Finite Group Databasebeta

← #28336817

Subgroups — #28336817

Order 13696.

374 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 101 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×#13133807C211yesmaximal, characteristic
2C2×#16812074C211yesmaximal, characteristic, complemented
3C2×#16812032C211yesmaximal, characteristic, complemented
4C2×#16812029C211yesmaximal, characteristic
5#20292528C211yesmaximal, characteristic
6#20292381C211yesmaximal, characteristic, complemented
7#340×C107C211yesmaximal, characteristic, complemented
8C22×Dic214C2211yescharacteristic
9C22×D428C2211yescharacteristic
10C2×#13133807—22no—
11C2×#13133807—22no—
12C2×#13133807C414yescomplemented
13C22×C107⋊C8C2211yescharacteristic
14#16812074—22no—
15#16812074—22no—
16C2×Dic107⋊C4—21no—
17C2×(C2×D107)⋊C4—21no—
18C2×(C2×D107)⋊C4—21no—
19C2×C4×Dic107—21no—
20C2×C4×Dic107C2211yescharacteristic
21#16812032—22no—
22#16812029—22no—
23C22×C8×C107C2211yescharacteristic
24C2×C4⋊C4×C107C2211yescharacteristic
25C2×C4⋊C4×C107C2211yescharacteristic
26C23×D107—21no—
27C2×Dic214—21no—
28C2×Dic214—21no—
29C2×Dic214—21no—
30C2×Dic214C2×C412yes—
31C2×Dic214D412yes—
32C2×Dic214D412yes—
33C2×D428—21no—
34C2×D428—21no—
35C2×D428—21no—
36C2×D428—21no—
37C2×D428—21no—
38C2×D428C2×C412yes—
39C22×Dic107—21no—
40C22×Dic107D411yescharacteristic
41C22×Dic107D411yescharacteristic
42#13133807—28no—
43C2×C107⋊C8—21no—
44C2×C107⋊C8—21no—
45C2×C107⋊C8—21no—
46Dic107⋊C4—24no—
47(C2×D107)⋊C4—42no—
48(C2×D107)⋊C4—42no—
49C4×Dic107—24no—
50C4×Dic107—21no—
51C4×Dic107—21no—
52C22×C4×C107—21no—
53C22×C4×C107—21no—
54C22×C4×C107C2311yescharacteristic
55C2×C8×C107—21no—
56C2×C8×C107—21no—
57C2×C8×C107C2×C412yes—
58C4⋊C4×C107—21no—
59C4⋊C4×C107—21no—
60C4⋊C4×C107—21no—
61C4⋊C4×C107D412yes—
62C23×C107C2×D411yescharacteristic
63C22×D107—22no—
64C22×D107—41no—
65C22×D107—41no—
66C22×D107—41no—
67C22×D107—41no—
68C22×D107—41no—
69C22×D107—41no—
70C2×C4×C107—22no—
71C2×C4×C107—41no—
72C2×C4×C107—41no—
73C2×C4×C107—41no—
74C2×C4×C107—41no—
75C2×C4×C107—41no—
76C2×C4×C107C22×C411yescharacteristic
77C2×C4×C107C2×D411yescharacteristic
78C2×C4×C107C2×D411yescharacteristic
79C2×C4×C107C2×D411yescharacteristic
80C2×C4×C107#4111yescharacteristic
81C2×C4×C107#4111yescharacteristic
82C2×Dic107—22no—
83C2×Dic107—21no—
84C2×Dic107—22no—
85C2×Dic107—22no—
86C2×Dic107—21no—
87C2×Dic107—21no—
88C2×Dic107—21no—
89C2×Dic107—21no—
90C2×Dic107SD1612yescomplemented
91D428—22no—
92D428—22no—
93D428—41no—
94D428—41no—
95Dic214—22no—
96Dic214—22no—
97Dic214—22no—
98Dic214C22⋊C414yescomplemented
99C8×C107—22no—
100C107⋊C8—41no—
101C22×C107C4×D411yescharacteristic
102C22×C107#12011yescharacteristic
103C22×C107#12111yescharacteristic
104C22×C107C2×SD1611yescharacteristic
105C22×C107#13511yescharacteristic
106C22×C107AGL(1,ℤ/8)11yescharacteristic
107C22×C107#13711yescharacteristic
108C2×D107—42no—
109C2×D107—42no—
110C2×D107—42no—
111C2×D107—81no—
112C2×D107—81no—
113C2×D107—81no—
114C2×D107—81no—
115C4×C107—42no—
116C4×C107—81no—
117C4×C107C2×C22⋊C411yescharacteristic
118C4×C107C4×D411yescharacteristic
119C4×C107#12111yescharacteristic
120C4×C107#12311yescharacteristic
121Dic107—24no—
122Dic107—22no—
123Dic107—41no—
124C2×C107#38611yescharacteristic
125C2×C107C4×SD1611yescharacteristic
126C2×C107#44011yescharacteristic
127C2×C107#44811yescharacteristic
128C2×C107#44911yescharacteristic
129C2×C107#46011yescharacteristic
