Finite Group Databasebeta

← #28364257

Subgroups — #28364257

Order 13952.

306 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 67 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×#16812625C212yesmaximal, complemented
3#20295028C212yesmaximal, complemented
4#20294895C211yesmaximal, characteristic, complemented
5#343×C109C211yesmaximal, characteristic, complemented
6C2×#13182029C2212yes—
7#16812679—22no—
8#16812674—22no—
9#16812673—22no—
10C2×C4⋊Dic109C2211yescharacteristic
11C2×#13182008—22no—
12C2×C4×Dic109C2211yescharacteristic
13#16812625—24no—
14#16812624—22no—
15#16812621—22no—
16C2×M16×C109C2212yes—
17C2×C4⋊C4×C109C2211yescharacteristic
18#13182029—22no—
19#13182029—22no—
20#13182029—22no—
21#13182029—22no—
22C2×Dic218D412yes—
23C2×D436—22no—
24C2×D436D412yes—
25C2×C22⋊D109—22no—
26C22×Dic109—21no—
27C22×Dic109—21no—
28C2×C4×D109—22no—
29C4⋊Dic109—22no—
30C4⋊Dic109—22no—
31#13182008—44no—
32#13182006—24no—
33C4×Dic109—21no—
34C4×Dic109D412yescomplemented
35C22×C4×C109D412yes—
36C22×C4×C109C2311yescharacteristic
37M16×C109—22no—
38M16×C109—22no—
39C2×C8×C109—22no—
40C4⋊C4×C109—22no—
41C23×C109C2×D411yescharacteristic
42C22×D109—22no—
43C2×C4×C109—22no—
44C2×C4×C109—42no—
45C2×C4×C109C2×D411yescharacteristic
46C2×C4×C109C2×D412yes—
47C2×C4×C109#4112yes—
48C2×C4×C109#4111yescharacteristic
49C22⋊D109—42no—
50C22⋊D109—42no—
51C2×Dic109—22no—
52C2×Dic109—22no—
53C2×Dic109—22no—
54C2×Dic109—22no—
55C2×Dic109—42no—
56C2×Dic109—41no—
57D436—22no—
58D436—22no—
59D436—82no—
60C4×D109—42no—
61C4×D109—42no—
62Dic218—22no—
63Dic218—22no—
64C8×C109—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—42no—
74C4×C109#12112yes—
75C4×C109#12312yes—
76Dic109—24no—
77Dic109—42no—
78Dic109—81no—
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#384—1091no—
91#343D10911yescharacteristic, complemented
92C2×#41—1092no—
93C2×M16C2×D10912yes—
94C22⋊Q8—2182no—
95#121—2182no—
96#121—2182no—
97C2×C4⋊C4—1091no—
98C2×C4⋊C4—2182no—
99C2×C4⋊C4C2×D10911yescharacteristic
100C2×C42—1091no—
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—2181no—
113C22×C4—2182no—
114C22×C4—2181no—
115C22×C4D43612yes—
116C22×C4C22×D10911yescharacteristic
117M16—22no—
118M16—22no—
119C2×C8—22no—
120C4⋊C4—22no—
121C4⋊C4—4364no—
122C4⋊C4—2182no—
123C4⋊C4—2182no—
124C22⋊C4—2184no—
125C42—1092no—
126C42—2181no—
127C23—2182no—
128C23C2×D43611yescharacteristic
129Q8—2182no—
130Q8—2182no—
131D4—8722no—
132D4—4362no—
133D4—4362no—
134D4—2182no—
135D4—2182no—
136C2×C4—22no—
137C2×C4—42no—
138C2×C4—4362no—
139C2×C4—4361no—
140C2×C4—4362no—
141C2×C4—4362no—
142C2×C4—2182no—
143C2×C4—2182no—
144C2×C4—2182no—
145C2×C4—2182no—
146C2×C4D4×D10911yescharacteristic
147C2×C4D4×D10912yes—
148C2×C4#1318202912yes—
149C2×C4#1318203011yescharacteristic
150C8—42no—
151C22—21no—
152C22—21no—
153C22—4362no—
154C22—4362no—
155C22—4362no—
156C22#1681266911yescharacteristic
157C22#1681267411yescharacteristic
158C22#1681268711yescharacteristic
159C4—42no—
160C4—8721no—
161C4—4362no—
162C4—2184no—
163C4#1681267212yes—
164C4#1681268412yes—
165C2—21no—
166C2—21no—
167C2—8722no—
168C2#2029494411yescharacteristic
169C2#2029519611yescharacteristic
170C2#2029521111yescharacteristic
subgroupG/H
centerC22#16812669
commutator, cosocleC22×C4×C109C23
FrattiniC22×C4C22×D109
Fitting#343×C109C2
socleC22×C109#120
prime radical#20295607C2
hypercenter#343D109
hypocenterC109#2039
LCSG ▷ C22×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
2. C2×#16812625C2semidirect
3. #20295028C2semidirect
4. #20294895C2semidirect
5. #343×C109C2semidirect
34. C4×Dic109D4semidirect
86. C109#2039semidirect
91. #343D109semidirect

Maximal subgroups

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

subgroupindex
#202956072
C2×#168126252
#202950282
#202948952
#343×C1092
#2039109