Finite Group Databasebeta

← #28345951

Subgroups — #28345951

Order 13696.

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

normal subgroups only

This group has 500 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
1C2×#16812140C211yesmaximal, characteristic, complemented
2C2×#16812135C211yesmaximal, characteristic
3C4×#13133818C211yesmaximal, characteristic, complemented
4#20292846C214yesmaximal, complemented
5#20292839C214yesmaximal, complemented
6#20292651C211yesmaximal, characteristic, complemented
7#20292636C211yesmaximal, characteristic, complemented
8#20292632C211yesmaximal, characteristic, complemented
9C2×#124×C107C211yesmaximal, characteristic, complemented
10C2×#13133818C2211yescharacteristic
11#16812140—24no—
12#16812135—24no—
13C2×#13133808C2212yescomplemented
14C2×#13133806C2212yes—
15C2×#13133805C2212yescomplemented
16C2×#13133803C2212yes—
17C2×#13133802C2211yescharacteristic
18C2×#13133802C2211yescharacteristic
19C4×Dic214C2212yescomplemented
20C4×D428C2212yescomplemented
21C4×C22⋊D107—22no—
22#16812068C2211yescharacteristic
23#16812066—21no—
24C42×D107—21no—
25C4⋊(C107⋊C8)C2214yescomplemented
26#16812033C2214yescomplemented
27#16812030C2214yes—
28C22×Q8×C107C2211yescharacteristic
29C22×D4×C107C2211yescharacteristic
30#124×C107—21no—
31#124×C107—21no—
32#124×C107C2214yes—
33C2×C22⋊C4×C107—22no—
34C2×C42×C107C2211yescharacteristic
35#13133818—22no—
36#13133818D414yes—
37C2×Dic214C2311yescharacteristic
38C2×D428C2311yescharacteristic
39C2×C22⋊D107—21no—
40C2×C4×D107—21no—
41C2×C4×D107—22no—
42#13133808—24no—
43#13133806—24no—
44#13133805—24no—
45#13133803—24no—
46#13133802—22no—
47#13133802—22no—
48C2×C107⋊C8C2312yes—
49C2×C107⋊C8C2312yes—
50C4⋊Dic107C2312yes—
51Dic107⋊C4—22no—
52C22⋊Dic107—21no—
53(C2×D107)⋊C4—22no—
54C4×Dic107—21no—
55C4×Dic107—22no—
56C24×C107—21no—
57C2×Q8×C107—21no—
58C2×Q8×C107—22no—
59C2×Q8×C107—22no—
60C2×Q8×C107C2312yes—
61C2×D4×C107—22no—
62C2×D4×C107—21no—
63C2×D4×C107—22no—
64C2×D4×C107C2312yes—
65C22×C4×C107—21no—
66C22×C4×C107D412yes—
67C22×C4×C107C2311yescharacteristic
68C22⋊C4×C107—24no—
69C22⋊C4×C107—24no—
70C42×C107D411yescharacteristic
71C42×C107D411yescharacteristic
72C42×C107C2312yes—
73C23×C107—22no—
74C23×C107—44no—
75C23×C107—42no—
76C23×C107C2×D411yescharacteristic
77C22×D107—21no—
78Q8×C107—22no—
79Q8×C107—22no—
80Q8×C107—42no—
81D4×C107—22no—
82D4×C107—22no—
83D4×C107—42no—
84C2×C4×C107—22no—
85C2×C4×C107—42no—
86C2×C4×C107—42no—
87C2×C4×C107C2×D412yes—
88C2×C4×C107C2×D412yes—
89C2×C4×C107C2×D412yes—
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107#4112yes—
92C2×C4×C107C2411yescharacteristic
93C22⋊D107—24no—
94C2×Dic107—21no—
95C2×Dic107—22no—
96D428—21no—
97D428C2×D412yes—
98C4×D107—24no—
99C4×D107—44no—
100Dic214—21no—
101Dic214C2×D412yes—
102C107⋊C8—22no—
103C107⋊C8—22no—
104C22×C107—22no—
105C22×C107—42no—
106C22×C107—44no—
107C22×C107—84no—
108C22×C107#12112yes—
109C22×C107C22×D411yescharacteristic
110C2×D107—22no—
111C2×D107—42no—
112C4×C107—22no—
113C4×C107—22no—
114C4×C107—42no—
115C4×C107#12112yes—
116C4×C107C22×D411yescharacteristic
117C4×C107C2×#4111yescharacteristic
118Dic107—22no—
119Dic107—42no—
120C2×C107—22no—
121C2×C107—82no—
122C2×C107C2×#12111yescharacteristic
