Finite Group Databasebeta

← #28365684

Subgroups — #28365684

Order 13952.

444 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 189 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
1C2×C4×Dic218C211yesmaximal, characteristic
2#20294932C212yesmaximal
3#20294924C212yesmaximal
4#20294908C214yesmaximal
5#20294893C212yesmaximal
6#20294890C212yesmaximal
7C4×C4⋊Dic109C211yesmaximal, characteristic
8#376×C109C211yesmaximal, characteristic
9C22×Dic218C2211yescharacteristic
10C4×Dic218C418yescomplemented
11C2×C4⋊Dic109C2212yes—
12C2×C4⋊Dic109C2211yescharacteristic
13C2×#13182008C2214yes—
14C2×#13182008C2212yes—
15C2×C4×Dic109C2212yes—
16C2×C4×Dic109C2212yes—
17#16812606C2214yes—
18#16812606C2214yes—
19#16812606C2214yes—
20#16812606C2212yes—
21C2×C42×C109C2212yes—
22C2×C42×C109C2211yescharacteristic
23#95×C109C2211yescharacteristic
24#95×C109C2211yescharacteristic
25#95×C109C2212yes—
26C2×Dic218—24no—
27C2×Dic218C2×C414yes—
28C22×Dic109C2314yes—
29C22×Dic109C2312yes—
30C22×Dic109C2312yes—
31C4⋊Dic109—24no—
32C4⋊Dic109C2×C414yes—
33#13182008—44no—
34#13182008C2×C418yes—
35C4×Dic109—42no—
36C4×Dic109C2×C418yes—
37C22×C4×C109C2312yes—
38C22×C4×C109C2312yes—
39C22×C4×C109C2311yescharacteristic
40C22×C4×C109C2311yescharacteristic
41C22×C4×C109C2311yescharacteristic
42C42×C109—24no—
43C42×C109C2×C414yes—
44C23×C109C2411yescharacteristic
45C2×C4×C109—21no—
46C2×C4×C109—22no—
47C2×C4×C109—21no—
48C2×C4×C109—22no—
49C2×C4×C109—22no—
50C2×C4×C109—22no—
51C2×C4×C109—22no—
52C2×C4×C109—22no—
53C2×C4×C109—22no—
54C2×C4×C109C22×C412yes—
55C2×C4×C109C22×C412yes—
56C2×C4×C109C22×C412yes—
57C2×C4×C109#4114yes—
58C2×Dic109—24no—
59C2×Dic109—22no—
60C2×Dic109—22no—
61C2×Dic109—24no—
62C2×Dic109—24no—
63C2×Dic109—24no—
64C2×Dic109C22×C414yes—
65C2×Dic109C22×C414yes—
66Dic218—28no—
67C22×C109C23×C411yescharacteristic
68C22×C109C2×#4112yes—
69C22×C109H(5,𝔽2)11yescharacteristic
70C22×C1092−1+412yes—
71C22×C1092−1+411yescharacteristic
72C4×C109—22no—
73C4×C109—22no—
74C4×C109—42no—
75C4×C109—42no—
76C4×C109#11714yes—
77Dic109—24no—
78Dic109—24no—
79Dic109—44no—
80C2×C109C2×#11711yescharacteristic
81C2×C109#51911yescharacteristic
82C2×C109#52011yescharacteristic
83C2×C109#54112yes—
84C2×C109#54212yes—
85#2320—1091nomaximal, Sylow
86C109#232011yescharacteristic, complemented, Sylow
87C2×C4×Q8—1091no—
88#391—1092no—
89#389—1092no—
90#387—1094no—
91#383—1092no—
92#382—1092no—
93C4×C4⋊C4—1091no—
94#376D10911yescharacteristic
95C22×Q8—1091no—
96C4×Q8—1098no—
97C2×C4⋊C4—1092no—
98C2×C4⋊C4—1091no—
99C2×C4⋊C4—1094no—
100C2×C4⋊C4—1092no—
101C2×C42—1092no—
102C2×C42—1092no—
103C2×C42C2×D10912yes—
104C2×C42C2×D10911yescharacteristic
105#95—1092no—
106#95—1094no—
107#95—1094no—
108#95—1094no—
109#95C2×D10911yescharacteristic
110#95C2×D10911yescharacteristic
111#95C2×D10912yes—
112C2×Q8—1094no—
113C2×Q8—2184no—
114C22×C4—1092no—
115C22×C4—1092no—
116C22×C4—1094no—
117C22×C4C22×D10912yes—
118C22×C4C22×D10912yes—
119C22×C4C22×D10911yescharacteristic
120C22×C4C22×D10911yescharacteristic
121C22×C4C22×D10911yescharacteristic
122C4⋊C4—1098no—
123C4⋊C4—1094no—
124C4⋊C4—4364no—
125C4⋊C4—2184no—
126C42—24no—
127C42—1098no—
128C42—4362no—
129C42C4×D10914yes—
130C23C23×D10911yescharacteristic
131Q8—2188no—
132C2×C4—21no—
133C2×C4—22no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—22no—
141C2×C4—1094no—
142C2×C4—1094no—
143C2×C4—2184no—
144C2×C4—2184no—
145C2×C4—2184no—
146C2×C4—2184no—
147C2×C4—2182no—
148C2×C4—2182no—
149C2×C4C2×C4×D10912yes—
150C2×C4C2×C4×D10912yes—
151C2×C4C2×C4×D10912yes—
152C2×C4#1318202914yes—
153C22C22×C4×D10911yescharacteristic
154C22C2×#1318202912yes—
155C22#1681274511yescharacteristic
156C22#1681274612yes—
157C22#1681274611yescharacteristic
158C4—22no—
159C4—22no—
160C4—42no—
161C4—42no—
162C4—4364no—
163C4—2184no—
164C4—2184no—
165C4#1681265614yes—
166C2C2×#1681265611yescharacteristic
167C2#2029550311yescharacteristic
168C2#2029550611yescharacteristic
169C2#2029563812yes—
170C2#2029564712yes—
subgroupG/H
center, FrattiniC23C23×D109
commutatorC22×C109C23×C4
Fitting#376×C109C2
socle, prime radical, cosocleC23×C109C24
hypercenter#376D109
hypocenterC109#2320
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C23 ◁ #376

Complements

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

NH ≅ G/Ntype
10. C4×Dic218C4semidirect
86. C109#2320semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic2182
#202949322
#202949242
#202949082
#202948932
#202948902
C4×C4⋊Dic1092
#376×C1092
#2320109