Subgroups — #36448223

Order 22784.

406 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26020412	C₂	1	1	yes	maximal, characteristic
2	#26020409	C₂	1	1	yes	maximal, characteristic
3	#26020380	C₂	1	1	yes	maximal, characteristic
4	#26020380	C₂	1	1	yes	maximal, characteristic
5	#26020380	C₂	1	1	yes	maximal, characteristic
6	#26020380	C₂	1	1	yes	maximal, characteristic
7	C₂×#322×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#15667946	—	2	2	no	—
9	C₂×#15667944	—	2	2	no	—
10	#19151468	C₂²	1	1	yes	characteristic
11	#19151468	C₂²	1	1	yes	characteristic
12	#19151468	C₂²	1	1	yes	characteristic
13	#19151468	C₂²	1	1	yes	characteristic
14	#19151361	—	2	2	no	—
15	#19151361	—	2	2	no	—
16	#19151361	—	2	2	no	—
17	#19151361	—	2	2	no	—
18	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
21	#322×C₈₉	C₄	1	4	yes	complemented
22	C₂×C₄⋊Dic₈₉	—	2	1	no	—
23	C₂×C₄⋊Dic₈₉	—	2	1	no	—
24	C₂×C₄⋊Dic₈₉	—	2	1	no	—
25	C₂×C₄⋊Dic₈₉	—	2	1	no	—
26	C₂×C₄⋊Dic₈₉	—	2	1	no	—
27	C₂×C₄⋊Dic₈₉	—	2	1	no	—
28	C₂×C₄⋊Dic₈₉	—	2	1	no	—
29	C₂×C₄⋊Dic₈₉	—	2	1	no	—
30	C₂×C₄⋊Dic₈₉	—	2	1	no	—
31	C₂×C₄⋊Dic₈₉	—	2	1	no	—
32	C₂×C₄⋊Dic₈₉	—	2	1	no	—
33	C₂×C₄⋊Dic₈₉	—	2	1	no	—
34	#15667946	—	4	4	no	—
35	#15667944	—	4	4	no	—
36	C₂²×C₈×C₈₉	D₄	1	2	yes	—
37	C₂²×C₈×C₈₉	D₄	1	2	yes	—
38	C₂²×C₈×C₈₉	D₄	1	2	yes	—
39	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
42	C₄×C₈×C₈₉	C₂×C₄	1	2	yes	—
43	C₄×C₈×C₈₉	Q₈	1	2	yes	—
44	C₄×C₈×C₈₉	Q₈	1	2	yes	—
45	C₄×C₈×C₈₉	Q₈	1	2	yes	—
46	C₂²×Dic₈₉	—	4	1	no	—
47	C₂²×Dic₈₉	—	4	1	no	—
48	C₂²×Dic₈₉	—	4	1	no	—
49	C₂²×Dic₈₉	—	4	1	no	—
50	C₄⋊Dic₈₉	—	4	2	no	—
51	C₄⋊Dic₈₉	—	4	2	no	—
52	C₄⋊Dic₈₉	—	4	2	no	—
53	C₄⋊Dic₈₉	—	4	2	no	—
54	C₄⋊Dic₈₉	—	4	2	no	—
55	C₄⋊Dic₈₉	—	4	2	no	—
56	C₄⋊Dic₈₉	—	4	2	no	—
57	C₄⋊Dic₈₉	—	4	2	no	—
58	C₄⋊Dic₈₉	—	4	2	no	—
59	C₄⋊Dic₈₉	—	4	2	no	—
60	C₄⋊Dic₈₉	—	4	2	no	—
61	C₄⋊Dic₈₉	—	4	2	no	—
62	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₈₉	—	2	2	no	—
66	C₂×C₈×C₈₉	—	2	2	no	—
67	C₂×C₈×C₈₉	—	2	2	no	—
68	C₂×C₈×C₈₉	—	2	2	no	—
69	C₂×C₈×C₈₉	—	2	2	no	—
70	C₂×C₈×C₈₉	—	2	2	no	—
71	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
72	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
73	C₂×C₈×C₈₉	C₄⋊C₄	1	4	yes	—
74	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
76	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
77	C₄²×C₈₉	C₂×Q₈	1	1	yes	characteristic
78	C₂³×C₈₉	#127	1	1	yes	characteristic
79	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₈₉	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
83	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
84	C₂×C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
85	C₂×C₄×C₈₉	#135	1	2	yes	—
86	C₂×C₄×C₈₉	#135	1	2	yes	—
87	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
