Finite Group Databasebeta

← #28372640

Subgroups — #28372640

Order 13952.

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

normal subgroups only

This group has 500 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×#16812719C211yesmaximal, characteristic, complemented
2C22×#13182012C211yesmaximal, characteristic, complemented
3#20295538C211yesmaximal, characteristic, complemented
4#20295533C211yesmaximal, characteristic, complemented
5#20295283C212yesmaximal, complemented
6#20295278C212yesmaximal, complemented
7#20295275C212yesmaximal, complemented
8#20295271C212yesmaximal, complemented
9#20295062C211yesmaximal, characteristic, complemented
10#20295023C211yesmaximal, characteristic, complemented
11#416×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13C22×C4×D109C2211yescharacteristic
14#16812719—24no—
15C2×#13182019C2211yescharacteristic, complemented
16C2×#13182017C2211yescharacteristic, complemented
17C2×#13182016C2211yescharacteristic, complemented
18C2×#13182014C2211yescharacteristic, complemented
19C2×#13182012—24no—
20C2×#13182012C2212yes—
21C2×#13182012C2212yes—
22C22×#11724784C2211yescharacteristic
23#16812683—21no—
24#16812680—21no—
25#16812679C2212yes—
26#16812675C2212yes—
27#16812674C2212yes—
28#16812670C2212yes—
29#16812669—21no—
30#16812667—21no—
31C4×D436—22no—
32C4×C22⋊D109—22no—
33#16812658C2211yescharacteristic
34#16812632C2212yescomplemented
35#16812630C2212yescomplemented
36#16812624C2212yescomplemented
37#16812621C2212yescomplemented
38C2×#41×C109C2211yescharacteristic
39C22×C8×C109C2211yescharacteristic
40#117×C109C2211yescharacteristic
41#103×C109C2212yescomplemented
42#102×C109C2212yescomplemented
43C23×D109D411yescharacteristic
44#13182029—22no—
45#13182029—22no—
46C2×Dic218C2311yescharacteristic
47C2×D436—22no—
48C2×D436C2311yescharacteristic
49C2×C22⋊D109—21no—
50C2×C22⋊D109—22no—
51C22×Dic109D411yescharacteristic
52C2×C4×D109—22no—
53C2×C4×D109—22no—
54C2×C4×D109—21no—
55C2×C4×D109D412yes—
56C2×C4×D109C2312yes—
57#13182019—22no—
58#13182017—22no—
59#13182016—22no—
60#13182014—22no—
61#13182012—24no—
62#13182012—24no—
63C2×#11724784D414yescomplemented
64C2×#11724784C2311yescharacteristic
65C2×#11724784C2311yescharacteristic
66C4⋊Dic109—22no—
67C4⋊Dic109C2312yes—
68#13182008—22no—
69C22⋊Dic109—21no—
70C22⋊Dic109—22no—
71#13182006—22no—
72#13182006—22no—
73#13182006—22no—
74C4×Dic109—21no—
75#41×C109—22no—
76#41×C109—22no—
77C2×Q8×C109C2311yescharacteristic
78C2×D4×C109—21no—
79C2×D4×C109C2311yescharacteristic
80C22×C4×C109—21no—
81C22×C4×C109C2311yescharacteristic
82C2×C8×C109—22no—
83C2×C8×C109C2312yes—
84C4⋊C4×C109C2312yes—
85C22⋊C4×C109—21no—
86C42×C109—21no—
87C23×C109—21no—
88C23×C109C2×D411yescharacteristic
89C22×D109—22no—
90C22×D109—21no—
91C22×D109—42no—
92C22×D109C2×D412yes—
93Q8×C109—21no—
94Q8×C109—21no—
95D4×C109—21no—
96D4×C109—21no—
97D4×C109—42no—
98C2×C4×C109—22no—
99C2×C4×C109—21no—
100C2×C4×C109—42no—
101C2×C4×C109C2×D412yes—
102C2×C4×C109C2×D411yescharacteristic
103C2×C4×C109#4112yes—
104C2×C4×C109C2411yescharacteristic
105C22⋊D109—44no—
106C22⋊D109—42no—
107C2×Dic109—22no—
108C2×Dic109—22no—
109C2×Dic109—21no—
110C2×Dic109C2×D412yes—
111D436—21no—
112D436—21no—
113D436—44no—
114C4×D109—24no—
115C4×D109—22no—
116C4×D109—22no—
117C4×D109—42no—
118Dic218—21no—
119Dic218—21no—
120C8×C109—22no—
121#11724784C2×D412yes—
122#11724784C2×D412yes—
123C22×C109—22no—
124C22×C109—41no—
125C22×C109—42no—
126C22×C109#12112yes—
127C22×C109C22×D411yescharacteristic
128C2×D109—22no—
129C2×D109—22no—
130C2×D109—22no—
131C2×D109—44no—
132C2×D109—41no—
133C2×D109—42no—
134C4×C109—41no—
135C4×C109—42no—
136C4×C109#12112yes—
137C4×C109C22×D411yescharacteristic
138C4×C109C2×#4111yescharacteristic
139Dic109—22no—
140Dic109—41no—
141Dic109—42no—
142C2×C109—22no—
143C2×C109—81no—
144C2×C109C2×#12111yescharacteristic
145C2×C109#57512yes—
146D109—42no—
