Subgroups — #36712425

Order 22784.

508 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033242	C₂	1	1	yes	maximal, characteristic, complemented
2	#26033238	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022512	C₂	1	2	yes	maximal
4	#26022504	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022317	C₂	1	2	yes	maximal, complemented
6	#26022312	C₂	1	2	yes	maximal, complemented
7	#26022298	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022154	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022108	C₂	1	2	yes	maximal, complemented
10	C₄×#15667953	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2208	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152322	—	2	1	no	—
13	#19152318	—	2	1	no	—
14	C₂×#15667958	C₂²	1	2	yes	—
15	C₂×#15667958	C₂²	1	2	yes	—
16	C₂×#15667957	C₂²	1	2	yes	—
17	C₂×#15667953	C₂²	1	1	yes	characteristic
18	#19151983	C₂²	1	2	yes	—
19	#19151981	C₂²	1	2	yes	—
20	#19151978	C₂²	1	1	yes	characteristic
21	#19151822	—	2	2	no	—
22	#19151819	—	2	2	no	—
23	#19151803	—	2	1	no	—
24	#19151800	—	2	1	no	—
25	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
26	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
27	#19151692	C₂²	1	2	yes	—
28	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
29	#19151423	C₂²	1	1	yes	characteristic
30	#19151416	C₂²	1	2	yes	—
31	#19151403	C₂²	1	2	yes	—
32	#19151399	C₂²	1	1	yes	characteristic
33	#19151398	C₂²	1	1	yes	characteristic
34	C₄×#13083549	C₂²	1	1	yes	characteristic
35	C₁₆×Dic₈₉	C₂²	1	2	yes	—
36	C₂×SD₃₂×C₈₉	C₂²	1	2	yes	—
37	#494×C₈₉	C₂²	1	1	yes	characteristic
38	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
40	#358×C₈₉	C₂²	1	2	yes	—
41	C₂×Dic₃₅₆	—	2	1	no	—
42	C₂×Dic₃₅₆	C₂³	1	2	yes	—
43	C₂×#13083562	—	2	2	no	—
44	C₂×#13083561	—	2	2	no	—
45	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
46	#15668012	—	2	1	no	—
47	#15668007	—	4	2	no	—
48	#15668005	—	4	2	no	—
49	#15668001	—	2	2	no	—
50	#15667999	—	2	2	no	—
51	#15667992	—	4	1	no	—
52	#15667991	—	4	1	no	—
53	Q₈×Dic₈₉	—	2	2	no	—
54	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
55	#15667977	—	2	2	no	—
56	C₄×D₃₅₆	—	2	1	no	—
57	#15667971	—	4	1	no	—
58	#15667970	—	4	1	no	—
59	#15667958	—	2	4	no	—
60	#15667958	—	4	2	no	—
61	#15667957	—	4	2	no	—
62	#15667953	—	2	2	no	—
63	C₂×#13083549	C₂³	1	1	yes	characteristic
64	C₂×#13083549	C₂³	1	1	yes	characteristic
65	#15667946	C₂³	1	1	yes	characteristic
66	C₄⋊#10623494	D₄	1	1	yes	characteristic
67	#15667941	D₄	1	2	yes	—
68	#15667936	—	2	2	no	—
69	#15667935	—	2	2	no	—
70	#15667934	—	2	1	no	—
71	C₈×Dic₈₉	C₂³	1	2	yes	—
72	C₂×Q₁₆×C₈₉	—	2	1	no	—
73	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
74	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₄⋊Q₈×C₈₉	—	2	1	no	—
