Finite Group Databasebeta

← #28336838

Subgroups — #28336838

Order 13696.

370 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16812137C211yesmaximal, characteristic
2#20293107C211yesmaximal, characteristic, complemented
3C2×#16812037C211yesmaximal, characteristic, complemented
4#20292631C211yesmaximal, characteristic, complemented
5#20292529C211yesmaximal, characteristic, complemented
6#20292395C211yesmaximal, characteristic, complemented
7#339×C107C211yesmaximal, characteristic, complemented
8C2×#13133818C2211yescharacteristic
9C22×D428C2211yescharacteristic
10#16812137—22no—
11#16812137C414yescomplemented
12C2×#13133807—21no—
13C2×D856—21no—
14C2×#13133802C2211yescharacteristic
15C4×Dic214—21no—
16C4×D428—21no—
17(C22⋊D107)⋊C4—22no—
18#16812067—21no—
19C2×C4⋊Dic107C2211yescharacteristic
20C4⋊C4×D107—21no—
21C2×(C2×D107)⋊C4—22no—
22#16812037—22no—
23#16812029—21no—
24#16812029—21no—
25C2×M16×C107C2211yescharacteristic
26#117×C107C2211yescharacteristic
27C2×C42×C107C2211yescharacteristic
28C23×D107—21no—
29#13133818—21no—
30#13133818—21no—
31#13133818C2×C412yes—
32#13133818D412yes—
33C2×Dic214D411yescharacteristic
34C2×D428—21no—
35C2×D428—21no—
36C2×D428—21no—
37C2×D428—21no—
38C2×D428—21no—
39C2×D428C2×C412yescomplemented
40C2×D428D411yescharacteristic
41C2×C22⋊D107—21no—
42C22×Dic107D412yes—
43C2×C4×D107—21no—
44C2×C4×D107—22no—
45#13133807—24no—
46D856—24no—
47#13133802—21no—
48#13133802—21no—
49C2×C107⋊C8—21no—
50C4⋊Dic107—21no—
51C4⋊Dic107—21no—
52Dic107⋊C4—22no—
53(C2×D107)⋊C4—22no—
54(C2×D107)⋊C4—44no—
55C4×Dic107—22no—
56C22×C4×C107—21no—
57C22×C4×C107C2311yescharacteristic
58M16×C107—21no—
59M16×C107C2×C412yescomplemented
60C2×C8×C107—21no—
61C4⋊C4×C107D411yescharacteristic
62C4⋊C4×C107D411yescharacteristic
63C22⋊C4×C107—21no—
64C42×C107—21no—
65C42×C107—21no—
66C42×C107—21no—
67C23×C107C2×D411yescharacteristic
68C22×D107—21no—
69C22×D107—22no—
70C22×D107—42no—
71C22×D107—42no—
72C22×D107—41no—
73C22×D107—41no—
74C2×C4×C107—22no—
75C2×C4×C107—21no—
76C2×C4×C107—21no—
77C2×C4×C107—41no—
78C2×C4×C107—41no—
79C2×C4×C107C22×C411yescharacteristic
80C2×C4×C107C2×D411yescharacteristic
81C2×C4×C107C2×D411yescharacteristic
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107#4111yescharacteristic
84C2×C4×C107#4111yescharacteristic
85C22⋊D107—22no—
86C22⋊D107—22no—
87C2×Dic107—21no—
88C2×Dic107—22no—
89C2×Dic107—42no—
90D428—22no—
91D428—22no—
92D428—21no—
93D428—41no—
94D428—41no—
95D428C22⋊C412yescomplemented
96C4×D107—22no—
97C4×D107—22no—
98C4×D107—82no—
99Dic214—21no—
100Dic214C22⋊C412yescomplemented
101C8×C107—22no—
102C107⋊C8—41no—
103C22×C107—21no—
104C22×C107—21no—
105C22×C107C4×D411yescharacteristic
106C22×C107#12011yescharacteristic
107C22×C107#12111yescharacteristic
108C2×D107—22no—
109C2×D107—41no—
110C2×D107—41no—
111C2×D107—44no—
112C2×D107—42no—
113C2×D107—82no—
114C2×D107—81no—
115C2×D107—81no—
116C4×C107—42no—
117C4×C107—41no—
118C4×C107—41no—
119C4×C107C2×C22⋊C411yescharacteristic
120C4×C107C4×D411yescharacteristic
121C4×C107#12111yescharacteristic
122C4×C107#12311yescharacteristic
123Dic107—22no—
124Dic107—42no—
125C2×C107—21no—
126C2×C107—21no—
127C2×C107#38611yescharacteristic
128C2×C107#45311yescharacteristic
129C2×C107#45411yescharacteristic
130D107—42no—
131D107—82no—
132#1883—1071nomaximal, Sylow
