Subgroups — #36542134

Order 22784.

706 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030839	C₂	1	2	yes	maximal
2	#26030836	C₂	1	2	yes	maximal
3	#26030763	C₂	1	2	yes	maximal
4	#26019870	C₂	1	1	yes	maximal, characteristic, complemented
5	#26019861	C₂	1	2	yes	maximal, complemented
6	#26019850	C₂	1	2	yes	maximal, complemented
7	#26019846	C₂	1	2	yes	maximal, complemented
8	C₈²×D₈₉	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#1713	C₂	1	1	yes	maximal, characteristic, complemented
10	#19152153	C₂²	1	4	yes	—
11	#19152003	C₂²	1	2	yes	—
12	#19152003	C₂²	1	2	yes	—
13	#19151999	—	2	4	no	—
14	#19151994	C₂²	1	4	yes	—
15	#19151978	C₂²	1	2	yes	—
16	#19151950	C₂²	1	4	yes	—
17	#19151948	C₂²	1	4	yes	—
18	#19151840	—	2	4	no	—
19	#19151836	—	2	4	no	—
20	Q₁₆×Dic₈₉	—	2	4	no	—
21	#19151692	—	2	4	no	—
22	#19151635	—	2	4	no	—
23	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
24	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
25	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
26	C₈×#10623494	C₂²	1	1	yes	characteristic
27	C₈×#10623494	C₂²	1	1	yes	characteristic
28	C₈×#10623494	C₂²	1	1	yes	characteristic
29	#500×C₈₉	C₂²	1	2	yes	—
30	#500×C₈₉	C₂²	1	2	yes	—
31	#494×C₈₉	C₂²	1	2	yes	—
32	C₈²×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×Dic₃₅₆	—	2	4	no	—
34	C₂×C₈×D₈₉	D₄	1	2	yes	—
35	C₂×C₈×D₈₉	D₄	1	2	yes	—
36	C₂×C₈×D₈₉	D₄	1	2	yes	—
37	#15668012	C₂³	1	4	yes	—
38	#15668003	—	2	4	no	—
39	#15667992	—	2	4	no	—
40	#15667988	—	2	4	no	—
41	Q₈×Dic₈₉	—	2	4	no	—
42	#15667977	—	2	4	no	—
43	#15667970	—	2	4	no	—
44	C₄²×D₈₉	C₂³	1	1	yes	characteristic
45	#15667946	—	2	4	no	—
46	#15667946	—	2	4	no	—
47	#15667943	—	2	8	no	—
48	#15667936	—	2	8	no	—
49	#15667935	—	2	8	no	—
50	C₄×#10623494	C₂³	1	1	yes	characteristic
51	C₄×#10623494	C₂³	1	1	yes	characteristic
52	C₄×#10623494	C₂³	1	1	yes	characteristic
53	C₈×Dic₈₉	D₄	1	2	yes	—
54	C₈×Dic₈₉	D₄	1	2	yes	—
55	C₈×Dic₈₉	D₄	1	2	yes	—
56	C₂×Q₁₆×C₈₉	—	2	4	no	—
57	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
58	#107×C₈₉	—	2	4	no	—
59	#107×C₈₉	—	2	4	no	—
60	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
63	C₂×Dic₁₇₈	—	2	4	no	—
64	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
66	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
67	Dic₃₅₆	—	4	8	no	—
68	C₂×#10623494	C₂×D₄	1	2	yes	—
69	C₂×#10623494	C₂×D₄	1	2	yes	—
70	C₂×#10623494	C₂×D₄	1	2	yes	—
71	C₈×D₈₉	—	2	4	no	—
72	C₈×D₈₉	—	2	4	no	—
73	C₈×D₈₉	Q₁₆	1	8	yes	complemented
74	C₄⋊Dic₈₉	—	2	4	no	—
75	C₄⋊Dic₈₉	—	2	4	no	—
76	C₄⋊Dic₈₉	—	4	4	no	—
77	#13083553	—	4	4	no	—
78	#13083551	—	4	4	no	—
79	C₄×Dic₈₉	—	4	4	no	—
80	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
81	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
