Finite Group Databasebeta

← #28372652

Subgroups — #28372652

Order 13952.

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

normal subgroups only

This group has 548 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×#16812727C211yesmaximal, characteristic, complemented
2C2×M16×D109C211yesmaximal, characteristic, complemented
3#20295538C211yesmaximal, characteristic, complemented
4C2×#16812674C211yesmaximal, characteristic, complemented
5#20295280C212yesmaximal, complemented
6#20295273C212yesmaximal, complemented
7#20295263C212yesmaximal, complemented
8#20295256C212yesmaximal, complemented
9#20295030C211yesmaximal, characteristic, complemented
10#20295025C211yesmaximal, characteristic, complemented
11#428×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13C22×D436—21no—
14C22×D436C2211yescharacteristic
15C22×C4×D109C2211yescharacteristic
16#16812727—24no—
17C2×#13182018C2212yescomplemented
18C2×D872C2212yescomplemented
19C2×#13182013C2211yescharacteristic
20C2×#13182012C2212yescomplemented
21M16×D109—24no—
22C2×C8×D109C2212yescomplemented
23#16812683—21no—
24#16812679C2211yescharacteristic, complemented
25#16812679C2211yescharacteristic, complemented
26#16812674—21no—
27#16812674—21no—
28#16812674C2212yes—
29#16812674C2212yes—
30#16812674C2211yescharacteristic, complemented
31#16812674C2211yescharacteristic, complemented
32#16812669—21no—
33C4×D436—21no—
34C4×D436—21no—
35C2×#13182006—21no—
36#16812622C2212yescomplemented
37#16812619C2212yescomplemented
38#16812619C2212yescomplemented
39#16812619C2212yescomplemented
40C2×M16×C109C2211yescharacteristic
41#117×C109C2211yescharacteristic
42C2×C4⋊C4×C109C2211yescharacteristic
43#107×C109C2212yescomplemented
44#106×C109C2212yescomplemented
45C23×D109—21no—
46C23×D109D411yescharacteristic
47#13182029—22no—
48#13182029—22no—
49C2×Dic218C2311yescharacteristic
50C2×D436—22no—
51C2×D436—22no—
52C2×D436—21no—
53C2×D436—21no—
54C2×D436—22no—
55C2×D436—22no—
56C2×D436—21no—
57C2×D436—44no—
58C2×D436C2312yes—
59C2×D436C2311yescharacteristic
60C2×C22⋊D109—21no—
61C22×Dic109D411yescharacteristic
62C2×C4×D109—22no—
63C2×C4×D109—22no—
64C2×C4×D109—21no—
65C2×C4×D109D411yescharacteristic
66C2×C4×D109D411yescharacteristic
67C2×C4×D109C2311yescharacteristic
68C2×C4×D109C2311yescharacteristic
69#13182018—24no—
70D872—24no—
71#13182013—22no—
72#13182012—24no—
73C2×#11724784C2312yes—
74C8×D109—24no—
75C4⋊Dic109—21no—
76C4⋊Dic109—21no—
77#13182006—21no—
78#13182006—21no—
79#13182006—22no—
80#13182006—22no—
81#13182006—21no—
82#13182006—21no—
83C22×C4×C109—21no—
84C22×C4×C109C2311yescharacteristic
85M16×C109D414yescomplemented
86C2×C8×C109C2312yes—
87C4⋊C4×C109—21no—
88C4⋊C4×C109C2311yescharacteristic
89C4⋊C4×C109C2312yes—
90C4⋊C4×C109C2311yescharacteristic
91C22⋊C4×C109—21no—
92C42×C109—21no—
93C23×C109C2×D411yescharacteristic
94C22×D109—22no—
95C22×D109—22no—
96C22×D109—21no—
97C22×D109—22no—
98C22×D109—41no—
99C22×D109—42no—
100C22×D109—42no—
101C22×D109—42no—
102C22×D109C2×D411yescharacteristic