76	C₄×D₄×C₈₉	—	2	1	no	—
77	SD₃₂×C₈₉	—	4	2	no	—
78	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
79	#107×C₈₉	C₂³	1	1	yes	characteristic
80	#102×C₈₉	—	2	1	no	—
81	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₂×Dic₁₇₈	—	2	2	no	—
83	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
84	C₂×D₃₅₆	—	2	1	no	—
85	C₂×C₄×D₈₉	—	4	1	no	—
86	Dic₃₅₆	—	2	2	no	—
87	Dic₃₅₆	—	2	2	no	—
88	Dic₃₅₆	—	8	1	no	—
89	#13083562	—	8	2	no	—
90	#13083561	—	8	2	no	—
91	D₇₁₂	—	2	1	no	—
92	D₇₁₂	—	2	1	no	—
93	C₂×#10623494	C₂×D₄	1	2	yes	—
94	C₄⋊Dic₈₉	—	2	1	no	—
95	C₄⋊Dic₈₉	—	4	1	no	—
96	C₄⋊Dic₈₉	—	4	1	no	—
97	C₄⋊Dic₈₉	—	4	2	no	—
98	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
99	#13083553	—	4	2	no	—
100	#13083553	D₈	1	2	yes	—
101	C₂²⋊Dic₈₉	—	4	2	no	—
102	C₂²⋊Dic₈₉	—	4	2	no	—
103	#13083551	—	4	1	no	—
104	#13083551	—	4	2	no	—
105	#13083551	—	4	2	no	—
106	C₄×Dic₈₉	—	4	2	no	—
107	C₄×Dic₈₉	—	4	2	no	—
108	C₄×Dic₈₉	C₂×D₄	1	2	yes	—
109	#13083549	—	2	1	no	—
110	#13083549	#41	1	2	yes	—
111	C₂×Q₈×C₈₉	—	2	2	no	—
112	C₂×D₄×C₈₉	—	2	1	no	—
113	C₂²×C₄×C₈₉	—	4	1	no	—
114	Q₁₆×C₈₉	—	2	2	no	—
115	Q₁₆×C₈₉	—	2	2	no	—
116	Q₁₆×C₈₉	—	8	1	no	—
117	D₈×C₈₉	—	2	1	no	—
118	D₈×C₈₉	—	2	1	no	—
119	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
121	C₄⋊C₄×C₈₉	—	2	1	no	—
122	C₄⋊C₄×C₈₉	—	4	1	no	—
123	C₄⋊C₄×C₈₉	—	4	1	no	—
124	C₂²⋊C₄×C₈₉	—	4	1	no	—
125	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
126	C₁₆×C₈₉	—	2	2	no	—
127	C₂³×C₈₉	—	4	1	no	—
128	C₂²×D₈₉	—	4	1	no	—
129	Q₈×C₈₉	—	4	2	no	—
130	Q₈×C₈₉	—	4	2	no	—
131	D₄×C₈₉	—	4	1	no	—
132	D₄×C₈₉	—	4	1	no	—
133	C₂×C₄×C₈₉	—	4	1	no	—
134	C₂×C₄×C₈₉	—	4	2	no	—
135	C₂×C₄×C₈₉	—	8	2	no	—
136	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×Dic₈₉	—	4	1	no	—
140	C₂×Dic₈₉	—	4	2	no	—
141	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
142	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
143	D₃₅₆	—	4	1	no	—
144	D₃₅₆	—	4	1	no	—
145	C₄×D₈₉	—	8	2	no	—
146	Dic₁₇₈	—	2	2	no	—
147	Dic₁₇₈	—	4	2	no	—
148	Dic₁₇₈	—	4	2	no	—
149	C₈×C₈₉	—	2	1	no	—
150	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
151	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
152	#10623494	—	2	2	no	—
153	C₂²×C₈₉	—	8	1	no	—
154	C₂²×C₈₉	—	8	1	no	—
155	C₂²×C₈₉	—	8	1	no	—
156	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
157	C₂×D₈₉	—	8	1	no	—
158	C₂×D₈₉	—	8	1	no	—
159	C₂×D₈₉	—	8	1	no	—
160	C₄×C₈₉	—	2	1	no	—
161	C₄×C₈₉	—	8	2	no	—
162	C₄×C₈₉	—	8	1	no	—
163	C₄×C₈₉	#508	1	2	yes	—
164	C₄×C₈₉	#550	1	1	yes	characteristic
165	C₄×C₈₉	#576	1	1	yes	characteristic
166	Dic₈₉	—	2	2	no	—
167	Dic₈₉	—	2	2	no	—
168	Dic₈₉	—	8	2	no	—
169	Dic₈₉	—	8	1	no	—
170	#33960	—	89	1	no	maximal, Sylow
171	C₂×C₈₉	—	16	1	no	—
