Subgroups — #36470468

Order 22784.

510 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	C₂×#19151680	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151416	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19151400	C₂	1	1	yes	maximal, characteristic
4	C₂×#19151398	C₂	1	1	yes	maximal, characteristic
5	#26020410	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018343	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
9	C₂×#15667974	—	2	1	no	—
10	C₂²×#13083549	C₂²	1	1	yes	characteristic
11	#19151680	—	2	2	no	—
12	#19151680	C₄	1	4	yes	complemented
13	C₂×#15667946	C₂²	1	1	yes	characteristic
14	C₂×#15667945	—	2	1	no	—
15	C₂×#15667932	—	2	1	no	—
16	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
17	#19151467	—	2	1	no	—
18	#19151416	—	2	2	no	—
19	#19151400	—	2	2	no	—
20	#19151398	—	2	2	no	—
21	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
24	C₂²×D₃₅₆	—	2	1	no	—
25	C₂²×C₄×D₈₉	—	4	1	no	—
26	C₂×D₇₁₂	—	2	1	no	—
27	C₂×D₇₁₂	—	2	1	no	—
28	C₂×D₇₁₂	—	2	1	no	—
29	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
30	C₂×D₇₁₂	D₄	1	2	yes	—
31	C₂×D₇₁₂	Q₈	1	2	yes	—
32	C₂²×#10623494	D₄	1	1	yes	characteristic
33	#15667974	—	2	4	no	—
34	#15667974	—	4	2	no	—
35	C₂×C₄⋊Dic₈₉	—	2	1	no	—
36	C₂×#13083553	—	4	1	no	—
37	C₂×#13083553	—	4	1	no	—
38	C₂×#13083551	—	4	1	no	—
39	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
40	C₂×#13083549	—	2	1	no	—
41	C₂×#13083549	—	2	1	no	—
42	C₂×#13083549	—	2	1	no	—
43	#15667946	—	2	1	no	—
44	#15667946	—	2	1	no	—
45	#15667945	—	4	2	no	—
46	#15667932	—	2	4	no	—
47	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
48	C₈×Dic₈₉	D₄	1	2	yes	—
49	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
51	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
52	C₂×C₁₆×C₈₉	—	2	1	no	—
53	C₂×C₁₆×C₈₉	—	2	1	no	—
54	C₂×C₁₆×C₈₉	—	2	1	no	—
55	#107×C₈₉	—	2	1	no	—
56	#107×C₈₉	—	2	1	no	—
57	#107×C₈₉	—	2	1	no	—
58	#107×C₈₉	C₂×C₄	1	2	yes	—
59	C₂³×D₈₉	—	4	1	no	—
60	C₂×D₃₅₆	—	2	2	no	—
61	C₂×D₃₅₆	—	2	2	no	—
62	C₂×D₃₅₆	—	2	2	no	—
63	C₂×D₃₅₆	—	4	1	no	—
64	C₂×D₃₅₆	—	4	1	no	—
65	C₂×D₃₅₆	—	4	1	no	—
66	C₂²×Dic₈₉	—	4	1	no	—
67	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
68	C₂²×Dic₈₉	D₈	1	1	yes	characteristic
69	C₂×C₄×D₈₉	—	4	4	no	—
70	C₂×C₄×D₈₉	—	8	2	no	—
71	C₂×C₄×D₈₉	—	8	2	no	—
72	D₇₁₂	—	2	2	no	—
73	D₇₁₂	—	2	2	no	—
74	D₇₁₂	—	2	2	no	—
75	D₇₁₂	C₄⋊C₄	1	4	yes	complemented
76	C₂×#10623494	—	2	1	no	—
77	C₂×#10623494	—	2	1	no	—
78	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
79	C₄⋊Dic₈₉	—	4	1	no	—
80	C₄⋊Dic₈₉	—	4	1	no	—
81	#13083553	—	8	2	no	—
82	#13083553	—	8	2	no	—
83	#13083551	—	4	4	no	—
84	C₄×Dic₈₉	C₂²⋊C₄	1	2	yes	—
