Subgroups — #36459496

Order 22784.

494 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26021058	C₂	1	1	yes	maximal, characteristic
2	#26021057	C₂	1	1	yes	maximal, characteristic
3	#26021057	C₂	1	1	yes	maximal, characteristic
4	#26020408	C₂	1	1	yes	maximal, characteristic, complemented
5	#26020379	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018079	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1834	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668009	C₂²	1	1	yes	characteristic
9	C₂×#15667943	—	2	2	no	—
10	C₂×#15667934	—	2	2	no	—
11	C₂×#15667932	—	2	2	no	—
12	C₂×#15667932	—	2	2	no	—
13	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
14	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
15	#19151468	C₂²	1	1	yes	characteristic
16	#19151360	—	2	2	no	—
17	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
18	#384×C₈₉	C₂²	1	1	yes	characteristic
19	#384×C₈₉	C₂²	1	1	yes	characteristic
20	#340×C₈₉	—	2	2	no	—
21	C₂²×D₃₅₆	—	2	1	no	—
22	C₂²×D₃₅₆	—	2	1	no	—
23	C₂²×D₃₅₆	—	2	1	no	—
24	C₂²×#10623494	D₄	1	2	yes	—
25	C₂²×#10623494	D₄	1	2	yes	—
26	#15668009	—	2	1	no	—
27	#15668009	—	2	1	no	—
28	#15668009	—	2	1	no	—
29	#15668009	Q₈	1	2	yes	—
30	C₂×C₄⋊Dic₈₉	—	2	1	no	—
31	C₂×C₄⋊Dic₈₉	—	2	1	no	—
32	C₂×C₄⋊Dic₈₉	—	2	1	no	—
33	#15667943	—	4	4	no	—
34	#15667934	—	4	4	no	—
35	#15667932	—	4	4	no	—
36	#15667932	—	4	4	no	—
37	C₄×#10623494	—	2	1	no	—
38	C₄×#10623494	—	2	1	no	—
39	C₄×#10623494	—	2	1	no	—
40	C₄×#10623494	—	2	1	no	—
41	C₂²×C₈×C₈₉	D₄	1	2	yes	—
42	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
43	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
44	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
45	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
46	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
47	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
48	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₈₉	—	2	1	no	—
50	C₄×C₈×C₈₉	—	2	1	no	—
51	C₂³×D₈₉	—	4	1	no	—
52	C₂×D₃₅₆	—	2	2	no	—
53	C₂×D₃₅₆	—	2	2	no	—
54	C₂×D₃₅₆	—	2	2	no	—
55	C₂×D₃₅₆	—	4	2	no	—
56	C₂×D₃₅₆	—	4	2	no	—
57	C₂×D₃₅₆	—	4	2	no	—
58	C₂×D₃₅₆	—	4	2	no	—
59	C₂×D₃₅₆	—	4	2	no	—
60	C₂×D₃₅₆	—	4	2	no	—
61	C₂×D₃₅₆	—	4	1	no	—
62	C₂×D₃₅₆	—	4	1	no	—
63	C₂×D₃₅₆	—	4	1	no	—
64	C₂²×Dic₈₉	—	4	1	no	—
65	C₂×#10623494	—	2	2	no	—
66	C₂×#10623494	—	2	2	no	—
67	C₂×#10623494	—	2	2	no	—
68	C₂×#10623494	—	2	2	no	—
69	C₂×#10623494	—	2	2	no	—
70	C₂×#10623494	SD₁₆	1	4	yes	—
71	C₄⋊Dic₈₉	—	4	2	no	—
72	C₄⋊Dic₈₉	—	4	2	no	—
