Finite Group Databasebeta

← #22598808

Subgroups — #22598808

Order 9088.

572 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 149 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
1C2×#14556158C212yesmaximal, complemented
2C2×#14556153C212yesmaximal, complemented
3C2×#14556116C211yesmaximal, characteristic, complemented
4#17986391C212yesmaximal, complemented
5#17986386C212yesmaximal, complemented
6#17986381C214yesmaximal, complemented
7C4×#11765849C211yesmaximal, characteristic, complemented
8#532×C71C211yesmaximal, characteristic, complemented
9C22×Dic142C2211yescharacteristic
10C22×D284C2211yescharacteristic
11#14556158—28no—
12#14556153—28no—
13C2×#11765855C2214yescomplemented
14C2×#11765853C2214yescomplemented
15C2×#11765852C2214yes—
16C2×#11765850C2214yes—
17C2×#11765849C2212yes—
18#14556116—22no—
19#14556116C2212yes—
20#14556116C2212yes—
21C2×(C2×D71)⋊C4—22no—
22#14556039C2212yescomplemented
23C4×C71⋊C8C2212yescomplemented
24C2×#41×C71C2212yes—
25C4⋊Q8×C71C2211yescharacteristic, complemented
26#127×C71C2211yescharacteristic, complemented
27#124×C71C2212yescomplemented
28#121×C71—22no—
29C4×D4×C71—22no—
30C2×C42×C71C2211yescharacteristic
31C23×D71—21no—
32C2×Dic142—24no—
33C2×Dic142—21no—
34C2×Dic142C2312yes—
35C2×D284—24no—
36C2×D284—21no—
37C2×D284C2312yes—
38C22×Dic71—21no—
39#11765855—28no—
40#11765853—28no—
41#11765852—28no—
42#11765850—28no—
43#11765849D418yescomplemented
44C2×C71⋊C8C2314yes—
45(C2×D71)⋊C4—24no—
46(C2×D71)⋊C4—24no—
47#41×C71—24no—
48#41×C71—24no—
49C2×Q8×C71C2312yes—
50C2×D4×C71—22no—
51C2×D4×C71—22no—
52C2×D4×C71C2312yes—
53C22×C4×C71—22no—
54C22×C4×C71D412yes—
55C22×C4×C71C2311yescharacteristic
56C4⋊C4×C71—22no—
57C22⋊C4×C71—24no—
58C42×C71D412yes—
59C42×C71C2311yescharacteristic
60C42×C71C2311yescharacteristic
61C23×C71—22no—
62C23×C71C2×D411yescharacteristic
63C22×D71—22no—
64C22×D71—44no—
65C22×D71—42no—
66Q8×C71—22no—
67Q8×C71—22no—
68D4×C71—22no—
69D4×C71—22no—
70D4×C71—44no—
71D4×C71—44no—
72C2×C4×C71—22no—
73C2×C4×C71—42no—
74C2×C4×C71—44no—
75C2×C4×C71C2×D414yes—
76C2×C4×C71C2×D412yes—
77C2×C4×C71C2×D412yes—
78C2×C4×C71C2×D411yescharacteristic
79C2×C4×C71C2411yescharacteristic
80C2×Dic71—22no—
81C2×Dic71—42no—
82D284—24no—
83D284—42no—
84Dic142—24no—
85Dic142—42no—
86C71⋊C8C2×D418yes—
87C22×C71—22no—
88C22×C71—42no—
89C22×C71—44no—
90C22×C71#12712yes—
91C22×C71C22×D411yescharacteristic
92C2×D71—42no—
93C2×D71—44no—
94C2×D71—84no—
95C4×C71—22no—
96C4×C71—22no—
97C4×C71—42no—
98C4×C71#12712yes—
99C4×C71C22×D412yes—
100Dic71—42no—
101C2×C71—22no—
102C2×C71—82no—
103C2×C71C2×#12711yescharacteristic
104C2×C71#57512yes—
105D71—82no—
106#3153—711nomaximal, Sylow
107C71#315311yescharacteristic, complemented, Sylow
108C2×#137—712no—