85	C₄×Dic₈₉	SD₁₆	1	2	yes	—
86	#13083549	—	4	1	no	—
87	C₂²×C₄×C₈₉	—	4	1	no	—
88	C₂²×C₄×C₈₉	—	4	1	no	—
89	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
95	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
96	C₄⋊C₄×C₈₉	—	2	2	no	—
97	C₄⋊C₄×C₈₉	—	4	1	no	—
98	C₄⋊C₄×C₈₉	—	4	1	no	—
99	C₄⋊C₄×C₈₉	—	4	1	no	—
100	C₁₆×C₈₉	—	4	1	no	—
101	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
102	C₂²×D₈₉	—	4	2	no	—
103	C₂²×D₈₉	—	4	2	no	—
104	C₂²×D₈₉	—	4	2	no	—
105	C₂²×D₈₉	—	8	1	no	—
106	C₂²×D₈₉	—	8	1	no	—
107	C₂²×D₈₉	—	8	1	no	—
108	C₂²×D₈₉	—	8	1	no	—
109	C₂×C₄×C₈₉	—	4	2	no	—
110	C₂×C₄×C₈₉	—	8	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	—	8	1	no	—
115	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
121	C₂×Dic₈₉	—	2	1	no	—
122	C₂×Dic₈₉	—	2	1	no	—
123	C₂×Dic₈₉	—	8	1	no	—
124	C₂×Dic₈₉	—	8	1	no	—
125	C₂×Dic₈₉	—	8	1	no	—
126	C₂×Dic₈₉	#102	1	2	yes	—
127	C₂×Dic₈₉	#102	1	2	yes	—
128	C₂×Dic₈₉	D₁₆	1	2	yes	—
129	C₂×Dic₈₉	SD₃₂	1	2	yes	—
130	D₃₅₆	—	2	4	no	—
131	D₃₅₆	—	4	2	no	—
132	D₃₅₆	—	4	2	no	—
133	D₃₅₆	—	4	2	no	—
134	C₄×D₈₉	—	8	8	no	—
135	C₈×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₈×C₈₉	#117	1	1	yes	characteristic
137	C₈×C₈₉	#121	1	1	yes	characteristic
138	C₈×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
139	#10623494	—	2	2	no	—
140	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
141	C₂²×C₈₉	#459	1	1	yes	characteristic
142	C₂²×C₈₉	#474	1	1	yes	characteristic
143	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
144	C₂²×C₈₉	C₂×SD₃₂	1	1	yes	characteristic
145	C₂²×C₈₉	#509	1	1	yes	characteristic
146	C₂²×C₈₉	#510	1	1	yes	characteristic
147	C₂×D₈₉	—	4	4	no	—
148	C₂×D₈₉	—	8	2	no	—
149	C₂×D₈₉	—	8	2	no	—
150	C₂×D₈₉	—	8	2	no	—
151	C₂×D₈₉	—	8	2	no	—
152	C₂×D₈₉	—	8	2	no	—
153	C₂×D₈₉	—	8	2	no	—
154	C₄×C₈₉	—	8	2	no	—
155	C₄×C₈₉	—	16	1	no	—
156	C₄×C₈₉	#380	1	1	yes	characteristic
157	C₄×C₈₉	#417	1	1	yes	characteristic
158	C₄×C₈₉	#461	1	1	yes	characteristic
159	C₄×C₈₉	#476	1	1	yes	characteristic
160	Dic₈₉	—	2	2	no	—
161	Dic₈₉	—	16	1	no	—
162	Dic₈₉	#357	1	4	yes	complemented
163	#12811	—	89	1	no	maximal, Sylow
164	C₂×C₈₉	#1802	1	1	yes	characteristic
165	C₂×C₈₉	C₂×#357	1	1	yes	characteristic
166	C₂×C₈₉	#2139	1	1	yes	characteristic
167	C₂×C₈₉	#2206	1	1	yes	characteristic
168	C₂×C₈₉	#2208	1	1	yes	characteristic
169	C₂×C₈₉	#2224	1	1	yes	characteristic
170	C₂×C₈₉	#2226	1	1	yes	characteristic
171	D₈₉	—	8	4	no	—
172	C₂×C₄×D₈	—	89	1	no	—
173	C₂×C₄⋊C₁₆	—	89	1	no	—
174	C₂×#357	—	89	1	no	—
175	C₂×#357	—	89	1	no	—
176	#1845	—	89	1	no	—
177	#1380	—	89	1	no	—
178	#1380	D₈₉	1	1	yes	characteristic, complemented
