Finite Group Databasebeta

← #28345825

Subgroups — #28345825

Order 13696.

698 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 161 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#20293425C212yesmaximal, complemented
2#20293201C211yesmaximal, characteristic, complemented
3#20292810C214yesmaximal, complemented
4#20292800C214yesmaximal, complemented
5#20292644C212yesmaximal, complemented
6C2×#16812028C211yesmaximal, characteristic, complemented
7C2×2−1+4×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9#16812140—22no—
10#16812139C2214yescomplemented
11#16812135—22no—
12#16812134C2214yescomplemented
13#16812129—24no—
14C2×#13133802C2212yes—
15#16812107C2212yescomplemented
16#16812098C2212yescomplemented
17#16812092—22no—
18#16812085—22no—
19#16812068C2211yescharacteristic
20#16812036C2218yescomplemented
21#16812028C2214yescomplemented
222−1+4×C107—24no—
232−1+4×C107—22no—
242−1+4×C107C2214yes—
25C2×#41×C107—22no—
26C2×#41×C107—22no—
27C2×#41×C107C2212yes—
28C22×Q8×C107—22no—
29C22×Q8×C107C2211yescharacteristic
30#13133818—22no—
31#13133818—22no—
32C2×Dic214C2311yescharacteristic
33C2×D428C2311yescharacteristic
34C2×C22⋊D107—21no—
35C2×C4×D107—21no—
36#13133808—24no—
37#13133806—24no—
38#13133805—24no—
39#13133803—24no—
40#13133802—22no—
41#13133802C2314yes—
42C2×C107⋊C8—22no—
43C4⋊Dic107—21no—
44Dic107⋊C4—24no—
45C22⋊Dic107—21no—
46(C2×D107)⋊C4—24no—
47C4×Dic107C2312yes—
48#41×C107—28no—
49#41×C107—28no—
50#41×C107—28no—
51#41×C107—28no—
52#41×C107—22no—
53#41×C107—22no—
54#41×C107D414yes—
55#41×C107C2314yes—
56C2×Q8×C107—22no—
57C2×Q8×C107—28no—
58C2×Q8×C107—28no—
59C2×Q8×C107—22no—
60C2×Q8×C107—22no—
61C2×Q8×C107D412yes—
62C2×Q8×C107D412yes—
63C2×Q8×C107C2312yes—
64C2×D4×C107—22no—
65C2×D4×C107—22no—
66C2×D4×C107D412yes—
67C22×C4×C107—24no—
68C22×C4×C107—22no—
69C22×C4×C107D412yes—
70C22×C4×C107C2311yescharacteristic
71C23×C107—22no—
72C23×C107C2×D411yescharacteristic
73C22×D107—21no—
74Q8×C107—28no—
75Q8×C107—22no—
76Q8×C107—24no—
77Q8×C107—24no—
78Q8×C107C2×D414yes—
79D4×C107—28no—
80D4×C107—24no—
81D4×C107—24no—
82D4×C107—22no—
83D4×C107C2×D414yes—
84C2×C4×C107—216no—
85C2×C4×C107—22no—
86C2×C4×C107—22no—
87C2×C4×C107—22no—
88C2×C4×C107—24no—
89C2×C4×C107—24no—
90C2×C4×C107C2×D414yes—
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107C2×D412yes—
93C2×C4×C107C2×D412yes—
94C2×C4×C107C2411yescharacteristic
95C22⋊D107—41no—
96C22⋊D107—41no—
97C2×Dic107—22no—
98C2×Dic107—21no—
99D428—22no—
100C4×D107—42no—
101Dic214—22no—
102C107⋊C8—24no—
103C22×C107—24no—
104C22×C107—41no—
105C22×C107—44no—
106C22×C107#12011yescharacteristic
107C22×C107#12011yescharacteristic
108C22×C107C22×D411yescharacteristic
109C2×D107—42no—
110C2×D107—41no—
111C4×C107—24no—
112C4×C107—24no—
113C4×C107#12012yes—
114C4×C107C22×D412yes—
115Dic107—42no—
116Dic107—41no—
117C2×C107—21no—
118C2×C107—21no—
119C2×C107—21no—
120C2×C107—44no—
121C2×C107C2×#12011yescharacteristic
122D107—81no—
123#3020—1071nomaximal, Sylow
124C107#302011yescharacteristic, complemented, Sylow
