Finite Group Databasebeta

← #28347061

Subgroups — #28347061

Order 13696.

364 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 125 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#20293161C211yesmaximal, characteristic
2#20293033C211yesmaximal, characteristic, complemented
3#20293019C211yesmaximal, characteristic, complemented
4#20292995C211yesmaximal, characteristic, complemented
5#20292989C211yesmaximal, characteristic, complemented
6#20292883C212yesmaximal, complemented
7#20292875C212yesmaximal
8C4×#13133808C212yesmaximal, complemented
9C4×#13133805C212yesmaximal, complemented
10C4×#13133802C211yesmaximal, characteristic, complemented
11#533×C107C211yesmaximal, characteristic, complemented
12C2×#13133808C2212yes—
13C2×#13133805C2212yes—
14C2×#13133802C2212yes—
15#16812103—21no—
16#16812101C2211yescharacteristic
17#16812097C2211yescharacteristic
18#16812092—21no—
19#16812088C2211yescharacteristic
20#16812083C2211yescharacteristic
21C4×Dic214C2211yescharacteristic
22C4×D428C2211yescharacteristic
23C4×C22⋊D107—21no—
24#16812066—21no—
25#16812042C2212yes—
26#16812040C2212yes—
27#16812033C2212yes—
28#16812033C2212yes—
29#16812032C2212yes—
30#16812029C2212yes—
31#16812020C2211yescharacteristic
32#16812020C2211yescharacteristic
33C4×C107⋊C8C2212yes—
34C4⋊Q8×C107C2211yescharacteristic
35C4⋊Q8×C107C2211yescharacteristic
36#125×C107—21no—
37C22⋊Q8×C107—21no—
38C22⋊Q8×C107C2212yes—
39C4×Q8×C107—21no—
40C4×Q8×C107C2212yes—
41#117×C107—21no—
42C2×C42×C107C2211yescharacteristic
43C2×Dic214C2311yescharacteristic
44C2×D428C2311yescharacteristic
45C2×C22⋊D107—21no—
46C2×C4×D107—21no—
47#13133808—24no—
48#13133805—24no—
49#13133802—24no—
50C2×C107⋊C8C2312yes—
51C2×C107⋊C8C2312yes—
52C4⋊Dic107C2311yescharacteristic
53C4⋊Dic107C2311yescharacteristic
54Dic107⋊C4—21no—
55Dic107⋊C4—21no—
56Dic107⋊C4—21no—
57Dic107⋊C4—21no—
58C22⋊Dic107—21no—
59(C2×D107)⋊C4—21no—
60(C2×D107)⋊C4—21no—
61(C2×D107)⋊C4—21no—
62(C2×D107)⋊C4—21no—
63C4×Dic107—21no—
64C2×Q8×C107—21no—
65C2×Q8×C107C2312yes—
66C22×C4×C107D411yescharacteristic
67C22×C4×C107D411yescharacteristic
68C22×C4×C107C2311yescharacteristic
69C4⋊C4×C107—21no—
70C4⋊C4×C107—21no—
71C4⋊C4×C107—21no—
72C4⋊C4×C107—22no—
73C4⋊C4×C107—22no—
74C4⋊C4×C107C2312yes—
75C22⋊C4×C107—22no—
76C42×C107—22no—
77C42×C107D411yescharacteristic
78C42×C107D411yescharacteristic
79C42×C107C2311yescharacteristic
80C42×C107C2311yescharacteristic
81C23×C107C2×D411yescharacteristic
82C22×D107—21no—
83Q8×C107—22no—
84Q8×C107—22no—
85Q8×C107—42no—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107—22no—
89C2×C4×C107—22no—
90C2×C4×C107—41no—
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107C2411yescharacteristic
97C22⋊D107—42no—
98C2×Dic107—21no—
99C2×Dic107—21no—
100C2×Dic107—21no—
101D428—22no—
102C4×D107—42no—
103Dic214—22no—
104C107⋊C8—22no—
105C107⋊C8#4114yes—
106C22×C107—21no—
107C22×C107—21no—
108C22×C107—21no—
109C22×C107C22×D411yescharacteristic
110C2×D107—41no—
111C2×D107—41no—
112C2×D107—41no—
113C4×C107—21no—
114C4×C107—21no—
115C4×C107—21no—
116C4×C107—21no—
117C4×C107—42no—
118C4×C107—42no—
119C4×C107#13712yes—
120C4×C107C2×#4112yes—
121Dic107—41no—
122Dic107—41no—
123Dic107—41no—
124C2×C107—41no—
125C2×C107#53211yescharacteristic
126C2×C107C2×#13711yescharacteristic
127C2×C107#57511yescharacteristic
