Finite Group Databasebeta

← #28347068

Subgroups — #28347068

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#20293184C211yesmaximal, characteristic, complemented
2#20293037C211yesmaximal, characteristic, complemented
3#20293022C211yesmaximal, characteristic, complemented
4#20292997C211yesmaximal, characteristic, complemented
5#20292986C211yesmaximal, characteristic, complemented
6#20292883C212yesmaximal, complemented
7#20292871C212yesmaximal
8C4×#13133808C212yesmaximal, complemented
9C4×#13133806C212yesmaximal, complemented
10C4×#13133802C211yesmaximal, characteristic, complemented
11#529×C107C211yesmaximal, characteristic
12C2×#13133808C2212yes—
13C2×#13133806C2212yes—
14C2×#13133802C2212yes—
15#16812109C2211yescharacteristic
16#16812109C2211yescharacteristic
17#16812101—21no—
18#16812088—21no—
19#16812088C2212yes—
20C4×Dic214—21no—
21C4×Dic214C2212yes—
22#16812068—21no—
23#16812042C2212yes—
24#16812040C2212yes—
25#16812033C2212yes—
26#16812032C2212yes—
27#16812032C2212yes—
28#16812030C2212yes—
29#16812020C2211yescharacteristic
30#16812020C2211yescharacteristic
31C4×C107⋊C8C2212yes—
32#126×C107—21no—
33#125×C107C2211yescharacteristic
34#124×C107C2211yescharacteristic
35#123×C107—21no—
36C22⋊Q8×C107C2211yescharacteristic
37#121×C107C2211yescharacteristic
38C4×Q8×C107C2211yescharacteristic
39C4×D4×C107—21no—
40C4×D4×C107C2211yescharacteristic
41#117×C107—21no—
42C2×C42×C107C2211yescharacteristic
43C2×Dic214—21no—
44C2×Dic214C2312yes—
45#13133808—24no—
46#13133806—24no—
47#13133802—24no—
48C2×C107⋊C8C2312yes—
49C2×C107⋊C8C2312yes—
50C4⋊Dic107—21no—
51C4⋊Dic107—21no—
52C4⋊Dic107—21no—
53C4⋊Dic107C2312yes—
54Dic107⋊C4—22no—
55Dic107⋊C4—22no—
56C22⋊Dic107—22no—
57C4×Dic107—22no—
58C2×Q8×C107C2311yescharacteristic
59C2×D4×C107—21no—
60C2×D4×C107C2311yescharacteristic
61C22×C4×C107—21no—
62C22×C4×C107D411yescharacteristic
63C22×C4×C107D411yescharacteristic
64C22×C4×C107C2311yescharacteristic
65C4⋊C4×C107—21no—
66C4⋊C4×C107—21no—
67C4⋊C4×C107—21no—
68C4⋊C4×C107—21no—
69C4⋊C4×C107C2311yescharacteristic
70C4⋊C4×C107C2311yescharacteristic
71C22⋊C4×C107—21no—
72C22⋊C4×C107—21no—
73C22⋊C4×C107—21no—
74C22⋊C4×C107—21no—
75C22⋊C4×C107—21no—
76C42×C107—21no—
77C42×C107D411yescharacteristic
78C42×C107D411yescharacteristic
79C42×C107C2311yescharacteristic
80C42×C107C2311yescharacteristic
81C23×C107—21no—
82C23×C107C2×D411yescharacteristic
83Q8×C107—22no—
84D4×C107—22no—
85D4×C107—42no—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107—21no—
89C2×C4×C107—21no—
90C2×C4×C107—21no—
91C2×C4×C107—41no—
92C2×C4×C107—42no—
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107C2×D411yescharacteristic
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2411yescharacteristic
99C2×Dic107—22no—
100C2×Dic107—22no—
101Dic214—22no—
102Dic214—22no—
103Dic214—42no—
104C107⋊C8—22no—
105C107⋊C8#4114yes—
106C22×C107—21no—
107C22×C107—21no—
108C22×C107—21no—
109C22×C107—41no—
110C22×C107—41no—
111C22×C107—41no—
112C22×C107C22×D411yescharacteristic
