Finite Group Databasebeta

← #28349089

Subgroups — #28349089

Order 13696.

354 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 117 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#20293342C211yesmaximal, characteristic
2#20293337C211yesmaximal, characteristic, complemented
3#20293032C211yesmaximal, characteristic, complemented
4#20292967C212yesmaximal, complemented
5#20292939C212yesmaximal, complemented
6#20292922C211yesmaximal, characteristic, complemented
7#20292917C211yesmaximal, characteristic, complemented
8#20292870C212yesmaximal, complemented
9#20292706C211yesmaximal, characteristic, complemented
10#20292666C212yesmaximal, complemented
11C4×SD16×C107C211yesmaximal, characteristic, complemented
12C2×#13133801C2212yes—
13#16812109C2211yescharacteristic
14#16812109C2211yescharacteristic
15#16812101—21no—
16#16812095—22no—
17#16812087—22no—
18#16812085C2212yes—
19#16812083—21no—
20#16812080C2212yes—
21C4×D428—21no—
22C4×D428C2211yescharacteristic
23#16812068—21no—
24#16812067C2212yes—
25C4⋊C4×D107C2212yes—
26#16812043C2212yes—
27#16812043C2212yes—
28#16812042C2212yes—
29#16812040C2212yes—
30C4⋊(C107⋊C8)C2211yescharacteristic
31#16812033C2212yes—
32#16812030C2212yes—
33(C2×D107)⋊C8C2212yes—
34C2×SD16×C107C2211yescharacteristic
35C4×Q8×C107C2211yescharacteristic
36C4×D4×C107C2211yescharacteristic
37#106×C107C2211yescharacteristic
38#103×C107C2211yescharacteristic
39#102×C107C2211yescharacteristic
40C4×C8×C107C2211yescharacteristic
41C2×Dic214—21no—
42C2×Dic214—21no—
43C2×D428D411yescharacteristic
44C2×C22⋊D107—22no—
45C2×C4×D107—22no—
46C2×C4×D107C2312yes—
47#13133801—24no—
48C2×C107⋊C8C2312yes—
49C4⋊Dic107—21no—
50C4⋊Dic107—21no—
51C4⋊Dic107D411yescharacteristic
52C4⋊Dic107C2312yes—
53C4⋊Dic107C2312yes—
54Dic107⋊C4—22no—
55Dic107⋊C4—22no—
56C22⋊Dic107—22no—
57(C2×D107)⋊C4—22no—
58(C2×D107)⋊C4—22no—
59(C2×D107)⋊C4D412yes—
60C4×Dic107—22no—
61C2×Q8×C107C2311yescharacteristic
62C2×D4×C107C2311yescharacteristic
63C22×C4×C107—21no—
64SD16×C107—22no—
65C2×C8×C107C2311yescharacteristic
66C2×C8×C107C2311yescharacteristic
67C4⋊C4×C107—21no—
68C4⋊C4×C107C2311yescharacteristic
69C4⋊C4×C107C2311yescharacteristic
70C22⋊C4×C107—21no—
71C42×C107—21no—
72C42×C107C2311yescharacteristic
73C23×C107—21no—
74C22×D107C2×D412yes—
75Q8×C107—21no—
76Q8×C107—21no—
77D4×C107—21no—
78D4×C107—21no—
79C2×C4×C107—21no—
80C2×C4×C107—21no—
81C2×C4×C107—21no—
82C2×C4×C107—42no—
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2411yescharacteristic
86C22⋊D107—82no—
87C2×Dic107—22no—
88C2×Dic107—22no—
89C2×Dic107C2×D412yes—
90D428—22no—
91C4×D107—22no—
92C4×D107—22no—
93C4×D107—82no—
94Dic214—81no—
95Dic214—81no—
96C8×C107—21no—
97C8×C107#4112yes—
98C107⋊C8—22no—
99C22×C107—41no—
100C22×C107—41no—
101C22×C107—41no—
102C22×C107C22×D411yescharacteristic
103C2×D107—22no—
104C2×D107—22no—
105C2×D107—22no—
106C4×C107—21no—
107C4×C107—41no—
108C4×C107—41no—
109C4×C107—41no—
110C4×C107#13712yes—
111C4×C107C2×#4111yescharacteristic
112C4×C1072−1+411yescharacteristic
113Dic107—22no—
114Dic107—42no—
115Dic107—42no—
116C2×C107—81no—
117C2×C107#54711yescharacteristic
118C2×C107C2×#13711yescharacteristic
119C2×C107#57611yescharacteristic
