Subgroups — #36709943

Order 22784.

474 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030789	C₂	1	2	yes	maximal
2	#26022539	C₂	1	2	yes	maximal, complemented
3	#26022536	C₂	1	2	yes	maximal, complemented
4	#26022534	C₂	1	1	yes	maximal, characteristic, complemented
5	#26022533	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022530	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022037	C₂	1	2	yes	maximal
8	#26022035	C₂	1	1	yes	maximal, characteristic
9	#367×D₈₉	C₂	1	1	yes	maximal, characteristic
10	#26021847	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2253	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152228	—	2	2	no	—
13	C₂×#15667951	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
15	#19152008	—	2	2	no	—
16	#19152005	—	2	2	no	—
17	#19152003	C₂²	1	2	yes	—
18	#19152002	C₂²	1	2	yes	—
19	#19151999	C₂²	1	2	yes	—
20	#19151637	—	2	2	no	—
21	#19151635	C₂²	1	2	yes	—
22	#107×D₈₉	C₂²	1	2	yes	—
23	#19151552	C₂²	1	1	yes	characteristic
24	#19151426	C₂²	1	1	yes	characteristic
25	#19151426	C₂²	1	1	yes	characteristic
26	#19151426	C₂²	1	2	yes	—
27	#19151425	C₂²	1	2	yes	—
28	#19151425	C₂²	1	2	yes	—
29	#19151423	C₂²	1	2	yes	—
30	#19151423	C₂²	1	2	yes	—
31	#19151374	C₂²	1	1	yes	characteristic
32	#19151373	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
34	#499×C₈₉	C₂²	1	2	yes	—
35	#367×C₈₉	C₂²	1	1	yes	characteristic
36	#367×C₈₉	C₂²	1	1	yes	characteristic
37	#366×C₈₉	C₂²	1	2	yes	—
38	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×#13083557	D₄	1	1	yes	characteristic
40	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
41	#15668012	—	2	2	no	—
42	#15668011	—	2	2	no	—
43	#15668011	—	4	2	no	—
44	#15668003	—	2	2	no	—
45	#15668002	—	4	2	no	—
46	#15667992	—	4	2	no	—
47	#15667990	—	4	2	no	—
48	#15667970	—	2	2	no	—
49	#15667969	D₄	1	1	yes	characteristic
50	C₄⋊C₄×D₈₉	—	2	2	no	—
51	#15667951	Q₈	1	4	yes	complemented
52	C₂×#13083549	C₂³	1	1	yes	characteristic
53	C₂×#13083549	C₂³	1	1	yes	characteristic
54	C₁₆×D₈₉	Q₈	1	4	yes	complemented
55	#15667946	—	2	2	no	—
56	#15667946	C₂³	1	2	yes	—
57	#15667946	C₂³	1	2	yes	—
58	#15667945	—	2	2	no	—
59	#15667945	—	2	2	no	—
60	#15667943	—	2	2	no	—
61	#15667943	—	2	2	no	—
62	#15667923	D₄	1	1	yes	characteristic
63	#15667922	D₄	1	1	yes	characteristic
64	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
65	#125×C₈₉	—	2	2	no	—
66	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
68	#107×C₈₉	C₂³	1	2	yes	—
69	#107×C₈₉	C₂³	1	2	yes	—
70	#106×C₈₉	—	2	2	no	—
71	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₂×Dic₁₇₈	—	4	2	no	—
73	C₂×Dic₁₇₈	—	4	2	no	—
74	C₂×C₄×D₈₉	—	4	2	no	—
75	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
76	#13083557	Q₁₆	1	4	yes	—
77	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
78	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
79	C₈×D₈₉	C₂×Q₈	1	2	yes	—
80	C₈×D₈₉	C₂×Q₈	1	2	yes	—
81	C₄⋊Dic₈₉	—	2	2	no	—
82	C₄⋊Dic₈₉	—	2	2	no	—
83	C₄⋊Dic₈₉	—	4	2	no	—
84	C₄⋊Dic₈₉	—	4	2	no	—
85	#13083553	—	4	2	no	—
