Finite Group Databasebeta

← #28372660

Subgroups — #28372660

Order 13952.

496 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 135 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×#16812730C211yesmaximal, characteristic, complemented
2C2×M16×D109C211yesmaximal, characteristic, complemented
3#20295538C211yesmaximal, characteristic
4C2×#16812675C211yesmaximal, characteristic, complemented
5#20295283C212yesmaximal, complemented
6#20295275C212yesmaximal, complemented
7#20295266C212yesmaximal, complemented
8#20295258C212yesmaximal, complemented
9#20295069C211yesmaximal, characteristic, complemented
10#20295028C211yesmaximal, characteristic, complemented
11#419×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13C22×C4×D109C2211yescharacteristic
14#16812730—24no—
15C2×#13182019C2212yescomplemented
16C2×#13182016C2212yescomplemented
17C2×#13182013C2211yescharacteristic
18C2×#13182012C2212yes—
19M16×D109—24no—
20C2×C8×D109C2212yes—
21#16812683—21no—
22#16812679C2211yescharacteristic
23#16812679C2211yescharacteristic
24#16812675—21no—
25#16812675—21no—
26#16812675C2212yes—
27#16812675C2212yes—
28#16812674C2211yescharacteristic
29#16812674C2211yescharacteristic
30#16812669—21no—
31C4×D436—21no—
32C4×D436—21no—
33C2×C4⋊Dic109C2211yescharacteristic
34C2×#13182008—21no—
35C2×#13182006—21no—
36#16812632C2212yescomplemented
37#16812630C2212yescomplemented
38#16812624C2212yescomplemented
39#16812621C2212yescomplemented
40C22×Q8×C109C2211yescharacteristic
41C2×M16×C109C2211yescharacteristic
42#117×C109C2211yescharacteristic
43#103×C109C2212yescomplemented
44#103×C109C2212yescomplemented
45C23×D109D411yescharacteristic
46#13182029—22no—
47#13182029—22no—
48C2×Dic218C2311yescharacteristic
49C2×D436—21no—
50C2×D436—21no—
51C2×D436C2311yescharacteristic
52C2×C22⋊D109—21no—
53C22×Dic109—21no—
54C22×Dic109D411yescharacteristic
55C2×C4×D109—22no—
56C2×C4×D109—22no—
57C2×C4×D109—21no—
58C2×C4×D109D411yescharacteristic
59C2×C4×D109D411yescharacteristic
60C2×C4×D109C2311yescharacteristic
61C2×C4×D109C2311yescharacteristic
62#13182019—24no—
63#13182016—24no—
64#13182013D414yescomplemented
65#13182012—24no—
66C2×#11724784C2312yes—
67C8×D109—24no—
68C4⋊Dic109—21no—
69C4⋊Dic109—21no—
70C4⋊Dic109—21no—
71C4⋊Dic109C2312yes—
72#13182008—22no—
73#13182008—22no—
74#13182006—22no—
75#13182006—22no—
76#13182006—21no—
77#13182006—21no—
78#13182006—21no—
79#13182006—21no—
80C2×Q8×C109—21no—
81C2×Q8×C109—22no—
82C2×Q8×C109—22no—
83C2×Q8×C109C2312yes—
84C22×C4×C109—21no—
85C22×C4×C109C2311yescharacteristic
86M16×C109—22no—
87C2×C8×C109C2312yes—
88C4⋊C4×C109C2311yescharacteristic
89C4⋊C4×C109C2311yescharacteristic
90C22⋊C4×C109—21no—
91C42×C109—21no—
92C23×C109C2×D411yescharacteristic
93C22×D109—22no—
94C22×D109—22no—
