Subgroups — #36554975

Order 22784.

894 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	#26022650	C₂	1	2	yes	maximal
2	#26020416	C₂	1	2	yes	maximal, complemented
3	#26020383	C₂	1	2	yes	maximal, complemented
4	#26020281	C₂	1	2	yes	maximal
5	#26020277	C₂	1	2	yes	maximal
6	C₄×#15667946	C₂	1	2	yes	maximal, complemented
7	C₄×#15667944	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#15667946	C₂²	1	4	yes	—
10	C₂×#15667944	C₂²	1	4	yes	—
11	#19151466	—	2	2	no	—
12	#19151464	C₂²	1	4	yes	—
13	#19151461	—	2	2	no	—
14	#19151458	C₂²	1	4	yes	—
15	C₄×C₄⋊Dic₈₉	C₂²	1	4	yes	—
16	C₄×C₂²⋊Dic₈₉	—	2	2	no	—
17	#19151444	—	2	2	no	—
18	#19151361	C₂²	1	4	yes	—
19	#19151361	C₂²	1	4	yes	—
20	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₈₉	C₄	1	8	yes	complemented
24	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊Dic₈₉	C₂³	1	8	yes	—
29	C₂×C₂²⋊Dic₈₉	—	2	4	no	—
30	C₂×C₄×Dic₈₉	—	2	4	no	—
31	#15667946	—	2	8	no	—
32	#15667944	—	2	8	no	—
33	#15667919	—	2	4	no	—
34	#15667919	—	2	4	no	—
35	#15667919	—	2	4	no	—
36	#15667919	—	2	4	no	—
37	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₈₉	—	2	4	no	—
41	C₂²×C₈×C₈₉	—	2	4	no	—
42	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
43	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₈₉	C₂×C₄	1	4	yes	—
49	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
50	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
51	C₂×C₄²×C₈₉	Q₈	1	4	yes	—
52	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
54	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
55	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
56	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
57	C₄×C₈×C₈₉	C₂×C₄	1	4	yes	—
58	C₂²×Dic₈₉	—	2	8	no	—
59	C₄⋊Dic₈₉	—	2	8	no	—
60	C₄⋊Dic₈₉	—	2	8	no	—
61	C₂²⋊Dic₈₉	—	4	8	no	—
62	C₄×Dic₈₉	—	4	8	no	—
63	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	—	2	4	no	—
65	C₂²×C₄×C₈₉	—	2	4	no	—
66	C₂²×C₄×C₈₉	—	2	2	no	—
67	C₂²×C₄×C₈₉	—	2	2	no	—
68	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
69	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
70	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
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×Q₈	1	2	yes	—
77	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₈₉	—	2	2	no	—
79	C₂×C₈×C₈₉	—	2	2	no	—
80	C₂×C₈×C₈₉	—	2	8	no	—
81	C₂×C₈×C₈₉	—	2	8	no	—
82	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₈₉	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₈₉	#41	1	4	yes	—
87	C₂×C₈×C₈₉	#41	1	4	yes	—
88	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
89	C₄²×C₈₉	C₄⋊C₄	1	4	yes	—
90	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
91	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
92	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
93	C₄²×C₈₉	C₂×Q₈	1	2	yes	—
94	C₂³×C₈₉	—	2	2	no	—
95	C₂³×C₈₉	—	2	2	no	—
96	C₂³×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
97	C₂³×C₈₉	#135	1	4	yes	—
98	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₈₉	—	2	2	no	—
100	C₂×C₄×C₈₉	—	2	2	no	—
101	C₂×C₄×C₈₉	—	2	2	no	—
102	C₂×C₄×C₈₉	—	2	2	no	—
103	C₂×C₄×C₈₉	—	2	8	no	—
104	C₂×C₄×C₈₉	—	2	2	no	—
105	C₂×C₄×C₈₉	—	2	2	no	—
106	C₂×C₄×C₈₉	—	2	4	no	—