125C2×2−1+4D10711yescharacteristic, complemented
126#575—1072no—
127#536—1071no—
128#456—1074no—
129#454—1074no—
130#421—1072no—
131C2×#101—1071no—
1322−1+4—24no—
1332−1+4—22no—
1342−1+4C2×D10714yes—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—1071no—
138C2×#41C2×D10712yes—
139C22×Q8—22no—
140C22×Q8C2×D10711yescharacteristic
141#137—1074no—
142#137—2142no—
143AGL(1,ℤ/8)—1074no—
144AGL(1,ℤ/8)—2142no—
145#135—2144no—
146C2×M16—1072no—
147C4⋊Q8—1072no—
148#124—1072no—
149#123—2142no—
150C22⋊Q8—2142no—
151#117—1071no—
152#104—1078no—
153#101—1074no—
154#41—28no—
155#41—28no—
156#41—28no—
157#41—28no—
158#41—22no—
159#41—22no—
160#41—2142no—
161#41—2142no—
162#41C22⋊D10714yes—
163#41C22×D10714yes—
164C2×Q8—22no—
165C2×Q8—28no—
166C2×Q8—28no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—1071no—
170C2×Q8C22⋊D10712yes—
171C2×Q8C22⋊D10712yes—
172C2×Q8C22×D10712yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4—1071no—
176C2×D4—2141no—
177C2×D4C22⋊D10712yes—
178C22×C4—24no—
179C22×C4—22no—
180C22×C4—2141no—
181C22×C4C22⋊D10712yes—
182C22×C4C22×D10711yescharacteristic
183Q16—2144no—
184SD16—2144no—
185SD16—2144no—
186D8—2144no—
187M16—1074no—
188M16—2142no—
189C2×C8—2142no—
190C4⋊C4—2141no—
191C4⋊C4—2144no—
192C22⋊C4—2141no—
193C22⋊C4—2144no—
194C42—1072no—
195C23—22no—
196C23—2141no—
197C23C2×C22⋊D10711yescharacteristic
198Q8—28no—
199Q8—22no—
200Q8—24no—
201Q8—24no—
202Q8—2142no—
203Q8C2×C22⋊D10714yes—
204D4—28no—
205D4—24no—
206D4—24no—
207D4—22no—
208D4—4281no—
209D4—4281no—
210D4—2142no—
211D4C2×C22⋊D10714yes—
212C2×C4—216no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—24no—
218C2×C4—4282no—
219C2×C4—2142no—
220C2×C4—2141no—
221C2×C4C2×C22⋊D10714yes—
222C2×C4C2×C22⋊D10711yescharacteristic
223C2×C4C2×C22⋊D10712yes—
224C2×C4C2×C22⋊D10712yes—
225C2×C4C23×D10711yescharacteristic
226C8—2144no—
227C22—24no—
228C22—41no—
229C22—44no—
230C22—4282no—
231C22—4281no—
232C22#1681207811yescharacteristic
233C22#1681207811yescharacteristic
234C22C22×C22⋊D10711yescharacteristic
235C4—24no—
236C4—24no—
237C4—4282no—
238C4—4281no—
239C4#1681207812yes—
240C4C22×C22⋊D10712yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—44no—
245C2—8561no—
246C2C2×#1681207811yescharacteristic
subgroupG/H
centerC2C2×#16812078
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×2−1+4×C107C2
socleC2×C107C2×#120
hypercenterC2×2−1+4D107
hypocenterC107#3020
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. #20293425C2semidirect
2. #20293201C2semidirect
3. #20292810C2semidirect
4. #20292800C2semidirect
5. #20292644C2semidirect
6. C2×#16812028C2semidirect
7. C2×2−1+4×C107C2semidirect
10. #16812139C22semidirect
12. #16812134C22semidirect
15. #16812107C22semidirect
16. #16812098C22semidirect
20. #16812036C22semidirect
21. #16812028C22semidirect
124. C107#3020semidirect
125. C2×2−1+4D107semidirect

Maximal subgroups

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

subgroupindex
#202934252
#202932012
#202928102
#202928002
#202926442
C2×#168120282
C2×2−1+4×C1072
#3020107