Subgroups — #36460115

Order 22784.

612 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028670	C₂	1	1	yes	maximal, characteristic
2	C₂×#19151370	C₂	1	2	yes	maximal
3	#26019978	C₂	1	2	yes	maximal
4	C₂×#19151356	C₂	1	1	yes	maximal, characteristic
5	C₂×#338×C₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	C₂²	1	1	yes	characteristic
7	C₂²×#13083558	C₂²	1	2	yes	—
8	C₂×#15667971	C₂²	1	1	yes	characteristic
9	#19151556	—	2	2	no	—
10	#19151553	C₄	1	4	yes	—
11	M₁₆×Dic₈₉	—	2	2	no	—
12	#19151370	—	2	4	no	—
13	#19151362	C₄	1	8	yes	complemented
14	#19151356	—	2	2	no	—
15	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
16	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	#338×C₈₉	—	2	2	no	—
18	C₂×#13083558	—	2	1	no	—
19	C₂×#13083558	—	2	2	no	—
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083558	—	2	2	no	—
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083558	—	2	2	no	—
25	C₂×#13083558	—	2	2	no	—
26	C₂×#13083558	C₂×C₄	1	4	yes	—
27	C₂×#13083558	C₂×C₄	1	2	yes	—
28	C₂²×#10623494	—	2	1	no	—
29	C₂²×#10623494	C₂×C₄	1	4	yes	—
30	#15667971	—	2	1	no	—
31	#15667971	—	2	1	no	—
32	#15667971	C₂×C₄	1	2	yes	—
33	#15667971	D₄	1	2	yes	—
34	C₂×C₄⋊Dic₈₉	—	2	1	no	—
35	C₂×C₂²⋊Dic₈₉	Q₈	1	2	yes	—
36	C₂×C₄×Dic₈₉	—	2	1	no	—
37	#15667922	—	2	4	no	—
38	C₈×Dic₈₉	—	2	4	no	—
39	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₈₉	—	2	2	no	—
41	C₂×M₁₆×C₈₉	—	2	2	no	—
42	C₂×M₁₆×C₈₉	—	2	2	no	—
43	C₂×M₁₆×C₈₉	—	2	2	no	—
44	C₂×M₁₆×C₈₉	—	2	2	no	—
45	C₂×M₁₆×C₈₉	—	2	2	no	—
46	C₂×M₁₆×C₈₉	—	2	2	no	—
47	C₂×M₁₆×C₈₉	C₂×C₄	1	4	yes	—
48	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
49	C₂×M₁₆×C₈₉	D₄	1	2	yes	—
50	C₂²×C₈×C₈₉	—	2	2	no	—
51	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
52	C₂²×Dic₈₉	—	2	2	no	—
53	#13083558	—	2	2	no	—
54	#13083558	—	2	4	no	—
55	#13083558	—	4	2	no	—
56	#13083558	—	4	2	no	—
57	#13083558	—	4	2	no	—
58	#13083558	—	4	2	no	—
59	#13083558	C₄²	1	4	yes	—
60	C₂×#10623494	—	2	2	no	—
61	C₂×#10623494	—	2	4	no	—
62	C₂×#10623494	—	2	4	no	—
63	C₂×#10623494	—	2	4	no	—
64	C₂×#10623494	—	2	4	no	—
65	C₄⋊Dic₈₉	—	2	2	no	—
66	C₄⋊Dic₈₉	—	2	2	no	—
67	C₂²⋊Dic₈₉	—	2	2	no	—
68	C₂²⋊Dic₈₉	C₄⋊C₄	1	4	yes	—
69	C₄×Dic₈₉	—	2	2	no	—
70	C₄×Dic₈₉	—	2	2	no	—
71	C₂⁴×C₈₉	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	#41	1	2	yes	—
80	C₂²×C₄×C₈₉	#41	1	2	yes	—
81	C₂²×C₄×C₈₉	#41	1	2	yes	—
82	M₁₆×C₈₉	—	2	4	no	—
83	M₁₆×C₈₉	—	2	4	no	—
84	M₁₆×C₈₉	—	2	4	no	—
85	M₁₆×C₈₉	—	2	4	no	—
86	M₁₆×C₈₉	—	4	2	no	—
87	M₁₆×C₈₉	—	4	2	no	—
88	C₂×C₈×C₈₉	—	2	2	no	—
89	C₂×C₈×C₈₉	—	2	2	no	—
90	C₂×C₈×C₈₉	—	2	4	no	—
91	C₂×C₈×C₈₉	—	4	2	no	—
92	C₂×C₈×C₈₉	—	4	2	no	—
93	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
94	C₂³×C₈₉	—	8	1	no	—
95	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂³×C₈₉	C₄×Q₈	1	2	yes	—
97	C₂³×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
98	C₂³×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₈₉	#125	1	2	yes	—
