Finite Group Databasebeta

← #28346097

Subgroups — #28346097

Order 13696.

492 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#13133806C211yesmaximal, characteristic, complemented
2C2×#16812116C211yesmaximal, characteristic, complemented
3#20293198C211yesmaximal, characteristic, complemented
4#20292871C214yesmaximal, complemented
5#20292854C214yesmaximal, complemented
6#20292663C211yesmaximal, characteristic, complemented
7#20292635C211yesmaximal, characteristic, complemented
8#20292632C211yesmaximal, characteristic, complemented
9#535×C107C211yesmaximal, characteristic
10C22×Dic214C2211yescharacteristic
11C2×#13133806—24no—
12C2×#13133806C2212yes—
13C2×#13133806C2212yes—
14#16812116—24no—
15C2×#13133802C2211yescharacteristic
16C2×#13133802C2212yescomplemented
17C22×C107⋊C8C2211yescharacteristic
18C22×C107⋊C8C2212yescomplemented
19#16812107—21no—
20#16812096—21no—
21#16812090—22no—
22#16812088C2214yescomplemented
23#16812068C2211yescharacteristic
24#16812042C2212yescomplemented
25#16812040C2212yescomplemented
26#16812032C2214yescomplemented
27#16812030C2214yes—
28C2×#41×C107C2211yescharacteristic
29C22×D4×C107C2211yescharacteristic
30#127×C107—21no—
31#124×C107—21no—
32#121×C107C2214yes—
33#120×C107—22no—
34#117×C107C2211yescharacteristic
35C2×Dic214—22no—
36C2×Dic214—22no—
37C2×Dic214—21no—
38C2×Dic214C2312yes—
39C22×Dic107—21no—
40#13133806—24no—
41#13133806—24no—
42#13133802—22no—
43#13133802—24no—
44C2×C107⋊C8—24no—
45C2×C107⋊C8D414yescomplemented
46C2×C107⋊C8C2312yes—
47C2×C107⋊C8C2311yescharacteristic
48C2×C107⋊C8C2311yescharacteristic
49C4⋊Dic107C2312yes—
50Dic107⋊C4—24no—
51C22⋊Dic107—21no—
52C22⋊Dic107—24no—
53C4×Dic107—21no—
54C24×C107—21no—
55#41×C107—22no—
56#41×C107—22no—
57C2×Q8×C107D411yescharacteristic
58C2×D4×C107—24no—
59C2×D4×C107—22no—
60C2×D4×C107—21no—
61C2×D4×C107—22no—
62C2×D4×C107D412yes—
63C2×D4×C107D411yescharacteristic
64C2×D4×C107C2312yes—
65C22×C4×C107C2312yes—
66C22×C4×C107C2311yescharacteristic
67C4⋊C4×C107C2312yes—
68C22⋊C4×C107—21no—
69C22⋊C4×C107—24no—
70C42×C107—21no—
71C23×C107—22no—
72C23×C107—42no—
73C23×C107—82no—
74C23×C107C2×D412yes—
75C23×C107C2×D411yescharacteristic
76Q8×C107—22no—
77D4×C107—24no—
78D4×C107—22no—
79D4×C107—22no—
80D4×C107—22no—
81D4×C107—42no—
82D4×C107—84no—
83C2×C4×C107—22no—
84C2×C4×C107—22no—
85C2×C4×C107—22no—
86C2×C4×C107C2×D412yes—
87C2×C4×C107C2×D412yes—
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107#4112yes—
90C2×C4×C107C2411yescharacteristic
91C2×Dic107—22no—
92C2×Dic107—22no—
93C2×Dic107—42no—
94Dic214—22no—
95Dic214—22no—
96Dic214—42no—
97C107⋊C8—22no—
98C107⋊C8C2×D412yes—
99C107⋊C8C2×D412yes—
100C22×C107—22no—
101C22×C107—22no—
102C22×C107—24no—
103C22×C107—42no—
104C22×C107—44no—
105C22×C107—84no—
106C22×C107#12112yes—
107C22×C107C22×D411yescharacteristic
108C4×C107—22no—
109C4×C107—42no—
110C4×C107#12112yes—
111C4×C107C22×D411yescharacteristic
112C4×C107C2×#4111yescharacteristic
113Dic107—42no—
114Dic107—42no—
115C2×C107—22no—
116C2×C107—42no—
117C2×C107—82no—
118C2×C107C2×#12111yescharacteristic
119C2×C107#57712yes—
120#3057—1071nomaximal, Sylow
