Subgroups — #36460324

Order 22784.

982 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	C₂²×#15667944	C₂	1	2	yes	maximal
2	C₂×#19151361	C₂	1	2	yes	maximal
3	#26018482	C₂	1	2	yes	maximal
4	C₂²×C₄×C₈×C₈₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₈₉	C₂²	1	4	yes	—
6	C₂×#15667944	—	2	4	no	—
7	C₂×#15667944	—	2	4	no	—
8	C₂×#15667944	C₄	1	8	yes	complemented
9	C₂×#15667919	—	2	4	no	—
10	#19151361	—	2	4	no	—
11	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₈₉	—	2	4	no	—
15	C₂×C₄×C₈×C₈₉	C₄	1	4	yes	complemented
16	C₂³×Dic₈₉	—	2	4	no	—
17	C₂×C₄⋊Dic₈₉	—	2	4	no	—
18	C₂×C₄⋊Dic₈₉	—	2	4	no	—
19	C₂×C₄⋊Dic₈₉	—	2	4	no	—
20	C₂×C₄⋊Dic₈₉	—	2	4	no	—
21	C₂×C₄⋊Dic₈₉	—	2	4	no	—
22	C₂×C₄⋊Dic₈₉	C₂×C₄	1	8	yes	—
23	#15667944	—	2	16	no	—
24	#15667919	—	4	8	no	—
25	C₂³×C₄×C₈₉	D₄	1	2	yes	—
26	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₈₉	—	2	2	no	—
28	C₂²×C₈×C₈₉	—	2	2	no	—
29	C₂²×C₈×C₈₉	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₈₉	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₈₉	D₄	1	2	yes	—
33	C₂²×C₈×C₈₉	D₄	1	2	yes	—
34	C₂²×C₈×C₈₉	D₄	1	4	yes	—
35	C₂²×C₈×C₈₉	Q₈	1	2	yes	—
36	C₂²×C₈×C₈₉	Q₈	1	2	yes	—
37	C₂×C₄²×C₈₉	—	2	2	no	—
38	C₂×C₄²×C₈₉	—	2	2	no	—
39	C₂×C₄²×C₈₉	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₈₉	Q₈	1	2	yes	—
41	C₄×C₈×C₈₉	—	2	8	no	—
42	C₂²×Dic₈₉	—	2	8	no	—
43	C₂²×Dic₈₉	—	4	4	no	—
44	C₂²×Dic₈₉	—	4	4	no	—
45	C₂²×Dic₈₉	—	4	8	no	—
46	C₂²×Dic₈₉	—	4	8	no	—
47	C₄⋊Dic₈₉	—	2	8	no	—
48	C₄⋊Dic₈₉	—	2	8	no	—
49	C₄⋊Dic₈₉	—	4	4	no	—
50	C₄⋊Dic₈₉	—	4	4	no	—
51	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₉	—	2	4	no	—
53	C₂²×C₄×C₈₉	—	2	4	no	—
54	C₂²×C₄×C₈₉	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₈₉	SD₁₆	1	8	yes	—
56	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₈₉	C₂²×C₄	1	2	yes	—
58	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₈₉	C₂×D₄	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	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	#41	1	2	yes	—
65	C₂²×C₄×C₈₉	#41	1	2	yes	—
66	C₂×C₈×C₈₉	—	2	2	no	—
67	C₂×C₈×C₈₉	—	2	2	no	—
68	C₂×C₈×C₈₉	—	2	4	no	—
69	C₂×C₈×C₈₉	—	2	4	no	—
70	C₂×C₈×C₈₉	C₄²	1	4	yes	—
71	C₂×C₈×C₈₉	C₂²⋊C₄	1	8	yes	—
72	C₂×C₈×C₈₉	C₂²⋊C₄	1	4	yes	—
73	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
74	C₂×C₈×C₈₉	C₄⋊C₄	1	8	yes	—
75	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
76	C₄²×C₈₉	—	2	2	no	—