123C2×C107#57512yes—
124D107—42no—
125#3040—1071nomaximal, Sylow
126C107#304011yescharacteristic, complemented, Sylow
127C2×#137—1071no—
128C2×AGL(1,ℤ/8)—1071no—
129C2×#124D10711yescharacteristic, complemented
130C4×#41—1071no—
131#464—1074no—
132#463—1074no—
133#423—1071no—
134#419—1071no—
135#418—1071no—
136C2×#41—1071no—
137C22×Q8C2×D10711yescharacteristic
138C22×D4C2×D10711yescharacteristic
139#137—2144no—
140AGL(1,ℤ/8)—2144no—
141C2×Q16—1072no—
142C2×SD16—1072no—
143C2×SD16—1072no—
144C2×D8—1072no—
145C2×M16—1071no—
146C2×M16—1071no—
147#124—21no—
148#124—21no—
149#124C2×D10714yes—
150C4×Q8—1072no—
151C4×D4—1072no—
152C4×D4—2142no—
153#117—1071no—
154#117—2141no—
155C2×C22⋊C4—22no—
156C2×C42—2141no—
157C2×C42C2×D10711yescharacteristic
158C4⋊C8—1074no—
159#103—1074no—
160#102—1074no—
161C24—21no—
162#41—1074no—
163#41—2142no—
164C2×Q8—21no—
165C2×Q8—22no—
166C2×Q8—22no—
167C2×Q8—1071no—
168C2×Q8C22×D10712yes—
169C2×D4—22no—
170C2×D4—21no—
171C2×D4—22no—
172C2×D4—1071no—
173C2×D4—2141no—
174C2×D4C22×D10712yes—
175C22×C4—21no—
176C22×C4—2142no—
177C22×C4—2141no—
178C22×C4C22⋊D10712yes—
179C22×C4C22×D10711yescharacteristic
180Q16—2144no—
181SD16—2144no—
182SD16—2144no—
183D8—2144no—
184M16—2142no—
185M16—2142no—
186C2×C8—1072no—
187C2×C8—1072no—
188C4⋊C4—1072no—
189C4⋊C4—2142no—
190C22⋊C4—24no—
191C22⋊C4—24no—
192C22⋊C4—2141no—
193C22⋊C4—2142no—
194C42—2141no—
195C42—2142no—
196C42C22⋊D10711yescharacteristic
197C42C22⋊D10711yescharacteristic
198C42C22×D10712yes—
199C23—22no—
200C23—44no—
201C23—42no—
202C23—2141no—
203C23C2×C22⋊D10711yescharacteristic
204Q8—22no—
205Q8—22no—
206Q8—42no—
207Q8—1072no—
208Q8—2141no—
209D4—22no—
210D4—22no—
211D4—42no—
212D4—1072no—
213D4—2144no—
214D4—2141no—
215C2×C4—22no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—4284no—
219C2×C4—2142no—
220C2×C4—2141no—
221C2×C4—2144no—
222C2×C4D4×D10712yes—
223C2×C4C2×C22⋊D10712yes—
224C2×C4C2×C22⋊D10712yes—
225C2×C4C2×C22⋊D10711yescharacteristic
226C2×C4#1313381712yes—
227C2×C4C23×D10711yescharacteristic
228C8—2142no—
229C8—2142no—
230C22—22no—
231C22—44no—
232C22—42no—
233C22—84no—
234C22—4282no—
235C22—2142no—
236C22#1681208012yes—
237C22C22×C22⋊D10711yescharacteristic
238C4—22no—
239C4—22no—
240C4—42no—
241C4—4282no—
242C4—2142no—
243C4#1681208012yes—
244C4C2×D4×D10711yescharacteristic
245C4C2×#1313381711yescharacteristic
246C2—22no—
247C2—82no—
248C2—4282no—
249C2C2×#1681208011yescharacteristic
250C2#2029342512yes—
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#124×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812135C2
hypercenterC2×#124D107
hypocenterC107#3040
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#16812140C2semidirect
3. C4×#13133818C2semidirect
4. #20292846C2semidirect
5. #20292839C2semidirect
6. #20292651C2semidirect
7. #20292636C2semidirect
8. #20292632C2semidirect
9. C2×#124×C107C2semidirect
13. C2×#13133808C22semidirect
15. C2×#13133805C22semidirect
19. C4×Dic214C22semidirect
20. C4×D428C22semidirect
25. C4⋊(C107⋊C8)C22semidirect
26. #16812033C22semidirect
126. C107#3040semidirect
129. C2×#124D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121402
C2×#168121352
C4×#131338182
#202928462
#202928392
#202926512
#202926362
#202926322
C2×#124×C1072
#3040107