
94	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
95	#13083575	C₂×C₈	1	2	yes	—
96	#13083574	—	2	2	no	—
97	#13083574	C₂²×C₄	1	2	yes	—
98	#13083573	—	2	2	no	—
99	#13083573	C₂×C₈	1	2	yes	—
100	Q₈×D₈₉	C₂×C₈	1	2	yes	—
101	D₄×D₈₉	—	2	2	no	—
102	D₄×D₈₉	C₂×C₈	1	2	yes	—
103	C₂×C₄×D₈₉	—	2	2	no	—
104	C₂×C₄×D₈₉	C₂²×C₄	1	1	yes	characteristic
105	Dic₃₅₆	C₂×C₈	1	1	yes	characteristic
106	#13083564	C₂×C₈	1	2	yes	—
107	#13083563	C₂×C₈	1	2	yes	—
108	#13083562	C₂×C₈	1	2	yes	—
109	#13083561	C₂×C₈	1	2	yes	—
110	D₇₁₂	C₂×C₈	1	1	yes	characteristic
111	#13083559	C₂×C₈	1	2	yes	—
112	#13083558	—	2	2	no	—
113	#13083558	C₂²×C₄	1	2	yes	—
114	#13083557	—	2	2	no	—
115	#13083557	—	2	2	no	—
116	#13083557	—	2	2	no	—
117	#13083557	C₂×C₈	1	2	yes	—
118	#13083557	C₂×C₈	1	2	yes	—
119	C₂×#10623494	—	2	2	no	—
120	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
121	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
122	C₈×D₈₉	—	2	2	no	—
123	C₈×D₈₉	—	2	2	no	—
124	C₈×D₈₉	—	2	2	no	—
125	C₈×D₈₉	C₂×D₄	1	2	yes	—
126	C₈×D₈₉	#41	1	2	yes	—
127	#13083553	—	2	1	no	—
128	#13083551	—	2	1	no	—
129	C₄×Dic₈₉	C₂²×C₄	1	1	yes	characteristic
130	#13083549	—	2	2	no	—
131	#13083549	—	2	1	no	—
132	#13083549	C₂×D₄	1	1	yes	characteristic
133	#13083549	C₂×D₄	1	1	yes	characteristic
134	#41×C₈₉	C₂²×C₄	1	2	yes	—
135	Q₁₆×C₈₉	C₂×C₈	1	1	yes	characteristic
136	SD₁₆×C₈₉	C₂×C₈	1	2	yes	—
137	D₈×C₈₉	C₂×C₈	1	1	yes	characteristic
138	M₁₆×C₈₉	—	2	2	no	—
139	M₁₆×C₈₉	C₂²×C₄	1	2	yes	—
140	C₂×C₈×C₈₉	—	2	2	no	—
141	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
143	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
144	C₁₆×C₈₉	—	2	2	no	—
145	C₁₆×C₈₉	—	2	1	no	—
146	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
147	C₂²×D₈₉	—	2	2	no	—
148	C₂²×D₈₉	#131	1	1	yes	characteristic
149	Q₈×C₈₉	C₂²×C₈	1	2	yes	—
150	D₄×C₈₉	—	2	2	no	—
151	D₄×C₈₉	C₂²×C₈	1	2	yes	—
152	C₂×C₄×C₈₉	—	2	2	no	—
153	C₂×C₄×C₈₉	—	2	1	no	—
154	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
155	C₂²⋊D₈₉	—	2	2	no	—
156	C₂²⋊D₈₉	—	2	2	no	—
157	C₂²⋊D₈₉	—	2	2	no	—
158	C₂×Dic₈₉	—	2	2	no	—
159	C₂×Dic₈₉	—	2	1	no	—
160	C₂×Dic₈₉	#131	1	1	yes	characteristic
161	D₃₅₆	—	2	2	no	—
162	D₃₅₆	C₂²×C₈	1	2	yes	—
163	C₄×D₈₉	—	2	2	no	—
164	C₄×D₈₉	—	2	2	no	—
165	C₄×D₈₉	—	2	2	no	—
166	C₄×D₈₉	C₄×D₄	1	2	yes	—
167	C₄×D₈₉	C₂²×C₈	1	2	yes	—
168	Dic₁₇₈	—	2	2	no	—
169	Dic₁₇₈	C₂²×C₈	1	2	yes	—
170	C₈×C₈₉	—	2	2	no	—
171	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