82	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂×Q₈×C₈₉	—	2	4	no	—
84	Q₁₆×C₈₉	—	4	8	no	—
85	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
88	C₄⋊C₄×C₈₉	—	2	4	no	—
89	C₄⋊C₄×C₈₉	—	2	4	no	—
90	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
91	C₂²×D₈₉	#127	1	1	yes	characteristic
92	Q₈×C₈₉	—	4	8	no	—
93	C₂×C₄×C₈₉	—	4	4	no	—
94	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
97	C₂×Dic₈₉	—	4	4	no	—
98	C₂×Dic₈₉	#127	1	1	yes	characteristic
99	C₂×Dic₈₉	#127	1	1	yes	characteristic
100	C₂×Dic₈₉	#127	1	1	yes	characteristic
101	C₄×D₈₉	—	2	2	no	—
102	C₄×D₈₉	—	2	2	no	—
103	C₄×D₈₉	C₂×Q₁₆	1	4	yes	—
104	Dic₁₇₈	—	4	8	no	—
105	C₈×C₈₉	C₂×Q₁₆	1	4	yes	—
106	C₈×C₈₉	#135	1	4	yes	—
107	C₈×C₈₉	#135	1	4	yes	—
108	#10623494	—	2	2	no	—
109	#10623494	—	2	2	no	—
110	#10623494	C₂×Q₁₆	1	4	yes	—
111	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
112	C₂×D₈₉	—	2	1	no	—
113	C₂×D₈₉	—	2	1	no	—
114	C₂×D₈₉	#494	1	2	yes	—
115	C₄×C₈₉	—	8	4	no	—
116	C₄×C₈₉	C₂²×Q₁₆	1	2	yes	—
117	C₄×C₈₉	C₂×#135	1	2	yes	—
118	C₄×C₈₉	C₂×#135	1	2	yes	—
119	Dic₈₉	—	2	1	no	—
120	Dic₈₉	—	2	1	no	—
121	Dic₈₉	—	8	4	no	—
122	Dic₈₉	#494	1	2	yes	—
123	#19604	—	89	1	no	maximal, Sylow
124	C₂×C₈₉	C₂×#494	1	1	yes	characteristic
125	C₂×C₈₉	#3147	1	1	yes	characteristic
126	C₂×C₈₉	#3147	1	1	yes	characteristic
127	D₈₉	—	2	2	no	—
128	#3170	—	89	2	no	—
129	#3170	—	89	2	no	—
130	#3160	—	89	2	no	—
131	#1713	—	89	1	no	—
132	#1713	D₈₉	1	1	yes	characteristic, complemented
133	#1710	—	89	2	no	—
134	#1707	—	89	2	no	—
135	#1707	—	89	2	no	—
136	C₂×C₈²	—	89	1	no	—
137	C₈₉	#19604	1	1	yes	characteristic, complemented, Sylow
138	#533	—	89	4	no	—
139	#500	—	89	2	no	—
140	#500	—	89	2	no	—
141	#500	C₂×D₈₉	1	2	yes	—
142	#500	C₂×D₈₉	1	2	yes	—
143	#499	—	178	4	no	—
144	#498	—	89	4	no	—
145	#494	—	89	2	no	—
146	#494	C₂×D₈₉	1	2	yes	—
147	#487	—	89	4	no	—
148	#487	—	89	4	no	—
149	#467	—	178	4	no	—
150	#467	—	178	4	no	—
151	C₄×Q₁₆	—	178	4	no	—
152	C₄×Q₁₆	—	178	4	no	—
153	#427	—	178	4	no	—
154	C₂×C₄×C₈	—	89	1	no	—
155	C₂×C₄×C₈	—	89	1	no	—
156	C₂×C₄×C₈	—	89	1	no	—
157	C₈²	—	89	1	no	—
158	C₈²	—	89	1	no	—
159	C₈²	—	89	1	no	—
160	C₈²	C₂×D₈₉	1	1	yes	characteristic
161	C₂×Q₁₆	—	2	4	no	—
162	C₂×Q₁₆	—	178	4	no	—
163	C₂²×C₈	—	89	2	no	—
164	C₂²×C₈	—	89	2	no	—
165	C₂²×C₈	—	89	2	no	—
166	C₄⋊Q₈	—	89	4	no	—
167	C₄⋊Q₈	C₂²×D₈₉	1	4	yes	—
168	#125	—	178	4	no	—
169	C₂²⋊Q₈	—	178	4	no	—
170	C₂²⋊Q₈	—	178	4	no	—
171	C₄×Q₈	—	178	4	no	—
172	C₄×Q₈	—	178	4	no	—
173	#117	—	178	4	no	—
174	C₂×C₄²	—	89	1	no	—
175	#107	—	2	4	no	—
176	#107	—	2	4	no	—
177	#107	—	178	4	no	—
178	#107	—	178	4	no	—
179	#106	—	178	8	no	—
180	#103	—	178	8	no	—
181	#103	—	178	8	no	—
182	C₄×C₈	—	89	2	no	—
183	C₄×C₈	—	89	2	no	—
