Finite Group Databasebeta

← #28346848

Subgroups — #28346848

Order 13696.

400 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 133 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#20293184C211yesmaximal, characteristic, complemented
2C2×#16812097C211yesmaximal, characteristic
3#20292987C214yesmaximal, complemented
4#20292982C214yesmaximal, complemented
5#20292637C212yesmaximal, complemented
6#20292633C212yesmaximal
7C4×M16×C107C211yesmaximal, characteristic, complemented
8C22×Dic214C2211yescharacteristic
9C22×D428C2211yescharacteristic
10#16812109C2212yescomplemented
11#16812101C2212yescomplemented
12#16812097—22no—
13#16812097C2212yes—
14#16812097C2212yes—
15#16812088—22no—
16C4×Dic214—22no—
17#16812068C2212yes—
18C2×(C2×D107)⋊C4—22no—
19#16812034C2218yescomplemented
20#16812031C2218yes—
21C2×M16×C107C2212yes—
22C2×C42×C107C2211yescharacteristic
23#97×C107C2212yescomplemented
24C4×C8×C107C2212yescomplemented
25C23×D107—21no—
26C2×Dic214—24no—
27C2×Dic214—22no—
28C2×Dic214—21no—
29C2×Dic214C2312yes—
30C2×D428—24no—
31C2×D428—21no—
32C2×D428C2312yes—
33C22×Dic107—21no—
34C4⋊Dic107—22no—
35C4⋊Dic107C2314yes—
36Dic107⋊C4—24no—
37C22⋊Dic107—22no—
38(C2×D107)⋊C4—24no—
39(C2×D107)⋊C4—24no—
40C4×Dic107—22no—
41C22×C4×C107D412yes—
42C22×C4×C107C2311yescharacteristic
43M16×C107—24no—
44C2×C8×C107C2314yes—
45C42×C107D412yes—
46C42×C107C2311yescharacteristic
47C42×C107C2311yescharacteristic
48C23×C107C2×D411yescharacteristic
49C22×D107—22no—
50C22×D107—44no—
51C22×D107—42no—
52C2×C4×C107—42no—
53C2×C4×C107C2×D412yes—
54C2×C4×C107C2×D412yes—
55C2×C4×C107C2×D411yescharacteristic
56C2×C4×C107#4114yes—
57C2×C4×C107C2411yescharacteristic
58C2×Dic107—24no—
59C2×Dic107—22no—
60C2×Dic107—42no—
61D428—24no—
62D428—42no—
63Dic214—24no—
64Dic214—44no—
65Dic214—42no—
66C8×C107—24no—
67C22×C107—22no—
68C22×C107#12412yes—
69C22×C107C22×D411yescharacteristic
70C2×D107—42no—
71C2×D107—44no—
72C2×D107—84no—
73C4×C107—22no—
74C4×C107—22no—
75C4×C107#12412yes—
76C4×C107C2×#4112yes—
77Dic107—42no—
78Dic107—44no—
79C2×C107—22no—
80C2×C107C2×#12411yescharacteristic
81C2×C107#57512yes—
82D107—82no—
83#3140—1071nomaximal, Sylow
84C107#314011yescharacteristic, complemented, Sylow
85#533—1071no—
86C2×#124—1071no—
87#489—1074no—
88#488—1074no—
89#419—1072no—
90#418—1072no—
91C4×M16D10711yescharacteristic, complemented
92C22×Q8—1071no—
93C22×D4—1071no—
94C2×M16C2×D10712yes—
95C4⋊Q8—1072no—
96#125—1072no—
97#124—1072no—
98#124—1072no—
99#124—2142no—
100C22⋊Q8—2142no—
101C4×Q8—2142no—
102#117—1072no—
103C2×C22⋊C4—2142no—
104C2×C42C2×D10711yescharacteristic
105#103—1078no—
106#102—1078no—
107#97C2×D10712yescomplemented
108C4×C8C2×D10712yescomplemented
109C24—2141no—
110C2×Q8—1072no—
111C2×Q8—2144no—
112C2×Q8—2142no—
113C2×Q8—2141no—
114C2×D4—1072no—
115C2×D4—2141no—
116C2×D4—2144no—
117C22×C4—2141no—
118C22×C4D42812yes—
119C22×C4C22×D10711yescharacteristic
120M16—24no—
121C2×C8C22×D10714yes—
122C4⋊C4—1074no—
123C4⋊C4—2144no—
124C4⋊C4—2142no—
125C22⋊C4—2142no—
126C22⋊C4—2144no—
127C22⋊C4—2144no—
128C42—2142no—
129C42D42812yes—
130C42C22×D10711yescharacteristic
131C42C22×D10711yescharacteristic
132C23—4284no—
133C23—4282no—
134C23—2142no—
135C23C2×D42811yescharacteristic
136Q8—4284no—
137Q8—4282no—
138Q8—2144no—
139D4—4282no—
140D4—2144no—
141C2×C4—42no—
142C2×C4—4282no—
143C2×C4—2144no—
144C2×C4—2142no—
145C2×C4C2×D42812yes—
146C2×C4C2×D42812yes—
147C2×C4C2×D42811yescharacteristic
148C2×C4#1313381814yes—
149C2×C4C23×D10711yescharacteristic
150C8—24no—
151C22—22no—
152C22—8564no—
153C22—4282no—
154C22—4284no—
155C22#1681209712yes—
156C22C22×D42811yescharacteristic
157C4—22no—
158C4—22no—
159C4—4282no—
160C4—4284no—
161C4#1681209712yes—
162C4C2×#1313381812yes—
163C2—22no—
164C2—8562no—
165C2C2×#1681209711yescharacteristic
166C2#2029342712yes—
subgroupG/H
centerC22C22×D428
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×M16×C107C2
socleC22×C107C22×D4
prime radicalC22×D428C22
hypercenterC4×M16D107
hypocenterC107#3140
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
1. #20293184C2semidirect
3. #20292987C2semidirect
4. #20292982C2semidirect
5. #20292637C2semidirect
7. C4×M16×C107C2semidirect
10. #16812109C22semidirect
11. #16812101C22semidirect
19. #16812034C22semidirect
23. #97×C107C22semidirect
24. C4×C8×C107C22semidirect
84. C107#3140semidirect
91. C4×M16D107semidirect
107. #97C2×D107semidirect
108. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202931842
C2×#168120972
#202929872
#202929822
#202926372
#202926332
C4×M16×C1072
#3140107