86	#13083553	—	4	2	no	—
87	#13083553	—	4	2	no	—
88	#13083553	D₈	1	2	yes	—
89	#13083551	—	4	2	no	—
90	#13083551	D₈	1	2	yes	—
91	C₄×Dic₈₉	—	4	2	no	—
92	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
93	#13083549	C₂×Q₈	1	2	yes	—
94	#13083549	C₂×Q₈	1	2	yes	—
95	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
97	C₄⋊C₄×C₈₉	—	2	2	no	—
98	C₄⋊C₄×C₈₉	—	2	2	no	—
99	C₄⋊C₄×C₈₉	—	4	2	no	—
100	C₄⋊C₄×C₈₉	—	4	2	no	—
101	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
103	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
104	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₈₉	—	4	2	no	—
106	C₂×C₄×C₈₉	—	4	2	no	—
107	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
109	C₂×Dic₈₉	—	4	2	no	—
110	C₂×Dic₈₉	—	4	2	no	—
111	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
112	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
113	C₄×D₈₉	—	16	2	no	—
114	C₄×D₈₉	C₄⋊Q₈	1	2	yes	—
115	C₄×D₈₉	C₂×Q₁₆	1	2	yes	—
116	Dic₁₇₈	—	16	2	no	—
117	Dic₁₇₈	—	16	2	no	—
118	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
119	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
120	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
121	#10623494	C₄⋊Q₈	1	2	yes	—
122	#10623494	C₂×Q₁₆	1	2	yes	—
123	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
124	C₂×D₈₉	—	2	2	no	—
125	C₂×D₈₉	#500	1	2	yes	—
126	C₄×C₈₉	—	2	2	no	—
127	C₄×C₈₉	—	8	2	no	—
128	C₄×C₈₉	—	8	2	no	—
129	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
130	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
131	Dic₈₉	—	2	2	no	—
132	Dic₈₉	—	8	2	no	—
133	Dic₈₉	—	8	2	no	—
134	Dic₈₉	#500	1	2	yes	—
135	#33730	—	89	1	no	maximal, Sylow
136	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
137	C₂×C₈₉	#3416	1	2	yes	—
138	D₈₉	—	2	2	no	—
139	#3164	—	89	2	no	—
140	#2255	—	89	2	no	—
141	#2255	—	89	2	no	—
142	#2254	—	89	1	no	—
143	#2254	—	89	1	no	—
144	#2253	—	89	1	no	—
145	#2253	D₈₉	1	1	yes	characteristic, complemented
146	#2159	—	89	2	no	—
147	#2158	—	89	1	no	—
148	C₂×#367	—	89	1	no	—
149	#2108	—	89	1	no	—
150	C₈₉	#33730	1	1	yes	characteristic, complemented, Sylow
151	#544	—	178	2	no	—
152	C₂×M₃₂	—	89	1	no	—
153	C₂²×C₁₆	—	89	1	no	—
154	#501	—	178	2	no	—
155	#501	—	178	2	no	—
156	#500	—	89	2	no	—
157	#500	—	89	2	no	—
158	#499	—	89	2	no	—
159	#499	C₂×D₈₉	1	2	yes	—
160	#428	—	178	2	no	—
161	#427	—	89	2	no	—
162	C₂×#107	—	89	2	no	—
163	#405	—	89	1	no	—
164	#367	—	89	1	no	—
165	#367	—	89	1	no	—
166	#367	—	89	2	no	—
167	#367	—	89	2	no	—
168	#367	—	89	2	no	—
169	#367	C₂×D₈₉	1	1	yes	characteristic
170	#367	C₂×D₈₉	1	1	yes	characteristic
171	#366	—	89	2	no	—
172	#366	—	89	2	no	—
173	#366	C₂×D₈₉	1	2	yes	—
174	#346	—	89	1	no	—
175	#346	—	89	1	no	—
176	C₄×C₁₆	—	89	1	no	—
177	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
178	C₂×M₁₆	—	89	1	no	—
179	C₂²×C₈	—	89	1	no	—
180	C₄⋊Q₈	—	356	2	no	—
181	C₄⋊Q₈	—	178	2	no	—
182	C₄⋊Q₈	—	178	2	no	—
183	#125	—	2	2	no	—
184	#125	—	356	2	no	—
185	#125	—	178	2	no	—
186	C₂²⋊Q₈	—	356	2	no	—
187	C₂²⋊Q₈	—	356	2	no	—
188	#117	—	89	1	no	—