179	C₈₉	#12811	1	1	yes	characteristic, complemented, Sylow
180	C₂²×D₈	—	89	1	no	—
181	C₂×C₄×D₄	—	178	1	no	—
182	C₂²×C₁₆	—	89	1	no	—
183	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
184	C₄×D₈	—	89	4	no	—
185	C₄×D₈	—	178	2	no	—
186	C₂×#107	—	89	1	no	—
187	C₂×#107	—	178	1	no	—
188	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
189	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
190	C₂×#102	—	178	1	no	—
191	C₂×C₄×C₈	—	89	1	no	—
192	#384	—	178	1	no	—
193	C₄⋊C₁₆	—	178	2	no	—
194	#357	—	178	2	no	—
195	#357	—	178	2	no	—
196	C₂²×D₄	—	178	1	no	—
197	C₂³×C₄	—	356	1	no	—
198	C₂×D₈	—	89	2	no	—
199	C₂×D₈	—	89	2	no	—
200	C₂×D₈	—	89	2	no	—
201	C₂×D₈	—	178	1	no	—
202	C₂×D₈	—	178	1	no	—
203	C₂×D₈	—	178	1	no	—
204	C₂²×C₈	—	89	1	no	—
205	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
206	C₄×D₄	—	356	2	no	—
207	C₄×D₄	—	178	4	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	356	1	no	—
211	C₂×C₄⋊C₄	—	356	1	no	—
212	C₂×C₄⋊C₄	—	178	1	no	—
213	C₂×C₂²⋊C₄	—	356	1	no	—
214	C₂×C₄²	—	89	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	178	1	no	—
219	C₂×C₁₆	—	178	1	no	—
220	C₂×C₁₆	—	178	1	no	—
221	#107	—	2	1	no	—
222	#107	—	2	1	no	—
223	#107	—	2	1	no	—
224	#107	—	356	2	no	—
225	#107	—	178	1	no	—
226	#107	—	178	1	no	—
227	#107	C₄×D₈₉	1	2	yes	—
228	#102	—	178	4	no	—
229	C₄×C₈	—	89	2	no	—
230	C₄×C₈	—	89	2	no	—
231	C₂⁴	—	356	1	no	—
232	C₂×D₄	—	356	1	no	—
233	C₂×D₄	—	356	1	no	—
234	C₂×D₄	—	356	1	no	—
235	C₂×D₄	—	178	2	no	—
236	C₂×D₄	—	178	2	no	—
237	C₂×D₄	—	178	2	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	89	1	no	—
241	C₂²×C₄	—	89	1	no	—
242	C₂²×C₄	—	356	4	no	—
243	C₂²×C₄	—	356	1	no	—
244	C₂²×C₄	—	712	2	no	—
245	C₂²×C₄	—	712	2	no	—
246	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
247	D₈	—	89	4	no	—
248	D₈	—	178	2	no	—
249	D₈	—	178	2	no	—
250	D₈	—	178	2	no	—
251	C₂×C₈	—	89	2	no	—
252	C₂×C₈	—	178	1	no	—
253	C₂×C₈	—	178	1	no	—
254	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
255	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
256	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
257	C₂×C₈	Q₈×D₈₉	1	1	yes	characteristic
258	C₂×C₈	#13083574	1	1	yes	characteristic
259	C₂×C₈	#13083574	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	356	1	no	—
265	C₄⋊C₄	—	356	1	no	—
266	C₄⋊C₄	—	712	2	no	—
267	C₄⋊C₄	—	712	2	no	—
268	C₂²⋊C₄	—	356	4	no	—
269	C₄²	—	89	2	no	—
270	C₄²	—	89	2	no	—
271	C₁₆	—	4	1	no	—
272	C₁₆	—	356	1	no	—
273	C₂³	—	356	2	no	—
274	C₂³	—	356	2	no	—
275	C₂³	—	356	2	no	—
276	C₂³	—	712	1	no	—
277	C₂³	—	712	1	no	—
278	C₂³	—	712	1	no	—
279	C₂³	—	712	1	no	—
280	C₂³	D₈×D₈₉	1	1	yes	characteristic
281	D₄	—	356	2	no	—
282	D₄	—	356	2	no	—
