Finite Group Databasebeta

← #28346150

Subgroups — #28346150

Order 13696.

472 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812140C211yesmaximal, characteristic, complemented
2C2×#16812115C211yesmaximal, characteristic, complemented
3#20293208C211yesmaximal, characteristic
4#20293201C211yesmaximal, characteristic, complemented
5#20292884C212yesmaximal, complemented
6#20292876C212yesmaximal, complemented
7#20292867C212yesmaximal, complemented
8#20292859C212yesmaximal, complemented
9#20292667C211yesmaximal, characteristic, complemented
10#20292624C211yesmaximal, characteristic, complemented
11C2×#103×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C2×#13133818C2211yescharacteristic
14#16812140—24no—
15C2×#13133808C2212yes—
16C2×#13133805C2212yes—
17#16812115—24no—
18C2×#13133802C2211yescharacteristic
19C2×#13133801C2212yes—
20C2×C8×D107C2212yes—
21#16812107—21no—
22#16812098—21no—
23#16812092—22no—
24#16812089—21no—
25#16812089C2212yes—
26#16812085C2212yes—
27#16812085C2212yes—
28#16812084—21no—
29#16812084C2212yes—
30#16812083—22no—
31#16812068C2211yescharacteristic
32#16812042C2212yes—
33#16812040C2212yes—
34#16812034C2211yescharacteristic
35#16812034C2211yescharacteristic
36#16812031C2211yescharacteristic
37#16812031C2211yescharacteristic
38C22×Q8×C107C2211yescharacteristic
39C22×C8×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2211yescharacteristic
41#103×C107C2212yes—
42#103×C107C2212yes—
43#13133818—22no—
44#13133818—22no—
45#13133818—22no—
46#13133818—22no—
47C2×Dic214D411yescharacteristic
48C2×Dic214C2311yescharacteristic
49C2×D428—22no—
50C2×D428D411yescharacteristic
51C2×D428C2311yescharacteristic
52C2×C22⋊D107—21no—
53C2×C22⋊D107D412yes—
54C2×C4×D107—21no—
55C2×C4×D107C2312yes—
56#13133808—24no—
57#13133805—24no—
58#13133802D414yescomplemented
59#13133801—24no—
60C2×C107⋊C8C2312yes—
61C8×D107—24no—
62C4⋊Dic107—22no—
63C4⋊Dic107C2311yescharacteristic
64C4⋊Dic107C2311yescharacteristic
65Dic107⋊C4—22no—
66Dic107⋊C4—22no—
67C22⋊Dic107—21no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4—22no—
71(C2×D107)⋊C4—22no—
72C4×Dic107—21no—
73C2×Q8×C107—21no—
74C2×Q8×C107—22no—
75C2×Q8×C107—22no—
76C2×Q8×C107C2312yes—
77C22×C4×C107—21no—
78C22×C4×C107—21no—
79C22×C4×C107C2311yescharacteristic
80C2×C8×C107—22no—
81C2×C8×C107C2311yescharacteristic
82C2×C8×C107C2311yescharacteristic
83C4⋊C4×C107—21no—
84C4⋊C4×C107C2312yes—
85C23×C107C2×D411yescharacteristic
86C22×D107—21no—
87C22×D107C2×D412yes—
88Q8×C107—22no—
89Q8×C107—22no—
90Q8×C107—42no—
91C2×C4×C107—22no—
92C2×C4×C107—22no—
93C2×C4×C107—42no—
94C2×C4×C107—81no—
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107C2×D412yes—
97C2×C4×C107#4112yes—
98C2×C4×C107C2411yescharacteristic
99C22⋊D107—24no—
100C22⋊D107—42no—
101C2×Dic107—21no—
102C2×Dic107—21no—
103C2×Dic107—21no—
104C2×Dic107C2×D412yes—
105D428—22no—
106D428—21no—
107D428—21no—
108D428—82no—
109C4×D107—22no—
110C4×D107—22no—
111C4×D107—42no—
112Dic214—22no—
113Dic214—21no—
114Dic214—21no—
115C8×C107—21no—
116C8×C107—21no—
117C107⋊C8C2×D414yes—
118C22×C107—21no—
119C22×C107—21no—
120C22×C107#12112yes—
121C22×C107C22×D411yescharacteristic
122C2×D107—22no—
123C2×D107—22no—
124C2×D107—22no—
125C2×D107—41no—
126C2×D107—41no—
127C2×D107—41no—
128C4×C107—42no—
129C4×C107—42no—
130C4×C107#12112yes—
131C4×C107C22×D411yescharacteristic
132C4×C107C2×#4111yescharacteristic
133Dic107—22no—
