Finite Group Databasebeta

← #28345996

Subgroups — #28345996

Order 13696.

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

normal subgroups only

This group has 504 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×#16812135C211yesmaximal, characteristic, complemented
2C2×#16812129C211yesmaximal, characteristic, complemented
3#20293107C211yesmaximal, characteristic, complemented
4#20292846C212yesmaximal, complemented
5#20292839C212yesmaximal, complemented
6#20292824C212yesmaximal, complemented
7#20292819C212yesmaximal, complemented
8#20292654C211yesmaximal, characteristic, complemented
9#20292628C211yesmaximal, characteristic, complemented
10C2×#16812030C211yesmaximal, characteristic, complemented
11#535×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13#16812135—24no—
14#16812129—24no—
15C2×#13133808C2211yescharacteristic, complemented
16C2×#13133806C2212yes—
17C2×#13133806C2211yescharacteristic, complemented
18C2×#13133805C2211yescharacteristic, complemented
19C2×#13133803C2212yes—
20C2×#13133803C2211yescharacteristic, complemented
21C2×#13133802C2211yescharacteristic
22C22×C107⋊C8C2211yescharacteristic
23C4×Dic214C2212yescomplemented
24C4×D428C2212yescomplemented
25(C22⋊D107)⋊C4—22no—
26#16812067—21no—
27C2×C4⋊Dic107C2211yescharacteristic
28C4⋊C4×D107—21no—
29C4⋊(C107⋊C8)C2212yescomplemented
30C4⋊(C107⋊C8)C2212yescomplemented
31#16812033C2212yescomplemented
32#16812030C2212yes—
33#16812030C2212yes—
34#16812030C2212yescomplemented
35C2×#41×C107C2211yescharacteristic
36C22×D4×C107C2211yescharacteristic
37#127×C107C2212yes—
38#124×C107C2212yes—
39#121×C107—21no—
40#121×C107—21no—
41#120×C107—22no—
42#117×C107C2211yescharacteristic
43#13133818—22no—
44#13133818D414yescomplemented
45C2×Dic214C2311yescharacteristic
46C2×D428C2311yescharacteristic
47C2×C22⋊D107—21no—
48C22×Dic107—21no—
49C2×C4×D107—21no—
50C2×C4×D107—22no—
51#13133808—22no—
52#13133806—24no—
53#13133806—22no—
54#13133805—22no—
55#13133803—24no—
56#13133803—22no—
57#13133802—22no—
58C2×C107⋊C8—22no—
59C2×C107⋊C8C2312yes—
60C2×C107⋊C8C2311yescharacteristic
61C2×C107⋊C8C2311yescharacteristic
62C4⋊Dic107—21no—
63C4⋊Dic107C2312yes—
64Dic107⋊C4—22no—
65(C2×D107)⋊C4—22no—
66C4×Dic107—22no—
67C24×C107—21no—
68#41×C107—22no—
69#41×C107—22no—
70C2×Q8×C107C2311yescharacteristic
71C2×D4×C107—21no—
72C2×D4×C107—22no—
73C2×D4×C107—21no—
74C2×D4×C107—22no—
75C2×D4×C107—22no—
76C2×D4×C107—22no—
77C2×D4×C107C2312yes—
78C2×D4×C107C2311yescharacteristic
79C22×C4×C107—21no—
80C22×C4×C107C2311yescharacteristic
81C4⋊C4×C107D411yescharacteristic
82C4⋊C4×C107D411yescharacteristic
83C22⋊C4×C107—22no—
84C22⋊C4×C107—22no—
85C22⋊C4×C107D412yes—
86C42×C107C2312yes—
87C23×C107—22no—
88C23×C107—21no—
89C23×C107—42no—
90C23×C107—81no—
91C23×C107—81no—
92C23×C107C2×D411yescharacteristic
93C22×D107—21no—
94Q8×C107—21no—
95Q8×C107—21no—
96D4×C107—22no—
97D4×C107—21no—
98D4×C107—22no—
99D4×C107—21no—
100D4×C107—42no—
101D4×C107—42no—
102D4×C107—82no—
103D4×C107—82no—
104C2×C4×C107—21no—
105C2×C4×C107—42no—
106C2×C4×C107C2×D412yes—
107C2×C4×C107C2×D412yes—
108C2×C4×C107C2×D412yes—
109C2×C4×C107C2×D411yescharacteristic
110C2×C4×C107#4112yes—
111C2×C4×C107C2411yescharacteristic
112C22⋊D107—24no—
113C2×Dic107—21no—
114C2×Dic107—22no—
115C2×Dic107—81no—
116D428—21no—
117D428C2×D412yes—
118C4×D107—24no—
119C4×D107—82no—
120Dic214—21no—
121Dic214C2×D412yes—
122C107⋊C8—21no—
123C107⋊C8—21no—
124C107⋊C8—22no—
125C22×C107—21no—
126C22×C107—21no—
127C22×C107—41no—
128C22×C107—42no—
129C22×C107—41no—
130C22×C107—42no—
131C22×C107—41no—
132C22×C107—42no—
133C22×C107—84no—
134C22×C107#12112yes—
135C22×C107C22×D411yescharacteristic
136C2×D107—22no—
137C2×D107—41no—
138C2×D107—41no—
139C4×C107—22no—
140C4×C107—22no—
141C4×C107—41no—
142C4×C107#12112yes—
143C4×C107C22×D411yescharacteristic
144C4×C107C2×#4111yescharacteristic
145Dic107—22no—
146Dic107—42no—
147C2×C107—22no—
148C2×C107—81no—
149C2×C107—82no—
150C2×C107C2×#12111yescharacteristic