100	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
103	C₂×C₄×C₈₉	#117	1	2	yes	—
104	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
105	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
106	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#117	1	2	yes	—
108	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
112	C₂×C₄×C₈₉	#121	1	1	yes	characteristic
113	C₂×C₄×C₈₉	#123	1	2	yes	—
114	C₂×C₄×C₈₉	#123	1	2	yes	—
115	C₂×C₄×C₈₉	#124	1	2	yes	—
116	C₂×C₄×C₈₉	#126	1	2	yes	—
117	C₂×C₄×C₈₉	#126	1	2	yes	—
118	C₂×Dic₈₉	—	2	4	no	—
119	C₂×Dic₈₉	—	8	2	no	—
120	C₈×C₈₉	—	2	4	no	—
121	C₈×C₈₉	—	4	4	no	—
122	#10623494	—	2	4	no	—
123	#10623494	—	4	4	no	—
124	C₂²×C₈₉	—	4	1	no	—
125	C₂²×C₈₉	—	4	1	no	—
126	C₂²×C₈₉	—	4	2	no	—
127	C₂²×C₈₉	—	4	1	no	—
128	C₂²×C₈₉	—	4	2	no	—
129	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₈₉	#380	1	1	yes	characteristic
131	C₂²×C₈₉	#382	1	1	yes	characteristic
132	C₂²×C₈₉	#387	1	2	yes	—
133	C₂²×C₈₉	#387	1	2	yes	—
134	C₄×C₈₉	#376	1	1	yes	characteristic
135	C₄×C₈₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
136	C₄×C₈₉	#381	1	1	yes	characteristic
137	C₄×C₈₉	#383	1	1	yes	characteristic
138	C₄×C₈₉	#388	1	2	yes	—
139	C₄×C₈₉	#388	1	2	yes	—
140	Dic₈₉	—	4	4	no	—
141	#10812	—	89	1	no	maximal, Sylow
142	C₂×C₈₉	—	2	1	no	—
143	C₂×C₈₉	—	2	1	no	—
144	C₂×C₈₉	—	2	1	no	—
145	C₂×C₈₉	—	2	2	no	—
146	C₂×C₈₉	—	2	2	no	—
147	C₂×C₈₉	#1434	1	1	yes	characteristic
148	#2873	—	89	1	no	—
149	C₂×#344	—	89	2	no	—
150	#1738	—	89	2	no	—
151	C₂×#338	—	89	1	no	—
152	C₂×#338	D₈₉	1	1	yes	characteristic
153	C₈₉	#10812	1	1	yes	characteristic, complemented, Sylow
154	C₂²×M₁₆	—	89	2	no	—
155	C₂²×M₁₆	—	89	1	no	—
156	C₂²×M₁₆	C₂×D₈₉	1	2	yes	—
157	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
158	C₂×#117	—	89	1	no	—
159	#405	—	89	4	no	—
160	#405	—	178	2	no	—
161	C₄×M₁₆	—	178	2	no	—
162	#344	—	178	4	no	—
163	#341	—	89	8	no	—
164	#338	—	2	2	no	—
165	#338	—	178	2	no	—
166	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	89	4	no	—
175	C₂×M₁₆	—	89	2	no	—
176	C₂×M₁₆	—	178	2	no	—
177	C₂×M₁₆	—	178	2	no	—
178	C₂×M₁₆	—	178	2	no	—
179	C₂×M₁₆	—	178	2	no	—
180	C₂×M₁₆	—	178	2	no	—
181	C₂×M₁₆	—	178	2	no	—
182	C₂×M₁₆	—	178	1	no	—
183	C₂×M₁₆	—	178	2	no	—
184	C₂×M₁₆	C₄×D₈₉	1	4	yes	—
185	C₂×M₁₆	D₃₅₆	1	2	yes	—
186	C₂×M₁₆	C₂²⋊D₈₉	1	2	yes	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	89	4	no	—
189	C₂²×C₈	—	178	1	no	—
190	C₂²×C₈	C₄×D₈₉	1	2	yes	—
191	#117	—	89	2	no	—
192	#117	—	89	2	no	—
193	#117	—	178	1	no	—
194	#117	—	178	1	no	—
195	C₂×C₄⋊C₄	—	178	1	no	—
196	C₂×C₂²⋊C₄	—	89	2	no	—
197	C₂×C₄²	—	178	1	no	—
198	#97	—	178	4	no	—
199	C₄×C₈	—	178	4	no	—
200	C₂⁴	Q₈×D₈₉	1	1	yes	characteristic
201	C₂²×C₄	—	178	2	no	—
202	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
204	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
207	C₂²×C₄	#13083573	1	1	yes	characteristic
208	C₂²×C₄	#13083573	1	2	yes	—
209	C₂²×C₄	#13083573	1	2	yes	—
