Finite Group Databasebeta

← #28372663

Subgroups — #28372663

Order 13952.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 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×#16812732C211yesmaximal, characteristic, complemented
2C2×M16×D109C211yesmaximal, characteristic, complemented
3#20295538C211yesmaximal, characteristic
4C2×#16812679C211yesmaximal, characteristic, complemented
5#20295285C212yesmaximal, complemented
6#20295273C212yesmaximal, complemented
7#20295268C212yesmaximal, complemented
8#20295256C212yesmaximal, complemented
9#20295034C211yesmaximal, characteristic, complemented
10#20295025C211yesmaximal, characteristic, complemented
11#428×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13C22×Dic218C2211yescharacteristic
14C22×C4×D109C2211yescharacteristic
15#16812732—24no—
16C2×Dic436C2212yescomplemented
17C2×#13182018C2212yescomplemented
18C2×#13182013C2211yescharacteristic
19C2×#13182012C2212yes—
20M16×D109—24no—
21C2×C8×D109C2212yes—
22#16812683—21no—
23#16812679—21no—
24#16812679—21no—
25#16812679C2212yes—
26#16812679C2211yescharacteristic
27#16812679C2212yes—
28#16812679C2211yescharacteristic
29#16812674C2211yescharacteristic
30#16812674C2211yescharacteristic
31#16812669—21no—
32C4×D436—21no—
33C4×D436—21no—
34C2×C4⋊Dic109—21no—
35C2×#13182006—21no—
36#16812622C2212yescomplemented
37#16812622C2212yescomplemented
38#16812622C2212yescomplemented
39#16812619C2212yescomplemented
40C2×M16×C109C2211yescharacteristic
41#117×C109C2211yescharacteristic
42C2×C4⋊C4×C109C2211yescharacteristic
43#107×C109C2212yescomplemented
44#106×C109C2212yescomplemented
45C23×D109D411yescharacteristic
46#13182029—22no—
47#13182029—22no—
48C2×Dic218—22no—
49C2×Dic218—22no—
50C2×Dic218—21no—
51C2×Dic218C2312yes—
52C2×Dic218C2311yescharacteristic
53C2×D436—21no—
54C2×D436—21no—
55C2×D436C2311yescharacteristic
56C2×C22⋊D109—21no—
57C22×Dic109—21no—
58C22×Dic109D411yescharacteristic
59C2×C4×D109—22no—
60C2×C4×D109—22no—
61C2×C4×D109—21no—
62C2×C4×D109D411yescharacteristic
63C2×C4×D109D411yescharacteristic
64C2×C4×D109C2311yescharacteristic
65C2×C4×D109C2311yescharacteristic
66Dic436—24no—
67#13182018—24no—
68#13182013—22no—
69#13182012—24no—
70C2×#11724784C2312yes—
71C8×D109—24no—
72C4⋊Dic109—22no—
73C4⋊Dic109—22no—
74C4⋊Dic109—21no—
75C4⋊Dic109—21no—
76#13182006—22no—
77#13182006—22no—
78#13182006—21no—
79#13182006—21no—
80#13182006—21no—
81#13182006—21no—
82C22×C4×C109—21no—
83C22×C4×C109C2311yescharacteristic
84M16×C109D414yescomplemented
85C2×C8×C109C2312yes—
86C4⋊C4×C109—21no—
87C4⋊C4×C109C2311yescharacteristic
88C4⋊C4×C109C2312yes—
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
99C2×C4×C109—21no—
100C2×C4×C109—21no—
101C2×C4×C109—22no—
102C2×C4×C109—42no—
103C2×C4×C109C2×D411yescharacteristic
104C2×C4×C109C2×D412yes—
105C2×C4×C109#4112yes—
106C2×C4×C109C2411yescharacteristic
107C22⋊D109—42no—
108C2×Dic109—22no—
109C2×Dic109—21no—
110C2×Dic109—22no—
111C2×Dic109—42no—
112C2×Dic109C2×D411yescharacteristic
113C2×Dic109C2×D411yescharacteristic
114D436—21no—
115D436—21no—
116D436—42no—
117D436—42no—
118C4×D109—22no—
119C4×D109—22no—
120C4×D109—21no—
