Finite Group Databasebeta

← #28372713

Subgroups — #28372713

Order 13952.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16812727C211yesmaximal, characteristic, complemented
2C2×#16812707C211yesmaximal, characteristic, complemented
3#20295607C211yesmaximal, characteristic, complemented
4#20295595C211yesmaximal, characteristic, complemented
5#20295280C212yesmaximal, complemented
6#20295273C212yesmaximal, complemented
7#20295263C212yesmaximal, complemented
8#20295256C212yesmaximal, complemented
9#20295020C211yesmaximal, characteristic, complemented
10C2×#16812619C211yesmaximal, characteristic, complemented
11#428×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13C2×#13182029C2211yescharacteristic
14C22×D436C2211yescharacteristic
15#16812727—24no—
16C2×#13182018C2212yes—
17C2×D872C2212yes—
18#16812707—24no—
19C2×#13182012C2212yescomplemented
20C22×#11724784C2211yescharacteristic
21C2×C8×D109C2212yescomplemented
22#16812696—21no—
23#16812687—21no—
24#16812679—21no—
25#16812679C2212yescomplemented
26#16812674—21no—
27#16812674C2212yes—
28#16812674C2212yes—
29#16812674C2212yescomplemented
30#16812673—22no—
31#16812669—22no—
32#16812622C2211yescharacteristic, complemented
33#16812622C2211yescharacteristic, complemented
34#16812619C2212yes—
35#16812619C2212yes—
36#16812619C2211yescharacteristic, complemented
37#16812619C2211yescharacteristic, complemented
38C2×M16×C109C2211yescharacteristic
39#117×C109C2211yescharacteristic
40C2×C4⋊C4×C109C2211yescharacteristic
41#107×C109C2212yescomplemented
42#106×C109C2212yescomplemented
43C23×D109—21no—
44#13182029—22no—
45#13182029—22no—
46#13182029—22no—
47#13182029—22no—
48C2×Dic218D411yescharacteristic
49C2×Dic218C2311yescharacteristic
50C2×D436—22no—
51C2×D436—22no—
52C2×D436—21no—
53C2×D436—22no—
54C2×D436—22no—
55C2×D436—22no—
56C2×D436D411yescharacteristic
57C2×D436C2312yes—
58C2×D436C2311yescharacteristic
59C2×C22⋊D109—21no—
60C2×C22⋊D109D412yes—
61C2×C4×D109—21no—
62C2×C4×D109C2312yes—
63#13182018—24no—
64D872—24no—
65#13182012—24no—
66C2×#11724784—22no—
67C2×#11724784C2311yescharacteristic
68C2×#11724784C2311yescharacteristic
69C8×D109—24no—
70C4⋊Dic109—22no—
71#13182006—22no—
72#13182006—22no—
73#13182006—22no—
74#13182006—22no—
75C22×C4×C109—21no—
76C22×C4×C109C2311yescharacteristic
77M16×C109D414yescomplemented
78C2×C8×C109C2312yes—
79C4⋊C4×C109—21no—
80C4⋊C4×C109C2311yescharacteristic
81C4⋊C4×C109C2312yes—
82C4⋊C4×C109C2311yescharacteristic
83C22⋊C4×C109—21no—
84C42×C109—21no—
85C23×C109C2×D411yescharacteristic
86C22×D109—21no—
87C22×D109—22no—
88C22×D109—42no—
89C22×D109—81no—
90C22×D109—81no—
91C22×D109C2×D412yes—
92C2×C4×C109—21no—
93C2×C4×C109—21no—
94C2×C4×C109—22no—
95C2×C4×C109—81no—
96C2×C4×C109C2×D411yescharacteristic
97C2×C4×C109C2×D412yes—
98C2×C4×C109#4112yes—
99C2×C4×C109C2411yescharacteristic
100C22⋊D109—24no—
101C22⋊D109—42no—
102C2×Dic109—21no—
103C2×Dic109C2×D412yes—
104D436—22no—
105D436—21no—
106D436—22no—
107D436—22no—
108D436—21no—
109D436—42no—
110D436—82no—
111D436—82no—
112D436—82no—
113C4×D109—22no—
114C4×D109—22no—
115C4×D109—42no—
116Dic218—22no—
117Dic218—21no—
118Dic218—21no—
119C8×C109C2×D414yes—
120#11724784—21no—
121#11724784—21no—
122C22×C109—21no—
123C22×C109—21no—
124C22×C109#12112yes—
125C22×C109C22×D411yescharacteristic
126C2×D109—22no—
127C2×D109—22no—
128C2×D109—22no—
129C2×D109—41no—
130C2×D109—42no—
131C2×D109—41no—
132C2×D109—42no—
133C2×D109—42no—
134C2×D109—41no—
135C2×D109—84no—
136C4×C109—41no—
137C4×C109—42no—
138C4×C109—41no—
139C4×C109#12112yes—
140C4×C109C22×D411yescharacteristic
141C4×C109C2×#4111yescharacteristic
142Dic109—22no—
143Dic109—41no—
144C2×C109—22no—
145C2×C109C2×#12111yescharacteristic
146C2×C109#57611yescharacteristic
147C2×C109#57711yescharacteristic
148D109—42no—
149D109—81no—
150D109—82no—
151#3062—1091nomaximal, Sylow
152C109#306211yescharacteristic, complemented, Sylow