73	C₄⋊Dic₈₉	—	4	2	no	—
74	C₂²×C₄×C₈₉	—	4	1	no	—
75	C₂²×C₄×C₈₉	—	4	1	no	—
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₈₉	—	2	2	no	—
80	C₂×C₈×C₈₉	—	2	2	no	—
81	C₂×C₈×C₈₉	—	2	2	no	—
82	C₄⋊C₄×C₈₉	—	4	2	no	—
83	C₄⋊C₄×C₈₉	—	4	2	no	—
84	C₄⋊C₄×C₈₉	—	4	2	no	—
85	C₄⋊C₄×C₈₉	—	4	2	no	—
86	C₄⋊C₄×C₈₉	—	4	2	no	—
87	C₄⋊C₄×C₈₉	—	4	2	no	—
88	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
89	C₄²×C₈₉	#41	1	1	yes	characteristic
90	C₄²×C₈₉	#41	1	1	yes	characteristic
91	C₄²×C₈₉	#41	1	1	yes	characteristic
92	C₂³×C₈₉	#127	1	1	yes	characteristic
93	C₂²×D₈₉	—	4	2	no	—
94	C₂²×D₈₉	—	8	1	no	—
95	C₂²×D₈₉	—	8	1	no	—
96	C₂²×D₈₉	—	8	1	no	—
97	C₂²×D₈₉	—	8	1	no	—
98	C₂²×D₈₉	—	8	1	no	—
99	C₂²×D₈₉	—	8	1	no	—
100	C₂×C₄×C₈₉	—	8	1	no	—
101	C₂×C₄×C₈₉	—	8	1	no	—
102	C₂×C₄×C₈₉	—	8	1	no	—
103	C₂×C₄×C₈₉	—	8	1	no	—
104	C₂×C₄×C₈₉	—	8	1	no	—
105	C₂×C₄×C₈₉	—	8	1	no	—
106	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
111	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
112	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
113	C₂×C₄×C₈₉	#135	1	2	yes	—
114	C₂×C₄×C₈₉	#135	1	2	yes	—
115	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
116	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
117	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
118	C₂×Dic₈₉	—	8	1	no	—
119	C₂×Dic₈₉	—	8	1	no	—
120	C₂×Dic₈₉	—	8	1	no	—
121	D₃₅₆	—	4	4	no	—
122	D₃₅₆	—	4	4	no	—
123	D₃₅₆	—	4	4	no	—
124	D₃₅₆	—	8	2	no	—
125	D₃₅₆	—	8	2	no	—
126	D₃₅₆	—	8	2	no	—
127	C₈×C₈₉	—	4	2	no	—
128	#10623494	—	2	4	no	—
129	#10623494	—	4	2	no	—
130	C₂²×C₈₉	#399	1	1	yes	characteristic
131	C₂²×C₈₉	#492	1	1	yes	characteristic
132	C₂²×C₈₉	#495	1	1	yes	characteristic
133	C₂²×C₈₉	#495	1	1	yes	characteristic
134	C₂²×C₈₉	#496	1	1	yes	characteristic
135	C₂²×C₈₉	#496	1	1	yes	characteristic
136	C₂²×C₈₉	#496	1	1	yes	characteristic
137	C₂×D₈₉	—	8	2	no	—
138	C₂×D₈₉	—	8	2	no	—
139	C₂×D₈₉	—	8	2	no	—
140	C₂×D₈₉	—	16	1	no	—
141	C₂×D₈₉	—	16	1	no	—
142	C₂×D₈₉	—	16	1	no	—
143	C₂×D₈₉	—	16	1	no	—
144	C₄×C₈₉	—	16	1	no	—
145	C₄×C₈₉	—	16	1	no	—
146	C₄×C₈₉	#476	1	2	yes	—
147	C₄×C₈₉	#478	1	2	yes	—
148	C₄×C₈₉	#478	1	2	yes	—
149	C₄×C₈₉	#481	1	2	yes	—
150	C₄×C₈₉	#482	1	2	yes	—
151	C₄×C₈₉	#482	1	2	yes	—
152	Dic₈₉	—	16	1	no	—
153	#10677	—	89	1	no	maximal, Sylow
154	C₂×C₈₉	#1708	1	1	yes	characteristic
155	C₂×C₈₉	#1715	1	1	yes	characteristic
156	C₂×C₈₉	#1717	1	1	yes	characteristic
157	C₂×C₈₉	#1717	1	1	yes	characteristic
158	C₂×C₈₉	#2092	1	1	yes	characteristic
159	C₂×C₈₉	#2092	1	1	yes	characteristic
