Finite Group Databasebeta

← #28489559

Subgroups — #28489559

Order 15104.

528 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 105 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#21434135C212yesmaximal
2#21425077C211yesmaximal, characteristic
3#21422525C212yesmaximal
4#21421925C211yesmaximal, characteristic
5C59×#2399C211yesmaximal, characteristic
6#16837959C2211yescharacteristic
7#16837737—22no—
8#16837697—22no—
9#16837612—22no—
10#16837558—22no—
11C4×#11745672—24no—
12#16837455C2212yes—
13#16837426C2211yescharacteristic
14#16837162—22no—
15Q8⋊(C59⋊C8)—24no—
16C2×C4×Q8×C59C2212yes—
17C59×#398—22no—
18C59×#395—22no—
19C4×C4⋊C4×C59—22no—
20C59×#376C2211yescharacteristic
21C2×#11745672—24no—
22C2×#11745664—21no—
23#14273727D412yes—
24#14273719—21no—
25#14273706—22no—
26C4×Dic118—22no—
27#14273686—21no—
28#14273660—24no—
29C4⋊(C59⋊C8)—22no—
30C4⋊(C59⋊C8)—22no—
31#14273651—24no—
32#14273650—24no—
33C22⋊(C59⋊C8)—22no—
34C4×C59⋊C8D414yes—
35C22×Q8×C59D412yes—
36C4×Q8×C59—22no—
37C4×Q8×C59—22no—
38C4×Q8×C59—22no—
39C4×Q8×C59—22no—
40C2×C4⋊C4×C59—24no—
41C2×C4⋊C4×C59—22no—
42C2×C4⋊C4×C59D412yes—
43C2×C42×C59—22no—
44C2×C42×C59D412yes—
45C2×C42×C59C2311yescharacteristic
46#95×C59—22no—
47#95×C59—22no—
48#95×C59—22no—
49C2×Q8×C59—22no—
50C2×Q8×C59—22no—
51C2×Q8×C59—44no—
52C22×C4×C59—22no—
53C22×C4×C59—22no—
54C22×C4×C59—22no—
55C22×C4×C59C2×D412yes—
56C22×C4×C59C2×D411yescharacteristic
57C2×Dic118—22no—
58C4⋊C4×C59—22no—
59C4⋊C4×C59—22no—
60C4⋊C4×C59—48no—
61C4⋊C4×C59—42no—
62C4⋊C4×C59—42no—
63C42×C59—22no—
64C42×C59—42no—
65C42×C59—42no—
66C42×C59Q1614yescomplemented
67C42×C59C2×D411yescharacteristic
68C42×C59C2×D412yes—
69C42×C59#4111yescharacteristic
70C22⋊Dic59—41no—
71#11745672—48no—
72C4⋊Dic59—22no—
73C4⋊Dic59—22no—
74Dic59⋊C4—42no—
75C4×Dic59—41no—
76#11745664—42no—
77C2×C59⋊C8—24no—
78C2×C59⋊C8—41no—
79C23×C59#12011yescharacteristic
80Q8×C59—24no—
81Q8×C59—24no—
82Q8×C59—42no—
83Q8×C59—42no—
84C2×C4×C59—24no—
85C2×C4×C59—24no—
86C2×C4×C59—42no—
87C2×C4×C59—44no—
88C2×C4×C59—42no—
89C2×C4×C59—42no—
90C2×C4×C59—44no—
91C2×C4×C59—42no—
92C2×C4×C59#12012yes—
93C2×C4×C59#12111yescharacteristic
94C2×C4×C59#12112yes—
95C2×C4×C59#12711yescharacteristic
96C2×C4×C59C2×Q1612yes—
97C2×C4×C59#13712yes—
98C2×Dic59—42no—
99Dic118—44no—
100C59⋊C8—44no—
101C59⋊C8—81no—
102#10017—591nomaximal, Sylow
103C22×C59—22no—
104C22×C59#39211yescharacteristic
105C22×C59#45112yes—
106C4×C59—22no—
107C4×C59—22no—
108C4×C59—44no—
109C4×C59—44no—
110C4×C59—82no—
111C4×C59#46212yes—
112C4×C59#49411yescharacteristic
113C4×C59#49711yescharacteristic
114Dic59—82no—
115#3104—592no—
116#2399D5911yescharacteristic
117#1995—591no—
