Finite Group Databasebeta

← #28335765

Subgroups — #28335765

Order 13696.

228 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 45 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#20293369C211yesmaximal, characteristic
2#20292989C212yesmaximal, complemented
3#20292367C211yesmaximal, characteristic, complemented
4#20292366C212yesmaximal, complemented
5#497×C107C211yesmaximal, characteristic, complemented
6#16812098—21no—
7#16812097C2212yes—
8#16812095—22no—
9#16812079—22no—
10#16812077—21no—
11#16812032—22no—
12#16812029—22no—
13#16812020C2212yescomplemented
14C2×Q16×C107—21no—
15C2×SD16×C107—21no—
16C4⋊Q8×C107C2211yescharacteristic, complemented
17#124×C107C2211yescharacteristic
18#97×C107C2211yescharacteristic, complemented
19C23×D107—21no—
20C2×Dic214D412yes—
21C2×D428—21no—
22C2×D428D412yes—
23C2×C22⋊D107—22no—
24C2×C107⋊C8—22no—
25C22⋊Dic107—21no—
26(C2×D107)⋊C4—22no—
27(C2×D107)⋊C4—22no—
28(C2×D107)⋊C4—22no—
29C4×Dic107—21no—
30C2×Q8×C107—21no—
31C2×Q8×C107D411yescharacteristic
32C2×D4×C107D411yescharacteristic
33Q16×C107—42no—
34SD16×C107—42no—
35C2×C8×C107—21no—
36C4⋊C4×C107—22no—
37C22⋊C4×C107—22no—
38C42×C107C2311yescharacteristic
39C23×C107—21no—
40C22×D107—22no—
41C22×D107—81no—
42C22×D107—82no—
43Q8×C107—41no—
44Q8×C107—41no—
45Q8×C107—41no—
46D4×C107—41no—
47C2×C4×C107—21no—
48C2×C4×C107—41no—
49C2×C4×C107C2×D412yes—
50C2×C4×C107C2×D411yescharacteristic
51C22⋊D107—82no—
52C2×Dic107—22no—
53D428—42no—
54D428—81no—
55Dic214—42no—
56C8×C107—22no—
57C107⋊C8—42no—
58C22×C107—41no—
59C22×C107—42no—
60C22×C107#12011yescharacteristic
61C2×D107—42no—
62C2×D107—42no—
63C2×D107—42no—
64C2×D107—84no—
65C4×C107—21no—
66C4×C107—22no—
67C4×C107—41no—
68C4×C107—81no—
69Dic107—42no—
70C2×C107—81no—
71C2×C107#45311yescharacteristic
72C2×C107#45512yes—
73D107—82no—
74#1659—1071nomaximal, Sylow
75C107#165911yescharacteristic, complemented, Sylow
76H(3,𝔽4)—1071no—
77#497D10711yescharacteristic, complemented
78#489—1072no—
79#329—1071no—
80#329—1072no—
81C2×Q16—21no—
82C2×SD16—21no—
83C4⋊Q8C2×D10711yescharacteristic, complemented
84#124—1072no—
85#124—2142no—
86#124—2141no—
87#124C2×D10711yescharacteristic
88#120—2142no—
89#120—2141no—
90#103—2142no—
91#102—2142no—
92#97—1072no—
93#97C2×D10711yescharacteristic, complemented
94C24—2141no—
95C2×Q8—21no—
96C2×Q8—1072no—
97C2×Q8C22⋊D10711yescharacteristic
98C2×D4—1072no—
99C2×D4—2141no—
100C2×D4—2142no—
101C2×D4C22⋊D10711yescharacteristic
102Q16—42no—
103SD16—42no—
104C2×C8—21no—
105C2×C8—2142no—
106C4⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—2142no—
109C22⋊C4—2141no—
110C22⋊C4—2142no—
111C22⋊C4—2142no—
112C42—2141no—
113C42C22×D10711yescharacteristic
114C23—21no—
115C23—8561no—
116C23—8562no—
117C23—2142no—
118Q8—41no—
119Q8—41no—
120Q8—41no—
121Q8—4282no—
122D4—41no—
123D4—8561no—
124D4—8562no—
125D4—4282no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—2142no—
129C2×C4D4×D10712yes—
130C2×C4C2×C22⋊D10711yescharacteristic
131C8—22no—
132C8—4282no—
133C22—41no—
134C22—42no—
135C22—8564no—
136C22—4282no—
137C22—4282no—
138C22—4282no—
139C22#1681207711yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—81no—
144C4—4282no—
145C2—81no—
146C2—8562no—
147C2#2029279411yescharacteristic
148C2#2029280312yes—
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#497×C107C2
socleC22×C107#120
prime radical#20293369C2
hypercenter#497D107
hypocenterC107#1659
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #20292989C2semidirect
3. #20292367C2semidirect
4. #20292366C2semidirect
5. #497×C107C2semidirect
13. #16812020C22semidirect
16. C4⋊Q8×C107C22semidirect
18. #97×C107C22semidirect
75. C107#1659semidirect
77. #497D107semidirect
83. C4⋊Q8C2×D107semidirect
93. #97C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202933692
#202929892
#202923672
#202923662
#497×C1072
#1659107