Finite Group Databasebeta

← #28373497

Subgroups — #28373497

Order 13952.

572 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 137 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×#16812728C212yesmaximal, complemented
2C2×#16812723C212yesmaximal, complemented
3#20295574C211yesmaximal, characteristic, complemented
4#124×D109C211yesmaximal, characteristic, complemented
5#20295421C211yesmaximal, characteristic, complemented
6#20295420C211yesmaximal, characteristic, complemented
7#20295413C211yesmaximal, characteristic, complemented
8#20295409C212yesmaximal, complemented
9#20295407C212yesmaximal, complemented
10#20294978C211yesmaximal, characteristic, complemented
11#496×C109C211yesmaximal, characteristic, complemented
12C2×#13182028C2212yes—
13C2×Q8×D109C2211yescharacteristic
14C2×D4×D109C2211yescharacteristic
15#16812728—28no—
16#16812723—28no—
17C2×Dic436C2212yescomplemented
18C2×#13182019C2212yescomplemented
19C2×#13182018C2212yes—
20C2×#13182017C2212yes—
21C2×#13182017C2212yescomplemented
22C2×#13182014C2212yes—
23C2×#13182012C2212yes—
24#16812699C2211yescharacteristic, complemented
25#16812688C2211yescharacteristic
26#16812687C2211yescharacteristic
27#16812686C2211yescharacteristic
28#16812686C2212yescomplemented
29#16812685—22no—
30#16812670—22no—
31D4×Dic109—22no—
32C22⋊C4×D109—22no—
33C42×D109C2211yescharacteristic
34#16812610C2211yescharacteristic, complemented
35C8×Dic109C2212yescomplemented
36C2×SD16×C109C2212yescomplemented
37C2×D8×C109C2212yes—
38#127×C109C2211yescharacteristic, complemented
39#124×C109C2211yescharacteristic
40#97×C109C2211yescharacteristic, complemented
41C23×D109—21no—
42#13182028—24no—
43#13182028—24no—
44C2×Dic218—21no—
45C2×Dic218C2311yescharacteristic
46C2×Dic218C2312yes—
47Q8×D109—24no—
48C2×D436C2311yescharacteristic
49C2×C22⋊D109—22no—
50C2×C22⋊D109—21no—
51D4×D109—24no—
52C22×Dic109—22no—
53C2×C4×D109—21no—
54C2×C4×D109D412yes—
55C2×C4×D109C2311yescharacteristic
56Dic436—24no—
57#13182019—24no—
58#13182018—24no—
59#13182017—24no—
60#13182017—24no—
61#13182014—24no—
62#13182012D418yescomplemented
63C2×#11724784C2312yes—
64C4⋊Dic109—22no—
65C22⋊Dic109—22no—
66C22⋊Dic109—24no—
67#13182006—22no—
68#13182006—22no—
69C4×Dic109D412yes—
70C4×Dic109C2311yescharacteristic
71C2×Q8×C109C2311yescharacteristic
72C2×D4×C109—22no—
73C2×D4×C109C2311yescharacteristic
74C2×D4×C109C2312yes—
75SD16×C109—24no—
76D8×C109—24no—
77C2×C8×C109C2312yes—
78C22⋊C4×C109—22no—
79C42×C109C2311yescharacteristic
80C23×C109—21no—
81C23×C109—22no—
82C22×D109—21no—
83C22×D109—44no—
84C22×D109—42no—
85C22×D109C2×D411yescharacteristic
86Q8×C109—22no—
87D4×C109—22no—
88D4×C109—22no—
89D4×C109—22no—
90D4×C109—44no—
91C2×C4×C109—21no—
92C2×C4×C109C2×D412yes—
93C2×C4×C109C2411yescharacteristic
94C22⋊D109—44no—
95C22⋊D109—42no—
96C2×Dic109—21no—
97C2×Dic109—22no—
98C2×Dic109—44no—
99C2×Dic109C2×D412yes—
100C2×Dic109C2×D411yescharacteristic
101D436—22no—
102C4×D109—42no—
103C4×D109—42no—
104C4×D109C2×D414yes—
105Dic218—22no—
106Dic218—22no—
107Dic218—22no—
108Dic218—42no—
109C8×C109C2×D414yes—
110#11724784C2×D414yes—
111C22×C109—42no—
112C22×C109—41no—
113C22×C109—44no—
114C22×C109—42no—
115C22×C109C22×D411yescharacteristic
116C2×D109—22no—
117C2×D109—41no—
118C2×D109—42no—
119C2×D109—84no—
120C2×D109#12712yes—
121C4×C109—22no—
122C4×C109—41no—
123C4×C109C22×D412yes—
124Dic109—22no—
125Dic109—41no—
126Dic109—42no—
127Dic109#12712yes—
128C2×C109—81no—
129C2×C109—82no—
130C2×C109C2×#12711yescharacteristic
131C2×C109#57512yes—
132D109—22no—
133D109—81no—