147D109—81no—
148#3054—1091nomaximal, Sylow
149C109#305411yescharacteristic, complemented, Sylow
150C2×#135—1091no—
151C22×M16—1091no—
152#525—1091no—
153#525—1091no—
154#470—1092no—
155#469—1092no—
156#468—1092no—
157#468—1092no—
158#427—1091no—
159#416—1091no—
160#416D10911yescharacteristic, complemented
161C2×#41—1091no—
162C2×#41C2×D10911yescharacteristic
163C23×C4—1091no—
164#135—2184no—
165C2×Q16—1091no—
166C2×SD16—1091no—
167C2×SD16—1091no—
168C2×D8—1091no—
169C2×M16—1092no—
170C2×M16—1092no—
171C2×M16—2184no—
172C22×C8—1091no—
173C22×C8C2×D10911yescharacteristic
174#123—2181no—
175#123—2181no—
176C22⋊Q8—1092no—
177C22⋊Q8—1092no—
178#121—1092no—
179#121—1092no—
180#120—2181no—
181#120—2181no—
182C4×D4—2182no—
183C4×D4—2182no—
184#117—1091no—
185#117C2×D10911yescharacteristic
186#107—1092no—
187#106—1092no—
188#103—1092no—
189#103C2×D10912yescomplemented
190#102—1092no—
191#102C2×D10912yescomplemented
192C24—1091no—
193#41—22no—
194#41—22no—
195#41—2182no—
196#41—2182no—
197C2×Q8—1091no—
198C2×Q8C22×D10911yescharacteristic
199C2×D4—21no—
200C2×D4—1091no—
201C2×D4—2182no—
202C2×D4—2182no—
203C2×D4—2181no—
204C2×D4C22×D10911yescharacteristic
205C22×C4—21no—
206C22×C4—1091no—
207C22×C4—1092no—
208C22×C4—1092no—
209C22×C4—2181no—
210C22×C4—2182no—
211C22×C4—2182no—
212C22×C4C22×D10911yescharacteristic
213Q16—2182no—
214SD16—2182no—
215SD16—2182no—
216D8—2182no—
217M16—2184no—
218M16—2184no—
219C2×C8—22no—
220C2×C8—1094no—
221C2×C8—1091no—
222C2×C8—1091no—
223C2×C8C22×D10912yes—
224C4⋊C4—1092no—
225C4⋊C4—2182no—
226C4⋊C4—2182no—
227C4⋊C4C22×D10912yes—
228C22⋊C4—21no—
229C22⋊C4—2182no—
230C22⋊C4—2182no—
231C22⋊C4—2182no—
232C22⋊C4—2182no—
233C22⋊C4—2181no—
234C42—21no—
235C42—2181no—
236C23—21no—
237C23—1092no—
238C23—4362no—
239C23—2181no—
240C23—2182no—
241C23D4×D10911yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—2181no—
245Q8—2181no—
246D4—21no—
247D4—21no—
248D4—42no—
249D4—4364no—
250D4—4364no—
251D4—4362no—
252D4—2181no—
253D4—2181no—
254C2×C4—22no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—1092no—
258C2×C4—4362no—
259C2×C4—2182no—
260C2×C4—2181no—
261C2×C4—2182no—
262C2×C4—2184no—
263C2×C4—2182no—
264C2×C4—2182no—
265C2×C4D4×D10912yes—
266C2×C4D4×D10911yescharacteristic
267C2×C4#1318202912yes—
268C2×C4C23×D10911yescharacteristic
269C8—22no—
270C8—1092no—
271C8—1092no—
272C22—22no—
273C22—41no—
274C22—42no—
275C22—4364no—
276C22—4361no—
277C22—4362no—
278C22—2182no—
279C22—2182no—
280C22—2182no—
281C22#1681267212yes—
282C22C2×D4×D10911yescharacteristic
283C4—41no—
284C4—42no—
285C4—4361no—
286C4—4362no—
287C4—2182no—
288C4#1681267212yes—
289C4C2×D4×D10911yescharacteristic
290C4C2×#1318202911yescharacteristic
291C2—22no—
292C2—81no—
293C2—8721no—
294C2—4362no—
295C2C2×#1681267211yescharacteristic
296C2#2029582312yes—
subgroupG/H
centerC2×C4D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#416×C109C2
socleC22×C109C22×D4
hypercenter#416D109
hypocenterC109#3054
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#16812719C2semidirect
2. C22×#13182012C2semidirect
3. #20295538C2semidirect
4. #20295533C2semidirect
5. #20295283C2semidirect
6. #20295278C2semidirect
7. #20295275C2semidirect
8. #20295271C2semidirect
9. #20295062C2semidirect
10. #20295023C2semidirect
11. #416×C109C2semidirect
15. C2×#13182019C22semidirect
16. C2×#13182017C22semidirect
17. C2×#13182016C22semidirect
18. C2×#13182014C22semidirect
34. #16812632C22semidirect
35. #16812630C22semidirect
36. #16812624C22semidirect
37. #16812621C22semidirect
41. #103×C109C22semidirect
42. #102×C109C22semidirect
63. C2×#11724784D4semidirect
149. C109#3054semidirect
160. #416D109semidirect
189. #103C2×D109semidirect
191. #102C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127192
C22×#131820122
#202955382
#202955332
#202952832
#202952782
#202952752
#202952712
#202950622
#202950232
#416×C1092
#3054109