Subgroups — #36447361

Order 22784.

454 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19151677	C₂	1	1	yes	maximal, characteristic
2	#26020241	C₂	1	2	yes	maximal
3	#26019011	C₂	1	1	yes	maximal, characteristic, complemented
4	#26018126	C₂	1	2	yes	maximal, complemented
5	C₈₉×#1566	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#13083557	C₄	1	2	yes	—
7	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
8	#19151677	—	2	1	no	—
9	#19151677	—	2	1	no	—
10	#19151677	—	2	1	no	—
11	#19151677	—	2	1	no	—
12	#19151622	C₂²	1	2	yes	—
13	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
14	C₂×#15667928	—	2	2	no	—
15	C₂×#15667924	C₄	1	2	yes	—
16	#19151366	—	2	2	no	—
17	#19151351	—	2	1	no	—
18	#19151351	—	2	1	no	—
19	#423×C₈₉	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	#334×C₈₉	—	2	1	no	—
22	#334×C₈₉	—	2	1	no	—
23	C₂²×D₃₅₆	D₄	1	1	yes	characteristic
24	C₂²×C₄×D₈₉	C₂×C₄	1	2	yes	—
25	C₂×#13083558	—	2	2	no	—
26	C₂×#13083557	—	2	2	no	—
27	C₂×#13083557	—	2	2	no	—
28	C₂×#13083557	—	4	4	no	—
29	C₂²×#10623494	C₂×C₄	1	2	yes	—
30	C₄×D₃₅₆	—	2	1	no	—
31	C₄×D₃₅₆	—	2	1	no	—
32	C₄×D₃₅₆	—	2	1	no	—
33	C₄×D₃₅₆	—	2	1	no	—
34	C₂×C₄⋊Dic₈₉	D₄	1	1	yes	characteristic
35	C₂×#13083551	Q₈	1	2	yes	—
36	C₄⋊#10623494	—	2	1	no	—
37	C₄⋊#10623494	—	2	1	no	—
38	C₄⋊#10623494	—	2	2	no	—
39	C₄⋊#10623494	—	2	2	no	—
40	#15667928	—	4	4	no	—
41	#15667924	—	2	2	no	—
42	#15667924	—	2	2	no	—
43	C₂×M₁₆×C₈₉	—	2	2	no	—
44	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
47	C₄⋊C₈×C₈₉	—	2	2	no	—
48	C₄⋊C₈×C₈₉	—	2	2	no	—
49	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
52	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
53	C₂³×D₈₉	C₄⋊C₄	1	2	yes	—
54	C₂×D₃₅₆	—	2	1	no	—
55	C₂×D₃₅₆	—	2	1	no	—
56	C₂×D₃₅₆	—	2	2	no	—
57	C₂×D₃₅₆	—	2	2	no	—
58	C₂²×Dic₈₉	C₄⋊C₄	1	2	yes	—
59	C₂×C₄×D₈₉	—	2	2	no	—
60	C₂×C₄×D₈₉	—	2	2	no	—
61	C₂×C₄×D₈₉	—	4	2	no	—
62	C₂×C₄×D₈₉	—	4	2	no	—
63	#13083558	—	4	2	no	—
64	#13083558	—	4	2	no	—
65	#13083557	—	4	4	no	—
66	#13083557	—	4	4	no	—
67	C₂×#10623494	—	2	2	no	—
68	C₂×#10623494	—	4	2	no	—
69	C₂×#10623494	—	4	2	no	—
70	C₄⋊Dic₈₉	—	2	1	no	—
71	C₄⋊Dic₈₉	—	2	1	no	—
72	#13083551	—	2	2	no	—
73	#13083551	—	2	2	no	—
74	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
77	M₁₆×C₈₉	—	4	2	no	—
78	M₁₆×C₈₉	—	4	2	no	—
79	C₂×C₈×C₈₉	—	2	2	no	—
80	C₂×C₈×C₈₉	—	4	1	no	—
81	C₂×C₈×C₈₉	—	4	2	no	—
82	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
86	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₈₉	#41	1	1	yes	characteristic
89	C₄²×C₈₉	#41	1	1	yes	characteristic
90	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂²×D₈₉	—	2	2	no	—
92	C₂²×D₈₉	—	2	4	no	—
93	C₂²×D₈₉	—	4	2	no	—
94	C₂²×D₈₉	—	4	2	no	—
95	C₂×C₄×C₈₉	—	2	1	no	—
96	C₂×C₄×C₈₉	—	2	1	no	—
97	C₂×C₄×C₈₉	—	4	1	no	—
98	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
100	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
101	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
