Subgroups — #36447641

Order 22784.

434 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028963	C₂	1	1	yes	maximal, characteristic
2	#26020414	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018998	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018813	C₂	1	1	yes	maximal, characteristic
5	#26018811	C₂	1	1	yes	maximal, characteristic
6	#26018112	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1286	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668020	—	2	1	no	—
9	#19152148	C₂²	1	1	yes	characteristic
10	#19151675	—	2	1	no	—
11	#19151675	C₄	1	2	yes	complemented
12	#19151663	—	2	1	no	—
13	#19151663	C₄	1	2	yes	complemented
14	#19151651	C₂²	1	1	yes	characteristic
15	C₂×#15667945	—	2	1	no	—
16	C₂×#15667943	—	2	1	no	—
17	#19151566	—	2	1	no	—
18	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
19	#19151464	C₂²	1	1	yes	characteristic
20	#19151350	—	2	1	no	—
21	#19151348	—	2	1	no	—
22	#19151333	—	2	2	no	—
23	#19151331	—	2	2	no	—
24	#431×C₈₉	C₂²	1	1	yes	characteristic
25	#423×C₈₉	C₂²	1	1	yes	characteristic
26	#382×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#13083574	—	2	1	no	—
28	#15668020	—	4	2	no	—
29	#15668020	—	4	2	no	—
30	C₂×#13083557	—	2	2	no	—
31	C₂²×#10623494	D₄	1	1	yes	characteristic
32	C₂²×#10623494	D₄	1	1	yes	characteristic
33	C₂×C₈×D₈₉	—	2	2	no	—
34	#15668012	Q₈	1	1	yes	characteristic
35	#15668009	Q₈	1	1	yes	characteristic
36	#15668000	—	2	1	no	—
37	#15667986	—	2	1	no	—
38	#15667986	D₄	1	2	yes	—
39	C₄×D₃₅₆	—	2	1	no	—
40	C₄×D₃₅₆	C₂×C₄	1	2	yes	—
41	C₂×C₄⋊Dic₈₉	—	2	1	no	—
42	#15667945	—	4	2	no	—
43	#15667943	—	4	2	no	—
44	C₄⋊#10623494	—	2	1	no	—
45	C₂²⋊#10623494	—	2	1	no	—
46	#15667924	—	2	2	no	—
47	#15667924	—	2	2	no	—
48	C₄×#10623494	—	2	1	no	—
49	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
50	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
51	C₄×#10623494	D₄	1	1	yes	characteristic
52	C₄×#10623494	D₄	1	1	yes	characteristic
53	#15667919	—	2	1	no	—
54	#15667919	—	2	1	no	—
55	C₂×M₁₆×C₈₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
57	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₈₉	—	2	1	no	—
59	C₄⋊C₈×C₈₉	—	2	1	no	—
60	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
61	C₂²⋊C₈×C₈₉	—	2	1	no	—
62	C₄×C₈×C₈₉	—	2	1	no	—
63	#95×C₈₉	—	2	1	no	—
64	#95×C₈₉	—	2	1	no	—
65	#13083574	—	4	1	no	—
66	#13083574	—	4	1	no	—
67	#13083574	—	4	1	no	—
68	#13083574	—	4	1	no	—
69	C₂×Dic₁₇₈	—	2	1	no	—
70	C₂×D₃₅₆	—	2	1	no	—
71	C₂×D₃₅₆	—	2	1	no	—
72	C₂×D₃₅₆	C₄⋊C₄	1	2	yes	—
73	C₂×C₂²⋊D₈₉	—	2	2	no	—
