Finite Group Databasebeta

← #28363564

Subgroups — #28363564

Order 13952.

362 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 89 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#20295593C211yesmaximal, characteristic
2#20295044C212yesmaximal, complemented
3C2×#16812616C211yesmaximal, characteristic, complemented
4#20294782C212yesmaximal, complemented
5#518×C109C211yesmaximal, characteristic, complemented
6C2×#13182029C2211yescharacteristic
7C2×#13182013C2212yes—
8#16812695C412yescomplemented
9#16812686C412yescomplemented
10#16812670—22no—
11#16812668—22no—
12#16812658C2211yescharacteristic
13#16812627—24no—
14#16812616—22no—
15C22×D4×C109C2211yescharacteristic
16C4×D4×C109—24no—
17#117×C109C2212yes—
18C2×C22⋊C4×C109—22no—
19#13182029—22no—
20#13182029—22no—
21C2×Dic218C2×C411yescharacteristic, complemented
22C2×D436C2×C411yescharacteristic, complemented
23C2×C22⋊D109—21no—
24C2×C22⋊D109—24no—
25C2×C4×D109—21no—
26#13182013—22no—
27#13182013—22no—
28C2×#11724784—22no—
29C4⋊Dic109—21no—
30C22⋊Dic109—24no—
31C22⋊Dic109—21no—
32C4×Dic109C2×C412yescomplemented
33C24×C109D412yes—
34C2×D4×C109—22no—
35C2×D4×C109—22no—
36C2×D4×C109C2×C412yes—
37C2×D4×C109D412yes—
38C22×C4×C109—24no—
39C22×C4×C109C2311yescharacteristic
40C4⋊C4×C109—22no—
41C22⋊C4×C109—24no—
42C22⋊C4×C109—22no—
43C42×C109D414yescomplemented
44C23×C109—22no—
45C23×C109—22no—
46C23×C109—82no—
47C23×C109C22⋊C414yes—
48C23×C109C2×D411yescharacteristic
49C22×D109—21no—
50D4×C109—24no—
51D4×C109—24no—
52C2×C4×C109—24no—
53C2×C4×C109—84no—
54C2×C4×C109C22×C411yescharacteristic
55C2×C4×C109C2×D412yes—
56C2×C4×C109C2×D411yescharacteristic
57C2×C4×C109#4112yes—
58C22⋊D109—41no—
59C22⋊D109—41no—
60C22⋊D109—84no—
61C2×Dic109—22no—
62C2×Dic109—21no—
63D436—22no—
64C4×D109—42no—
65Dic218—22no—
66#11724784—42no—
67C22×C109—24no—
68C22×C109—41no—
69C22×C109—44no—
70C22×C109—44no—
71C22×C109—82no—
72C22×C109C2×C22⋊C411yescharacteristic
73C22×C109#12011yescharacteristic
74C22×C109#12311yescharacteristic
75C2×D109—42no—
76C2×D109—41no—
77C4×C109—44no—
78C4×C109C4×D412yes—
79C4×C109#12112yes—
80Dic109—42no—
81Dic109—41no—
82C2×C109—21no—
83C2×C109—21no—
84C2×C109—21no—
85C2×C109—44no—
86C2×C109#38611yescharacteristic
87D109—81no—
88#1911—1091nomaximal, Sylow
89C109#191111yescharacteristic, complemented, Sylow
90#535—1091no—
91#518D10911yescharacteristic, complemented
92#421—1092no—
93C2×#100—1091no—
94H(3,ℤ/4)—1092no—
95C2×#41—1091no—
96C22×D4C2×D10911yescharacteristic
97C2×M16—1092no—
98#127—1092no—
99#124—1092no—
100#121—2182no—
101#120—2182no—
102C4×D4—24no—
103#117—1091no—
104#117C2×D10912yes—
105C2×C22⋊C4—22no—
106#104—2184no—
107#100—2182no—
108C24C22⋊D10912yes—
109#41—2182no—
110#41—2182no—
111C2×Q8—1091no—
112C2×D4—22no—
113C2×D4—22no—
114C2×D4—1091no—
115C2×D4—2184no—
116C2×D4—2181no—
117C2×D4C4×D10912yes—
118C2×D4D43612yes—
119C22×C4—24no—
120C22×C4—2181no—
121C22×C4C22×D10911yescharacteristic
122M16—2182no—
123M16—2182no—
124C2×C8—2182no—
125C4⋊C4—22no—
126C4⋊C4—2181no—
127C22⋊C4—24no—
128C22⋊C4—22no—
129C22⋊C4—2184no—
130C22⋊C4—2181no—
131C42—1092no—
132C42C22⋊D10914yescomplemented
133C23—22no—
134C23—22no—
135C23—82no—
136C23—2181no—
137C23#1318200614yes—
138C23C2×C22⋊D10911yescharacteristic
139Q8—2182no—
140D4—24no—
141D4—24no—
142D4—8724no—
143D4—4361no—
144D4—4361no—
145D4—2182no—
146C2×C4—24no—
147C2×C4—84no—
148C2×C4—4362no—
149C2×C4—2182no—
150C2×C4—2181no—
151C2×C4C2×C4×D10911yescharacteristic
152C2×C4C2×C22⋊D10912yes—
153C2×C4C2×D43611yescharacteristic
154C2×C4#1318202912yes—
155C8—4362no—
156C22—24no—
157C22—41no—
158C22—44no—
159C22—44no—
160C22—82no—
161C22—4362no—
162C22—4361no—
163C22C2×#1318200611yescharacteristic
164C22#1681266811yescharacteristic
165C22#1681268211yescharacteristic
166C4—44no—
167C4—4362no—
168C4—4361no—
169C4C4×C22⋊D10912yes—
170C4#1681267312yes—
171C2—21no—
172C2—21no—
173C2—21no—
174C2—44no—
175C2—8721no—
176C2#2029490511yescharacteristic
subgroupG/H
centerC2#20294905
commutatorC2×C4×C109C22×C4
FrattiniC22×C4C22×D109
Fitting#518×C109C2
socleC2×C109#386
prime radical#20295593C2
cosocleC22×C4×C109C23
hypercenter#518D109
hypocenterC109#1911
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22×C4 ◁ #518

Complements

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

NH ≅ G/Ntype
2. #20295044C2semidirect
3. C2×#16812616C2semidirect
4. #20294782C2semidirect
5. #518×C109C2semidirect
8. #16812695C4semidirect
9. #16812686C4semidirect
21. C2×Dic218C2×C4semidirect
22. C2×D436C2×C4semidirect
32. C4×Dic109C2×C4semidirect
43. C42×C109D4semidirect
89. C109#1911semidirect
91. #518D109semidirect
132. C42C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
#202955932
#202950442
C2×#168126162
#202947822
#518×C1092
#1911109