Finite Group Databasebeta

← #28486248

Subgroups — #28486248

Order 15104.

508 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 99 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#21434553C211yesmaximal, characteristic
2#21422822C211yesmaximal, characteristic, complemented
3#21422642C212yesmaximal
4#21421960C212yesmaximal, complemented
5C59×#2381C211yesmaximal, characteristic, complemented
6C2×#14273758—22no—
7#16837952C2211yescharacteristic
8#16837794—22no—
9#16837781C412yescomplemented
10#16837776C412yescomplemented
11#16837459C2212yes—
12C4×#11745664C2211yescharacteristic
13#16837151—22no—
14#16837135—24no—
15C2×C4⋊Q8×C59C2211yescharacteristic
16C59×#391—22no—
17C59×#387—24no—
18C59×#383C2212yes—
19C2×#11745685—21no—
20#14273758—44no—
21#14273758—44no—
22C2×#11745672—24no—
23C2×#11745671—24no—
24C2×#11745664—21no—
25#14273727C2×C411yescharacteristic
26#14273724C2×C411yescharacteristic
27#14273715—21no—
28#14273701—22no—
29C4×D236—22no—
30C4⋊(C59⋊C8)—22no—
31C22⋊(C59⋊C8)—22no—
32#14273638—21no—
33#14273638—22no—
34C4×C59⋊C8C2×C412yes—
35C22×Q8×C59D412yes—
36C4⋊Q8×C59—22no—
37C4⋊Q8×C59C2×C412yes—
38C4⋊Q8×C59D412yes—
39C2×C4⋊C4×C59—22no—
40C2×C4⋊C4×C59—22no—
41C2×C42×C59—22no—
42C2×C42×C59C2311yescharacteristic
43#95×C59—24no—
44#95×C59—22no—
45#95×C59—22no—
46#95×C59D414yes—
47C2×Q8×C59—22no—
48C2×Q8×C59—42no—
49C2×Q8×C59—42no—
50C2×Q8×C59C22⋊C414yes—
51C22×C4×C59—22no—
52C22×C4×C59—24no—
53C22×C4×C59C2×D411yescharacteristic
54C22×C4×C59C2×D412yes—
55C2×C22⋊D59—41no—
56#11745685—42no—
57#11745685—42no—
58C2×D236—21no—
59C2×D236—22no—
60C2×C4×D59—41no—
61C2×Dic118—21no—
62C4⋊C4×C59—24no—
63C4⋊C4×C59—24no—
64C4⋊C4×C59—82no—
65C42×C59—82no—
66C42×C59C22×C411yescharacteristic
67C42×C59C2×D411yescharacteristic
68C42×C59#4112yes—
69#11745672—48no—
70#11745671—48no—
71(C2×D59)⋊C4—42no—
72C4⋊Dic59—22no—
73#11745664—24no—
74C2×C59⋊C8—22no—
75C2×C59⋊C8—42no—
76C23×C59#12011yescharacteristic
77C22×D59—41no—
78Q8×C59—44no—
79Q8×C59—82no—
80C2×C4×C59—22no—
81C2×C4×C59—24no—
82C2×C4×C59—42no—
83C2×C4×C59—42no—
84C2×C4×C59—42no—
85C2×C4×C59—48no—
86C2×C4×C59—44no—
87C2×C4×C59C2×C22⋊C411yescharacteristic
88C2×C4×C59C4×D412yes—
89C2×C4×C59#12112yes—
90C2×C4×C59#12311yescharacteristic
91C22⋊D59—82no—
92C2×Dic59—41no—
93D236—42no—
94D236—82no—
95C4×D59—81no—
96C4×D59—81no—
97Dic118—42no—
98C59⋊C8—24no—
99C59⋊C8—82no—
100#9429—591nomaximal, Sylow
101C22×C59—22no—
102C22×C59#38611yescharacteristic
103C22×C59#45312yes—
104C2×D59—81no—
105C2×D59—82no—
106C4×C59—21no—
107C4×C59—21no—
108C4×C59—22no—
109C4×C59—22no—
110C4×C59—44no—
111C4×C59—84no—
112Dic59—81no—
113#3152—591no—
114#2381D5911yescharacteristic, complemented
115#1558—591no—
116#1514—592no—
117#1288—592no—
118C2×C59—22no—
