Finite Group Databasebeta

← #22598583

Subgroups — #22598583

Order 9088.

418 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 125 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#17986604C211yesmaximal, characteristic, complemented
2#17986323C212yesmaximal, complemented
3#17986299C212yesmaximal, complemented
4#17986194C212yesmaximal, complemented
5#17986133C212yesmaximal, complemented
6#17986109C212yesmaximal, complemented
7C4×#11765850C212yesmaximal, complemented
8#17986060C211yesmaximal, characteristic, complemented
9#518×C71C211yesmaximal, characteristic, complemented
10C2×#11765850C2212yes—
11C2×#11765850C2212yes—
12#14556125C2211yescharacteristic, complemented
13#14556120C2211yescharacteristic, complemented
14#14556113—22no—
15#14556102C2212yes—
16#14556101—22no—
17C4×D284C2212yescomplemented
18#14556061C2212yescomplemented
19#14556059C2212yescomplemented
20C4⋊(C71⋊C8)C2212yescomplemented
21#14556049C2212yes—
22#14556049C2212yes—
23#14556048C2212yescomplemented
24#14556048C2212yescomplemented
25C22⋊(C71⋊C8)C2212yes—
26#14556039C2211yescharacteristic, complemented
27C4×C71⋊C8C2211yescharacteristic, complemented
28C22×D4×C71C2211yescharacteristic
29C4×D4×C71—22no—
30C4×D4×C71C2212yes—
31C4×D4×C71C2212yes—
32#117×C71C2211yescharacteristic
33#117×C71C2211yescharacteristic
34C2×C22⋊C4×C71—22no—
35C2×D284—22no—
36C2×D284C2312yes—
37C2×C22⋊D71—22no—
38C2×C4×D71—22no—
39#11765850—24no—
40#11765850—24no—
41C2×C71⋊C8C2312yes—
42C2×C71⋊C8C2312yes—
43C4⋊Dic71C2312yes—
44Dic71⋊C4—22no—
45(C2×D71)⋊C4—22no—
46(C2×D71)⋊C4—22no—
47C24×C71—21no—
48C2×D4×C71—24no—
49C2×D4×C71—21no—
50C2×D4×C71C2312yes—
51C22×C4×C71—22no—
52C22×C4×C71—22no—
53C22×C4×C71C2311yescharacteristic
54C4⋊C4×C71D412yes—
55C4⋊C4×C71C2312yes—
56C22⋊C4×C71—21no—
57C22⋊C4×C71—22no—
58C22⋊C4×C71—22no—
59C22⋊C4×C71D412yes—
60C42×C71—21no—
61C42×C71C2311yescharacteristic
62C42×C71C2311yescharacteristic
63C23×C71—24no—
64C23×C71—22no—
65C23×C71—82no—
66C23×C71C2×D411yescharacteristic
67C22×D71—22no—
68D4×C71—24no—
69D4×C71—22no—
70D4×C71#4114yes—
71C2×C4×C71—22no—
72C2×C4×C71—22no—
73C2×C4×C71—82no—
74C2×C4×C71—82no—
75C2×C4×C71C2×D411yescharacteristic
76C2×C4×C71C2×D412yes—
77C2×C4×C71C2×D412yes—
78C2×C4×C71C2411yescharacteristic
79C22⋊D71—82no—
80C2×Dic71—22no—
81D284—22no—
82D284—22no—
83D284—82no—
84C4×D71—82no—
85C71⋊C8—22no—
86C71⋊C8—22no—
87C22×C71—21no—
88C22×C71—24no—
89C22×C71—22no—
90C22×C71—42no—
91C22×C71—42no—
92C22×C71—84no—
93C22×C71C22×D411yescharacteristic
94C2×D71—42no—
95C2×D71—42no—
96C2×D71—42no—
97C4×C71—21no—
98C4×C71—22no—
99C4×C71—22no—
100C4×C71—42no—
101C4×C71C2×#4112yes—
102Dic71—42no—
103C2×C71—41no—
104C2×C71—44no—
105C2×C71#52511yescharacteristic
106C2×C71#57612yes—
107D71—82no—
108#3122—711nomaximal, Sylow
109C71#312211yescharacteristic, complemented, Sylow
110#540—711no—
111#518D7111yescharacteristic, complemented
112#482—712no—
113#478—712no—
114#459—712no—
115#447—712no—
116#442—712no—
117C4×D8—712no—
118#433—711no—
119C22×D4C2×D7111yescharacteristic
120C2×D8—712no—
121C2×D8—712no—
122#127—711no—
123#125—711no—
124#123—1422no—
125#121—712no—
126#121—1422no—
127C4×D4—22no—
128C4×D4—712no—
129C4×D4C2×D7112yes—
130C4×D4C2×D7112yes—
131#117C2×D7111yescharacteristic
132#117C2×D7111yescharacteristic
133C2×C22⋊C4—22no—
134#107—712no—
135#106—712no—
136C4⋊C8—712no—
137#102—712no—
138#102—712no—
139#102—712no—
140#102—712no—
141C22⋊C8—712no—
142#97—711no—
143C4×C8—711no—
144C24—21no—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—712no—
148C2×D4—1422no—
149C2×D4—1422no—
150C2×D4C22×D7112yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—1422no—
154C22×C4C22×D7111yescharacteristic
155D8—1424no—
156D8—1424no—
157C2×C8—712no—
158C2×C8—712no—
159C4⋊C4—712no—
160C4⋊C4—1422no—
161C4⋊C4C22⋊D7112yes—
162C4⋊C4C22×D7112yes—
163C22⋊C4—21no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—1422no—
167C22⋊C4—1422no—
168C22⋊C4C22⋊D7112yes—
169C42—21no—
170C42C22×D7111yescharacteristic
171C42C22×D7111yescharacteristic
172C23—24no—
173C23—22no—
174C23—82no—
175C23—1422no—
176C23C2×C22⋊D7111yescharacteristic
177D4—24no—
178D4—22no—
179D4—5682no—
180D4—5682no—
181D4—1422no—
182D4—1422no—
183D4#1176586514yes—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—5682no—
189C2×C4—1422no—
190C2×C4C2×C22⋊D7111yescharacteristic
191C2×C4C2×C22⋊D7112yes—
192C2×C4C2×C22⋊D7112yes—
193C2×C4C23×D7111yescharacteristic
194C8—1422no—
195C8—1422no—
196C22—21no—
197C22—24no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22—84no—
202C22—2842no—
203C22—2842no—
204C22—2842no—
205C22C22×C22⋊D7111yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—2842no—
211C4C2×#1176586512yes—
212C2—41no—
213C2—44no—
214C2—5682no—
215C2#1798651011yescharacteristic
216C2#1798680212yes—
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#518×C71C2
socleC22×C71C22×D4
hypercenter#518D71
hypocenterC71#3122
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #17986604C2semidirect
2. #17986323C2semidirect
3. #17986299C2semidirect
4. #17986194C2semidirect
5. #17986133C2semidirect
6. #17986109C2semidirect
7. C4×#11765850C2semidirect
8. #17986060C2semidirect
9. #518×C71C2semidirect
12. #14556125C22semidirect
13. #14556120C22semidirect
17. C4×D284C22semidirect
18. #14556061C22semidirect
19. #14556059C22semidirect
20. C4⋊(C71⋊C8)C22semidirect
23. #14556048C22semidirect
24. #14556048C22semidirect
26. #14556039C22semidirect
27. C4×C71⋊C8C22semidirect
109. C71#3122semidirect
111. #518D71semidirect