283	D₄	—	356	2	no	—
284	D₄	—	178	4	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	8	1	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	89	2	no	—
292	C₂×C₄	—	89	2	no	—
293	C₂×C₄	—	89	2	no	—
294	C₂×C₄	—	89	2	no	—
295	C₂×C₄	—	178	1	no	—
296	C₂×C₄	—	178	1	no	—
297	C₂×C₄	—	712	8	no	—
298	C₂×C₄	—	712	1	no	—
299	C₂×C₄	—	712	1	no	—
300	C₂×C₄	—	712	1	no	—
301	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
302	C₂×C₄	#15667985	1	1	yes	characteristic
303	C₂×C₄	#15667990	1	1	yes	characteristic
304	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
305	C₂×C₄	#15668039	1	1	yes	characteristic
306	C₂×C₄	#15668044	1	1	yes	characteristic
307	C₈	—	178	2	no	—
308	C₈	C₄⋊C₄×D₈₉	1	1	yes	characteristic
309	C₈	#15667969	1	1	yes	characteristic
310	C₈	#15667985	1	1	yes	characteristic
311	C₈	#15667990	1	1	yes	characteristic
312	C₂²	—	356	4	no	—
313	C₂²	—	712	2	no	—
314	C₂²	—	712	2	no	—
315	C₂²	—	712	2	no	—
316	C₂²	—	712	2	no	—
317	C₂²	—	712	2	no	—
318	C₂²	—	712	2	no	—
319	C₂²	#102×D₈₉	1	1	yes	characteristic
320	C₂²	#19151792	1	1	yes	characteristic
321	C₂²	#19151877	1	1	yes	characteristic
322	C₂²	D₁₆×D₈₉	1	1	yes	characteristic
323	C₂²	SD₃₂×D₈₉	1	1	yes	characteristic
324	C₂²	#19152035	1	1	yes	characteristic
325	C₂²	#19152040	1	1	yes	characteristic
326	C₄	—	8	2	no	—
327	C₄	—	16	1	no	—
328	C₄	—	89	4	no	—
329	C₄	—	1424	1	no	—
330	C₄	—	178	2	no	—
331	C₄	#19151457	1	1	yes	characteristic
332	C₄	#19151598	1	1	yes	characteristic
333	C₄	#19151802	1	1	yes	characteristic
334	C₄	#19151887	1	1	yes	characteristic
335	C₂	—	712	4	no	—
336	C₂	#26020251	1	1	yes	characteristic
337	C₂	#357×D₈₉	1	1	yes	characteristic
338	C₂	#26021963	1	1	yes	characteristic
339	C₂	#26022286	1	1	yes	characteristic
340	C₂	#26022296	1	1	yes	characteristic
341	C₂	#26022388	1	1	yes	characteristic
342	C₂	#26022398	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₈₉
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₈₉×#1380	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂²×D₇₁₂	C₂²
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	#1380	D₈₉
hypocenter	C₈₉	#12811
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

Complements

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

N	H ≅ G/N	type
2. C₂×#19151416	C₂	semidirect
5. #26020410	C₂	semidirect
6. #26018343	C₂	semidirect
7. C₈₉×#1380	C₂	semidirect
12. #19151680	C₄	semidirect
75. D₇₁₂	C₄⋊C₄	semidirect
162. Dic₈₉	#357	semidirect
178. #1380	D₈₉	semidirect
179. C₈₉	#12811	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151680	2
C₂×#19151416	2
C₂×#19151400	2
C₂×#19151398	2
#26020410	2
#26018343	2
C₈₉×#1380	2
#12811	89