Subgroups — #36709342

Order 22784.

546 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029910	C₂	1	1	yes	maximal, characteristic
2	#26028860	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022470	C₂	1	4	yes	maximal, complemented
4	#26022465	C₂	1	4	yes	maximal, complemented
5	C₂×#19151422	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#19151421	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022001	C₂	1	1	yes	maximal, characteristic, complemented
8	#26021872	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2148	C₂	1	1	yes	maximal, characteristic, complemented
10	#19152364	—	2	1	no	—
11	C₂×#15668018	C₂²	1	1	yes	characteristic
12	#19152176	—	2	1	no	—
13	C₂×#15667952	C₂²	1	1	yes	characteristic
14	#19151853	C₂²	1	2	yes	—
15	#19151848	C₂²	1	2	yes	—
16	#19151824	—	2	2	no	—
17	#19151646	—	2	1	no	—
18	#19151645	C₂²	1	4	yes	—
19	C₂×#15667948	C₂²	1	1	yes	characteristic
20	#108×D₈₉	—	2	2	no	—
21	C₂×#15667944	C₂²	1	1	yes	characteristic
22	#19151599	—	2	1	no	—
23	#19151422	C₂²	1	4	yes	—
24	#19151421	C₂²	1	4	yes	—
25	#19151414	C₂²	1	2	yes	—
26	#19151411	C₂²	1	2	yes	—
27	#19151379	C₂²	1	4	yes	—
28	C₂×#135×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×#108×C₈₉	C₂²	1	1	yes	characteristic
31	#362×C₈₉	C₂²	1	2	yes	—
32	#361×C₈₉	C₂²	1	2	yes	—
33	C₂×#13083574	D₄	1	1	yes	characteristic
34	#15668020	—	4	2	no	—
35	#15668018	—	2	1	no	—
36	#15668018	—	2	1	no	—
37	#15668018	—	2	1	no	—
38	#15668018	—	2	1	no	—
39	#15668018	—	4	2	no	—
40	C₂×#13083558	—	2	1	no	—
41	C₂×#13083558	D₄	1	1	yes	characteristic
42	C₂×#13083557	C₂³	1	2	yes	—
43	M₁₆×D₈₉	—	2	4	no	—
44	C₂×C₈×D₈₉	D₄	1	2	yes	—
45	#15667991	—	4	1	no	—
46	#15667988	—	2	2	no	—
47	#15667986	—	4	1	no	—
48	#15667984	—	4	2	no	—
49	#15667983	—	2	2	no	—
50	C₂×C₄⋊Dic₈₉	—	2	1	no	—
51	#15667952	—	2	1	no	—
52	#15667952	C₂³	1	2	yes	—
53	C₂×#13083549	—	2	1	no	—
54	#15667948	—	2	1	no	—
55	#15667948	C₂³	1	2	yes	—
56	#15667947	—	2	4	no	—
57	#15667944	—	2	1	no	—
58	#15667944	C₂³	1	2	yes	—
59	#15667936	—	2	2	no	—
60	#15667933	—	2	2	no	—
61	C₂×#41×C₈₉	—	2	1	no	—
62	#135×C₈₉	—	2	1	no	—
63	#135×C₈₉	—	2	1	no	—
64	#135×C₈₉	—	2	1	no	—
65	#135×C₈₉	—	2	1	no	—
66	C₂×Q₁₆×C₈₉	C₂³	1	1	yes	characteristic
67	C₂×SD₁₆×C₈₉	—	2	1	no	—
68	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
69	C₂×M₁₆×C₈₉	—	2	1	no	—
70	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	M₃₂×C₈₉	—	2	1	no	—
72	M₃₂×C₈₉	C₂³	1	2	yes	—
73	C₂×C₁₆×C₈₉	—	2	1	no	—
74	#108×C₈₉	—	2	1	no	—
75	#108×C₈₉	C₂³	1	2	yes	—
76	#13083574	—	2	1	no	—
77	#13083574	—	2	1	no	—
78	#13083574	—	2	1	no	—
79	#13083574	—	2	1	no	—
80	C₂×Dic₁₇₈	SD₁₆	1	1	yes	characteristic
81	C₂×D₃₅₆	SD₁₆	1	1	yes	characteristic
82	C₂×C₂²⋊D₈₉	—	4	2	no	—
83	C₂×C₂²⋊D₈₉	SD₁₆	1	2	yes	—
84	C₂²×Dic₈₉	—	4	1	no	—
85	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
86	#13083558	—	2	1	no	—
87	#13083558	—	2	1	no	—
88	#13083558	—	2	2	no	—
89	#13083558	—	4	1	no	—
90	#13083557	—	2	2	no	—
91	#13083557	—	2	2	no	—