172	C₈×C₈₉	C₂²×C₈	1	1	yes	characteristic
173	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
174	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
175	#10623494	—	2	2	no	—
176	#10623494	C₄×D₄	1	2	yes	—
177	#10623494	C₂²×C₈	1	2	yes	—
178	C₂²×C₈₉	—	2	2	no	—
179	C₂²×C₈₉	C₂×#131	1	1	yes	characteristic
180	C₂×D₈₉	—	2	2	no	—
181	C₂×D₈₉	—	4	1	no	—
182	C₂×D₈₉	—	4	4	no	—
183	C₂×D₈₉	C₈×D₄	1	2	yes	—
184	C₄×C₈₉	—	2	2	no	—
185	C₄×C₈₉	—	4	1	no	—
186	C₄×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
187	C₄×C₈₉	C₂³×C₈	1	1	yes	characteristic
188	Dic₈₉	—	2	2	no	—
189	Dic₈₉	—	4	1	no	—
190	Dic₈₉	C₈×D₄	1	2	yes	—
191	#33441	—	89	1	no	maximal, Sylow
192	C₂×C₈₉	—	2	1	no	—
193	C₂×C₈₉	—	4	2	no	—
194	C₂×C₈₉	C₂×C₈×D₄	1	1	yes	characteristic
195	D₈₉	—	2	2	no	—
196	D₈₉	—	4	2	no	—
197	C₂×#504	—	89	2	no	—
198	#2955	—	89	1	no	—
199	#2171	—	89	2	no	—
200	#2171	—	89	1	no	—
201	#2171	D₈₉	1	1	yes	characteristic, complemented
202	#2170	—	89	1	no	—
203	#2170	—	89	1	no	—
204	#2170	—	89	2	no	—
205	#2153	—	89	1	no	—
206	#2117	—	89	2	no	—
207	#2108	—	89	1	no	—
208	C₈₉	#33441	1	1	yes	characteristic, complemented, Sylow
209	#575	—	89	1	no	—
210	C₂×#131	—	89	2	no	—
211	#504	—	89	2	no	—
212	#504	—	89	2	no	—
213	#504	—	89	2	no	—
214	#504	—	178	2	no	—
215	#504	—	178	2	no	—
216	#504	C₂×D₈₉	1	2	yes	complemented
217	C₂×M₃₂	—	89	1	no	—
218	C₂×M₃₂	—	89	2	no	—
219	C₂×M₃₂	—	178	2	no	—
220	C₂²×C₁₆	—	89	1	no	—
221	C₂²×C₁₆	—	178	2	no	—
222	#444	—	89	1	no	—
223	#444	—	89	1	no	—
224	#444	—	89	2	no	—
225	#443	—	89	2	no	—
226	#443	—	89	1	no	—
227	#443	C₂×D₈₉	1	1	yes	characteristic
228	#430	—	89	1	no	—
229	#421	—	89	2	no	—
230	#405	—	89	1	no	—
231	#364	—	89	2	no	—
232	#364	—	89	1	no	—
233	#364	C₂×D₈₉	1	1	yes	characteristic, complemented
234	#350	—	89	2	no	—
235	#350	—	89	2	no	—
236	#350	—	89	2	no	—
237	#350	C₂×D₈₉	1	2	yes	complemented
238	#346	—	89	1	no	—
239	#346	C₂×D₈₉	1	1	yes	characteristic, complemented
240	C₄×C₁₆	—	89	1	no	—
241	C₄×C₁₆	—	89	1	no	—
242	C₂×#41	—	89	2	no	—
243	#137	—	89	2	no	—
244	#137	—	89	2	no	—
245	AGL(1,ℤ/8)	—	89	2	no	—
246	AGL(1,ℤ/8)	—	89	2	no	—
247	#135	—	89	2	no	—
248	#135	—	89	1	no	—
249	#135	C₄×D₈₉	1	1	yes	characteristic
250	#131	—	89	2	no	—
251	#131	—	89	2	no	—
252	#131	—	89	2	no	—
253	#131	—	178	2	no	—
254	#131	—	178	2	no	—
255	#131	C₂²×D₈₉	1	2	yes	—
256	C₂×M₁₆	—	89	1	no	—
257	C₂×M₁₆	—	89	2	no	—
258	C₂×M₁₆	—	178	2	no	—
259	C₂²×C₈	—	89	1	no	—