210	C₂²×C₄	#13083574	1	2	yes	—
211	C₂²×C₄	#13083575	1	1	yes	characteristic
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	89	4	no	—
219	M₁₆	—	356	2	no	—
220	M₁₆	—	356	2	no	—
221	M₁₆	—	356	2	no	—
222	M₁₆	—	356	2	no	—
223	M₁₆	—	178	2	no	—
224	M₁₆	—	178	4	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	4	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	178	4	no	—
231	C₂×C₈	—	178	4	no	—
232	C₂×C₈	—	178	4	no	—
233	C₂×C₈	—	178	4	no	—
234	C₂×C₈	—	178	2	no	—
235	C₂×C₈	#13083551	1	4	yes	—
236	C₄⋊C₄	—	178	2	no	—
237	C₄⋊C₄	—	178	2	no	—
238	C₂²⋊C₄	—	89	4	no	—
239	C₂²⋊C₄	—	178	2	no	—
240	C₄²	—	178	2	no	—
241	C₄²	—	178	2	no	—
242	C₂³	—	8	1	no	—
243	C₂³	C₄⋊C₄×D₈₉	1	1	yes	characteristic
244	C₂³	#15667977	1	2	yes	—
245	C₂³	#15667988	1	1	yes	characteristic
246	C₂³	#15667992	1	1	yes	characteristic
247	C₂³	#15668002	1	2	yes	—
248	C₂×C₄	—	178	4	no	—
249	C₂×C₄	—	712	2	no	—
250	C₂×C₄	C₄²×D₈₉	1	1	yes	characteristic
251	C₂×C₄	C₂×#13083551	1	1	yes	characteristic
252	C₂×C₄	#15667968	1	1	yes	characteristic
253	C₂×C₄	#15667968	1	2	yes	—
254	C₂×C₄	#15667968	1	1	yes	characteristic
255	C₂×C₄	#15667969	1	1	yes	characteristic
256	C₂×C₄	#15667970	1	1	yes	characteristic
257	C₂×C₄	#15667970	1	2	yes	—
258	C₂×C₄	#15667972	1	2	yes	—
259	C₂×C₄	C₄×C₂²⋊D₈₉	1	2	yes	—
260	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
261	C₂×C₄	#15667983	1	1	yes	characteristic
262	C₂×C₄	#15667987	1	1	yes	characteristic
263	C₂×C₄	#15667993	1	2	yes	—
264	C₂×C₄	#15667996	1	2	yes	—
265	C₂×C₄	#15667998	1	2	yes	—
266	C₂×C₄	#15668005	1	2	yes	—
267	C₂×C₄	#15668007	1	2	yes	—
268	C₈	—	2	4	no	—
269	C₈	—	4	4	no	—
270	C₈	—	356	4	no	—
271	C₈	—	178	4	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	1	no	—
275	C₂²	—	4	2	no	—
276	C₂²	—	4	1	no	—
277	C₂²	#19151449	1	1	yes	characteristic
278	C₂²	#19151455	1	1	yes	characteristic
279	C₂²	#19151462	1	1	yes	characteristic
280	C₂²	#19151479	1	2	yes	—
281	C₂²	#19151483	1	2	yes	—
282	C₄	—	356	4	no	—
283	C₄	#19151443	1	1	yes	characteristic
284	C₄	C₄×#13083551	1	1	yes	characteristic
285	C₄	#19151459	1	1	yes	characteristic
286	C₄	#19151465	1	1	yes	characteristic
287	C₄	#19151486	1	2	yes	—
288	C₄	#19151490	1	2	yes	—
289	C₂	—	2	1	no	—
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	2	no	—
293	C₂	—	2	2	no	—
294	C₂	#26018476	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×#13083551
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₂×#338×C₈₉	C₂
socle	C₂×C₈₉	#1434
prime radical	C₂⁴×C₈₉	C₂×Q₈
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₂×#338	D₈₉
hypocenter	C₈₉	#10812
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂×#338

Complements

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

N	H ≅ G/N	type
13. #19151362	C₄	semidirect
153. C₈₉	#10812	semidirect

Maximal subgroups

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

subgroup	index
#26028670	2
C₂×#19151370	2
#26019978	2
C₂×#19151356	2
C₂×#338×C₈₉	2
#10812	89