Finite Group Databasebeta

← #22599393

Subgroups — #22599393

Order 9088.

420 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986605C211yesmaximal, characteristic, complemented
2#17986567C211yesmaximal, characteristic, complemented
3#17986389C211yesmaximal, characteristic, complemented
4#17986376C211yesmaximal, characteristic, complemented
5#17986365C211yesmaximal, characteristic, complemented
6#17986217C212yesmaximal, complemented
7#17986193C212yesmaximal, complemented
8#17986147C212yesmaximal, complemented
9#17986134C212yesmaximal, complemented
10#17986058C211yesmaximal, characteristic, complemented
11#488×C71C211yesmaximal, characteristic, complemented
12C2×#11765866C2211yescharacteristic
13C2×D4×D71C2211yescharacteristic
14C2×#11765852C2212yescomplemented
15C2×D568C2212yes—
16C2×D568C2212yescomplemented
17C2×#11765850C2212yes—
18#14556125C2211yescharacteristic
19#14556125C2211yescharacteristic, complemented
20#14556124C2211yescharacteristic
21#14556117C2211yescharacteristic
22#14556113—22no—
23#14556103—22no—
24#14556103C2212yes—
25#14556101—22no—
26#14556098—22no—
27#14556090C2212yescomplemented
28#14556085C2211yescharacteristic
29Dic71⋊C8C2212yescomplemented
30#14556048C2212yes—
31#14556048C2212yescomplemented
32(C2×D71)⋊C8C2212yes—
33#14556039C2211yescharacteristic, complemented
34#125×C71C2211yescharacteristic, complemented
35#124×C71C2211yescharacteristic
36#103×C71C2212yescomplemented
37#102×C71C2212yes—
38C4×C8×C71C2211yescharacteristic, complemented
39C23×D71—21no—
40#11765866—42no—
41C2×D284—21no—
42C2×D284—22no—
43C2×D284—22no—
44C2×D284—22no—
45C2×D284C2311yescharacteristic
46C2×D284C2311yescharacteristic
47C2×D284C2312yes—
48C2×C22⋊D71—21no—
49C2×C22⋊D71—22no—
50D4×D71—42no—
51C2×C4×D71—21no—
52C2×C4×D71—22no—
53C2×C4×D71C2311yescharacteristic
54#11765852—42no—
55D568—42no—
56D568—42no—
57#11765850—42no—
58C2×C71⋊C8C2312yes—
59Dic71⋊C4D412yes—
60(C2×D71)⋊C4—22no—
61(C2×D71)⋊C4—22no—
62(C2×D71)⋊C4—22no—
63(C2×D71)⋊C4D412yes—
64C4×Dic71C2311yescharacteristic
65C2×Q8×C71C2311yescharacteristic
66C2×D4×C71C2311yescharacteristic
67C2×C8×C71C2312yes—
68C4⋊C4×C71—22no—
69C4⋊C4×C71C2312yes—
70C22⋊C4×C71—22no—
71C42×C71C2311yescharacteristic
72C23×C71—21no—
73C22×D71—22no—
74C22×D71—21no—
75C22×D71—21no—
76C22×D71—42no—
77C22×D71—82no—
78C22×D71C2×D411yescharacteristic
79Q8×C71—22no—
80D4×C71—22no—
81C2×C4×C71—21no—
82C2×C4×C71—22no—
83C2×C4×C71C2×D412yes—
84C2×C4×C71C2411yescharacteristic
85C22⋊D71—42no—
86C22⋊D71—82no—
87C2×Dic71C2×D412yes—
88C2×Dic71C2×D411yescharacteristic
89D284—22no—
90D284—22no—
91D284—22no—
92D284—22no—
93D284—42no—
94D284—82no—
95D284—82no—
96D284—82no—
97C4×D71—22no—
98C4×D71—42no—
99C4×D71—82no—
100C8×C71—22no—
101C71⋊C8—22no—
102C22×C71—41no—
103C22×C71—42no—
104C22×C71C22×D411yescharacteristic
105C2×D71—21no—
106C2×D71—22no—
107C2×D71—41no—
108C2×D71—41no—
109C2×D71—42no—
110C2×D71—42no—
111C2×D71—42no—
112C2×D71—42no—
113C2×D71—42no—
114C2×D71—84no—
115C4×C71—22no—
116C4×C71—41no—
117C4×C71—42no—
118C4×C71H(5,𝔽2)12yes—
119Dic71—21no—
120Dic71—22no—
121C2×C71—81no—
122C2×C71#53511yescharacteristic
123C2×C71#57612yes—
124D71—41no—
125D71—81no—
126D71—82no—
127D71—81no—
128#3220—711nomaximal, Sylow
