Finite Group Databasebeta

← #28489633

Subgroups — #28489633

Order 15104.

512 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 83 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#21435406C211yesmaximal, characteristic
2#21434218C212yesmaximal, complemented
3#21422706C212yesmaximal, complemented
4#21421966C211yesmaximal, characteristic, complemented
5C59×#1985C211yesmaximal, characteristic, complemented
6#16837952C2212yes—
7#16837715—22no—
8#16837697—22no—
9#16837683—22no—
10#16837638—22no—
11#16837602—22no—
12#16837598—22no—
13#16837541—22no—
14#16837500—24no—
15#16837459C2212yes—
16#16837141—24no—
17C2×C4×Q8×C59C2211yescharacteristic
18C2×C4⋊C8×C59C2211yescharacteristic
19C2×#97×C59C2211yescharacteristic
20C59×#336—22no—
21C2×#11745685—22no—
22C2×#11745671—24no—
23C2×D472—22no—
24#14273727D412yes—
25#14273724D412yes—
26#14273715—22no—
27#14273701—22no—
28#14273701—22no—
29C4×D236—22no—
30C4×D236—22no—
31#14273661—22no—
32#14273660—24no—
33C4⋊(C59⋊C8)—22no—
34#14273651—24no—
35#14273649—24no—
36#14273647—24no—
37C22⋊(C59⋊C8)—22no—
38#14273638D414yes—
39C22×Q8×C59D411yescharacteristic
40C22×C8×C59—21no—
41C22×C8×C59D412yes—
42C4×Q8×C59—22no—
43C4×Q8×C59—22no—
44C2×C4⋊C4×C59—21no—
45C2×C4⋊C4×C59D411yescharacteristic
46C2×C42×C59—21no—
47C2×C42×C59C2311yescharacteristic
48C4⋊C8×C59—22no—
49#97×C59—22no—
50C2×Q8×C59—22no—
51C2×Q8×C59—42no—
52C22×C4×C59—21no—
53C22×C4×C59—21no—
54C22×C4×C59C2×D411yescharacteristic
55C22×C4×C59C2×D411yescharacteristic
56C22×C4×C59#4111yescharacteristic
57C2×C22⋊D59—42no—
58#11745685—84no—
59C2×D236—22no—
60C2×D236—22no—
61C2×D236—22no—
62C2×C4×D59—42no—
63C2×Dic118—22no—
64C2×C8×C59—22no—
65C2×C8×C59—22no—
66C2×C8×C59—41no—
67C2×C8×C59—41no—
68C2×C8×C59—42no—
69C4⋊C4×C59—22no—
70C4⋊C4×C59—42no—
71C42×C59—41no—
72C42×C59—41no—
73C42×C59C2×D412yes—
74C42×C59C2×D412yes—
75#11745671—48no—
76(C2×D59)⋊C4—42no—
77(C2×D59)⋊C4—42no—
78C4⋊Dic59—22no—
79C4⋊Dic59—22no—
80C2×C59⋊C8—24no—
81D472—82no—
82C23×C59#12111yescharacteristic
83C22×D59—42no—
84Q8×C59—24no—
85Q8×C59—42no—
86C2×C4×C59—22no—
87C2×C4×C59—21no—
88C2×C4×C59—21no—
89C2×C4×C59—22no—
90C2×C4×C59—41no—
91C2×C4×C59—41no—
92C2×C4×C59—41no—
93C2×C4×C59—41no—
94C2×C4×C59—42no—
95C2×C4×C59#12011yescharacteristic
96C2×C4×C59#12012yes—
97C2×C4×C59#12112yes—
98C2×C4×C59#12711yescharacteristic
99C22⋊D59—84no—
100C2×Dic59—42no—
101D236—44no—
102D236—42no—
103D236—42no—
104D236—82no—
105C4×D59—84no—
106Dic118—82no—
107C8×C59—42no—
108C8×C59—42no—
109C59⋊C8—44no—
110#10028—591nomaximal, Sylow
111C22×C59—21no—
112C22×C59—21no—
113C22×C59#39211yescharacteristic
114C22×C59#47212yes—
115C2×D59—82no—
116C2×D59—82no—
117C2×D59—82no—
118C4×C59—21no—
119C4×C59—22no—
120C4×C59—22no—
121C4×C59—42no—
122C4×C59—81no—
123C4×C59#45511yescharacteristic
124C4×C59#45611yescharacteristic
125Dic59—82no—
126#3245—591no—
127#3112—592no—
128#1985D5911yescharacteristic, complemented
129#1530—592no—
130#1290—591no—
131C2×C59—22no—
132C2×C59#200711yescharacteristic
133C2×C59#201011yescharacteristic
