Subgroups — #36447363

Order 22784.

454 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151675	C₂	1	1	yes	maximal, characteristic
2	#26020241	C₂	1	1	yes	maximal, characteristic
3	#26020241	C₂	1	1	yes	maximal, characteristic
4	#26019010	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019007	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018127	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1292	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#13083557	C₄	1	2	yes	—
9	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
10	#19151675	—	2	1	no	—
11	#19151675	—	2	1	no	—
12	#19151675	—	2	1	no	—
13	#19151675	—	2	1	no	—
14	#19151622	C₂²	1	1	yes	characteristic
15	#19151622	C₂²	1	1	yes	characteristic
16	C₂×#15667928	—	2	1	no	—
17	C₂×#15667928	—	2	1	no	—
18	C₂×#15667924	C₄	1	2	yes	—
19	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
20	#19151366	—	2	1	no	—
21	#19151366	—	2	1	no	—
22	#19151350	—	2	1	no	—
23	#19151350	—	2	1	no	—
24	#19151348	—	2	1	no	—
25	#19151348	—	2	1	no	—
26	#423×C₈₉	C₂²	1	1	yes	characteristic
27	#423×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₂²×D₃₅₆	D₄	1	1	yes	characteristic
30	C₂²×C₄×D₈₉	C₂×C₄	1	2	yes	—
31	C₂×#13083558	—	2	1	no	—
32	C₂×#13083558	—	2	1	no	—
33	C₂×#13083557	—	2	2	no	—
34	C₂×#13083557	—	2	2	no	—
35	C₂×#13083557	—	4	4	no	—
36	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
37	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
38	C₄×D₃₅₆	—	2	1	no	—
39	C₄×D₃₅₆	—	2	1	no	—
40	C₄×D₃₅₆	—	2	1	no	—
41	C₄×D₃₅₆	—	2	1	no	—
42	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
43	C₂×#13083551	Q₈	1	2	yes	—
44	C₄⋊#10623494	—	2	1	no	—
45	C₄⋊#10623494	—	2	1	no	—
46	C₄⋊#10623494	—	2	1	no	—
47	C₄⋊#10623494	—	2	1	no	—
48	#15667928	—	4	2	no	—
49	#15667928	—	4	2	no	—
50	#15667924	—	2	2	no	—
51	#15667924	—	2	2	no	—
52	C₄×#10623494	D₄	1	1	yes	characteristic
53	C₄×#10623494	D₄	1	1	yes	characteristic
54	C₄×#10623494	D₄	1	1	yes	characteristic
55	C₄×#10623494	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₈₉	—	2	1	no	—
57	C₂×M₁₆×C₈₉	—	2	1	no	—
58	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₈₉	—	2	1	no	—
61	C₄⋊C₈×C₈₉	—	2	1	no	—
62	C₄⋊C₈×C₈₉	—	2	1	no	—
63	C₄⋊C₈×C₈₉	—	2	1	no	—
64	C₄⋊C₈×C₈₉	—	2	1	no	—
65	C₄⋊C₈×C₈₉	—	2	1	no	—
66	C₂³×D₈₉	C₄⋊C₄	1	2	yes	—
67	C₂×D₃₅₆	—	2	1	no	—
68	C₂×D₃₅₆	—	2	1	no	—
69	C₂×D₃₅₆	—	2	1	no	—
70	C₂×D₃₅₆	—	2	1	no	—
71	C₂×D₃₅₆	—	2	1	no	—
72	C₂×D₃₅₆	—	2	1	no	—
73	C₂²×Dic₈₉	C₄⋊C₄	1	2	yes	—
74	C₂×C₄×D₈₉	—	2	2	no	—
75	C₂×C₄×D₈₉	—	2	2	no	—
76	C₂×C₄×D₈₉	—	4	2	no	—
77	C₂×C₄×D₈₉	—	4	2	no	—
78	#13083558	—	4	1	no	—
79	#13083558	—	4	1	no	—
80	#13083558	—	4	1	no	—
81	#13083558	—	4	1	no	—
