Finite Group Databasebeta

← #28335558

Subgroups — #28335558

Order 13696.

210 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 55 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#20293352C211yesmaximal, characteristic, complemented
2#20292975C211yesmaximal, characteristic
3#20292828C211yesmaximal, characteristic
4#20292828C211yesmaximal, characteristic
5#20292357C211yesmaximal, characteristic, complemented
6#20292357C211yesmaximal, characteristic, complemented
7#332×C107C211yesmaximal, characteristic, complemented
8C2×#13133807—21no—
9C2×#13133807—21no—
10#16812107—21no—
11#16812106C2211yescharacteristic
12#16812100—21no—
13#16812099—22no—
14Q8×Dic107—21no—
15C4×Dic214C2211yescharacteristic
16C4×Dic214C2211yescharacteristic
17#16812074—21no—
18#16812031—21no—
19#16812031—21no—
20#16812029—22no—
21C4×C107⋊C8C2211yescharacteristic
22C4⋊Q8×C107C2211yescharacteristic
23C4⋊C8×C107C2211yescharacteristic
24C4⋊C8×C107C2211yescharacteristic
25C2×Dic214D411yescharacteristic
26C2×Dic214D411yescharacteristic
27C2×D428—21no—
28C2×D428—21no—
29C2×D428—21no—
30#13133807—42no—
31#13133807—42no—
32C2×C107⋊C8—21no—
33C4⋊Dic107—22no—
34C4⋊Dic107D411yescharacteristic
35C4⋊Dic107D411yescharacteristic
36Dic107⋊C4—22no—
37Dic107⋊C4—22no—
38Dic107⋊C4—21no—
39Dic107⋊C4—21no—
40C4×Dic107—22no—
41C4×Dic107—21no—
42C4×Dic107—21no—
43C2×Q8×C107—21no—
44C2×C8×C107—21no—
45C2×C8×C107—21no—
46C4⋊C4×C107—21no—
47C4⋊C4×C107D412yes—
48C42×C107C2311yescharacteristic
49C22×D107—41no—
50Q8×C107—41no—
51Q8×C107—41no—
52C2×C4×C107—22no—
53C2×C4×C107C2×D411yescharacteristic
54C2×C4×C107C2×D411yescharacteristic
55C2×C4×C107C2×D411yescharacteristic
56C2×Dic107—21no—
57C2×Dic107—21no—
58C2×Dic107—21no—
59C2×Dic107—21no—
60D428—42no—
61D428—41no—
62D428—41no—
63D428—41no—
64D428—41no—
65Dic214—21no—
66Dic214—21no—
67Dic214—21no—
68Dic214SD1612yescomplemented
69C8×C107—41no—
70C8×C107—41no—
71C107⋊C8—41no—
72C22×C107#12011yescharacteristic
73C2×D107—81no—
74C2×D107—81no—
75C2×D107—81no—
76C4×C107—21no—
77C4×C107—42no—
78C4×C107C2×SD1611yescharacteristic
79C4×C107#13511yescharacteristic
80C4×C107AGL(1,ℤ/8)11yescharacteristic
81C4×C107#13711yescharacteristic
82Dic107—22no—
83Dic107—41no—
84Dic107—41no—
85Dic107—41no—
86C2×C107#44811yescharacteristic
87C2×C107#44911yescharacteristic
88C2×C107#45411yescharacteristic
89D107—161no—
90#1626—1071nomaximal, Sylow
91C107#162611yescharacteristic, complemented, Sylow
92#557—1071no—
93#486—1071no—
94#460—1071no—
95#460—1071no—
96#332D10711yescharacteristic, complemented
97Q8⋊C8—1071no—
98Q8⋊C8—1071no—
99C2×SD16—2141no—
100C2×SD16—2141no—
101C4⋊Q8—2141no—
102C4⋊Q8C2×D10711yescharacteristic
103#127—1071no—
104#125—2141no—
105#125—2142no—
106C4×Q8—1071no—
107C4×Q8—1071no—
108C4×Q8—2141no—
109C4×Q8—2141no—
110C4⋊C8C2×D10711yescharacteristic
111C4⋊C8C2×D10711yescharacteristic
112#102—2141no—
113#102—2141no—
114#102—2142no—
115C4×C8—1071no—
116C2×Q8—21no—
117C2×Q8—1071no—
118C2×Q8—1071no—
119C2×D4—2141no—
120C2×D4—2141no—
121C2×D4—2141no—
122SD16—4282no—
123SD16—4282no—
124C2×C8—21no—
125C2×C8—21no—
126C2×C8—2141no—
127C4⋊C4—21no—
128C4⋊C4—1071no—
129C4⋊C4—1071no—
130C4⋊C4—2142no—
131C4⋊C4—2142no—
132C4⋊C4—2142no—
133C4⋊C4—2141no—
134C4⋊C4—2141no—
135C4⋊C4D42812yes—
136C42—2142no—
137C42—2141no—
138C42—2141no—
139C42C22×D10711yescharacteristic
140C23—4281no—
141Q8—41no—
142Q8—41no—
143Q8—1072no—
144Q8—2141no—
145Q8—2141no—
146Q8—2141no—
147D4—4282no—
148D4—4281no—
149D4—4281no—
150D4—4281no—
151D4—4281no—
152C2×C4—22no—
153C2×C4—2141no—
154C2×C4—2141no—
155C2×C4—2141no—
156C2×C4—2141no—
157C2×C4D4×D10711yescharacteristic
158C2×C4D4×D10711yescharacteristic
159C2×C4C2×D42811yescharacteristic
160C8—41no—
161C8—41no—
162C8—4281no—
163C22—8561no—
164C22—8561no—
165C22—8561no—
166C22#1681207911yescharacteristic
167C4—21no—
168C4—42no—
169C4—4281no—
170C4—4281no—
171C4—4281no—
172C4—2142no—
173C4SD16×D10711yescharacteristic
174C4#1681213011yescharacteristic
175C4#1681213311yescharacteristic
176C4#1681214111yescharacteristic
177C2—17121no—
178C2#2029276711yescharacteristic
179C2#2029277311yescharacteristic
180C2#2029280211yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#332×C107C2
socleC22×C107#120
prime radical#16812106C22
hypercenter#332D107
hypocenterC107#1626
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
1. #20293352C2semidirect
5. #20292357C2semidirect
6. #20292357C2semidirect
7. #332×C107C2semidirect
68. Dic214SD16semidirect
91. C107#1626semidirect
96. #332D107semidirect

Maximal subgroups

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

subgroupindex
#202933522
#202929752
#202928282
#202923572
#332×C1072
#1626107