Finite Group Databasebeta

← #28378455

Subgroups — #28378455

Order 13952.

208 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 56 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#20296049C211yesmaximal, characteristic
2#20295971C211yesmaximal, characteristic
3#20295879C214yesmaximal
4#20295581C211yesmaximal, characteristic
5#16812766—21no—
6#16812756C2212yes—
7#16812754—21no—
8(C2×#89137)⋊C4—21no—
9(C2×#89137)⋊C4—21no—
10(C2×#89137)⋊C4C2212yes—
11#16812699C411yescharacteristic, complemented
12#16812690—21no—
13#16812679C2212yes—
14#16812675—21no—
15Q8×Dic109—21no—
16#16812664C412yescomplemented
17#16812657—21no—
18C42×D109C2211yescharacteristic
19C4⋊Q8×C109C411yescharacteristic, complemented
20C22×#89137—22no—
21C2×#11724786—22no—
22C2×Dic218C2×C412yes—
23C2×C4×D109D411yescharacteristic
24C2×C4×D109D411yescharacteristic
25C2×C4×D109C2311yescharacteristic
26C4⋊Dic109—21no—
27C4⋊Dic109—21no—
28C4⋊Dic109—22no—
29#13182008—22no—
30#13182006—22no—
31C4×Dic109—22no—
32C4×Dic109C2×C411yescharacteristic
33C4×Dic109D411yescharacteristic
34C4×Dic109D411yescharacteristic
35C2×Q8×C109—21no—
36C4⋊C4×C109—21no—
37C4⋊C4×C109C2×C412yes—
38C42×C109C2×C411yescharacteristic
39C22×D109C2×D411yescharacteristic
40C2×#89137—42no—
41C2×#89137—42no—
42C2×#89137—42no—
43Q8×C109—42no—
44C2×C4×C109—22no—
45C2×C4×C109C22⋊C411yescharacteristic
46C2×C4×C109C22⋊C411yescharacteristic
47C2×C4×C109C22×C411yescharacteristic
48C2×Dic109—22no—
49C2×Dic109C22⋊C411yescharacteristic
50C2×Dic109C22⋊C411yescharacteristic
51C2×Dic109C2×D411yescharacteristic
52C4×D109—22no—
53C4×D109—41no—
54C4×D109—41no—
55Dic218—42no—
56#11724786—42no—
57C22×C109C2×C22⋊C411yescharacteristic
58C2×D109—21no—
59C2×D109—21no—
60C2×D109#13712yes—
61#89137—82no—
62C4×C109—21no—
63C4×C109—21no—
64C4×C109—21no—
65C4×C109—42no—
66Dic109—21no—
67Dic109—21no—
68Dic109—21no—
69Dic109—42no—
70C2×C109#41011yescharacteristic
71C2×C109#41911yescharacteristic
72C2×C109#42111yescharacteristic
73D109—22no—
74#1515—1091nomaximal, Sylow
75C109#151511yescharacteristic, complemented, Sylow
76#533—1091no—
77#432—1091no—
78#383—1091no—
79#328—1094no—
80C4⋊Q8—1091no—
81C4⋊Q8#8913711yescharacteristic, complemented
82#125—2181no—
83C22⋊Q8—1092no—
84C22⋊Q8—2181no—
85C4×Q8—1092no—
86C4×Q8—2181no—
87#117—2181no—
88C2×C42—1091no—
89C4⋊C8—2181no—
90C22⋊C8—1092no—
91#97—2181no—
92#95—1092no—
93#95—2181no—
94#95—2181no—
95C2×Q8—21no—
96C2×Q8—1092no—
97C22×C4—1091no—
98C22×C4—1091no—
99C22×C4—1091no—
100C22×C4—2182no—
101C2×C8—2182no—
102C4⋊C4—21no—
103C4⋊C4—2182no—
104C4⋊C4—2182no—
105C4⋊C4—2181no—
106C4⋊C4—2181no—
107C4⋊C4C2×#8913712yes—
108C22⋊C4—2182no—
109C42—1091no—
110C42—1091no—
111C42—1091no—
112C42—2182no—
113C42C2×#8913711yescharacteristic
114C23—1091no—
115Q8—42no—
116Q8—4362no—
117C2×C4—22no—
118C2×C4—1091no—
119C2×C4—1091no—
120C2×C4—1091no—
121C2×C4—4361no—
122C2×C4—4361no—
123C2×C4—4362no—
124C2×C4—4362no—
125C2×C4—4362no—
126C2×C4—2182no—
127C2×C4—2182no—
128C2×C4C22⋊#8913711yescharacteristic
129C2×C4C22⋊#8913711yescharacteristic
130C2×C4C22×#8913711yescharacteristic
131C8—4362no—
132C22—1092no—
133C22—2181no—
134C22—2181no—
135C22C2×C22⋊#8913711yescharacteristic
136C4—21no—
137C4—21no—
138C4—21no—
139C4—42no—
140C4—8722no—
141C4—4362no—
142C4—2181no—
143C4—2181no—
144C4—2181no—
145C2—2182no—
146C2#2029600711yescharacteristic
147C2#2029601911yescharacteristic
148C2#2029602411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
FittingC4⋊Q8×C109C4
socleC22×C109C2×C22⋊C4
prime radicalC22×D109C2×D4
cosocleC2×C4×D109C23
hypercenterC4⋊Q8#89137
hypocenterC109#1515
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
11. #16812699C4semidirect
16. #16812664C4semidirect
19. C4⋊Q8×C109C4semidirect
75. C109#1515semidirect
81. C4⋊Q8#89137semidirect

Maximal subgroups

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

subgroupindex
#202960492
#202959712
#202958792
#202955812
#1515109