Finite Group Databasebeta

← #28345759

Subgroups — #28345759

Order 13696.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16812156C211yesmaximal, characteristic
2C2×#16812139C211yesmaximal, characteristic, complemented
3C22×Dic428C211yesmaximal, characteristic, complemented
4#20293203C211yesmaximal, characteristic, complemented
5#20292790C214yesmaximal, complemented
6#20292784C214yesmaximal, complemented
7#20292635C211yesmaximal, characteristic, complemented
8#20292593C211yesmaximal, characteristic, complemented
9#416×C107C211yesmaximal, characteristic, complemented
10#16812156—24no—
11#16812156—24no—
12C2×#13133818—21no—
13C2×#13133818C2211yescharacteristic
14C2×#13133817—22no—
15C2×#13133817C2212yes—
16C22×Dic214—21no—
17C22×Dic214C2211yescharacteristic
18C22×Dic214C2211yescharacteristic
19C2×Q8×D107C2212yes—
20#16812139—24no—
21C2×Dic428—24no—
22C2×Dic428C2214yes—
23C2×#13133808C2212yes—
24C2×#13133806C2212yes—
25C2×#13133802C2211yescharacteristic
26#16812109—21no—
27#16812097—21no—
28#16812093—22no—
29#16812089C2214yes—
30#16812032C2214yes—
31(C2×D107)⋊C8C2214yes—
32C2×#41×C107C2211yescharacteristic
33C22×C8×C107C2211yescharacteristic
34#117×C107C2211yescharacteristic
35#103×C107C2212yes—
36#102×C107C2212yes—
37#13133818—28no—
38#13133818—22no—
39#13133818—22no—
40#13133817—216no—
41#13133817—24no—
42#13133817—24no—
43C2×Dic214—28no—
44C2×Dic214—21no—
45C2×Dic214—22no—
46C2×Dic214—22no—
47C2×Dic214—22no—
48C2×Dic214—22no—
49C2×Dic214—21no—
50C2×Dic214D411yescharacteristic
51C2×Dic214D414yes—
52C2×Dic214C2312yes—
53C2×Dic214C2312yes—
54Q8×D107—24no—
55Q8×D107—24no—
56C2×D428D411yescharacteristic
57C2×C22⋊D107—22no—
58C2×C22⋊D107D412yes—
59C22×Dic107—21no—
60C22×Dic107—22no—
61C22×Dic107—21no—
62C2×C4×D107—22no—
63C2×C4×D107C2312yes—
64Dic428—28no—
65#13133808—24no—
66#13133806—24no—
67#13133802—22no—
68C2×C107⋊C8C2312yes—
69C4⋊Dic107—24no—
70(C2×D107)⋊C4—24no—
71#41×C107—22no—
72#41×C107D414yes—
73C2×Q8×C107C2311yescharacteristic
74C2×D4×C107—21no—
75C2×D4×C107C2311yescharacteristic
76C22×C4×C107—21no—
77C22×C4×C107C2311yescharacteristic
78C2×C8×C107—22no—
79C2×C8×C107C2312yes—
80C4⋊C4×C107C2312yes—
81C22⋊C4×C107—21no—
82C42×C107—21no—
83C23×C107—21no—
84C23×C107C2×D411yescharacteristic
85C22×D107C2×D412yes—
86Q8×C107—21no—
87Q8×C107C2×D412yes—
88D4×C107—24no—
89D4×C107—21no—
90D4×C107C2×D412yes—
91C2×C4×C107—21no—
92C2×C4×C107—24no—
93C2×C4×C107—22no—
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107C2×D412yes—
96C2×C4×C107C2×D412yes—
97C2×C4×C107C2411yescharacteristic
98C22⋊D107—28no—
99C22⋊D107—24no—
100C2×Dic107—28no—
101C2×Dic107—24no—
102C2×Dic107—22no—
103C2×Dic107—22no—
104C2×Dic107—42no—
105C2×Dic107C2×D412yes—
106D428—22no—
107C4×D107—28no—
108C4×D107—22no—
109C4×D107—22no—
110Dic214—28no—
111Dic214—22no—
112Dic214—24no—
113Dic214—22no—
114Dic214—22no—
115Dic214—22no—
116Dic214—42no—
117Dic214C2×D414yes—
118C8×C107—22no—
119C107⋊C8—22no—
120C22×C107—22no—
121C22×C107—22no—
122C22×C107—42no—
123C22×C107#12012yes—
124C22×C107C22×D411yescharacteristic
125C2×D107—22no—
126C2×D107—24no—
127C4×C107—22no—
128C4×C107—42no—
129C4×C107#12012yes—
130C4×C107C22×D411yescharacteristic
131C4×C107C22×D411yescharacteristic
132Dic107—22no—
133Dic107—24no—
134Dic107—42no—
135C2×C107—22no—
136C2×C107—42no—
137C2×C107C2×#12011yescharacteristic
