Finite Group Databasebeta

← #28335718

Subgroups — #28335718

Order 13696.

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

normal subgroups only

This group has 200 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#20293351C211yesmaximal, characteristic
2#20293030C211yesmaximal, characteristic, complemented
3#20292913C211yesmaximal, characteristic, complemented
4#20292833C211yesmaximal, characteristic
5#20292357C211yesmaximal, characteristic, complemented
6#20292354C211yesmaximal, characteristic
7#332×C107C211yesmaximal, characteristic, complemented
8C2×#13133807—21no—
9#16812109C2211yescharacteristic
10#16812104—22no—
11#16812098—21no—
12#16812089—21no—
13Q8×Dic107—21no—
14C4×Dic214C2211yescharacteristic
15C4×D428C2211yescharacteristic
16#16812067—21no—
17#16812042—22no—
18#16812041—21no—
19#16812034—21no—
20#16812034—21no—
21C4×C107⋊C8C2211yescharacteristic
22C4⋊Q8×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4⋊C8×C107C2211yescharacteristic
25C2×Dic214—21no—
26C2×Dic214D411yescharacteristic
27C2×D428D411yescharacteristic
28C2×C4×D107—21no—
29#13133807—42no—
30C2×C107⋊C8—21no—
31C4⋊Dic107—22no—
32C4⋊Dic107—21no—
33C4⋊Dic107—21no—
34C4⋊Dic107D411yescharacteristic
35C4⋊Dic107D411yescharacteristic
36Dic107⋊C4—21no—
37(C2×D107)⋊C4—22no—
38(C2×D107)⋊C4—22no—
39(C2×D107)⋊C4—21no—
40C4×Dic107—22no—
41C4×Dic107—21no—
42C2×Q8×C107—21no—
43C2×C8×C107—21no—
44C2×C8×C107—21no—
45C4⋊C4×C107—21no—
46C4⋊C4×C107D412yes—
47C42×C107C2311yescharacteristic
48C22×D107—21no—
49Q8×C107—41no—
50Q8×C107—41no—
51C2×C4×C107—22no—
52C2×C4×C107C2×D411yescharacteristic
53C2×C4×C107C2×D411yescharacteristic
54C2×C4×C107C2×D411yescharacteristic
55C2×Dic107—21no—
56C2×Dic107—21no—
57C2×Dic107—21no—
58C2×Dic107—41no—
59D428—21no—
60D428—21no—
61C4×D107—41no—
62C4×D107—41no—
63Dic214—21no—
64Dic214—21no—
65Dic214—41no—
66Dic214—41no—
67C8×C107—41no—
68C8×C107—41no—
69C107⋊C8—22no—
70C22×C107#12011yescharacteristic
71C2×D107—41no—
72C2×D107—41no—
73C2×D107—41no—
74C4×C107—21no—
75C4×C107—42no—
76C4×C107#13511yescharacteristic
77C4×C107#13511yescharacteristic
78C4×C107AGL(1,ℤ/8)11yescharacteristic
79C4×C107#13711yescharacteristic
80Dic107—41no—
81Dic107—41no—
82Dic107—41no—
83Dic107—81no—
84C2×C107#44911yescharacteristic
85C2×C107#45211yescharacteristic
86C2×C107#45611yescharacteristic
87D107—81no—
88#1650—1071nomaximal, Sylow
89C107#165011yescharacteristic, complemented, Sylow
90#556—1071no—
91#500—1071no—
92#475—1071no—
93#461—1071no—
94#332D10711yescharacteristic, complemented
95Q8⋊C8—1071no—
96D4⋊C8—1071no—
97C2×SD16—2141no—
98C4⋊Q8—1071no—
99C4⋊Q8C2×D10711yescharacteristic
100#126—2142no—
101#124—2141no—
102C22⋊Q8—2141no—
103C4×Q8—1071no—
104C4×Q8—2141no—
105C4×D4—1071no—
106#117—2141no—
107#107—2142no—
108#106—2141no—
109C4⋊C8C2×D10711yescharacteristic
110C4⋊C8C2×D10711yescharacteristic
111#103—2141no—
112#103—2141no—
113C4×C8—1071no—
114C2×Q8—21no—
115C2×Q8—1071no—
116C2×Q8—2141no—
117C2×D4—1071no—
118C22×C4—2141no—
119SD16—4282no—
120C2×C8—21no—
121C2×C8—21no—
122C2×C8—2141no—
123C4⋊C4—21no—
124C4⋊C4—1071no—
125C4⋊C4—1071no—
126C4⋊C4—2142no—
127C4⋊C4—2141no—
128C4⋊C4—2141no—
129C4⋊C4—2141no—
130C4⋊C4D42812yes—
131C22⋊C4—2142no—
132C22⋊C4—2142no—
133C22⋊C4—2141no—
134C42—2142no—
135C42—2141no—
136C42C22×D10711yescharacteristic
137C23—2141no—
138Q8—41no—
139Q8—41no—
140Q8—4281no—
141Q8—4281no—
142Q8—2141no—
143Q8—2141no—
144D4—2141no—
145D4—2141no—
146C2×C4—22no—
147C2×C4—4281no—
148C2×C4—4281no—
149C2×C4—4281no—
150C2×C4—2141no—
151C2×C4—2141no—
152C2×C4—2141no—
153C2×C4D4×D10711yescharacteristic
154C2×C4D4×D10711yescharacteristic
155C2×C4C2×D42811yescharacteristic
156C8—41no—
157C8—41no—
158C8—2142no—
159C22—4281no—
160C22—4281no—
161C22—4281no—
162C22#1681207911yescharacteristic
163C4—21no—
164C4—42no—
165C4—8561no—
166C4—4281no—
167C4—4281no—
168C4—4281no—
169C4#1681213011yescharacteristic
170C4#1681213011yescharacteristic
171C4#1681213311yescharacteristic
172C4#1681214111yescharacteristic
173C2—8561no—
174C2#2029277311yescharacteristic
175C2#2029279211yescharacteristic
176C2#2029281211yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#332×C107C2
socleC22×C107#120
prime radicalC2×D428D4
hypercenter#332D107
hypocenterC107#1650
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
2. #20293030C2semidirect
3. #20292913C2semidirect
5. #20292357C2semidirect
7. #332×C107C2semidirect
89. C107#1650semidirect
94. #332D107semidirect

Maximal subgroups

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

subgroupindex
#202933512
#202930302
#202929132
#202928332
#202923572
#202923542
#332×C1072
#1650107