Finite Group Databasebeta

← #28379316

Subgroups — #28379316

Order 13952.

238 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 54 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#20296005C212yesmaximal
2#20295933C214yesmaximal, complemented
3#20295602C211yesmaximal, characteristic
4#16812778—22no—
5#16812758C2214yes—
6#16812757—22no—
7C2×#13182029C2211yescharacteristic
8C22×Dic218C411yescharacteristic, complemented
9#16812699C412yescomplemented
10#16812687C2212yes—
11#16812683—22no—
12#16812679—22no—
13#117×C109C411yescharacteristic, complemented
14C4⋊#89137—42no—
15C22⋊#89137—24no—
16C4×#89137—42no—
17#13182029—42no—
18C2×Dic218—21no—
19C2×Dic218—42no—
20C2×Dic218C2×C412yes—
21C2×Dic218C2×C411yescharacteristic
22C2×D436C2311yescharacteristic
23C2×C22⋊D109D412yes—
24C22×Dic109—21no—
25C2×C4×D109D412yes—
26C4⋊Dic109—24no—
27#13182006—24no—
28C22×C4×C109C2×C411yescharacteristic
29C4⋊C4×C109—21no—
30C22⋊C4×C109—21no—
31C42×C109C2×C412yes—
32C23×C109C22⋊C411yescharacteristic
33C22×D109C2×D412yes—
34C2×#89137—44no—
35C2×C4×C109—22no—
36C2×C4×C109—22no—
37C2×C4×C109C22⋊C411yescharacteristic
38C2×C4×C109C22×C411yescharacteristic
39C22⋊D109—42no—
40C2×Dic109—22no—
41C2×Dic109—42no—
42C2×Dic109C22⋊C412yes—
43D436—41no—
44C4×D109—22no—
45C4×D109—22no—
46Dic218—41no—
47Dic218—42no—
48Dic218—42no—
49C22×C109—21no—
50C22×C109—41no—
51C22×C109C2×C22⋊C411yescharacteristic
52C2×D109—22no—
53C2×D109—42no—
54#89137—84no—
55C4×C109—21no—
56C4×C109—42no—
57C4×C109#9912yes—
58Dic109—22no—
59Dic109—42no—
60C2×C109—21no—
61C2×C109—41no—
62C2×C109C2×#9911yescharacteristic
63D109—42no—
64#2129—1091nomaximal, Sylow
65C109#212911yescharacteristic, complemented, Sylow
66#536—1091no—
67#410—1092no—
68#354—1094no—
69C2×#41—1091no—
70C22×Q8—1091no—
71C4⋊Q8—1092no—
72#124—1092no—
73#123—2182no—
74C22⋊Q8—2182no—
75#117—2182no—
76#117#8913711yescharacteristic, complemented
77#99—1094no—
78#99—2182no—
79#41—4362no—
80C2×Q8—1091no—
81C2×Q8—1092no—
82C2×Q8—4362no—
83C2×Q8—2181no—
84C2×D4—1092no—
85C2×D4—1091no—
86C22×C4—1092no—
87C22×C4—2181no—
88C22×C4C2×#8913711yescharacteristic
89C4⋊C4—21no—
90C4⋊C4—4362no—
91C4⋊C4—2184no—
92C22⋊C4—21no—
93C22⋊C4—2184no—
94C22⋊C4—2184no—
95C42—4362no—
96C42C2×#8913712yes—
97C23—1092no—
98C23C22⋊#8913711yescharacteristic
99Q8—4362no—
100Q8—4361no—
101Q8—4362no—
102D4—4361no—
103D4—4362no—
104C2×C4—22no—
105C2×C4—22no—
106C2×C4—1092no—
107C2×C4—4364no—
108C2×C4—4362no—
109C2×C4—2182no—
110C2×C4—2182no—
111C2×C4—2182no—
112C2×C4C22⋊#8913711yescharacteristic
113C2×C4C22×#8913711yescharacteristic
114C22—21no—
115C22—41no—
116C22—4362no—
117C22—2182no—
118C22C2×C22⋊#8913711yescharacteristic
119C4—21no—
120C4—42no—
121C4—8724no—
122C4—4362no—
123C4—2182no—
124C4#1681275812yes—
125C2—21no—
126C2—41no—
127C2—4362no—
128C2C2×#1681275811yescharacteristic
subgroupG/H
centerC2C2×#16812758
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
Fitting#117×C109C4
socleC2×C109C2×#99
prime radicalC2×#13182029C22
cosocleC2×D436C23
hypercenter#117#89137
hypocenterC109#2129
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #20295933C2semidirect
8. C22×Dic218C4semidirect
9. #16812699C4semidirect
13. #117×C109C4semidirect
65. C109#2129semidirect
76. #117#89137semidirect

Maximal subgroups

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

subgroupindex
#202960052
#202959332
#202956022
#2129109