189	#117	—	178	2	no	—
190	C₂×C₄⋊C₄	—	178	2	no	—
191	M₃₂	—	89	4	no	—
192	C₂×C₁₆	—	89	1	no	—
193	C₂×C₁₆	—	89	1	no	—
194	C₂×C₁₆	—	89	4	no	—
195	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
196	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
197	#107	—	89	2	no	—
198	#107	—	89	2	no	—
199	#107	—	178	2	no	—
200	#107	—	178	2	no	—
201	#107	—	178	2	no	—
202	#107	C₂²×D₈₉	1	2	yes	—
203	#107	C₂²×D₈₉	1	2	yes	—
204	#106	—	2	2	no	—
205	#106	—	178	2	no	—
206	#106	—	178	2	no	—
207	#97	—	89	1	no	—
208	#97	—	89	1	no	—
209	C₄×C₈	—	89	1	no	—
210	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
211	C₂×Q₈	—	356	2	no	—
212	C₂×Q₈	—	356	2	no	—
213	C₂²×C₄	—	89	1	no	—
214	C₂²×C₄	—	356	2	no	—
215	M₁₆	—	89	4	no	—
216	C₂×C₈	—	89	1	no	—
217	C₂×C₈	—	89	1	no	—
218	C₂×C₈	—	89	2	no	—
219	C₂×C₈	—	89	2	no	—
220	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
221	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	—	4	2	no	—
226	C₄⋊C₄	—	89	2	no	—
227	C₄⋊C₄	—	356	2	no	—
228	C₄⋊C₄	—	356	2	no	—
229	C₄⋊C₄	—	356	2	no	—
230	C₄⋊C₄	—	356	2	no	—
231	C₄⋊C₄	—	356	2	no	—
232	C₄⋊C₄	—	178	2	no	—
233	C₄⋊C₄	—	178	2	no	—
234	C₂²⋊C₄	—	89	2	no	—
235	C₂²⋊C₄	—	356	2	no	—
236	C₄²	—	89	1	no	—
237	C₄²	—	356	2	no	—
238	C₄²	D₄×D₈₉	1	1	yes	characteristic
239	C₁₆	—	89	2	no	—
240	C₁₆	—	89	2	no	—
241	C₁₆	Q₈×D₈₉	1	2	yes	—
242	C₁₆	Q₈×D₈₉	1	2	yes	—
243	C₂³	—	89	1	no	—
244	Q₈	—	1424	2	no	—
245	Q₈	—	1424	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	89	1	no	—
249	C₂×C₄	—	89	2	no	—
250	C₂×C₄	—	89	2	no	—
251	C₂×C₄	—	89	2	no	—
252	C₂×C₄	—	356	2	no	—
253	C₂×C₄	—	356	2	no	—
254	C₂×C₄	—	1424	2	no	—
255	C₂×C₄	D₈×D₈₉	1	2	yes	—
256	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
257	C₈	—	89	2	no	—
258	C₈	—	89	2	no	—
259	C₈	Q₁₆×D₈₉	1	2	yes	—
260	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
261	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
262	C₂²	—	89	2	no	—
263	C₂²	—	178	2	no	—
264	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	8	2	no	—
267	C₄	—	8	2	no	—
268	C₄	—	89	2	no	—
269	C₄	—	178	2	no	—
270	C₄	—	712	2	no	—
271	C₄	—	712	2	no	—
272	C₄	C₄⋊Q₈×D₈₉	1	1	yes	characteristic
273	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
274	C₂	—	178	2	no	—
275	C₂	#500×D₈₉	1	1	yes	characteristic
276	C₂	#26033395	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2253	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂²×D₈₉	C₂×D₈
hypercenter	#2253	D₈₉
hypocenter	C₈₉	#33730
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
2. #26022539	C₂	semidirect
3. #26022536	C₂	semidirect
4. #26022534	C₂	semidirect
5. #26022533	C₂	semidirect
6. #26022530	C₂	semidirect
11. C₈₉×#2253	C₂	semidirect
51. #15667951	Q₈	semidirect
54. C₁₆×D₈₉	Q₈	semidirect
145. #2253	D₈₉	semidirect
150. C₈₉	#33730	semidirect

Maximal subgroups

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

subgroup	index
#26030789	2
#26022539	2
#26022536	2
#26022534	2
#26022533	2
#26022530	2
#26022037	2
#26022035	2
#367×D₈₉	2
#26021847	2
C₈₉×#2253	2
#33730	89