88	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
89	C₂×C₄×C₈₉	#137	1	2	yes	—
90	C₂×C₄×C₈₉	#137	1	2	yes	—
91	C₂×Dic₈₉	—	8	1	no	—
92	C₂×Dic₈₉	—	8	1	no	—
93	C₂×Dic₈₉	—	8	1	no	—
94	C₂×Dic₈₉	—	8	1	no	—
95	C₂×Dic₈₉	—	8	1	no	—
96	C₂×Dic₈₉	—	8	1	no	—
97	C₂×Dic₈₉	—	8	1	no	—
98	C₂×Dic₈₉	—	8	1	no	—
99	C₂×Dic₈₉	—	8	1	no	—
100	C₂×Dic₈₉	—	8	1	no	—
101	C₂×Dic₈₉	—	8	1	no	—
102	C₂×Dic₈₉	—	8	1	no	—
103	C₈×C₈₉	—	2	4	no	—
104	C₈×C₈₉	—	2	4	no	—
105	C₈×C₈₉	—	2	4	no	—
106	C₂²×C₈₉	#384	1	1	yes	characteristic
107	C₂²×C₈₉	#495	1	1	yes	characteristic
108	C₂²×C₈₉	#495	1	1	yes	characteristic
109	C₂²×C₈₉	#496	1	1	yes	characteristic
110	C₂²×C₈₉	#496	1	1	yes	characteristic
111	C₂²×C₈₉	#497	1	1	yes	characteristic
112	C₂²×C₈₉	#497	1	1	yes	characteristic
113	C₄×C₈₉	#427	1	2	yes	—
114	C₄×C₈₉	#428	1	2	yes	—
115	C₄×C₈₉	#428	1	2	yes	—
116	C₄×C₈₉	#499	1	2	yes	—
117	C₄×C₈₉	#501	1	2	yes	—
118	C₄×C₈₉	#501	1	2	yes	—
119	Dic₈₉	—	16	1	no	—
120	Dic₈₉	—	16	1	no	—
121	Dic₈₉	—	16	1	no	—
122	Dic₈₉	—	16	1	no	—
123	#8821	—	89	1	no	maximal, Sylow
124	C₂×C₈₉	#1717	1	1	yes	characteristic
125	C₂×C₈₉	#1718	1	1	yes	characteristic
126	C₂×C₈₉	#1719	1	1	yes	characteristic
127	C₂×C₈₉	#1720	1	1	yes	characteristic
128	C₂×C₈₉	#1846	1	1	yes	characteristic
129	C₂×C₈₉	#1847	1	1	yes	characteristic
130	C₂×C₈₉	#1847	1	1	yes	characteristic
131	#1845	—	89	1	no	—
132	#1844	—	89	1	no	—
133	#1834	—	89	1	no	—
134	#1834	—	89	1	no	—
135	#1834	—	89	1	no	—
136	#1834	—	89	1	no	—
137	C₂×#322	D₈₉	1	1	yes	characteristic
138	C₈₉	#8821	1	1	yes	characteristic, complemented, Sylow
139	C₂×#107	—	178	2	no	—
140	C₂×#106	—	178	2	no	—
141	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
144	#384	—	89	1	no	—
145	#384	—	89	1	no	—
146	#384	—	89	1	no	—
147	#384	—	89	1	no	—
148	#340	—	178	2	no	—
149	#340	—	178	2	no	—
150	#340	—	178	2	no	—
151	#340	—	178	2	no	—
152	#322	Dic₈₉	1	4	yes	complemented
153	C₂²×C₈	D₃₅₆	1	2	yes	—
154	C₂²×C₈	D₃₅₆	1	2	yes	—
155	C₂²×C₈	D₃₅₆	1	2	yes	—
156	C₂×C₄⋊C₄	—	178	1	no	—
157	C₂×C₄⋊C₄	—	178	1	no	—
158	C₂×C₄⋊C₄	—	178	1	no	—
159	C₂×C₄⋊C₄	—	178	1	no	—
160	C₂×C₄⋊C₄	—	178	1	no	—
161	C₂×C₄⋊C₄	—	178	1	no	—
162	C₂×C₄⋊C₄	—	178	1	no	—
163	C₂×C₄⋊C₄	—	178	1	no	—
164	C₂×C₄⋊C₄	—	178	1	no	—
165	C₂×C₄⋊C₄	—	178	1	no	—
166	C₂×C₄⋊C₄	—	178	1	no	—
167	C₂×C₄⋊C₄	—	178	1	no	—
168	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
169	#107	—	356	4	no	—
170	#106	—	356	4	no	—
171	C₄×C₈	Dic₁₇₈	1	2	yes	—
172	C₄×C₈	Dic₁₇₈	1	2	yes	—
173	C₄×C₈	Dic₁₇₈	1	2	yes	—
174	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
175	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
176	C₄×C₈	C₂×Dic₈₉	1	2	yes	—
177	C₂²×C₄	—	356	1	no	—
178	C₂²×C₄	—	356	1	no	—
179	C₂²×C₄	—	356	1	no	—
180	C₂²×C₄	—	356	1	no	—
181	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