82	#13083557	—	4	4	no	—
83	#13083557	—	4	4	no	—
84	C₂×#10623494	—	2	2	no	—
85	C₂×#10623494	—	4	1	no	—
86	C₂×#10623494	—	4	1	no	—
87	C₂×#10623494	—	4	1	no	—
88	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×#10623494	C₂²⋊C₄	1	1	yes	characteristic
92	C₄⋊Dic₈₉	—	2	1	no	—
93	C₄⋊Dic₈₉	—	2	1	no	—
94	#13083551	—	2	2	no	—
95	#13083551	—	2	2	no	—
96	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
99	M₁₆×C₈₉	—	4	1	no	—
100	M₁₆×C₈₉	—	4	1	no	—
101	M₁₆×C₈₉	—	4	1	no	—
102	M₁₆×C₈₉	—	4	1	no	—
103	C₂×C₈×C₈₉	—	2	2	no	—
104	C₂×C₈×C₈₉	—	4	2	no	—
105	C₂×C₈×C₈₉	—	4	1	no	—
106	C₂×C₈×C₈₉	—	4	1	no	—
107	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₈₉	#41	1	1	yes	characteristic
110	C₄²×C₈₉	#41	1	1	yes	characteristic
111	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂²×D₈₉	—	2	2	no	—
113	C₂²×D₈₉	—	2	2	no	—
114	C₂²×D₈₉	—	2	2	no	—
115	C₂²×D₈₉	—	4	2	no	—
116	C₂²×D₈₉	—	4	2	no	—
117	C₂×C₄×C₈₉	—	2	1	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	4	1	no	—
120	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
122	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
123	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
124	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
126	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	1	yes	characteristic
128	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
129	C₂×C₄×C₈₉	#137	1	1	yes	characteristic
130	C₂×Dic₈₉	—	2	2	no	—
131	C₂×Dic₈₉	—	4	2	no	—
132	D₃₅₆	—	4	1	no	—
133	D₃₅₆	—	4	1	no	—
134	D₃₅₆	—	4	1	no	—
135	D₃₅₆	—	4	1	no	—
136	C₄×D₈₉	—	4	4	no	—
137	C₄×D₈₉	—	4	2	no	—
138	C₄×D₈₉	—	4	2	no	—
139	C₈×C₈₉	—	4	2	no	—
140	C₈×C₈₉	—	8	1	no	—
141	C₈×C₈₉	—	8	1	no	—
142	#10623494	—	2	1	no	—
143	#10623494	—	2	1	no	—
144	#10623494	—	2	1	no	—
145	#10623494	—	8	1	no	—
146	#10623494	—	8	1	no	—
147	#10623494	#100	1	2	yes	complemented
148	C₂²×C₈₉	—	2	1	no	—
149	C₂²×C₈₉	—	2	1	no	—
150	C₂²×C₈₉	#380	1	1	yes	characteristic
151	C₂²×C₈₉	#428	1	1	yes	characteristic
152	C₂²×C₈₉	#428	1	1	yes	characteristic
153	C₂×D₈₉	—	4	2	no	—
154	C₂×D₈₉	—	4	2	no	—
155	C₂×D₈₉	—	4	2	no	—
156	C₂×D₈₉	—	8	2	no	—
157	C₂×D₈₉	—	8	2	no	—
158	C₄×C₈₉	—	2	1	no	—
159	C₄×C₈₉	—	2	1	no	—
160	C₄×C₈₉	—	2	1	no	—
161	C₄×C₈₉	C₂×#100	1	1	yes	characteristic
162	C₄×C₈₉	#413	1	1	yes	characteristic
163	C₄×C₈₉	#468	1	1	yes	characteristic
164	C₄×C₈₉	#468	1	1	yes	characteristic
165	C₄×C₈₉	#469	1	1	yes	characteristic
166	C₄×C₈₉	#470	1	1	yes	characteristic
167	Dic₈₉	—	4	2	no	—
168	#8618	—	89	1	no	maximal, Sylow
169	C₂×C₈₉	—	2	1	no	—
170	C₂×C₈₉	—	2	1	no	—
171	C₂×C₈₉	#1800	1	1	yes	characteristic
172	C₂×C₈₉	#1809	1	1	yes	characteristic
173	C₂×C₈₉	#1813	1	1	yes	characteristic
174	D₈₉	—	8	2	no	—