119C2×C59#188711yescharacteristic
120C2×C59#190512yes—
121D59—161no—
122C2×#137—1182no—
123#532—591no—
124C2×C4⋊Q8C2×D5911yescharacteristic
125#497—1182no—
126#494—592no—
127#492—592no—
128#432—592no—
129C4×M16—591no—
130#391—22no—
131#387—24no—
132#383C2×D5912yes—
133#333—1182no—
134#327—1184no—
135C59#942911yescharacteristic, complemented, Sylow
136C2×#41—1181no—
137C22×Q8C22⋊D5912yes—
138#137—2364no—
139#137—2364no—
140C2×Q16—1184no—
141C2×SD16—1184no—
142C2×M16—1181no—
143C4⋊Q8—22no—
144C4⋊Q8—591no—
145C4⋊Q8C4×D5912yes—
146C4⋊Q8D23612yes—
147#127—591no—
148#124—1181no—
149#121—1182no—
150C4×D4—1182no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C42—22no—
154C2×C42C22×D5911yescharacteristic
155C4⋊C8—1182no—
156C22⋊C8—1182no—
157#97—1181no—
158#97—1182no—
159C4×C8—592no—
160#95—24no—
161#95—22no—
162#95—22no—
163#95C22⋊D5914yes—
164#41—2362no—
165#41—2362no—
166C2×Q8—22no—
167C2×Q8—42no—
168C2×Q8—42no—
169C2×Q8—1181no—
170C2×Q8(C2×D59)⋊C414yes—
171C2×D4—1181no—
172C2×D4—1182no—
173C2×D4—2361no—
174C22×C4—22no—
175C22×C4—24no—
176C22×C4—2361no—
177C22×C4C2×C22⋊D5911yescharacteristic
178C22×C4C2×C22⋊D5912yes—
179Q16—2368no—
180SD16—2368no—
181M16—1184no—
182C2×C8—1182no—
183C2×C8—2362no—
184C4⋊C4—24no—
185C4⋊C4—24no—
186C4⋊C4—82no—
187C4⋊C4—1182no—
188C22⋊C4—2362no—
189C42—82no—
190C42C2×C4×D5911yescharacteristic
191C42C2×D23611yescharacteristic
192C42#1174568512yes—
193C23—2361no—
194C23#1427369611yescharacteristic
195Q8—44no—
196Q8—82no—
197Q8—2362no—
198D4—4722no—
199D4—4722no—
200D4—2362no—
201C2×C4—22no—
202C2×C4—24no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—48no—
207C2×C4—44no—
208C2×C4—4721no—
209C2×C4—4721no—
210C2×C4—2361no—
211C2×C4C2×(C2×D59)⋊C411yescharacteristic
212C2×C4C4×C22⋊D5912yes—
213C2×C4#1427370112yes—
214C2×C4#1427371011yescharacteristic
215C8—1184no—
216C8—4722no—
217C22—22no—
218C22—4721no—
219C22—4722no—
220C22#1683728111yescharacteristic
221C22#1683757012yes—
222C4—21no—
223C4—21no—
224C4—22no—
225C4—22no—
226C4—44no—
227C4—84no—
228C4—4721no—
229C2—22no—
230C2—9441no—
231C2#2142449411yescharacteristic
232C2#2142459012yes—
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2381C2
socleC22×C59#386
prime radical#16837952C22
cosocleC2×C42×C59C23
hypercenter#2381D59
hypocenterC59#9429
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2381

Complements

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

NH ≅ G/Ntype
2. #21422822C2semidirect
4. #21421960C2semidirect
5. C59×#2381C2semidirect
9. #16837781C4semidirect
10. #16837776C4semidirect
114. #2381D59semidirect
135. C59#9429semidirect

Maximal subgroups

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

subgroupindex
#214345532
#214228222
#214226422
#214219602
C59×#23812
#942959