Subgroups — #36707476

Order 22784.

658 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152016	C₂	1	1	yes	maximal, characteristic
2	#26028849	C₂	1	1	yes	maximal, characteristic
3	#26028848	C₂	1	1	yes	maximal, characteristic
4	#26022040	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022039	C₂	1	2	yes	maximal, complemented
6	#26022037	C₂	1	2	yes	maximal, complemented
7	#26022034	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022033	C₂	1	2	yes	maximal, complemented
9	C₂×#19151425	C₂	1	1	yes	maximal, characteristic, complemented
10	#367×D₈₉	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2159	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#15668018	C₂²	1	1	yes	characteristic
13	#19152079	—	2	1	no	—
14	#19152078	—	2	1	no	—
15	#19152016	C₄	1	8	yes	complemented
16	C₂×#15667951	C₂²	1	2	yes	—
17	C₂²×#13083549	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₈₉	C₂²	1	2	yes	—
19	#19151641	—	2	1	no	—
20	#19151639	—	2	2	no	—
21	#19151638	—	2	1	no	—
22	#19151637	—	2	2	no	—
23	#19151636	C₂²	1	1	yes	characteristic
24	#19151635	C₂²	1	2	yes	—
25	#19151635	C₂²	1	2	yes	—
26	C₂×#15667946	C₂²	1	1	yes	characteristic
27	#107×D₈₉	C₂²	1	2	yes	—
28	#107×D₈₉	C₂²	1	2	yes	—
29	#19151426	C₂²	1	2	yes	—
30	#19151425	C₂²	1	1	yes	characteristic
31	#19151425	C₂²	1	2	yes	—
32	#19151425	C₂²	1	2	yes	—
33	#19151425	C₂²	1	1	yes	characteristic
34	#19151424	C₂²	1	2	yes	—
35	#19151423	C₂²	1	1	yes	characteristic
36	#19151423	C₂²	1	1	yes	characteristic
37	C₂×M₃₂×C₈₉	C₂²	1	1	yes	characteristic
38	#427×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
40	#367×C₈₉	C₂²	1	2	yes	—
41	#366×C₈₉	C₂²	1	2	yes	—
42	C₂×#13083574	D₄	1	1	yes	characteristic
43	#15668018	C₂×C₄	1	4	yes	—
44	#15668018	Q₈	1	4	yes	—
45	C₂×#13083558	D₄	1	1	yes	characteristic
46	C₂×#13083557	D₄	1	2	yes	—
47	C₂×C₈×D₈₉	C₂³	1	2	yes	—
48	C₄×Dic₁₇₈	—	4	1	no	—
49	#15667977	—	4	1	no	—
50	C₄×D₃₅₆	—	4	1	no	—
51	#15667974	—	4	1	no	—
52	C₄×C₂²⋊D₈₉	—	4	2	no	—
53	#15667972	—	4	2	no	—
54	#15667971	—	2	1	no	—
55	#15667970	—	2	2	no	—
56	#15667970	—	2	2	no	—
57	C₂×C₄⋊Dic₈₉	—	2	1	no	—
58	C₄⋊C₄×D₈₉	—	2	2	no	—
59	C₄⋊C₄×D₈₉	—	2	2	no	—
60	#15667951	C₂×C₄	1	8	yes	complemented
61	C₂×#13083549	C₂×C₄	1	4	yes	complemented
62	C₂×#13083549	C₂³	1	1	yes	characteristic
63	C₂×#13083549	C₂³	1	1	yes	characteristic
64	C₁₆×D₈₉	C₂×C₄	1	8	yes	complemented
65	#15667946	—	2	1	no	—
66	#15667946	C₂³	1	1	yes	characteristic
67	#15667946	C₂³	1	2	yes	—
68	#15667946	C₂³	1	1	yes	characteristic
69	#15667945	—	2	2	no	—
70	#15667945	—	2	2	no	—
71	#15667944	—	2	1	no	—
72	#15667943	—	2	2	no	—
73	#15667943	—	2	2	no	—
74	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	#117×C₈₉	—	2	1	no	—
76	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
77	M₃₂×C₈₉	C₂×C₄	1	4	yes	complemented