118#1487—592no—
119#1274—591no—
120C2×C59—22no—
121C2×C59#201012yes—
122C2×C59#201611yescharacteristic
123#533—591no—
124C2×C4×Q8C2×D5912yes—
125#484—1182no—
126#477—1182no—
127#462—1182no—
128#451—1182no—
129C4×Q16—1184no—
130#431—592no—
131#423—591no—
132#398—22no—
133#395—22no—
134C4×C4⋊C4—22no—
135#376C2×D5911yescharacteristic
136#339—1182no—
137Q8⋊C8—1184no—
138C59#1001711yescharacteristic, complemented, Sylow
139C22×Q8C22⋊D5912yes—
140C2×Q16—1184no—
141C2×M16—1181no—
142C4⋊Q8—592no—
143#125—1181no—
144C22⋊Q8—1182no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—1182no—
150#117—1181no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4C22⋊D5912yes—
154C2×C42—22no—
155C2×C42C22⋊D5912yes—
156C2×C42C22×D5911yescharacteristic
157#107—1184no—
158C4⋊C8—1182no—
159C4⋊C8—1182no—
160#103—1184no—
161#103—1184no—
162C22⋊C8—1182no—
163C4×C8—594no—
164#95—22no—
165#95—22no—
166#95—22no—
167C2×Q8—22no—
168C2×Q8—22no—
169C2×Q8—44no—
170C2×Q8—1182no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4C2×C22⋊D5912yes—
175C22×C4C2×C22⋊D5911yescharacteristic
176Q16—2368no—
177M16—2362no—
178C2×C8—1184no—
179C2×C8—2361no—
180C4⋊C4—22no—
181C4⋊C4—22no—
182C4⋊C4—48no—
183C4⋊C4—42no—
184C4⋊C4—42no—
185C4⋊C4—1182no—
186C4⋊C4—1182no—
187C4⋊C4—2362no—
188C22⋊C4—2361no—
189C42—22no—
190C42—42no—
191C42—42no—
192C42—2361no—
193C42#1174567214yescomplemented
194C42D4×D5911yescharacteristic
195C42D4×D5912yes—
196C42#1174568711yescharacteristic
197C23#1427369611yescharacteristic
198Q8—24no—
199Q8—24no—
200Q8—42no—
201Q8—42no—
202Q8—2364no—
203C2×C4—24no—
204C2×C4—24no—
205C2×C4—42no—
206C2×C4—44no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—44no—
210C2×C4—42no—
211C2×C4—2362no—
212C2×C4#1427369512yes—
213C2×C4#1427369811yescharacteristic
214C2×C4#1427369912yes—
215C2×C4#1427372311yescharacteristic
216C2×C4C2×#1174567212yes—
217C2×C4#1427375812yes—
218C8—4721no—
219C8—2364no—
220C22—22no—
221C22#1683731011yescharacteristic
222C22#1683755812yes—
223C4—22no—
224C4—22no—
225C4—44no—
226C4—44no—
227C4—82no—
228C4—4722no—
229C4#1683760912yes—
230C4#1683778111yescharacteristic
231C4#1683779411yescharacteristic
232C2—22no—
233C2#2142516112yes—
234C2#2142519611yescharacteristic
subgroupG/H
centerC2×C4#14273723
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2399C2
socleC22×C59#392
prime radicalC23×C59#120
hypercenter#2399D59
hypocenterC59#10017
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
66. C42×C59Q16semidirect
138. C59#10017semidirect
193. C42#11745672semidirect

Maximal subgroups

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

subgroupindex
#214341352
#214250772
#214225252
#214219252
C59×#23992
#1001759