Finite Group Databasebeta

← #28489667

Subgroups — #28489667

Order 15104.

584 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#21434542C211yesmaximal, characteristic
2#21434127C212yesmaximal, complemented
3#21422494C212yesmaximal, complemented
4#21421927C211yesmaximal, characteristic, complemented
5C59×#1985C211yesmaximal, characteristic, complemented
6C2×#14273749—22no—
7#16837952C2212yes—
8#16837796—22no—
9#16837791—22no—
10#16837720—22no—
11#16837687—22no—
12#16837602—22no—
13#16837541—22no—
14C4×#11745671—24no—
15#16837455C2212yes—
16#16837129—24no—
17C2×C4×Q8×C59C2211yescharacteristic
18C2×C4⋊C8×C59C2211yescharacteristic
19C2×#97×C59C2211yescharacteristic
20C59×#336—22no—
21C2×#11745685—22no—
22#14273749—44no—
23#14273749—44no—
24C2×Dic236—22no—
25C2×#11745660—24no—
26C2×#11745671—24no—
27C2×D472—22no—
28#14273727D412yes—
29#14273724D412yes—
30#14273715—22no—
31#14273701—22no—
32#14273701—22no—
33C4×D236—22no—
34C4×D236—22no—
35#14273658—24no—
36C4⋊(C59⋊C8)—22no—
37#14273651—24no—
38#14273647—24no—
39C22⋊(C59⋊C8)—22no—
40C4×C59⋊C8D414yes—
41C22×Q8×C59D411yescharacteristic
42C22×C8×C59—21no—
43C22×C8×C59D412yes—
44C4×Q8×C59—22no—
45C4×Q8×C59—22no—
46C2×C4⋊C4×C59—21no—
47C2×C4⋊C4×C59D411yescharacteristic
48C2×C42×C59—21no—
49C2×C42×C59C2311yescharacteristic
50C4⋊C8×C59—22no—
51#97×C59—22no—
52C2×Q8×C59—22no—
53C2×Q8×C59—42no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59C2×D411yescharacteristic
57C22×C4×C59C2×D411yescharacteristic
58C22×C4×C59#4111yescharacteristic
59C2×C22⋊D59—42no—
60#11745685—44no—
61#11745685—44no—
62C2×D236—22no—
63C2×D236—22no—
64C2×D236—22no—
65C2×C4×D59—42no—
66C2×Dic118—22no—
67C2×C8×C59—22no—
68C2×C8×C59—22no—
69C2×C8×C59—41no—
70C2×C8×C59—42no—
71C2×C8×C59SD1614yescomplemented
72C4⋊C4×C59—22no—
73C4⋊C4×C59—42no—
74C42×C59—41no—
75C42×C59—41no—
76C42×C59C2×D412yes—
77C42×C59C2×D412yes—
78#11745671—48no—
79(C2×D59)⋊C4—42no—
80(C2×D59)⋊C4—42no—
81C4⋊Dic59—22no—
82C4⋊Dic59—22no—
83C2×C59⋊C8—24no—
84Dic236—44no—
85D472—44no—
86#11745660—48no—
87C23×C59#12111yescharacteristic
88C22×D59—42no—
89Q8×C59—24no—
90Q8×C59—42no—
91C2×C4×C59—22no—
92C2×C4×C59—41no—
93C2×C4×C59—41no—
94C2×C4×C59—41no—
95C2×C4×C59—41no—
96C2×C4×C59—42no—
97C2×C4×C59—41no—
98C2×C4×C59#12011yescharacteristic
99C2×C4×C59#12012yes—
100C2×C4×C59#12112yes—
101C2×C4×C59#12711yescharacteristic
102C2×C4×C59C2×SD1612yes—
103C2×C4×C59AGL(1,ℤ/8)12yes—
104C22⋊D59—84no—
105C2×Dic59—42no—
106D236—44no—
107D236—44no—
108D236—44no—
109C4×D59—82no—
110C4×D59—82no—
111Dic118—44no—
112C8×C59—24no—
113C8×C59—81no—
114C59⋊C8—44no—
115#10034—591nomaximal, Sylow
116C22×C59—21no—
117C22×C59—21no—
118C22×C59#39211yescharacteristic
119C22×C59#46012yes—
120C2×D59—82no—
121C2×D59—82no—
122C2×D59—82no—
123C4×C59—22no—
124C4×C59—22no—
125C4×C59—42no—
126C4×C59—81no—
127C4×C59#45012yes—
128C4×C59#49211yescharacteristic
129C4×C59#49611yescharacteristic
130Dic59—82no—
131#3150—591no—
132#3103—592no—
133#1985D5911yescharacteristic, complemented
134#1480—592no—
135#1275—591no—
136C2×C59—22no—
137C2×C59#201011yescharacteristic
138C2×C59#201411yescharacteristic
139C2×C59#201711yescharacteristic