78	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
79	#107×C₈₉	—	2	1	no	—
80	#107×C₈₉	C₂³	1	1	yes	characteristic
81	#107×C₈₉	C₂³	1	2	yes	—
82	#107×C₈₉	C₂³	1	1	yes	characteristic
83	#106×C₈₉	—	2	1	no	—
84	#13083574	C₄⋊C₄	1	4	yes	—
85	#13083574	Q₁₆	1	4	yes	—
86	C₂×Dic₁₇₈	D₈	1	1	yes	characteristic
87	C₂×D₃₅₆	D₈	1	1	yes	characteristic
88	C₂×C₂²⋊D₈₉	D₈	1	2	yes	—
89	C₂²×Dic₈₉	—	4	1	no	—
90	C₂×C₄×D₈₉	—	4	2	no	—
91	C₂×C₄×D₈₉	—	4	2	no	—
92	C₂×C₄×D₈₉	C₂×D₄	1	2	yes	—
93	#13083558	C₄⋊C₄	1	4	yes	—
94	#13083557	C₄⋊C₄	1	8	yes	—
95	C₂×#10623494	C₂×D₄	1	2	yes	—
96	C₈×D₈₉	C₂²×C₄	1	4	yes	—
97	C₈×D₈₉	C₂×Q₈	1	4	yes	—
98	C₄⋊Dic₈₉	—	2	1	no	—
99	C₄⋊Dic₈₉	—	2	2	no	—
100	C₄⋊Dic₈₉	—	2	1	no	—
101	C₄⋊Dic₈₉	—	4	1	no	—
102	#13083553	—	4	2	no	—
103	#13083553	—	4	2	no	—
104	C₂²⋊Dic₈₉	—	4	1	no	—
105	#13083551	—	4	2	no	—
106	#13083551	—	4	2	no	—
107	C₄×Dic₈₉	—	4	1	no	—
108	C₄×Dic₈₉	—	4	2	no	—
109	C₄×Dic₈₉	—	4	2	no	—
110	#13083549	C₂²×C₄	1	2	yes	—
111	#13083549	C₂²×C₄	1	2	yes	—
112	C₂²×C₄×C₈₉	—	4	1	no	—
113	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
115	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
117	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
118	C₄⋊C₄×C₈₉	—	2	1	no	—
119	C₄⋊C₄×C₈₉	—	2	2	no	—
120	C₄⋊C₄×C₈₉	—	2	1	no	—
121	C₄⋊C₄×C₈₉	—	4	1	no	—
122	C₂²⋊C₄×C₈₉	—	4	1	no	—
123	C₄²×C₈₉	—	4	1	no	—
124	C₁₆×C₈₉	C₂²×C₄	1	4	yes	—
125	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
126	C₂²×D₈₉	C₂×D₈	1	2	yes	—
127	C₂×C₄×C₈₉	—	4	1	no	—
128	C₂×C₄×C₈₉	—	4	1	no	—
129	C₂×C₄×C₈₉	—	4	2	no	—
130	C₂×C₄×C₈₉	—	16	1	no	—
131	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
132	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
134	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
135	C₂²⋊D₈₉	#107	1	8	yes	—
136	C₂×Dic₈₉	—	4	1	no	—
137	C₂×Dic₈₉	—	4	2	no	—
138	C₂×Dic₈₉	—	4	1	no	—
139	C₂×Dic₈₉	—	16	1	no	—
140	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
141	D₃₅₆	#107	1	4	yes	—
142	C₄×D₈₉	—	16	2	no	—
143	C₄×D₈₉	—	16	2	no	—
144	C₄×D₈₉	C₂×C₄⋊C₄	1	4	yes	—
145	C₄×D₈₉	C₂×Q₁₆	1	4	yes	—
146	Dic₁₇₈	#107	1	4	yes	—
147	C₈×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
148	C₈×C₈₉	C₂³×C₄	1	1	yes	characteristic
149	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
150	#10623494	C₂×C₄⋊C₄	1	4	yes	—
151	C₂²×C₈₉	—	2	1	no	—
152	C₂²×C₈₉	—	2	1	no	—
153	C₂²×C₈₉	C₂×#107	1	2	yes	—
154	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
155	C₂×D₈₉	—	2	2	no	—
156	C₂×D₈₉	—	2	2	no	—
157	C₂×D₈₉	C₂×#107	1	4	yes	—
158	C₄×C₈₉	—	8	1	no	—
159	C₄×C₈₉	—	8	2	no	—
160	C₄×C₈₉	—	8	1	no	—
161	C₄×C₈₉	C₂×#107	1	2	yes	—
162	C₄×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
163	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
164	Dic₈₉	—	8	1	no	—
165	Dic₈₉	—	8	2	no	—
166	Dic₈₉	—	8	1	no	—
167	Dic₈₉	C₂×#107	1	4	yes	—
168	#33411	—	89	1	no	maximal, Sylow