138C2×C107#57812yes—
139D107—42no—
140#3011—1071nomaximal, Sylow
141C107#301111yescharacteristic, complemented, Sylow
142C2×2−1+4—1071no—
143C2×#137—1071no—
144C22×Q16—1071no—
145#536—1071no—
146#452—1074no—
147#451—1074no—
148#419—1071no—
149#416D10711yescharacteristic, complemented
150#408—1071no—
1512−1+4—2144no—
1522−1+4—2144no—
153C2×#41—1071no—
154C2×#41—1072no—
155C2×#41—2141no—
156C2×#41—2142no—
157C2×#41C2×D10711yescharacteristic
158C22×Q8—1072no—
159C22×Q8—1071no—
160C22×Q8—1071no—
161C22×Q8—2141no—
162#137—2144no—
163C2×Q16—1072no—
164C2×Q16—1074no—
165C2×Q16—2144no—
166C2×SD16—1072no—
167C2×M16—1071no—
168C22×C8C2×D10711yescharacteristic
169C4⋊Q8—2141no—
170#124—2141no—
171#123—2142no—
172C22⋊Q8—1074no—
173#117C2×D10711yescharacteristic
174#103—1074no—
175#103C2×D10712yes—
176#102C2×D10712yes—
177C22⋊C8—1074no—
178#41—22no—
179#41—2148no—
180#41—2144no—
181#41—21416no—
182#41—2144no—
183#41—2142no—
184#41—2142no—
185#41D42814yes—
186C2×Q8—1071no—
187C2×Q8—1074no—
188C2×Q8—1072no—
189C2×Q8—1072no—
190C2×Q8—2142no—
191C2×Q8—2148no—
192C2×Q8—2142no—
193C2×Q8—2142no—
194C2×Q8—2144no—
195C2×Q8—2144no—
196C2×Q8—2141no—
197C2×Q8—2141no—
198C2×Q8—2142no—
199C2×Q8C22×D10711yescharacteristic
200C2×D4—21no—
201C2×D4—1072no—
202C2×D4—1071no—
203C2×D4—2142no—
204C2×D4C22×D10711yescharacteristic
205C22×C4—21no—
206C22×C4—1072no—
207C22×C4—2142no—
208C22×C4—2142no—
209C22×C4—2141no—
210C22×C4—2141no—
211C22×C4C22×D10711yescharacteristic
212Q16—2148no—
213Q16—2144no—
214SD16—2144no—
215M16—2142no—
216C2×C8—22no—
217C2×C8—1072no—
218C2×C8C22×D10712yes—
219C4⋊C4—2144no—
220C4⋊C4C22×D10712yes—
221C22⋊C4—21no—
222C22⋊C4—2144no—
223C42—21no—
224C23—21no—
225C23—1072no—
226C23D4×D10711yescharacteristic
227Q8—21no—
228Q8—1074no—
229Q8—4282no—
230Q8—2148no—
231Q8—2142no—
232Q8—2142no—
233Q8—2142no—
234Q8—2144no—
235Q8—2142no—
236Q8C2×D42812yes—
237D4—24no—
238D4—21no—
239D4—2148no—
240D4—2144no—
241D4—2142no—
242D4C2×D42812yes—
243C2×C4—21no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—1072no—
247C2×C4—4282no—
248C2×C4—2148no—
249C2×C4—2142no—
250C2×C4—2148no—
251C2×C4—2144no—
252C2×C4—2142no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4D4×D10711yescharacteristic
256C2×C4D4×D10712yes—
257C2×C4C2×D42812yes—
258C2×C4C23×D10711yescharacteristic
259C8—22no—
260C8—2142no—
261C22—22no—
262C22—22no—
263C22—42no—
264C22—2144no—
265C22—2142no—
266C22#1681207912yes—
267C22C2×D4×D10711yescharacteristic
268C4—22no—
269C4—42no—
270C4—4282no—
271C4—2142no—
272C4—2144no—
273C4#1681207912yes—
274C4C2×D4×D10711yescharacteristic
275C4C22×D42811yescharacteristic
276C2—22no—
277C2—42no—
278C2—4282no—
279C2C2×#1681207911yescharacteristic
280C2#2029344112yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#416×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812156C2
hypercenter#416D107
hypocenterC107#3011
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C2×#16812139C2semidirect
3. C22×Dic428C2semidirect
4. #20293203C2semidirect
5. #20292790C2semidirect
6. #20292784C2semidirect
7. #20292635C2semidirect
8. #20292593C2semidirect
9. #416×C107C2semidirect
141. C107#3011semidirect
149. #416D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121562
C2×#168121392
C22×Dic4282
#202932032
#202927902
#202927842
#202926352
#202925932
#416×C1072
#3011107