Finite Group Databasebeta

← #28372507

Subgroups — #28372507

Order 13952.

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

normal subgroups only

This group has 584 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×#16812723C212yesmaximal, complemented
2#127×D109C211yesmaximal, characteristic, complemented
3C4×#13182028C211yesmaximal, characteristic, complemented
4#20295227C211yesmaximal, characteristic, complemented
5#20295225C212yesmaximal, complemented
6#20295223C211yesmaximal, characteristic, complemented
7#20295221C211yesmaximal, characteristic, complemented
8#20295219C212yesmaximal, complemented
9#20295048C211yesmaximal, characteristic, complemented
10#20295029C212yesmaximal, complemented
11#459×C109C211yesmaximal, characteristic, complemented
12C2×#13182028C2211yescharacteristic
13C2×D4×D109—21no—
14C2×D4×D109—21no—
15C2×D4×D109C2212yes—
16#16812723—28no—
17C2×#13182018C2212yescomplemented
18C2×#13182017C2212yescomplemented
19C2×#13182014C2212yescomplemented
20C2×#13182012C2212yes—
21#16812696C2211yescharacteristic
22#16812695—21no—
23#16812695—21no—
24#16812695C2212yes—
25C4×Dic218C2211yescharacteristic, complemented
26#16812664C2211yescharacteristic, complemented
27D4×Dic109C2211yescharacteristic, complemented
28C4×C22⋊D109—21no—
29#16812659—21no—
30#16812657C2211yescharacteristic
31#16812655—21no—
32C2×C4×Dic109—21no—
33C42×D109C2211yescharacteristic
34C4⋊#11724784C2211yescharacteristic, complemented
35#16812628C2212yescomplemented
36#16812621C2212yescomplemented
37#16812620C2212yescomplemented
38#16812619C2212yescomplemented
39C2×D8×C109C2212yescomplemented
40#127×C109C2211yescharacteristic
41C4×D4×C109C2211yescharacteristic, complemented
42C4⋊C8×C109C2211yescharacteristic, complemented
43#102×C109C2212yescomplemented
44C23×D109—22no—
45#13182028—22no—
46#13182028D414yescomplemented
47C2×Dic218C2311yescharacteristic
48C2×D436—21no—
49C2×D436—21no—
50C2×D436C2312yes—
51C2×C22⋊D109—21no—
52C2×C22⋊D109—22no—
53C2×C22⋊D109—22no—
54C2×C22⋊D109—22no—
55D4×D109—24no—
56D4×D109—24no—
57D4×D109—44no—
58D4×D109—44no—
59C22×Dic109—21no—
60C22×Dic109—21no—
61C2×C4×D109D411yescharacteristic
62C2×C4×D109D411yescharacteristic
63C2×C4×D109C2311yescharacteristic
64#13182018—24no—
65#13182017—24no—
66#13182014—24no—
67#13182012—24no—
68C2×#11724784C2312yes—
69C4⋊Dic109C2311yescharacteristic
70#13182008—21no—
71#13182008—21no—
72C22⋊Dic109—21no—
73#13182006—21no—
74C4×Dic109—21no—
75C4×Dic109—21no—
76C4×Dic109—21no—
77C4×Dic109D411yescharacteristic
78C4×Dic109D411yescharacteristic
79C4×Dic109C2311yescharacteristic
80C2×D4×C109—21no—
81C2×D4×C109—21no—
82C2×D4×C109C2312yes—
83C2×D4×C109C2311yescharacteristic
84C22×C4×C109—21no—
85D8×C109—24no—
86C2×C8×C109C2312yes—
87C4⋊C4×C109C2311yescharacteristic
88C22⋊C4×C109—21no—
89C42×C109C2311yescharacteristic
90C23×C109—21no—
91C23×C109—22no—
92C22×D109—22no—
93C22×D109—44no—
94C22×D109—44no—
95C22×D109—44no—
96C22×D109C2×D411yescharacteristic
97D4×C109—22no—