172	C₂×C₈₉	#3391	1	1	yes	characteristic
173	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
174	C₂×C₈₉	#3417	1	1	yes	characteristic
175	D₈₉	—	16	1	no	—
176	#3393	—	89	1	no	—
177	#3393	—	89	1	no	—
178	#2249	—	89	2	no	—
179	#2247	—	89	1	no	—
180	#2211	—	89	2	no	—
181	#2211	—	89	2	no	—
182	#2208	—	89	1	no	—
183	#2208	D₈₉	1	1	yes	characteristic, complemented
184	Q₈×C₁₆	—	89	1	no	—
185	C₄×Q₃₂	—	89	2	no	—
186	C₄×D₁₆	—	89	1	no	—
187	C₈₉	#33960	1	1	yes	characteristic, complemented, Sylow
188	#556	—	178	1	no	—
189	#556	—	178	1	no	—
190	C₂×Q₃₂	—	89	2	no	—
191	C₂×Q₃₂	—	89	2	no	—
192	C₂×SD₃₂	—	89	2	no	—
193	C₂×SD₃₂	C₂×D₈₉	1	2	yes	—
194	C₂×D₁₆	—	89	1	no	—
195	#495	—	89	2	no	—
196	#495	—	89	2	no	—
197	#494	—	89	1	no	—
198	#494	C₂×D₈₉	1	1	yes	characteristic
199	#464	—	178	2	no	—
200	#464	—	178	2	no	—
201	#461	—	178	1	no	—
202	#461	—	178	1	no	—
203	C₈×Q₈	—	89	1	no	—
204	C₄×Q₁₆	—	89	2	no	—
205	C₄×Q₁₆	—	89	2	no	—
206	C₄×D₈	—	89	1	no	—
207	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
208	#366	—	89	1	no	—
209	C₄⋊C₁₆	—	89	2	no	—
210	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
211	#358	—	89	2	no	—
212	#358	C₂×D₈₉	1	2	yes	—
213	#357	—	89	1	no	—
214	#357	—	89	1	no	—
215	C₄×C₁₆	—	89	2	no	—
216	C₄×C₁₆	—	89	1	no	—
217	C₂×Q₁₆	—	2	1	no	—
218	C₂×Q₁₆	—	89	2	no	—
219	C₂×Q₁₆	—	178	1	no	—
220	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
221	C₂×SD₁₆	—	178	2	no	—
222	C₂×SD₁₆	—	178	2	no	—
223	C₂×D₈	—	89	1	no	—
224	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
225	C₄⋊Q₈	—	2	1	no	—
226	C₄⋊Q₈	—	178	1	no	—
227	#126	—	356	2	no	—
228	#126	—	356	2	no	—
229	#124	—	178	2	no	—
230	#124	—	178	2	no	—
231	C₂²⋊Q₈	—	356	1	no	—
232	C₂²⋊Q₈	—	356	1	no	—
233	C₄×Q₈	—	89	1	no	—
234	C₄×Q₈	—	178	2	no	—
235	C₄×Q₈	—	178	2	no	—
236	C₄×D₄	—	2	1	no	—
237	C₄×D₄	—	178	1	no	—
238	#117	—	356	1	no	—
239	#117	—	356	1	no	—
240	Q₃₂	—	356	2	no	—
241	Q₃₂	—	178	4	no	—
242	SD₃₂	—	4	2	no	—
243	SD₃₂	—	356	2	no	—
244	D₁₆	—	178	2	no	—
245	C₂×C₁₆	—	89	1	no	—
246	C₂×C₁₆	—	89	1	no	—
247	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
248	#107	—	89	1	no	—
249	#107	C₂²×D₈₉	1	1	yes	characteristic
250	C₄⋊C₈	—	89	2	no	—
251	C₄⋊C₈	—	89	1	no	—
252	#103	—	178	2	no	—
253	#103	—	178	2	no	—
254	#102	—	2	1	no	—
255	#102	—	178	1	no	—
256	C₄×C₈	—	89	2	no	—
257	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	—	89	1	no	—
260	C₂×Q₈	—	178	2	no	—
261	C₂×D₄	—	2	1	no	—
262	C₂×D₄	—	178	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	356	1	no	—
265	Q₁₆	—	2	2	no	—
266	Q₁₆	—	2	2	no	—
267	Q₁₆	—	8	1	no	—
268	Q₁₆	—	178	2	no	—
269	Q₁₆	—	178	2	no	—
270	Q₁₆	—	712	1	no	—
271	SD₁₆	—	712	2	no	—
272	SD₁₆	—	712	2	no	—
273	D₈	—	2	1	no	—
274	D₈	—	2	1	no	—
275	D₈	—	178	1	no	—
