Subgroups — #36472655

Order 22784.

522 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	C₂×#19151983	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151402	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	C₂×#19151373	C₂	1	1	yes	maximal, characteristic, complemented
6	#26020410	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#358×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#13083561	—	2	1	no	—
9	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
10	C₂×#15668001	—	2	1	no	—
11	C₂²×#13083549	C₂²	1	1	yes	characteristic
12	#19151983	—	2	2	no	—
13	#19151983	C₄	1	4	yes	complemented
14	C₂×#15667946	C₂²	1	1	yes	characteristic
15	C₂×#15667945	—	2	1	no	—
16	C₂×C₈×Dic₈₉	C₂²	1	1	yes	characteristic
17	#19151467	—	2	1	no	—
18	#19151402	—	2	2	no	—
19	#19151400	—	2	2	no	—
20	#19151398	—	2	2	no	—
21	#19151373	—	2	2	no	—
22	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
25	#358×C₈₉	—	2	2	no	—
26	C₂²×D₃₅₆	—	2	1	no	—
27	C₂×#13083561	—	2	4	no	—
28	C₂×#13083561	—	4	2	no	—
29	C₂×#13083561	—	4	2	no	—
30	C₂×D₇₁₂	—	2	1	no	—
31	C₂×D₇₁₂	—	2	1	no	—
32	C₂×D₇₁₂	—	2	1	no	—
33	C₂×D₇₁₂	—	2	1	no	—
34	C₂×D₇₁₂	—	2	1	no	—
35	C₂×D₇₁₂	C₂×C₄	1	2	yes	—
36	C₂²×#10623494	D₄	1	1	yes	characteristic
37	#15668001	—	2	4	no	—
38	#15668001	—	4	2	no	—
39	C₂×C₄⋊Dic₈₉	—	2	1	no	—
40	C₂×#13083553	—	4	2	no	—
41	C₂×#13083551	—	4	2	no	—
42	C₂×C₄×Dic₈₉	D₄	1	1	yes	characteristic
43	C₂×#13083549	—	2	1	no	—
44	C₂×#13083549	—	2	1	no	—
45	C₂×#13083549	D₄	1	2	yes	—
46	#15667946	—	2	1	no	—
47	#15667946	—	2	1	no	—
48	#15667945	—	4	2	no	—
49	C₈×Dic₈₉	C₂×C₄	1	2	yes	—
50	C₈×Dic₈₉	D₄	1	2	yes	—
51	C₂²×Q₈×C₈₉	—	2	1	no	—
52	C₂×Q₁₆×C₈₉	—	2	1	no	—
53	C₂×Q₁₆×C₈₉	—	2	1	no	—
54	C₂×Q₁₆×C₈₉	—	2	1	no	—
55	C₂×Q₁₆×C₈₉	—	2	1	no	—
56	C₂×Q₁₆×C₈₉	—	2	1	no	—
57	C₂×Q₁₆×C₈₉	C₂×C₄	1	2	yes	—
58	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
60	C₂×C₁₆×C₈₉	—	2	1	no	—
61	C₂×C₁₆×C₈₉	—	2	1	no	—
62	C₂×C₁₆×C₈₉	D₄	1	2	yes	—
63	#107×C₈₉	—	2	1	no	—
64	#107×C₈₉	—	2	1	no	—
65	C₂³×D₈₉	—	4	1	no	—
66	C₂×D₃₅₆	—	2	2	no	—
67	C₂×D₃₅₆	—	4	1	no	—
68	C₂×D₃₅₆	—	4	1	no	—
69	C₂×D₃₅₆	—	4	1	no	—
70	C₂×D₃₅₆	—	4	1	no	—
71	C₂×D₃₅₆	—	4	1	no	—
72	C₂²×Dic₈₉	—	4	1	no	—
73	C₂²×Dic₈₉	D₈	1	2	yes	—
74	#13083561	—	4	8	no	—
75	D₇₁₂	—	2	2	no	—
76	D₇₁₂	—	2	2	no	—
77	D₇₁₂	—	4	1	no	—
78	D₇₁₂	—	4	1	no	—
79	C₂×#10623494	C₂²⋊C₄	1	2	yes	—
80	C₂×#10623494	D₈	1	2	yes	—
81	C₂×#10623494	SD₁₆	1	2	yes	—
82	C₄⋊Dic₈₉	—	4	1	no	—
83	C₄⋊Dic₈₉	—	4	1	no	—
84	#13083553	—	8	4	no	—
85	#13083551	—	4	8	no	—
86	C₄×Dic₈₉	C₂²⋊C₄	1	2	yes	—
87	C₄×Dic₈₉	SD₁₆	1	2	yes	—
88	#13083549	—	2	2	no	—
89	C₂×Q₈×C₈₉	—	2	2	no	—
