Subgroups — #36584398

Order 22784.

664 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26028815	C₂	1	1	yes	maximal, characteristic
2	#26021826	C₂	1	8	yes	maximal, complemented
3	C₂×#19151369	C₂	1	2	yes	maximal
4	#26019963	C₂	1	2	yes	maximal
5	C₂×#19151355	C₂	1	1	yes	maximal, characteristic
6	C₂×#430×C₈₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#13083558	C₂²	1	1	yes	characteristic
8	C₂×#15667971	C₂²	1	2	yes	—
9	C₂×#15667971	C₂²	1	1	yes	characteristic
10	#19151626	—	2	2	no	—
11	#19151624	—	2	2	no	—
12	#19151621	C₂²	1	4	yes	—
13	#19151369	C₂²	1	8	yes	—
14	#19151359	C₂²	1	8	yes	—
15	#19151355	C₂²	1	4	yes	—
16	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	#430×C₈₉	—	2	2	no	—
19	#430×C₈₉	C₂²	1	4	yes	—
20	C₂×#108×C₈₉	—	2	2	no	—
21	C₂×#13083558	—	2	1	no	—
22	C₂×#13083558	—	2	2	no	—
23	C₂×#13083558	—	2	2	no	—
24	C₂×#13083558	C₂³	1	2	yes	—
25	C₂²×#10623494	—	2	1	no	—
26	#15667971	—	2	1	no	—
27	#15667971	—	2	1	no	—
28	#15667971	—	2	2	no	—
29	#15667971	—	2	2	no	—
30	#15667971	—	2	2	no	—
31	#15667971	Q₈	1	2	yes	—
32	#15667971	Q₈	1	2	yes	—
33	#15667971	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₈₉	—	2	1	no	—
35	C₂×C₄⋊Dic₈₉	—	2	2	no	—
36	C₂×C₂²⋊Dic₈₉	—	2	1	no	—
37	C₂×C₂²⋊Dic₈₉	—	2	2	no	—
38	C₂×C₄×Dic₈₉	—	2	2	no	—
39	C₂×C₄×Dic₈₉	C₂³	1	2	yes	—
40	#15667941	—	2	8	no	—
41	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
42	C₂×M₁₆×C₈₉	—	2	2	no	—
43	C₂×M₁₆×C₈₉	—	2	2	no	—
44	C₂×M₁₆×C₈₉	—	2	2	no	—
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	D₄	1	2	yes	—
50	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
51	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
52	C₂²×C₈×C₈₉	—	2	2	no	—
53	C₂²×C₈×C₈₉	D₄	1	2	yes	—
54	#108×C₈₉	—	2	8	no	—
55	C₂²×Dic₈₉	—	2	2	no	—
56	C₂²×Dic₈₉	—	4	2	no	—
57	#13083558	—	2	4	no	—
58	#13083558	—	4	1	no	—
59	#13083558	—	4	1	no	—
60	C₂×#10623494	—	2	2	no	—
61	C₂×#10623494	—	4	2	no	—
62	C₄⋊Dic₈₉	—	2	2	no	—
63	C₄⋊Dic₈₉	—	2	2	no	—
64	C₄⋊Dic₈₉	—	2	4	no	—
65	C₄⋊Dic₈₉	—	4	2	no	—
66	C₂²⋊Dic₈₉	—	2	2	no	—
67	C₂²⋊Dic₈₉	—	2	2	no	—
68	C₂²⋊Dic₈₉	—	2	4	no	—
69	C₂²⋊Dic₈₉	—	4	2	no	—
70	C₄×Dic₈₉	—	2	2	no	—
71	C₄×Dic₈₉	—	2	4	no	—
72	C₄×Dic₈₉	—	4	2	no	—
73	C₄×Dic₈₉	C₂×Q₈	1	4	yes	—
74	C₂⁴×C₈₉	#41	1	1	yes	characteristic
75	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₈₉	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
80	C₂²×C₄×C₈₉	#41	1	2	yes	—
81	C₂²×C₄×C₈₉	#41	1	2	yes	—
82	C₂²×C₄×C₈₉	#41	1	2	yes	—
83	C₂²×C₄×C₈₉	#41	1	2	yes	—
84	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
85	M₁₆×C₈₉	—	2	4	no	—
86	M₁₆×C₈₉	—	2	4	no	—
87	M₁₆×C₈₉	—	2	4	no	—
88	M₁₆×C₈₉	—	2	4	no	—
89	M₁₆×C₈₉	—	4	2	no	—
90	M₁₆×C₈₉	—	4	2	no	—
91	C₂×C₈×C₈₉	—	2	2	no	—
92	C₂×C₈×C₈₉	—	2	2	no	—
93	C₂×C₈×C₈₉	—	2	4	no	—
94	C₂×C₈×C₈₉	—	4	2	no	—
95	C₂×C₈×C₈₉	—	4	2	no	—
96	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
97	C₂³×C₈₉	—	8	1	no	—
98	C₂³×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂³×C₈₉	#126	1	2	yes	—
101	C₂³×C₈₉	#126	1	2	yes	—
102	C₂³×C₈₉	C₂×#41	1	1	yes	characteristic
103	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₈₉	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