134C2×C59#201811yescharacteristic
135D59—162no—
136#532—592no—
137C2×C4×Q8C2×D5911yescharacteristic
138#480—1182no—
139#477—1182no—
140#474—1182no—
141#466—1182no—
142#460—1182no—
143#459—1182no—
144#448—1182no—
145#440—1184no—
146#432—592no—
147C2×C4⋊C8C2×D5911yescharacteristic
148C2×#97C2×D5911yescharacteristic
149#336—22no—
150#329—1184no—
151C59#1002811yescharacteristic, complemented, Sylow
152C2×#41—1182no—
153C22×Q8D23611yescharacteristic
154C2×SD16—1184no—
155C2×D8—1182no—
156C22×C8—21no—
157C22×C8C22⋊D5912yes—
158C4⋊Q8—592no—
159#127—592no—
160#124—1182no—
161#121—1182no—
162#121—1182no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×D4—1182no—
166C4×D4—1182no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4D23611yescharacteristic
169C2×C42—21no—
170C2×C42C22×D5911yescharacteristic
171#107—1184no—
172#107—1182no—
173C4⋊C8—22no—
174C4⋊C8—1182no—
175#103—1184no—
176#102—1184no—
177#102—1184no—
178C22⋊C8—1182no—
179#97—22no—
180#97—594no—
181#41—4724no—
182C2×Q8—22no—
183C2×Q8—42no—
184C2×Q8—1182no—
185C2×D4—1182no—
186C2×D4—1182no—
187C2×D4—1182no—
188C2×D4—2362no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—2362no—
192C22×C4C2×D23611yescharacteristic
193C22×C4#1174568511yescharacteristic
194C22×C4C2×C22⋊D5911yescharacteristic
195SD16—2368no—
196D8—4722no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—42no—
202C2×C8—1184no—
203C4⋊C4—22no—
204C4⋊C4—42no—
205C4⋊C4—1182no—
206C4⋊C4—1182no—
207C22⋊C4—2362no—
208C22⋊C4—2362no—
209C42—41no—
210C42—41no—
211C42D4×D5912yes—
212C42D4×D5912yes—
213C23—2362no—
214C23#1427370111yescharacteristic
215Q8—24no—
216Q8—42no—
217Q8—4722no—
218D4—4722no—
219D4—4724no—
220D4—2364no—
221D4—2362no—
222D4—2362no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—22no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—4724no—
233C2×C4—2362no—
234C2×C4#1427369511yescharacteristic
235C2×C4#1427369712yes—
236C2×C4#1427370012yes—
237C2×C4#1427372311yescharacteristic
238C8—42no—
239C8—42no—
240C8—2364no—
241C22—21no—
242C22—21no—
243C22—4722no—
244C22—4722no—
245C22—4722no—
246C22#1683731111yescharacteristic
247C22#1683767212yes—
248C4—22no—
249C4—22no—
250C4—21no—
251C4—42no—
252C4—81no—
253C4—4722no—
254C4#1683757911yescharacteristic
255C4#1683758411yescharacteristic
256C2—22no—
257C2—9442no—
258C2#2142514211yescharacteristic
259C2#2142516311yescharacteristic
260C2#2142520511yescharacteristic
subgroupG/H
centerC22#16837311
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1985C2
socleC22×C59#392
prime radical#21435406C2
hypercenter#1985D59
hypocenterC59#10028
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1985

Complements

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

NH ≅ G/Ntype
2. #21434218C2semidirect
3. #21422706C2semidirect
4. #21421966C2semidirect
5. C59×#1985C2semidirect
128. #1985D59semidirect
151. C59#10028semidirect

Maximal subgroups

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

subgroupindex
#214354062
#214342182
#214227062
#214219662
C59×#19852
#1002859