103C22×D109C2×D411yescharacteristic
104C2×C4×C109—21no—
105C2×C4×C109—21no—
106C2×C4×C109—22no—
107C2×C4×C109—42no—
108C2×C4×C109C2×D411yescharacteristic
109C2×C4×C109C2×D412yes—
110C2×C4×C109#4112yes—
111C2×C4×C109C2411yescharacteristic
112C22⋊D109—42no—
113C2×Dic109—22no—
114C2×Dic109—21no—
115C2×Dic109C2×D411yescharacteristic
116C2×Dic109C2×D411yescharacteristic
117D436—21no—
118D436—22no—
119D436—22no—
120D436—21no—
121D436—42no—
122D436—44no—
123D436—42no—
124D436—44no—
125D436—42no—
126C4×D109—22no—
127C4×D109—22no—
128C4×D109—21no—
129C4×D109—21no—
130C4×D109—21no—
131C4×D109—21no—
132C4×D109—42no—
133Dic218—21no—
134Dic218—21no—
135C8×C109C2×D414yescomplemented
136#11724784—22no—
137C22×C109—21no—
138C22×C109—21no—
139C22×C109#12112yes—
140C22×C109C22×D411yescharacteristic
141C2×D109—21no—
142C2×D109—21no—
143C2×D109—21no—
144C2×D109—21no—
145C2×D109—21no—
146C2×D109—44no—
147C2×D109—41no—
148C2×D109—42no—
149C2×D109—42no—
150C2×D109—41no—
151C2×D109—41no—
152C2×D109—42no—
153C2×D109—84no—
154C2×D109AGL(1,ℤ/8)12yescomplemented
155C4×C109—42no—
156C4×C109—41no—
157C4×C109—41no—
158C4×C109#12112yes—
159C4×C109C22×D411yescharacteristic
160C4×C109C2×#4111yescharacteristic
161Dic109—21no—
162Dic109—21no—
163Dic109—41no—
164C2×C109—22no—
165C2×C109C2×#12111yescharacteristic
166C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
167C2×C109#57511yescharacteristic
168D109—22no—
169D109—41no—
170D109—81no—
171D109—82no—
172#3055—1091nomaximal, Sylow
173C109#305511yescharacteristic, complemented, Sylow
174C2×AGL(1,ℤ/8)—1091no—
175C22×M16—1091no—
176#525—1091no—
177C2×#121—1091no—
178#469—1092no—
179#468—1092no—
180#466—1092no—
181#465—1092no—
182#428D10911yescharacteristic, complemented
183#418—1091no—
184#417—1091no—
185C2×#41—1091no—
186C22×D4—1091no—
187C22×D4—2181no—
188C23×C4—1091no—
189AGL(1,ℤ/8)—2184no—
190C2×SD16—1092no—
191C2×D8—1092no—
192C2×M16—1091no—
193C2×M16—1092no—
194C2×M16—2184no—
195C2×M16C2×D10911yescharacteristic
196C22×C8—1092no—
197#123—2181no—
198C22⋊Q8—1091no—
199C22⋊Q8—1091no—
200#121—1092no—
201#121—1092no—
202#121—1091no—
203#121—1091no—
204#121—2181no—
205#121—2181no—
206#120—2181no—
207C4×D4—2181no—
208C4×D4—2181no—
209#117C2×D10911yescharacteristic
210C2×C4⋊C4C2×D10911yescharacteristic
211C2×C22⋊C4—2181no—
212#107C2×D10912yescomplemented
213#106C2×D10912yescomplemented
214#103—1092no—
215#102—1092no—
216#102—1092no—
217#102—1092no—
218C24—1091no—
219C24—2181no—
220#41—2182no—
221#41—2182no—
222C2×Q8—1091no—
223C2×D4—1091no—
224C2×D4—1092no—
225C2×D4—4364no—
226C2×D4—2181no—
227C2×D4—2181no—
228C2×D4—2182no—
229C2×D4—2182no—
230C2×D4—2181no—
231C2×D4—2181no—
232C2×D4—2182no—
233C2×D4—2182no—
234C22×C4—21no—
235C22×C4—1091no—
236C22×C4—1091no—
237C22×C4—1091no—
238C22×C4—1091no—
239C22×C4—1091no—
240C22×C4—2181no—
241C22×C4—2182no—
242C22×C4—2182no—
243C22×C4C22×D10911yescharacteristic
244SD16—2184no—
245D8—2184no—
246M16—2182no—
247M16—2184no—
248M16D43614yescomplemented
249C2×C8—1092no—