92	#13083557	—	4	4	no	—
93	C₂×#10623494	—	4	1	no	—
94	C₂×#10623494	C₂×D₄	1	2	yes	—
95	C₈×D₈₉	—	2	2	no	—
96	C₈×D₈₉	—	2	2	no	—
97	C₈×D₈₉	—	4	4	no	—
98	C₄⋊Dic₈₉	—	2	2	no	—
99	C₄⋊Dic₈₉	—	4	1	no	—
100	#13083553	—	4	2	no	—
101	C₂²⋊Dic₈₉	—	4	2	no	—
102	#13083551	—	4	2	no	—
103	#13083549	—	2	2	no	—
104	#41×C₈₉	—	4	1	no	—
105	#41×C₈₉	—	4	1	no	—
106	#41×C₈₉	—	4	1	no	—
107	#41×C₈₉	—	4	1	no	—
108	C₂×Q₈×C₈₉	—	2	1	no	—
109	C₂×D₄×C₈₉	—	2	1	no	—
110	C₂×D₄×C₈₉	—	4	1	no	—
111	C₂²×C₄×C₈₉	—	4	1	no	—
112	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
113	Q₁₆×C₈₉	—	2	1	no	—
114	Q₁₆×C₈₉	—	2	1	no	—
115	SD₁₆×C₈₉	—	4	1	no	—
116	SD₁₆×C₈₉	—	4	1	no	—
117	D₈×C₈₉	—	2	1	no	—
118	D₈×C₈₉	—	2	1	no	—
119	M₁₆×C₈₉	—	2	2	no	—
120	M₁₆×C₈₉	—	4	1	no	—
121	C₂×C₈×C₈₉	—	4	1	no	—
122	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
124	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
125	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
126	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
127	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
128	C₁₆×C₈₉	—	2	2	no	—
129	C₂³×C₈₉	—	4	1	no	—
130	C₂³×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
131	C₂²×D₈₉	C₂×SD₁₆	1	2	yes	—
132	Q₈×C₈₉	—	4	1	no	—
133	Q₈×C₈₉	—	4	1	no	—
134	D₄×C₈₉	—	4	1	no	—
135	D₄×C₈₉	—	4	1	no	—
136	D₄×C₈₉	—	8	1	no	—
137	D₄×C₈₉	—	8	1	no	—
138	C₂×C₄×C₈₉	—	4	1	no	—
139	C₂×C₄×C₈₉	—	8	1	no	—
140	C₂×C₄×C₈₉	—	8	1	no	—
141	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
142	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
143	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
144	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
145	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
146	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
147	C₂²⋊D₈₉	—	2	2	no	—
148	C₂²⋊D₈₉	—	2	2	no	—
149	C₂²⋊D₈₉	—	16	2	no	—
150	C₂×Dic₈₉	—	4	2	no	—
151	C₂×Dic₈₉	—	16	1	no	—
152	C₂×Dic₈₉	C₂×SD₁₆	1	2	yes	—
153	D₃₅₆	—	2	1	no	—
154	D₃₅₆	—	2	1	no	—
155	C₄×D₈₉	—	2	2	no	—
156	C₄×D₈₉	—	2	2	no	—
157	Dic₁₇₈	—	2	1	no	—
158	Dic₁₇₈	—	2	1	no	—
159	C₈×C₈₉	—	4	2	no	—
160	C₈×C₈₉	#123	1	1	yes	characteristic
161	C₈×C₈₉	#123	1	1	yes	characteristic
162	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
163	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
164	#10623494	—	2	2	no	—
165	#10623494	—	4	2	no	—
166	C₂²×C₈₉	—	4	1	no	—
167	C₂²×C₈₉	—	8	1	no	—
168	C₂²×C₈₉	—	8	1	no	—
169	C₂²×C₈₉	—	8	1	no	—
170	C₂²×C₈₉	#481	1	1	yes	characteristic
171	C₂²×C₈₉	#481	1	1	yes	characteristic
172	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
173	C₂×D₈₉	—	2	2	no	—
174	C₂×D₈₉	—	4	2	no	—
175	C₄×C₈₉	—	8	1	no	—
176	C₄×C₈₉	#481	1	1	yes	characteristic
177	C₄×C₈₉	#481	1	1	yes	characteristic
178	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
179	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
180	Dic₈₉	—	2	2	no	—
181	Dic₈₉	—	8	2	no	—
182	#33659	—	89	1	no	maximal, Sylow
183	C₂×C₈₉	—	2	1	no	—
184	C₂×C₈₉	—	2	1	no	—
185	C₂×C₈₉	—	2	1	no	—