120D107—42no—
121#3357—1071nomaximal, Sylow
122C107#335711yescharacteristic, complemented, Sylow
123#555—1071no—
124#554—1071no—
125#500—1071no—
126#485—1072no—
127#480—1072no—
128#477—1071no—
129#476—1071no—
130#468—1072no—
131C4×SD16D10711yescharacteristic, complemented
132#436—1071no—
133#428—1072no—
134C2×SD16C2×D10711yescharacteristic
135C2×M16—1072no—
136C4⋊Q8—1071no—
137C4⋊Q8—1071no—
138#125—2141no—
139#124—2142no—
140C22⋊Q8—1072no—
141C22⋊Q8—2142no—
142#121—1072no—
143#121—2141no—
144C4×Q8C2×D10711yescharacteristic
145C4×D4—1071no—
146C4×D4—2141no—
147C4×D4C2×D10711yescharacteristic
148#117—1072no—
149#117—2141no—
150C2×C4⋊C4—1072no—
151#107—1072no—
152#107—1072no—
153#107—1072no—
154#106—1072no—
155#106C2×D10711yescharacteristic
156C4⋊C8—1071no—
157#103—1072no—
158#103C2×D10711yescharacteristic
159#102—1072no—
160#102C2×D10711yescharacteristic
161C22⋊C8—1072no—
162C4×C8C2×D10711yescharacteristic
163C2×Q8—2141no—
164C2×Q8—2141no—
165C2×Q8C22×D10711yescharacteristic
166C2×D4—1071no—
167C2×D4—2142no—
168C2×D4C22×D10711yescharacteristic
169C22×C4—21no—
170C22×C4—1072no—
171C22×C4—2142no—
172SD16—22no—
173M16—2144no—
174C2×C8—1072no—
175C2×C8C22×D10711yescharacteristic
176C2×C8C22×D10711yescharacteristic
177C4⋊C4—21no—
178C4⋊C4—1071no—
179C4⋊C4—1072no—
180C4⋊C4—1072no—
181C4⋊C4—2142no—
182C4⋊C4—2142no—
183C4⋊C4—2141no—
184C4⋊C4—2141no—
185C4⋊C4C22×D10711yescharacteristic
186C4⋊C4C22×D10711yescharacteristic
187C22⋊C4—21no—
188C22⋊C4—1072no—
189C22⋊C4—2142no—
190C22⋊C4—2142no—
191C22⋊C4—2142no—
192C42—21no—
193C42—2142no—
194C42C22×D10711yescharacteristic
195C23—21no—
196C23—1072no—
197Q8—21no—
198Q8—21no—
199Q8—8561no—
200Q8—8561no—
201D4—21no—
202D4—21no—
203D4—8562no—
204D4—2142no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—42no—
209C2×C4—8562no—
210C2×C4—1072no—
211C2×C4—2142no—
212C2×C4—2142no—
213C2×C4—2142no—
214C2×C4—2142no—
215C2×C4D4×D10711yescharacteristic
216C2×C4D4×D10711yescharacteristic
217C2×C4C23×D10711yescharacteristic
218C8—21no—
219C8—2142no—
220C8#1313381812yes—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—2142no—
225C22—2142no—
226C22—2142no—
227C22C2×D4×D10711yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—4282no—
233C4—4282no—
234C4—2142no—
235C4#1681213812yes—
236C4C2×#1313381811yescharacteristic
237C4#1681215611yescharacteristic
238C2—81no—
239C2—4282no—
240C2#2029328211yescharacteristic
241C2C2×#1681213811yescharacteristic
242C2#2029342911yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC4×SD16×C107C2
socleC22×C107C22×D4
prime radical#20293342C2
hypercenterC4×SD16D107
hypocenterC107#3357
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
2. #20293337C2semidirect
3. #20293032C2semidirect
4. #20292967C2semidirect
5. #20292939C2semidirect
6. #20292922C2semidirect
7. #20292917C2semidirect
8. #20292870C2semidirect
9. #20292706C2semidirect
10. #20292666C2semidirect
11. C4×SD16×C107C2semidirect
122. C107#3357semidirect
131. C4×SD16D107semidirect

Maximal subgroups

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

subgroupindex
#202933422
#202933372
#202930322
#202929672
#202929392
#202929222
#202929172
#202928702
#202927062
#202926662
C4×SD16×C1072
#3357107