134#3153—1091nomaximal, Sylow
135C109#315311yescharacteristic, complemented, Sylow
136C2×#137—1092no—
137C2×AGL(1,ℤ/8)—1092no—
138#532—1091no—
139C2×#124—1091no—
140#497—1091no—
141#497—1091no—
142#496—1091no—
143#496D10911yescharacteristic, complemented
144#495—1092no—
145#495—1092no—
146C4×M16—1091no—
147C2×#41—1092no—
148C22×Q8—1091no—
149C22×D4—1091no—
150#137—2188no—
151AGL(1,ℤ/8)—2188no—
152C2×Q16—1092no—
153C2×Q16—1092no—
154C2×SD16—1092no—
155C2×SD16—1092no—
156C2×SD16—1092no—
157C2×SD16C2×D10912yescomplemented
158C2×D8—1092no—
159C2×D8C2×D10912yes—
160C2×M16—1092no—
161C4⋊Q8—1091no—
162#127C2×D10911yescharacteristic, complemented
163#124—1091no—
164#124—1091no—
165#124—1092no—
166#124—1091no—
167#124—2182no—
168#124C2×D10911yescharacteristic
169#121—2182no—
170C4×D4—2182no—
171C2×C22⋊C4—2182no—
172C2×C42—1091no—
173#97—1091no—
174#97C2×D10911yescharacteristic, complemented
175C4×C8—1092no—
176C24—2181no—
177#41—2184no—
178#41—2184no—
179C2×Q8—1091no—
180C2×Q8—1092no—
181C2×Q8—2181no—
182C2×Q8—2184no—
183C2×Q8C22×D10911yescharacteristic
184C2×D4—22no—
185C2×D4—1091no—
186C2×D4—2184no—
187C2×D4—2181no—
188C2×D4—2182no—
189C2×D4C22×D10911yescharacteristic
190C2×D4C22×D10912yes—
191C22×C4—1091no—
192C22×C4—1092no—
193C22×C4—2181no—
194C22×C4—2182no—
195Q16—2184no—
196Q16—2184no—
197SD16—24no—
198SD16—2184no—
199SD16—2184no—
200SD16—2184no—
201D8—24no—
202D8—2184no—
203M16—1098no—
204C2×C8—1092no—
205C2×C8C22×D10912yes—
206C4⋊C4—2182no—
207C22⋊C4—22no—
208C22⋊C4—2182no—
209C22⋊C4—2182no—
210C22⋊C4—2184no—
211C22⋊C4—2182no—
212C42—1092no—
213C42—1091no—
214C42C22×D10911yescharacteristic
215C23—21no—
216C23—22no—
217C23—1091no—
218C23—4362no—
219C23—4364no—
220C23—2181no—
221Q8—22no—
222Q8—4362no—
223Q8—2182no—
224Q8—2182no—
225Q8—2182no—
226D4—22no—
227D4—22no—
228D4—22no—
229D4—44no—
230D4—4362no—
231D4—4364no—
232D4—2182no—
233C2×C4—21no—
234C2×C4—1091no—
235C2×C4—1094no—
236C2×C4—1092no—
237C2×C4—4362no—
238C2×C4—4362no—
239C2×C4—4364no—
240C2×C4—2181no—
241C2×C4—2182no—
242C2×C4D4×D10912yes—
243C2×C4C23×D10911yescharacteristic
244C8—1094no—
245C8D4×D10914yes—
246C22—42no—
247C22—41no—
248C22—44no—
249C22—42no—
250C22—8724no—
251C22—1092no—
252C22—4361no—
253C22—4362no—
254C22—2182no—
255C22C2×D4×D10911yescharacteristic
256C4—22no—
257C4—41no—
258C4—1092no—
259C4—4362no—
260C4—4361no—
261C4—2182no—
262C4C2×D4×D10912yes—
263C2—81no—
264C2—82no—
265C2—8721no—
266C2—2182no—
267C2#127×D10911yescharacteristic
268C2#2029582212yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#496×C109C2
socleC22×C109C22×D4
hypercenter#496D109
hypocenterC109#3153
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#16812728C2semidirect
2. C2×#16812723C2semidirect
3. #20295574C2semidirect
4. #124×D109C2semidirect
5. #20295421C2semidirect
6. #20295420C2semidirect
7. #20295413C2semidirect
8. #20295409C2semidirect
9. #20295407C2semidirect
10. #20294978C2semidirect
11. #496×C109C2semidirect
17. C2×Dic436C22semidirect
18. C2×#13182019C22semidirect
21. C2×#13182017C22semidirect
24. #16812699C22semidirect
28. #16812686C22semidirect
34. #16812610C22semidirect
35. C8×Dic109C22semidirect
36. C2×SD16×C109C22semidirect
38. #127×C109C22semidirect
40. #97×C109C22semidirect
62. #13182012D4semidirect
135. C109#3153semidirect
143. #496D109semidirect
157. C2×SD16C2×D109semidirect
162. #127C2×D109semidirect
174. #97C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127282
C2×#168127232
#202955742
#124×D1092
#202954212
#202954202
#202954132
#202954092
#202954072
#202949782
#496×C1092
#3153109