Subgroups — #36711856

Order 22784.

492 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032675	C₂	1	1	yes	maximal, characteristic, complemented
2	#26022525	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022443	C₂	1	2	yes	maximal, complemented
4	#26022440	C₂	1	2	yes	maximal, complemented
5	#26022399	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022396	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022336	C₂	1	2	yes	maximal, complemented
8	C₄×#15667953	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×#13083549	C₂	1	1	yes	maximal, characteristic, complemented
10	#26022034	C₂	1	2	yes	maximal, complemented
11	C₈₉×#3341	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152313	—	2	1	no	—
13	C₂×#15667953	C₂²	1	1	yes	characteristic
14	C₂²×#13083549	C₂²	1	2	yes	—
15	#19152003	C₂²	1	1	yes	characteristic
16	#19151923	—	2	2	no	—
17	#19151889	—	2	1	no	—
18	#19151834	C₂²	1	2	yes	—
19	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
20	#19151636	C₂²	1	2	yes	—
21	#19151425	C₂²	1	2	yes	—
22	#19151425	C₂²	1	2	yes	—
23	#19151423	C₂²	1	2	yes	—
24	#19151423	C₂²	1	2	yes	—
25	#19151423	C₂²	1	1	yes	characteristic
26	C₄⋊#13083549	C₂²	1	1	yes	characteristic
27	#19151399	C₂²	1	1	yes	characteristic
28	#19151399	C₂²	1	2	yes	—
29	#19151398	C₂²	1	1	yes	characteristic
30	#19151398	C₂²	1	2	yes	—
31	C₂²⋊#13083549	C₂²	1	2	yes	—
32	C₄×#13083549	C₂²	1	1	yes	characteristic
33	#555×C₈₉	—	2	1	no	—
34	#500×C₈₉	C₂²	1	1	yes	characteristic
35	#482×C₈₉	—	2	2	no	—
36	#476×C₈₉	—	2	1	no	—
37	#466×C₈₉	C₂²	1	2	yes	—
38	C₄×D₈×C₈₉	C₂²	1	1	yes	characteristic
39	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
40	#427×C₈₉	C₂²	1	2	yes	—
41	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
42	#15668012	—	2	1	no	—
43	#15667998	—	4	2	no	—
44	#15667986	—	2	2	no	—
45	#15667983	—	4	1	no	—
46	C₄×D₃₅₆	—	2	1	no	—
47	#15667971	—	2	2	no	—
48	#15667970	—	4	1	no	—
49	#15667953	—	2	2	no	—
50	C₂×#13083549	—	2	4	no	—
51	C₂×#13083549	C₂³	1	1	yes	characteristic
52	C₂×#13083549	C₂³	1	2	yes	—
53	C₂×#13083549	C₂³	1	1	yes	characteristic
54	#15667946	—	2	1	no	—
55	#15667946	C₂³	1	1	yes	characteristic
56	#15667946	C₂³	1	2	yes	—
57	#15667944	—	2	2	no	—
58	#15667934	—	2	2	no	—
59	#15667934	—	2	1	no	—
60	C₂×D₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
62	C₄⋊Q₈×C₈₉	—	2	1	no	—
63	#124×C₈₉	—	4	2	no	—
64	#121×C₈₉	—	2	2	no	—
65	#121×C₈₉	—	4	1	no	—
66	C₄×D₄×C₈₉	—	2	1	no	—
67	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
68	#117×C₈₉	—	2	2	no	—
69	#117×C₈₉	—	4	1	no	—
70	#107×C₈₉	—	2	1	no	—
71	#107×C₈₉	C₂³	1	1	yes	characteristic
72	#107×C₈₉	C₂³	1	2	yes	—
73	#106×C₈₉	—	2	2	no	—