260	C₂²×C₈	—	178	2	no	—
261	#117	—	89	1	no	—
262	M₃₂	—	2	2	no	—
263	M₃₂	—	89	2	no	—
264	M₃₂	—	178	2	no	—
265	M₃₂	—	178	2	no	—
266	M₃₂	—	178	2	no	—
267	M₃₂	—	178	2	no	—
268	M₃₂	—	178	2	no	—
269	M₃₂	C₂²×D₈₉	1	2	yes	—
270	C₂×C₁₆	—	2	2	no	—
271	C₂×C₁₆	—	89	1	no	—
272	C₂×C₁₆	—	89	2	no	—
273	C₂×C₁₆	—	89	2	no	—
274	C₂×C₁₆	—	89	1	no	—
275	C₂×C₁₆	—	178	2	no	—
276	C₂×C₁₆	—	178	2	no	—
277	C₂×C₁₆	—	178	2	no	—
278	C₂×C₁₆	—	178	2	no	—
279	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
280	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
281	#108	—	89	2	no	—
282	#108	—	89	1	no	—
283	#108	C₄×D₈₉	1	1	yes	characteristic
284	#104	—	89	2	no	—
285	#104	—	89	2	no	—
286	#104	—	89	2	no	—
287	#104	C₄×D₈₉	1	2	yes	—
288	#97	—	89	1	no	—
289	#97	—	89	1	no	—
290	C₄×C₈	—	89	1	no	—
291	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
292	#41	—	89	2	no	—
293	#41	—	89	2	no	—
294	#41	—	89	2	no	—
295	#41	—	178	2	no	—
296	#41	—	178	2	no	—
297	#41	C₂×C₄×D₈₉	1	2	yes	—
298	C₂×Q₈	—	89	2	no	—
299	C₂×D₄	—	89	2	no	—
300	C₂×D₄	—	178	2	no	—
301	C₂²×C₄	—	89	1	no	—
302	C₂²×C₄	—	178	2	no	—
303	Q₁₆	—	89	2	no	—
304	Q₁₆	—	89	1	no	—
305	Q₁₆	C₈×D₈₉	1	1	yes	characteristic
306	SD₁₆	—	89	2	no	—
307	SD₁₆	—	89	2	no	—
308	SD₁₆	—	89	2	no	—
309	SD₁₆	C₈×D₈₉	1	2	yes	—
310	D₈	—	89	2	no	—
311	D₈	—	89	1	no	—
312	D₈	C₈×D₈₉	1	1	yes	characteristic
313	M₁₆	—	2	2	no	—
314	M₁₆	—	89	2	no	—
315	M₁₆	—	89	2	no	—
316	M₁₆	—	89	2	no	—
317	M₁₆	—	178	2	no	—
318	M₁₆	—	178	2	no	—
319	M₁₆	—	178	2	no	—
320	M₁₆	—	178	2	no	—
321	M₁₆	C₂×C₄×D₈₉	1	2	yes	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	89	1	no	—
324	C₂×C₈	—	89	1	no	—
325	C₂×C₈	—	89	2	no	—
326	C₂×C₈	—	89	2	no	—
327	C₂×C₈	—	178	2	no	—
328	C₂×C₈	—	178	2	no	—
329	C₂×C₈	—	178	2	no	—
330	C₂×C₈	—	178	2	no	—
331	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
332	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
333	C₄⋊C₄	—	178	1	no	—
334	C₂²⋊C₄	—	178	1	no	—
335	C₄²	—	89	1	no	—
336	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
337	C₁₆	—	2	2	no	—
338	C₁₆	—	2	1	no	—
339	C₁₆	—	89	1	no	—
340	C₁₆	—	89	1	no	—
341	C₁₆	—	178	2	no	—
342	C₁₆	—	178	1	no	—
343	C₁₆	D₄×D₈₉	1	2	yes	—
344	C₂³	—	89	1	no	—
345	C₂³	—	178	2	no	—
346	Q₈	—	89	2	no	—
347	Q₈	—	178	2	no	—
348	Q₈	C₂×C₈×D₈₉	1	2	yes	—
349	D₄	—	2	2	no	—
350	D₄	—	89	2	no	—
351	D₄	—	178	2	no	—
352	D₄	—	178	2	no	—
353	D₄	—	178	2	no	—
354	D₄	—	178	2	no	—
