Finite Group Databasebeta

← #28372639

Subgroups — #28372639

Order 13952.

500 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 127 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×#16812718C211yesmaximal, characteristic, complemented
2C2×M16×D109C211yesmaximal, characteristic, complemented
3#20295536C212yesmaximal, complemented
4#20295285C211yesmaximal, characteristic, complemented
5#20295284C211yesmaximal, characteristic, complemented
6#20295276C211yesmaximal, characteristic, complemented
7#20295271C212yesmaximal, complemented
8#20295270C212yesmaximal, complemented
9#20295061C211yesmaximal, characteristic, complemented
10#20295019C212yesmaximal, complemented
11#469×C109C211yesmaximal, characteristic, complemented
12C2×#13182028C2212yes—
13C22×C4×D109C2211yescharacteristic
14#16812718—24no—
15C2×Dic436C2211yescharacteristic, complemented
16C2×#13182017C2212yescomplemented
17C2×#13182013C2211yescharacteristic
18C2×#13182012C2211yescharacteristic
19C2×#13182012C2211yescharacteristic
20M16×D109—24no—
21C2×C8×D109C2211yescharacteristic
22C2×C8×D109C2211yescharacteristic
23#16812681—22no—
24#16812679C2212yes—
25#16812678C2212yes—
26#16812670C2212yes—
27#16812667—22no—
28D4×Dic109—22no—
29#16812659—22no—
30#16812657C2212yes—
31#16812632C2212yescomplemented
32#16812631C2211yescharacteristic, complemented
33#16812624C2212yescomplemented
34#16812622C2212yescomplemented
35#16812620C2212yescomplemented
36C2×D8×C109C2211yescharacteristic, complemented
37C2×M16×C109C2211yescharacteristic
38#121×C109C2212yes—
39#106×C109C2211yescharacteristic, complemented
40#102×C109C2212yescomplemented
41C23×D109D411yescharacteristic
42#13182028—24no—
43#13182028—24no—
44C2×Dic218C2312yes—
45C2×C22⋊D109—22no—
46C2×C22⋊D109—22no—
47C22×Dic109—22no—
48C22×Dic109D411yescharacteristic
49C2×C4×D109—22no—
50C2×C4×D109—22no—
51C2×C4×D109D411yescharacteristic
52C2×C4×D109D411yescharacteristic
53C2×C4×D109C2311yescharacteristic
54C2×C4×D109C2311yescharacteristic
55Dic436—22no—
56#13182017—24no—
57#13182013—22no—
58#13182012—22no—
59#13182012—22no—
60C2×#11724784C2311yescharacteristic
61C2×#11724784C2311yescharacteristic
62C8×D109—22no—
63C8×D109D414yescomplemented
64C4⋊Dic109—22no—
65C4⋊Dic109C2312yes—
66#13182008—22no—
67C22⋊Dic109—22no—
68C22⋊Dic109—22no—
69#13182006—22no—
70#13182006—22no—
71C4×Dic109—22no—
72C2×D4×C109—22no—
73C2×D4×C109C2312yes—
74C22×C4×C109C2311yescharacteristic
75D8×C109—22no—
76M16×C109—22no—
77C2×C8×C109C2311yescharacteristic
78C2×C8×C109C2311yescharacteristic
79C4⋊C4×C109C2312yes—
80C22⋊C4×C109—22no—
81C23×C109—22no—
82C23×C109C2×D411yescharacteristic
83C22×D109—22no—
84C22×D109—42no—
85C22×D109C2×D411yescharacteristic
86C22×D109C2×D411yescharacteristic
87D4×C109—22no—
88D4×C109—22no—
89D4×C109—44no—
90C2×C4×C109—21no—
91C2×C4×C109—21no—
92C2×C4×C109—22no—
93C2×C4×C109C2×D411yescharacteristic
94C2×C4×C109C2411yescharacteristic
95C22⋊D109—44no—
96C22⋊D109—44no—
97C2×Dic109—22no—
98C2×Dic109—22no—
99C2×Dic109—22no—
100C2×Dic109—44no—
101C2×Dic109C2×D411yescharacteristic
102C2×Dic109C2×D411yescharacteristic
103C4×D109—22no—
104C4×D109—21no—
105C4×D109—21no—
106C4×D109—22no—
107C4×D109C2×D412yes—
108C4×D109#4112yes—
109Dic218—22no—
110Dic218—22no—
111C8×C109—21no—
112C8×C109C2×D412yes—
113#11724784—21no—
114#11724784C2×D412yes—
115C22×C109—21no—
116C22×C109—22no—
117C22×C109—42no—
118C22×C109—42no—
119C22×C109—42no—
120C22×C109C22×D411yescharacteristic
121C2×D109—22no—
122C2×D109—21no—
123C2×D109—22no—
124C2×D109—44no—
125C2×D109#12112yes—
126C4×C109—21no—
127C4×C109—42no—
128C4×C109C22×D411yescharacteristic
129C4×C109C2×#4111yescharacteristic
130Dic109—21no—
131Dic109—42no—
132Dic109—42no—
133Dic109#12112yes—
134C2×C109—41no—
135C2×C109—82no—
136C2×C109C2×#12111yescharacteristic