74	C₂²×Dic₈₉	—	4	1	no	—
75	C₂×C₄×D₈₉	—	2	2	no	—
76	#13083557	—	4	4	no	—
77	C₂×#10623494	—	2	1	no	—
78	C₂×#10623494	—	2	1	no	—
79	C₂×#10623494	—	4	1	no	—
80	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×#10623494	D₈	1	2	yes	—
85	C₂×#10623494	SD₁₆	1	2	yes	—
86	C₈×D₈₉	—	4	4	no	—
87	C₄⋊Dic₈₉	—	2	1	no	—
88	C₄⋊Dic₈₉	—	4	1	no	—
89	C₄⋊Dic₈₉	—	4	1	no	—
90	C₄⋊Dic₈₉	C₄⋊C₄	1	2	yes	—
91	#13083551	—	2	2	no	—
92	#13083551	—	2	2	no	—
93	C₂²×C₄×C₈₉	—	4	1	no	—
94	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
97	M₁₆×C₈₉	—	4	1	no	—
98	M₁₆×C₈₉	—	4	1	no	—
99	C₂×C₈×C₈₉	—	2	2	no	—
100	C₂×C₈×C₈₉	—	4	1	no	—
101	C₄⋊C₄×C₈₉	—	4	1	no	—
102	C₄⋊C₄×C₈₉	—	4	1	no	—
103	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
106	C₄²×C₈₉	#41	1	1	yes	characteristic
107	C₂³×C₈₉	#121	1	1	yes	characteristic
108	C₂²×D₈₉	—	2	2	no	—
109	C₂×C₄×C₈₉	—	4	1	no	—
110	C₂×C₄×C₈₉	—	4	1	no	—
111	C₂×C₄×C₈₉	—	8	1	no	—
112	C₂×C₄×C₈₉	—	8	1	no	—
113	C₂×C₄×C₈₉	—	8	1	no	—
114	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
117	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
124	C₂²⋊D₈₉	—	4	2	no	—
125	C₂²⋊D₈₉	—	4	2	no	—
126	C₂×Dic₈₉	—	2	2	no	—
127	C₂×Dic₈₉	—	8	1	no	—
128	C₂×Dic₈₉	—	8	1	no	—
129	C₂×Dic₈₉	—	8	1	no	—
130	D₃₅₆	—	4	2	no	—
131	D₃₅₆	—	4	2	no	—
132	D₃₅₆	—	4	1	no	—
133	D₃₅₆	—	4	1	no	—
134	C₄×D₈₉	—	4	2	no	—
135	C₄×D₈₉	—	4	2	no	—
136	Dic₁₇₈	—	4	1	no	—
137	Dic₁₇₈	—	4	1	no	—
138	C₈×C₈₉	—	4	2	no	—
139	C₈×C₈₉	—	8	1	no	—
140	#10623494	—	2	1	no	—
141	#10623494	—	2	1	no	—
142	#10623494	—	8	1	no	—
143	#10623494	#102	1	2	yes	—
144	#10623494	#102	1	2	yes	—
145	C₂²×C₈₉	—	2	1	no	—
146	C₂²×C₈₉	—	2	1	no	—
147	C₂²×C₈₉	#380	1	1	yes	characteristic
148	C₂²×C₈₉	#459	1	1	yes	characteristic
149	C₂²×C₈₉	#461	1	1	yes	characteristic
150	C₂×D₈₉	—	4	2	no	—
151	C₂×D₈₉	—	4	2	no	—
152	C₂×D₈₉	—	4	2	no	—
153	C₄×C₈₉	—	2	1	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	—	2	1	no	—
156	C₄×C₈₉	—	2	1	no	—
157	C₄×C₈₉	—	16	1	no	—
158	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
159	C₄×C₈₉	#417	1	1	yes	characteristic
160	C₄×C₈₉	#474	1	1	yes	characteristic
161	C₄×C₈₉	#476	1	1	yes	characteristic
162	Dic₈₉	—	4	2	no	—
163	Dic₈₉	—	16	1	no	—
164	#8676	—	89	1	no	maximal, Sylow
165	C₂×C₈₉	—	2	1	no	—
166	C₂×C₈₉	—	2	1	no	—
167	C₂×C₈₉	#1802	1	1	yes	characteristic
168	C₂×C₈₉	#1812	1	1	yes	characteristic
169	C₂×C₈₉	#1814	1	1	yes	characteristic
170	D₈₉	—	8	2	no	—
171	#2931	—	89	1	no	—
172	#1846	—	89	1	no	—
173	#1564	—	89	1	no	—
174	#1517	—	89	1	no	—
175	#1517	—	89	1	no	—
176	#1286	—	89	1	no	—
177	#1286	D₈₉	1	1	yes	characteristic, complemented