77	C₄²×C₈₉	—	2	2	no	—
78	C₄²×C₈₉	—	2	4	no	—
79	C₂³×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂³×C₈₉	C₄×D₄	1	2	yes	—
81	C₂³×C₈₉	#121	1	2	yes	—
82	C₂³×C₈₉	#127	1	1	yes	characteristic
83	C₂³×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
84	C₂³×C₈₉	C₂×SD₁₆	1	4	yes	—
85	C₂³×C₈₉	#135	1	4	yes	—
86	C₂×C₄×C₈₉	—	2	4	no	—
87	C₂×C₄×C₈₉	—	2	4	no	—
88	C₂×C₄×C₈₉	—	2	16	no	—
89	C₂×C₄×C₈₉	#106	1	8	yes	complemented
90	C₂×C₄×C₈₉	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
92	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
97	C₂×C₄×C₈₉	#117	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
100	C₂×C₄×C₈₉	C₄×Q₈	1	2	yes	—
101	C₂×C₄×C₈₉	#121	1	2	yes	—
102	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₈₉	#124	1	2	yes	—
105	C₂×C₄×C₈₉	#125	1	2	yes	—
106	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
108	C₂×Dic₈₉	—	4	8	no	—
109	C₂×Dic₈₉	—	4	16	no	—
110	C₂×Dic₈₉	—	8	8	no	—
111	C₂×Dic₈₉	—	8	4	no	—
112	C₂×Dic₈₉	—	8	4	no	—
113	C₈×C₈₉	—	2	4	no	—
114	C₈×C₈₉	#95	1	8	yes	complemented
115	C₂²×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×C₈₉	#380	1	1	yes	characteristic
117	C₂²×C₈₉	#384	1	1	yes	characteristic
118	C₂²×C₈₉	#389	1	2	yes	—
119	C₂²×C₈₉	#390	1	2	yes	—
120	C₂²×C₈₉	C₂×#106	1	2	yes	—
121	C₂²×C₈₉	#427	1	2	yes	—
122	C₂²×C₈₉	C₄×SD₁₆	1	8	yes	—
123	C₂²×C₈₉	#465	1	8	yes	—
124	C₂²×C₈₉	#492	1	2	yes	—
125	C₂²×C₈₉	#495	1	2	yes	—
126	C₂²×C₈₉	#498	1	2	yes	—
127	C₂²×C₈₉	#499	1	2	yes	—
128	C₄×C₈₉	—	2	8	no	—
129	C₄×C₈₉	C₂×#95	1	1	yes	characteristic
130	C₄×C₈₉	C₄×C₄⋊C₄	1	1	yes	characteristic
131	C₄×C₈₉	#380	1	1	yes	characteristic
132	C₄×C₈₉	#382	1	1	yes	characteristic
133	C₄×C₈₉	#389	1	2	yes	—
134	C₄×C₈₉	#391	1	2	yes	—
135	Dic₈₉	—	8	8	no	—
136	#10861	—	89	1	no	maximal, Sylow
137	C₂×C₈₉	C₄⋊#95	1	1	yes	characteristic
138	C₂×C₈₉	C₄×#106	1	2	yes	—
139	C₂×C₈₉	#1807	1	2	yes	—
140	C₂×C₈₉	#1844	1	1	yes	characteristic
141	C₂×C₈₉	#1846	1	1	yes	characteristic
142	C₂×C₈₉	#1941	1	4	yes	—
143	C₂×C₈₉	#1968	1	4	yes	—
144	C₂²×#106	—	89	2	no	—
145	C₂²×C₄×C₈	D₈₉	1	1	yes	characteristic
146	C₂×#340	—	89	2	no	—
147	C₄⋊#95	—	89	2	no	—
148	C₈₉	#10861	1	1	yes	characteristic, complemented, Sylow
149	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
150	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
151	C₂²×C₄⋊C₄	—	89	4	no	—
152	C₂²×C₄²	C₂×D₈₉	1	1	yes	characteristic
153	C₂×#106	—	89	8	no	—
