Finite Group Databasebeta

← #28335637

Subgroups — #28335637

Order 13696.

308 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 57 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#20293262C211yesmaximal, characteristic
2#20292904C211yesmaximal, characteristic, complemented
3#20292829C211yesmaximal, characteristic, complemented
4#20292371C211yesmaximal, characteristic, complemented
5#20292356C211yesmaximal, characteristic, complemented
6#20292356C211yesmaximal, characteristic, complemented
7#486×C107C211yesmaximal, characteristic, complemented
8C2×D4×D107—21no—
9C2×D4×D107—21no—
10C2×#13133805—21no—
11#16812080—21no—
12#16812080—21no—
13#16812077—22no—
14C4×D428C2211yescharacteristic
15C4×D428C2211yescharacteristic
16#16812042—21no—
17C4⋊(C107⋊C8)C2211yescharacteristic, complemented
18C4⋊(C107⋊C8)C2211yescharacteristic, complemented
19#16812033—21no—
20#16812029—21no—
21C4⋊Q8×C107C2211yescharacteristic, complemented
22#127×C107C2211yescharacteristic
23#102×C107—22no—
24C4×C8×C107C2211yescharacteristic, complemented
25C23×D107—22no—
26C2×D428D411yescharacteristic
27C2×D428D411yescharacteristic
28C2×C22⋊D107—22no—
29C2×C22⋊D107—22no—
30D4×D107—42no—
31D4×D107—42no—
32D4×D107—42no—
33D4×D107—42no—
34C2×C4×D107—21no—
35C2×C4×D107—21no—
36#13133805—42no—
37C2×C107⋊C8—21no—
38C2×C107⋊C8—21no—
39C4⋊Dic107D411yescharacteristic
40C4⋊Dic107D411yescharacteristic
41C22⋊Dic107—22no—
42(C2×D107)⋊C4—21no—
43(C2×D107)⋊C4—21no—
44C2×Q8×C107—21no—
45C2×D4×C107—21no—
46C2×D4×C107—21no—
47C2×D4×C107D412yes—
48C2×C8×C107—21no—
49C4⋊C4×C107—21no—
50C4⋊C4×C107—21no—
51C42×C107C2311yescharacteristic
52C23×C107—22no—
53C22×D107—21no—
54C22×D107—21no—
55C22×D107—44no—
56C22×D107—44no—
57C22×D107—82no—
58Q8×C107—41no—
59Q8×C107—41no—
60D4×C107—42no—
61D4×C107—41no—
62D4×C107—41no—
63D4×C107—41no—
64D4×C107—41no—
65C2×C4×C107—41no—
66C2×C4×C107C2×D411yescharacteristic
67C2×C4×C107C2×D411yescharacteristic
68C2×C4×C107C2×D411yescharacteristic
69C22⋊D107—44no—
70C22⋊D107—44no—
71C2×Dic107—21no—
72C2×Dic107—21no—
73D428—21no—
74D428—21no—
75D428D812yes—
76D428SD1612yescomplemented
77C4×D107—41no—
78C4×D107—41no—
79C4×D107—41no—
80C4×D107—41no—
81C8×C107—41no—
82C107⋊C8—41no—
83C107⋊C8—41no—
84C22×C107—42no—
85C22×C107—42no—
86C22×C107—42no—
87C22×C107#12011yescharacteristic
88C2×D107—22no—
89C2×D107—22no—
90C2×D107—41no—
91C2×D107—41no—
92C2×D107—41no—
93C2×D107—41no—
94C2×D107—88no—
95C4×C107—21no—
96C4×C107—81no—
97C4×C107C2×D811yescharacteristic
98C4×C107C2×SD1611yescharacteristic
99C4×C107AGL(1,ℤ/8)11yescharacteristic
100C4×C107AGL(1,ℤ/8)11yescharacteristic
101Dic107—41no—
102Dic107—41no—
103C2×C107—82no—
104C2×C107#44711yescharacteristic
105C2×C107#45011yescharacteristic
106C2×C107#45511yescharacteristic
107D107—42no—
108D107—42no—
109#1639—1071nomaximal, Sylow
110C107#163911yescharacteristic, complemented, Sylow
