Subgroups — #36712447

Order 22784.

532 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033261	C₂	1	2	yes	maximal, complemented
2	#26022514	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022500	C₂	1	1	yes	maximal, characteristic
4	#26022309	C₂	1	2	yes	maximal, complemented
5	#26022304	C₂	1	2	yes	maximal, complemented
6	#26022290	C₂	1	2	yes	maximal, complemented
7	C₁₆×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic, complemented
8	Q₃₂×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×D₁₄₂₄	C₂	1	1	yes	maximal, characteristic, complemented
10	D₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2249	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152330	—	2	2	no	—
13	C₂×#15667955	C₂²	1	2	yes	—
14	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
15	C₂×#15667953	C₂²	1	2	yes	—
16	#19151983	C₂²	1	2	yes	—
17	#19151973	C₂²	1	2	yes	—
18	#19151815	—	2	2	no	—
19	#19151810	—	2	2	no	—
20	#19151796	—	2	2	no	—
21	C₈×Dic₁₇₈	C₂²	1	1	yes	characteristic
22	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
23	C₄×D₇₁₂	C₂²	1	2	yes	—
24	D₈×Dic₈₉	C₂²	1	2	yes	—
25	#19151424	C₂²	1	1	yes	characteristic
26	C₄⋊#13083549	C₂²	1	1	yes	characteristic
27	#19151416	C₂²	1	1	yes	characteristic
28	#19151416	C₂²	1	1	yes	characteristic
29	#19151402	C₂²	1	2	yes	—
30	#19151400	C₂²	1	2	yes	—
31	#19151399	C₂²	1	2	yes	—
32	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
34	C₂×Q₃₂×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×SD₃₂×C₈₉	C₂²	1	2	yes	—
36	C₂×D₁₆×C₈₉	C₂²	1	1	yes	characteristic
37	#495×C₈₉	C₂²	1	2	yes	—
38	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×#13083561	—	2	2	no	—
40	C₂×D₇₁₂	C₂³	1	2	yes	—
41	C₂×#13083559	—	2	2	no	—
42	#15668008	—	2	2	no	—
43	#15668001	—	2	2	no	—
44	#15667998	—	4	2	no	—
45	#15667996	—	4	2	no	—
46	#15667994	—	4	2	no	—
47	Q₈×Dic₈₉	—	2	2	no	—
48	C₄×Dic₁₇₈	D₄	1	1	yes	characteristic
49	C₄×D₃₅₆	—	2	2	no	—
50	D₄×Dic₈₉	—	2	2	no	—
51	#15667972	—	4	2	no	—
52	#15667955	—	4	2	no	—
53	D₁₄₂₄	—	2	2	no	—
54	#15667953	—	4	2	no	—
55	C₂×#13083549	C₂³	1	1	yes	characteristic
56	C₂×#13083549	C₂³	1	1	yes	characteristic
57	#15667946	C₂³	1	2	yes	—
58	C₄⋊#10623494	D₄	1	1	yes	characteristic
59	#15667941	D₄	1	1	yes	characteristic
60	#15667941	D₄	1	1	yes	characteristic
61	#15667936	—	2	2	no	—
62	#15667934	—	2	2	no	—
63	#15667933	—	2	2	no	—
64	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
65	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
66	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
67	C₂×SD₁₆×C₈₉	—	2	2	no	—
68	C₂×D₈×C₈₉	C₂³	1	2	yes	—
69	#124×C₈₉	—	2	2	no	—
70	Q₃₂×C₈₉	—	2	2	no	—
