Finite Group Databasebeta

← #28335500

Subgroups — #28335500

Order 13696.

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

normal subgroups only

This group has 246 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#20292946C212yesmaximal, complemented
3#20292566C211yesmaximal, characteristic, complemented
4#20292362C212yesmaximal, complemented
5#324×C107C211yesmaximal, characteristic, complemented
6C2×#13133818—21no—
7#16812089—22no—
8#16812084—22no—
9#16812083—21no—
10#16812083C2212yes—
11C2×C4⋊Dic107C2211yescharacteristic
12#16812041—22no—
13#16812031—22no—
14C107⋊#95—21no—
15C107⋊#95—21no—
16C107⋊#95C2211yescharacteristic
17C2×C4⋊C4×C107C2211yescharacteristic
18C22⋊C8×C107C2212yes—
19#13133818—42no—
20#13133818—42no—
21C2×Dic214—21no—
22C2×D428—21no—
23C2×D428—21no—
24C2×D428D412yes—
25C2×C22⋊D107—22no—
26C22×Dic107—21no—
27C22×Dic107—21no—
28C2×C4×D107—22no—
29C4⋊Dic107—21no—
30C4⋊Dic107—22no—
31C4⋊Dic107D412yes—
32(C2×D107)⋊C4—22no—
33(C2×D107)⋊C4—22no—
34C22×C4×C107D411yescharacteristic
35C22×C4×C107D411yescharacteristic
36C22×C4×C107C2311yescharacteristic
37C2×C8×C107—22no—
38C4⋊C4×C107—22no—
39C23×C107C2×D411yescharacteristic
40C22×D107—22no—
41C2×C4×C107—21no—
42C2×C4×C107—21no—
43C2×C4×C107—21no—
44C2×C4×C107—41no—
45C2×C4×C107—41no—
46C2×C4×C107SD1612yes—
47C2×C4×C107C2×D412yes—
48C22⋊D107—42no—
49C22⋊D107—42no—
50C2×Dic107—22no—
51C2×Dic107—41no—
52C2×Dic107—41no—
53C2×Dic107—42no—
54C2×Dic107—41no—
55C2×Dic107—41no—
56D428—42no—
57D428—42no—
58D428—81no—
59C4×D107—44no—
60Dic214—42no—
61C8×C107—42no—
62C22×C107—21no—
63C22×C107—21no—
64C22×C107#12011yescharacteristic
65C22×C107C2×SD1611yescharacteristic
66C22×C107AGL(1,ℤ/8)11yescharacteristic
67C2×D107—42no—
68C2×D107—42no—
69C2×D107—42no—
70C4×C107—22no—
71C4×C107—22no—
72C4×C107—41no—
73Dic107—42no—
74Dic107—81no—
75Dic107—81no—
76C2×C107—21no—
77C2×C107—21no—
78C2×C107#45011yescharacteristic
79C2×C107#45411yescharacteristic
80C2×C107#45811yescharacteristic
81D107—82no—
82#1616—1071nomaximal, Sylow
83C107#161611yescharacteristic, complemented, Sylow
84#537—1071no—
85#481—1072no—
86#400—1071no—
87#327—1072no—
88#324D10711yescharacteristic, complemented
89C2×#41—2141no—
90C22⋊Q8—2142no—
91#121—1072no—
92#121—2142no—
93#121—2141no—
94C2×C4⋊C4—1071no—
95C2×C4⋊C4C2×D10711yescharacteristic
96#106—2142no—
97#102—2142no—
98C22⋊C8C2×D10712yes—
99#95—1071no—
100#95—2141no—
101#95—2141no—
102#41—4282no—
103#41—4282no—
104C2×Q8—2141no—
105C2×D4—1072no—
106C2×D4—2141no—
107C2×D4—2141no—
108C2×D4—2142no—
109C22×C4—2142no—
110C22×C4—2141no—
111C22×C4—2141no—
112C22×C4D42811yescharacteristic
113C22×C4D42811yescharacteristic
114C22×C4C22×D10711yescharacteristic
115C2×C8—22no—
116C4⋊C4—22no—
117C4⋊C4—1072no—
118C4⋊C4—2141no—
119C4⋊C4—2142no—
120C22⋊C4—2142no—
121C22⋊C4—2142no—
122C23—2142no—
123C23C2×D42811yescharacteristic
124Q8—4282no—
125D4—8561no—
126D4—4282no—
127D4—4282no—
128D4—4282no—
129D4—4282no—
130C2×C4—21no—
131C2×C4—21no—
132C2×C4—21no—
133C2×C4—41no—
134C2×C4—41no—
135C2×C4—4284no—
136C2×C4—4281no—
137C2×C4—4281no—
138C2×C4—4282no—
139C2×C4—4281no—
140C2×C4—4281no—
141C2×C4—2142no—
142C2×C4#1313380712yes—
143C2×C4D4×D10712yes—
144C8—42no—
145C22—21no—
146C22—21no—
147C22—4282no—
148C22—4282no—
149C22—4282no—
150C22#1681207911yescharacteristic
151C22C2×#1313380711yescharacteristic
152C22#1681213711yescharacteristic
153C4—22no—
154C4—22no—
155C4—41no—
156C4—8561no—
157C4—8561no—
158C4—4282no—
159C2—21no—
160C2—21no—
161C2—8562no—
162C2#2029278011yescharacteristic
163C2#2029280211yescharacteristic
164C2#2029281811yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#324×C107C2
socleC22×C107#120
prime radical#20293208C2
hypercenter#324D107
hypocenterC107#1616
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. #20292946C2semidirect
3. #20292566C2semidirect
4. #20292362C2semidirect
5. #324×C107C2semidirect
83. C107#1616semidirect
88. #324D107semidirect

Maximal subgroups

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

subgroupindex
#202932082
#202929462
#202925662
#202923622
#324×C1072
#1616107