102	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×C₄×C₈₉	#137	1	2	yes	—
106	C₂×Dic₈₉	—	2	2	no	—
107	C₂×Dic₈₉	—	4	2	no	—
108	D₃₅₆	—	4	2	no	—
109	D₃₅₆	—	4	1	no	—
110	D₃₅₆	—	4	1	no	—
111	C₄×D₈₉	—	4	4	no	—
112	C₄×D₈₉	—	4	2	no	—
113	C₄×D₈₉	—	4	2	no	—
114	C₈×C₈₉	—	2	1	no	—
115	C₈×C₈₉	—	2	1	no	—
116	C₈×C₈₉	—	2	1	no	—
117	C₈×C₈₉	—	8	2	no	—
118	C₈×C₈₉	#100	1	2	yes	complemented
119	#10623494	—	4	2	no	—
120	#10623494	—	8	2	no	—
121	C₂²×C₈₉	—	2	1	no	—
122	C₂²×C₈₉	—	2	1	no	—
123	C₂²×C₈₉	#380	1	1	yes	characteristic
124	C₂²×C₈₉	#428	1	2	yes	—
125	C₂×D₈₉	—	4	2	no	—
126	C₂×D₈₉	—	4	2	no	—
127	C₂×D₈₉	—	4	2	no	—
128	C₂×D₈₉	—	8	4	no	—
129	C₄×C₈₉	—	2	1	no	—
130	C₄×C₈₉	—	2	1	no	—
131	C₄×C₈₉	—	2	1	no	—
132	C₄×C₈₉	C₂×#100	1	1	yes	characteristic
133	C₄×C₈₉	#413	1	1	yes	characteristic
134	C₄×C₈₉	#468	1	2	yes	—
135	C₄×C₈₉	#469	1	1	yes	characteristic
136	C₄×C₈₉	#470	1	1	yes	characteristic
137	Dic₈₉	—	4	2	no	—
138	#8618	—	89	1	no	maximal, Sylow
139	C₂×C₈₉	—	2	2	no	—
140	C₂×C₈₉	#1800	1	1	yes	characteristic
141	C₂×C₈₉	#1809	1	1	yes	characteristic
142	C₂×C₈₉	#1813	1	1	yes	characteristic
143	D₈₉	—	8	2	no	—
144	C₂×#436	—	89	1	no	—
145	#1800	—	89	2	no	—
146	#1567	—	89	1	no	—
147	#1566	D₈₉	1	1	yes	characteristic, complemented
148	#1292	—	89	2	no	—
149	C₈₉	#8618	1	1	yes	characteristic, complemented, Sylow
150	C₂²×M₁₆	—	89	2	no	—
151	C₂×C₄×D₄	—	89	1	no	—
152	#436	—	178	1	no	—
153	#436	—	178	1	no	—
154	#436	—	178	1	no	—
155	#436	—	178	1	no	—
156	#423	—	89	2	no	—
157	#423	C₂×D₈₉	1	2	yes	—
158	C₂×C₄⋊C₈	—	89	1	no	—
159	C₂×#100	—	178	2	no	—
160	C₂×C₂²⋊C₈	—	89	2	no	—
161	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
162	#343	—	178	2	no	—
163	#335	—	178	1	no	—
164	#335	—	178	1	no	—
165	#334	—	2	1	no	—
166	#334	—	2	1	no	—
167	C₂²×D₄	—	89	1	no	—
168	C₂³×C₄	—	89	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	356	4	no	—
171	C₂×M₁₆	—	178	2	no	—
172	C₂×M₁₆	—	178	2	no	—
173	C₂×M₁₆	—	178	2	no	—
174	C₂²×C₈	—	89	2	no	—
175	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
176	C₂²×C₈	C₄×D₈₉	1	1	yes	characteristic
177	C₄×D₄	—	178	1	no	—
178	C₄×D₄	—	178	1	no	—
179	C₄×D₄	—	178	1	no	—
180	C₄×D₄	—	178	1	no	—
181	C₂×C₄⋊C₄	—	89	1	no	—
182	C₂×C₂²⋊C₄	—	89	2	no	—
183	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
184	C₄⋊C₈	—	2	2	no	—
185	C₄⋊C₈	—	2	2	no	—
186	C₄⋊C₈	—	178	1	no	—
187	C₄⋊C₈	—	178	2	no	—
188	C₄⋊C₈	—	178	1	no	—
189	C₄⋊C₈	—	178	2	no	—
190	#100	—	356	4	no	—
191	C₂²⋊C₈	—	178	2	no	—
192	C₂²⋊C₈	—	178	2	no	—
193	C₄×C₈	D₃₅₆	1	1	yes	characteristic
194	C₄×C₈	D₃₅₆	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
196	C₄×C₈	C₂²⋊D₈₉	1	1	yes	characteristic
197	C₂⁴	—	89	2	no	—
198	C₂×D₄	—	178	1	no	—
199	C₂×D₄	—	178	1	no	—
200	C₂×D₄	—	178	2	no	—
201	C₂×D₄	—	178	2	no	—
202	C₂²×C₄	—	89	2	no	—
203	C₂²×C₄	—	356	2	no	—
204	C₂²×C₄	—	356	2	no	—
205	C₂²×C₄	—	178	2	no	—
206	C₂²×C₄	—	178	2	no	—
207	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
