Subgroups — #36463379

Order 22784.

842 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033657	C₂	1	1	yes	maximal, characteristic
2	#26019990	C₂	1	2	yes	maximal
3	#26019950	C₂	1	2	yes	maximal
4	#26019949	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1742	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13083558	C₂²	1	1	yes	characteristic
7	C₂³×#10623494	C₂²	1	1	yes	characteristic
8	#19152229	—	2	2	no	—
9	#19152226	—	2	4	no	—
10	#19152224	C₄	1	4	yes	—
11	#19152155	—	2	2	no	—
12	#19152151	—	2	4	no	—
13	C₂×#15667971	C₂²	1	2	yes	—
14	#19151366	—	2	4	no	—
15	#19151353	—	2	2	no	—
16	#19151352	—	2	4	no	—
17	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
18	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
19	#343×C₈₉	—	2	2	no	—
20	C₂×#13083558	—	2	2	no	—
21	C₂×#13083558	—	2	1	no	—
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	Q₈	1	2	yes	complemented
24	C₂²×#10623494	—	2	2	no	—
25	C₂²×#10623494	—	2	1	no	—
26	C₂²×#10623494	—	2	1	no	—
27	C₂²×#10623494	—	2	1	no	—
28	C₂²×#10623494	—	2	2	no	—
29	C₂²×#10623494	Q₈	1	2	yes	complemented
30	#15668012	—	2	4	no	—
31	#15668011	—	2	8	no	—
32	#15668004	—	2	4	no	—
33	#15668003	—	2	8	no	—
34	#15667991	—	2	8	no	—
35	#15667989	—	2	16	no	—
36	C₄×Dic₁₇₈	—	2	8	no	—
37	#15667971	—	2	2	no	—
38	#15667971	—	2	2	no	—
39	#15667971	—	2	2	no	—
40	#15667971	—	2	2	no	—
41	C₂×C₄⋊Dic₈₉	D₄	1	4	yes	—
42	C₂×C₂²⋊Dic₈₉	C₂×C₄	1	4	yes	—
43	C₂×C₄×Dic₈₉	D₄	1	4	yes	—
44	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
45	C₂×M₁₆×C₈₉	—	2	2	no	—
46	C₂×M₁₆×C₈₉	—	2	2	no	—
47	C₂×M₁₆×C₈₉	—	2	2	no	—
48	C₂×M₁₆×C₈₉	—	2	2	no	—
49	C₂×M₁₆×C₈₉	—	2	2	no	—
50	C₂×M₁₆×C₈₉	Q₈	1	4	yes	complemented
51	C₂²×C₈×C₈₉	—	2	2	no	—
52	#117×C₈₉	—	2	2	no	—
53	#117×C₈₉	—	2	2	no	—
54	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₈₉	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₈₉	D₄	1	2	yes	—
57	C₂×Dic₁₇₈	—	2	8	no	—
58	C₂²×Dic₈₉	C₂²⋊C₄	1	8	yes	—
59	#13083558	—	2	4	no	—
60	#13083558	—	4	2	no	—
61	C₂×#10623494	—	2	4	no	—
62	C₂×#10623494	—	2	2	no	—
63	C₂×#10623494	—	2	2	no	—
64	C₂×#10623494	—	4	1	no	—
65	C₂×#10623494	—	4	1	no	—
66	C₂×#10623494	—	4	2	no	—
67	C₂×#10623494	—	4	1	no	—
68	C₂×#10623494	—	4	1	no	—
69	C₄⋊Dic₈₉	—	2	8	no	—
70	C₄⋊Dic₈₉	—	2	4	no	—
71	C₄⋊Dic₈₉	—	2	4	no	—
72	#13083553	—	2	16	no	—
73	C₂²⋊Dic₈₉	—	2	4	no	—
74	C₂²⋊Dic₈₉	—	2	4	no	—
75	C₄×Dic₈₉	—	2	4	no	—
76	C₄×Dic₈₉	—	2	4	no	—
77	C₂⁴×C₈₉	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂²⋊C₄	1	4	yes	—
79	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₄×C₈₉	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
87	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
88	C₂²×C₄×C₈₉	#41	1	2	yes	—
89	M₁₆×C₈₉	—	2	4	no	—
90	M₁₆×C₈₉	—	2	4	no	—
91	M₁₆×C₈₉	—	4	2	no	—
92	M₁₆×C₈₉	—	4	2	no	—
93	C₂×C₈×C₈₉	—	2	4	no	—
94	C₂×C₈×C₈₉	—	4	2	no	—
95	C₂×C₈×C₈₉	—	4	2	no	—
96	C₄⋊C₄×C₈₉	—	2	4	no	—
97	C₂²⋊C₄×C₈₉	—	2	4	no	—
98	C₄²×C₈₉	—	2	2	no	—
99	C₄²×C₈₉	—	2	2	no	—
100	C₂³×C₈₉	—	4	2	no	—
101	C₂³×C₈₉	C₂×C₂²⋊C₄	1	2	yes	—
102	C₂³×C₈₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂³×C₈₉	C₄×Q₈	1	1	yes	characteristic
104	C₂³×C₈₉	C₄×Q₈	1	1	yes	characteristic
105	C₂³×C₈₉	C₄×Q₈	1	2	yes	—
106	C₂×C₄×C₈₉	—	2	4	no	—