74	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
75	#102×C₈₉	—	2	1	no	—
76	#102×C₈₉	—	2	2	no	—
77	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂×Dic₁₇₈	—	4	1	no	—
80	C₂×D₃₅₆	—	2	1	no	—
81	C₂×C₂²⋊D₈₉	—	4	2	no	—
82	C₂×C₄×D₈₉	—	4	1	no	—
83	D₇₁₂	—	2	1	no	—
84	D₇₁₂	—	2	1	no	—
85	C₄⋊Dic₈₉	—	2	1	no	—
86	C₄⋊Dic₈₉	—	2	2	no	—
87	C₄⋊Dic₈₉	—	4	1	no	—
88	C₂²⋊Dic₈₉	—	4	2	no	—
89	#13083551	—	4	2	no	—
90	#13083551	—	4	1	no	—
91	C₄×Dic₈₉	—	4	2	no	—
92	#13083549	—	2	2	no	—
93	#13083549	—	2	1	no	—
94	#13083549	#41	1	2	yes	—
95	C₂×Q₈×C₈₉	—	4	1	no	—
96	C₂×D₄×C₈₉	—	2	1	no	—
97	C₂×D₄×C₈₉	—	4	2	no	—
98	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	—	4	1	no	—
100	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
101	D₈×C₈₉	—	2	1	no	—
102	D₈×C₈₉	—	2	1	no	—
103	C₂×C₈×C₈₉	—	2	2	no	—
104	C₂×C₈×C₈₉	—	2	2	no	—
105	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	—	2	1	no	—
109	C₄⋊C₄×C₈₉	—	2	2	no	—
110	C₄⋊C₄×C₈₉	—	4	1	no	—
111	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
112	C₂²⋊C₄×C₈₉	—	4	2	no	—
113	C₂²⋊C₄×C₈₉	—	4	2	no	—
114	C₂²⋊C₄×C₈₉	—	4	1	no	—
115	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
116	C₄²×C₈₉	—	4	2	no	—
117	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂³×C₈₉	—	4	1	no	—
119	C₂³×C₈₉	C₂×D₈	1	2	yes	—
120	C₂²×D₈₉	—	4	1	no	—
121	Q₈×C₈₉	—	16	1	no	—
122	D₄×C₈₉	—	2	2	no	—
123	D₄×C₈₉	—	4	1	no	—
124	D₄×C₈₉	—	4	1	no	—
125	D₄×C₈₉	—	16	2	no	—
126	C₂×C₄×C₈₉	—	2	2	no	—
127	C₂×C₄×C₈₉	—	2	2	no	—
128	C₂×C₄×C₈₉	—	4	1	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	8	2	no	—
131	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
132	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
135	C₂²⋊D₈₉	—	16	2	no	—
136	C₂×Dic₈₉	—	4	1	no	—
137	C₂×Dic₈₉	—	4	2	no	—
138	D₃₅₆	—	4	1	no	—
139	D₃₅₆	—	4	1	no	—
140	C₄×D₈₉	—	8	2	no	—
141	Dic₁₇₈	—	16	1	no	—
142	C₈×C₈₉	—	2	2	no	—
143	C₈×C₈₉	—	2	1	no	—
144	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
145	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂²×C₈₉	—	2	2	no	—
147	C₂²×C₈₉	—	2	2	no	—
148	C₂²×C₈₉	—	2	2	no	—
149	C₂²×C₈₉	—	8	1	no	—
150	C₂²×C₈₉	—	8	1	no	—
151	C₂²×C₈₉	—	8	1	no	—
152	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
153	C₂×D₈₉	—	8	1	no	—
154	C₂×D₈₉	—	8	1	no	—
155	C₂×D₈₉	—	8	1	no	—
156	C₄×C₈₉	—	2	2	no	—
157	C₄×C₈₉	—	2	1	no	—
158	C₄×C₈₉	—	8	1	no	—
159	C₄×C₈₉	—	8	2	no	—
160	C₄×C₈₉	#508	1	2	yes	—
161	C₄×C₈₉	#547	1	1	yes	characteristic
162	C₄×C₈₉	#578	1	1	yes	characteristic
163	Dic₈₉	—	8	1	no	—
164	Dic₈₉	—	8	2	no	—
165	#33908	—	89	1	no	maximal, Sylow
166	C₂×C₈₉	—	4	2	no	—
167	C₂×C₈₉	—	16	1	no	—
168	C₂×C₈₉	#3334	1	1	yes	characteristic
