Finite Group Databasebeta

← #28335716

Subgroups — #28335716

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#20293350C211yesmaximal, characteristic
2#20293030C211yesmaximal, characteristic, complemented
3#20292829C211yesmaximal, characteristic
4#20292372C211yesmaximal, characteristic, complemented
5Q8⋊(C107⋊C8)C211yesmaximal, characteristic, complemented
6#20292354C211yesmaximal, characteristic
7#475×C107C211yesmaximal, characteristic, complemented
8C2×#13133805—21no—
9#16812109C2211yescharacteristic
10#16812104—22no—
11#16812097—21no—
12#16812085—21no—
13C4×Dic214—21no—
14C4×D428C2211yescharacteristic
15#16812067—21no—
16#16812042—22no—
17C4⋊(C107⋊C8)C2211yescharacteristic
18#16812033—21no—
19C4×C107⋊C8C2211yescharacteristic
20C4⋊Q8×C107C2211yescharacteristic
21C4×Q8×C107C2211yescharacteristic
22#106×C107—21no—
23C4⋊C8×C107C2211yescharacteristic
24#103×C107—21no—
25C2×Dic214—21no—
26C2×D428D411yescharacteristic
27C2×C4×D107—21no—
28#13133805—42no—
29C2×C107⋊C8—21no—
30C2×C107⋊C8—21no—
31C4⋊Dic107—21no—
32C4⋊Dic107D411yescharacteristic
33C4⋊Dic107D412yes—
34Dic107⋊C4—22no—
35(C2×D107)⋊C4—22no—
36(C2×D107)⋊C4—22no—
37(C2×D107)⋊C4—21no—
38C4×Dic107—22no—
39C2×Q8×C107—21no—
40C2×Q8×C107D411yescharacteristic
41C2×C8×C107—21no—
42C4⋊C4×C107—21no—
43C4⋊C4×C107—21no—
44C4⋊C4×C107—21no—
45C4⋊C4×C107D411yescharacteristic
46C42×C107—21no—
47C42×C107C2311yescharacteristic
48C22×D107—21no—
49Q8×C107—21no—
50Q8×C107—21no—
51Q8×C107—41no—
52Q8×C107—41no—
53C2×C4×C107—21no—
54C2×C4×C107—21no—
55C2×C4×C107—41no—
56C2×C4×C107C2×D411yescharacteristic
57C2×C4×C107C2×D411yescharacteristic
58C2×C4×C107C2×D411yescharacteristic
59C2×Dic107—21no—
60C2×Dic107—22no—
61D428—21no—
62D428—21no—
63C4×D107—41no—
64C4×D107—41no—
65Dic214—41no—
66Dic214—41no—
67C8×C107—41no—
68C107⋊C8—22no—
69C107⋊C8—41no—
70C22×C107#12011yescharacteristic
71C2×D107—41no—
72C2×D107—41no—
73C2×D107—41no—
74C4×C107—21no—
75C4×C107—41no—
76C4×C107—41no—
77C4×C107—81no—
78C4×C107#13511yescharacteristic
79C4×C107#13511yescharacteristic
80C4×C107AGL(1,ℤ/8)11yescharacteristic
81C4×C107#13711yescharacteristic
82Dic107—42no—
83Dic107—41no—
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#475D10711yescharacteristic, complemented
93#461—1071no—
94#332—1071no—
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—2141no—
104C4×Q8C2×D10711yescharacteristic
105C4×D4—1071no—
106#117—2141no—
107#107—2142no—
108#106—21no—
109C4⋊C8—1071no—
110C4⋊C8C2×D10711yescharacteristic
111#103—21no—
112#103—2141no—
113C4×C8—1071no—
114C2×Q8—21no—
115C2×Q8—2141no—
116C2×Q8C22⋊D10711yescharacteristic
117C2×D4—1071no—
118C22×C4—2141no—
119SD16—4282no—
120C2×C8—21no—
121C2×C8—2141no—
122C2×C8—2141no—
123C4⋊C4—21no—
124C4⋊C4—21no—
125C4⋊C4—21no—
126C4⋊C4—1071no—
127C4⋊C4—1072no—
128C4⋊C4—2142no—
129C4⋊C4—2141no—
130C4⋊C4C22⋊D10711yescharacteristic
131C22⋊C4—2142no—
132C22⋊C4—2142no—
133C22⋊C4—2141no—
134C42—21no—
135C42—2142no—
136C42C22×D10711yescharacteristic
137C23—2141no—
138Q8—21no—
139Q8—21no—
140Q8—41no—
141Q8—41no—
142Q8—4281no—
143Q8—4281no—
144D4—2141no—
145D4—2141no—
146C2×C4—21no—
147C2×C4—21no—
148C2×C4—41no—
149C2×C4—4281no—
150C2×C4—4281no—
151C2×C4—2141no—
152C2×C4—2142no—
153C2×C4D4×D10711yescharacteristic
154C2×C4D4×D10711yescharacteristic
155C2×C4C2×C22⋊D10711yescharacteristic
156C8—41no—
157C8—4281no—
158C8—2142no—
159C22—4281no—
160C22—4281no—
161C22—4281no—
162C22#1681207711yescharacteristic
163C4—21no—
164C4—41no—
165C4—41no—
166C4—81no—
167C4—4282no—
168C4—4281no—
169C4#1681212911yescharacteristic
170C4#1681213011yescharacteristic
171C4#1681213611yescharacteristic
172C4#1681214011yescharacteristic
173C2—8561no—
174C2#2029277111yescharacteristic
175C2#2029278711yescharacteristic
176C2#2029280811yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#475×C107C2
socleC22×C107#120
prime radicalC2×D428D4
hypercenter#475D107
hypocenterC107#1650
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #475

Complements

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

NH ≅ G/Ntype
2. #20293030C2semidirect
4. #20292372C2semidirect
5. Q8⋊(C107⋊C8)C2semidirect
7. #475×C107C2semidirect
89. C107#1650semidirect
92. #475D107semidirect

Maximal subgroups

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

subgroupindex
#202933502
#202930302
#202928292
#202923722
Q8⋊(C107⋊C8)2
#202923542
#475×C1072
#1650107