Finite Group Databasebeta

← #28364256

Subgroups — #28364256

Order 13952.

306 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 69 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#20295607C211yesmaximal, characteristic
2C2×#16812627C212yesmaximal, complemented
3#20295025C212yesmaximal, complemented
4#20294795C211yesmaximal, characteristic, complemented
5#384×C109C211yesmaximal, characteristic, complemented
6C2×#13182029C2212yes—
7C2×#13182013C2212yes—
8#16812679—22no—
9#16812674—22no—
10#16812673—22no—
11#16812627—24no—
12#16812622—22no—
13#16812619—22no—
14C2×C4⋊C4×C109—22no—
15C2×C4⋊C4×C109C2211yescharacteristic
16C2×C4⋊C4×C109C2211yescharacteristic
17C2×C42×C109C2211yescharacteristic
18#13182029—22no—
19#13182029—22no—
20#13182029—22no—
21#13182029—22no—
22C2×Dic218D412yes—
23C2×D436—22no—
24C2×D436D412yes—
25C2×C22⋊D109—22no—
26C2×C4×D109—22no—
27#13182013—22no—
28#13182013—22no—
29C2×#11724784—22no—
30C4⋊Dic109—22no—
31#13182006—24no—
32C22×C4×C109—21no—
33C22×C4×C109—21no—
34C22×C4×C109D412yes—
35C22×C4×C109C2311yescharacteristic
36C4⋊C4×C109—22no—
37C4⋊C4×C109—22no—
38C4⋊C4×C109—44no—
39C42×C109—21no—
40C42×C109D412yescomplemented
41C23×C109C2×D411yescharacteristic
42C22×D109—22no—
43C2×C4×C109—22no—
44C2×C4×C109—22no—
45C2×C4×C109—22no—
46C2×C4×C109—22no—
47C2×C4×C109—42no—
48C2×C4×C109—42no—
49C2×C4×C109—41no—
50C2×C4×C109C2×D412yes—
51C2×C4×C109C2×D411yescharacteristic
52C2×C4×C109#4111yescharacteristic
53C2×C4×C109#4112yes—
54C22⋊D109—42no—
55C22⋊D109—42no—
56C2×Dic109—22no—
57D436—22no—
58D436—22no—
59D436—82no—
60C4×D109—42no—
61C4×D109—42no—
62Dic218—22no—
63Dic218—22no—
64#11724784—42no—
65C22×C109—21no—
66C22×C109—21no—
67C22×C109#12011yescharacteristic
68C22×C109#12111yescharacteristic
69C22×C109#12411yescharacteristic
70C2×D109—42no—
71C2×D109—42no—
72C2×D109—42no—
73C4×C109—24no—
74C4×C109—42no—
75C4×C109—81no—
76C4×C109#12112yes—
77C4×C109#12312yes—
78Dic109—42no—
79C2×C109—21no—
80C2×C109—21no—
81C2×C109#39411yescharacteristic
82C2×C109#45311yescharacteristic
83C2×C109#45611yescharacteristic
84D109—82no—
85#2039—1091nomaximal, Sylow
86C109#203911yescharacteristic, complemented, Sylow
87#537—1091no—
88C2×#104—1092no—
89#417—1092no—
90#384D10911yescharacteristic, complemented
91#343—1091no—
92C2×#41—1092no—
93C2×M16—1092no—
94C22⋊Q8—2182no—
95#121—2182no—
96#121—2182no—
97C2×C4⋊C4—22no—
98C2×C4⋊C4C2×D10911yescharacteristic
99C2×C4⋊C4C2×D10911yescharacteristic
100C2×C42C2×D10911yescharacteristic
101#104—2184no—
102#103—2182no—
103#102—2182no—
104#41—2182no—
105#41—2182no—
106#41—2182no—
107#41—2182no—
108C2×Q8—1092no—
109C2×D4—1092no—
110C2×D4—2182no—
111C2×D4—2182no—
112C22×C4—21no—
113C22×C4—21no—
114C22×C4—2182no—
115C22×C4C22⋊D10912yes—
116C22×C4C22×D10911yescharacteristic
117M16—2182no—
118M16—2182no—
119C2×C8—2182no—
120C4⋊C4—22no—
121C4⋊C4—22no—
122C4⋊C4—44no—
123C4⋊C4—2182no—
124C22⋊C4—2184no—
125C42—21no—
126C42D43612yescomplemented
127C23—2182no—
128C23C2×C22⋊D10911yescharacteristic
129Q8—2182no—
130Q8—2182no—
131D4—8722no—
132D4—4362no—
133D4—4362no—
134D4—2182no—
135D4—2182no—
136C2×C4—22no—
137C2×C4—22no—
138C2×C4—22no—
139C2×C4—22no—
140C2×C4—42no—
141C2×C4—42no—
142C2×C4—41no—
143C2×C4—4362no—
144C2×C4—4362no—
145C2×C4—2182no—
146C2×C4D4×D10912yes—
147C2×C4C2×D43611yescharacteristic
148C2×C4#1318202911yescharacteristic
149C2×C4#1318203012yes—
150C8—4362no—
151C22—21no—
152C22—21no—
153C22—4362no—
154C22—4362no—
155C22—4362no—
156C22#1681266711yescharacteristic
157C22#1681267311yescharacteristic
158C22#1681268811yescharacteristic
159C4—24no—
160C4—42no—
161C4—81no—
162C4—4362no—
163C4#1681267412yes—
164C4#1681268412yes—
165C2—21no—
166C2—21no—
167C2—8722no—
168C2#2029494311yescharacteristic
169C2#2029519311yescharacteristic
170C2#2029520811yescharacteristic
subgroupG/H
centerC22#16812667
commutator, cosocleC22×C4×C109C23
FrattiniC22×C4C22×D109
Fitting#384×C109C2
socleC22×C109#120
prime radical#20295607C2
hypercenter#384D109
hypocenterC109#2039
LCSG ▷ C22×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #384

Complements

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

NH ≅ G/Ntype
2. C2×#16812627C2semidirect
3. #20295025C2semidirect
4. #20294795C2semidirect
5. #384×C109C2semidirect
40. C42×C109D4semidirect
86. C109#2039semidirect
90. #384D109semidirect
126. C42D436semidirect

Maximal subgroups

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

subgroupindex
#202956072
C2×#168126272
#202950252
#202947952
#384×C1092
#2039109