90	C₂×Q₈×C₈₉	—	4	1	no	—
91	C₂×Q₈×C₈₉	—	4	1	no	—
92	C₂×Q₈×C₈₉	—	4	1	no	—
93	C₂×Q₈×C₈₉	—	4	1	no	—
94	C₂×Q₈×C₈₉	—	4	1	no	—
95	C₂²×C₄×C₈₉	—	4	1	no	—
96	C₂²×C₄×C₈₉	—	4	1	no	—
97	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
98	Q₁₆×C₈₉	—	2	2	no	—
99	Q₁₆×C₈₉	—	2	2	no	—
100	Q₁₆×C₈₉	—	4	1	no	—
101	Q₁₆×C₈₉	—	4	1	no	—
102	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
107	C₂×C₈×C₈₉	#41	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	—	4	1	no	—
109	C₄⋊C₄×C₈₉	—	4	1	no	—
110	C₁₆×C₈₉	—	2	2	no	—
111	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
112	C₂²×D₈₉	—	4	2	no	—
113	C₂²×D₈₉	—	8	1	no	—
114	C₂²×D₈₉	—	8	1	no	—
115	C₂²×D₈₉	—	8	2	no	—
116	C₂²×D₈₉	—	8	2	no	—
117	Q₈×C₈₉	—	4	2	no	—
118	Q₈×C₈₉	—	4	2	no	—
119	Q₈×C₈₉	—	8	1	no	—
120	Q₈×C₈₉	—	8	1	no	—
121	C₂×C₄×C₈₉	—	4	2	no	—
122	C₂×C₄×C₈₉	—	8	1	no	—
123	C₂×C₄×C₈₉	—	8	1	no	—
124	C₂×C₄×C₈₉	—	8	1	no	—
125	C₂×C₄×C₈₉	—	8	2	no	—
126	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
128	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
129	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
131	C₂×C₄×C₈₉	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×Dic₈₉	—	2	2	no	—
133	C₂×Dic₈₉	—	2	2	no	—
134	C₂×Dic₈₉	—	8	1	no	—
135	C₂×Dic₈₉	—	8	2	no	—
136	C₂×Dic₈₉	#102	1	4	yes	—
137	D₃₅₆	—	4	2	no	—
138	D₃₅₆	—	4	2	no	—
139	D₃₅₆	—	8	1	no	—
140	D₃₅₆	—	8	1	no	—
141	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
142	C₈×C₈₉	C₄×D₄	1	1	yes	characteristic
143	C₈×C₈₉	#121	1	1	yes	characteristic
144	C₈×C₈₉	#121	1	1	yes	characteristic
145	#10623494	#102	1	4	yes	—
146	C₂²×C₈₉	C₂×#102	1	1	yes	characteristic
147	C₂²×C₈₉	#486	1	1	yes	characteristic
148	C₂²×C₈₉	#493	1	1	yes	characteristic
149	C₂²×C₈₉	#509	1	2	yes	—
150	C₂²×C₈₉	#510	1	2	yes	—
151	C₂×D₈₉	—	8	4	no	—
152	C₂×D₈₉	—	8	2	no	—
153	C₂×D₈₉	—	16	2	no	—
154	C₂×D₈₉	—	16	1	no	—
155	C₂×D₈₉	—	16	1	no	—
156	C₄×C₈₉	—	8	2	no	—
157	C₄×C₈₉	—	16	1	no	—
158	C₄×C₈₉	#390	1	1	yes	characteristic
159	C₄×C₈₉	C₂×#102	1	1	yes	characteristic
160	C₄×C₈₉	#486	1	1	yes	characteristic
161	C₄×C₈₉	#492	1	1	yes	characteristic
162	Dic₈₉	—	2	4	no	—
163	Dic₈₉	—	16	1	no	—
164	#13205	—	89	1	no	maximal, Sylow
165	C₂×C₈₉	#1957	1	1	yes	characteristic
166	C₂×C₈₉	#2139	1	2	yes	—
167	C₂×C₈₉	#2243	1	2	yes	—
168	C₂×C₈₉	#2250	1	1	yes	characteristic
169	C₂×C₈₉	#2251	1	1	yes	characteristic
170	D₈₉	—	16	2	no	—
171	C₂×#495	—	89	1	no	—
172	C₂×#358	—	89	1	no	—
173	C₂×#358	D₈₉	1	1	yes	characteristic, complemented
174	C₂×#357	—	89	1	no	—
175	C₂×#357	—	89	1	no	—
176	C₂×#346	—	89	1	no	—
177	#1845	—	89	1	no	—
178	C₈₉	#13205	1	1	yes	characteristic, complemented, Sylow
