Finite Group Databasebeta

← #28346108

Subgroups — #28346108

Order 13696.

524 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 133 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×#13133803C211yesmaximal, characteristic, complemented
2C2×#16812117C211yesmaximal, characteristic, complemented
3#20293196C211yesmaximal, characteristic, complemented
4#20293194C211yesmaximal, characteristic, complemented
5#20292879C214yesmaximal, complemented
6#20292862C214yesmaximal, complemented
7#20292663C211yesmaximal, characteristic, complemented
8#20292633C211yesmaximal, characteristic, complemented
9#418×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11C22×D428C2211yescharacteristic
12C2×#13133803—24no—
13C2×#13133803C2212yes—
14C2×#13133803C2212yes—
15#16812117—24no—
16C2×#13133801C2212yescomplemented
17C22×C107⋊C8C2211yescharacteristic
18C2×C8×D107C2212yescomplemented
19#16812106—21no—
20#16812105—21no—
21#16812097—21no—
22#16812096—21no—
23#16812084C2214yescomplemented
24#16812080C2214yescomplemented
25#16812079—22no—
26#16812078—22no—
27#16812068C2211yescharacteristic
28#16812042C2212yescomplemented
29#16812040C2212yescomplemented
30#16812031C2214yescomplemented
31C22×D4×C107C2211yescharacteristic
32C2×M16×C107C2211yescharacteristic
33#117×C107C2211yescharacteristic
34#102×C107C2214yescomplemented
35C23×D107—21no—
36#13133818—22no—
37#13133818—22no—
38C2×Dic214D411yescharacteristic
39C2×D428—22no—
40C2×D428—22no—
41C2×D428—21no—
42C2×D428—24no—
43C2×D428D411yescharacteristic
44C2×D428C2312yes—
45C2×C22⋊D107—24no—
46C2×C22⋊D107D412yes—
47C2×C4×D107C2312yes—
48#13133803—24no—
49#13133803—24no—
50#13133801—24no—
51C2×C107⋊C8D414yescomplemented
52C2×C107⋊C8C2311yescharacteristic
53C2×C107⋊C8C2311yescharacteristic
54C8×D107—24no—
55C4⋊Dic107C2312yes—
56C22⋊Dic107—21no—
57C22⋊Dic107—24no—
58(C2×D107)⋊C4—24no—
59C4×Dic107—21no—
60C24×C107—21no—
61C2×D4×C107—22no—
62C2×D4×C107—21no—
63C2×D4×C107—22no—
64C2×D4×C107C2312yes—
65C22×C4×C107C2311yescharacteristic
66M16×C107—22no—
67C2×C8×C107C2312yes—
68C4⋊C4×C107C2312yes—
69C22⋊C4×C107—21no—
70C42×C107—21no—
71C23×C107—22no—
72C23×C107—42no—
73C23×C107—82no—
74C23×C107C2×D411yescharacteristic
75C22×D107—22no—
76C22×D107—42no—
77C22×D107—82no—
78C22×D107C2×D412yes—
79D4×C107—22no—
80D4×C107—22no—
81D4×C107—42no—
82C2×C4×C107—22no—
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107C2×D412yes—
85C2×C4×C107#4112yes—
86C2×C4×C107C2411yescharacteristic
87C22⋊D107—24no—
88C22⋊D107—84no—
89C2×Dic107—22no—
90C2×Dic107C2×D412yes—
91D428—22no—
92D428—22no—
93D428—22no—
94D428—42no—
95D428—84no—
96C4×D107—22no—
97C4×D107—22no—
98Dic214—22no—
99C8×C107—22no—
100C107⋊C8C2×D412yes—
101C107⋊C8C2×D412yes—
102C22×C107—22no—
103C22×C107—42no—
104C22×C107—44no—
105C22×C107—84no—
106C22×C107#12112yes—
107C22×C107C22×D411yescharacteristic
108C2×D107—22no—
109C2×D107—24no—
110C2×D107—42no—
111C2×D107—44no—
112C2×D107—84no—
113C4×C107—42no—
114C4×C107#12112yes—
115C4×C107C22×D411yescharacteristic
116C4×C107C2×#4111yescharacteristic
117Dic107—22no—
118Dic107—42no—
119C2×C107—22no—
120C2×C107—82no—
121C2×C107C2×#12111yescharacteristic
122C2×C107#57612yes—
123D107—42no—
124D107—82no—