129C71#322011yescharacteristic, complemented, Sylow
130#540—711no—
131#535—711no—
132#496—711no—
133#493—711no—
134#490—711no—
135#488D7111yescharacteristic, complemented
136#463—712no—
137#459—712no—
138#449—712no—
139#447—712no—
140#433—711no—
141C2×#41—711no—
142C22×D4—711no—
143C2×SD16—712no—
144C2×D8—712no—
145C2×D8—712no—
146C2×D8—712no—
147#127—711no—
148#127—711no—
149#127—711no—
150#125C2×D7111yescharacteristic, complemented
151#124—711no—
152#124C2×D7111yescharacteristic
153#123—1422no—
154#121—712no—
155#121—1422no—
156#121—1422no—
157#120—1422no—
158C4×D4—712no—
159#117—711no—
160C4⋊C8—712no—
161#103C2×D7112yescomplemented
162#102—712no—
163#102—712no—
164#102C2×D7112yes—
165C22⋊C8—712no—
166#97—711no—
167C4×C8C2×D7111yescharacteristic, complemented
168C24—1421no—
169#41—2842no—
170C2×Q8C22×D7111yescharacteristic
171C2×D4—2842no—
172C2×D4—711no—
173C2×D4—711no—
174C2×D4—712no—
175C2×D4—1421no—
176C2×D4—1422no—
177C2×D4—1421no—
178C2×D4—1422no—
179C2×D4—1422no—
180C2×D4—1422no—
181C2×D4C22×D7111yescharacteristic
182C22×C4—711no—
183C22×C4—1421no—
184C22×C4—1422no—
185SD16—2842no—
186D8—2842no—
187D8—2842no—
188D8—2842no—
189C2×C8—712no—
190C2×C8C22×D7112yes—
191C4⋊C4—22no—
192C4⋊C4—712no—
193C4⋊C4C22×D7112yes—
194C22⋊C4—22no—
195C22⋊C4—712no—
196C22⋊C4—1422no—
197C22⋊C4—1422no—
198C22⋊C4—1422no—
199C42—711no—
200C42C22×D7111yescharacteristic
201C23—21no—
202C23—2842no—
203C23—5682no—
204C23—711no—
205C23—1422no—
206C23—1421no—
207C23—1421no—
208Q8—22no—
209D4—22no—
210D4—2842no—
211D4—2842no—
212D4—5682no—
213D4—5682no—
214D4—5682no—
215D4—5682no—
216D4—1422no—
217D4—1422no—
218D4—1422no—
219D4—1422no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—2842no—
223C2×C4—5682no—
224C2×C4—712no—
225C2×C4—711no—
226C2×C4—1422no—
227C2×C4D4×D7112yes—
228C2×C4C23×D7111yescharacteristic
229C8—22no—
230C8—1422no—
231C22—41no—
232C22—42no—
233C22—2841no—
234C22—2841no—
235C22—2842no—
236C22—2842no—
237C22—2842no—
238C22—2842no—
239C22—2842no—
240C22—5684no—
241C22—1421no—
242C22—1422no—
243C22C2×D4×D7111yescharacteristic
244C4—22no—
245C4—41no—
246C4—42no—
247C4—1421no—
248C4—1422no—
249C4#1455617412yes—
250C2—81no—
251C2—2841no—
252C2—5681no—
253C2—5681no—
254C2—5682no—
255C2#1798656711yescharacteristic
256C2#1798680112yes—
subgroupG/H
centerC22C2×D4×D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#488×C71C2
socleC22×C71C22×D4
hypercenter#488D71
hypocenterC71#3220
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C42 ◁ #488

Complements

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

NH ≅ G/Ntype
1. #17986605C2semidirect
2. #17986567C2semidirect
3. #17986389C2semidirect
4. #17986376C2semidirect
5. #17986365C2semidirect
6. #17986217C2semidirect
7. #17986193C2semidirect
8. #17986147C2semidirect
9. #17986134C2semidirect
10. #17986058C2semidirect
11. #488×C71C2semidirect
14. C2×#11765852C22semidirect
16. C2×D568C22semidirect
19. #14556125C22semidirect
27. #14556090C22semidirect
29. Dic71⋊C8C22semidirect
31. #14556048C22semidirect
33. #14556039C22semidirect
34. #125×C71C22semidirect
36. #103×C71C22semidirect
38. C4×C8×C71C22semidirect
129. C71#3220semidirect
135. #488D71semidirect
150. #125C2×D71semidirect
161. #103C2×D71semidirect
167. C4×C8C2×D71semidirect