169	C₂×C₈₉	—	2	2	no	—
170	C₂×C₈₉	C₂²×#107	1	1	yes	characteristic
171	C₂×C₈₉	#3416	1	1	yes	characteristic
172	C₂×C₈₉	#3416	1	1	yes	characteristic
173	D₈₉	—	2	4	no	—
174	C₂×#504	—	89	1	no	—
175	#2913	—	89	1	no	—
176	#2913	—	89	1	no	—
177	#2159	—	89	1	no	—
178	#2159	—	89	2	no	—
179	#2159	—	89	2	no	—
180	#2159	D₈₉	1	1	yes	characteristic, complemented
181	#2158	—	89	1	no	—
182	#2158	—	89	2	no	—
183	C₂×#367	—	89	1	no	—
184	C₂×#367	—	89	2	no	—
185	C₈₉	#33411	1	1	yes	characteristic, complemented, Sylow
186	C₂×#131	—	89	1	no	—
187	#520	—	178	1	no	—
188	#520	—	178	1	no	—
189	#504	—	89	8	no	—
190	C₂×M₃₂	—	89	2	no	—
191	C₂×M₃₂	C₂×D₈₉	1	1	yes	characteristic
192	C₂²×C₁₆	—	89	1	no	—
193	C₂²×C₁₆	—	89	2	no	—
194	#428	—	178	2	no	—
195	#428	—	178	2	no	—
196	#428	—	178	1	no	—
197	#428	—	178	1	no	—
198	#427	—	89	1	no	—
199	#427	—	89	2	no	—
200	#427	—	89	2	no	—
201	#427	C₂×D₈₉	1	1	yes	characteristic
202	C₂×#107	—	89	1	no	—
203	C₂×#107	—	89	2	no	—
204	C₂×#107	—	89	2	no	—
205	C₂×#107	C₂×D₈₉	1	1	yes	characteristic
206	#367	—	89	2	no	—
207	#367	—	89	1	no	—
208	#367	—	89	2	no	—
209	#367	—	89	2	no	—
210	#367	—	89	1	no	—
211	#367	C₂×D₈₉	1	2	yes	—
212	#366	—	89	2	no	—
213	#366	—	89	1	no	—
214	#366	—	89	1	no	—
215	#366	C₂×D₈₉	1	2	yes	—
216	C₂×#41	—	89	1	no	—
217	#131	—	89	4	no	—
218	#131	—	89	4	no	—
219	C₂×M₁₆	—	89	1	no	—
220	C₂×M₁₆	—	89	2	no	—
221	C₂²×C₈	—	89	2	no	—
222	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
223	C₄×Q₈	—	356	1	no	—
224	C₄×Q₈	—	356	1	no	—
225	C₄×D₄	—	356	1	no	—
226	C₄×D₄	—	356	2	no	—
227	C₄×D₄	—	356	1	no	—
228	C₄×D₄	—	356	2	no	—
229	#117	—	2	1	no	—
230	#117	—	178	2	no	—
231	#117	—	178	2	no	—
232	#117	—	178	1	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	178	2	no	—
235	C₂×C₄⋊C₄	—	178	2	no	—
236	C₂×C₄⋊C₄	—	178	1	no	—
237	M₃₂	—	89	8	no	—
238	M₃₂	C₄×D₈₉	1	4	yes	complemented
239	C₂×C₁₆	—	89	4	no	—
240	C₂×C₁₆	—	89	1	no	—
241	C₂×C₁₆	—	89	1	no	—
242	C₂×C₁₆	—	89	8	no	—
243	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
244	#107	—	2	1	no	—
245	#107	—	89	1	no	—
246	#107	—	89	2	no	—
247	#107	—	89	1	no	—
248	#107	—	178	2	no	—
249	#107	—	178	2	no	—
250	#107	—	178	1	no	—
251	#107	C₂²×D₈₉	1	1	yes	characteristic
252	#107	C₂²×D₈₉	1	1	yes	characteristic
253	#107	C₂²×D₈₉	1	2	yes	—
254	#106	—	2	1	no	—
255	#106	—	178	2	no	—
256	#106	—	178	2	no	—
257	#106	—	178	1	no	—
258	#41	—	89	4	no	—
259	#41	—	89	4	no	—
260	C₂×Q₈	—	89	1	no	—
261	C₂×D₄	—	89	2	no	—
262	C₂×D₄	—	89	1	no	—
263	C₂²×C₄	—	4	1	no	—
264	C₂²×C₄	—	89	2	no	—
265	C₂²×C₄	—	356	2	no	—
266	C₂²×C₄	—	356	2	no	—
267	C₂²×C₄	—	356	1	no	—
268	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
269	M₁₆	—	89	4	no	—
270	M₁₆	—	89	8	no	—
271	C₂×C₈	—	89	2	no	—
272	C₂×C₈	—	89	4	no	—
273	C₂×C₈	—	89	4	no	—
274	C₂×C₈	C₂×C₄×D₈₉	1	2	yes	—