178	C₈₉	#8676	1	1	yes	characteristic, complemented, Sylow
179	C₂×#131	—	178	1	no	—
180	#532	—	89	1	no	—
181	#436	—	89	2	no	—
182	#436	—	178	1	no	—
183	C₈×D₄	—	89	2	no	—
184	C₈×D₄	—	178	1	no	—
185	#431	—	89	1	no	—
186	#431	C₂×D₈₉	1	1	yes	characteristic
187	C₂×#107	—	178	1	no	—
188	C₂×#106	—	178	1	no	—
189	#423	C₂×D₈₉	1	1	yes	characteristic
190	#408	—	178	1	no	—
191	C₂×C₄×C₈	—	89	1	no	—
192	#382	—	89	1	no	—
193	#382	C₂×D₈₉	1	1	yes	characteristic
194	#335	—	178	1	no	—
195	#334	—	178	1	no	—
196	#327	—	178	2	no	—
197	#327	—	178	2	no	—
198	C₂×#41	—	178	1	no	—
199	#131	—	356	2	no	—
200	#131	—	356	2	no	—
201	C₂×M₁₆	—	2	1	no	—
202	C₂×M₁₆	—	178	2	no	—
203	C₂²×C₈	—	89	1	no	—
204	C₂²×C₈	—	89	1	no	—
205	C₂²×C₈	—	178	2	no	—
206	C₄⋊Q₈	—	89	1	no	—
207	#127	—	89	1	no	—
208	#124	—	178	1	no	—
209	#121	—	89	2	no	—
210	#121	—	178	1	no	—
211	C₄×D₄	—	89	2	no	—
212	C₄×D₄	—	178	1	no	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	178	1	no	—
215	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
216	#107	—	356	2	no	—
217	#106	—	356	2	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	178	1	no	—
221	C₄⋊C₈	C₄×D₈₉	1	2	yes	—
222	C₂²⋊C₈	—	2	1	no	—
223	C₂²⋊C₈	—	178	2	no	—
224	C₂²⋊C₈	—	178	2	no	—
225	C₂²⋊C₈	—	178	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₄×C₈	—	89	1	no	—
228	C₄×C₈	—	89	1	no	—
229	C₄×C₈	—	89	1	no	—
230	C₄×C₈	—	89	1	no	—
231	C₄×C₈	—	178	1	no	—
232	#95	—	2	1	no	—
233	#95	—	2	1	no	—
234	#95	—	178	1	no	—
235	#95	—	178	1	no	—
236	#41	—	356	1	no	—
237	#41	—	356	1	no	—
238	#41	—	356	1	no	—
239	#41	—	356	1	no	—
240	C₂×Q₈	—	178	1	no	—
241	C₂×D₄	—	89	2	no	—
242	C₂×D₄	—	178	2	no	—
243	C₂×D₄	—	178	1	no	—
244	C₂×D₄	—	178	1	no	—
245	C₂²×C₄	—	4	1	no	—
246	C₂²×C₄	—	356	1	no	—
247	C₂²×C₄	—	178	2	no	—
248	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
249	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
250	C₂²×C₄	#13083574	1	1	yes	characteristic
251	M₁₆	—	4	1	no	—
252	M₁₆	—	4	1	no	—
253	M₁₆	—	356	4	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	89	2	no	—
257	C₂×C₈	—	89	2	no	—
258	C₂×C₈	—	89	1	no	—
259	C₂×C₈	—	89	1	no	—
260	C₂×C₈	—	89	1	no	—
261	C₂×C₈	—	89	1	no	—
262	C₂×C₈	—	356	4	no	—
263	C₂×C₈	—	356	1	no	—
264	C₂×C₈	—	178	1	no	—
265	C₂×C₈	—	178	1	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	89	2	no	—
269	C₄⋊C₄	—	356	1	no	—
270	C₄⋊C₄	—	356	1	no	—
271	C₄⋊C₄	—	178	1	no	—
272	C₂²⋊C₄	—	178	2	no	—
273	C₂²⋊C₄	—	178	2	no	—
274	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
275	C₄²	D₄×D₈₉	1	1	yes	characteristic