154	C₂×#106	—	178	4	no	—
155	C₂×#106	—	178	4	no	—
156	C₂×C₄×C₈	—	2	4	no	—
157	C₂×C₄×C₈	Dic₈₉	1	4	yes	complemented
158	C₂×#95	—	178	4	no	—
159	#340	—	178	4	no	—
160	C₂³×C₄	—	178	4	no	—
161	C₂³×C₄	D₃₅₆	1	2	yes	—
162	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	2	2	no	—
165	C₂²×C₈	Dic₁₇₈	1	2	yes	—
166	C₂²×C₈	Dic₁₇₈	1	2	yes	—
167	C₂²×C₈	C₄×D₈₉	1	4	yes	—
168	C₂²×C₈	D₃₅₆	1	2	yes	—
169	C₂²×C₈	D₃₅₆	1	2	yes	—
170	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
171	C₂²×C₈	C₂×Dic₈₉	1	2	yes	—
172	C₂²×C₈	C₂²⋊D₈₉	1	4	yes	—
173	C₂×C₄⋊C₄	—	89	8	no	—
174	C₂×C₄⋊C₄	—	178	4	no	—
175	C₂×C₄⋊C₄	—	178	4	no	—
176	C₂×C₄⋊C₄	—	178	4	no	—
177	C₂×C₄⋊C₄	—	178	4	no	—
178	C₂×C₄⋊C₄	—	178	4	no	—
179	C₂×C₄²	—	2	2	no	—
180	C₂×C₄²	—	2	2	no	—
181	C₂×C₄²	Dic₁₇₈	1	2	yes	—
182	C₂×C₄²	C₂×Dic₈₉	1	2	yes	—
183	#106	—	178	16	no	—
184	C₄×C₈	—	2	8	no	—
185	#95	—	356	8	no	—
186	C₂⁴	C₂×D₃₅₆	1	1	yes	characteristic
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	4	no	—
189	C₂²×C₄	—	356	4	no	—
190	C₂²×C₄	—	356	4	no	—
191	C₂²×C₄	—	356	8	no	—
192	C₂²×C₄	—	356	8	no	—
193	C₂²×C₄	—	178	8	no	—
194	C₂²×C₄	C₄⋊Dic₈₉	1	4	yes	—
195	C₂²×C₄	#13083563	1	8	yes	—
196	C₂²×C₄	C₂×C₄×D₈₉	1	2	yes	—
197	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₂²⋊D₈₉	1	2	yes	—
199	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
201	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
202	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
203	C₂²×C₄	C₂×Dic₁₇₈	1	1	yes	characteristic
204	C₂²×C₄	#13083574	1	2	yes	—
205	C₂²×C₄	#13083574	1	2	yes	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	4	no	—
209	C₂×C₈	—	2	2	no	—
210	C₂×C₈	C₄×Dic₈₉	1	4	yes	—
211	C₂×C₈	#13083551	1	8	yes	—
212	C₂×C₈	C₂²⋊Dic₈₉	1	4	yes	—
213	C₂×C₈	#13083553	1	8	yes	—
214	C₂×C₈	C₄⋊Dic₈₉	1	4	yes	—
215	C₂×C₈	C₄⋊Dic₈₉	1	4	yes	—
216	C₄⋊C₄	—	356	4	no	—
217	C₄⋊C₄	—	356	4	no	—
218	C₄⋊C₄	—	178	8	no	—
219	C₄⋊C₄	—	178	8	no	—
220	C₄²	—	2	2	no	—
221	C₄²	—	2	4	no	—
222	C₄²	—	2	2	no	—
223	C₂³	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
224	C₂³	C₄×D₃₅₆	1	2	yes	—
225	C₂³	#15667986	1	2	yes	—
226	C₂³	#15668009	1	1	yes	characteristic
227	C₂³	#15668012	1	1	yes	characteristic
228	C₂³	C₂×#13083563	1	4	yes	—
229	C₂³	#15668034	1	4	yes	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	16	no	—
233	C₂×C₄	—	356	16	no	—
234	C₂×C₄	—	356	8	no	—
235	C₂×C₄	—	712	4	no	—
236	C₂×C₄	—	712	4	no	—