95C22×D109—21no—
96C22×D109—41no—
97C22×D109C2×D411yescharacteristic
98C22×D109C2×D411yescharacteristic
99Q8×C109—22no—
100Q8×C109—22no—
101Q8×C109—42no—
102C2×C4×C109—21no—
103C2×C4×C109—21no—
104C2×C4×C109—22no—
105C2×C4×C109—42no—
106C2×C4×C109C2×D411yescharacteristic
107C2×C4×C109C2×D412yes—
108C2×C4×C109#4112yes—
109C2×C4×C109C2411yescharacteristic
110C22⋊D109—42no—
111C2×Dic109—22no—
112C2×Dic109—22no—
113C2×Dic109—21no—
114C2×Dic109—42no—
115C2×Dic109C2×D411yescharacteristic
116C2×Dic109C2×D411yescharacteristic
117D436—21no—
118D436—21no—
119D436—42no—
120D436—42no—
121C4×D109—22no—
122C4×D109—22no—
123C4×D109—21no—
124C4×D109—21no—
125C4×D109—21no—
126C4×D109—21no—
127C4×D109—42no—
128Dic218—21no—
129Dic218—21no—
130C8×C109—22no—
131#11724784C2×D414yes—
132C22×C109—21no—
133C22×C109—21no—
134C22×C109#12112yes—
135C22×C109C22×D411yescharacteristic
136C2×D109—21no—
137C2×D109—21no—
138C2×D109—21no—
139C2×D109—21no—
140C2×D109—21no—
141C2×D109—44no—
142C2×D109—41no—
143C2×D109—41no—
144C2×D109—41no—
145C2×D109#13712yescomplemented
146C4×C109—42no—
147C4×C109—41no—
148C4×C109—41no—
149C4×C109#12112yes—
150C4×C109C22×D411yescharacteristic
151C4×C109C2×#4111yescharacteristic
152Dic109—21no—
153Dic109—21no—
154Dic109—41no—
155Dic109—42no—
156C2×C109—22no—
157C2×C109C2×#12111yescharacteristic
158C2×C109C2×#13711yescharacteristic
159C2×C109#57511yescharacteristic
160D109—22no—
161D109—41no—
162D109—81no—
163#3056—1091nomaximal, Sylow
164C109#305611yescharacteristic, complemented, Sylow
165C2×#137—1091no—
166C22×M16—1091no—
167#525—1091no—
168C2×C22⋊Q8—1091no—
169#470—1092no—
170#468—1092no—
171#467—1092no—
172#465—1092no—
173#428—1091no—
174#419D10911yescharacteristic, complemented
175#417—1091no—
176C2×#41—1091no—
177C22×Q8C2×D10911yescharacteristic
178C23×C4—1091no—
179#137—2184no—
180C2×Q16—1092no—
181C2×SD16—1092no—
182C2×M16—1091no—
183C2×M16—1092no—
184C2×M16—2184no—
185C2×M16C2×D10911yescharacteristic
186C22×C8—1092no—
187#123—2181no—
188C22⋊Q8—1092no—
189C22⋊Q8—1092no—
190C22⋊Q8—1091no—
191C22⋊Q8—1091no—
192C22⋊Q8—2181no—
193C22⋊Q8—2181no—
194#121—1091no—
195#121—1091no—
196#120—2181no—
197C4×D4—2181no—
198C4×D4—2181no—
199#117C2×D10911yescharacteristic
200C2×C4⋊C4—1091no—
201C2×C4⋊C4—2181no—
202C2×C22⋊C4—2181no—
203#107—1092no—
204#106—1092no—
205#103—1092no—
206#103C2×D10912yescomplemented
207#103C2×D10912yescomplemented
208#102—1092no—
209C24—1091no—
210#41—2182no—
211#41—2182no—
212C2×Q8—21no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—1091no—
216C2×Q8C22×D10912yes—
217C2×D4—1091no—
218C2×D4—2181no—
219C2×D4—2181no—
220C2×D4—2181no—
221C22×C4—21no—
222C22×C4—1091no—
223C22×C4—1091no—
224C22×C4—1091no—
225C22×C4—1091no—
226C22×C4—1091no—
227C22×C4—2181no—
228C22×C4—2181no—
229C22×C4—2182no—