130C2×C107#46411yescharacteristic
131D107—82no—
132#1880—1071nomaximal, Sylow
133C107#188011yescharacteristic, complemented, Sylow
134C22×SD16—1071no—
135C2×C4×Q8—1071no—
136C2×#103—1071no—
137C2×#102—1071no—
138#390—1071no—
139#340D10711yescharacteristic, complemented
140#336—1071no—
141C22×Q8—1071no—
142C22×D4—1071no—
143C2×SD16—1074no—
144C2×SD16—2142no—
145C2×SD16—2142no—
146C22×C8—1071no—
147C22×C8C2×D10711yescharacteristic
148C4×Q8—2142no—
149C4×Q8—2142no—
150C2×C4⋊C4—2141no—
151C2×C4⋊C4C2×D10711yescharacteristic
152C2×C4⋊C4C2×D10711yescharacteristic
153C2×C22⋊C4—2141no—
154C2×C22⋊C4—2141no—
155C2×C42—1071no—
156C2×C42—2141no—
157#103—2142no—
158#102—2142no—
159C24—2141no—
160C2×Q8—1072no—
161C2×Q8—1072no—
162C2×Q8—1072no—
163C2×Q8—2141no—
164C2×Q8—2141no—
165C2×Q8—2141no—
166C2×D4—1072no—
167C2×D4—2141no—
168C2×D4—2141no—
169C2×D4—2141no—
170C2×D4—2141no—
171C2×D4—2141no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—1071no—
175C22×C4—1071no—
176C22×C4—2141no—
177C22×C4C22×D10711yescharacteristic
178SD16—2148no—
179C2×C8—21no—
180C2×C8—21no—
181C2×C8—2141no—
182C2×C8—2141no—
183C2×C8—2141no—
184C2×C8C4×D10712yes—
185C4⋊C4—21no—
186C4⋊C4—21no—
187C4⋊C4—21no—
188C4⋊C4—2144no—
189C4⋊C4D42812yes—
190C22⋊C4—4282no—
191C22⋊C4—4282no—
192C42—2144no—
193C42—2141no—
194C42—2141no—
195C23—4281no—
196C23—4281no—
197C23—4281no—
198C23—4281no—
199C23—4281no—
200C23—4281no—
201C23—2142no—
202C23D4×D10711yescharacteristic
203Q8—1074no—
204Q8—2142no—
205Q8—2142no—
206Q8—2142no—
207D4—4281no—
208D4—4281no—
209D4—2142no—
210D4—2142no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—1072no—
218C2×C4—2141no—
219C2×C4—2141no—
220C2×C4—2141no—
221C2×C4—2142no—
222C2×C4—2141no—
223C2×C4—2142no—
224C2×C4—2142no—
225C2×C4—2141no—
226C2×C4C2×C4×D10711yescharacteristic
227C2×C4D4×D10711yescharacteristic
228C2×C4D4×D10711yescharacteristic
229C2×C4C2×D42811yescharacteristic
230C2×C4#1313381811yescharacteristic
231C2×C4#1313381911yescharacteristic
232C8—22no—
233C8—4281no—
234C22—8561no—
235C22—8561no—
236C22—8561no—
237C22—8561no—
238C22—4282no—
239C22—4282no—
240C22—4282no—
241C22#1681207111yescharacteristic
242C22#1681207911yescharacteristic
243C22#1681208211yescharacteristic
244C22SD16×D10711yescharacteristic
245C22#1681213011yescharacteristic
246C22#1681213311yescharacteristic
247C22#1681214111yescharacteristic
248C4—42no—
249C4—81no—
250C4—4281no—
251C4—2144no—
252C4—2142no—
253C4C22⋊C4×D10711yescharacteristic
254C4C4×D42811yescharacteristic
255C4#1681208411yescharacteristic
256C4#1681209411yescharacteristic
257C2—8562no—
258C2#2029250511yescharacteristic
259C2#2029271411yescharacteristic
260C2#2029272611yescharacteristic
261C2#2029276711yescharacteristic
262C2#2029277311yescharacteristic
263C2#2029282611yescharacteristic
264C2#2029285211yescharacteristic
subgroupG/H
centerC23D4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
Fitting#340×C107C2
socleC23×C107C2×D4
prime radicalC22×D428C22
cosocleC22×C4×C107C23
hypercenter#340D107
hypocenterC107#1880
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#16812074C2semidirect
3. C2×#16812032C2semidirect
6. #20292381C2semidirect
7. #340×C107C2semidirect
12. C2×#13133807C4semidirect
90. C2×Dic107SD16semidirect
98. Dic214C22⋊C4semidirect
133. C107#1880semidirect
139. #340D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338072
C2×#168120742
C2×#168120322
C2×#168120292
#202925282
#202923812
#340×C1072
#1880107