107	C₂×C₄×C₈₉	—	4	4	no	—
108	C₂×C₄×C₈₉	#120	1	2	yes	—
109	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
110	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
111	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₈₉	#124	1	2	yes	—
119	C₂×C₄×C₈₉	#124	1	2	yes	—
120	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
121	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
122	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₈₉	C₄⋊Q₈	1	2	yes	—
125	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
126	C₂×Dic₈₉	—	2	8	no	—
127	C₂×Dic₈₉	—	4	8	no	—
128	Dic₁₇₈	—	8	8	no	—
129	C₈×C₈₉	—	4	4	no	—
130	#10623494	—	4	2	no	—
131	#10623494	—	4	2	no	—
132	C₂²×C₈₉	—	2	4	no	—
133	C₂²×C₈₉	—	2	2	no	—
134	C₂²×C₈₉	—	4	1	no	—
135	C₂²×C₈₉	—	4	1	no	—
136	C₂²×C₈₉	—	4	2	no	—
137	C₂²×C₈₉	#379	1	1	yes	characteristic
138	C₂²×C₈₉	#391	1	2	yes	—
139	C₂²×C₈₉	#391	1	2	yes	—
140	C₂²×C₈₉	#391	1	1	yes	characteristic
141	C₂²×C₈₉	#391	1	1	yes	characteristic
142	C₄×C₈₉	—	4	4	no	—
143	C₄×C₈₉	#393	1	2	yes	—
144	C₄×C₈₉	#393	1	1	yes	characteristic
145	C₄×C₈₉	#393	1	1	yes	characteristic
146	C₄×C₈₉	#395	1	2	yes	—
147	C₄×C₈₉	#395	1	1	yes	characteristic
148	C₄×C₈₉	#395	1	1	yes	characteristic
149	Dic₈₉	—	4	8	no	—
150	#11391	—	89	1	no	maximal, Sylow
151	C₂×C₈₉	—	2	2	no	—
152	C₂×C₈₉	—	2	2	no	—
153	C₂×C₈₉	—	2	1	no	—
154	C₂×C₈₉	—	2	1	no	—
155	C₂×C₈₉	#1446	1	1	yes	characteristic
156	C₂×C₈₉	#1791	1	2	yes	—
157	#3458	—	89	1	no	—
158	#1742	—	89	2	no	—
159	#1742	D₈₉	1	1	yes	characteristic
160	#1729	—	89	2	no	—
161	#1729	—	89	1	no	—
162	C₈₉	#11391	1	1	yes	characteristic, complemented, Sylow
163	C₂²×M₁₆	—	89	1	no	—
164	C₂²×M₁₆	C₂×D₈₉	1	2	yes	—
165	C₂³×C₈	—	89	1	no	—
166	#544	—	178	4	no	—
167	#544	—	178	2	no	—
168	#543	—	89	4	no	—
169	#533	—	178	4	no	—
170	#533	—	178	2	no	—
171	C₂×#117	—	89	2	no	—
172	C₂×#117	C₂×D₈₉	1	1	yes	characteristic
173	#343	—	2	2	no	—
174	#343	—	178	4	no	—
175	#336	—	178	4	no	—
176	#336	—	178	2	no	—
177	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	89	2	no	—
184	C₂×M₁₆	—	178	2	no	—
185	C₂×M₁₆	—	178	2	no	—
186	C₂×M₁₆	—	178	1	no	—
187	C₂×M₁₆	Dic₁₇₈	1	4	yes	complemented
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	89	2	no	—
190	C₂²×C₈	—	178	2	no	—
191	C₂²×C₈	—	178	1	no	—
192	C₂²×C₈	—	178	1	no	—
193	C₂²×C₈	—	178	1	no	—
194	C₂²×C₈	—	178	2	no	—
195	C₄⋊Q₈	—	178	4	no	—
196	C₄⋊Q₈	—	178	8	no	—
197	#125	—	178	4	no	—
198	#125	—	178	8	no	—
199	C₂²⋊Q₈	—	178	16	no	—
200	C₂²⋊Q₈	—	178	8	no	—
201	C₄×Q₈	—	178	8	no	—
202	#117	—	2	2	no	—
203	#117	—	2	2	no	—
204	#117	—	178	2	no	—
205	#117	—	178	2	no	—
206	#117	—	178	2	no	—
207	#117	—	178	2	no	—
208	C₂×C₄⋊C₄	—	89	4	no	—
209	C₂×C₄⋊C₄	C₂²⋊D₈₉	1	2	yes	—
210	C₂×C₂²⋊C₄	—	89	4	no	—
211	C₂×C₂²⋊C₄	C₄×D₈₉	1	2	yes	—
212	C₂×C₄²	—	89	4	no	—
213	C₂×C₄²	D₃₅₆	1	2	yes	—
214	C₂⁴	C₂×C₄×D₈₉	1	1	yes	characteristic
215	C₂×Q₈	—	178	8	no	—
216	C₂²×C₄	—	89	8	no	—
217	C₂²×C₄	#13083551	1	4	yes	—
218	C₂²×C₄	D₄×D₈₉	1	2	yes	—
219	C₂²×C₄	D₄×D₈₉	1	2	yes	—
220	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
222	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
223	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
224	C₂²×C₄	C₂×Dic₁₇₈	1	2	yes	—