111D42—1071no—
112#486D10711yescharacteristic, complemented
113#474—1071no—
114#461—1071no—
115#331—1071no—
116D4⋊C8—1071no—
117D4⋊C8—1071no—
118C22×D4—2141no—
119C22×D4—2141no—
120C2×SD16—2141no—
121C4⋊Q8C2×D10711yescharacteristic, complemented
122#127C2×D10711yescharacteristic
123#121—2141no—
124#121—2141no—
125#120—2142no—
126C4×D4—1071no—
127C4×D4—1071no—
128#107—2141no—
129C4⋊C8—1071no—
130C4⋊C8—1071no—
131#103—2141no—
132#102—22no—
133#102—2141no—
134C4×C8C2×D10711yescharacteristic, complemented
135C24—2142no—
136C2×Q8—21no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—1071no—
140C2×D4—1071no—
141C2×D4—4282no—
142C2×D4—4282no—
143C2×D4—4282no—
144C2×D4—4282no—
145C2×D4—2142no—
146C2×D4—2142no—
147C2×D4C22⋊D10712yes—
148C22×C4—2141no—
149C22×C4—2141no—
150SD16—4282no—
151C2×C8—21no—
152C2×C8—2141no—
153C2×C8—2141no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—1071no—
157C4⋊C4—1071no—
158C22⋊C4—2142no—
159C22⋊C4—2141no—
160C22⋊C4—2141no—
161C42C22×D10711yescharacteristic
162C23—22no—
163C23—8562no—
164C23—4284no—
165C23—4284no—
166C23—2141no—
167C23—2141no—
168Q8—41no—
169Q8—41no—
170D4—42no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—1072no—
176D4—1072no—
177D4—4284no—
178D4—4284no—
179D4—2141no—
180D4—2141no—
181C2×C4—41no—
182C2×C4—4281no—
183C2×C4—4281no—
184C2×C4—4281no—
185C2×C4—4281no—
186C2×C4—2141no—
187C2×C4—2141no—
188C2×C4D4×D10711yescharacteristic
189C2×C4D4×D10711yescharacteristic
190C2×C4C2×C22⋊D10711yescharacteristic
191C8—41no—
192C8—4281no—
193C8—4281no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—8568no—
198C22—4281no—
199C22—4281no—
200C22—4281no—
201C22—4281no—
202C22—2142no—
203C22—2142no—
204C22#1681207711yescharacteristic
205C4—21no—
206C4—81no—
207C4—4281no—
208C4—4281no—
209C4D8×D10711yescharacteristic
210C4SD16×D10711yescharacteristic
211C4#1681213311yescharacteristic
212C4#1681213611yescharacteristic
213C2—82no—
214C2—4282no—
215C2—4282no—
216C2#2029275911yescharacteristic
217C2#2029277611yescharacteristic
218C2#2029280411yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC42×C107C23
FrattiniC42C22×D107
Fitting#486×C107C2
socleC22×C107#120
prime radical#20293262C2
hypercenter#486D107
hypocenterC107#1639
LCSG ▷ C42×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. #20292904C2semidirect
3. #20292829C2semidirect
4. #20292371C2semidirect
5. #20292356C2semidirect
6. #20292356C2semidirect
7. #486×C107C2semidirect
17. C4⋊(C107⋊C8)C22semidirect
18. C4⋊(C107⋊C8)C22semidirect
21. C4⋊Q8×C107C22semidirect
24. C4×C8×C107C22semidirect
76. D428SD16semidirect
110. C107#1639semidirect
112. #486D107semidirect
121. C4⋊Q8C2×D107semidirect
134. C4×C8C2×D107semidirect

Maximal subgroups

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

subgroupindex
#202932622
#202929042
#202928292
#202923712
#202923562
#486×C1072
#1639107