179	C₂²×Q₁₆	C₂×D₈₉	1	1	yes	characteristic
180	C₂²×SD₁₆	—	178	1	no	—
181	C₂²×D₈	—	89	1	no	—
182	C₂×#124	—	178	1	no	—
183	C₂²×C₁₆	—	89	1	no	—
184	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
185	#495	—	89	4	no	—
186	#495	—	178	2	no	—
187	C₂×#107	—	89	1	no	—
188	C₂×#107	—	178	1	no	—
189	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
190	C₂×C₄×C₈	—	89	1	no	—
191	#384	—	178	1	no	—
192	#358	—	2	2	no	—
193	#358	—	178	2	no	—
194	#357	—	178	2	no	—
195	#357	—	178	2	no	—
196	#346	—	178	2	no	—
197	C₂²×Q₈	—	2	1	no	—
198	C₂²×D₄	—	178	1	no	—
199	C₂×Q₁₆	—	2	1	no	—
200	C₂×Q₁₆	—	2	1	no	—
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	—	2	1	no	—
203	C₂×Q₁₆	—	2	1	no	—
204	C₂×Q₁₆	C₄×D₈₉	1	2	yes	—
205	C₂×SD₁₆	—	356	2	no	—
206	C₂×SD₁₆	—	356	2	no	—
207	C₂×SD₁₆	—	178	4	no	—
208	C₂×D₈	—	89	2	no	—
209	C₂×D₈	—	178	1	no	—
210	C₂×D₈	—	178	1	no	—
211	C₂×D₈	—	178	1	no	—
212	C₂×D₈	—	178	1	no	—
213	C₂×D₈	—	178	1	no	—
214	C₂²×C₈	—	89	1	no	—
215	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
216	#124	—	356	2	no	—
217	#124	—	178	4	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	356	2	no	—
220	C₂×C₄⋊C₄	—	178	1	no	—
221	C₂×C₂²⋊C₄	—	356	2	no	—
222	C₂×C₄²	—	89	1	no	—
223	C₂×C₁₆	—	2	1	no	—
224	C₂×C₁₆	—	2	1	no	—
225	C₂×C₁₆	—	89	2	no	—
226	C₂×C₁₆	—	178	1	no	—
227	C₂×C₁₆	—	178	1	no	—
228	C₂×C₁₆	C₂²⋊D₈₉	1	2	yes	—
229	#107	—	2	1	no	—
230	#107	—	2	1	no	—
231	#107	—	356	2	no	—
232	#107	—	178	1	no	—
233	#107	—	178	1	no	—
234	C₄×C₈	—	89	2	no	—
235	C₄×C₈	—	89	2	no	—
236	C₂⁴	—	356	1	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	1	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	4	1	no	—
243	C₂×D₄	—	356	1	no	—
244	C₂×D₄	—	356	1	no	—
245	C₂×D₄	—	356	1	no	—
246	C₂×D₄	—	356	1	no	—
247	C₂×D₄	—	356	1	no	—
248	C₂×D₄	—	178	2	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	4	1	no	—
251	C₂²×C₄	—	89	2	no	—
252	C₂²×C₄	—	356	1	no	—
253	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	2	2	no	—
256	Q₁₆	—	4	1	no	—
257	Q₁₆	—	4	1	no	—
258	SD₁₆	—	356	8	no	—
259	D₈	—	356	1	no	—
260	D₈	—	356	1	no	—
261	D₈	—	178	2	no	—
262	D₈	—	178	2	no	—
263	C₂×C₈	—	89	2	no	—
264	C₂×C₈	—	89	2	no	—
265	C₂×C₈	—	89	2	no	—
266	C₂×C₈	C₂×C₄×D₈₉	1	1	yes	characteristic
267	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
268	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
269	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
270	C₂×C₈	#13083573	1	1	yes	characteristic
271	C₂×C₈	#13083574	1	1	yes	characteristic
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	356	1	no	—
275	C₄⋊C₄	—	356	1	no	—