140D59—162no—
141C2×#135—1182no—
142#532—592no—
143C2×C4×Q8C2×D5911yescharacteristic
144#497—1182no—
145#496—1182no—
146#481—1182no—
147#475—1182no—
148#460—1182no—
149#448—1182no—
150C4×SD16—1184no—
151#431—592no—
152C2×C4⋊C8C2×D5911yescharacteristic
153C2×#97C2×D5911yescharacteristic
154#336—22no—
155D4⋊C8—1184no—
156C59#1003411yescharacteristic, complemented, Sylow
157C2×#41—1182no—
158C22×Q8D23611yescharacteristic
159#135—2364no—
160#135—2364no—
161C2×Q16—1182no—
162C2×SD16—1184no—
163C2×SD16—1184no—
164C2×D8—1182no—
165C22×C8—21no—
166C22×C8C22⋊D5912yes—
167C4⋊Q8—592no—
168#127—592no—
169#124—1182no—
170#121—1182no—
171#121—1182no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×D4—1182no—
175C4×D4—1182no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4D23611yescharacteristic
178C2×C42—21no—
179C2×C42C22×D5911yescharacteristic
180#106—1184no—
181C4⋊C8—22no—
182C4⋊C8—1182no—
183#103—1184no—
184#102—1184no—
185C22⋊C8—1182no—
186#97—22no—
187C4×C8—594no—
188#41—2364no—
189#41—2364no—
190C2×Q8—22no—
191C2×Q8—42no—
192C2×Q8—1182no—
193C2×D4—1182no—
194C2×D4—1182no—
195C2×D4—1182no—
196C2×D4—2362no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—2362no—
200C22×C4C2×D23611yescharacteristic
201C22×C4#1174568511yescharacteristic
202C22×C4C2×C22⋊D5911yescharacteristic
203Q16—2364no—
204SD16—2368no—
205SD16—2368no—
206D8—2364no—
207C2×C8—22no—
208C2×C8—22no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—1184no—
212C2×C8#1174567114yescomplemented
213C4⋊C4—22no—
214C4⋊C4—42no—
215C4⋊C4—1182no—
216C4⋊C4—1182no—
217C22⋊C4—2362no—
218C22⋊C4—2362no—
219C42—41no—
220C42—41no—
221C42D4×D5912yes—
222C42D4×D5912yes—
223C23—2362no—
224C23#1427370111yescharacteristic
225Q8—24no—
226Q8—42no—
227Q8—2364no—
228D4—4724no—
229D4—2364no—
230D4—2364no—
231D4—2364no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—41no—
239C2×C4—4722no—
240C2×C4—4722no—
241C2×C4—2362no—
242C2×C4#1427369511yescharacteristic
243C2×C4#1427369712yes—
244C2×C4#1427370012yes—
245C2×C4#1427372311yescharacteristic
246C2×C4C2×#1174567112yes—
247C2×C4#1427375212yes—
248C8—24no—
249C8—81no—
250C8—2364no—
251C22—21no—
252C22—21no—
253C22—4722no—
254C22—4722no—
255C22—4722no—
256C22#1683731111yescharacteristic
257C22#1683760212yes—
258C4—22no—
259C4—22no—
260C4—42no—
261C4—81no—
262C4—4722no—
263C4#1683755212yes—
264C4#1683777611yescharacteristic
265C4#1683779211yescharacteristic
266C2—22no—
267C2—9442no—
268C2#2142516311yescharacteristic
269C2#2142518311yescharacteristic
270C2#2142520011yescharacteristic
subgroupG/H
centerC2×C4#14273723
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1985C2
socleC22×C59#392
prime radical#21434542C2
hypercenter#1985D59
hypocenterC59#10034
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. #21434127C2semidirect
3. #21422494C2semidirect
4. #21421927C2semidirect
5. C59×#1985C2semidirect
71. C2×C8×C59SD16semidirect
133. #1985D59semidirect
156. C59#10034semidirect
212. C2×C8#11745671semidirect

Maximal subgroups

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

subgroupindex
#214345422
#214341272
#214224942
#214219272
C59×#19852
#1003459