237	C₂×C₄	—	712	8	no	—
238	C₂×C₄	#15667944	1	8	yes	complemented
239	C₂×C₄	C₂×C₄×Dic₈₉	1	1	yes	characteristic
240	C₂×C₄	C₂×#13083551	1	2	yes	—
241	C₂×C₄	C₂×C₂²⋊Dic₈₉	1	1	yes	characteristic
242	C₂×C₄	C₂×#13083553	1	2	yes	—
243	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
244	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
245	C₂×C₄	#15667971	1	1	yes	characteristic
246	C₂×C₄	#15667971	1	1	yes	characteristic
247	C₂×C₄	C₄×D₃₅₆	1	2	yes	—
248	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
249	C₂×C₄	C₄×Dic₁₇₈	1	2	yes	—
250	C₂×C₄	#15667986	1	2	yes	—
251	C₂×C₄	#15667991	1	2	yes	—
252	C₂×C₄	#15667991	1	2	yes	—
253	C₂×C₄	#15668000	1	2	yes	—
254	C₂×C₄	#15668004	1	2	yes	—
255	C₂×C₄	#15668012	1	1	yes	characteristic
256	C₂×C₄	#15668012	1	1	yes	characteristic
257	C₈	—	2	4	no	—
258	C₈	#15667919	1	8	yes	complemented
259	C₂²	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
260	C₂²	#19151458	1	1	yes	characteristic
261	C₂²	#19151468	1	1	yes	characteristic
262	C₂²	#19151496	1	2	yes	—
263	C₂²	#19151500	1	2	yes	—
264	C₂²	C₂×#15667944	1	2	yes	—
265	C₂²	#19151636	1	2	yes	—
266	C₂²	C₄×#13083563	1	8	yes	—
267	C₂²	#19151827	1	8	yes	—
268	C₂²	#19151971	1	2	yes	—
269	C₂²	#19151982	1	2	yes	—
270	C₂²	#19151996	1	2	yes	—
271	C₂²	#19152000	1	2	yes	—
272	C₄	—	2	8	no	—
273	C₄	—	712	8	no	—
274	C₄	C₂×#15667919	1	1	yes	characteristic
275	C₄	C₄×C₄⋊Dic₈₉	1	1	yes	characteristic
276	C₄	#19151458	1	1	yes	characteristic
277	C₄	#19151464	1	1	yes	characteristic
278	C₄	#19151496	1	2	yes	—
279	C₄	#19151504	1	2	yes	—
280	C₂	#26018482	1	1	yes	characteristic
281	C₂	C₄×#15667944	1	2	yes	—
282	C₂	#26020277	1	2	yes	—
283	C₂	#26020409	1	1	yes	characteristic
284	C₂	#26020416	1	1	yes	characteristic
285	C₂	#26020980	1	4	yes	—
286	C₂	#26021106	1	4	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₂²×C₄×C₈×C₈₉	C₂
socle, prime radical	C₂⁴×C₈₉	C₂×D₄
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	C₂²×C₄×C₈	D₈₉
hypocenter	C₈₉	#10861
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
8. C₂×#15667944	C₄	semidirect
15. C₂×C₄×C₈×C₈₉	C₄	semidirect
89. C₂×C₄×C₈₉	#106	semidirect
114. C₈×C₈₉	#95	semidirect
148. C₈₉	#10861	semidirect
157. C₂×C₄×C₈	Dic₈₉	semidirect
238. C₂×C₄	#15667944	semidirect
258. C₈	#15667919	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#15667944	2
C₂×#19151361	2
#26018482	2
C₂²×C₄×C₈×C₈₉	2
#10861	89