Finite Group Databasebeta

← #28346704

Subgroups — #28346704

Order 13696.

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#20293232C211yesmaximal, characteristic, complemented
2#20292951C212yesmaximal, complemented
3#20292927C212yesmaximal, complemented
4#20292822C212yesmaximal, complemented
5#20292761C212yesmaximal, complemented
6#20292737C212yesmaximal, complemented
7C4×#13133803C212yesmaximal, complemented
8#20292688C211yesmaximal, characteristic, complemented
9#518×C107C211yesmaximal, characteristic, complemented
10C2×#13133803C2212yes—
11C2×#13133803C2212yes—
12#16812106C2211yescharacteristic, complemented
13#16812101C2211yescharacteristic, complemented
14#16812094—22no—
15#16812083C2212yes—
16#16812082—22no—
17C4×D428C2212yescomplemented
18#16812042C2212yescomplemented
19#16812040C2212yescomplemented
20C4⋊(C107⋊C8)C2212yescomplemented
21#16812030C2212yes—
22#16812030C2212yes—
23#16812029C2212yescomplemented
24#16812029C2212yescomplemented
25C22⋊(C107⋊C8)C2212yes—
26#16812020C2211yescharacteristic, complemented
27C4×C107⋊C8C2211yescharacteristic, complemented
28C22×D4×C107C2211yescharacteristic
29C4×D4×C107—22no—
30C4×D4×C107C2212yes—
31C4×D4×C107C2212yes—
32#117×C107C2211yescharacteristic
33#117×C107C2211yescharacteristic
34C2×C22⋊C4×C107—22no—
35C2×D428—22no—
36C2×D428C2312yes—
37C2×C22⋊D107—22no—
38C2×C4×D107—22no—
39#13133803—24no—
40#13133803—24no—
41C2×C107⋊C8C2312yes—
42C2×C107⋊C8C2312yes—
43C4⋊Dic107C2312yes—
44Dic107⋊C4—22no—
45(C2×D107)⋊C4—22no—
46(C2×D107)⋊C4—22no—
47C24×C107—21no—
48C2×D4×C107—24no—
49C2×D4×C107—21no—
50C2×D4×C107C2312yes—
51C22×C4×C107—22no—
52C22×C4×C107—22no—
53C22×C4×C107C2311yescharacteristic
54C4⋊C4×C107D412yes—
55C4⋊C4×C107C2312yes—
56C22⋊C4×C107—21no—
57C22⋊C4×C107—22no—
58C22⋊C4×C107—22no—
59C22⋊C4×C107D412yes—
60C42×C107—21no—
61C42×C107C2311yescharacteristic
62C42×C107C2311yescharacteristic
63C23×C107—24no—
64C23×C107—22no—
65C23×C107—82no—
66C23×C107C2×D411yescharacteristic
67C22×D107—22no—
68D4×C107—24no—
69D4×C107—22no—
70D4×C107#4114yes—
71C2×C4×C107—22no—
72C2×C4×C107—22no—
73C2×C4×C107—82no—
74C2×C4×C107—82no—
75C2×C4×C107C2×D411yescharacteristic
76C2×C4×C107C2×D412yes—
77C2×C4×C107C2×D412yes—
78C2×C4×C107C2411yescharacteristic
79C22⋊D107—82no—
80C2×Dic107—22no—
81D428—22no—
82D428—22no—
83D428—82no—
84C4×D107—82no—
85C107⋊C8—22no—
86C107⋊C8—22no—
87C22×C107—21no—
88C22×C107—24no—
89C22×C107—22no—
90C22×C107—42no—
91C22×C107—42no—
92C22×C107—84no—
93C22×C107C22×D411yescharacteristic
94C2×D107—42no—
95C2×D107—42no—
96C2×D107—42no—
97C4×C107—21no—
98C4×C107—22no—
99C4×C107—22no—
100C4×C107—42no—
101C4×C107C2×#4112yes—
102Dic107—42no—
103C2×C107—41no—
104C2×C107—44no—
105C2×C107#52511yescharacteristic
106C2×C107#57612yes—
107D107—82no—
108#3122—1071nomaximal, Sylow
109C107#312211yescharacteristic, complemented, Sylow
110#540—1071no—
111#518D10711yescharacteristic, complemented