184	C₄×C₈	—	89	2	no	—
185	C₄×C₈	—	89	1	no	—
186	C₄×C₈	—	89	1	no	—
187	C₄×C₈	—	89	1	no	—
188	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
189	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
190	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	—	178	4	no	—
193	C₂²×C₄	—	89	1	no	—
194	C₂²×C₄	—	89	1	no	—
195	C₂²×C₄	—	89	1	no	—
196	Q₁₆	—	4	8	no	—
197	Q₁₆	—	356	8	no	—
198	C₂×C₈	—	89	2	no	—
199	C₂×C₈	—	89	8	no	—
200	C₂×C₈	—	89	2	no	—
201	C₂×C₈	—	89	2	no	—
202	C₂×C₈	—	178	4	no	—
203	C₂×C₈	—	178	4	no	—
204	C₂×C₈	D₄×D₈₉	1	2	yes	—
205	C₂×C₈	D₄×D₈₉	1	2	yes	—
206	C₂×C₈	D₄×D₈₉	1	2	yes	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	—	2	4	no	—
209	C₄⋊C₄	—	356	4	no	—
210	C₄⋊C₄	—	356	4	no	—
211	C₄⋊C₄	—	178	4	no	—
212	C₄⋊C₄	—	178	4	no	—
213	C₂²⋊C₄	—	356	4	no	—
214	C₄²	—	89	1	no	—
215	C₄²	—	89	1	no	—
216	C₄²	—	89	1	no	—
217	C₄²	—	356	4	no	—
218	C₄²	C₂³×D₈₉	1	1	yes	characteristic
219	C₂³	—	89	1	no	—
220	Q₈	—	4	8	no	—
221	Q₈	—	356	8	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	89	1	no	—
224	C₂×C₄	—	89	4	no	—
225	C₂×C₄	—	89	1	no	—
226	C₂×C₄	—	89	1	no	—
227	C₂×C₄	—	356	4	no	—
228	C₂×C₄	—	178	2	no	—
229	C₂×C₄	—	178	2	no	—
230	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
231	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
232	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
233	C₈	—	89	4	no	—
234	C₈	—	178	2	no	—
235	C₈	—	178	2	no	—
236	C₈	Q₁₆×D₈₉	1	4	yes	—
237	C₈	#15668031	1	4	yes	—
238	C₈	#15668035	1	4	yes	—
239	C₂²	—	89	2	no	—
240	C₂²	—	178	1	no	—
241	C₂²	—	178	1	no	—
242	C₂²	#127×D₈₉	1	1	yes	characteristic
243	C₄	—	8	4	no	—
244	C₄	—	89	2	no	—
245	C₄	—	178	1	no	—
246	C₄	—	178	1	no	—
247	C₄	—	712	4	no	—
248	C₄	C₂×Q₁₆×D₈₉	1	2	yes	—
249	C₄	C₂×#15668031	1	2	yes	—
250	C₄	C₂×#15668035	1	2	yes	—
251	C₂	—	178	2	no	—
252	C₂	#494×D₈₉	1	1	yes	characteristic
253	C₂	#26030679	1	1	yes	characteristic
254	C₂	#26030681	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#127×D₈₉
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#1713	C₂
socle	C₂²×C₈₉	C₂×#127
prime radical	C₂²×D₈₉	#127
hypercenter	#1713	D₈₉
hypocenter	C₈₉	#19604
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1713

Complements

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

N	H ≅ G/N	type
4. #26019870	C₂	semidirect
5. #26019861	C₂	semidirect
6. #26019850	C₂	semidirect
7. #26019846	C₂	semidirect
9. C₈₉×#1713	C₂	semidirect
73. C₈×D₈₉	Q₁₆	semidirect
132. #1713	D₈₉	semidirect
137. C₈₉	#19604	semidirect

Maximal subgroups

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

subgroup	index
#26030839	2
#26030836	2
#26030763	2
#26019870	2
#26019861	2
#26019850	2
#26019846	2
C₈²×D₈₉	2
C₈₉×#1713	2
#19604	89