107	C₂×C₄×C₈₉	—	2	8	no	—
108	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₈₉	#135	1	4	yes	—
114	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
116	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
117	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
118	C₂×Dic₈₉	—	4	8	no	—
119	C₂×Dic₈₉	—	4	8	no	—
120	C₂×Dic₈₉	—	4	8	no	—
121	C₈×C₈₉	—	2	2	no	—
122	C₈×C₈₉	—	2	2	no	—
123	C₈×C₈₉	#117	1	4	yes	—
124	C₈×C₈₉	#117	1	4	yes	—
125	C₂²×C₈₉	—	2	2	no	—
126	C₂²×C₈₉	—	2	8	no	—
127	C₂²×C₈₉	—	2	2	no	—
128	C₂²×C₈₉	#427	1	4	yes	—
129	C₂²×C₈₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₂²×C₈₉	#532	1	2	yes	—
131	C₂²×C₈₉	C₂×#135	1	2	yes	—
132	C₂²×C₈₉	C₂×#135	1	2	yes	—
133	C₄×C₈₉	—	2	2	no	—
134	C₄×C₈₉	—	2	2	no	—
135	C₄×C₈₉	—	2	2	no	—
136	C₄×C₈₉	—	2	2	no	—
137	C₄×C₈₉	#427	1	4	yes	—
138	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
139	C₄×C₈₉	C₂×#117	1	1	yes	characteristic
140	C₄×C₈₉	#533	1	2	yes	—
141	Dic₈₉	—	8	8	no	—
142	#20818	—	89	1	no	maximal, Sylow
143	C₂×C₈₉	—	2	4	no	—
144	C₂×C₈₉	#2285	1	1	yes	characteristic
145	C₂×C₈₉	C₂×#427	1	1	yes	characteristic
146	C₂×C₈₉	C₂×#427	1	1	yes	characteristic
147	C₂×C₈₉	#3168	1	4	yes	—
148	C₂²×C₄×C₈	D₈₉	1	1	yes	characteristic
149	#2283	—	89	2	no	—
150	#1846	—	89	2	no	—
151	#1835	—	89	2	no	—
152	#1808	—	89	2	no	—
153	#1807	—	89	2	no	—
154	C₄×#107	—	89	2	no	—
155	C₄×#106	—	89	2	no	—
156	C₈₉	#20818	1	1	yes	characteristic, complemented, Sylow
157	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
158	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
159	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
160	C₂×#107	—	89	4	no	—
161	C₂×#106	—	89	4	no	—
162	C₂×C₄×C₈	Dic₈₉	1	8	yes	complemented
163	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
164	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
165	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
167	#383	—	178	2	no	—
168	#382	—	89	4	no	—
169	#381	—	178	2	no	—
170	#380	—	89	4	no	—
171	C₄×C₄⋊C₄	—	89	4	no	—
172	C₄×C₂²⋊C₄	—	178	2	no	—
173	#376	—	178	2	no	—
174	#340	—	89	4	no	—
175	#340	—	89	4	no	—
176	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
177	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
178	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
179	C₂²×C₈	—	2	4	no	—
180	C₂²×C₈	—	2	4	no	—
181	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
182	C₂²×C₈	C₂×Dic₈₉	1	4	yes	—
183	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
184	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
186	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
187	C₂×C₄⋊C₄	—	89	8	no	—
188	C₂×C₂²⋊C₄	—	178	4	no	—
189	C₂×C₄²	—	178	4	no	—
190	C₂×C₄²	Dic₁₇₈	1	4	yes	—
191	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
192	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
193	C₂×C₄²	C₂×Dic₈₉	1	4	yes	—
194	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
195	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
196	#107	—	178	8	no	—
197	#106	—	178	8	no	—
198	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
199	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
200	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
201	C₄×C₈	C₂×Dic₈₉	1	4	yes	—