175	C₂×#436	—	89	1	no	—
176	#1800	—	89	1	no	—
177	#1800	—	89	1	no	—
178	#1567	—	89	1	no	—
179	#1566	—	89	1	no	—
180	#1292	—	89	1	no	—
181	#1292	D₈₉	1	1	yes	characteristic, complemented
182	C₈₉	#8618	1	1	yes	characteristic, complemented, Sylow
183	C₂²×M₁₆	—	89	2	no	—
184	C₂×C₄×D₄	—	89	1	no	—
185	#436	—	178	1	no	—
186	#436	—	178	1	no	—
187	#436	—	178	1	no	—
188	#436	—	178	1	no	—
189	#423	—	89	1	no	—
190	#423	—	89	1	no	—
191	#423	C₂×D₈₉	1	1	yes	characteristic
192	#423	C₂×D₈₉	1	1	yes	characteristic
193	C₂×C₄⋊C₈	C₂×D₈₉	1	1	yes	characteristic
194	C₂×#100	—	178	1	no	—
195	C₂×#100	—	178	1	no	—
196	C₂×C₂²⋊C₈	—	89	2	no	—
197	C₂×C₄×C₈	—	89	1	no	—
198	#343	—	178	1	no	—
199	#343	—	178	1	no	—
200	#335	—	178	1	no	—
201	#335	—	178	1	no	—
202	#334	—	178	1	no	—
203	#334	—	178	1	no	—
204	C₂²×D₄	—	89	1	no	—
205	C₂³×C₄	—	89	2	no	—
206	C₂×M₁₆	—	2	1	no	—
207	C₂×M₁₆	—	2	1	no	—
208	C₂×M₁₆	—	356	4	no	—
209	C₂×M₁₆	—	178	2	no	—
210	C₂×M₁₆	—	178	2	no	—
211	C₂×M₁₆	—	178	1	no	—
212	C₂×M₁₆	—	178	1	no	—
213	C₂²×C₈	—	89	1	no	—
214	C₂²×C₈	—	89	1	no	—
215	C₂²×C₈	C₄×D₈₉	1	2	yes	—
216	C₄×D₄	—	178	1	no	—
217	C₄×D₄	—	178	1	no	—
218	C₄×D₄	—	178	1	no	—
219	C₄×D₄	—	178	1	no	—
220	C₂×C₄⋊C₄	—	89	1	no	—
221	C₂×C₂²⋊C₄	—	89	2	no	—
222	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
223	C₄⋊C₈	—	2	1	no	—
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	—	178	1	no	—
230	C₄⋊C₈	—	178	1	no	—
231	C₄⋊C₈	—	178	1	no	—
232	C₄⋊C₈	—	178	1	no	—
233	#100	—	356	2	no	—
234	#100	—	356	2	no	—
235	C₂²⋊C₈	—	178	2	no	—
236	C₂²⋊C₈	—	178	2	no	—
237	C₄×C₈	—	89	1	no	—
238	C₄×C₈	—	89	1	no	—
239	C₄×C₈	—	89	1	no	—
240	C₄×C₈	—	89	1	no	—
241	C₂⁴	—	89	2	no	—
242	C₂×D₄	—	178	1	no	—
243	C₂×D₄	—	178	1	no	—
244	C₂×D₄	—	178	1	no	—
245	C₂×D₄	—	178	1	no	—
246	C₂×D₄	—	178	1	no	—
247	C₂×D₄	—	178	1	no	—
248	C₂²×C₄	—	89	2	no	—
249	C₂²×C₄	—	356	2	no	—
250	C₂²×C₄	—	356	2	no	—
251	C₂²×C₄	—	178	2	no	—
252	C₂²×C₄	—	178	2	no	—
253	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
254	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
255	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
256	M₁₆	—	4	1	no	—
257	M₁₆	—	4	1	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	4	1	no	—
260	M₁₆	—	356	4	no	—
261	M₁₆	—	356	4	no	—
262	M₁₆	—	356	1	no	—
263	M₁₆	—	356	1	no	—
264	M₁₆	—	356	1	no	—
265	M₁₆	—	356	1	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	4	2	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	—	89	1	no	—
271	C₂×C₈	—	89	1	no	—
272	C₂×C₈	—	89	1	no	—
273	C₂×C₈	—	89	1	no	—
274	C₂×C₈	—	356	1	no	—
275	C₂×C₈	—	356	1	no	—
276	C₂×C₈	—	356	1	no	—
277	C₂×C₈	—	178	2	no	—
278	C₄⋊C₄	—	178	1	no	—
279	C₄⋊C₄	—	178	1	no	—