186	C₂×C₈₉	—	16	1	no	—
187	C₂×C₈₉	C₂×#481	1	1	yes	characteristic
188	D₈₉	—	4	2	no	—
189	#3060	—	89	1	no	—
190	#2916	—	89	1	no	—
191	#2239	—	89	4	no	—
192	#2238	—	89	4	no	—
193	C₂×#365	—	89	1	no	—
194	C₂×#365	—	89	1	no	—
195	#2148	—	89	1	no	—
196	#2148	D₈₉	1	1	yes	characteristic, complemented
197	#2115	—	89	1	no	—
198	C₈₉	#33659	1	1	yes	characteristic, complemented, Sylow
199	C₂×#135	C₂×D₈₉	1	1	yes	characteristic
200	#568	—	178	1	no	—
201	C₂×#131	—	89	1	no	—
202	#537	—	178	1	no	—
203	C₂×M₃₂	—	89	1	no	—
204	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
205	#470	—	89	2	no	—
206	#469	—	89	2	no	—
207	#465	—	178	2	no	—
208	#430	—	89	4	no	—
209	#430	—	178	1	no	—
210	C₂×#108	—	89	1	no	—
211	C₂×#108	—	178	2	no	—
212	C₂×#108	C₂×D₈₉	1	1	yes	characteristic
213	C₂×#106	—	89	1	no	—
214	#417	—	178	1	no	—
215	#365	—	89	4	no	—
216	#365	—	89	4	no	—
217	#362	—	89	2	no	—
218	#362	C₂×D₈₉	1	2	yes	—
219	#361	—	89	2	no	—
220	#361	C₂×D₈₉	1	2	yes	—
221	#349	—	89	4	no	—
222	C₂×#41	—	2	1	no	—
223	C₂×#41	—	89	1	no	—
224	#135	—	2	1	no	—
225	#135	—	2	1	no	—
226	#135	—	2	1	no	—
227	#135	—	2	1	no	—
228	C₂×Q₁₆	C₂²×D₈₉	1	1	yes	characteristic
229	C₂×SD₁₆	—	2	1	no	—
230	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
231	#131	—	356	2	no	—
232	#131	—	356	2	no	—
233	#131	—	178	1	no	—
234	#131	—	178	1	no	—
235	#131	—	178	1	no	—
236	#131	—	178	1	no	—
237	C₂×M₁₆	—	2	1	no	—
238	C₂×M₁₆	—	89	1	no	—
239	C₂×M₁₆	—	89	2	no	—
240	C₂×M₁₆	—	178	4	no	—
241	C₂×M₁₆	—	178	1	no	—
242	C₂²×C₈	—	89	2	no	—
243	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
244	C₂²⋊Q₈	—	356	1	no	—
245	C₂²⋊Q₈	—	178	2	no	—
246	#121	—	356	1	no	—
247	#121	—	356	2	no	—
248	#121	—	178	2	no	—
249	C₂×C₄⋊C₄	—	178	1	no	—
250	M₃₂	—	2	1	no	—
251	M₃₂	—	89	2	no	—
252	M₃₂	—	178	1	no	—
253	M₃₂	C₂²×D₈₉	1	2	yes	—
254	C₂×C₁₆	—	2	1	no	—
255	C₂×C₁₆	—	178	1	no	—
256	#108	—	2	1	no	—
257	#108	—	89	2	no	—
258	#108	—	178	4	no	—
259	#108	—	178	1	no	—
260	#108	C₂²×D₈₉	1	2	yes	—
261	#106	—	89	2	no	—
262	#106	—	178	1	no	—
263	#103	—	178	2	no	—
264	#102	—	178	2	no	—
265	#41	—	4	1	no	—
266	#41	—	4	1	no	—
267	#41	—	4	1	no	—
268	#41	—	4	1	no	—
269	#41	—	178	1	no	—
270	#41	—	178	1	no	—
271	#41	—	178	1	no	—
272	#41	—	178	1	no	—
273	C₂×Q₈	—	2	1	no	—
274	C₂×Q₈	—	89	1	no	—
275	C₂×D₄	—	2	1	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	89	2	no	—
278	C₂×D₄	—	89	1	no	—
279	C₂×D₄	—	356	2	no	—
280	C₂²×C₄	—	4	1	no	—
281	C₂²×C₄	—	89	2	no	—
282	C₂²×C₄	—	356	1	no	—
283	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
284	Q₁₆	—	2	1	no	—
285	Q₁₆	—	2	1	no	—
286	SD₁₆	—	4	1	no	—
287	SD₁₆	—	4	1	no	—
288	D₈	—	2	1	no	—
289	D₈	—	2	1	no	—
290	M₁₆	—	2	2	no	—
291	M₁₆	—	4	1	no	—
292	M₁₆	—	356	4	no	—
293	M₁₆	—	356	1	no	—
294	M₁₆	—	178	2	no	—
295	M₁₆	—	178	1	no	—
296	M₁₆	—	178	1	no	—
297	M₁₆	—	178	2	no	—
298	M₁₆	—	178	2	no	—
299	C₂×C₈	—	4	1	no	—
300	C₂×C₈	—	89	2	no	—
301	C₂×C₈	—	356	4	no	—
302	C₂×C₈	—	356	1	no	—
303	C₂×C₈	—	178	2	no	—
304	C₂×C₈	—	178	2	no	—