106	C₂×C₄×C₈₉	#123	1	1	yes	characteristic
107	C₂×C₄×C₈₉	#123	1	2	yes	—
108	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
109	C₂×C₄×C₈₉	#124	1	2	yes	—
110	C₂×C₄×C₈₉	#124	1	1	yes	characteristic
111	C₂×C₄×C₈₉	#125	1	2	yes	—
112	C₂×C₄×C₈₉	#125	1	2	yes	—
113	C₂×C₄×C₈₉	#125	1	2	yes	—
114	C₂×C₄×C₈₉	#125	1	2	yes	—
115	C₂×C₄×C₈₉	#126	1	2	yes	—
116	C₂×C₄×C₈₉	#126	1	2	yes	—
117	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
120	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
121	C₂×Dic₈₉	—	2	4	no	—
122	C₂×Dic₈₉	—	4	4	no	—
123	C₂×Dic₈₉	—	8	2	no	—
124	C₂×Dic₈₉	—	8	4	no	—
125	C₈×C₈₉	—	2	4	no	—
126	C₈×C₈₉	—	4	4	no	—
127	#10623494	—	4	2	no	—
128	C₂²×C₈₉	—	4	2	no	—
129	C₂²×C₈₉	—	4	1	no	—
130	C₂²×C₈₉	—	4	1	no	—
131	C₂²×C₈₉	—	4	1	no	—
132	C₂²×C₈₉	—	4	2	no	—
133	C₂²×C₈₉	#400	1	1	yes	characteristic
134	C₂²×C₈₉	#400	1	1	yes	characteristic
135	C₂²×C₈₉	#400	1	2	yes	—
136	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
137	C₂²×C₈₉	C₂×#126	1	2	yes	—
138	C₄×C₈₉	#400	1	1	yes	characteristic
139	C₄×C₈₉	#400	1	1	yes	characteristic
140	C₄×C₈₉	#400	1	2	yes	—
141	C₄×C₈₉	C₂×#123	1	1	yes	characteristic
142	C₄×C₈₉	C₂×#124	1	1	yes	characteristic
143	C₄×C₈₉	C₂×#125	1	2	yes	—
144	Dic₈₉	—	4	4	no	—
145	Dic₈₉	—	8	4	no	—
146	#23751	—	89	1	no	maximal, Sylow
147	C₂×C₈₉	—	2	1	no	—
148	C₂×C₈₉	—	2	2	no	—
149	C₂×C₈₉	—	2	1	no	—
150	C₂×C₈₉	—	2	2	no	—
151	C₂×C₈₉	—	2	1	no	—
152	C₂×C₈₉	C₂×#400	1	1	yes	characteristic
153	C₂×#430	D₈₉	1	1	yes	characteristic
154	#2905	—	89	1	no	—
155	#2102	—	89	8	no	—
156	C₂×#344	—	89	2	no	—
157	#1733	—	89	2	no	—
158	C₂×H(3,ℤ/4)	—	89	1	no	—
159	C₈₉	#23751	1	1	yes	characteristic, complemented, Sylow
160	C₂²×M₁₆	—	89	1	no	—
161	C₂²×M₁₆	C₂×D₈₉	1	2	yes	—
162	C₂²×M₁₆	C₂×D₈₉	1	1	yes	characteristic
163	C₂×#117	—	89	1	no	—
164	C₂×#117	—	89	2	no	—
165	#430	—	2	2	no	—
166	#430	C₂×D₈₉	1	4	yes	—
167	C₂×#108	—	2	2	no	—
168	#424	—	178	2	no	—
169	#424	—	178	2	no	—
170	#423	—	89	4	no	—
171	#344	—	89	8	no	—
172	#339	—	89	8	no	—
173	H(3,ℤ/4)	—	89	4	no	—
174	C₂³×C₄	C₂²×D₈₉	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	2	2	no	—
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	—	89	2	no	—
183	C₂×M₁₆	—	178	2	no	—
184	C₂×M₁₆	—	178	2	no	—
185	C₂×M₁₆	—	178	1	no	—
186	C₂×M₁₆	C₂²⋊D₈₉	1	2	yes	—
187	C₂×M₁₆	C₂²×D₈₉	1	4	yes	—
188	C₂×M₁₆	C₂²×D₈₉	1	2	yes	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	—	178	1	no	—
191	C₂²×C₈	C₂²⋊D₈₉	1	2	yes	—
192	#117	—	89	2	no	—
193	#117	—	89	2	no	—
194	#117	—	89	4	no	—
195	#117	—	178	2	no	—
196	#117	—	178	2	no	—
197	#117	—	178	2	no	—
198	#117	—	178	1	no	—
199	#117	—	178	1	no	—
200	C₂×C₄⋊C₄	—	178	2	no	—
201	C₂×C₄⋊C₄	—	178	1	no	—
202	C₂×C₂²⋊C₄	—	178	2	no	—
203	C₂×C₂²⋊C₄	—	178	1	no	—
204	C₂×C₄²	—	89	2	no	—
205	C₂×C₄²	—	178	2	no	—
206	#108	—	2	8	no	—
207	C₄⋊C₈	—	178	8	no	—
208	C₂⁴	#13083575	1	1	yes	characteristic
209	C₂²×C₄	—	356	2	no	—
210	C₂²×C₄	—	178	2	no	—
211	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
213	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
