Finite Group Databasebeta

← #28346146

Subgroups — #28346146

Order 13696.

472 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16812117C211yesmaximal, characteristic, complemented
2#20293205C211yesmaximal, characteristic
3#20293201C211yesmaximal, characteristic, complemented
4#20292882C212yesmaximal, complemented
5#20292870C212yesmaximal, complemented
6#20292865C212yesmaximal, complemented
7#20292853C212yesmaximal, complemented
8#20292669C211yesmaximal, characteristic, complemented
9#20292623C211yesmaximal, characteristic, complemented
10C2×#16812033C211yesmaximal, characteristic, complemented
11C2×#137×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13#16812117—24no—
14C2×#13133801C2212yes—
15C22×C107⋊C8C2211yescharacteristic
16C2×C8×D107C2212yes—
17#16812107—21no—
18#16812098—21no—
19#16812092—22no—
20#16812088—21no—
21#16812085C2212yes—
22#16812085C2212yes—
23#16812085C2212yes—
24#16812083—21no—
25#16812081—22no—
26#16812080C2212yes—
27#16812068C2211yescharacteristic
28C2×C4⋊Dic107C2211yescharacteristic
29#16812043C2212yes—
30#16812041C2212yes—
31#16812033C2211yescharacteristic
32#16812033C2212yes—
33#16812033C2212yes—
34#16812033C2211yescharacteristic
35#16812030C2211yescharacteristic
36#16812030C2211yescharacteristic
37C2×#41×C107C2211yescharacteristic
38C22×Q8×C107C2211yescharacteristic
39#137×C107—24no—
40C2×Q16×C107C2212yes—
41C2×SD16×C107C2212yes—
42C2×M16×C107C2211yescharacteristic
43#13133818—22no—
44#13133818—22no—
45C2×Dic214D411yescharacteristic
46C2×D428D411yescharacteristic
47C2×C22⋊D107—22no—
48C2×C22⋊D107D412yes—
49C22×Dic107—21no—
50C2×C4×D107C2312yes—
51#13133801—24no—
52C2×C107⋊C8—22no—
53C2×C107⋊C8C2311yescharacteristic
54C2×C107⋊C8C2311yescharacteristic
55C8×D107—24no—
56C4⋊Dic107—21no—
57C4⋊Dic107C2312yes—
58C4⋊Dic107C2311yescharacteristic
59C4⋊Dic107C2311yescharacteristic
60Dic107⋊C4—22no—
61Dic107⋊C4—22no—
62Dic107⋊C4—22no—
63C22⋊Dic107—21no—
64C22⋊Dic107—22no—
65(C2×D107)⋊C4—22no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—22no—
68C4×Dic107—21no—
69#41×C107—22no—
70#41×C107—22no—
71C2×Q8×C107—21no—
72C2×Q8×C107—22no—
73C2×Q8×C107—22no—
74C2×Q8×C107C2312yes—
75C2×Q8×C107C2311yescharacteristic
76C2×D4×C107—21no—
77C2×D4×C107C2311yescharacteristic
78C22×C4×C107—21no—
79C22×C4×C107—21no—
80C22×C4×C107C2311yescharacteristic
81Q16×C107—24no—
82SD16×C107—24no—
83M16×C107D414yescomplemented
84C2×C8×C107C2312yes—
85C23×C107—21no—
86C23×C107C2×D411yescharacteristic
87C22×D107C2×D412yes—
88Q8×C107—22no—
89Q8×C107—21no—
90Q8×C107—22no—
91Q8×C107—21no—
92Q8×C107—42no—
93D4×C107—21no—
94D4×C107—21no—
95D4×C107—42no—
96C2×C4×C107—21no—
97C2×C4×C107—22no—
98C2×C4×C107—42no—
99C2×C4×C107—42no—
100C2×C4×C107C2×D411yescharacteristic
101C2×C4×C107C2×D412yes—
102C2×C4×C107#4112yes—
103C2×C4×C107C2411yescharacteristic
104C22⋊D107—24no—
105C22⋊D107—82no—
106C2×Dic107—21no—
107C2×Dic107—21no—
108C2×Dic107—22no—
109C2×Dic107—81no—
110C2×Dic107C2×D412yes—
111D428—22no—
112C4×D107—22no—
113C4×D107—22no—
114Dic214—22no—
115C8×C107C2×D414yes—
116C107⋊C8—21no—
117C107⋊C8—21no—
118C22×C107—21no—
119C22×C107—21no—
120C22×C107—41no—
121C22×C107—41no—
122C22×C107—41no—
123C22×C107#12112yes—
124C22×C107C22×D411yescharacteristic
125C2×D107—22no—
126C2×D107—22no—
127C2×D107—22no—
128C4×C107—41no—
129C4×C107—42no—
130C4×C107#12112yes—
131C4×C107C22×D411yescharacteristic
132C4×C107C2×#4111yescharacteristic
133Dic107—22no—
134Dic107—41no—
135Dic107—42no—
136Dic107—41no—
137C2×C107—22no—
138C2×C107—81no—
139C2×C107C2×#12111yescharacteristic
