Finite Group Databasebeta

← #28346708

Subgroups — #28346708

Order 13696.

394 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 121 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#20293101C211yesmaximal, characteristic
2#20292965C212yesmaximal, complemented
3#20292924C212yesmaximal, complemented
4#20292829C212yesmaximal, complemented
5#20292777C212yesmaximal
6#20292725C212yesmaximal, complemented
7C4×#13133805C212yesmaximal, complemented
8#20292688C211yesmaximal, characteristic, complemented
9#541×C107C211yesmaximal, characteristic, complemented
10C22×D428C2211yescharacteristic
11C2×#13133805C2212yes—
12C2×#13133805C2212yes—
13C4×D428C2212yes—
14C4×D428C2212yes—
15#16812071—22no—
16#16812068C2211yescharacteristic
17C22⋊C4×D107—22no—
18#16812042C2212yescomplemented
19#16812040C2212yescomplemented
20C4⋊(C107⋊C8)C2212yescomplemented
21#16812033C2212yescomplemented
22#16812033C2212yescomplemented
23#16812029C2212yes—
24#16812029C2212yes—
25C22⋊(C107⋊C8)C2212yes—
26#16812020C2211yescharacteristic, complemented
27C4×C107⋊C8C2211yescharacteristic, complemented
28C4⋊Q8×C107C2211yescharacteristic, complemented
29#126×C107—22no—
30#125×C107—22no—
31#125×C107C2211yescharacteristic, complemented
32C22⋊Q8×C107C2212yes—
33C4×Q8×C107C2212yescomplemented
34#117×C107C2211yescharacteristic
35C23×D107—21no—
36C2×D428—21no—
37C2×D428—24no—
38C2×D428C2312yes—
39C2×C4×D107—22no—
40C2×C4×D107—22no—
41#13133805—24no—
42#13133805—24no—
43C2×C107⋊C8C2312yes—
44C2×C107⋊C8C2312yes—
45C4⋊Dic107C2312yes—
46C22⋊Dic107—21no—
47(C2×D107)⋊C4—22no—
48(C2×D107)⋊C4—22no—
49C4×Dic107—21no—
50C2×Q8×C107C2312yes—
51C22×C4×C107C2311yescharacteristic
52C4⋊C4×C107—22no—
53C4⋊C4×C107—22no—
54C4⋊C4×C107—22no—
55C4⋊C4×C107—22no—
56C4⋊C4×C107D412yes—
57C4⋊C4×C107C2312yes—
58C22⋊C4×C107—22no—
59C22⋊C4×C107D412yes—
60C42×C107—22no—
61C42×C107C2311yescharacteristic
62C42×C107C2311yescharacteristic
63C23×C107C2×D411yescharacteristic
64C22×D107—22no—
65C22×D107—24no—
66C22×D107—82no—
67Q8×C107—22no—
68Q8×C107—22no—
69C2×C4×C107—22no—
70C2×C4×C107—22no—
71C2×C4×C107—22no—
72C2×C4×C107C2×D412yes—
73C2×C4×C107C2×D412yes—
74C2×C4×C107C2×D411yescharacteristic
75C2×C4×C107C2411yescharacteristic
76C2×Dic107—22no—
77D428—24no—
78D428—22no—
79D428#4114yes—
80C4×D107—82no—
81C4×D107—82no—
82C107⋊C8—22no—
83C107⋊C8—22no—
84C22×C107—21no—
85C22×C107—22no—
86C22×C107C22×D411yescharacteristic
87C2×D107—24no—
88C2×D107—42no—
89C2×D107—42no—
90C2×D107—84no—
91C4×C107—21no—
92C4×C107—22no—
93C4×C107—22no—
94C4×C107—42no—
95C4×C107—42no—
96C4×C107C2×#4112yes—
97Dic107—42no—
98C2×C107—41no—
99C2×C107#52511yescharacteristic
100C2×C107#57712yes—
101D107—44no—
102#3123—1071nomaximal, Sylow
103C107#312311yescharacteristic, complemented, Sylow
104#541D10711yescharacteristic, complemented
105#518—1071no—
106#485—1072no—