208	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
209	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
210	M₁₆	—	4	2	no	—
211	M₁₆	—	4	2	no	—
212	M₁₆	—	356	4	no	—
213	M₁₆	—	356	4	no	—
214	M₁₆	—	356	2	no	—
215	M₁₆	—	356	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	2	no	—
219	C₂×C₈	—	356	2	no	—
220	C₂×C₈	—	356	2	no	—
221	C₂×C₈	—	178	2	no	—
222	C₂×C₈	#13083551	1	1	yes	characteristic
223	C₂×C₈	#13083551	1	1	yes	characteristic
224	C₂×C₈	#13083551	1	1	yes	characteristic
225	C₂×C₈	#13083551	1	1	yes	characteristic
226	C₄⋊C₄	—	178	1	no	—
227	C₄⋊C₄	—	178	1	no	—
228	C₂²⋊C₄	—	178	2	no	—
229	C₂²⋊C₄	—	178	2	no	—
230	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
231	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
232	C₄²	#13083573	1	1	yes	characteristic
233	C₄²	#13083575	1	1	yes	characteristic
234	C₂³	—	356	2	no	—
235	C₂³	—	356	2	no	—
236	C₂³	—	178	2	no	—
237	C₂³	—	178	4	no	—
238	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
239	D₄	—	356	2	no	—
240	D₄	—	356	1	no	—
241	D₄	—	356	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	356	4	no	—
246	C₂×C₄	—	356	2	no	—
247	C₂×C₄	—	356	2	no	—
248	C₂×C₄	—	356	2	no	—
249	C₂×C₄	—	178	2	no	—
250	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
251	C₂×C₄	#15667970	1	1	yes	characteristic
252	C₂×C₄	#15667983	1	1	yes	characteristic
253	C₂×C₄	#15667987	1	1	yes	characteristic
254	C₂×C₄	#15667988	1	1	yes	characteristic
255	C₂×C₄	#15667992	1	1	yes	characteristic
256	C₂×C₄	#15668039	1	2	yes	—
257	C₂×C₄	#15668041	1	2	yes	—
258	C₈	—	2	1	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	8	2	no	—
262	C₈	—	356	2	no	—
263	C₈	—	712	2	no	—
264	C₈	#15667928	1	2	yes	complemented
265	C₂²	—	2	1	no	—
266	C₂²	—	2	1	no	—
267	C₂²	—	356	2	no	—
268	C₂²	—	356	2	no	—
269	C₂²	—	356	2	no	—
270	C₂²	—	712	4	no	—
271	C₂²	#19151455	1	1	yes	characteristic
272	C₂²	#19151637	1	2	yes	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	356	2	no	—
277	C₄	C₂×#15667928	1	1	yes	characteristic
278	C₄	#19151580	1	1	yes	characteristic
279	C₄	#19151841	1	2	yes	—
280	C₄	#19151852	1	1	yes	characteristic
281	C₄	#19151857	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	712	2	no	—
284	C₂	#26020241	1	1	yes	characteristic
285	C₂	#26020282	1	1	yes	characteristic
286	C₂	#26020299	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19151455
commutator	C₂²×C₄×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1566	C₂
socle	C₂²×C₈₉	#380
prime radical	C₂²×D₃₅₆	D₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	#1566	D₈₉
hypocenter	C₈₉	#8618
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
3. #26019011	C₂	semidirect
4. #26018126	C₂	semidirect
5. C₈₉×#1566	C₂	semidirect
118. C₈×C₈₉	#100	semidirect
147. #1566	D₈₉	semidirect
149. C₈₉	#8618	semidirect
264. C₈	#15667928	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19151677	2
#26020241	2
#26019011	2
#26018126	2
C₈₉×#1566	2
#8618	89