113C4×C107—21no—
114C4×C107—21no—
115C4×C107—21no—
116C4×C107—21no—
117C4×C107—41no—
118C4×C107—41no—
119C4×C107—41no—
120C4×C107#13712yes—
121C4×C107C2×#4112yes—
122Dic107—42no—
123Dic107—42no—
124C2×C107—41no—
125C2×C107—81no—
126C2×C107#53211yescharacteristic
127C2×C107C2×#13711yescharacteristic
128C2×C107#57511yescharacteristic
129#3172—1071nomaximal, Sylow
130C107#317211yescharacteristic, complemented, Sylow
131#533—1071no—
132#529D10711yescharacteristic
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⋊Q8—1071no—
146C4⋊Q8—1071no—
147#126—21no—
148#125—2141no—
149#125C2×D10711yescharacteristic
150#124C2×D10711yescharacteristic
151#123—21no—
152C22⋊Q8—1072no—
153C22⋊Q8—2141no—
154C22⋊Q8C2×D10711yescharacteristic
155#121C2×D10711yescharacteristic
156C4×Q8—1072no—
157C4×Q8—2141no—
158C4×Q8C2×D10711yescharacteristic
159C4×D4—21no—
160C4×D4C2×D10711yescharacteristic
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—1072no—
174C2×Q8—2141no—
175C2×Q8C22×D10711yescharacteristic
176C2×D4—21no—
177C2×D4C22×D10711yescharacteristic
178C22×C4—21no—
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—21no—
191C4⋊C4—1072no—
192C4⋊C4—2141no—
193C4⋊C4—2141no—
194C4⋊C4—2142no—
195C4⋊C4—2142no—
196C4⋊C4—2141no—
197C4⋊C4C22×D10711yescharacteristic
198C4⋊C4C22×D10711yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—21no—
202C22⋊C4—21no—
203C22⋊C4—21no—
204C22⋊C4—2142no—
205C42—21no—
206C42—2142no—
207C42C22⋊D10711yescharacteristic
208C42C22⋊D10711yescharacteristic
209C42C22×D10711yescharacteristic
210C42C22×D10711yescharacteristic
211C23—21no—
212C23C2×C22⋊D10711yescharacteristic
213Q8—22no—
214Q8—4282no—
215Q8—2142no—
216Q8—2142no—
217D4—22no—
218D4—42no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—21no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4—2142no—
227C2×C4—2142no—
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—41no—
240C22—41no—
241C22—41no—
242C22C22×C22⋊D10711yescharacteristic
243C4—21no—
244C4—21no—
245C4—21no—
246C4—21no—
247C4—41no—
248C4—41no—
249C4—41no—
250C4—4282no—
251C4—4282no—
252C4#1681213912yes—
253C4#41×D10712yes—
254C2—41no—
255C2—81no—
256C2#2029317311yescharacteristic
257C2C2×#1681213911yescharacteristic
258C2#2029342511yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#529×C107C2
socleC22×C107C22×D4
prime radical#121×C107C22
hypercenter#529D107
hypocenterC107#3172
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #529

Complements

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

NH ≅ G/Ntype
1. #20293184C2semidirect
2. #20293037C2semidirect
3. #20293022C2semidirect
4. #20292997C2semidirect
5. #20292986C2semidirect
6. #20292883C2semidirect
8. C4×#13133808C2semidirect
9. C4×#13133806C2semidirect
10. C4×#13133802C2semidirect
130. C107#3172semidirect

Maximal subgroups

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

subgroupindex
#202931842
#202930372
#202930222
#202929972
#202929862
#202928832
#202928712
C4×#131338082
C4×#131338062
C4×#131338022
#529×C1072
#3172107