183	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	C₄⋊Dic₈₉	1	4	yes	—
191	C₂×C₈	C₄⋊Dic₈₉	1	4	yes	—
192	C₂×C₈	C₄⋊Dic₈₉	1	4	yes	—
193	C₄⋊C₄	—	356	2	no	—
194	C₄⋊C₄	—	356	2	no	—
195	C₄⋊C₄	—	356	2	no	—
196	C₄⋊C₄	—	356	2	no	—
197	C₄⋊C₄	—	356	2	no	—
198	C₄⋊C₄	—	356	2	no	—
199	C₄⋊C₄	—	356	2	no	—
200	C₄⋊C₄	—	356	2	no	—
201	C₄⋊C₄	—	356	2	no	—
202	C₄⋊C₄	—	356	2	no	—
203	C₄⋊C₄	—	356	2	no	—
204	C₄⋊C₄	—	356	2	no	—
205	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
206	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
207	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
208	C₄²	C₂×Dic₁₇₈	1	1	yes	characteristic
209	C₂³	#15668009	1	1	yes	characteristic
210	C₂×C₄	—	712	1	no	—
211	C₂×C₄	—	712	1	no	—
212	C₂×C₄	—	712	1	no	—
213	C₂×C₄	—	712	1	no	—
214	C₂×C₄	—	712	1	no	—
215	C₂×C₄	—	712	1	no	—
216	C₂×C₄	—	712	1	no	—
217	C₂×C₄	—	712	1	no	—
218	C₂×C₄	—	712	1	no	—
219	C₂×C₄	—	712	1	no	—
220	C₂×C₄	—	712	1	no	—
221	C₂×C₄	—	712	1	no	—
222	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
224	C₂×C₄	C₂×C₄⋊Dic₈₉	1	1	yes	characteristic
225	C₂×C₄	#15668012	1	1	yes	characteristic
226	C₂×C₄	#15668012	1	1	yes	characteristic
227	C₂×C₄	#15668012	1	1	yes	characteristic
228	C₂×C₄	#15668034	1	2	yes	—
229	C₂×C₄	#15668034	1	2	yes	—
230	C₂×C₄	#15668040	1	2	yes	—
231	C₂×C₄	#15668040	1	2	yes	—
232	C₂×C₄	#15668045	1	2	yes	—
233	C₂×C₄	#15668045	1	2	yes	—
234	C₈	—	2	4	no	—
235	C₈	—	2	4	no	—
236	C₈	—	2	4	no	—
237	C₂²	#19151468	1	1	yes	characteristic
238	C₂²	#19151982	1	1	yes	characteristic
239	C₂²	#19151982	1	1	yes	characteristic
240	C₂²	#19151987	1	1	yes	characteristic
241	C₂²	#19151987	1	1	yes	characteristic
242	C₂²	#19151992	1	1	yes	characteristic
243	C₂²	#19151992	1	1	yes	characteristic
244	C₄	—	1424	1	no	—
245	C₄	—	1424	1	no	—
246	C₄	—	1424	1	no	—
247	C₄	—	1424	1	no	—
248	C₄	#19151636	1	2	yes	—
249	C₄	#19151640	1	2	yes	—
250	C₄	#19151640	1	2	yes	—
251	C₄	#19152000	1	2	yes	—
252	C₄	#19152007	1	2	yes	—
253	C₄	#19152007	1	2	yes	—
254	C₂	#26019891	1	1	yes	characteristic
255	C₂	#26019898	1	1	yes	characteristic
256	C₂	#26019905	1	1	yes	characteristic
257	C₂	#26019912	1	1	yes	characteristic
258	C₂	#26020416	1	1	yes	characteristic
259	C₂	#26020420	1	1	yes	characteristic
260	C₂	#26020420	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#15668009
commutator	C₄²×C₈₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₉
Fitting	C₂×#322×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	#127
cosocle	C₂×C₄²×C₈₉	C₂³
hypercenter	C₂×#322	D₈₉
hypocenter	C₈₉	#8821
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
21. #322×C₈₉	C₄	semidirect
138. C₈₉	#8821	semidirect
152. #322	Dic₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26020412	2
#26020409	2
#26020380	2
C₂×#322×C₈₉	2
#8821	89