121C107#305711yescharacteristic, complemented, Sylow
122C22×SD16—1071no—
123C2×#131—1071no—
124#536—1071no—
125#535D10711yescharacteristic
126#468—1074no—
127#465—1074no—
128#427—1071no—
129#419—1071no—
130#418—1071no—
131C2×#41C2×D10711yescharacteristic
132C22×Q8—1071no—
133C22×D4C2×D10711yescharacteristic
134C2×SD16—1072no—
135C2×SD16—1072no—
136C2×SD16—2144no—
137#131—2144no—
138C2×M16—1071no—
139C2×M16—1072no—
140C22×C8—1071no—
141C22×C8—1072no—
142C4⋊Q8—2141no—
143#127—21no—
144#124—21no—
145#124—2141no—
146#123—2142no—
147C22⋊Q8—1074no—
148#121C2×D10714yes—
149#120—22no—
150#117—1071no—
151#117C2×D10711yescharacteristic
152#107—1072no—
153#106—1072no—
154#103—1074no—
155#102—1074no—
156C24—21no—
157#41—22no—
158#41—22no—
159C2×Q8—1072no—
160C2×Q8—2142no—
161C2×Q8—2142no—
162C2×Q8—2141no—
163C2×Q8C22⋊D10711yescharacteristic
164C2×D4—24no—
165C2×D4—22no—
166C2×D4—21no—
167C2×D4—22no—
168C2×D4C22⋊D10712yes—
169C2×D4C22⋊D10711yescharacteristic
170C2×D4C22×D10712yes—
171C22×C4—2141no—
172C22×C4C22×D10712yes—
173C22×C4C22×D10711yescharacteristic
174SD16—2144no—
175SD16—2144no—
176M16—2144no—
177M16—2142no—
178C2×C8—1074no—
179C2×C8—1071no—
180C2×C8—1071no—
181C2×C8—1072no—
182C2×C8—2144no—
183C4⋊C4—1072no—
184C4⋊C4—2144no—
185C4⋊C4C22×D10712yes—
186C22⋊C4—21no—
187C22⋊C4—24no—
188C22⋊C4—2141no—
189C22⋊C4—2144no—
190C42—21no—
191C42—2141no—
192C23—22no—
193C23—42no—
194C23—82no—
195C23C2×C22⋊D10712yes—
196C23C2×C22⋊D10711yescharacteristic
197Q8—22no—
198Q8—4282no—
199Q8—2142no—
200Q8—2142no—
201D4—24no—
202D4—22no—
203D4—22no—
204D4—22no—
205D4—42no—
206D4—84no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—4282no—
211C2×C4—2142no—
212C2×C4—2142no—
213C2×C4D4×D10712yes—
214C2×C4C2×C22⋊D10712yes—
215C2×C4C2×C22⋊D10711yescharacteristic
216C2×C4#1313381712yes—
217C2×C4C23×D10711yescharacteristic
218C8—1072no—
219C8—1072no—
220C8—2142no—
221C22—22no—
222C22—24no—
223C22—22no—
224C22—44no—
225C22—42no—
226C22—84no—
227C22#1681208112yes—
228C22C22×C22⋊D10711yescharacteristic
229C4—22no—
230C4—42no—
231C4—4282no—
232C4—4282no—
233C4#1681208112yes—
234C4C2×D4×D10711yescharacteristic
235C4C2×#1313381711yescharacteristic
236C2—22no—
237C2—42no—
238C2—82no—
239C2C2×#1681208111yescharacteristic
240C2#2029343812yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting, prime radical#535×C107C2
socleC22×C107C22×D4
hypercenter#535D107
hypocenterC107#3057
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C22×#13133806C2semidirect
2. C2×#16812116C2semidirect
3. #20293198C2semidirect
4. #20292871C2semidirect
5. #20292854C2semidirect
6. #20292663C2semidirect
7. #20292635C2semidirect
8. #20292632C2semidirect
16. C2×#13133802C22semidirect
18. C22×C107⋊C8C22semidirect
22. #16812088C22semidirect
24. #16812042C22semidirect
25. #16812040C22semidirect
26. #16812032C22semidirect
45. C2×C107⋊C8D4semidirect
121. C107#3057semidirect

Maximal subgroups

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

subgroupindex
C22×#131338062
C2×#168121162
#202931982
#202928712
#202928542
#202926632
#202926352
#202926322
#535×C1072
#3057107