Finite Group Databasebeta

← #28345824

Subgroups — #28345824

Order 13696.

698 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 145 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×#16812157C211yesmaximal, characteristic, complemented
2#20293425C212yesmaximal, complemented
3#20292809C214yesmaximal, complemented
4#20292799C214yesmaximal, complemented
5#20292645C212yesmaximal, complemented
6C2×#16812028C211yesmaximal, characteristic, complemented
7#536×C107C211yesmaximal, characteristic, complemented
8#16812157—24no—
9#16812157—22no—
10#16812157C2214yes—
11C2×#13133819—22no—
12C2×#13133818—22no—
13C2×#13133818C2212yes—
14C2×Q8×D107—22no—
15#16812140C2214yescomplemented
16#16812139—22no—
17#16812135C2214yescomplemented
18#16812134—22no—
19#16812129—24no—
20C2×#13133802C2212yes—
21#16812037C2218yescomplemented
22#16812028C2214yescomplemented
23C2×#41×C107C2211yescharacteristic
24C22×Q8×C107C2211yescharacteristic
25C4⋊Q8×C107C2212yescomplemented
26#124×C107C2212yescomplemented
27#123×C107—22no—
28C22⋊Q8×C107—22no—
29#117×C107C2211yescharacteristic
30#13133819—28no—
31#13133819—28no—
32#13133818—28no—
33#13133818—28no—
34#13133818—22no—
35#13133818—22no—
36#13133818D414yes—
37#13133818C2314yes—
38C2×Dic214—22no—
39C2×Dic214D412yes—
40Q8×D107—28no—
41Q8×D107—28no—
42C2×D428—22no—
43C2×D428D412yes—
44C2×C22⋊D107—22no—
45C2×C4×D107—24no—
46C2×C4×D107—22no—
47#13133808—24no—
48#13133806—24no—
49#13133805—24no—
50#13133803—24no—
51#13133802—22no—
52#13133802C2314yes—
53C2×C107⋊C8—22no—
54#41×C107—22no—
55#41×C107—22no—
56C2×Q8×C107—22no—
57C2×Q8×C107—22no—
58C2×Q8×C107D412yes—
59C2×Q8×C107C2311yescharacteristic
60C2×Q8×C107C2312yes—
61C2×D4×C107—21no—
62C2×D4×C107C2311yescharacteristic
63C22×C4×C107—21no—
64C22×C4×C107D412yes—
65C22×C4×C107C2311yescharacteristic
66C4⋊C4×C107—21no—
67C4⋊C4×C107—24no—
68C22⋊C4×C107—21no—
69C22⋊C4×C107—24no—
70C42×C107C2312yes—
71C23×C107—21no—
72C23×C107C2×D411yescharacteristic
73C22×D107—22no—
74Q8×C107—24no—
75Q8×C107—24no—
76Q8×C107—22no—
77D4×C107—22no—
78D4×C107—41no—
79D4×C107—41no—
80C2×C4×C107—22no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—21no—
84C2×C4×C107—42no—
85C2×C4×C107C2×D412yes—
86C2×C4×C107C2×D412yes—
87C2×C4×C107C2×D414yes—
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2411yescharacteristic
90C22⋊D107—24no—
91C22⋊D107—24no—
92C2×Dic107—22no—
93D428—28no—
94D428—22no—
95D428C2×D414yes—
96C4×D107—216no—
97C4×D107—24no—
98C4×D107—24no—
99Dic214—28no—
100Dic214—22no—
101Dic214C2×D414yes—
102C107⋊C8—24no—
103C22×C107—41no—
104C22×C107—42no—
105C22×C107—41no—
106C22×C107#12011yescharacteristic
107C22×C107#12011yescharacteristic
108C22×C107C22×D411yescharacteristic
109C2×D107—24no—
110C2×D107—44no—
111C4×C107—24no—
112C4×C107—42no—
113C4×C107—41no—
114C4×C107#12012yes—
115C4×C107C22×D412yes—
116Dic107—24no—
117C2×C107—21no—
118C2×C107—21no—
119C2×C107—21no—
120C2×C107—81no—
121C2×C107C2×#12011yescharacteristic
122D107—44no—
123#3020—1071nomaximal, Sylow
