Subgroups — #36447293

Order 22784.

462 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26019000	C₂	1	2	yes	maximal, complemented
2	#26018566	C₂	1	2	yes	maximal
3	#26018083	C₂	1	2	yes	maximal, complemented
4	C₂×#405×C₈₉	C₂	1	1	yes	maximal, characteristic
5	#19151651	C₂²	1	4	yes	—
6	#19151351	—	2	2	no	—
7	#19151349	—	2	2	no	—
8	#19151324	—	2	2	no	—
9	#19151322	—	2	2	no	—
10	C₂²×M₁₆×C₈₉	C₄	1	1	yes	characteristic
11	C₂³×C₈×C₈₉	C₄	1	1	yes	characteristic
12	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
13	#405×C₈₉	—	2	2	no	—
14	#405×C₈₉	—	2	2	no	—
15	C₂×#97×C₈₉	C₄	1	1	yes	characteristic
16	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₈₉	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₄⋊#10623494	—	2	4	no	—
20	C₂²⋊#10623494	—	2	4	no	—
21	C₄×#10623494	—	2	4	no	—
22	C₂³×C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
23	C₂×M₁₆×C₈₉	—	2	2	no	—
24	C₂×M₁₆×C₈₉	—	2	2	no	—
25	C₂×M₁₆×C₈₉	C₈	1	4	yes	complemented
26	C₂²×C₈×C₈₉	—	2	2	no	—
27	C₂²×C₈×C₈₉	—	2	1	no	—
28	C₂²×C₈×C₈₉	—	2	1	no	—
29	C₂²×C₈×C₈₉	—	2	2	no	—
30	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₉	C₂×C₄	1	1	yes	characteristic
34	#117×C₈₉	—	2	2	no	—
35	#117×C₈₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₈₉	Q₈	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₈₉	Q₈	1	1	yes	characteristic
40	C₂×C₄²×C₈₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
42	#97×C₈₉	—	2	2	no	—
43	#97×C₈₉	—	2	2	no	—
44	C₄×C₈×C₈₉	—	2	1	no	—
45	C₄×C₈×C₈₉	—	2	1	no	—
46	C₄×C₈×C₈₉	D₄	1	2	yes	—
47	C₄×C₈×C₈₉	D₄	1	2	yes	—
48	C₂×#10623494	—	4	4	no	—
49	C₂⁴×C₈₉	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₈₉	—	2	1	no	—
51	C₂²×C₄×C₈₉	—	2	1	no	—
52	C₂²×C₄×C₈₉	—	2	1	no	—
53	C₂²×C₄×C₈₉	—	2	1	no	—
54	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₈₉	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₂×C₈	1	2	yes	—
58	C₂²×C₄×C₈₉	M₁₆	1	2	yes	—
59	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
62	M₁₆×C₈₉	—	2	8	no	—
63	C₂×C₈×C₈₉	—	2	4	no	—
64	C₂×C₈×C₈₉	—	2	4	no	—
65	C₂×C₈×C₈₉	—	2	1	no	—
66	C₂×C₈×C₈₉	—	2	1	no	—
67	C₂×C₈×C₈₉	—	2	1	no	—
68	C₂×C₈×C₈₉	—	2	1	no	—
69	C₂×C₈×C₈₉	—	2	1	no	—
70	C₂×C₈×C₈₉	—	2	1	no	—
71	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₈₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₈₉	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₈₉	C₂×C₈	1	2	yes	—
76	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
77	C₂×C₈×C₈₉	M₁₆	1	2	yes	—
78	C₄⋊C₄×C₈₉	—	2	2	no	—
79	C₄⋊C₄×C₈₉	—	2	2	no	—
80	C₂²⋊C₄×C₈₉	—	2	2	no	—
81	C₂²⋊C₄×C₈₉	—	2	2	no	—
82	C₄²×C₈₉	—	2	1	no	—
83	C₄²×C₈₉	—	2	1	no	—
84	C₄²×C₈₉	C₂×D₄	1	2	yes	—
85	C₄²×C₈₉	#41	1	2	yes	—
86	C₂³×C₈₉	—	2	1	no	—
87	C₂³×C₈₉	—	2	1	no	—
88	C₂³×C₈₉	—	4	1	no	—
89	C₂³×C₈₉	—	4	1	no	—
90	C₂³×C₈₉	C₄⋊C₈	1	2	yes	—
91	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₈₉	—	2	1	no	—
93	C₂×C₄×C₈₉	—	2	1	no	—
94	C₂×C₄×C₈₉	—	2	2	no	—
95	C₂×C₄×C₈₉	—	2	2	no	—