250C2×C8—2184no—
251C2×C8C22×D10912yes—
252C4⋊C4—21no—
253C4⋊C4—2181no—
254C4⋊C4—2181no—
255C4⋊C4C22×D10911yescharacteristic
256C4⋊C4C22×D10911yescharacteristic
257C4⋊C4C22×D10912yes—
258C22⋊C4—21no—
259C22⋊C4—2182no—
260C22⋊C4—2182no—
261C22⋊C4—2181no—
262C22⋊C4—2181no—
263C22⋊C4—2181no—
264C22⋊C4—2181no—
265C42—21no—
266C23—1091no—
267C23—1091no—
268C23—4362no—
269C23—4362no—
270C23—4361no—
271C23—4362no—
272C23—2182no—
273C23—2181no—
274C23—2182no—
275C23—2182no—
276C23D4×D10911yescharacteristic
277Q8—2181no—
278Q8—2181no—
279D4—4364no—
280D4—4362no—
281D4—4364no—
282D4—4362no—
283D4—4362no—
284D4—4362no—
285D4—2181no—
286D4—2182no—
287D4—2182no—
288D4—2181no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—42no—
293C2×C4—1091no—
294C2×C4—1091no—
295C2×C4—4362no—
296C2×C4—2181no—
297C2×C4—2182no—
298C2×C4—2182no—
299C2×C4—2182no—
300C2×C4—2181no—
301C2×C4—2181no—
302C2×C4—2181no—
303C2×C4—2181no—
304C2×C4D4×D10911yescharacteristic
305C2×C4C2×D43612yes—
306C2×C4#1318203012yes—
307C2×C4C23×D10911yescharacteristic
308C8—2182no—
309C8C2×D43614yescomplemented
310C22—21no—
311C22—21no—
312C22—8724no—
313C22—1092no—
314C22—4364no—
315C22—4361no—
316C22—4362no—
317C22—4362no—
318C22—4361no—
319C22—4361no—
320C22—4362no—
321C22—2181no—
322C22—2181no—
323C22—2181no—
324C22—2181no—
325C22—2181no—
326C22#1681267412yes—
327C22C2×D4×D10911yescharacteristic
328C4—42no—
329C4—41no—
330C4—41no—
331C4—4361no—
332C4—2181no—
333C4—2181no—
334C4#1681267412yes—
335C4C22×D43611yescharacteristic
336C4C2×#1318203011yescharacteristic
337C2—22no—
338C2—8722no—
339C2—8721no—
340C2—4361no—
341C2—2182no—
342C2C2×#1681267411yescharacteristic
343C2AGL(1,ℤ/8)×D10911yescharacteristic
344C2#2029582511yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#428×C109C2
socleC22×C109C22×D4
hypercenter#428D109
hypocenterC109#3055
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×M16×D109C2semidirect
3. #20295538C2semidirect
4. C2×#16812674C2semidirect
5. #20295280C2semidirect
6. #20295273C2semidirect
7. #20295263C2semidirect
8. #20295256C2semidirect
9. #20295030C2semidirect
10. #20295025C2semidirect
11. #428×C109C2semidirect
17. C2×#13182018C22semidirect
18. C2×D872C22semidirect
20. C2×#13182012C22semidirect
22. C2×C8×D109C22semidirect
24. #16812679C22semidirect
25. #16812679C22semidirect
30. #16812674C22semidirect
31. #16812674C22semidirect
36. #16812622C22semidirect
37. #16812619C22semidirect
38. #16812619C22semidirect
39. #16812619C22semidirect
43. #107×C109C22semidirect
44. #106×C109C22semidirect
85. M16×C109D4semidirect
135. C8×C109C2×D4semidirect
154. C2×D109AGL(1,ℤ/8)semidirect
173. C109#3055semidirect
182. #428D109semidirect
212. #107C2×D109semidirect
213. #106C2×D109semidirect
248. M16D436semidirect
309. C8C2×D436semidirect

Maximal subgroups

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

subgroupindex
C2×#168127272
C2×M16×D1092
#202955382
C2×#168126742
#202952802
#202952732
#202952632
#202952562
#202950302
#202950252
#428×C1092
#3055109