305	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
306	C₂×C₈	#13083573	1	1	yes	characteristic
307	C₂×C₈	#13083573	1	1	yes	characteristic
308	C₂×C₈	#13083574	1	1	yes	characteristic
309	C₂×C₈	#13083574	1	1	yes	characteristic
310	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
311	C₄⋊C₄	—	356	2	no	—
312	C₄⋊C₄	—	356	1	no	—
313	C₄⋊C₄	—	178	2	no	—
314	C₂²⋊C₄	—	356	2	no	—
315	C₂²⋊C₄	—	356	2	no	—
316	C₁₆	—	2	2	no	—
317	C₁₆	—	178	2	no	—
318	C₂³	—	4	1	no	—
319	C₂³	—	89	2	no	—
320	C₂³	SD₁₆×D₈₉	1	1	yes	characteristic
321	Q₈	—	4	1	no	—
322	Q₈	—	4	1	no	—
323	Q₈	—	178	1	no	—
324	Q₈	—	178	1	no	—
325	D₄	—	4	1	no	—
326	D₄	—	4	1	no	—
327	D₄	—	8	1	no	—
328	D₄	—	8	1	no	—
329	D₄	—	1424	2	no	—
330	D₄	—	178	2	no	—
331	D₄	—	178	2	no	—
332	D₄	—	178	1	no	—
333	D₄	—	178	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	8	1	no	—
336	C₂×C₄	—	8	1	no	—
337	C₂×C₄	—	89	2	no	—
338	C₂×C₄	—	356	2	no	—
339	C₂×C₄	—	1424	1	no	—
340	C₂×C₄	—	178	2	no	—
341	C₂×C₄	—	178	2	no	—
342	C₂×C₄	#15667994	1	1	yes	characteristic
343	C₂×C₄	#15667994	1	1	yes	characteristic
344	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
345	C₂×C₄	#15668036	1	1	yes	characteristic
346	C₂×C₄	#15668036	1	1	yes	characteristic
347	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
348	C₈	—	4	2	no	—
349	C₈	—	356	2	no	—
350	C₈	—	178	2	no	—
351	C₈	#15667994	1	1	yes	characteristic
352	C₈	#15667994	1	1	yes	characteristic
353	C₈	C₂×#13083573	1	1	yes	characteristic
354	C₈	C₂×#13083574	1	1	yes	characteristic
355	C₂²	—	4	1	no	—
356	C₂²	—	8	1	no	—
357	C₂²	—	8	1	no	—
358	C₂²	—	8	1	no	—
359	C₂²	—	356	2	no	—
360	C₂²	—	178	2	no	—
361	C₂²	#19151915	1	1	yes	characteristic
362	C₂²	#19151915	1	1	yes	characteristic
363	C₂²	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
364	C₄	—	8	1	no	—
365	C₄	—	178	2	no	—
366	C₄	—	712	2	no	—
367	C₄	#19151915	1	1	yes	characteristic
368	C₄	#19151915	1	1	yes	characteristic
369	C₄	C₂×#15667994	1	1	yes	characteristic
370	C₄	C₂×#15668036	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	2	1	no	—
374	C₂	—	16	1	no	—
375	C₂	—	356	2	no	—
376	C₂	C₂×#19151915	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#19151915
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2148	C₂
socle	C₂×C₈₉	C₂×#481
prime radical	#26029910	C₂
hypercenter	#2148	D₈₉
hypocenter	C₈₉	#33659
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2148

Complements

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

N	H ≅ G/N	type
2. #26028860	C₂	semidirect
3. #26022470	C₂	semidirect
4. #26022465	C₂	semidirect
5. C₂×#19151422	C₂	semidirect
6. C₂×#19151421	C₂	semidirect
7. #26022001	C₂	semidirect
8. #26021872	C₂	semidirect
9. C₈₉×#2148	C₂	semidirect
196. #2148	D₈₉	semidirect
198. C₈₉	#33659	semidirect

Maximal subgroups

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

subgroup	index
#26029910	2
#26028860	2
#26022470	2
#26022465	2
C₂×#19151422	2
C₂×#19151421	2
#26022001	2
#26021872	2
C₈₉×#2148	2
#33659	89