169	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
170	C₂×C₈₉	#3415	1	1	yes	characteristic
171	D₈₉	—	16	1	no	—
172	#3341	—	89	1	no	—
173	#3341	D₈₉	1	1	yes	characteristic, complemented
174	#2252	—	89	1	no	—
175	#2234	—	89	2	no	—
176	#2234	—	89	2	no	—
177	#2226	—	89	1	no	—
178	#2226	—	89	1	no	—
179	#2215	—	89	2	no	—
180	C₄×D₁₆	—	89	1	no	—
181	D₄×C₁₆	—	89	1	no	—
182	#2158	—	89	2	no	—
183	C₈₉	#33908	1	1	yes	characteristic, complemented, Sylow
184	#555	—	2	1	no	—
185	#555	—	178	1	no	—
186	C₂×D₁₆	—	89	1	no	—
187	C₂²×C₁₆	—	89	2	no	—
188	#500	—	89	1	no	—
189	#500	C₂×D₈₉	1	1	yes	characteristic
190	#482	—	2	2	no	—
191	#482	—	178	2	no	—
192	#476	—	2	1	no	—
193	#476	—	178	1	no	—
194	#466	—	89	2	no	—
195	#466	C₂×D₈₉	1	2	yes	—
196	C₄×D₈	—	89	1	no	—
197	C₄×D₈	C₂×D₈₉	1	1	yes	characteristic
198	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
199	#427	—	89	2	no	—
200	#427	C₂×D₈₉	1	2	yes	—
201	#367	—	89	2	no	—
202	#367	—	89	2	no	—
203	#366	—	89	1	no	—
204	#366	—	89	2	no	—
205	#366	—	89	2	no	—
206	C₄⋊C₁₆	—	89	1	no	—
207	#357	—	89	1	no	—
208	#357	—	89	2	no	—
209	#357	—	89	1	no	—
210	#357	—	89	2	no	—
211	C₂²⋊C₁₆	—	89	2	no	—
212	C₄×C₁₆	—	89	1	no	—
213	C₂×D₈	—	89	1	no	—
214	C₂×D₈	C₂²×D₈₉	1	1	yes	characteristic
215	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	178	1	no	—
218	#124	—	4	2	no	—
219	#124	—	356	2	no	—
220	#121	—	2	2	no	—
221	#121	—	4	1	no	—
222	#121	—	356	1	no	—
223	#121	—	178	2	no	—
224	C₄×D₄	—	2	1	no	—
225	C₄×D₄	—	178	1	no	—
226	C₄×D₄	C₂²⋊D₈₉	1	1	yes	characteristic
227	#117	—	2	2	no	—
228	#117	—	4	1	no	—
229	#117	—	356	1	no	—
230	#117	—	178	2	no	—
231	D₁₆	—	178	2	no	—
232	C₂×C₁₆	—	89	1	no	—
233	C₂×C₁₆	—	89	1	no	—
234	C₂×C₁₆	—	89	2	no	—
235	C₂×C₁₆	—	178	4	no	—
236	#107	—	2	1	no	—
237	#107	—	89	2	no	—
238	#107	—	89	1	no	—
239	#107	—	178	1	no	—
240	#107	C₂²×D₈₉	1	2	yes	—
241	#107	C₂²×D₈₉	1	1	yes	characteristic
242	#106	—	2	2	no	—
243	#106	—	178	2	no	—
244	C₄⋊C₈	C₂²⋊D₈₉	1	1	yes	characteristic
245	#102	—	2	1	no	—
246	#102	—	2	2	no	—
247	#102	—	178	1	no	—
248	#102	—	178	2	no	—
249	C₂²⋊C₈	C₂²⋊D₈₉	1	2	yes	—
250	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
251	C₂×Q₈	—	4	1	no	—
252	C₂×Q₈	—	356	1	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	—	356	2	no	—
256	C₂×D₄	—	178	1	no	—
257	C₂×D₄	#13083559	1	1	yes	characteristic
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	—	356	1	no	—
260	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
261	D₈	—	2	1	no	—
262	D₈	—	2	1	no	—
263	D₈	—	178	1	no	—
264	D₈	—	178	1	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
268	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
269	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	356	1	no	—
