Finite Group Databasebeta

← #28489631

Subgroups — #28489631

Order 15104.

512 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 85 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#21434214C212yesmaximal, complemented
2#21425035C211yesmaximal, characteristic, complemented
3#21422707C212yesmaximal, complemented
4#21421966C211yesmaximal, characteristic, complemented
5C59×#3245C211yesmaximal, characteristic
6C2×C4×Dic118C2211yescharacteristic
7#16837713—22no—
8#16837694—22no—
9#16837599—22no—
10#16837540—22no—
11#16837502—24no—
12#16837459C2212yes—
13C2×#14273638C2211yescharacteristic
14#16837156—22no—
15#16837142—24no—
16C59×#532C2212yes—
17C59×#474—22no—
18C59×#466—22no—
19C59×#459—22no—
20C2×C4⋊C8×C59C2211yescharacteristic
21C22×Dic118D411yescharacteristic
22C2×#11745670—24no—
23C22×C59⋊C8—21no—
24C4×Dic118—22no—
25C4×Dic118—22no—
26C2×C4⋊Dic59D411yescharacteristic
27C2×Dic59⋊C4—21no—
28C2×C4×Dic59—21no—
29#14273660—24no—
30C4⋊(C59⋊C8)—22no—
31#14273650—24no—
32#14273648—24no—
33C22⋊(C59⋊C8)—22no—
34#14273638—22no—
35#14273638D414yes—
36C2×#41×C59—22no—
37C2×D8×C59—22no—
38C22×C8×C59D412yes—
39C4⋊Q8×C59D412yes—
40#127×C59D412yes—
41#124×C59—22no—
42#121×C59—22no—
43#121×C59—22no—
44C4×D4×C59—22no—
45C4×D4×C59—22no—
46C2×C42×C59C2311yescharacteristic
47#107×C59—22no—
48C4⋊C8×C59—22no—
49#102×C59—24no—
50#41×C59—84no—
51C2×Q8×C59—22no—
52C2×D4×C59—22no—
53C2×D4×C59—22no—
54C2×D4×C59—22no—
55C2×D4×C59—42no—
56C22×C4×C59—42no—
57C22×C4×C59C2×D411yescharacteristic
58C22×C4×C59C2×D411yescharacteristic
59C22×C4×C59#4111yescharacteristic
60C22×Dic59—21no—
61C22×Dic59—21no—
62C2×Dic118—22no—
63C2×Dic118—42no—
64D8×C59—82no—
65C2×C8×C59—22no—
66C2×C8×C59—42no—
67C4⋊C4×C59—22no—
68C4⋊C4×C59—22no—
69C22⋊C4×C59—42no—
70C22⋊C4×C59—42no—
71C42×C59C2×D412yes—
72C42×C59C2×D412yes—
73#11745670—48no—
74C4⋊Dic59—22no—
75Dic59⋊C4—42no—
76C4×Dic59—41no—
77C4×Dic59—41no—
78C2×C59⋊C8—22no—
79C2×C59⋊C8—24no—
80C2×C59⋊C8—41no—
81C2×C59⋊C8—41no—
82C23×C59—42no—
83C23×C59#12111yescharacteristic
84Q8×C59—82no—
85D4×C59—44no—
86D4×C59—42no—
87D4×C59—42no—
88D4×C59—82no—
89D4×C59—84no—
90C2×C4×C59—22no—
91C2×C4×C59—21no—
92C2×C4×C59—21no—
93C2×C4×C59—41no—
94C2×C4×C59—42no—
95C2×C4×C59—84no—
96C2×C4×C59#12012yes—
97C2×C4×C59#12011yescharacteristic
98C2×C4×C59#12112yes—
99C2×C4×C59#12711yescharacteristic
100C2×Dic59—22no—
101C2×Dic59—41no—
102C2×Dic59—41no—
103C2×Dic59—41no—
104C2×Dic59—42no—
105Dic118—24no—
106Dic118—42no—
107C8×C59—42no—
108C59⋊C8—42no—
109C59⋊C8—44no—
110#10028—591nomaximal, Sylow
111C22×C59—21no—
112C22×C59—21no—
113C22×C59—82no—
114C22×C59—82no—
115C22×C59—82no—
116C22×C59#39211yescharacteristic
117C22×C59#47212yes—
118C4×C59—21no—
119C4×C59—22no—
120C4×C59—22no—
121C4×C59—82no—
122C4×C59#45511yescharacteristic
123C4×C59#45611yescharacteristic
124Dic59—42no—
125Dic59—81no—
126#3245D5911yescharacteristic
127#3112—592no—
128#1985—591no—
129#1530—592no—
