Finite Group Databasebeta

← #28346850

Subgroups — #28346850

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
1C2×#16812097C211yesmaximal, characteristic
2#20292989C214yesmaximal, complemented
3#20292984C214yesmaximal, complemented
4#20292635C212yesmaximal, complemented
5#20292631C212yesmaximal
6C4×#13133802C211yesmaximal, characteristic, complemented
7#533×C107C211yesmaximal, characteristic, complemented
8C22×Dic214C2211yescharacteristic
9C22×D428C2211yescharacteristic
10C2×#13133802C2212yes—
11#16812097—22no—
12#16812097C2212yes—
13#16812097C2212yes—
14C2×(C2×D107)⋊C4—22no—
15#16812032C2218yescomplemented
16#16812029C2218yes—
17#16812020C2212yescomplemented
18C4×C107⋊C8C2212yescomplemented
19C4⋊Q8×C107C2212yescomplemented
20#125×C107C2212yescomplemented
21C22⋊Q8×C107—22no—
22C4×Q8×C107—22no—
23#117×C107C2212yes—
24C2×C42×C107C2211yescharacteristic
25C23×D107—21no—
26C2×Dic214—24no—
27C2×Dic214—21no—
28C2×Dic214C2312yes—
29C2×D428—24no—
30C2×D428—21no—
31C2×D428C2312yes—
32C22×Dic107—21no—
33#13133802—24no—
34C2×C107⋊C8C2314yes—
35(C2×D107)⋊C4—24no—
36(C2×D107)⋊C4—24no—
37C2×Q8×C107—22no—
38C22×C4×C107D412yes—
39C22×C4×C107C2311yescharacteristic
40C4⋊C4×C107—24no—
41C4⋊C4×C107—22no—
42C4⋊C4×C107C2314yes—
43C22⋊C4×C107—22no—
44C42×C107—22no—
45C42×C107D412yes—
46C42×C107C2311yescharacteristic
47C42×C107C2311yescharacteristic
48C23×C107C2×D411yescharacteristic
49C22×D107—22no—
50C22×D107—44no—
51C22×D107—42no—
52Q8×C107—44no—
53C2×C4×C107—24no—
54C2×C4×C107—42no—
55C2×C4×C107C2×D412yes—
56C2×C4×C107C2×D412yes—
57C2×C4×C107C2×D411yescharacteristic
58C2×C4×C107#4114yes—
59C2×C4×C107C2411yescharacteristic
60C2×Dic107—22no—
61C2×Dic107—42no—
62D428—24no—
63D428—42no—
64Dic214—24no—
65Dic214—42no—
66C107⋊C8—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—44no—
76C4×C107#12412yes—
77C4×C107C2×#4112yes—
78Dic107—42no—
79C2×C107—22no—
80C2×C107C2×#12411yescharacteristic
81C2×C107#57512yes—
82D107—82no—
83#3140—1071nomaximal, Sylow
84C107#314011yescharacteristic, complemented, Sylow
85#533D10711yescharacteristic, complemented
86C2×#124—1071no—
87#489—1074no—
88#488—1074no—
89#419—1072no—
90#418—1072no—
91C4×M16—1071no—
92C22×Q8—1071no—
93C22×D4—1071no—
94C2×M16—1072no—
95C4⋊Q8C2×D10712yescomplemented
96#125C2×D10712yescomplemented
97#124—1072no—
98#124—1072no—
99#124—2142no—
100C22⋊Q8—22no—
101C4×Q8—22no—
102#117C2×D10712yes—
103C2×C22⋊C4—2142no—
104C2×C42C2×D10711yescharacteristic
105#103—1078no—
106#102—1078no—
107#97—1072no—
108C4×C8—1072no—
109C24—2141no—
110C2×Q8—22no—
111C2×Q8—1072no—
112C2×Q8—2144no—
113C2×Q8—2141no—
114C2×D4—1072no—
115C2×D4—2141no—
116C2×D4—2144no—
117C22×C4—2141no—
118C22×C4C22⋊D10712yes—
119C22×C4C22×D10711yescharacteristic
120M16—2144no—
121C2×C8—1074no—
122C4⋊C4—24no—
123C4⋊C4—22no—
124C4⋊C4C22×D10714yes—
125C22⋊C4—22no—
126C22⋊C4—2144no—
127C22⋊C4—2144no—
128C42—22no—
129C42C22⋊D10712yes—
130C42C22×D10711yescharacteristic
131C42C22×D10711yescharacteristic
132C23—4284no—
133C23—4282no—
134C23—2142no—
135C23C2×C22⋊D10711yescharacteristic
136Q8—44no—
137Q8—4282no—
138Q8—2144no—
139D4—4282no—
140D4—2144no—
141C2×C4—24no—
142C2×C4—42no—
143C2×C4—4282no—
144C2×C4—2142no—
145C2×C4C2×C22⋊D10712yes—
146C2×C4C2×C22⋊D10712yes—
147C2×C4C2×C22⋊D10711yescharacteristic
148C2×C4#1313381914yes—
149C2×C4C23×D10711yescharacteristic
150C8—2144no—
151C22—22no—
152C22—8564no—
153C22—4282no—
154C22—4284no—
155C22#1681209812yes—
156C22C22×C22⋊D10711yescharacteristic
157C4—22no—
158C4—22no—
159C4—44no—
160C4—4282no—
161C4#1681209812yes—
162C4C2×#1313381912yes—
163C2—22no—
164C2—8562no—
165C2C2×#1681209811yescharacteristic
166C2#2029342512yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#533×C107C2
socleC22×C107C22×D4
prime radicalC22×D428C22
hypercenter#533D107
hypocenterC107#3140
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #20292989C2semidirect
3. #20292984C2semidirect
4. #20292635C2semidirect
6. C4×#13133802C2semidirect
7. #533×C107C2semidirect
15. #16812032C22semidirect
17. #16812020C22semidirect
18. C4×C107⋊C8C22semidirect
19. C4⋊Q8×C107C22semidirect
20. #125×C107C22semidirect
84. C107#3140semidirect
85. #533D107semidirect
95. C4⋊Q8C2×D107semidirect
96. #125C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120972
#202929892
#202929842
#202926352
#202926312
C4×#131338022
#533×C1072
#3140107