98D4×C109—21no—
99D4×C109—22no—
100D4×C109—42no—
101D4×C109—42no—
102D4×C109C2×D412yescomplemented
103C2×C4×C109—21no—
104C2×C4×C109—42no—
105C2×C4×C109C2×D411yescharacteristic
106C2×C4×C109C2×D411yescharacteristic
107C2×C4×C109C2411yescharacteristic
108C22⋊D109—24no—
109C22⋊D109—44no—
110C22⋊D109—44no—
111C22⋊D109—44no—
112C2×Dic109—21no—
113C2×Dic109—24no—
114C2×Dic109—21no—
115C2×Dic109—42no—
116C2×Dic109C2×D411yescharacteristic
117C2×Dic109C2×D411yescharacteristic
118C2×Dic109C2×D411yescharacteristic
119D436—22no—
120D436—22no—
121D436—42no—
122D436—42no—
123C4×D109—22no—
124C4×D109—22no—
125C4×D109C2×D412yes—
126C4×D109#4112yes—
127Dic218—21no—
128Dic218C2×D412yescomplemented
129C8×C109—22no—
130#11724784—22no—
131C22×C109—22no—
132C22×C109—41no—
133C22×C109—41no—
134C22×C109—42no—
135C22×C109—42no—
136C22×C109—42no—
137C22×C109C22×D411yescharacteristic
138C2×D109—21no—
139C2×D109—21no—
140C2×D109—42no—
141C2×D109—42no—
142C2×D109—42no—
143C2×D109—88no—
144C2×D109#12112yes—
145C4×C109—21no—
146C4×C109—41no—
147C4×C109AGL(1,ℤ/8)12yescomplemented
148C4×C109C22×D411yescharacteristic
149C4×C109C2×#4111yescharacteristic
150Dic109—22no—
151Dic109—21no—
152Dic109—41no—
153Dic109#12112yes—
154Dic109AGL(1,ℤ/8)12yescomplemented
155C2×C109—42no—
156C2×C109—82no—
157C2×C109C2×#12111yescharacteristic
158C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
159C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
160D109—22no—
161D109—82no—
162#3038—1091nomaximal, Sylow
163C109#303811yescharacteristic, complemented, Sylow
164C2×AGL(1,ℤ/8)—1092no—
165C2×#127—1091no—
166C4×#41—1091no—
167#460—1091no—
168#460—1091no—
169#460—1092no—
170#459—1092no—
171#459—1091no—
172#459D10911yescharacteristic, complemented
173#423—1091no—
174#418—1092no—
175C2×#41—1091no—
176C22×D4—1092no—
177C22×D4—2181no—
178C22×D4—2181no—
179AGL(1,ℤ/8)—2188no—
180C2×SD16—1092no—
181C2×SD16—1092no—
182C2×D8—1092no—
183C2×D8C2×D10912yescomplemented
184C2×M16—1092no—
185#127—1092no—
186#127—1091no—
187#127—2181no—
188#127—2181no—
189#127C2×D10911yescharacteristic
190C4×Q8—1091no—
191C4×Q8—1091no—
192C4×D4—1091no—
193C4×D4—2181no—
194C4×D4—2181no—
195C4×D4C2×D10911yescharacteristic, complemented
196#117—1091no—
197#117—2181no—
198C2×C42—1091no—
199C2×C42—2181no—
200C4⋊C8—1092no—
201C4⋊C8—1091no—
202C4⋊C8C2×D10911yescharacteristic, complemented
203#102—1092no—
204#102—1092no—
205#102—1092no—
206#102C2×D10912yescomplemented
207C24—2182no—
208#41—1094no—
209#41—2182no—
210C2×Q8—1091no—
211C2×D4—21no—
212C2×D4—21no—
213C2×D4—1092no—
214C2×D4—4364no—
215C2×D4—4364no—
216C2×D4—2184no—
217C2×D4—2184no—
218C2×D4—2182no—
219C2×D4—2182no—
220C2×D4—2182no—
221C2×D4—2181no—
222C2×D4—2181no—
223C2×D4—2181no—
224C2×D4C22×D10912yes—
225C2×D4C22×D10911yescharacteristic
226C22×C4—21no—
227C22×C4—1091no—
228C22×C4—1091no—
229C22×C4—1091no—
230C22×C4—2181no—
231C22×C4—2181no—
232SD16—2184no—
233SD16—2184no—
234D8—24no—
235D8—2184no—
236M16—2184no—
237C2×C8—1092no—
