Finite Group Databasebeta

← #28362336

Subgroups — #28362336

Order 13952.

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#20295768C211yesmaximal, characteristic
2#20295388C212yesmaximal, complemented
3#20294766C211yesmaximal, characteristic, complemented
4#20294765C212yesmaximal, complemented
5#497×C109C211yesmaximal, characteristic, complemented
6#16812688—21no—
7#16812687C2212yes—
8#16812685—22no—
9#16812669—22no—
10#16812667—21no—
11#16812622—22no—
12#16812619—22no—
13#16812610C2212yescomplemented
14C2×Q16×C109—21no—
15C2×SD16×C109—21no—
16C4⋊Q8×C109C2211yescharacteristic, complemented
17#124×C109C2211yescharacteristic
18#97×C109C2211yescharacteristic, complemented
19C23×D109—21no—
20C2×Dic218D412yes—
21C2×D436—21no—
22C2×D436D412yes—
23C2×C22⋊D109—22no—
24C2×#11724784—22no—
25C22⋊Dic109—21no—
26#13182006—22no—
27#13182006—22no—
28#13182006—22no—
29C4×Dic109—21no—
30C2×Q8×C109—21no—
31C2×Q8×C109D411yescharacteristic
32C2×D4×C109D411yescharacteristic
33Q16×C109—42no—
34SD16×C109—42no—
35C2×C8×C109—21no—
36C4⋊C4×C109—22no—
37C22⋊C4×C109—22no—
38C42×C109C2311yescharacteristic
39C23×C109—21no—
40C22×D109—22no—
41C22×D109—81no—
42C22×D109—82no—
43Q8×C109—41no—
44Q8×C109—41no—
45Q8×C109—41no—
46D4×C109—41no—
47C2×C4×C109—21no—
48C2×C4×C109—41no—
49C2×C4×C109C2×D412yes—
50C2×C4×C109C2×D411yescharacteristic
51C22⋊D109—82no—
52C2×Dic109—22no—
53D436—42no—
54D436—81no—
55Dic218—42no—
56C8×C109—22no—
57#11724784—42no—
58C22×C109—41no—
59C22×C109—42no—
60C22×C109#12011yescharacteristic
61C2×D109—42no—
62C2×D109—42no—
63C2×D109—42no—
64C2×D109—84no—
65C4×C109—21no—
66C4×C109—22no—
67C4×C109—41no—
68C4×C109—81no—
69Dic109—42no—
70C2×C109—81no—
71C2×C109#45311yescharacteristic
72C2×C109#45512yes—
73D109—82no—
74#1659—1091nomaximal, Sylow
75C109#165911yescharacteristic, complemented, Sylow
76H(3,𝔽4)—1091no—
77#497D10911yescharacteristic, complemented
78#489—1092no—
79#329—1091no—
80#329—1092no—
81C2×Q16—21no—
82C2×SD16—21no—
83C4⋊Q8C2×D10911yescharacteristic, complemented
84#124—1092no—
85#124—2182no—
86#124—2181no—
87#124C2×D10911yescharacteristic
88#120—2182no—
89#120—2181no—
90#103—2182no—
91#102—2182no—
92#97—1092no—
93#97C2×D10911yescharacteristic, complemented
94C24—2181no—
95C2×Q8—21no—
96C2×Q8—1092no—
97C2×Q8C22⋊D10911yescharacteristic
98C2×D4—1092no—
99C2×D4—2181no—
100C2×D4—2182no—
101C2×D4C22⋊D10911yescharacteristic
102Q16—42no—
103SD16—42no—
104C2×C8—21no—
105C2×C8—2182no—
106C4⋊C4—22no—
107C22⋊C4—22no—
108C22⋊C4—2182no—
109C22⋊C4—2181no—
110C22⋊C4—2182no—
111C22⋊C4—2182no—
112C42—2181no—
113C42C22×D10911yescharacteristic
114C23—21no—
115C23—8721no—
116C23—8722no—
117C23—2182no—
118Q8—41no—
119Q8—41no—
120Q8—41no—
121Q8—4362no—
122D4—41no—
123D4—8721no—
124D4—8722no—
125D4—4362no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—2182no—
129C2×C4D4×D10912yes—
130C2×C4C2×C22⋊D10911yescharacteristic
131C8—22no—
132C8—4362no—
133C22—41no—
134C22—42no—
135C22—8724no—
136C22—4362no—
137C22—4362no—
138C22—4362no—
139C22#1681266711yescharacteristic
140C4—21no—
141C4—22no—
142C4—41no—
143C4—81no—
144C4—4362no—
145C2—81no—
146C2—8722no—
147C2#2029519311yescharacteristic
148C2#2029520212yes—
subgroupG/H
centerC22#16812667
commutator, cosocleC42×C109C23
FrattiniC42C22×D109
Fitting#497×C109C2
socleC22×C109#120
prime radical#20295768C2
hypercenter#497D109
hypocenterC109#1659
LCSG ▷ C42×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #497

Complements

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

NH ≅ G/Ntype
2. #20295388C2semidirect
3. #20294766C2semidirect
4. #20294765C2semidirect
5. #497×C109C2semidirect
13. #16812610C22semidirect
16. C4⋊Q8×C109C22semidirect
18. #97×C109C22semidirect
75. C109#1659semidirect
77. #497D109semidirect
83. C4⋊Q8C2×D109semidirect
93. #97C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202957682
#202953882
#202947662
#202947652
#497×C1092
#1659109