276	C₄⋊C₄	—	712	4	no	—
277	C₂²⋊C₄	—	356	8	no	—
278	C₄²	—	89	2	no	—
279	C₄²	—	89	2	no	—
280	C₁₆	—	2	2	no	—
281	C₁₆	—	178	2	no	—
282	C₂³	—	356	2	no	—
283	C₂³	—	712	1	no	—
284	C₂³	—	712	1	no	—
285	C₂³	—	712	2	no	—
286	C₂³	—	712	2	no	—
287	C₂³	D₈×D₈₉	1	1	yes	characteristic
288	Q₈	—	4	2	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	8	1	no	—
291	Q₈	—	8	1	no	—
292	D₄	—	356	2	no	—
293	D₄	—	356	2	no	—
294	D₄	—	712	1	no	—
295	D₄	—	712	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	—	89	4	no	—
302	C₂×C₄	—	178	2	no	—
303	C₂×C₄	—	178	2	no	—
304	C₂×C₄	—	712	1	no	—
305	C₂×C₄	—	712	2	no	—
306	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
307	C₂×C₄	#15667998	1	1	yes	characteristic
308	C₂×C₄	#15668008	1	1	yes	characteristic
309	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
310	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
311	C₂×C₄	SD₁₆×D₈₉	1	1	yes	characteristic
312	C₈	—	89	4	no	—
313	C₈	#15667972	1	1	yes	characteristic
314	C₈	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
315	C₈	#15667984	1	1	yes	characteristic
316	C₈	#15667985	1	1	yes	characteristic
317	C₂²	—	1424	2	no	—
318	C₂²	—	1424	1	no	—
319	C₂²	—	1424	1	no	—
320	C₂²	—	712	4	no	—
321	C₂²	—	712	2	no	—
322	C₂²	#102×D₈₉	1	1	yes	characteristic
323	C₂²	#19151943	1	1	yes	characteristic
324	C₂²	#19151973	1	1	yes	characteristic
325	C₂²	#19152035	1	2	yes	—
326	C₂²	#19152040	1	2	yes	—
327	C₄	—	8	2	no	—
328	C₄	—	16	1	no	—
329	C₄	—	1424	1	no	—
330	C₄	—	178	4	no	—
331	C₄	#19151498	1	1	yes	characteristic
332	C₄	#102×D₈₉	1	1	yes	characteristic
333	C₄	#19151943	1	1	yes	characteristic
334	C₄	#19151969	1	1	yes	characteristic
335	C₂	—	1424	2	no	—
336	C₂	#26021056	1	1	yes	characteristic
337	C₂	#26021963	1	2	yes	—
338	C₂	#26022490	1	2	yes	—
339	C₂	#26022517	1	1	yes	characteristic
340	C₂	#26022520	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₈₉
commutator	C₂×C₈×C₈₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₉
Fitting	C₂×#358×C₈₉	C₂
socle	C₂³×C₈₉	C₂×D₈
prime radical	C₂²×D₇₁₂	C₂²
cosocle	C₂²×C₈×C₈₉	C₂³
hypercenter	C₂×#358	D₈₉
hypocenter	C₈₉	#13205
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#358

Complements

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

N	H ≅ G/N	type
2. C₂×#19151402	C₂	semidirect
5. C₂×#19151373	C₂	semidirect
6. #26020410	C₂	semidirect
7. C₂×#358×C₈₉	C₂	semidirect
13. #19151983	C₄	semidirect
173. C₂×#358	D₈₉	semidirect
178. C₈₉	#13205	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151983	2
C₂×#19151402	2
C₂×#19151400	2
C₂×#19151398	2
C₂×#19151373	2
#26020410	2
C₂×#358×C₈₉	2
#13205	89