280	C₂²⋊C₄	—	178	2	no	—
281	C₂²⋊C₄	—	178	2	no	—
282	C₄²	D₄×D₈₉	1	1	yes	characteristic
283	C₄²	D₄×D₈₉	1	1	yes	characteristic
284	C₄²	#13083574	1	1	yes	characteristic
285	C₄²	#13083574	1	1	yes	characteristic
286	C₂³	—	356	2	no	—
287	C₂³	—	356	2	no	—
288	C₂³	—	178	2	no	—
289	C₂³	—	178	2	no	—
290	C₂³	—	178	2	no	—
291	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
292	D₄	—	356	1	no	—
293	D₄	—	356	1	no	—
294	D₄	—	356	1	no	—
295	D₄	—	356	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	356	4	no	—
300	C₂×C₄	—	356	2	no	—
301	C₂×C₄	—	356	2	no	—
302	C₂×C₄	—	356	2	no	—
303	C₂×C₄	—	178	2	no	—
304	C₂×C₄	C₂²⋊C₄×D₈₉	1	1	yes	characteristic
305	C₂×C₄	#15667969	1	1	yes	characteristic
306	C₂×C₄	#15667985	1	1	yes	characteristic
307	C₂×C₄	#15667985	1	1	yes	characteristic
308	C₂×C₄	#15667990	1	1	yes	characteristic
309	C₂×C₄	#15667990	1	1	yes	characteristic
310	C₂×C₄	#15668036	1	1	yes	characteristic
311	C₂×C₄	#15668039	1	1	yes	characteristic
312	C₂×C₄	#15668041	1	1	yes	characteristic
313	C₂×C₄	#15668044	1	1	yes	characteristic
314	C₈	—	4	2	no	—
315	C₈	—	8	1	no	—
316	C₈	—	8	1	no	—
317	C₈	—	89	2	no	—
318	C₈	—	178	1	no	—
319	C₈	—	178	1	no	—
320	C₈	—	178	1	no	—
321	C₈	—	712	1	no	—
322	C₈	—	712	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	356	2	no	—
326	C₂²	—	356	2	no	—
327	C₂²	—	356	2	no	—
328	C₂²	—	712	2	no	—
329	C₂²	—	712	2	no	—
330	C₂²	#19151457	1	1	yes	characteristic
331	C₂²	#19151637	1	1	yes	characteristic
332	C₂²	#19151639	1	1	yes	characteristic
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	356	2	no	—
337	C₄	#100×D₈₉	1	1	yes	characteristic
338	C₄	#19151581	1	1	yes	characteristic
339	C₄	#19151843	1	1	yes	characteristic
340	C₄	#19151847	1	1	yes	characteristic
341	C₄	#19151850	1	1	yes	characteristic
342	C₄	#19151855	1	1	yes	characteristic
343	C₂	—	2	1	no	—
344	C₂	—	2	1	no	—
345	C₂	—	712	2	no	—
346	C₂	#26020243	1	1	yes	characteristic
347	C₂	#26020284	1	1	yes	characteristic
348	C₂	#26020301	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151457
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1292	C₂
socle	C₂²×C₈₉	#380
prime radical	C₂²×D₃₅₆	D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1292	D₈₉
hypocenter	C₈₉	#8618
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1292

Complements

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

N	H ≅ G/N	type
4. #26019010	C₂	semidirect
5. #26019007	C₂	semidirect
6. #26018127	C₂	semidirect
7. C₈₉×#1292	C₂	semidirect
147. #10623494	#100	semidirect
181. #1292	D₈₉	semidirect
182. C₈₉	#8618	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151675	2
#26020241	2
#26019010	2
#26019007	2
#26018127	2
C₈₉×#1292	2
#8618	89