124C107#302011yescharacteristic, complemented, Sylow
125C2×2−1+4—1071no—
126#575—1072no—
127#536D10711yescharacteristic, complemented
128#456—1074no—
129#454—1074no—
130#421—1072no—
131C2×#101—1071no—
1322−1+4—1074no—
1332−1+4—2144no—
1342−1+4—2142no—
135C2×#41—1072no—
136C2×#41—2142no—
137C2×#41—2142no—
138C2×#41C2×D10711yescharacteristic
139C22×Q8—2142no—
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⋊Q8C2×D10712yescomplemented
148#124C2×D10712yescomplemented
149#123—22no—
150C22⋊Q8—22no—
151#117C2×D10711yescharacteristic
152#104—1078no—
153#101—1074no—
154#41—22no—
155#41—22no—
156#41—1074no—
157#41—1074no—
158#41—2148no—
159#41—2148no—
160#41—2148no—
161#41—2142no—
162#41—2142no—
163#41—2148no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—1072no—
167C2×Q8—2142no—
168C2×Q8—2148no—
169C2×Q8—2148no—
170C2×Q8C22⋊D10712yes—
171C2×Q8C22×D10711yescharacteristic
172C2×Q8C22×D10712yes—
173C2×D4—21no—
174C2×D4—1072no—
175C2×D4—2142no—
176C2×D4—2142no—
177C2×D4C22×D10711yescharacteristic
178C22×C4—21no—
179C22×C4—2144no—
180C22×C4—2142no—
181C22×C4C22⋊D10712yes—
182C22×C4C22×D10711yescharacteristic
183Q16—2144no—
184SD16—2144no—
185SD16—2144no—
186D8—2144no—
187M16—1074no—
188M16—2142no—
189C2×C8—2142no—
190C4⋊C4—21no—
191C4⋊C4—24no—
192C22⋊C4—21no—
193C22⋊C4—24no—
194C42C22×D10712yes—
195C23—21no—
196C23—2142no—
197C23C2×C22⋊D10711yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—22no—
201Q8—1074no—
202Q8—2148no—
203Q8—2142no—
204D4—22no—
205D4—41no—
206D4—41no—
207D4—1074no—
208D4—2148no—
209D4—2144no—
210D4—2144no—
211D4—2142no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—21416no—
218C2×C4—2142no—
219C2×C4—2144no—
220C2×C4—2144no—
221C2×C4D4×D10712yes—
222C2×C4D4×D10712yes—
223C2×C4C2×C22⋊D10714yes—
224C2×C4C2×C22⋊D10711yescharacteristic
225C2×C4C23×D10711yescharacteristic
226C8—2144no—
227C22—41no—
228C22—42no—
229C22—41no—
230C22—4284no—
231C22—2144no—
232C22#1681207711yescharacteristic
233C22#1681207711yescharacteristic
234C22C22×C22⋊D10711yescharacteristic
235C4—24no—
236C4—42no—
237C4—41no—
238C4—2144no—
239C4#1681207712yes—
240C4C2×D4×D10712yes—
241C2—21no—
242C2—21no—
243C2—21no—
244C2—81no—
245C2—4284no—
246C2C2×#1681207711yescharacteristic
subgroupG/H
centerC2C2×#16812077
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#536×C107C2
socleC2×C107C2×#120
hypercenter#536D107
hypocenterC107#3020
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#16812157C2semidirect
2. #20293425C2semidirect
3. #20292809C2semidirect
4. #20292799C2semidirect
5. #20292645C2semidirect
6. C2×#16812028C2semidirect
7. #536×C107C2semidirect
15. #16812140C22semidirect
17. #16812135C22semidirect
21. #16812037C22semidirect
22. #16812028C22semidirect
25. C4⋊Q8×C107C22semidirect
26. #124×C107C22semidirect
124. C107#3020semidirect
127. #536D107semidirect
147. C4⋊Q8C2×D107semidirect
148. #124C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121572
#202934252
#202928092
#202927992
#202926452
C2×#168120282
#536×C1072
#3020107