121C4×D109—21no—
122C4×D109—21no—
123C4×D109—21no—
124C4×D109—42no—
125Dic218—22no—
126Dic218—21no—
127Dic218—22no—
128Dic218—21no—
129Dic218—42no—
130C8×C109C2×D414yes—
131#11724784—22no—
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—41no—
147C4×C109—41no—
148C4×C109—42no—
149C4×C109#12112yes—
150C4×C109C22×D411yescharacteristic
151C4×C109C2×#4111yescharacteristic
152Dic109—21no—
153Dic109—21no—
154Dic109—42no—
155Dic109—41no—
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#428D10911yescharacteristic, complemented
174#419—1091no—
175#417—1091no—
176C2×#41—1091no—
177C22×Q8—1091no—
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—2181no—
201C2×C4⋊C4C2×D10911yescharacteristic
202C2×C22⋊C4—2181no—
203#107C2×D10912yescomplemented
204#106C2×D10912yescomplemented
205#103—1092no—
206#103—1092no—
207#103—1092no—
208#102—1092no—
209C24—1091no—
210#41—2182no—
211#41—2182no—
212C2×Q8—1092no—
213C2×Q8—1091no—
214C2×Q8—2182no—
215C2×Q8—2182no—
216C2×Q8—2181no—
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—2182no—
235M16—2184no—
236M16D43614yescomplemented
237C2×C8—1092no—
238C2×C8—2184no—
239C2×C8C22×D10912yes—
240C4⋊C4—21no—
241C4⋊C4—2182no—
242C4⋊C4—2182no—
243C4⋊C4—2181no—
244C4⋊C4—2181no—
245C4⋊C4C22×D10912yes—
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—4362no—
264Q8—2182no—
265Q8—2181no—
266Q8—2182no—
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—2181no—
282C2×C4—2182no—
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×C4C2×D43612yes—
292C2×C4#1318203012yes—
293C2×C4C23×D10911yescharacteristic
294C8—2182no—
295C8C2×D43614yes—
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#1681267412yes—
309C22C2×D4×D10911yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—4361no—
314C4—4362no—
315C4—2181no—
316C4—2181no—
317C4#1681267412yes—
318C4C22×D43611yescharacteristic
319C4C2×#1318203011yescharacteristic
320C2—22no—
321C2—8721no—
322C2—4361no—
323C2—2182no—
324C2C2×#1681267411yescharacteristic
325C2#137×D10911yescharacteristic
326C2#2029582211yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#428×C109C2
socleC22×C109C22×D4
prime radical#20295538C2
hypercenter#428D109
hypocenterC109#3056
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#16812732C2semidirect
2. C2×M16×D109C2semidirect
4. C2×#16812679C2semidirect
5. #20295285C2semidirect
6. #20295273C2semidirect
7. #20295268C2semidirect
8. #20295256C2semidirect
9. #20295034C2semidirect
10. #20295025C2semidirect
11. #428×C109C2semidirect
16. C2×Dic436C22semidirect
17. C2×#13182018C22semidirect
36. #16812622C22semidirect
37. #16812622C22semidirect
38. #16812622C22semidirect
39. #16812619C22semidirect
43. #107×C109C22semidirect
44. #106×C109C22semidirect
84. M16×C109D4semidirect
145. C2×D109#137semidirect
164. C109#3056semidirect
173. #428D109semidirect
203. #107C2×D109semidirect
204. #106C2×D109semidirect
236. M16D436semidirect

Maximal subgroups

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

subgroupindex
C2×#168127322
C2×M16×D1092
#202955382
C2×#168126792
#202952852
#202952732
#202952682
#202952562
#202950342
#202950252
#428×C1092
#3056109