230C22×C4—2182no—
231C22×C4C22×D10911yescharacteristic
232Q16—2184no—
233SD16—2184no—
234M16—22no—
235M16—1094no—
236M16—2184no—
237C2×C8—1092no—
238C2×C8—2184no—
239C2×C8C22×D10912yes—
240C4⋊C4—1092no—
241C4⋊C4—2182no—
242C4⋊C4—2182no—
243C4⋊C4—2181no—
244C4⋊C4—2181no—
245C4⋊C4—2181no—
246C4⋊C4C22×D10911yescharacteristic
247C4⋊C4C22×D10911yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—2181no—
250C22⋊C4—2181no—
251C22⋊C4—2182no—
252C22⋊C4—2182no—
253C22⋊C4—2181no—
254C22⋊C4—2181no—
255C42—21no—
256C23—1091no—
257C23—1091no—
258C23—4361no—
259C23—2182no—
260C23—2181no—
261C23—2182no—
262C23D4×D10911yescharacteristic
263Q8—22no—
264Q8—22no—
265Q8—42no—
266Q8—2181no—
267Q8—2181no—
268D4—4362no—
269D4—4362no—
270D4—4362no—
271D4—2181no—
272D4—2181no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—1091no—
278C2×C4—1091no—
279C2×C4—4362no—
280C2×C4—4362no—
281C2×C4—2182no—
282C2×C4—2181no—
283C2×C4—2182no—
284C2×C4—2182no—
285C2×C4—2182no—
286C2×C4—2181no—
287C2×C4—2181no—
288C2×C4—2181no—
289C2×C4—2181no—
290C2×C4D4×D10911yescharacteristic
291C2×C4D4×D10912yes—
292C2×C4#1318202912yes—
293C2×C4C23×D10911yescharacteristic
294C8—22no—
295C8—1094no—
296C22—21no—
297C22—21no—
298C22—1092no—
299C22—4364no—
300C22—4361no—
301C22—4361no—
302C22—4361no—
303C22—2181no—
304C22—2181no—
305C22—2181no—
306C22—2181no—
307C22—2181no—
308C22#1681267212yes—
309C22C2×D4×D10911yescharacteristic
310C4—42no—
311C4—41no—
312C4—41no—
313C4—4361no—
314C4—4362no—
315C4—2181no—
316C4—2181no—
317C4#1681267212yes—
318C4C2×D4×D10911yescharacteristic
319C4C2×#1318202911yescharacteristic
320C2—22no—
321C2—8721no—
322C2—4361no—
323C2—2182no—
324C2C2×#1681267211yescharacteristic
325C2#137×D10911yescharacteristic
326C2#2029582511yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#419×C109C2
socleC22×C109C22×D4
prime radical#20295538C2
hypercenter#419D109
hypocenterC109#3056
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C2×#16812730C2semidirect
2. C2×M16×D109C2semidirect
4. C2×#16812675C2semidirect
5. #20295283C2semidirect
6. #20295275C2semidirect
7. #20295266C2semidirect
8. #20295258C2semidirect
9. #20295069C2semidirect
10. #20295028C2semidirect
11. #419×C109C2semidirect
15. C2×#13182019C22semidirect
16. C2×#13182016C22semidirect
36. #16812632C22semidirect
37. #16812630C22semidirect
38. #16812624C22semidirect
39. #16812621C22semidirect
43. #103×C109C22semidirect
44. #103×C109C22semidirect
64. #13182013D4semidirect
145. C2×D109#137semidirect
164. C109#3056semidirect
174. #419D109semidirect
206. #103C2×D109semidirect
207. #103C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127302
C2×M16×D1092
#202955382
C2×#168126752
#202952832
#202952752
#202952662
#202952582
#202950692
#202950282
#419×C1092
#3056109