153C2×AGL(1,ℤ/8)—1091no—
154C2×#131—1091no—
155#537—1091no—
156#535—1091no—
157#469—1092no—
158#468—1092no—
159#466—1092no—
160#465—1092no—
161#428D10911yescharacteristic, complemented
162#416—1091no—
163C2×#102—1091no—
164C2×#41—1091no—
165C2×#41—1091no—
166C22×D4—1091no—
167AGL(1,ℤ/8)—2184no—
168C2×SD16—1092no—
169C2×D8—1092no—
170#131—2184no—
171C2×M16—1092no—
172C2×M16C2×D10911yescharacteristic
173C22×C8—1091no—
174C22×C8—1092no—
175#127—2181no—
176#124—2181no—
177C22⋊Q8—1092no—
178C22⋊Q8—2181no—
179#121—1092no—
180#121—1092no—
181#121—1092no—
182#121—2181no—
183#121—2182no—
184#120—2182no—
185#117C2×D10911yescharacteristic
186C2×C4⋊C4C2×D10911yescharacteristic
187#107C2×D10912yescomplemented
188#106C2×D10912yescomplemented
189#103—1091no—
190#103—1091no—
191#102—1092no—
192#102—1092no—
193#102—1091no—
194#102—1091no—
195C24—2181no—
196#41—2182no—
197#41—2182no—
198#41—2182no—
199#41—2182no—
200C2×Q8—1091no—
201C2×Q8—1091no—
202C2×D4—1092no—
203C2×D4—1091no—
204C2×D4—1091no—
205C2×D4—1092no—
206C2×D4—2182no—
207C2×D4—2182no—
208C2×D4—2182no—
209C2×D4—2181no—
210C2×D4—2181no—
211C2×D4—2182no—
212C2×D4—2182no—
213C22×C4—21no—
214C22×C4—1092no—
215C22×C4—2181no—
216C22×C4C22×D10911yescharacteristic
217SD16—2184no—
218D8—2184no—
219M16—2184no—
220M16D43614yescomplemented
221C2×C8—1091no—
222C2×C8—1091no—
223C2×C8—2182no—
224C2×C8—2184no—
225C2×C8C22×D10912yes—
226C4⋊C4—21no—
227C4⋊C4—2182no—
228C4⋊C4C22×D10912yes—
229C4⋊C4C22×D10911yescharacteristic
230C4⋊C4C22×D10911yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—2182no—
233C22⋊C4—2182no—
234C22⋊C4—2182no—
235C22⋊C4—2182no—
236C42—21no—
237C23—8721no—
238C23—8721no—
239C23—1092no—
240C23—4362no—
241C23—2181no—
242C23—2182no—
243C23D4×D10911yescharacteristic
244Q8—2181no—
245Q8—2181no—
246Q8—2182no—
247D4—8722no—
248D4—8722no—
249D4—8722no—
250D4—4362no—
251D4—4362no—
252D4—2184no—
253D4—2182no—
254D4—2181no—
255D4—2182no—
256D4—2182no—
257D4—2181no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—81no—
262C2×C4—1092no—
263C2×C4—4362no—
264C2×C4—2181no—
265C2×C4—2182no—
266C2×C4—2182no—
267C2×C4D4×D10911yescharacteristic
268C2×C4C2×D43612yes—
269C2×C4#1318203012yes—
270C2×C4C23×D10911yescharacteristic
271C8—2181no—
272C8—2181no—
273C8C2×D43614yes—
274C22—21no—
275C22—21no—
276C22—8724no—
277C22—4361no—
278C22—4362no—
279C22—4361no—
280C22—4362no—
281C22—4362no—
282C22—4361no—
283C22—2182no—
284C22—2182no—
285C22—2182no—
286C22#1681267412yes—
287C22C2×D4×D10911yescharacteristic
288C4—42no—
289C4—41no—
290C4—41no—
291C4—4361no—
292C4—2182no—
293C4#1681267412yes—
294C4C22×D43611yescharacteristic
295C4C2×#1318203011yescharacteristic
296C2—22no—
297C2—8722no—
298C2—8721no—
299C2—4362no—
300C2C2×#1681267411yescharacteristic
301C2#2029583011yescharacteristic
302C2#2029583511yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#428×C109C2
socleC22×C109C22×D4
hypercenter#428D109
hypocenterC109#3062
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×#16812727C2semidirect
2. C2×#16812707C2semidirect
3. #20295607C2semidirect
4. #20295595C2semidirect
5. #20295280C2semidirect
6. #20295273C2semidirect
7. #20295263C2semidirect
8. #20295256C2semidirect
9. #20295020C2semidirect
10. C2×#16812619C2semidirect
11. #428×C109C2semidirect
19. C2×#13182012C22semidirect
21. C2×C8×D109C22semidirect
25. #16812679C22semidirect
29. #16812674C22semidirect
32. #16812622C22semidirect
33. #16812622C22semidirect
36. #16812619C22semidirect
37. #16812619C22semidirect
41. #107×C109C22semidirect
42. #106×C109C22semidirect
77. M16×C109D4semidirect
152. C109#3062semidirect
161. #428D109semidirect
187. #107C2×D109semidirect
188. #106C2×D109semidirect
220. M16D436semidirect

Maximal subgroups

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

subgroupindex
C2×#168127272
C2×#168127072
#202956072
#202955952
#202952802
#202952732
#202952632
#202952562
#202950202
C2×#168126192
#428×C1092
#3062109