Finite Group Databasebeta

← #28372576

Subgroups — #28372576

Order 13952.

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

normal subgroups only

This group has 504 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×#16812725C211yesmaximal, characteristic, complemented
2C2×#16812721C211yesmaximal, characteristic, complemented
3#20295593C211yesmaximal, characteristic, complemented
4#20295507C211yesmaximal, characteristic, complemented
5#20295247C212yesmaximal, complemented
6#20295240C212yesmaximal, complemented
7#20295225C212yesmaximal, complemented
8#20295220C212yesmaximal, complemented
9#20295052C211yesmaximal, characteristic, complemented
10#20295025C211yesmaximal, characteristic, complemented
11C2×#102×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13C2×#13182029C2211yescharacteristic
14#16812725—24no—
15#16812721—24no—
16C2×Dic436C2211yescharacteristic, complemented
17C2×#13182018C2211yescharacteristic, complemented
18C2×#13182018C2211yescharacteristic, complemented
19C2×#13182017C2212yes—
20C2×D872C2211yescharacteristic, complemented
21C2×#13182014C2212yes—
22C2×#13182013C2211yescharacteristic
23#16812695C2212yes—
24#16812686C2212yes—
25#16812670—21no—
26#16812670—21no—
27#16812668—22no—
28#16812664C2212yescomplemented
29#16812661C2212yescomplemented
30C4×C22⋊D109—22no—
31#16812658C2211yescharacteristic
32#16812657—21no—
33C4⋊C4×D109—21no—
34#16812628C2212yescomplemented
35#16812628C2212yescomplemented
36#16812622C2212yescomplemented
37#16812619C2212yescomplemented
38C22×D4×C109C2211yescharacteristic
39C22×C8×C109C2211yescharacteristic
40C2×C4⋊C4×C109C2211yescharacteristic
41#102×C109C2212yes—
42#102×C109C2212yes—
43#13182029—22no—
44#13182029—22no—
45#13182029—22no—
46#13182029D414yescomplemented
47C2×Dic218C2311yescharacteristic
48C2×Dic218C2311yescharacteristic
49C2×D436C2311yescharacteristic
50C2×D436C2311yescharacteristic
51C2×C22⋊D109—21no—
52C2×C22⋊D109—21no—
53C2×C22⋊D109—22no—
54C2×C22⋊D109—22no—
55C2×C4×D109—21no—
56C2×C4×D109—21no—
57C2×C4×D109—22no—
58Dic436—22no—
59#13182018—22no—
60#13182018—22no—
61#13182017—24no—
62D872—22no—
63#13182014—24no—
64#13182013—22no—
65C2×#11724784C2312yes—
66C4⋊Dic109D411yescharacteristic
67C4⋊Dic109D411yescharacteristic
68#13182008—22no—
69C22⋊Dic109—22no—
70C22⋊Dic109—22no—
71C22⋊Dic109D412yes—
72#13182006—22no—
73C4×Dic109—22no—
74C4×Dic109C2312yes—
75C24×C109—21no—
76C2×D4×C109—22no—
77C2×D4×C109—21no—
78C2×D4×C109—22no—
79C2×D4×C109C2312yes—
80C22×C4×C109—21no—
81C22×C4×C109C2311yescharacteristic
82C2×C8×C109—22no—
83C2×C8×C109C2311yescharacteristic
84C2×C8×C109C2311yescharacteristic
85C4⋊C4×C109—21no—
86C4⋊C4×C109C2312yes—
87C23×C109—22no—
88C23×C109—42no—
89C23×C109—81no—
90C23×C109—81no—
91C23×C109C2×D411yescharacteristic
92C22×D109—21no—
93C22×D109—21no—
94D4×C109—22no—
95D4×C109—22no—
96D4×C109—42no—
97C2×C4×C109—22no—
98C2×C4×C109—81no—
99C2×C4×C109C2×D411yescharacteristic
100C2×C4×C109C2×D412yes—
101C2×C4×C109#4112yes—
102C2×C4×C109C2411yescharacteristic
103C22⋊D109—24no—
104C22⋊D109—42no—
105C22⋊D109—82no—
106C22⋊D109—82no—
107C2×Dic109—21no—
108C2×Dic109—21no—
109C2×Dic109C2×D412yes—
110C2×Dic109C2×D412yes—
111D436—21no—
112D436—21no—
113D436—21no—
114D436C2×D412yes—
115C4×D109—24no—
116C4×D109—42no—
117C4×D109—82no—
118Dic218—21no—
119Dic218—21no—
120Dic218—21no—
121Dic218C2×D412yes—
122C8×C109—21no—
123C8×C109—21no—
124#11724784—22no—
125C22×C109—21no—
126C22×C109—21no—
127C22×C109—42no—
128C22×C109—42no—
129C22×C109—42no—
130C22×C109—84no—
131C22×C109#12112yes—
132C22×C109C22×D411yescharacteristic
133C2×D109—22no—
134C2×D109—41no—
135C2×D109—41no—
136C2×D109—41no—
137C2×D109—41no—
138C2×D109—41no—
139C4×C109—42no—
140C4×C109#12112yes—
141C4×C109C22×D411yescharacteristic
142C4×C109C2×#4111yescharacteristic
143Dic109—22no—
144Dic109—22no—
145Dic109—22no—
146Dic109—41no—
147C2×C109—22no—
148C2×C109—82no—
149C2×C109C2×#12111yescharacteristic
150C2×C109#57611yescharacteristic