225	C₂²×C₄	#13083573	1	1	yes	characteristic
226	C₂²×C₄	#13083573	1	1	yes	characteristic
227	C₂²×C₄	#13083574	1	2	yes	—
228	M₁₆	—	2	4	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	4	2	no	—
231	M₁₆	—	4	2	no	—
232	M₁₆	—	356	2	no	—
233	M₁₆	—	178	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	2	no	—
237	C₂×C₈	—	356	1	no	—
238	C₂×C₈	—	356	1	no	—
239	C₂×C₈	—	356	2	no	—
240	C₂×C₈	—	356	1	no	—
241	C₂×C₈	—	356	1	no	—
242	C₂×C₈	—	178	4	no	—
243	C₂×C₈	—	178	2	no	—
244	C₂×C₈	—	178	2	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	178	16	no	—
247	C₄⋊C₄	—	178	8	no	—
248	C₄⋊C₄	—	178	4	no	—
249	C₄⋊C₄	—	178	4	no	—
250	C₂²⋊C₄	—	2	4	no	—
251	C₂²⋊C₄	—	178	4	no	—
252	C₂²⋊C₄	—	178	4	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	178	4	no	—
256	C₄²	—	178	4	no	—
257	C₂³	—	4	2	no	—
258	C₂³	C₂²⋊C₄×D₈₉	1	2	yes	—
259	C₂³	C₂×#13083551	1	1	yes	characteristic
260	C₂³	#15667977	1	1	yes	characteristic
261	C₂³	#15667977	1	1	yes	characteristic
262	C₂³	C₄×Dic₁₇₈	1	2	yes	—
263	Q₈	—	712	8	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	356	8	no	—
267	C₂×C₄	—	178	8	no	—
268	C₂×C₄	#15667980	1	2	yes	—
269	C₂×C₄	#15667982	1	1	yes	characteristic
270	C₂×C₄	#15667982	1	1	yes	characteristic
271	C₂×C₄	#15667989	1	2	yes	—
272	C₂×C₄	#15667989	1	2	yes	—
273	C₂×C₄	#15667990	1	2	yes	—
274	C₂×C₄	#15667990	1	2	yes	—
275	C₂×C₄	#15667991	1	2	yes	—
276	C₂×C₄	#15667992	1	1	yes	characteristic
277	C₂×C₄	#15667992	1	1	yes	characteristic
278	C₂×C₄	#15667998	1	2	yes	—
279	C₂×C₄	#15667998	1	2	yes	—
280	C₂×C₄	#15667999	1	1	yes	characteristic
281	C₂×C₄	#15668000	1	1	yes	characteristic
282	C₂×C₄	#15668010	1	1	yes	characteristic
283	C₂×C₄	#15668011	1	2	yes	—
284	C₂×C₄	#15668011	1	2	yes	—
285	C₂×C₄	#15668012	1	1	yes	characteristic
286	C₈	—	4	4	no	—
287	C₈	—	356	2	no	—
288	C₈	—	356	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	4	1	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	2	no	—
294	C₂²	#19151453	1	1	yes	characteristic
295	C₂²	#19151502	1	2	yes	—
296	C₂²	#19151502	1	2	yes	—
297	C₂²	#19151503	1	1	yes	characteristic
298	C₂²	#19151504	1	1	yes	characteristic
299	C₄	—	4	4	no	—
300	C₄	—	356	8	no	—
301	C₄	#19151510	1	2	yes	—
302	C₄	#19151511	1	1	yes	characteristic
303	C₄	#19151511	1	1	yes	characteristic
304	C₄	#19151518	1	2	yes	—
305	C₄	#19151519	1	1	yes	characteristic
306	C₄	#19151519	1	1	yes	characteristic
307	C₂	—	2	2	no	—
308	C₂	—	2	2	no	—
309	C₂	—	2	1	no	—
310	C₂	—	2	1	no	—
311	C₂	#26018522	1	1	yes	characteristic
312	C₂	#26020201	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#15667982
commutator, prime radical	C₂⁴×C₈₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₈₉
Fitting	C₈₉×#1742	C₂
socle	C₂²×C₈₉	#379
cosocle	C₂³×C₄×C₈₉	C₂³
hypercenter	#1742	D₈₉
hypocenter	C₈₉	#11391
LCS	G ▷ C₂⁴×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1742

Complements

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

N	H ≅ G/N	type
23. C₂×#13083558	Q₈	semidirect
29. C₂²×#10623494	Q₈	semidirect
50. C₂×M₁₆×C₈₉	Q₈	semidirect
162. C₈₉	#11391	semidirect
187. C₂×M₁₆	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26033657	2
#26019990	2
#26019950	2
#26019949	2
C₈₉×#1742	2
#11391	89