109C2×AGL(1,ℤ/8)—712no—
110#532D7111yescharacteristic, complemented
111C2×#124—711no—
112#497—712no—
113#496—712no—
114#495—714no—
115C4×M16—711no—
116C2×#41C2×D7112yes—
117C22×Q8—711no—
118C22×D4—711no—
119#137—1428no—
120AGL(1,ℤ/8)—1428no—
121C2×Q16—714no—
122C2×SD16—714no—
123C2×SD16—714no—
124C2×D8—714no—
125C2×M16—712no—
126C4⋊Q8C2×D7111yescharacteristic, complemented
127#127C2×D7111yescharacteristic, complemented
128#124—712no—
129#124—712no—
130#124—1422no—
131#124C2×D7112yescomplemented
132#121—22no—
133C4×D4—22no—
134C2×C22⋊C4—1422no—
135C2×C42C2×D7111yescharacteristic
136#97—712no—
137C4×C8—712no—
138C24—1421no—
139#41—24no—
140#41—24no—
141C2×Q8—712no—
142C2×Q8—1424no—
143C2×Q8—1421no—
144C2×Q8C22×D7112yes—
145C2×D4—22no—
146C2×D4—22no—
147C2×D4—712no—
148C2×D4—1424no—
149C2×D4—1421no—
150C2×D4C22×D7112yes—
151C22×C4—22no—
152C22×C4—1421no—
153C22×C4C22⋊D7112yes—
154C22×C4C22×D7111yescharacteristic
155Q16—1428no—
156SD16—1428no—
157SD16—1428no—
158D8—1428no—
159M16—718no—
160C2×C8—714no—
161C4⋊C4—22no—
162C22⋊C4—24no—
163C22⋊C4—1424no—
164C22⋊C4—1424no—
165C42C22⋊D7112yes—
166C42C22×D7111yescharacteristic
167C42C22×D7111yescharacteristic
168C23—22no—
169C23—2844no—
170C23—2842no—
171C23—1422no—
172C23C2×C22⋊D7111yescharacteristic
173Q8—22no—
174Q8—22no—
175Q8—2842no—
176Q8—1424no—
177D4—22no—
178D4—22no—
179D4—44no—
180D4—44no—
181D4—2842no—
182D4—1424no—
183C2×C4—22no—
184C2×C4—42no—
185C2×C4—44no—
186C2×C4—2842no—
187C2×C4—1422no—
188C2×C4D4×D7114yes—
189C2×C4C2×C22⋊D7112yes—
190C2×C4C2×C22⋊D7112yes—
191C2×C4C2×C22⋊D7111yescharacteristic
192C2×C4C23×D7111yescharacteristic
193C8—718no—
194C22—22no—
195C22—42no—
196C22—44no—
197C22—2842no—
198C22—2844no—
199C22—5684no—
200C22#1455612412yes—
201C22C22×C22⋊D7111yescharacteristic
202C4—22no—
203C4—22no—
204C4—42no—
205C4—2842no—
206C4#1455612412yes—
207C4C2×D4×D7112yes—
208C2—22no—
209C2—82no—
210C2—5682no—
211C2C2×#1455612411yescharacteristic
212C2#1798679712yes—
subgroupG/H
centerC22C22×C22⋊D71
commutatorC2×C4×C71C24
FrattiniC2×C4C23×D71
Fitting#532×C71C2
socleC22×C71C22×D4
hypercenter#532D71
hypocenterC71#3153
LCSG ▷ C2×C4×C71 ▷ C2×C71 ▷ C71
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C2×#14556158C2semidirect
2. C2×#14556153C2semidirect
3. C2×#14556116C2semidirect
4. #17986391C2semidirect
5. #17986386C2semidirect
6. #17986381C2semidirect
7. C4×#11765849C2semidirect
8. #532×C71C2semidirect
13. C2×#11765855C22semidirect
14. C2×#11765853C22semidirect
22. #14556039C22semidirect
23. C4×C71⋊C8C22semidirect
25. C4⋊Q8×C71C22semidirect
26. #127×C71C22semidirect
27. #124×C71C22semidirect
43. #11765849D4semidirect
107. C71#3153semidirect
110. #532D71semidirect
126. C4⋊Q8C2×D71semidirect
127. #127C2×D71semidirect
131. #124C2×D71semidirect