125#3058—1071nomaximal, Sylow
126C107#305811yescharacteristic, complemented, Sylow
127C22×D8—1071no—
128C2×#131—1071no—
129#535—1071no—
130#535—1071no—
131#469—1074no—
132#466—1074no—
133#427—1071no—
134#418—1071no—
135#418D10711yescharacteristic, complemented
136C2×#41—1071no—
137C22×D4—1071no—
138C22×D4C2×D10711yescharacteristic
139C2×D8—1072no—
140C2×D8—1072no—
141C2×D8—2144no—
142#131—2144no—
143C2×M16—1072no—
144C2×M16C2×D10711yescharacteristic
145C22×C8—1071no—
146C22×C8—1072no—
147#127—2141no—
148#127—2141no—
149#124—2141no—
150#124—2141no—
151#121—1074no—
152#121—1074no—
153#120—2142no—
154#120—2142no—
155#117—1071no—
156#117C2×D10711yescharacteristic
157#107—1072no—
158#106—1072no—
159#102—1074no—
160#102C2×D10714yescomplemented
161C24—21no—
162C24—2141no—
163#41—2142no—
164#41—2142no—
165C2×Q8—1071no—
166C2×D4—22no—
167C2×D4—21no—
168C2×D4—22no—
169C2×D4—1072no—
170C2×D4—1071no—
171C2×D4—1072no—
172C2×D4—2144no—
173C2×D4—2144no—
174C2×D4—2141no—
175C2×D4—2142no—
176C2×D4—2142no—
177C2×D4C22×D10712yes—
178C22×C4—1072no—
179C22×C4C22×D10711yescharacteristic
180D8—2144no—
181D8—2144no—
182M16—22no—
183M16—2144no—
184C2×C8—1074no—
185C2×C8—1071no—
186C2×C8—1071no—
187C2×C8—2144no—
188C2×C8C22×D10712yes—
189C4⋊C4—1072no—
190C4⋊C4C22×D10712yes—
191C22⋊C4—21no—
192C22⋊C4—2144no—
193C22⋊C4—2144no—
194C22⋊C4—2141no—
195C42—21no—
196C42—2141no—
197C23—22no—
198C23—42no—
199C23—82no—
200C23—8562no—
201C23—1072no—
202C23—4282no—
203C23—2142no—
204C23D4×D10711yescharacteristic
205Q8—2142no—
206D4—22no—
207D4—22no—
208D4—42no—
209D4—8564no—
210D4—8564no—
211D4—4282no—
212D4—2144no—
213D4—2142no—
214D4—2142no—
215D4—2142no—
216C2×C4—22no—
217C2×C4—1072no—
218C2×C4—2142no—
219C2×C4—2142no—
220C2×C4—2142no—
221C2×C4D4×D10711yescharacteristic
222C2×C4D4×D10712yes—
223C2×C4#1313381812yes—
224C2×C4C23×D10711yescharacteristic
225C8—22no—
226C8—1072no—
227C8—1072no—
228C22—22no—
229C22—44no—
230C22—42no—
231C22—84no—
232C22—8564no—
233C22—4284no—
234C22—4282no—
235C22—2144no—
236C22—2142no—
237C22#1681208212yes—
238C22C2×D4×D10711yescharacteristic
239C4—42no—
240C4—4282no—
241C4—2142no—
242C4#1681208212yes—
243C4C2×D4×D10711yescharacteristic
244C4C2×#1313381811yescharacteristic
245C2—22no—
246C2—82no—
247C2—8562no—
248C2—4282no—
249C2C2×#1681208211yescharacteristic
250C2#2029343112yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#418×C107C2
socleC22×C107C22×D4
hypercenter#418D107
hypocenterC107#3058
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
1. C22×#13133803C2semidirect
2. C2×#16812117C2semidirect
3. #20293196C2semidirect
4. #20293194C2semidirect
5. #20292879C2semidirect
6. #20292862C2semidirect
7. #20292663C2semidirect
8. #20292633C2semidirect
9. #418×C107C2semidirect
16. C2×#13133801C22semidirect
18. C2×C8×D107C22semidirect
23. #16812084C22semidirect
24. #16812080C22semidirect
28. #16812042C22semidirect
29. #16812040C22semidirect
30. #16812031C22semidirect
34. #102×C107C22semidirect
51. C2×C107⋊C8D4semidirect
126. C107#3058semidirect
135. #418D107semidirect
160. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338032
C2×#168121172
#202931962
#202931942
#202928792
#202928622
#202926632
#202926332
#418×C1072
#3058107