133C107#188311yescharacteristic, complemented, Sylow
134C2×AGL(1,ℤ/8)—1071no—
135#520—1071no—
136C2×#104—1071no—
137#418—1071no—
138#390—1071no—
139#343—1071no—
140#339D10711yescharacteristic, complemented
141C2×#41—1071no—
142C22×D4—1071no—
143AGL(1,ℤ/8)—1074no—
144AGL(1,ℤ/8)—2142no—
145C2×SD16—2141no—
146C2×D8—2141no—
147C2×M16—1071no—
148C2×M16C2×D10711yescharacteristic
149C4×Q8—2141no—
150C4×D4—2141no—
151C4×D4—2142no—
152#117—2141no—
153#117C2×D10711yescharacteristic
154C2×C4⋊C4—1071no—
155C2×C4⋊C4—2141no—
156C2×C22⋊C4—2142no—
157C2×C42C2×D10711yescharacteristic
158#104—2142no—
159#102—2141no—
160#102—2141no—
161C24—2141no—
162#41—1072no—
163#41—1072no—
164#41—2141no—
165#41—2141no—
166C2×Q8—1071no—
167C2×D4—1072no—
168C2×D4—1071no—
169C2×D4—2141no—
170C2×D4—2141no—
171C2×D4—2141no—
172C2×D4—2141no—
173C2×D4—2141no—
174C2×D4—2141no—
175C22×C4—21no—
176C22×C4—1072no—
177C22×C4—2142no—
178C22×C4—2141no—
179C22×C4C22×D10711yescharacteristic
180SD16—2144no—
181D8—2144no—
182M16—21no—
183M16—2141no—
184M16—2141no—
185M16C4×D10712yescomplemented
186C2×C8—21no—
187C2×C8—2141no—
188C4⋊C4—2142no—
189C4⋊C4—2141no—
190C4⋊C4—2141no—
191C4⋊C4D42811yescharacteristic
192C4⋊C4D42811yescharacteristic
193C22⋊C4—21no—
194C22⋊C4—4284no—
195C22⋊C4—2142no—
196C42—21no—
197C42—21no—
198C42—21no—
199C42—2142no—
200C23—4282no—
201C23—4282no—
202C23—4281no—
203C23—4281no—
204C23—2142no—
205C23—2141no—
206C23D4×D10711yescharacteristic
207Q8—1072no—
208Q8—2141no—
209D4—1072no—
210D4—4281no—
211D4—4281no—
212D4—2142no—
213D4—2142no—
214D4—2142no—
215D4—2142no—
216D4—2141no—
217C2×C4—22no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—8562no—
223C2×C4—4282no—
224C2×C4—2142no—
225C2×C4—2141no—
226C2×C4—2142no—
227C2×C4—2142no—
228C2×C4C2×C4×D10711yescharacteristic
229C2×C4D4×D10711yescharacteristic
230C2×C4D4×D10711yescharacteristic
231C2×C4C2×D42811yescharacteristic
232C2×C4#1313381811yescharacteristic
233C2×C4#1313381911yescharacteristic
234C8—22no—
235C8—4281no—
236C22—21no—
237C22—21no—
238C22—8562no—
239C22—8561no—
240C22—8561no—
241C22—4284no—
242C22—4282no—
243C22—4281no—
244C22—4281no—
245C22—2142no—
246C22#1681207111yescharacteristic
247C22#1681207911yescharacteristic
248C22#1681208211yescharacteristic
249C4—42no—
250C4—41no—
251C4—41no—
252C4—4282no—
253C4—2142no—
254C4C22⋊C4×D10711yescharacteristic
255C4C4×D42811yescharacteristic
256C4#1681208411yescharacteristic
257C4#1681209411yescharacteristic
258C2—21no—
259C2—21no—
260C2—8562no—
261C2—4282no—
262C2#2029250511yescharacteristic
263C2#2029279611yescharacteristic
264C2#2029280111yescharacteristic
subgroupG/H
centerC22#16812079
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#339×C107C2
socleC22×C107#120
prime radicalC2×#16812137C2
cosocleC22×C4×C107C23
hypercenter#339D107
hypocenterC107#1883
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. #20293107C2semidirect
3. C2×#16812037C2semidirect
4. #20292631C2semidirect
5. #20292529C2semidirect
6. #20292395C2semidirect
7. #339×C107C2semidirect
11. #16812137C4semidirect
39. C2×D428C2×C4semidirect
59. M16×C107C2×C4semidirect
95. D428C22⋊C4semidirect
100. Dic214C22⋊C4semidirect
133. C107#1883semidirect
140. #339D107semidirect
185. M16C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
#202931072
C2×#168120372
#202926312
#202925292
#202923952
#339×C1072
#1883107