238C2×C8C22×D10912yes—
239C4⋊C4—1091no—
240C4⋊C4—2181no—
241C4⋊C4—2181no—
242C4⋊C4C22×D10911yescharacteristic
243C22⋊C4—21no—
244C22⋊C4—2181no—
245C22⋊C4—2181no—
246C42—1091no—
247C42—1091no—
248C42—1091no—
249C42—2181no—
250C42—2181no—
251C42—2181no—
252C42C22×D10911yescharacteristic
253C23—21no—
254C23—22no—
255C23—1091no—
256C23—4364no—
257C23—4364no—
258C23—4364no—
259C23—2182no—
260Q8—1092no—
261Q8—2181no—
262D4—22no—
263D4—21no—
264D4—22no—
265D4—42no—
266D4—42no—
267D4—4362no—
268D4—4364no—
269D4—4364no—
270D4—4364no—
271D4—4362no—
272D4—2184no—
273D4—2182no—
274D4—2182no—
275D4D4×D10912yescomplemented
276C2×C4—21no—
277C2×C4—42no—
278C2×C4—1091no—
279C2×C4—1091no—
280C2×C4—1092no—
281C2×C4—1092no—
282C2×C4—1091no—
283C2×C4—4362no—
284C2×C4—2182no—
285C2×C4—2184no—
286C2×C4—2181no—
287C2×C4—2181no—
288C2×C4—2182no—
289C2×C4D4×D10911yescharacteristic
290C2×C4D4×D10911yescharacteristic
291C2×C4C23×D10911yescharacteristic
292C8—22no—
293C8—2182no—
294C22—22no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—42no—
300C22—8728no—
301C22—1092no—
302C22—4362no—
303C22—4362no—
304C22—4362no—
305C22—2181no—
306C22—2181no—
307C22C2×D4×D10911yescharacteristic
308C4—21no—
309C4—41no—
310C4—1092no—
311C4—1092no—
312C4—4361no—
313C4—2182no—
314C4—2181no—
315C4#1681272312yescomplemented
316C4C2×D4×D10911yescharacteristic
317C4#41×D10911yescharacteristic
318C2—42no—
319C2—82no—
320C2—8722no—
321C2—2182no—
322C2#121×D10911yescharacteristic
323C2AGL(1,ℤ/8)×D10911yescharacteristic
324C2C2×#1681272311yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#459×C109C2
socleC22×C109C22×D4
hypercenter#459D109
hypocenterC109#3038
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#16812723C2semidirect
2. #127×D109C2semidirect
3. C4×#13182028C2semidirect
4. #20295227C2semidirect
5. #20295225C2semidirect
6. #20295223C2semidirect
7. #20295221C2semidirect
8. #20295219C2semidirect
9. #20295048C2semidirect
10. #20295029C2semidirect
11. #459×C109C2semidirect
17. C2×#13182018C22semidirect
18. C2×#13182017C22semidirect
19. C2×#13182014C22semidirect
25. C4×Dic218C22semidirect
26. #16812664C22semidirect
27. D4×Dic109C22semidirect
34. C4⋊#11724784C22semidirect
35. #16812628C22semidirect
36. #16812621C22semidirect
37. #16812620C22semidirect
38. #16812619C22semidirect
39. C2×D8×C109C22semidirect
41. C4×D4×C109C22semidirect
42. C4⋊C8×C109C22semidirect
43. #102×C109C22semidirect
46. #13182028D4semidirect
102. D4×C109C2×D4semidirect
128. Dic218C2×D4semidirect
147. C4×C109AGL(1,ℤ/8)semidirect
154. Dic109AGL(1,ℤ/8)semidirect
163. C109#3038semidirect
172. #459D109semidirect
183. C2×D8C2×D109semidirect
195. C4×D4C2×D109semidirect
202. C4⋊C8C2×D109semidirect
206. #102C2×D109semidirect
275. D4D4×D109semidirect
315. C4#16812723semidirect

Maximal subgroups

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

subgroupindex
C2×#168127232
#127×D1092
C4×#131820282
#202952272
#202952252
#202952232
#202952212
#202952192
#202950482
#202950292
#459×C1092
#3038109