130#1290—591no—
131C2×C59—22no—
132C2×C59—162no—
133C2×C59#200711yescharacteristic
134C2×C59#201011yescharacteristic
135C2×C59#201811yescharacteristic
136#532C2×D5912yes—
137C2×C4×Q8—591no—
138#480—1182no—
139#477—1182no—
140#474—22no—
141#466—22no—
142#460—1182no—
143#459—22no—
144#448—1182no—
145#440—1184no—
146#432—592no—
147C2×C4⋊C8C2×D5911yescharacteristic
148C2×#97—591no—
149#336—1182no—
150#329—1184no—
151C59#1002811yescharacteristic, complemented, Sylow
152C2×#41—22no—
153C22×Q8—591no—
154C2×SD16—1184no—
155C2×D8—22no—
156C22×C8—1181no—
157C22×C8C22⋊D5912yes—
158C4⋊Q8C22⋊D5912yes—
159#127C22⋊D5912yes—
160#124—22no—
161#121—22no—
162#121—22no—
163C4×Q8—1182no—
164C4×Q8—1182no—
165C4×D4—22no—
166C4×D4—22no—
167C2×C4⋊C4—591no—
168C2×C4⋊C4—1181no—
169C2×C42—1181no—
170C2×C42C22×D5911yescharacteristic
171#107—22no—
172#107—1184no—
173C4⋊C8—22no—
174C4⋊C8—1182no—
175#103—1184no—
176#102—24no—
177#102—1184no—
178C22⋊C8—1182no—
179#97—594no—
180#97—1182no—
181#41—84no—
182C2×Q8—22no—
183C2×Q8—1182no—
184C2×Q8—2362no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—42no—
189C22×C4—42no—
190C22×C4—1181no—
191C22×C4—1181no—
192C22×C4D4×D5911yescharacteristic
193C22×C4#1174568711yescharacteristic
194C22×C4C2×C22⋊D5911yescharacteristic
195SD16—2368no—
196D8—82no—
197C2×C8—22no—
198C2×C8—42no—
199C2×C8—1182no—
200C2×C8—1184no—
201C2×C8—2361no—
202C2×C8—2361no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—1182no—
206C4⋊C4—2362no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C42—2361no—
210C42—2361no—
211C42D4×D5912yes—
212C42C2×C22⋊D5912yes—
213C23—42no—
214C23#1427369811yescharacteristic
215Q8—82no—
216Q8—1184no—
217Q8—2362no—
218D4—44no—
219D4—42no—
220D4—42no—
221D4—82no—
222D4—84no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—84no—
229C2×C4—1182no—
230C2×C4—2362no—
231C2×C4—2361no—
232C2×C4—2361no—
233C2×C4—2361no—
234C2×C4#1427369512yes—
235C2×C4#1427369611yescharacteristic
236C2×C4#1427369912yes—
237C2×C4#1427372311yescharacteristic
238C8—42no—
239C8—2362no—
240C8—2364no—
241C22—21no—
242C22—21no—
243C22—82no—
244C22—82no—
245C22—82no—
246C22#1683731011yescharacteristic
247C22#1683766912yes—
248C4—22no—
249C4—22no—
250C4—21no—
251C4—82no—
252C4—4721no—
253C4—2362no—
254C4#1683758011yescharacteristic
255C4#1683758511yescharacteristic
256C2—22no—
257C2—162no—
258C2#2142514311yescharacteristic
259C2#2142516011yescharacteristic
260C2#2142520311yescharacteristic
subgroupG/H
centerC22#16837310
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#3245C2
socleC22×C59#392
hypercenter#3245D59
hypocenterC59#10028
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #21434214C2semidirect
2. #21425035C2semidirect
3. #21422707C2semidirect
4. #21421966C2semidirect
151. C59#10028semidirect

Maximal subgroups

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

subgroupindex
#214342142
#214250352
#214227072
#214219662
C59×#32452
#1002859