71	SD₃₂×C₈₉	—	4	2	no	—
72	D₁₆×C₈₉	—	2	2	no	—
73	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
77	C₂×D₃₅₆	—	2	2	no	—
78	C₂×C₂²⋊D₈₉	—	4	2	no	—
79	C₂×C₂²⋊D₈₉	—	4	2	no	—
80	C₂²×Dic₈₉	—	4	2	no	—
81	C₂×C₄×D₈₉	—	4	2	no	—
82	#13083561	—	8	2	no	—
83	D₇₁₂	—	2	2	no	—
84	D₇₁₂	—	2	2	no	—
85	#13083559	—	8	2	no	—
86	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
87	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
88	C₄⋊Dic₈₉	—	2	2	no	—
89	C₄⋊Dic₈₉	—	4	2	no	—
90	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
91	#13083553	D₈	1	1	yes	characteristic
92	#13083553	D₈	1	1	yes	characteristic
93	C₂²⋊Dic₈₉	—	4	2	no	—
94	#13083551	—	4	2	no	—
95	#13083551	—	4	2	no	—
96	#13083551	—	4	2	no	—
97	C₄×Dic₈₉	—	4	2	no	—
98	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
99	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
100	#13083549	—	2	1	no	—
101	#13083549	—	2	1	no	—
102	C₂×Q₈×C₈₉	—	2	2	no	—
103	C₂×D₄×C₈₉	—	2	2	no	—
104	Q₁₆×C₈₉	—	2	2	no	—
105	Q₁₆×C₈₉	—	2	2	no	—
106	SD₁₆×C₈₉	—	8	2	no	—
107	D₈×C₈₉	—	2	2	no	—
108	D₈×C₈₉	—	2	2	no	—
109	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
111	C₂²⋊C₄×C₈₉	—	4	2	no	—
112	C₂²⋊C₄×C₈₉	—	4	2	no	—
113	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₁₆×C₈₉	—	2	1	no	—
115	C₁₆×C₈₉	#41	1	2	yes	—
116	C₂³×C₈₉	—	4	2	no	—
117	C₂²×D₈₉	—	4	2	no	—
118	Q₈×C₈₉	—	4	2	no	—
119	Q₈×C₈₉	—	4	2	no	—
120	D₄×C₈₉	—	4	2	no	—
121	D₄×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
126	C₂²⋊D₈₉	—	8	4	no	—
127	C₂²⋊D₈₉	—	16	2	no	—
128	C₂×Dic₈₉	—	4	2	no	—
129	C₂×Dic₈₉	—	16	2	no	—
130	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
132	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
133	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
134	D₃₅₆	—	4	2	no	—
135	D₃₅₆	—	4	2	no	—
136	C₄×D₈₉	—	16	2	no	—
137	Dic₁₇₈	—	2	2	no	—
138	C₈×C₈₉	—	2	1	no	—
139	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
140	C₈×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
141	#10623494	—	2	1	no	—
142	#10623494	—	2	1	no	—
143	C₂²×C₈₉	—	8	2	no	—
144	C₂²×C₈₉	—	8	2	no	—
145	C₂²×C₈₉	—	8	2	no	—
146	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₈₉	—	8	2	no	—
148	C₂×D₈₉	—	8	2	no	—
149	C₂×D₈₉	—	8	2	no	—
150	C₄×C₈₉	—	2	1	no	—
151	C₄×C₈₉	—	2	1	no	—
152	C₄×C₈₉	—	8	2	no	—
153	C₄×C₈₉	#550	1	1	yes	characteristic
154	C₄×C₈₉	#576	1	1	yes	characteristic
155	Dic₈₉	—	2	1	no	—
156	Dic₈₉	—	2	1	no	—
157	Dic₈₉	—	2	1	no	—
158	Dic₈₉	—	8	2	no	—
159	Dic₈₉	#508	1	2	yes	—
160	#33963	—	89	1	no	maximal, Sylow
161	C₂×C₈₉	—	16	2	no	—
162	C₂×C₈₉	#3391	1	1	yes	characteristic
163	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
