Finite Group Databasebeta

← #28362003

Subgroups — #28362003

Order 13952.

282 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 53 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
2#20295167C212yesmaximal, complemented
3#20294946C211yesmaximal, characteristic, complemented
4#20294761C212yesmaximal, complemented
5#324×C109C211yesmaximal, characteristic, complemented
6C2×#13182029—21no—
7C22×Dic218C2211yescharacteristic
8C2×#13182018—22no—
9#16812679—22no—
10#16812674—22no—
11#16812673—21no—
12#16812673C2212yes—
13C2×#13182008—21no—
14#16812624—22no—
15#16812606—21no—
16#16812606C2211yescharacteristic
17C2×C4⋊C4×C109C2211yescharacteristic
18C22⋊C8×C109C2212yes—
19#13182029—42no—
20#13182029—42no—
21C2×Dic218—22no—
22C2×Dic218—21no—
23C2×Dic218—42no—
24C2×D436—21no—
25C2×D436—21no—
26C2×D436D412yes—
27C2×C22⋊D109—22no—
28C22×Dic109—21no—
29C22×Dic109—21no—
30C2×C4×D109—22no—
31#13182018—44no—
32C4⋊Dic109—21no—
33C4⋊Dic109D412yes—
34#13182008—42no—
35#13182006—22no—
36#13182006—22no—
37C22×C4×C109D411yescharacteristic
38C22×C4×C109D411yescharacteristic
39C22×C4×C109C2311yescharacteristic
40C2×C8×C109—22no—
41C4⋊C4×C109—22no—
42C23×C109C2×D411yescharacteristic
43C22×D109—22no—
44C2×C4×C109—21no—
45C2×C4×C109—21no—
46C2×C4×C109—21no—
47C2×C4×C109—41no—
48C2×C4×C109—41no—
49C2×C4×C109SD1612yescomplemented
50C2×C4×C109C2×D412yes—
51C22⋊D109—42no—
52C22⋊D109—42no—
53C2×Dic109—22no—
54C2×Dic109—22no—
55C2×Dic109—22no—
56C2×Dic109—41no—
57C2×Dic109—41no—
58C2×Dic109—41no—
59C2×Dic109—41no—
60D436—42no—
61D436—42no—
62D436—81no—
63C4×D109—44no—
64Dic218—24no—
65Dic218—42no—
66Dic218—42no—
67C8×C109—42no—
68C22×C109—21no—
69C22×C109—21no—
70C22×C109#12011yescharacteristic
71C22×C109C2×SD1611yescharacteristic
72C22×C109AGL(1,ℤ/8)11yescharacteristic
73C2×D109—42no—
74C2×D109—42no—
75C2×D109—42no—
76C4×C109—22no—
77C4×C109—22no—
78C4×C109—41no—
79Dic109—42no—
80Dic109—42no—
81Dic109—42no—
82C2×C109—21no—
83C2×C109—21no—
84C2×C109#45011yescharacteristic
85C2×C109#45611yescharacteristic
86C2×C109B(4,𝔽2)11yescharacteristic
87D109—82no—
88#1604—1091nomaximal, Sylow
89C109#160411yescharacteristic, complemented, Sylow
90#537—1091no—
91#448—1092no—
92#395—1091no—
93#327—1092no—
94#324D10911yescharacteristic, complemented
95C2×#41—2181no—
96C22×Q8—1091no—
97C2×SD16—2182no—
98C22⋊Q8—2182no—
99#121—1092no—
100#121—2182no—
101#121—2181no—
102C2×C4⋊C4—2181no—
103C2×C4⋊C4C2×D10911yescharacteristic
104#103—2182no—
105C22⋊C8C2×D10912yes—
106#95—1091no—
107#95—2181no—
108#41—4362no—
109#41—4362no—
110C2×Q8—4362no—
111C2×Q8—2181no—
112C2×Q8—2182no—
113C2×D4—1092no—
114C2×D4—2181no—
115C2×D4—2181no—
116C2×D4—2182no—
117C22×C4—2182no—
118C22×C4—2181no—
119C22×C4—2181no—
120C22×C4D43611yescharacteristic
121C22×C4D43611yescharacteristic
122C22×C4C22×D10911yescharacteristic
123SD16—4364no—
124C2×C8—22no—
125C4⋊C4—22no—
126C4⋊C4—1092no—
127C4⋊C4—4362no—
128C4⋊C4—2181no—
129C22⋊C4—2182no—
130C22⋊C4—2182no—
131C23—2182no—
132C23C2×D43611yescharacteristic
133Q8—4362no—
134Q8—4362no—
135Q8—2184no—
136D4—8721no—
137D4—4362no—
138D4—4362no—
139D4—4362no—
140D4—4362no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—4364no—
147C2×C4—4361no—
148C2×C4—4361no—
149C2×C4—4361no—
150C2×C4—4361no—
151C2×C4—2182no—
152C2×C4—2182no—
153C2×C4—2182no—
154C2×C4#1318201812yescomplemented
155C2×C4D4×D10912yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—4362no—
160C22—4362no—
161C22—4362no—
162C22#1681266911yescharacteristic
163C22C2×#1318201811yescharacteristic
164C22#1681272711yescharacteristic
165C4—22no—
166C4—22no—
167C4—41no—
168C4—4362no—
169C4—4362no—
170C4—4362no—
171C2—21no—
172C2—21no—
173C2—8722no—
174C2#2029517911yescharacteristic
175C2#2029521111yescharacteristic
176C2#2029521411yescharacteristic
subgroupG/H
centerC22#16812669
commutator, cosocleC22×C4×C109C23
FrattiniC22×C4C22×D109
Fitting#324×C109C2
socleC22×C109#120
prime radical#20295607C2
hypercenter#324D109
hypocenterC109#1604
LCSG ▷ C22×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #20295167C2semidirect
3. #20294946C2semidirect
4. #20294761C2semidirect
5. #324×C109C2semidirect
49. C2×C4×C109SD16semidirect
89. C109#1604semidirect
94. #324D109semidirect
154. C2×C4#13182018semidirect

Maximal subgroups

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

subgroupindex
#202956072
#202951672
#202949462
#202947612
#324×C1092
#1604109