107#478—1072no—
108#461—1072no—
109#450—1072no—
110#440—1072no—
111C4×SD16—1072no—
112#433—1071no—
113C22×D4—1071no—
114C2×SD16—1072no—
115C2×SD16—1072no—
116C4⋊Q8C2×D10711yescharacteristic, complemented
117#126—22no—
118#125—22no—
119#125C2×D10711yescharacteristic, complemented
120C22⋊Q8C2×D10712yes—
121C4×Q8C2×D10712yescomplemented
122C4×D4—1072no—
123C4×D4—1072no—
124C4×D4—2142no—
125#117—1071no—
126#117C2×D10711yescharacteristic
127C2×C22⋊C4—2142no—
128#107—1072no—
129#106—1072no—
130C4⋊C8—1072no—
131#103—1072no—
132#103—1072no—
133#102—1072no—
134#102—1072no—
135C22⋊C8—1072no—
136#97—1071no—
137C4×C8—1071no—
138C24—2141no—
139C2×Q8C22×D10712yes—
140C2×D4—1072no—
141C2×D4—2144no—
142C2×D4—2141no—
143C22×C4—2142no—
144C22×C4—2142no—
145C22×C4C22×D10711yescharacteristic
146SD16—2144no—
147SD16—2144no—
148C2×C8—1072no—
149C2×C8—1072no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—22no—
153C4⋊C4—22no—
154C4⋊C4—1072no—
155C4⋊C4C22⋊D10712yes—
156C4⋊C4C22×D10712yes—
157C22⋊C4—22no—
158C22⋊C4—2141no—
159C22⋊C4—2142no—
160C22⋊C4—2142no—
161C22⋊C4C22⋊D10712yes—
162C42—22no—
163C42—2141no—
164C42C22×D10711yescharacteristic
165C42C22×D10711yescharacteristic
166C23—8562no—
167C23—2144no—
168C23—2142no—
169C23C2×C22⋊D10711yescharacteristic
170Q8—22no—
171Q8—22no—
172D4—1074no—
173D4—2144no—
174D4—2142no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—8562no—
179C2×C4—8562no—
180C2×C4—2142no—
181C2×C4C2×C22⋊D10712yes—
182C2×C4C2×C22⋊D10711yescharacteristic
183C2×C4C2×C22⋊D10712yes—
184C2×C4C23×D10711yescharacteristic
185C8—2142no—
186C8—2142no—
187C22—21no—
188C22—22no—
189C22—8564no—
190C22—4282no—
191C22—4282no—
192C22—2144no—
193C22C22×C22⋊D10711yescharacteristic
194C4—21no—
195C4—22no—
196C4—22no—
197C4—42no—
198C4—42no—
199C4—4282no—
200C4#41×D10712yes—
201C2—41no—
202C2—4284no—
203C2#2029313411yescharacteristic
204C2#2029343512yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#541×C107C2
socleC22×C107C22×D4
prime radicalC22×D428C22
hypercenter#541D107
hypocenterC107#3123
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #20292965C2semidirect
3. #20292924C2semidirect
4. #20292829C2semidirect
6. #20292725C2semidirect
7. C4×#13133805C2semidirect
8. #20292688C2semidirect
9. #541×C107C2semidirect
18. #16812042C22semidirect
19. #16812040C22semidirect
20. C4⋊(C107⋊C8)C22semidirect
21. #16812033C22semidirect
22. #16812033C22semidirect
26. #16812020C22semidirect
27. C4×C107⋊C8C22semidirect
28. C4⋊Q8×C107C22semidirect
31. #125×C107C22semidirect
33. C4×Q8×C107C22semidirect
103. C107#3123semidirect
104. #541D107semidirect
116. C4⋊Q8C2×D107semidirect
119. #125C2×D107semidirect
121. C4×Q8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931012
#202929652
#202929242
#202928292
#202927772
#202927252
C4×#131338052
#202926882
#541×C1072
#3123107