137C2×C109#57512yes—
138D109—22no—
139D109—41no—
140#3054—1091nomaximal, Sylow
141C109#305411yescharacteristic, complemented, Sylow
142C2×#135—1091no—
143C22×M16—1091no—
144#525—1092no—
145#470—1091no—
146#470—1091no—
147#469—1091no—
148#469D10911yescharacteristic, complemented
149#468—1092no—
150#468—1092no—
151#427—1091no—
152#416—1092no—
153C2×#41—1092no—
154C23×C4—1091no—
155#135—2184no—
156C2×Q16—1091no—
157C2×SD16—1092no—
158C2×D8C2×D10911yescharacteristic, complemented
159C2×M16—1091no—
160C2×M16—1091no—
161C2×M16—1091no—
162C2×M16—2184no—
163C2×M16C2×D10911yescharacteristic
164C22×C8—1091no—
165C22×C8—1091no—
166#123—2182no—
167C22⋊Q8—1092no—
168C22⋊Q8—1092no—
169#121—1092no—
170#121C2×D10912yes—
171#120—2182no—
172C4×D4—2182no—
173C4×D4—2182no—
174#117—1092no—
175#107—1092no—
176#106—1091no—
177#106C2×D10911yescharacteristic, complemented
178#103—1092no—
179#103—1092no—
180#102—1092no—
181#102C2×D10912yescomplemented
182C24—1091no—
183#41—2184no—
184#41—2184no—
185C2×Q8—1092no—
186C2×D4—22no—
187C2×D4—2182no—
188C2×D4—2182no—
189C2×D4C22×D10912yes—
190C22×C4—1091no—
191C22×C4—1091no—
192C22×C4—1091no—
193C22×C4—1091no—
194C22×C4—1091no—
195C22×C4—2182no—
196C22×C4—2182no—
197C22×C4—2182no—
198C22×C4C22×D10911yescharacteristic
199Q16—2182no—
200SD16—2184no—
201D8—22no—
202M16—22no—
203M16—2182no—
204M16—2182no—
205M16—2182no—
206C2×C8—1091no—
207C2×C8—1094no—
208C2×C8—1091no—
209C2×C8—2182no—
210C2×C8C22×D10911yescharacteristic
211C2×C8C22×D10911yescharacteristic
212C4⋊C4—1092no—
213C4⋊C4—2182no—
214C4⋊C4—2182no—
215C4⋊C4C22×D10912yes—
216C22⋊C4—22no—
217C22⋊C4—2182no—
218C22⋊C4—2182no—
219C22⋊C4—2182no—
220C22⋊C4—2182no—
221C42—2182no—
222C23—22no—
223C23—1091no—
224C23—1091no—
225C23—4362no—
226C23—2182no—
227C23D4×D10911yescharacteristic
228Q8—2182no—
229Q8—2182no—
230D4—22no—
231D4—22no—
232D4—44no—
233D4—4364no—
234D4—4364no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—1091no—
239C2×C4—1092no—
240C2×C4—1092no—
241C2×C4—1091no—
242C2×C4—4364no—
243C2×C4—2182no—
244C2×C4—2182no—
245C2×C4—2182no—
246C2×C4—2182no—
247C2×C4—2182no—
248C2×C4—2181no—
249C2×C4—2181no—
250C2×C4D4×D10911yescharacteristic
251C2×C4C23×D10911yescharacteristic
252C8—21no—
253C8—1092no—
254C8—2181no—
255C8D4×D10912yes—
256C22—21no—
257C22—22no—
258C22—42no—
259C22—42no—
260C22—42no—
261C22—1092no—
262C22—4364no—
263C22—2181no—
264C22—2182no—
265C22—2182no—
266C22C2×D4×D10911yescharacteristic
267C4—21no—
268C4—42no—
269C4—1092no—
270C4—4362no—
271C4—4362no—
272C4—2181no—
273C4C2×D4×D10911yescharacteristic
274C4#41×D10911yescharacteristic
275C2—41no—
276C2—82no—
277C2—4361no—
278C2—2182no—
279C2#121×D10911yescharacteristic
280C2#2029582212yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#469×C109C2
socleC22×C109C22×D4
hypercenter#469D109
hypocenterC109#3054
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#16812718C2semidirect
2. C2×M16×D109C2semidirect
3. #20295536C2semidirect
4. #20295285C2semidirect
5. #20295284C2semidirect
6. #20295276C2semidirect
7. #20295271C2semidirect
8. #20295270C2semidirect
9. #20295061C2semidirect
10. #20295019C2semidirect
11. #469×C109C2semidirect
15. C2×Dic436C22semidirect
16. C2×#13182017C22semidirect
31. #16812632C22semidirect
32. #16812631C22semidirect
33. #16812624C22semidirect
34. #16812622C22semidirect
35. #16812620C22semidirect
36. C2×D8×C109C22semidirect
39. #106×C109C22semidirect
40. #102×C109C22semidirect
63. C8×D109D4semidirect
141. C109#3054semidirect
148. #469D109semidirect
158. C2×D8C2×D109semidirect
177. #106C2×D109semidirect
181. #102C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127182
C2×M16×D1092
#202955362
#202952852
#202952842
#202952762
#202952712
#202952702
#202950612
#202950192
#469×C1092
#3054109