160	C₂×C₈₉	#2094	1	1	yes	characteristic
161	D₈₉	—	16	2	no	—
162	#1957	—	89	1	no	—
163	#1957	—	89	1	no	—
164	#1957	—	89	1	no	—
165	#1844	—	89	1	no	—
166	#1834	—	89	1	no	—
167	#1834	D₈₉	1	1	yes	characteristic, complemented
168	#1273	—	89	1	no	—
169	C₈₉	#10677	1	1	yes	characteristic, complemented, Sylow
170	C₂×#127	—	89	1	no	—
171	C₂×#106	—	178	2	no	—
172	C₂×#102	—	178	2	no	—
173	C₂×#102	—	178	2	no	—
174	C₂×#102	—	178	2	no	—
175	C₂×C₄×C₈	—	89	1	no	—
176	C₂×C₄×C₈	—	89	1	no	—
177	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
178	#384	—	89	1	no	—
179	#384	C₂×D₈₉	1	1	yes	characteristic
180	#384	C₂×D₈₉	1	1	yes	characteristic
181	#340	—	2	2	no	—
182	#340	—	178	2	no	—
183	C₂²×D₄	—	178	1	no	—
184	C₂²×D₄	—	178	1	no	—
185	C₂²×D₄	—	178	1	no	—
186	C₂²×C₈	—	89	2	no	—
187	C₂²×C₈	—	89	2	no	—
188	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
189	#127	—	89	2	no	—
190	#127	—	178	1	no	—
191	#127	—	178	1	no	—
192	#127	—	178	1	no	—
193	C₂×C₄⋊C₄	—	2	1	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	2	1	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	178	1	no	—
200	C₂×C₄⋊C₄	—	178	1	no	—
201	C₂×C₄⋊C₄	—	178	1	no	—
202	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
203	#106	—	356	4	no	—
204	#102	—	356	4	no	—
205	#102	—	356	4	no	—
206	#102	—	356	4	no	—
207	C₄×C₈	—	2	1	no	—
208	C₄×C₈	—	2	1	no	—
209	C₄×C₈	—	178	1	no	—
210	C₄×C₈	—	178	1	no	—
211	C₄×C₈	—	178	1	no	—
212	C₄×C₈	—	178	1	no	—
213	C₂⁴	—	356	1	no	—
214	C₂×D₄	—	356	1	no	—
215	C₂×D₄	—	356	1	no	—
216	C₂×D₄	—	356	1	no	—
217	C₂×D₄	—	356	2	no	—
218	C₂×D₄	—	356	2	no	—
219	C₂×D₄	—	356	2	no	—
220	C₂×D₄	—	356	2	no	—
221	C₂×D₄	—	356	2	no	—
222	C₂×D₄	—	356	2	no	—
223	C₂×D₄	—	178	2	no	—
224	C₂×D₄	—	178	2	no	—
225	C₂×D₄	—	178	2	no	—
226	C₂²×C₄	—	4	1	no	—
227	C₂²×C₄	—	4	1	no	—
228	C₂²×C₄	—	356	1	no	—
229	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
230	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	89	4	no	—
236	C₂×C₈	—	178	2	no	—
237	C₂×C₈	—	178	2	no	—
238	C₂×C₈	—	178	2	no	—
239	C₂×C₈	—	178	2	no	—
240	C₂×C₈	—	178	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	2	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	356	2	no	—
248	C₄⋊C₄	—	356	2	no	—
249	C₄⋊C₄	—	356	2	no	—
250	C₄²	Q₈×D₈₉	1	1	yes	characteristic
251	C₄²	#13083574	1	1	yes	characteristic
252	C₄²	#13083574	1	1	yes	characteristic
253	C₄²	#13083575	1	1	yes	characteristic
254	C₂³	—	356	2	no	—
255	C₂³	—	712	1	no	—
256	C₂³	—	712	1	no	—
257	C₂³	—	712	1	no	—
258	C₂³	—	712	1	no	—