96	C₂×C₄×C₈₉	—	2	2	no	—
97	C₂×C₄×C₈₉	—	2	2	no	—
98	C₂×C₄×C₈₉	—	2	1	no	—
99	C₂×C₄×C₈₉	—	2	1	no	—
100	C₂×C₄×C₈₉	—	4	1	no	—
101	C₂×C₄×C₈₉	—	4	1	no	—
102	C₂×C₄×C₈₉	—	4	1	no	—
103	C₂×C₄×C₈₉	—	4	1	no	—
104	C₂×C₄×C₈₉	C₄⋊C₈	1	2	yes	—
105	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#121	1	2	yes	—
108	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂×M₁₆	1	1	yes	characteristic
113	C₈×C₈₉	—	2	2	no	—
114	C₈×C₈₉	—	2	1	no	—
115	C₈×C₈₉	—	2	1	no	—
116	C₈×C₈₉	—	2	2	no	—
117	C₈×C₈₉	C₂²⋊C₈	1	4	yes	—
118	#10623494	—	8	4	no	—
119	C₂²×C₈₉	—	2	1	no	—
120	C₂²×C₈₉	—	2	1	no	—
121	C₂²×C₈₉	—	2	2	no	—
122	C₂²×C₈₉	—	4	2	no	—
123	C₂²×C₈₉	—	4	2	no	—
124	C₂²×C₈₉	—	4	1	no	—
125	C₂²×C₈₉	—	4	1	no	—
126	C₂²×C₈₉	#380	1	1	yes	characteristic
127	C₂²×C₈₉	C₂×C₄⋊C₈	1	1	yes	characteristic
128	C₂²×C₈₉	#423	1	1	yes	characteristic
129	C₄×C₈₉	—	2	2	no	—
130	C₄×C₈₉	—	2	2	no	—
131	C₄×C₈₉	—	2	2	no	—
132	C₄×C₈₉	—	4	1	no	—
133	C₄×C₈₉	—	4	1	no	—
134	C₄×C₈₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
135	C₄×C₈₉	#407	1	1	yes	characteristic
136	C₄×C₈₉	#431	1	1	yes	characteristic
137	C₄×C₈₉	#432	1	1	yes	characteristic
138	#8602	—	89	1	no	maximal, Sylow
139	C₂×C₈₉	—	2	1	no	—
140	C₂×C₈₉	—	2	1	no	—
141	C₂×C₈₉	—	4	2	no	—
142	C₂×C₈₉	#1797	1	1	yes	characteristic
143	C₂×C₈₉	#1812	1	1	yes	characteristic
144	C₂×C₈₉	#1813	1	1	yes	characteristic
145	C₂×#405	D₈₉	1	1	yes	characteristic
146	#1564	—	89	2	no	—
147	#1458	—	89	2	no	—
148	#1275	—	89	2	no	—
149	C₈₉	#8602	1	1	yes	characteristic, complemented, Sylow
150	C₂²×M₁₆	Dic₈₉	1	1	yes	characteristic
151	C₂³×C₈	Dic₈₉	1	1	yes	characteristic
152	C₂×#117	C₂×D₈₉	1	1	yes	characteristic
153	#431	—	89	4	no	—
154	#405	—	2	2	no	—
155	#405	—	2	2	no	—
156	C₂×#97	Dic₈₉	1	1	yes	characteristic
157	C₂×#97	C₂×D₈₉	1	1	yes	characteristic
158	C₂×C₄×C₈	Dic₈₉	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
160	#335	—	178	2	no	—
161	#334	—	178	2	no	—
162	#324	—	178	2	no	—
163	#323	—	178	2	no	—
164	C₂³×C₄	C₂×Dic₈₉	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	#10623494	1	4	yes	complemented
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	—	2	1	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
173	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
174	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
175	C₂²×C₈	C₂×Dic₈₉	1	1	yes	characteristic
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	C₂×C₄⋊C₄	Dic₁₇₈	1	1	yes	characteristic
179	C₂×C₄⋊C₄	D₃₅₆	1	1	yes	characteristic
180	C₂×C₂²⋊C₄	Dic₁₇₈	1	1	yes	characteristic
181	C₂×C₂²⋊C₄	D₃₅₆	1	1	yes	characteristic
182	C₂×C₄²	C₂×Dic₈₉	1	1	yes	characteristic
183	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
184	C₄⋊C₈	—	178	4	no	—
185	C₂²⋊C₈	—	178	4	no	—
186	#97	—	2	2	no	—
187	#97	—	2	2	no	—
188	C₄×C₈	—	2	1	no	—
189	C₄×C₈	—	2	1	no	—
190	C₄×C₈	—	178	4	no	—
191	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
192	C₄×C₈	C₂²⋊D₈₉	1	2	yes	—