355	D₄	C₂×C₈×D₈₉	1	2	yes	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	89	1	no	—
359	C₂×C₄	—	89	2	no	—
360	C₂×C₄	—	89	2	no	—
361	C₂×C₄	—	178	2	no	—
362	C₂×C₄	—	178	2	no	—
363	C₂×C₄	—	178	2	no	—
364	C₂×C₄	—	178	1	no	—
365	C₂×C₄	—	178	2	no	—
366	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
367	C₈	—	2	2	no	—
368	C₈	—	89	2	no	—
369	C₈	—	89	2	no	—
370	C₈	—	178	2	no	—
371	C₈	C₂×C₈×D₈₉	1	1	yes	characteristic
372	C₈	C₂×C₈×D₈₉	1	1	yes	characteristic
373	C₈	C₂×D₄×D₈₉	1	1	yes	characteristic
374	C₈	#41×D₈₉	1	1	yes	characteristic
375	C₂²	—	2	2	no	—
376	C₂²	—	89	2	no	—
377	C₂²	—	356	1	no	—
378	C₂²	—	356	4	no	—
379	C₂²	—	178	2	no	—
380	C₂²	#131×D₈₉	1	1	yes	characteristic
381	C₄	—	2	2	no	—
382	C₄	—	4	1	no	—
383	C₄	—	89	2	no	—
384	C₄	—	356	1	no	—
385	C₄	—	178	2	no	—
386	C₄	C₄×D₄×D₈₉	1	1	yes	characteristic
387	C₄	C₂²×C₈×D₈₉	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	4	2	no	—
390	C₂	—	356	2	no	—
391	C₂	—	178	2	no	—
392	C₂	C₈×D₄×D₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₈₉
commutator	C₄×C₈₉	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2171	C₂
socle	C₂×C₈₉	C₂×C₈×D₄
prime radical	#19152395	C₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2171	D₈₉
hypocenter	C₈₉	#33441
LCS	G ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #2171

Complements

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

N	H ≅ G/N	type
1. #504×D₈₉	C₂	semidirect
3. #26022095	C₂	semidirect
4. #26022092	C₂	semidirect
5. #26022089	C₂	semidirect
6. #26022088	C₂	semidirect
7. #26022086	C₂	semidirect
8. #26022016	C₂	semidirect
9. #26021878	C₂	semidirect
10. #26021850	C₂	semidirect
11. C₈₉×#2171	C₂	semidirect
15. #19152016	C₂²	semidirect
16. #19152015	C₂²	semidirect
18. #19152014	C₂²	semidirect
32. #19151419	C₂²	semidirect
33. #19151418	C₂²	semidirect
34. #19151383	C₂²	semidirect
35. #19151382	C₂²	semidirect
36. #19151381	C₂²	semidirect
37. #19151373	C₂²	semidirect
38. C₄×#13083549	C₂²	semidirect
39. C₁₆×Dic₈₉	C₂²	semidirect
40. #504×C₈₉	C₂²	semidirect
42. #364×C₈₉	C₂²	semidirect
43. #350×C₈₉	C₂²	semidirect
44. #346×C₈₉	C₂²	semidirect
201. #2171	D₈₉	semidirect
208. C₈₉	#33441	semidirect
216. #504	C₂×D₈₉	semidirect
233. #364	C₂×D₈₉	semidirect
237. #350	C₂×D₈₉	semidirect
239. #346	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#504×D₈₉	2
#26029150	2
#26022095	2
#26022092	2
#26022089	2
#26022088	2
#26022086	2
#26022016	2
#26021878	2
#26021850	2
C₈₉×#2171	2
#33441	89