259	C₂³	—	712	1	no	—
260	C₂³	—	712	1	no	—
261	C₂³	#15668008	1	1	yes	characteristic
262	D₄	—	356	4	no	—
263	D₄	—	356	4	no	—
264	D₄	—	356	4	no	—
265	D₄	—	712	2	no	—
266	D₄	—	712	2	no	—
267	D₄	—	712	2	no	—
268	C₂×C₄	—	8	1	no	—
269	C₂×C₄	—	8	1	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	8	1	no	—
274	C₂×C₄	—	712	1	no	—
275	C₂×C₄	—	712	1	no	—
276	C₂×C₄	—	712	1	no	—
277	C₂×C₄	#15667988	1	1	yes	characteristic
278	C₂×C₄	#15667990	1	1	yes	characteristic
279	C₂×C₄	#15667990	1	1	yes	characteristic
280	C₂×C₄	#15667995	1	1	yes	characteristic
281	C₂×C₄	#15667997	1	1	yes	characteristic
282	C₂×C₄	#15667997	1	1	yes	characteristic
283	C₂×C₄	SD₁₆×D₈₉	1	2	yes	—
284	C₂×C₄	#15668032	1	2	yes	—
285	C₂×C₄	#15668035	1	2	yes	—
286	C₂×C₄	#15668036	1	2	yes	—
287	C₂×C₄	#15668037	1	2	yes	—
288	C₂×C₄	#15668039	1	2	yes	—
289	C₈	—	4	2	no	—
290	C₈	—	356	2	no	—
291	C₈	—	178	4	no	—
292	C₂²	—	1424	1	no	—
293	C₂²	—	1424	1	no	—
294	C₂²	—	1424	1	no	—
295	C₂²	—	1424	1	no	—
296	C₂²	—	712	2	no	—
297	C₂²	—	712	2	no	—
298	C₂²	—	712	2	no	—
299	C₂²	#19151533	1	1	yes	characteristic
300	C₂²	#19151969	1	1	yes	characteristic
301	C₂²	#19151980	1	1	yes	characteristic
302	C₂²	#19151983	1	1	yes	characteristic
303	C₂²	#19151984	1	1	yes	characteristic
304	C₂²	#19151986	1	1	yes	characteristic
305	C₂²	#19151988	1	1	yes	characteristic
306	C₄	—	16	1	no	—
307	C₄	—	16	1	no	—
308	C₄	—	1424	1	no	—
309	C₄	#19151887	1	2	yes	—
310	C₄	#19151895	1	2	yes	—
311	C₄	#19151901	1	2	yes	—
312	C₄	#19151918	1	2	yes	—
313	C₄	#19151922	1	2	yes	—
314	C₄	#19151925	1	2	yes	—
315	C₂	—	1424	2	no	—
316	C₂	#26019855	1	1	yes	characteristic
317	C₂	#26019882	1	1	yes	characteristic
318	C₂	#26019888	1	1	yes	characteristic
319	C₂	#26019894	1	1	yes	characteristic
320	C₂	#26021770	1	1	yes	characteristic
321	C₂	#26021776	1	1	yes	characteristic
322	C₂	#26021782	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15668008
commutator, cosocle	C₂×C₄²×C₈₉	C₂³
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₈₉×#1834	C₂
socle	C₂³×C₈₉	#127
prime radical	C₂×#15668009	C₂²
hypercenter	#1834	D₈₉
hypocenter	C₈₉	#10677
LCS	G ▷ C₂×C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
4. #26020408	C₂	semidirect
5. #26020379	C₂	semidirect
6. #26018079	C₂	semidirect
7. C₈₉×#1834	C₂	semidirect
167. #1834	D₈₉	semidirect
169. C₈₉	#10677	semidirect

Maximal subgroups

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

subgroup	index
#26021058	2
#26021057	2
#26020408	2
#26020379	2
#26018079	2
C₈₉×#1834	2
#10677	89