151C2×C109#57711yescharacteristic
152D109—42no—
153D109—81no—
154#3045—1091nomaximal, Sylow
155C109#304511yescharacteristic, complemented, Sylow
156C2×AGL(1,ℤ/8)—1091no—
157C2×#135—1091no—
158#535—1091no—
159#520—1091no—
160#464—1092no—
161#463—1092no—
162#460—1092no—
163#459—1092no—
164#424—1091no—
165#417—1091no—
166C2×#102D10911yescharacteristic, complemented
167C2×#41—1091no—
168C2×#41—1091no—
169C22×D4C2×D10911yescharacteristic
170AGL(1,ℤ/8)—2184no—
171#135—2184no—
172C2×Q16—1091no—
173C2×SD16—1092no—
174C2×SD16—1091no—
175C2×SD16—1091no—
176C2×D8—1092no—
177C2×D8—1091no—
178C2×M16—1091no—
179C22×C8C2×D10911yescharacteristic
180#127—1092no—
181#124—1092no—
182#121—2181no—
183#121—2181no—
184#120—2182no—
185C4×Q8—1092no—
186C4×D4—1092no—
187C4×D4—2182no—
188#117—1091no—
189#117—2181no—
190C2×C4⋊C4—2181no—
191C2×C4⋊C4C2×D10911yescharacteristic
192C4⋊C8—1092no—
193C4⋊C8—1092no—
194#103—1092no—
195#102—1092no—
196#102C2×D10912yes—
197#102C2×D10912yes—
198C24—21no—
199#41—1094no—
200#41—2182no—
201#41—2182no—
202#41—2182no—
203C2×Q8—1091no—
204C2×Q8—1091no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—22no—
208C2×D4—1091no—
209C2×D4—1091no—
210C2×D4—2182no—
211C2×D4—2182no—
212C2×D4—2181no—
213C2×D4—2181no—
214C2×D4C22×D10912yes—
215C22×C4—21no—
216C22×C4—2182no—
217C22×C4—2181no—
218C22×C4—2181no—
219C22×C4C22×D10911yescharacteristic
220Q16—2182no—
221SD16—2182no—
222SD16—2182no—
223SD16—2184no—
224D8—2182no—
225D8—2184no—
226M16—2182no—
227C2×C8—22no—
228C2×C8—1092no—
229C2×C8C22×D10911yescharacteristic
230C2×C8C22×D10911yescharacteristic
231C4⋊C4—21no—
232C4⋊C4—1091no—
233C4⋊C4—1091no—
234C4⋊C4—2182no—
235C4⋊C4C22×D10912yes—
236C22⋊C4—1092no—
237C22⋊C4—2182no—
238C22⋊C4—2182no—
239C22⋊C4—2182no—
240C42—1092no—
241C42—2182no—
242C23—22no—
243C23—42no—
244C23—81no—
245C23—81no—
246C23—2181no—
247C23—2181no—
248C23D4×D10911yescharacteristic
249Q8—1092no—
250Q8—2181no—
251Q8—2181no—
252Q8—2181no—
253D4—22no—
254D4—22no—
255D4—42no—
256D4—8722no—
257D4—8722no—
258D4—1092no—
259D4—4362no—
260D4—2184no—
261D4—2181no—
262D4—2181no—
263D4—2181no—
264C2×C4—22no—
265C2×C4—81no—
266C2×C4—8722no—
267C2×C4—1092no—
268C2×C4—1092no—
269C2×C4—4362no—
270C2×C4—2181no—
271C2×C4—2181no—
272C2×C4—2184no—
273C2×C4D4×D10911yescharacteristic
274C2×C4D4×D10912yes—
275C2×C4#1318202912yes—
276C2×C4C23×D10911yescharacteristic
277C8—21no—
278C8—21no—
279C8—2182no—
280C22—21no—
281C22—21no—
282C22—42no—
283C22—42no—
284C22—42no—
285C22—84no—
286C22—4361no—
287C22—4361no—
288C22—4361no—
289C22—4361no—
290C22—4361no—
291C22—2182no—
292C22#1681267212yes—
293C22C2×D4×D10911yescharacteristic
294C4—42no—
295C4—4361no—
296C4—2182no—
297C4—2182no—
298C4—2182no—
299C4#1681267212yes—
300C4C2×D4×D10911yescharacteristic
301C4C2×#1318202911yescharacteristic
302C2—22no—
303C2—82no—
304C2—8721no—
305C2—4362no—
306C2C2×#1681267211yescharacteristic
307C2#2029582711yescharacteristic
308C2#2029583211yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC2×#102×C109C2
socleC22×C109C22×D4
hypercenterC2×#102D109
hypocenterC109#3045
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
1. C2×#16812725C2semidirect
2. C2×#16812721C2semidirect
3. #20295593C2semidirect
4. #20295507C2semidirect
5. #20295247C2semidirect
6. #20295240C2semidirect
7. #20295225C2semidirect
8. #20295220C2semidirect
9. #20295052C2semidirect
10. #20295025C2semidirect
11. C2×#102×C109C2semidirect
16. C2×Dic436C22semidirect
17. C2×#13182018C22semidirect
18. C2×#13182018C22semidirect
20. C2×D872C22semidirect
28. #16812664C22semidirect
29. #16812661C22semidirect
34. #16812628C22semidirect
35. #16812628C22semidirect
36. #16812622C22semidirect
37. #16812619C22semidirect
46. #13182029D4semidirect
155. C109#3045semidirect
166. C2×#102D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127252
C2×#168127212
#202955932
#202955072
#202952472
#202952402
#202952252
#202952202
#202950522
#202950252
C2×#102×C1092
#3045109