134Dic107—41no—
135Dic107—41no—
136Dic107—41no—
137C2×C107—22no—
138C2×C107C2×#12111yescharacteristic
139C2×C107#57711yescharacteristic
140C2×C107#57811yescharacteristic
141D107—42no—
142D107—81no—
143#3063—1071nomaximal, Sylow
144C107#306311yescharacteristic, complemented, Sylow
145C2×#137—1071no—
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×#103D10711yescharacteristic, complemented
156C2×#41—1071no—
157C2×#41—1071no—
158C22×Q8C2×D10711yescharacteristic
159#137—2144no—
160C2×Q16—1072no—
161C2×SD16—1072no—
162#131—2144no—
163C2×M16—1071no—
164C2×M16—1072no—
165C22×C8—1072no—
166C22×C8C2×D10711yescharacteristic
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⋊C4C2×D10711yescharacteristic
179#107—1072no—
180#106—1072no—
181#103—1071no—
182#103—1071no—
183#103C2×D10712yes—
184#103C2×D10712yes—
185#102—1071no—
186#102—1071no—
187#41—2142no—
188#41—2142no—
189#41—2142no—
190#41—2142no—
191C2×Q8—21no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—1071no—
195C2×Q8—1071no—
196C2×Q8C22×D10712yes—
197C2×D4—1072no—
198C2×D4—1071no—
199C2×D4—1071no—
200C2×D4—2142no—
201C2×D4—2141no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—1072no—
205C22×C4—2141no—
206C22×C4C22×D10711yescharacteristic
207Q16—2144no—
208SD16—2144no—
209M16—1074no—
210M16—2144no—
211C2×C8—22no—
212C2×C8—1072no—
213C2×C8—2144no—
214C2×C8C22×D10711yescharacteristic
215C2×C8C22×D10711yescharacteristic
216C4⋊C4—21no—
217C4⋊C4—1071no—
218C4⋊C4—1071no—
219C4⋊C4—2142no—
220C4⋊C4—2142no—
221C4⋊C4—2142no—
222C4⋊C4C22×D10712yes—
223C22⋊C4—2142no—
224C22⋊C4—2142no—
225C22⋊C4—2142no—
226C22⋊C4—2142no—
227C22⋊C4—2141no—
228C42—2141no—
229C23—1072no—
230C23—2141no—
231C23D4×D10711yescharacteristic
232Q8—22no—
233Q8—22no—
234Q8—42no—
235Q8—2141no—
236Q8—2141no—
237Q8—2142no—
238D4—8562no—
239D4—4282no—
240D4—2144no—
241D4—2142no—
242D4—2141no—
243D4—2141no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—81no—
248C2×C4—1072no—
249C2×C4—4282no—
250C2×C4—2141no—
251C2×C4—2141no—
252C2×C4—2141no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4D4×D10711yescharacteristic
256C2×C4D4×D10712yes—
257C2×C4#1313381812yes—
258C2×C4C23×D10711yescharacteristic
259C8—21no—
260C8—21no—
261C8—1074no—
262C22—21no—
263C22—21no—
264C22—4281no—
265C22—4281no—
266C22—4281no—
267C22—2142no—
268C22—2142no—
269C22—2142no—
270C22#1681208212yes—
271C22C2×D4×D10711yescharacteristic
272C4—42no—
273C4—42no—
274C4—4281no—
275C4—4281no—
276C4—4281no—
277C4—2142no—
278C4#1681208212yes—
279C4C2×D4×D10711yescharacteristic
280C4C2×#1313381811yescharacteristic
281C2—22no—
282C2—8561no—
283C2—4282no—
284C2C2×#1681208211yescharacteristic
285C2#2029343311yescharacteristic
286C2#2029344011yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#103×C107C2
socleC22×C107C22×D4
prime radical#20293208C2
hypercenterC2×#103D107
hypocenterC107#3063
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C2×#16812140C2semidirect
2. C2×#16812115C2semidirect
4. #20293201C2semidirect
5. #20292884C2semidirect
6. #20292876C2semidirect
7. #20292867C2semidirect
8. #20292859C2semidirect
9. #20292667C2semidirect
10. #20292624C2semidirect
11. C2×#103×C107C2semidirect
58. #13133802D4semidirect
144. C107#3063semidirect
155. C2×#103D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121402
C2×#168121152
#202932082
#202932012
#202928842
#202928762
#202928672
#202928592
#202926672
#202926242
C2×#103×C1072
#3063107