164	C₂×C₈₉	#3415	1	1	yes	characteristic
165	D₈₉	—	16	2	no	—
166	#3396	—	89	2	no	—
167	#2249	—	89	1	no	—
168	#2249	D₈₉	1	1	yes	characteristic, complemented
169	#2246	—	89	1	no	—
170	#2210	—	89	2	no	—
171	#2210	—	89	2	no	—
172	#2207	—	89	2	no	—
173	Q₈×C₁₆	—	89	1	no	—
174	C₄×Q₃₂	—	89	1	no	—
175	C₄×D₁₆	—	89	1	no	—
176	C₄×D₁₆	—	89	1	no	—
177	C₈₉	#33963	1	1	yes	characteristic, complemented, Sylow
178	#559	—	178	2	no	—
179	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic
180	C₂×SD₃₂	—	89	2	no	—
181	C₂×SD₃₂	C₂×D₈₉	1	2	yes	—
182	C₂×D₁₆	—	89	2	no	—
183	C₂×D₁₆	—	89	1	no	—
184	C₂×D₁₆	C₂×D₈₉	1	1	yes	characteristic
185	#495	—	89	2	no	—
186	#495	C₂×D₈₉	1	2	yes	—
187	#493	—	89	2	no	—
188	#463	—	178	2	no	—
189	#463	—	178	2	no	—
190	#460	—	178	2	no	—
191	C₈×Q₈	—	89	1	no	—
192	C₄×Q₁₆	—	89	2	no	—
193	C₄×D₈	—	89	2	no	—
194	C₄×D₈	—	89	2	no	—
195	#366	—	89	1	no	—
196	C₄⋊C₁₆	—	89	1	no	—
197	C₄⋊C₁₆	—	89	1	no	—
198	C₄⋊C₁₆	—	89	1	no	—
199	#358	—	89	2	no	—
200	#357	—	89	2	no	—
201	#357	—	89	2	no	—
202	C₄×C₁₆	—	89	1	no	—
203	C₄×C₁₆	—	89	1	no	—
204	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
205	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
206	C₂×SD₁₆	—	2	2	no	—
207	C₂×SD₁₆	—	178	2	no	—
208	C₂×D₈	—	89	2	no	—
209	C₂×D₈	—	178	2	no	—
210	C₂×D₈	C₂²×D₈₉	1	2	yes	—
211	#127	—	178	2	no	—
212	#124	—	2	2	no	—
213	#124	—	356	2	no	—
214	#124	—	178	2	no	—
215	#123	—	356	2	no	—
216	#123	—	356	2	no	—
217	C₄×Q₈	—	89	1	no	—
218	C₄×Q₈	—	178	2	no	—
219	C₄×D₄	—	356	2	no	—
220	C₄×D₄	—	178	2	no	—
221	C₄×D₄	—	178	2	no	—
222	Q₃₂	—	2	2	no	—
223	SD₃₂	—	4	2	no	—
224	SD₃₂	—	356	2	no	—
225	D₁₆	—	2	2	no	—
226	D₁₆	—	356	2	no	—
227	D₁₆	—	178	2	no	—
228	C₂×C₁₆	—	89	1	no	—
229	C₂×C₁₆	—	89	1	no	—
230	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
231	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
232	#107	—	89	2	no	—
233	C₄⋊C₈	—	89	1	no	—
234	C₄⋊C₈	—	89	1	no	—
235	C₄⋊C₈	—	89	1	no	—
236	#103	—	178	2	no	—
237	#102	—	178	2	no	—
238	#102	—	178	2	no	—
239	C₄×C₈	—	89	1	no	—
240	C₄×C₈	—	89	1	no	—
241	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
242	C₂×Q₈	—	2	2	no	—
243	C₂×Q₈	—	89	1	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	356	2	no	—
246	C₂×D₄	—	356	2	no	—
247	C₂×D₄	—	178	2	no	—
248	C₂²×C₄	—	356	2	no	—
249	C₂²×C₄	—	356	2	no	—
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	2	2	no	—
252	SD₁₆	—	8	2	no	—
253	SD₁₆	—	712	2	no	—
254	D₈	—	2	2	no	—
255	D₈	—	2	2	no	—
256	D₈	—	178	2	no	—
257	D₈	—	178	2	no	—
258	D₈	—	712	2	no	—
259	C₂×C₈	—	89	1	no	—
260	C₂×C₈	—	89	1	no	—
261	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