112#482—1072no—
113#478—1072no—
114#459—1072no—
115#447—1072no—
116#442—1072no—
117C4×D8—1072no—
118#433—1071no—
119C22×D4C2×D10711yescharacteristic
120C2×D8—1072no—
121C2×D8—1072no—
122#127—1071no—
123#125—1071no—
124#123—2142no—
125#121—1072no—
126#121—2142no—
127C4×D4—22no—
128C4×D4—1072no—
129C4×D4C2×D10712yes—
130C4×D4C2×D10712yes—
131#117C2×D10711yescharacteristic
132#117C2×D10711yescharacteristic
133C2×C22⋊C4—22no—
134#107—1072no—
135#106—1072no—
136C4⋊C8—1072no—
137#102—1072no—
138#102—1072no—
139#102—1072no—
140#102—1072no—
141C22⋊C8—1072no—
142#97—1071no—
143C4×C8—1071no—
144C24—21no—
145C2×D4—24no—
146C2×D4—21no—
147C2×D4—1072no—
148C2×D4—2142no—
149C2×D4—2142no—
150C2×D4C22×D10712yes—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—2142no—
154C22×C4C22×D10711yescharacteristic
155D8—2144no—
156D8—2144no—
157C2×C8—1072no—
158C2×C8—1072no—
159C4⋊C4—1072no—
160C4⋊C4—2142no—
161C4⋊C4C22⋊D10712yes—
162C4⋊C4C22×D10712yes—
163C22⋊C4—21no—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—2142no—
167C22⋊C4—2142no—
168C22⋊C4C22⋊D10712yes—
169C42—21no—
170C42C22×D10711yescharacteristic
171C42C22×D10711yescharacteristic
172C23—24no—
173C23—22no—
174C23—82no—
175C23—2142no—
176C23C2×C22⋊D10711yescharacteristic
177D4—24no—
178D4—22no—
179D4—8562no—
180D4—8562no—
181D4—2142no—
182D4—2142no—
183D4#1313381814yes—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—82no—
187C2×C4—82no—
188C2×C4—8562no—
189C2×C4—2142no—
190C2×C4C2×C22⋊D10711yescharacteristic
191C2×C4C2×C22⋊D10712yes—
192C2×C4C2×C22⋊D10712yes—
193C2×C4C23×D10711yescharacteristic
194C8—2142no—
195C8—2142no—
196C22—21no—
197C22—24no—
198C22—22no—
199C22—42no—
200C22—42no—
201C22—84no—
202C22—4282no—
203C22—4282no—
204C22—4282no—
205C22C22×C22⋊D10711yescharacteristic
206C4—21no—
207C4—22no—
208C4—22no—
209C4—42no—
210C4—4282no—
211C4C2×#1313381812yes—
212C2—41no—
213C2—44no—
214C2—8562no—
215C2#2029313811yescharacteristic
216C2#2029343012yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#518×C107C2
socleC22×C107C22×D4
hypercenter#518D107
hypocenterC107#3122
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #20293232C2semidirect
2. #20292951C2semidirect
3. #20292927C2semidirect
4. #20292822C2semidirect
5. #20292761C2semidirect
6. #20292737C2semidirect
7. C4×#13133803C2semidirect
8. #20292688C2semidirect
9. #518×C107C2semidirect
12. #16812106C22semidirect
13. #16812101C22semidirect
17. C4×D428C22semidirect
18. #16812042C22semidirect
19. #16812040C22semidirect
20. C4⋊(C107⋊C8)C22semidirect
23. #16812029C22semidirect
24. #16812029C22semidirect
26. #16812020C22semidirect
27. C4×C107⋊C8C22semidirect
109. C107#3122semidirect
111. #518D107semidirect

Maximal subgroups

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

subgroupindex
#202932322
#202929512
#202929272
#202928222
#202927612
#202927372
C4×#131338032
#202926882
#518×C1072
#3122107