Finite Group Databasebeta

← #28335432

Subgroups — #28335432

Order 13696.

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#20293208C211yesmaximal, characteristic
2#20292768C212yesmaximal, complemented
3#20292547C211yesmaximal, characteristic, complemented
4#20292362C212yesmaximal, complemented
5#324×C107C211yesmaximal, characteristic, complemented
6C2×#13133818—21no—
7C22×Dic214C2211yescharacteristic
8C2×#13133807—22no—
9#16812089—22no—
10#16812084—22no—
11#16812083—21no—
12#16812083C2212yes—
13C2×Dic107⋊C4—21no—
14#16812034—22no—
15C107⋊#95—21no—
16C107⋊#95C2211yescharacteristic
17C2×C4⋊C4×C107C2211yescharacteristic
18C22⋊C8×C107C2212yes—
19#13133818—42no—
20#13133818—42no—
21C2×Dic214—22no—
22C2×Dic214—21no—
23C2×Dic214—42no—
24C2×D428—21no—
25C2×D428—21no—
26C2×D428D412yes—
27C2×C22⋊D107—22no—
28C22×Dic107—21no—
29C22×Dic107—21no—
30C2×C4×D107—22no—
31#13133807—44no—
32C4⋊Dic107—21no—
33C4⋊Dic107D412yes—
34Dic107⋊C4—42no—
35(C2×D107)⋊C4—22no—
36(C2×D107)⋊C4—22no—
37C22×C4×C107D411yescharacteristic
38C22×C4×C107D411yescharacteristic
39C22×C4×C107C2311yescharacteristic
40C2×C8×C107—22no—
41C4⋊C4×C107—22no—
42C23×C107C2×D411yescharacteristic
43C22×D107—22no—
44C2×C4×C107—21no—
45C2×C4×C107—21no—
46C2×C4×C107—21no—
47C2×C4×C107—41no—
48C2×C4×C107—41no—
49C2×C4×C107SD1612yescomplemented
50C2×C4×C107C2×D412yes—
51C22⋊D107—42no—
52C22⋊D107—42no—
53C2×Dic107—22no—
54C2×Dic107—22no—
55C2×Dic107—22no—
56C2×Dic107—41no—
57C2×Dic107—41no—
58C2×Dic107—41no—
59C2×Dic107—41no—
60D428—42no—
61D428—42no—
62D428—81no—
63C4×D107—44no—
64Dic214—24no—
65Dic214—42no—
66Dic214—42no—
67C8×C107—42no—
68C22×C107—21no—
69C22×C107—21no—
70C22×C107#12011yescharacteristic
71C22×C107C2×SD1611yescharacteristic
72C22×C107AGL(1,ℤ/8)11yescharacteristic
73C2×D107—42no—
74C2×D107—42no—
75C2×D107—42no—
76C4×C107—22no—
77C4×C107—22no—
78C4×C107—41no—
79Dic107—42no—
80Dic107—42no—
81Dic107—42no—
82C2×C107—21no—
83C2×C107—21no—
84C2×C107#45011yescharacteristic
85C2×C107#45611yescharacteristic
86C2×C107B(4,𝔽2)11yescharacteristic
87D107—82no—
88#1604—1071nomaximal, Sylow
89C107#160411yescharacteristic, complemented, Sylow
90#537—1071no—
91#448—1072no—
92#395—1071no—
93#327—1072no—
94#324D10711yescharacteristic, complemented
95C2×#41—2141no—
96C22×Q8—1071no—
97C2×SD16—2142no—
98C22⋊Q8—2142no—
99#121—1072no—
100#121—2142no—
101#121—2141no—
102C2×C4⋊C4—2141no—
103C2×C4⋊C4C2×D10711yescharacteristic
104#103—2142no—
105C22⋊C8C2×D10712yes—
106#95—1071no—
107#95—2141no—
108#41—4282no—
109#41—4282no—
110C2×Q8—4282no—
111C2×Q8—2141no—
112C2×Q8—2142no—
113C2×D4—1072no—
114C2×D4—2141no—
115C2×D4—2141no—
116C2×D4—2142no—
117C22×C4—2142no—
118C22×C4—2141no—
119C22×C4—2141no—
120C22×C4D42811yescharacteristic
121C22×C4D42811yescharacteristic
122C22×C4C22×D10711yescharacteristic
123SD16—4284no—
124C2×C8—22no—
125C4⋊C4—22no—
126C4⋊C4—1072no—
127C4⋊C4—4282no—
128C4⋊C4—2141no—
129C22⋊C4—2142no—
130C22⋊C4—2142no—
131C23—2142no—
132C23C2×D42811yescharacteristic
133Q8—4282no—
134Q8—4282no—
135Q8—2144no—
136D4—8561no—
137D4—4282no—
138D4—4282no—
139D4—4282no—
140D4—4282no—
141C2×C4—21no—
142C2×C4—21no—
143C2×C4—21no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—4284no—
147C2×C4—4281no—
148C2×C4—4281no—
149C2×C4—4281no—
150C2×C4—4281no—
151C2×C4—2142no—
152C2×C4—2142no—
153C2×C4—2142no—
154C2×C4#1313380712yescomplemented
155C2×C4D4×D10712yes—
156C8—42no—
157C22—21no—
158C22—21no—
159C22—4282no—
160C22—4282no—
161C22—4282no—
162C22#1681207911yescharacteristic
163C22C2×#1313380711yescharacteristic
164C22#1681213711yescharacteristic
165C4—22no—
166C4—22no—
167C4—41no—
168C4—4282no—
169C4—4282no—
170C4—4282no—
171C2—21no—
172C2—21no—
173C2—8562no—
174C2#2029278011yescharacteristic
175C2#2029281211yescharacteristic
176C2#2029281511yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#324×C107C2
socleC22×C107#120
prime radical#20293208C2
hypercenter#324D107
hypocenterC107#1604
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #20292768C2semidirect
3. #20292547C2semidirect
4. #20292362C2semidirect
5. #324×C107C2semidirect
49. C2×C4×C107SD16semidirect
89. C107#1604semidirect
94. #324D107semidirect
154. C2×C4#13133807semidirect

Maximal subgroups

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

subgroupindex
#202932082
#202927682
#202925472
#202923622
#324×C1072
#1604107