140C2×C107#57711yescharacteristic
141C2×C107#57811yescharacteristic
142D107—42no—
143#3063—1071nomaximal, Sylow
144C107#306311yescharacteristic, complemented, Sylow
145C2×#137D10711yescharacteristic, complemented
146C2×#131—1071no—
147#537—1071no—
148#536—1071no—
149#470—1072no—
150#468—1072no—
151#467—1072no—
152#465—1072no—
153#428—1071no—
154#416—1071no—
155C2×#103—1071no—
156C2×#41—1071no—
157C2×#41C2×D10711yescharacteristic
158C22×Q8C2×D10711yescharacteristic
159#137—24no—
160C2×Q16C2×D10712yes—
161C2×SD16C2×D10712yes—
162#131—2144no—
163C2×M16—1072no—
164C2×M16C2×D10711yescharacteristic
165C22×C8—1071no—
166C22×C8—1072no—
167C4⋊Q8—2141no—
168#124—2141no—
169#123—2142no—
170C22⋊Q8—1072no—
171C22⋊Q8—1072no—
172C22⋊Q8—1072no—
173C22⋊Q8—2141no—
174#121—1072no—
175#121—2141no—
176#121—2142no—
177#117—1071no—
178C2×C4⋊C4—1071no—
179#107—1072no—
180#106—1072no—
181#103—1072no—
182#103—1072no—
183#103—1071no—
184#103—1071no—
185#102—1071no—
186#102—1071no—
187#41—22no—
188#41—22no—
189#41—2142no—
190#41—2142no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—1071no—
195C2×Q8C22×D10712yes—
196C2×Q8C22×D10711yescharacteristic
197C2×D4—21no—
198C2×D4—1072no—
199C2×D4—1071no—
200C2×D4—2142no—
201C2×D4C22×D10711yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—1072no—
205C22×C4—2141no—
206C22×C4C22×D10711yescharacteristic
207Q16—24no—
208SD16—24no—
209M16—2144no—
210M16C22⋊D10714yescomplemented
211C2×C8—1071no—
212C2×C8—1071no—
213C2×C8—2142no—
214C2×C8—2144no—
215C2×C8C22×D10712yes—
216C4⋊C4—1071no—
217C4⋊C4—1071no—
218C4⋊C4—1072no—
219C4⋊C4—2142no—
220C4⋊C4—2142no—
221C4⋊C4—2142no—
222C4⋊C4—2141no—
223C22⋊C4—2142no—
224C22⋊C4—2142no—
225C22⋊C4—2142no—
226C22⋊C4—2142no—
227C22⋊C4—2141no—
228C42—2141no—
229C23—21no—
230C23—1072no—
231C23D4×D10711yescharacteristic
232Q8—22no—
233Q8—21no—
234Q8—22no—
235Q8—21no—
236Q8—42no—
237Q8—2142no—
238D4—21no—
239D4—21no—
240D4—42no—
241D4—8562no—
242D4—2144no—
243D4—2142no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—8561no—
249C2×C4—1072no—
250C2×C4—2141no—
251C2×C4—2141no—
252C2×C4—2142no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4D4×D10711yescharacteristic
256C2×C4C2×C22⋊D10712yes—
257C2×C4#1313381712yes—
258C2×C4C23×D10711yescharacteristic
259C8—2141no—
260C8—2141no—
261C8C2×C22⋊D10714yes—
262C22—21no—
263C22—21no—
264C22—41no—
265C22—41no—
266C22—41no—
267C22—2142no—
268C22—2142no—
269C22—2142no—
270C22#1681208012yes—
271C22C2×D4×D10711yescharacteristic
272C4—41no—
273C4—42no—
274C4—4282no—
275C4—4281no—
276C4—4281no—
277C4—2142no—
278C4#1681208012yes—
279C4C22×C22⋊D10711yescharacteristic
280C4C2×#1313381711yescharacteristic
281C2—22no—
282C2—81no—
283C2—4282no—
284C2C2×#1681208011yescharacteristic
285C2#2029343611yescharacteristic
286C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#137×C107C2
socleC22×C107C22×D4
prime radical#20293205C2
hypercenterC2×#137D107
hypocenterC107#3063
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#16812117C2semidirect
3. #20293201C2semidirect
4. #20292882C2semidirect
5. #20292870C2semidirect
6. #20292865C2semidirect
7. #20292853C2semidirect
8. #20292669C2semidirect
9. #20292623C2semidirect
10. C2×#16812033C2semidirect
11. C2×#137×C107C2semidirect
83. M16×C107D4semidirect
144. C107#3063semidirect
145. C2×#137D107semidirect
210. M16C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121172
#202932052
#202932012
#202928822
#202928702
#202928652
#202928532
#202926692
#202926232
C2×#168120332
C2×#137×C1072
#3063107