Finite Group Databasebeta

← #28362325

Subgroups — #28362325

Order 13952.

230 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 49 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#20295416C212yesmaximal, complemented
2#20295392C211yesmaximal, characteristic, complemented
3#20294768C211yesmaximal, characteristic, complemented
4#20294766C212yesmaximal, complemented
5#562×C109C211yesmaximal, characteristic, complemented
6C2×#13182016—22no—
7C2×#13182014—22no—
8#16812696C2211yescharacteristic
9#16812619—22no—
10#16812610C2211yescharacteristic
11#16812610C2212yes—
12#125×C109C2211yescharacteristic
13#124×C109C2212yes—
14#123×C109—22no—
15C22⋊Q8×C109—22no—
16#121×C109—22no—
17C2×D436—21no—
18C2×D436—22no—
19#13182016—44no—
20#13182014—44no—
21C2×#11724784—22no—
22C2×#11724784—21no—
23C2×Q8×C109D412yes—
24C2×D4×C109—22no—
25C2×D4×C109D412yes—
26C22×C4×C109—22no—
27C4⋊C4×C109—22no—
28C4⋊C4×C109—22no—
29C4⋊C4×C109D412yes—
30C22⋊C4×C109—22no—
31C22⋊C4×C109—22no—
32C22⋊C4×C109—22no—
33C42×C109C2311yescharacteristic
34C23×C109—22no—
35C22×D109—41no—
36Q8×C109—42no—
37D4×C109—42no—
38D4×C109—82no—
39C2×C4×C109—22no—
40C2×C4×C109—22no—
41C2×C4×C109—82no—
42C2×C4×C109C2×D411yescharacteristic
43C2×C4×C109C2×D412yes—
44D436—42no—
45D436—42no—
46D436—42no—
47#11724784—24no—
48#11724784—41no—
49C22×C109—42no—
50C22×C109—42no—
51C22×C109—42no—
52C22×C109#12011yescharacteristic
53C2×D109—82no—
54C2×D109—81no—
55C4×C109—22no—
56C4×C109—21no—
57C4×C109—42no—
58C4×C109—42no—
59C2×C109—82no—
60C2×C109#45312yes—
61C2×C109#45411yescharacteristic
62D109—161no—
63#1656—1091nomaximal, Sylow
64C109#165611yescharacteristic, complemented, Sylow
65#562D10911yescharacteristic, complemented
66#496—1092no—
67#490—1091no—
68#330—1091no—
69#329—1092no—
70C2×SD16—2182no—
71C2×D8—2182no—
72#127—1091no—
73#125C2×D10911yescharacteristic
74#124C2×D10912yes—
75#123—22no—
76C22⋊Q8—22no—
77#121—22no—
78#102—2182no—
79#97—1091no—
80#97—1092no—
81C2×Q8C22⋊D10912yes—
82C2×D4—22no—
83C2×D4—2181no—
84C2×D4—2182no—
85C2×D4C22⋊D10912yes—
86C22×C4—22no—
87SD16—4364no—
88D8—4364no—
89C2×C8—2181no—
90C2×C8—2182no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4C22⋊D10912yes—
94C22⋊C4—22no—
95C22⋊C4—22no—
96C22⋊C4—22no—
97C42C22×D10911yescharacteristic
98C23—22no—
99C23—4361no—
100Q8—42no—
101D4—42no—
102D4—82no—
103D4—4362no—
104D4—4362no—
105D4—4362no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—82no—
109C2×C4C2×C22⋊D10911yescharacteristic
110C2×C4C2×C22⋊D10912yes—
111C8—4361no—
112C8—2184no—
113C22—42no—
114C22—42no—
115C22—42no—
116C22—8722no—
117C22—8721no—
118C22#1681266811yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—42no—
123C2—82no—
124C2—17441no—
125C2#2029519412yes—
126C2#2029519911yescharacteristic
subgroupG/H
centerC22#16812668
commutator, cosocleC42×C109C23
FrattiniC42C22×D109
Fitting#562×C109C2
socleC22×C109#120
hypercenter#562D109
hypocenterC109#1656
LCSG ▷ C42×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #20295416C2semidirect
2. #20295392C2semidirect
3. #20294768C2semidirect
4. #20294766C2semidirect
5. #562×C109C2semidirect
64. C109#1656semidirect
65. #562D109semidirect

Maximal subgroups

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

subgroupindex
#202954162
#202953922
#202947682
#202947662
#562×C1092
#1656109