274	C₄⋊C₄	—	178	1	no	—
275	C₄⋊C₄	—	178	2	no	—
276	C₄⋊C₄	#13083559	1	1	yes	characteristic
277	C₂²⋊C₄	—	4	2	no	—
278	C₂²⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	356	2	no	—
281	C₂²⋊C₄	—	356	2	no	—
282	C₂²⋊C₄	—	356	1	no	—
283	C₂²⋊C₄	#13083559	1	2	yes	—
284	C₄²	—	4	2	no	—
285	C₄²	—	356	2	no	—
286	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
287	C₁₆	—	89	2	no	—
288	C₁₆	—	178	2	no	—
289	C₁₆	—	178	1	no	—
290	C₂³	—	4	1	no	—
291	C₂³	—	356	1	no	—
292	C₂³	C₂×#13083559	1	2	yes	—
293	Q₈	—	16	1	no	—
294	Q₈	—	1424	1	no	—
295	D₄	—	2	2	no	—
296	D₄	—	4	1	no	—
297	D₄	—	4	1	no	—
298	D₄	—	16	2	no	—
299	D₄	—	356	1	no	—
300	D₄	—	356	1	no	—
301	D₄	—	1424	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	—	356	1	no	—
308	C₂×C₄	—	356	2	no	—
309	C₂×C₄	—	712	2	no	—
310	C₂×C₄	C₂×#13083559	1	2	yes	—
311	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
312	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
313	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
314	C₈	—	2	1	no	—
315	C₈	—	2	2	no	—
316	C₈	#41×D₈₉	1	1	yes	characteristic
317	C₈	#15668059	1	1	yes	characteristic
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	8	1	no	—
322	C₂²	—	8	1	no	—
323	C₂²	—	8	1	no	—
324	C₂²	—	712	1	no	—
325	C₂²	—	712	1	no	—
326	C₂²	—	712	1	no	—
327	C₂²	C₂²×#13083559	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	1	no	—
330	C₄	—	8	1	no	—
331	C₄	—	8	2	no	—
332	C₄	—	712	1	no	—
333	C₄	—	712	2	no	—
334	C₄	#19152028	1	2	yes	—
335	C₄	#19152247	1	1	yes	characteristic
336	C₄	#19152413	1	1	yes	characteristic
337	C₂	—	4	2	no	—
338	C₂	—	16	1	no	—
339	C₂	—	1424	1	no	—
340	C₂	#26032595	1	1	yes	characteristic
341	C₂	C₂×#19152028	1	1	yes	characteristic
342	C₂	#26033391	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3341	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	#3341	D₈₉
hypocenter	C₈₉	#33908
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #3341

Complements

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

N	H ≅ G/N	type
1. #26032675	C₂	semidirect
2. #26022525	C₂	semidirect
3. #26022443	C₂	semidirect
4. #26022440	C₂	semidirect
5. #26022399	C₂	semidirect
6. #26022396	C₂	semidirect
7. #26022336	C₂	semidirect
8. C₄×#15667953	C₂	semidirect
9. D₄×#13083549	C₂	semidirect
10. #26022034	C₂	semidirect
11. C₈₉×#3341	C₂	semidirect
173. #3341	D₈₉	semidirect
183. C₈₉	#33908	semidirect

Maximal subgroups

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

subgroup	index
#26032675	2
#26022525	2
#26022443	2
#26022440	2
#26022399	2
#26022396	2
#26022336	2
C₄×#15667953	2
D₄×#13083549	2
#26022034	2
C₈₉×#3341	2
#33908	89