193	C₂⁴	C₄⋊Dic₈₉	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
199	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
200	C₂²×C₄	C₄⋊Dic₈₉	1	1	yes	characteristic
201	C₂²×C₄	C₂×#10623494	1	2	yes	—
202	C₂²×C₄	#13083558	1	2	yes	—
203	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
205	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
206	M₁₆	—	2	8	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	1	no	—
213	C₂×C₈	—	2	1	no	—
214	C₂×C₈	—	2	1	no	—
215	C₂×C₈	—	356	4	no	—
216	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
217	C₂×C₈	C₂²⋊Dic₈₉	1	1	yes	characteristic
218	C₂×C₈	C₂²⋊Dic₈₉	1	2	yes	—
219	C₂×C₈	C₂×#10623494	1	2	yes	—
220	C₂×C₈	C₂×#10623494	1	2	yes	—
221	C₂×C₈	#13083558	1	2	yes	—
222	C₂×C₈	#13083558	1	2	yes	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	2	no	—
226	C₂²⋊C₄	—	2	2	no	—
227	C₄²	—	2	1	no	—
228	C₄²	—	2	1	no	—
229	C₄²	C₂×C₂²⋊D₈₉	1	2	yes	—
230	C₄²	#13083574	1	2	yes	—
231	C₂³	—	2	1	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	4	1	no	—
234	C₂³	—	4	1	no	—
235	C₂³	C₄⋊#10623494	1	2	yes	—
236	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	C₄⋊#10623494	1	2	yes	—
250	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
251	C₂×C₄	#15667971	1	1	yes	characteristic
252	C₂×C₄	#15667986	1	2	yes	—
253	C₂×C₄	#15667991	1	2	yes	—
254	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
255	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
256	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
257	C₂×C₄	C₂×#13083558	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	2	no	—
262	C₈	—	712	4	no	—
263	C₈	C₂²⋊#10623494	1	4	yes	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	4	2	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	#19151458	1	1	yes	characteristic
272	C₂²	C₂×C₄⋊#10623494	1	1	yes	characteristic
273	C₂²	#19151622	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	2	2	no	—
276	C₄	—	2	2	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	C₂×C₂²⋊#10623494	1	1	yes	characteristic
280	C₄	#19151562	1	1	yes	characteristic
281	C₄	#19151651	1	1	yes	characteristic
282	C₄	#19151655	1	1	yes	characteristic
283	C₂	—	2	1	no	—
284	C₂	—	2	1	no	—
285	C₂	—	4	2	no	—
286	C₂	#26020232	1	1	yes	characteristic
287	C₂	#26020298	1	1	yes	characteristic
288	C₂	#26020302	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#15667971
commutator	C₂×C₄×C₈₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#405×C₈₉	C₂
socle	C₂²×C₈₉	#380
prime radical	C₂⁴×C₈₉	C₄⋊C₄
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#405	D₈₉
hypocenter	C₈₉	#8602
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
1. #26019000	C₂	semidirect
3. #26018083	C₂	semidirect
25. C₂×M₁₆×C₈₉	C₈	semidirect
149. C₈₉	#8602	semidirect
167. C₂×M₁₆	#10623494	semidirect

Maximal subgroups

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

subgroup	index
#26019000	2
#26018566	2
#26018083	2
C₂×#405×C₈₉	2
#8602	89