128D107—81no—
129#3172—1071nomaximal, Sylow
130C107#317211yescharacteristic, complemented, Sylow
131#533D10711yescharacteristic, complemented
132#529—1071no—
133#501—1071no—
134#497—1071no—
135#491—1071no—
136#489—1071no—
137#470—1072no—
138#468—1072no—
139C4×Q16—1072no—
140C4×SD16—1072no—
141C4×M16—1071no—
142C2×Q16—1072no—
143C2×SD16—1072no—
144C2×M16—1072no—
145C4⋊Q8C2×D10711yescharacteristic
146C4⋊Q8C2×D10711yescharacteristic
147#126—2141no—
148#125—21no—
149#125—1071no—
150#124—1071no—
151#123—2141no—
152C22⋊Q8—21no—
153C22⋊Q8—1071no—
154C22⋊Q8C2×D10712yes—
155#121—1071no—
156C4×Q8—21no—
157C4×Q8—1071no—
158C4×Q8C2×D10712yes—
159C4×D4—1071no—
160C4×D4—2141no—
161#117—21no—
162#117—2141no—
163C2×C42C2×D10711yescharacteristic
164#107—1072no—
165#106—1072no—
166#103—1072no—
167#103—1072no—
168#103—1072no—
169#102—1072no—
170#97—1071no—
171#97—1071no—
172C4×C8—1072no—
173C2×Q8—21no—
174C2×Q8—1071no—
175C2×Q8C22×D10712yes—
176C2×D4—1071no—
177C2×D4—2141no—
178C22×C4—2141no—
179C22×C4C22⋊D10711yescharacteristic
180C22×C4C22⋊D10711yescharacteristic
181C22×C4C22×D10711yescharacteristic
182Q16—2144no—
183SD16—2144no—
184M16—2144no—
185C2×C8—1072no—
186C2×C8—1072no—
187C4⋊C4—21no—
188C4⋊C4—21no—
189C4⋊C4—21no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—1071no—
193C4⋊C4—1071no—
194C4⋊C4—2141no—
195C4⋊C4—2141no—
196C4⋊C4—2141no—
197C4⋊C4—2141no—
198C4⋊C4C22×D10712yes—
199C22⋊C4—22no—
200C22⋊C4—2141no—
201C22⋊C4—2141no—
202C22⋊C4—2141no—
203C22⋊C4—2141no—
204C22⋊C4—2141no—
205C42—22no—
206C42—2141no—
207C42C22⋊D10711yescharacteristic
208C42C22⋊D10711yescharacteristic
209C42C22×D10711yescharacteristic
210C42C22×D10711yescharacteristic
211C23—2141no—
212C23C2×C22⋊D10711yescharacteristic
213Q8—22no—
214Q8—22no—
215Q8—42no—
216Q8—2142no—
217D4—4282no—
218D4—2142no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—41no—
224C2×C4—4282no—
225C2×C4—2141no—
226C2×C4—2141no—
227C2×C4—2141no—
228C2×C4C2×C22⋊D10711yescharacteristic
229C2×C4C2×C22⋊D10711yescharacteristic
230C2×C4C2×C22⋊D10711yescharacteristic
231C2×C4C2×C22⋊D10711yescharacteristic
232C2×C4C2×C22⋊D10711yescharacteristic
233C2×C4C23×D10711yescharacteristic
234C8—1074no—
235C8—2142no—
236C22—21no—
237C22—21no—
238C22—21no—
239C22—4281no—
240C22—4281no—
241C22—4281no—
242C22C22×C22⋊D10711yescharacteristic
243C4—21no—
244C4—21no—
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—4281no—
250C4—4281no—
251C4—4281no—
252C4#1681214012yes—
253C4#41×D10712yes—
254C2—41no—
255C2—8561no—
256C2#2029317311yescharacteristic
257C2C2×#1681214011yescharacteristic
258C2#2029342511yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#533×C107C2
socleC22×C107C22×D4
prime radical#16812083C22
hypercenter#533D107
hypocenterC107#3172
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #20293033C2semidirect
3. #20293019C2semidirect
4. #20292995C2semidirect
5. #20292989C2semidirect
6. #20292883C2semidirect
8. C4×#13133808C2semidirect
9. C4×#13133805C2semidirect
10. C4×#13133802C2semidirect
11. #533×C107C2semidirect
130. C107#3172semidirect
131. #533D107semidirect

Maximal subgroups

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

subgroupindex
#202931612
#202930332
#202930192
#202929952
#202929892
#202928832
#202928752
C4×#131338082
C4×#131338052
C4×#131338022
#533×C1072
#3172107