Finite Group Databasebeta

← #28334901

Subgroups — #28334901

Order 13696.

266 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 81 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
1C4×#13133818C211yesmaximal, characteristic
2#20292651C211yesmaximal, characteristic, complemented
3#20292359C212yesmaximal, complemented
4#20292356C212yesmaximal, complemented
5C4×M16×C107C211yesmaximal, characteristic, complemented
6C2×#13133818C411yescharacteristic
7C2×#13133802—21no—
8C4×Dic214C411yescharacteristic
9C4×Dic214C2211yescharacteristic
10C4×D428C411yescharacteristic
11C4×D428C2211yescharacteristic
12C4×C22⋊D107—21no—
13C4×C22⋊D107—21no—
14#16812068C411yescharacteristic
15#16812066—21no—
16C42×D107—21no—
17C4⋊(C107⋊C8)—21no—
18C4⋊(C107⋊C8)C2212yes—
19C2×M16×C107—21no—
20C2×C42×C107C2211yescharacteristic
21#97×C107—21no—
22C4×C8×C107C2212yes—
23#13133818—22no—
24#13133818—22no—
25C2×Dic214C2×C411yescharacteristic
26C2×D428C2×C411yescharacteristic
27C2×C22⋊D107—21no—
28C2×C4×D107—21no—
29C2×C4×D107—21no—
30C2×C4×D107—21no—
31#13133802—42no—
32C2×C107⋊C8—22no—
33C4⋊Dic107C2×C411yescharacteristic
34C4⋊Dic107C2×C411yescharacteristic
35Dic107⋊C4—21no—
36Dic107⋊C4—21no—
37C22⋊Dic107—21no—
38(C2×D107)⋊C4—21no—
39(C2×D107)⋊C4—21no—
40C4×Dic107—21no—
41C4×Dic107—21no—
42C4×Dic107—21no—
43C22×C4×C107C2×C411yescharacteristic
44C22×C4×C107D411yescharacteristic
45C22×C4×C107D411yescharacteristic
46M16×C107—24no—
47C2×C8×C107—22no—
48C42×C107C2×C411yescharacteristic
49C42×C107D411yescharacteristic
50C42×C107D411yescharacteristic
51C42×C107C2311yescharacteristic
52C23×C107C22⋊C411yescharacteristic
53C22×D107—21no—
54C2×C4×C107—22no—
55C2×C4×C107—21no—
56C2×C4×C107—21no—
57C2×C4×C107—21no—
58C2×C4×C107—21no—
59C2×C4×C107C22⋊C411yescharacteristic
60C2×C4×C107C22⋊C411yescharacteristic
61C2×C4×C107C22⋊C411yescharacteristic
62C2×C4×C107C22×C411yescharacteristic
63C2×C4×C107C2×D411yescharacteristic
64C2×C4×C107C2×D411yescharacteristic
65C22⋊D107—24no—
66C2×Dic107—21no—
67C2×Dic107—21no—
68C2×Dic107—21no—
69D428—21no—
70D428M1612yescomplemented
71C4×D107—24no—
72C4×D107—42no—
73C4×D107—41no—
74C4×D107—41no—
75Dic214—21no—
76Dic214M1612yescomplemented
77C8×C107—24no—
78C107⋊C8—42no—
79C22×C107—21no—
80C22×C107—21no—
81C22×C107—21no—
82C22×C107C2×C22⋊C411yescharacteristic
83C2×D107—22no—
84C2×D107—41no—
85C2×D107—41no—
86C4×C107—21no—
87C4×C107—21no—
88C4×C107—21no—
89C4×C107#10412yes—
90C4×C107C2×M1611yescharacteristic
91C4×C107C2×M1611yescharacteristic
92C4×C107AGL(1,ℤ/8)11yescharacteristic
93C4×C107#13711yescharacteristic
94Dic107—22no—
95Dic107—41no—
96Dic107—41no—
97C2×C107—41no—
98C2×C107#40711yescharacteristic
99C2×C107#41711yescharacteristic
100C2×C107C2×#10411yescharacteristic
101D107—42no—
102#1489—1071nomaximal, Sylow
103C107#148911yescharacteristic, complemented, Sylow