262	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
263	C₄⋊C₄	—	89	1	no	—
264	C₄⋊C₄	—	89	1	no	—
265	C₄⋊C₄	—	89	1	no	—
266	C₄⋊C₄	—	356	2	no	—
267	C₄⋊C₄	—	178	2	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	356	2	no	—
271	C₂²⋊C₄	—	356	2	no	—
272	C₂²⋊C₄	—	356	2	no	—
273	C₂²⋊C₄	—	356	2	no	—
274	C₄²	—	89	1	no	—
275	C₄²	—	89	1	no	—
276	C₄²	—	356	2	no	—
277	C₄²	D₄×D₈₉	1	1	yes	characteristic
278	C₁₆	—	2	1	no	—
279	C₁₆	—	178	1	no	—
280	C₁₆	—	178	1	no	—
281	C₁₆	#13083573	1	2	yes	—
282	C₂³	—	4	2	no	—
283	C₂³	—	356	2	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	Q₈	—	178	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	4	2	no	—
289	D₄	—	356	2	no	—
290	D₄	—	356	2	no	—
291	D₄	—	1424	2	no	—
292	D₄	—	712	4	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	89	1	no	—
295	C₂×C₄	—	89	1	no	—
296	C₂×C₄	—	89	1	no	—
297	C₂×C₄	—	89	1	no	—
298	C₂×C₄	—	356	2	no	—
299	C₂×C₄	—	1424	2	no	—
300	C₂×C₄	—	1424	2	no	—
301	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
302	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
303	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	178	1	no	—
306	C₈	—	178	1	no	—
307	C₈	C₂×#13083573	1	1	yes	characteristic
308	C₈	#15668057	1	1	yes	characteristic
309	C₂²	—	8	2	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	712	2	no	—
313	C₂²	—	712	2	no	—
314	C₂²	—	712	2	no	—
315	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	8	2	no	—
319	C₄	—	89	2	no	—
320	C₄	—	178	1	no	—
321	C₄	—	178	1	no	—
322	C₄	—	178	1	no	—
323	C₄	—	712	2	no	—
324	C₄	#19152268	1	1	yes	characteristic
325	C₄	#19152399	1	1	yes	characteristic
326	C₂	—	16	2	no	—
327	C₂	—	1424	2	no	—
328	C₂	#26033225	1	1	yes	characteristic
329	C₂	#508×D₈₉	1	1	yes	characteristic
330	C₂	#26033390	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×#2249	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26022500	C₂
hypercenter	#2249	D₈₉
hypocenter	C₈₉	#33963
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2249

Complements

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

N	H ≅ G/N	type
1. #26033261	C₂	semidirect
2. #26022514	C₂	semidirect
4. #26022309	C₂	semidirect
5. #26022304	C₂	semidirect
6. #26022290	C₂	semidirect
7. C₁₆×Dic₁₇₈	C₂	semidirect
8. Q₃₂×Dic₈₉	C₂	semidirect
9. C₄×D₁₄₂₄	C₂	semidirect
10. D₁₆×Dic₈₉	C₂	semidirect
11. C₈₉×#2249	C₂	semidirect
168. #2249	D₈₉	semidirect
177. C₈₉	#33963	semidirect

Maximal subgroups

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

subgroup	index
#26033261	2
#26022514	2
#26022500	2
#26022309	2
#26022304	2
#26022290	2
C₁₆×Dic₁₇₈	2
Q₃₂×Dic₈₉	2
C₄×D₁₄₂₄	2
D₁₆×Dic₈₉	2
C₈₉×#2249	2
#33963	89