276	C₄²	Q₈×D₈₉	1	1	yes	characteristic
277	C₄²	#13083574	1	1	yes	characteristic
278	C₂³	—	178	2	no	—
279	C₂³	#15667985	1	1	yes	characteristic
280	Q₈	—	356	1	no	—
281	Q₈	—	356	1	no	—
282	D₄	—	356	2	no	—
283	D₄	—	356	2	no	—
284	D₄	—	356	2	no	—
285	D₄	—	356	2	no	—
286	D₄	—	356	1	no	—
287	D₄	—	356	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	356	2	no	—
294	C₂×C₄	—	356	2	no	—
295	C₂×C₄	—	178	2	no	—
296	C₂×C₄	—	712	1	no	—
297	C₂×C₄	—	712	1	no	—
298	C₂×C₄	—	712	1	no	—
299	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
300	C₂×C₄	C₄⋊C₄×D₈₉	1	1	yes	characteristic
301	C₂×C₄	#15667969	1	1	yes	characteristic
302	C₂×C₄	#15667985	1	1	yes	characteristic
303	C₂×C₄	#15667990	1	1	yes	characteristic
304	C₂×C₄	#15667990	1	1	yes	characteristic
305	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
306	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
307	C₂×C₄	#15668039	1	1	yes	characteristic
308	C₂×C₄	#15668044	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	—	8	1	no	—
311	C₈	—	89	2	no	—
312	C₈	—	89	2	no	—
313	C₈	—	178	1	no	—
314	C₈	—	178	1	no	—
315	C₈	—	712	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	356	2	no	—
319	C₂²	—	356	2	no	—
320	C₂²	—	356	2	no	—
321	C₂²	#19151457	1	1	yes	characteristic
322	C₂²	#19151792	1	1	yes	characteristic
323	C₂²	#19151802	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	16	1	no	—
329	C₄	—	356	2	no	—
330	C₄	—	1424	1	no	—
331	C₄	#102×D₈₉	1	1	yes	characteristic
332	C₄	#19151598	1	1	yes	characteristic
333	C₄	#19151877	1	1	yes	characteristic
334	C₄	#19151887	1	1	yes	characteristic
335	C₂	—	2	1	no	—
336	C₂	—	2	1	no	—
337	C₂	—	712	2	no	—
338	C₂	#26020251	1	1	yes	characteristic
339	C₂	#26020297	1	1	yes	characteristic
340	C₂	#26020305	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151457
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1286	C₂
socle	C₂²×C₈₉	#380
prime radical	#19152148	C₂²
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1286	D₈₉
hypocenter	C₈₉	#8676
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1286

Complements

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

N	H ≅ G/N	type
2. #26020414	C₂	semidirect
3. #26018998	C₂	semidirect
6. #26018112	C₂	semidirect
7. C₈₉×#1286	C₂	semidirect
11. #19151675	C₄	semidirect
13. #19151663	C₄	semidirect
177. #1286	D₈₉	semidirect
178. C₈₉	#8676	semidirect

Maximal subgroups

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

subgroup	index
#26028963	2
#26020414	2
#26018998	2
#26018813	2
#26018811	2
#26018112	2
C₈₉×#1286	2
#8676	89