104C4×#41—1071no—
105#423—1071no—
106C4×M16D10711yescharacteristic, complemented
107Q8⋊C8—1072no—
108D4⋊C8—1072no—
109C2×#41—1071no—
110C2×M16—21no—
111C2×M16—2141no—
112C4×Q8—1071no—
113C4×Q8—1071no—
114C4×D4—1071no—
115C4×D4—1071no—
116C4×D4—2141no—
117C4×D4—2141no—
118#117—1071no—
119#117—2141no—
120C2×C42—2141no—
121C2×C42C2×D10711yescharacteristic
122C4⋊C8—1072no—
123C4⋊C8—2141no—
124#97—21no—
125C4×C8C2×D10712yes—
126#41—2142no—
127#41—2142no—
128C2×Q8—1071no—
129C2×D4—1071no—
130C2×D4—2141no—
131C22×C4—2141no—
132C22×C4—2141no—
133C22×C4—2141no—
134C22×C4C4×D10711yescharacteristic
135C22×C4D42811yescharacteristic
136C22×C4C22⋊D10711yescharacteristic
137M16—24no—
138M16—4282no—
139C2×C8—22no—
140C2×C8—2142no—
141C4⋊C4—1071no—
142C4⋊C4—1071no—
143C4⋊C4—2141no—
144C4⋊C4—2141no—
145C22⋊C4—2141no—
146C22⋊C4—2141no—
147C22⋊C4—2141no—
148C42—2141no—
149C42—2141no—
150C42—2141no—
151C42C4×D10711yescharacteristic
152C42D42811yescharacteristic
153C42C22⋊D10711yescharacteristic
154C42C22×D10711yescharacteristic
155C23—2141no—
156C23(C2×D107)⋊C411yescharacteristic
157Q8—1072no—
158Q8—2141no—
159D4—1072no—
160D4—2144no—
161D4—2141no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—4282no—
168C2×C4—4281no—
169C2×C4—4281no—
170C2×C4—2144no—
171C2×C4—2141no—
172C2×C4—2141no—
173C2×C4—2141no—
174C2×C4(C2×D107)⋊C411yescharacteristic
175C2×C4(C2×D107)⋊C411yescharacteristic
176C2×C4(C2×D107)⋊C411yescharacteristic
177C2×C4C2×C4×D10711yescharacteristic
178C2×C4C2×C22⋊D10711yescharacteristic
179C2×C4C2×D42811yescharacteristic
180C8—24no—
181C8—4282no—
182C22—21no—
183C22—21no—
184C22—21no—
185C22—4281no—
186C22—4281no—
187C22—2142no—
188C22C2×(C2×D107)⋊C411yescharacteristic
189C4—21no—
190C4—21no—
191C4—21no—
192C4—4281no—
193C4—4281no—
194C4—2142no—
195C4#1681203712yes—
196C4M16×D10711yescharacteristic
197C4M16×D10711yescharacteristic
198C4#1681213711yescharacteristic
199C4#1681214211yescharacteristic
200C2—41no—
201C2—4282no—
202C2#2029259011yescharacteristic
203C2#2029262911yescharacteristic
204C2C2×#1681203711yescharacteristic
subgroupG/H
centerC2×C4C2×D428
commutatorC2×C4×C107C22×C4
FrattiniC42C22×D107
FittingC4×M16×C107C2
socleC22×C107C2×C22⋊C4
prime radicalC2×#13133818C4
cosocleC42×C107C23
hypercenterC4×M16D107
hypocenterC107#1489
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×M16

Complements

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

NH ≅ G/Ntype
2. #20292651C2semidirect
3. #20292359C2semidirect
4. #20292356C2semidirect
5. C4×M16×C107C2semidirect
70. D428M16semidirect
76. Dic214M16semidirect
103. C107#1489semidirect
106. C4×M16D107semidirect

Maximal subgroups

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

subgroupindex
C4×#131338182
#202926512
#202923592
#202923562
C4×M16×C1072
#1489107