202	#95	—	178	4	no	—
203	#95	—	178	4	no	—
204	#95	—	178	4	no	—
205	#95	—	178	4	no	—
206	C₂⁴	C₂×D₃₅₆	1	1	yes	characteristic
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	178	8	no	—
212	C₂²×C₄	C₄⋊Dic₈₉	1	4	yes	—
213	C₂²×C₄	C₄⋊Dic₈₉	1	4	yes	—
214	C₂²×C₄	C₂²×Dic₈₉	1	2	yes	—
215	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
218	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
219	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
220	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
221	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	8	no	—
225	C₂×C₈	—	2	8	no	—
226	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
227	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
228	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
229	C₂×C₈	C₂²×Dic₈₉	1	2	yes	—
230	C₂×C₈	#13083574	1	4	yes	—
231	C₂×C₈	#13083574	1	4	yes	—
232	C₄⋊C₄	—	178	8	no	—
233	C₄⋊C₄	—	178	8	no	—
234	C₂²⋊C₄	—	356	8	no	—
235	C₄²	—	356	8	no	—
236	C₄²	C₄⋊Dic₈₉	1	4	yes	—
237	C₄²	C₄⋊Dic₈₉	1	4	yes	—
238	C₄²	C₂²×Dic₈₉	1	2	yes	—
239	C₄²	C₂²×Dic₈₉	1	2	yes	—
240	C₄²	C₂×Dic₁₇₈	1	2	yes	—
241	C₄²	C₂×Dic₁₇₈	1	2	yes	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	C₂×C₄⋊Dic₈₉	1	2	yes	—
245	C₂³	#15668034	1	4	yes	—
246	C₂³	C₂²×D₃₅₆	1	1	yes	characteristic
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	356	8	no	—
257	C₂×C₄	—	356	8	no	—
258	C₂×C₄	—	356	8	no	—
259	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
260	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
261	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
262	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
263	C₂×C₄	C₂×C₄⋊Dic₈₉	1	2	yes	—
264	C₂×C₄	#15668034	1	4	yes	—
265	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
266	C₂×C₄	C₂²×Dic₁₇₈	1	1	yes	characteristic
267	C₂×C₄	C₂×#13083574	1	2	yes	—
268	C₂×C₄	C₂×#13083574	1	2	yes	—
269	C₈	—	2	2	no	—
270	C₈	—	2	2	no	—
271	C₈	#15667971	1	4	yes	—
272	C₈	#15667971	1	4	yes	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	8	no	—
276	C₂²	#19151636	1	4	yes	—
277	C₂²	C₂²×C₄⋊Dic₈₉	1	1	yes	characteristic
278	C₂²	#19152148	1	2	yes	—
279	C₂²	C₂×#15668034	1	2	yes	—
280	C₂²	C₂×#15668034	1	2	yes	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	712	8	no	—
286	C₄	#19151636	1	4	yes	—
287	C₄	C₂×#15667971	1	1	yes	characteristic
288	C₄	C₂×#15667971	1	1	yes	characteristic
289	C₄	#19152155	1	2	yes	—
290	C₂	—	2	4	no	—
291	C₂	#26022659	1	1	yes	characteristic
292	C₂	C₂×#19151636	1	1	yes	characteristic
293	C₂	C₂×#19151636	1	1	yes	characteristic
294	C₂	#26030822	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂²×C₄×C₈×C₈₉	C₂
socle	C₂³×C₈₉	C₂²×D₄
prime radical	C₂⁴×C₈₉	C₂×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂²×C₄×C₈	D₈₉
hypocenter	C₈₉	#20818
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #26020416	C₂	semidirect
3. #26020383	C₂	semidirect
6. C₄×#15667946	C₂	semidirect
7. C₄×#15667944	C₂	semidirect
23. C₂×C₄×C₈×C₈₉	C₄	semidirect
156. C₈₉	#20818	semidirect
162. C₂×C₄×C₈	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26022650	2
#26020416	2
#26020383	2
#26020281	2
#26020277	2
C₄×#15667946	2
C₄×#15667944	2
C₂²×C₄×C₈×C₈₉	2
#20818	89