276	D₈	—	178	1	no	—
277	C₂×C₈	—	89	2	no	—
278	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
279	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	89	1	no	—
284	C₄⋊C₄	—	89	2	no	—
285	C₄⋊C₄	—	356	2	no	—
286	C₄⋊C₄	—	356	2	no	—
287	C₄⋊C₄	—	356	1	no	—
288	C₄⋊C₄	—	356	1	no	—
289	C₄⋊C₄	—	178	1	no	—
290	C₂²⋊C₄	—	4	1	no	—
291	C₂²⋊C₄	—	356	2	no	—
292	C₂²⋊C₄	—	356	2	no	—
293	C₂²⋊C₄	—	356	2	no	—
294	C₂²⋊C₄	—	356	2	no	—
295	C₂²⋊C₄	—	356	1	no	—
296	C₄²	—	89	2	no	—
297	C₄²	—	356	2	no	—
298	C₄²	—	356	2	no	—
299	C₄²	D₄×D₈₉	1	1	yes	characteristic
300	C₁₆	—	2	2	no	—
301	C₁₆	—	89	2	no	—
302	C₁₆	—	178	1	no	—
303	C₂³	—	4	1	no	—
304	C₂³	—	356	1	no	—
305	Q₈	—	4	2	no	—
306	Q₈	—	4	2	no	—
307	Q₈	—	356	2	no	—
308	Q₈	—	356	2	no	—
309	Q₈	—	178	2	no	—
310	D₄	—	4	1	no	—
311	D₄	—	4	1	no	—
312	D₄	—	356	1	no	—
313	D₄	—	356	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	8	2	no	—
317	C₂×C₄	—	89	2	no	—
318	C₂×C₄	—	89	2	no	—
319	C₂×C₄	—	356	1	no	—
320	C₂×C₄	—	356	2	no	—
321	C₂×C₄	—	712	2	no	—
322	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
323	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
324	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
325	C₈	—	2	1	no	—
326	C₈	—	178	2	no	—
327	C₈	#41×D₈₉	1	1	yes	characteristic
328	C₈	#15668057	1	1	yes	characteristic
329	C₂²	—	8	1	no	—
330	C₂²	—	8	1	no	—
331	C₂²	—	8	1	no	—
332	C₂²	—	712	1	no	—
333	C₂²	—	712	1	no	—
334	C₂²	—	712	1	no	—
335	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	8	2	no	—
338	C₄	—	8	1	no	—
339	C₄	—	178	2	no	—
340	C₄	—	178	2	no	—
341	C₄	—	712	2	no	—
342	C₄	—	712	1	no	—
343	C₄	#19152029	1	2	yes	—
344	C₄	#19152266	1	1	yes	characteristic
345	C₄	#19152402	1	1	yes	characteristic
346	C₂	—	16	1	no	—
347	C₂	—	1424	1	no	—
348	C₂	#26033223	1	1	yes	characteristic
349	C₂	C₂×#19152029	1	1	yes	characteristic
350	C₂	#26033404	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2208	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#15667953	C₂²
hypercenter	#2208	D₈₉
hypocenter	C₈₉	#33960
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2208

Complements

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

N	H ≅ G/N	type
1. #26033242	C₂	semidirect
2. #26033238	C₂	semidirect
4. #26022504	C₂	semidirect
5. #26022317	C₂	semidirect
6. #26022312	C₂	semidirect
7. #26022298	C₂	semidirect
8. #26022154	C₂	semidirect
9. #26022108	C₂	semidirect
11. C₈₉×#2208	C₂	semidirect
183. #2208	D₈₉	semidirect
187. C₈₉	#33960	semidirect

Maximal subgroups

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

subgroup	index
#26033242	2
#26033238	2
#26022512	2
#26022504	2
#26022317	2
#26022312	2
#26022298	2
#26022154	2
#26022108	2
C₄×#15667953	2
C₈₉×#2208	2
#33960	89