275	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
276	C₂×C₈	Q₈×D₈₉	1	2	yes	—
277	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	356	2	no	—
283	C₄⋊C₄	—	356	2	no	—
284	C₄⋊C₄	—	356	1	no	—
285	C₄⋊C₄	—	178	1	no	—
286	C₄⋊C₄	—	178	2	no	—
287	C₄⋊C₄	—	178	1	no	—
288	C₂²⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	356	2	no	—
290	C₂²⋊C₄	—	356	2	no	—
291	C₂²⋊C₄	—	356	1	no	—
292	C₄²	—	4	1	no	—
293	C₄²	—	356	2	no	—
294	C₄²	—	356	2	no	—
295	C₄²	—	356	1	no	—
296	C₁₆	—	89	2	no	—
297	C₁₆	—	89	2	no	—
298	C₁₆	C₂×C₄×D₈₉	1	4	yes	—
299	C₂³	—	89	2	no	—
300	C₂³	D₈×D₈₉	1	1	yes	characteristic
301	Q₈	—	89	4	no	—
302	D₄	—	89	8	no	—
303	D₄	—	89	4	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	16	1	no	—
308	C₂×C₄	—	89	2	no	—
309	C₂×C₄	—	89	4	no	—
310	C₂×C₄	—	89	4	no	—
311	C₂×C₄	—	356	1	no	—
312	C₂×C₄	—	356	2	no	—
313	C₂×C₄	—	356	1	no	—
314	C₂×C₄	—	1424	2	no	—
315	C₂×C₄	—	1424	2	no	—
316	C₂×C₄	—	1424	1	no	—
317	C₂×C₄	C₄⋊C₄×D₈₉	1	2	yes	—
318	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
319	C₂×C₄	Q₁₆×D₈₉	1	2	yes	—
320	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
321	C₈	—	89	4	no	—
322	C₈	C₄⋊C₄×D₈₉	1	2	yes	—
323	C₈	C₂²×C₄×D₈₉	1	1	yes	characteristic
324	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	89	4	no	—
328	C₂²	—	178	2	no	—
329	C₂²	—	178	2	no	—
330	C₂²	#107×D₈₉	1	2	yes	—
331	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
332	C₄	—	8	1	no	—
333	C₄	—	8	2	no	—
334	C₄	—	8	1	no	—
335	C₄	—	89	4	no	—
336	C₄	—	712	2	no	—
337	C₄	—	712	1	no	—
338	C₄	—	712	1	no	—
339	C₄	#107×D₈₉	1	2	yes	—
340	C₄	C₂×C₄⋊C₄×D₈₉	1	1	yes	characteristic
341	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	178	4	no	—
344	C₂	C₂×#107×D₈₉	1	1	yes	characteristic
345	C₂	#26033397	1	1	yes	characteristic
346	C₂	#26033398	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator	C₈×C₈₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2159	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×#13083574	D₄
cosocle	C₂×C₈×C₈₉	C₂⁴
hypercenter	#2159	D₈₉
hypocenter	C₈₉	#33411
LCS	G ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2159

Complements

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

N	H ≅ G/N	type
4. #26022040	C₂	semidirect
5. #26022039	C₂	semidirect
6. #26022037	C₂	semidirect
7. #26022034	C₂	semidirect
8. #26022033	C₂	semidirect
9. C₂×#19151425	C₂	semidirect
10. #367×D₈₉	C₂	semidirect
11. C₈₉×#2159	C₂	semidirect
15. #19152016	C₄	semidirect
60. #15667951	C₂×C₄	semidirect
61. C₂×#13083549	C₂×C₄	semidirect
64. C₁₆×D₈₉	C₂×C₄	semidirect
77. M₃₂×C₈₉	C₂×C₄	semidirect
180. #2159	D₈₉	semidirect
185. C₈₉	#33411	semidirect
238. M₃₂	C₄×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152016	2
#26028849	2
#26028848	2
#26022040	2
#26022039	2
#26022037	2
#26022034	2
#26022033	2
C₂×#19151425	2
#367×D₈₉	2
C₈₉×#2159	2
#33411	89