214	C₂²×C₄	Q₈×D₈₉	1	1	yes	characteristic
215	C₂²×C₄	#13083573	1	2	yes	—
216	C₂²×C₄	#13083573	1	2	yes	—
217	C₂²×C₄	#13083574	1	2	yes	—
218	C₂²×C₄	#13083574	1	2	yes	—
219	C₂²×C₄	#13083575	1	1	yes	characteristic
220	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	356	1	no	—
228	M₁₆	—	356	1	no	—
229	M₁₆	—	178	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	4	2	no	—
234	C₂×C₈	—	4	2	no	—
235	C₂×C₈	—	356	2	no	—
236	C₂×C₈	—	178	2	no	—
237	C₂×C₈	C₂×C₂²⋊D₈₉	1	4	yes	—
238	C₄⋊C₄	—	356	2	no	—
239	C₄⋊C₄	—	178	4	no	—
240	C₄⋊C₄	—	178	2	no	—
241	C₄⋊C₄	—	178	2	no	—
242	C₂²⋊C₄	—	356	2	no	—
243	C₂²⋊C₄	—	178	4	no	—
244	C₂²⋊C₄	—	178	2	no	—
245	C₂²⋊C₄	—	178	2	no	—
246	C₄²	—	89	4	no	—
247	C₄²	—	356	2	no	—
248	C₄²	—	178	4	no	—
249	C₄²	—	178	2	no	—
250	C₂³	—	8	1	no	—
251	C₂³	#15667988	1	1	yes	characteristic
252	C₂³	#15667988	1	1	yes	characteristic
253	C₂³	#15668007	1	2	yes	—
254	C₂³	#15668007	1	2	yes	—
255	C₂³	C₂×#13083575	1	1	yes	characteristic
256	C₂×C₄	—	356	4	no	—
257	C₂×C₄	—	178	4	no	—
258	C₂×C₄	—	712	2	no	—
259	C₂×C₄	—	712	4	no	—
260	C₂×C₄	#15667988	1	1	yes	characteristic
261	C₂×C₄	#15667988	1	1	yes	characteristic
262	C₂×C₄	#15667995	1	1	yes	characteristic
263	C₂×C₄	#15667995	1	1	yes	characteristic
264	C₂×C₄	#15667995	1	2	yes	—
265	C₂×C₄	#15667999	1	1	yes	characteristic
266	C₂×C₄	#15667999	1	2	yes	—
267	C₂×C₄	#15667999	1	1	yes	characteristic
268	C₂×C₄	#15668002	1	2	yes	—
269	C₂×C₄	#15668002	1	2	yes	—
270	C₂×C₄	#15668002	1	2	yes	—
271	C₂×C₄	#15668002	1	2	yes	—
272	C₂×C₄	#15668007	1	2	yes	—
273	C₂×C₄	#15668007	1	2	yes	—
274	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
275	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
276	C₂×C₄	C₂×#13083573	1	2	yes	—
277	C₂×C₄	C₂×#13083574	1	2	yes	—
278	C₈	—	2	4	no	—
279	C₈	—	4	4	no	—
280	C₈	—	356	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	2	no	—
286	C₂²	#19151536	1	1	yes	characteristic
287	C₂²	#19151536	1	1	yes	characteristic
288	C₂²	#19151536	1	2	yes	—
289	C₂²	C₂×#15667988	1	1	yes	characteristic
290	C₂²	C₂×#15668007	1	2	yes	—
291	C₄	—	356	4	no	—
292	C₄	—	712	4	no	—
293	C₄	#19151536	1	1	yes	characteristic
294	C₄	#19151536	1	1	yes	characteristic
295	C₄	#19151536	1	2	yes	—
296	C₄	C₂×#15667995	1	1	yes	characteristic
297	C₄	C₂×#15667999	1	1	yes	characteristic
298	C₄	C₂×#15668002	1	2	yes	—
299	C₂	—	2	1	no	—
300	C₂	—	2	2	no	—
301	C₂	—	2	1	no	—
302	C₂	—	2	2	no	—
303	C₂	—	2	1	no	—
304	C₂	C₂×#19151536	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#15667999
commutator, cosocle	C₂²×C₄×C₈₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×#430×C₈₉	C₂
socle	C₂×C₈₉	C₂×#400
prime radical	C₂⁴×C₈₉	#41
hypercenter	C₂×#430	D₈₉
hypocenter	C₈₉	#23751
LCS	G ▷ C₂²×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#430

Complements

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

N	H ≅ G/N	type
2. #26021826	C₂	semidirect
159. C₈₉	#23751	semidirect

Maximal subgroups

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

subgroup	index
#26028815	2
#26021826	2
C₂×#19151369	2
#26019963	2
C₂×#19151355	2
C₂×#430×C₈₉	2
#23751	89