Finite Group Databasebeta

← #28362330

Subgroups — #28362330

Order 13952.

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

normal subgroups only

This group has 206 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#20295416C211yesmaximal, characteristic
2#20295392C212yesmaximal, complemented
3#20294768C212yesmaximal, complemented
4#20294766C211yesmaximal, characteristic, complemented
5#563×C109C211yesmaximal, characteristic, complemented
6C2×#13182016—21no—
7C2×#13182014—21no—
8#16812696C2211yescharacteristic
9#16812619—22no—
10#16812619—22no—
11#16812610C2212yes—
12#16812610C2211yescharacteristic
13C4⋊Q8×C109—21no—
14#126×C109—22no—
15#125×C109C2212yes—
16#124×C109C2211yescharacteristic
17#123×C109—21no—
18C22⋊Q8×C109—22no—
19C2×D436—22no—
20C2×D436—21no—
21#13182016—42no—
22#13182014—42no—
23C2×#11724784—21no—
24C2×#11724784—22no—
25C2×Q8×C109—21no—
26C2×Q8×C109D411yescharacteristic
27C2×D4×C109D411yescharacteristic
28C22×C4×C109—21no—
29C4⋊C4×C109—22no—
30C4⋊C4×C109—22no—
31C4⋊C4×C109—22no—
32C4⋊C4×C109D412yes—
33C4⋊C4×C109D412yes—
34C22⋊C4×C109—22no—
35C22⋊C4×C109—21no—
36C22⋊C4×C109—22no—
37C42×C109—21no—
38C42×C109C2311yescharacteristic
39C23×C109—21no—
40C22×D109—41no—
41Q8×C109—41no—
42Q8×C109—81no—
43D4×C109—41no—
44C2×C4×C109—22no—
45C2×C4×C109—21no—
46C2×C4×C109—22no—
47C2×C4×C109—81no—
48C2×C4×C109C2×D412yes—
49C2×C4×C109C2×D411yescharacteristic
50D436—41no—
51D436—41no—
52D436—44no—
53#11724784—22no—
54#11724784—42no—
55C22×C109—41no—
56C22×C109—42no—
57C22×C109#12011yescharacteristic
58C2×D109—81no—
59C2×D109—82no—
60C4×C109—21no—
61C4×C109—22no—
62C4×C109—42no—
63C4×C109—41no—
64C4×C109—42no—
65C2×C109—81no—
66C2×C109#45311yescharacteristic
67C2×C109#45412yes—
68D109—161no—
69#1657—1091nomaximal, Sylow
70C109#165711yescharacteristic, complemented, Sylow
71#563D10911yescharacteristic, complemented
72#496—1091no—
73#490—1092no—
74#330—1092no—
75#329—1091no—
76C2×SD16—2181no—
77C2×D8—2181no—
78C4⋊Q8—21no—
79#127—1091no—
80#126—22no—
81#125C2×D10912yes—
82#124C2×D10911yescharacteristic
83#123—21no—
84C22⋊Q8—22no—
85#102—2182no—
86#102—2182no—
87#97—1092no—
88#97—1091no—
89C2×Q8—21no—
90C2×Q8C22⋊D10911yescharacteristic
91C2×D4—2182no—
92C2×D4—2181no—
93C2×D4C22⋊D10911yescharacteristic
94C22×C4—21no—
95SD16—4362no—
96D8—4362no—
97C2×C8—2182no—
98C2×C8—2181no—
99C4⋊C4—22no—
100C4⋊C4—22no—
101C4⋊C4—22no—
102C4⋊C4C22⋊D10912yes—
103C4⋊C4C22⋊D10912yes—
104C22⋊C4—22no—
105C22⋊C4—21no—
106C22⋊C4—22no—
107C42—21no—
108C42C22×D10911yescharacteristic
109C23—21no—
110C23—4361no—
111Q8—41no—
112Q8—81no—
113D4—41no—
114D4—4361no—
115D4—4361no—
116D4—4364no—
117C2×C4—22no—
118C2×C4—21no—
119C2×C4—22no—
120C2×C4—81no—
121C2×C4C2×C22⋊D10912yes—
122C2×C4C2×C22⋊D10911yescharacteristic
123C8—4362no—
124C8—2182no—
125C22—41no—
126C22—42no—
127C22—8721no—
128C22—8722no—
129C22#1681266811yescharacteristic
130C4—21no—
131C4—22no—
132C4—42no—
133C4—41no—
134C4—42no—
135C2—81no—
136C2—17441no—
137C2#2029519411yescharacteristic
138C2#2029519912yes—
subgroupG/H
centerC22#16812668
commutator, cosocleC42×C109C23
FrattiniC42C22×D109
Fitting#563×C109C2
socleC22×C109#120
prime radical#20295416C2
hypercenter#563D109
hypocenterC109#1657
LCSG ▷ C42×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
2. #20295392C2semidirect
3. #20294768C2semidirect
4. #20294766C2semidirect
5. #563×C109C2semidirect
70. C109#1657semidirect
71. #563D109semidirect

Maximal subgroups

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

subgroupindex
#202954162
#202953922
#202947682
#202947662
#563×C1092
#1657109