151C2×C107#57611yescharacteristic
152C2×C107#57711yescharacteristic
153D107—42no—
154#3045—1071nomaximal, Sylow
155C107#304511yescharacteristic, complemented, Sylow
156C2×AGL(1,ℤ/8)—1071no—
157C2×#135—1071no—
158#535D10711yescharacteristic, complemented
159#520—1071no—
160#464—1072no—
161#463—1072no—
162#460—1072no—
163#459—1072no—
164#424—1071no—
165#417—1071no—
166C2×#102—1071no—
167C2×#41—1071no—
168C2×#41C2×D10711yescharacteristic
169C22×D4C2×D10711yescharacteristic
170AGL(1,ℤ/8)—2144no—
171#135—2144no—
172C2×Q16—1071no—
173C2×SD16—1071no—
174C2×SD16—1071no—
175C2×SD16—1072no—
176C2×D8—1071no—
177C2×D8—1072no—
178C2×M16—1071no—
179C22×C8—1071no—
180#127C2×D10712yes—
181#124C2×D10712yes—
182#121—21no—
183#121—21no—
184#120—22no—
185C4×Q8—1072no—
186C4×D4—1072no—
187C4×D4—2142no—
188#117—2141no—
189#117C2×D10711yescharacteristic
190C2×C4⋊C4—1071no—
191C2×C4⋊C4—2141no—
192C4⋊C8—1072no—
193C4⋊C8—1072no—
194#103—1072no—
195#102—1072no—
196#102—1072no—
197#102—1072no—
198C24—21no—
199#41—22no—
200#41—22no—
201#41—1074no—
202#41—2142no—
203C2×Q8—1071no—
204C2×Q8C22×D10711yescharacteristic
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—21no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—1071no—
212C2×D4—2141no—
213C2×D4C22×D10712yes—
214C2×D4C22×D10711yescharacteristic
215C22×C4—21no—
216C22×C4—2141no—
217C22×C4—2142no—
218C22×C4—2141no—
219C22×C4C22×D10711yescharacteristic
220Q16—2142no—
221SD16—2142no—
222SD16—2144no—
223SD16—2142no—
224D8—2142no—
225D8—2144no—
226M16—2142no—
227C2×C8—1071no—
228C2×C8—1071no—
229C2×C8—1072no—
230C2×C8—2142no—
231C4⋊C4—1072no—
232C4⋊C4—2141no—
233C4⋊C4—2142no—
234C4⋊C4C22⋊D10711yescharacteristic
235C4⋊C4C22⋊D10711yescharacteristic
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—2142no—
239C22⋊C4C22⋊D10712yes—
240C42—2142no—
241C42C22×D10712yes—
242C23—22no—
243C23—21no—
244C23—42no—
245C23—81no—
246C23—81no—
247C23—2141no—
248C23C2×C22⋊D10711yescharacteristic
249Q8—21no—
250Q8—21no—
251Q8—1072no—
252Q8—2141no—
253D4—22no—
254D4—21no—
255D4—22no—
256D4—21no—
257D4—42no—
258D4—42no—
259D4—82no—
260D4—82no—
261D4—1072no—
262D4—2144no—
263D4—2141no—
264C2×C4—21no—
265C2×C4—42no—
266C2×C4—8562no—
267C2×C4—8561no—
268C2×C4—2142no—
269C2×C4—2141no—
270C2×C4—2144no—
271C2×C4D4×D10712yes—
272C2×C4C2×C22⋊D10712yes—
273C2×C4C2×C22⋊D10712yes—
274C2×C4C2×C22⋊D10711yescharacteristic
275C2×C4#1313381712yes—
276C2×C4C23×D10711yescharacteristic
277C8—2141no—
278C8—2141no—
279C8—2142no—
280C22—21no—
281C22—21no—
282C22—41no—
283C22—42no—
284C22—41no—
285C22—42no—
286C22—41no—
287C22—42no—
288C22—84no—
289C22—4281no—
290C22—4281no—
291C22—2142no—
292C22#1681208012yes—
293C22C22×C22⋊D10711yescharacteristic
294C4—22no—
295C4—22no—
296C4—41no—
297C4—4282no—
298C4—2142no—
299C4#1681208012yes—
300C4C2×D4×D10711yescharacteristic
301C4C2×#1313381711yescharacteristic
302C2—22no—
303C2—82no—
304C2—81no—
305C2—4282no—
306C2C2×#1681208011yescharacteristic
307C2#2029343011yescharacteristic
308C2#2029343811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#535×C107C2
socleC22×C107C22×D4
hypercenter#535D107
hypocenterC107#3045
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#16812135C2semidirect
2. C2×#16812129C2semidirect
3. #20293107C2semidirect
4. #20292846C2semidirect
5. #20292839C2semidirect
6. #20292824C2semidirect
7. #20292819C2semidirect
8. #20292654C2semidirect
9. #20292628C2semidirect
10. C2×#16812030C2semidirect
11. #535×C107C2semidirect
15. C2×#13133808C22semidirect
17. C2×#13133806C22semidirect
18. C2×#13133805C22semidirect
20. C2×#13133803C22semidirect
23. C4×Dic214C22semidirect
24. C4×D428C22semidirect
29. C4⋊(C107⋊C8)C22semidirect
30. C4⋊(C107⋊C8)C22semidirect
31. #16812033C22semidirect
34. #16812030C22semidirect
44. #13133818D4semidirect
155. C107#3045semidirect
158. #535D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121352
C2×#168121292
#202931072
#202928462
#202928392
#202928242
#202928192
#202926542
#202926282
C2×#168120302
#535×C1072
#3045107