Finite Group Databasebeta

← #28486298

Subgroups — #28486298

Order 15104.

556 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 107 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#21434549C211yesmaximal, characteristic
2#21422831C211yesmaximal, characteristic, complemented
3#21422763C212yesmaximal, complemented
4#21421987C212yesmaximal, complemented
5C59×#2979C211yesmaximal, characteristic, complemented
6C2×#14273752—22no—
7C2×#14273724C2211yescharacteristic
8#16837792C414yes—
9#16837778—21no—
10#16837776—21no—
11#16837426C2212yes—
12C4×#11745664C2211yescharacteristic
13#16837151—21no—
14#16837146—21no—
15#16837145—22no—
16#16837145—22no—
17C59×#532C2211yescharacteristic
18C59×#436—22no—
19C8×D4×C59—22no—
20C59×#431C2212yes—
21C59×#408—22no—
22C22×D236—21no—
23C22×D236—22no—
24#14273752—44no—
25#14273752—44no—
26C2×#11745671—24no—
27C2×#11745669—24no—
28C2×#11745664—21no—
29C2×#11745664—22no—
30#14273724—22no—
31#14273724—21no—
32#14273724C2×C412yes—
33C4⋊(C59⋊C8)—22no—
34C4⋊(C59⋊C8)—22no—
35#14273638C2×C412yes—
36C4×C59⋊C8—21no—
37C2×#41×C59D412yes—
38C2×M16×C59—22no—
39C22×C8×C59—22no—
40C4⋊Q8×C59D411yescharacteristic
41#127×C59D411yescharacteristic
42#124×C59C2×C412yes—
43#121×C59—22no—
44C4×D4×C59—22no—
45C2×C42×C59C2311yescharacteristic
46C4⋊C8×C59D412yes—
47C4⋊C8×C59D412yes—
48C22⋊C8×C59—24no—
49C22⋊C8×C59—22no—
50C4×C8×C59—22no—
51C23×D59—41no—
52#41×C59—42no—
53#41×C59—42no—
54C2×Q8×C59C22⋊C412yes—
55C2×D4×C59—22no—
56C2×D4×C59—22no—
57C2×D4×C59C22⋊C412yes—
58C22×C4×C59—22no—
59C22×C4×C59C2×D411yescharacteristic
60C22×C4×C59#4112yes—
61C2×D236—22no—
62C2×D236—24no—
63C2×D236—42no—
64C2×D236—42no—
65C2×D236—41no—
66C2×D236—42no—
67C2×D236—82no—
68C2×D236—82no—
69M16×C59—82no—
70C2×C8×C59—22no—
71C2×C8×C59—22no—
72C2×C8×C59—82no—
73C4⋊C4×C59—22no—
74C22⋊C4×C59—24no—
75C42×C59C22×C411yescharacteristic
76C42×C59C2×D411yescharacteristic
77C42×C59C2×D412yes—
78#11745671—48no—
79#11745669—48no—
80#11745664—22no—
81#11745664—22no—
82#11745664—82no—
83C2×C59⋊C8—22no—
84C2×C59⋊C8—42no—
85C23×C59—22no—
86C23×C59#12311yescharacteristic
87C22×D59—42no—
88C22×D59—82no—
89C22×D59—84no—
90Q8×C59—42no—
91D4×C59—42no—
92D4×C59—42no—
93D4×C59—42no—
94D4×C59—82no—
95C2×C4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—22no—
100C2×C4×C59—42no—
101C2×C4×C59—42no—
102C2×C4×C59C2×C22⋊C411yescharacteristic
103C2×C4×C59C4×D412yes—
104C2×C4×C59#12011yescharacteristic
105C2×C4×C59#12112yes—
106D236—44no—
107D236—84no—
108D236—81no—
109D236—81no—
110D236—82no—
111D236—82no—
112C8×C59—42no—
113C8×C59—42no—
114C59⋊C8—24no—
115C59⋊C8—82no—
116#9437—591nomaximal, Sylow
117C22×C59—21no—
118C22×C59—22no—
119C22×C59—42no—
120C22×C59—44no—
121C22×C59#38611yescharacteristic
122C2×D59—82no—
123C2×D59—84no—
124C2×D59—164no—
125C4×C59—21no—
126C4×C59—22no—
127C4×C59—42no—
128C4×C59#45311yescharacteristic
129C4×C59#45311yescharacteristic
130C4×C59#47112yes—
131C4×C59#47212yes—
132#3151—591no—
133#2979D5911yescharacteristic, complemented
134#1560—591no—
135#1545—592no—
136#1299—592no—
137C2×C59—41no—
138C2×C59—82no—
139C2×C59#188711yescharacteristic
140C2×C59#190712yes—
141D59—162no—
142C2×AGL(1,ℤ/8)—1182no—
143#532C2×D5911yescharacteristic
144C2×#127—591no—
145#496—594no—
146#493—1181no—
147#492—1181no—
148#436—22no—
149C8×D4—22no—
150#431C2×D5912yes—
151#423—592no—
152#408—22no—
153C4×M16—591no—
154#333—1181no—
155#331—1181no—
156#331—1182no—
157#331—1182no—
158C59#943711yescharacteristic, complemented, Sylow
159C2×#41C22⋊D5912yes—
160C22×D4—1182no—
161C22×D4—1181no—
162AGL(1,ℤ/8)—2364no—
163AGL(1,ℤ/8)—2364no—
164C2×SD16—1184no—
165C2×D8—1184no—
166C2×M16—22no—
167C2×M16—1181no—
168C2×M16—1182no—
169C22×C8—22no—
170C4⋊Q8D23611yescharacteristic
171#127—592no—
172#127—1181no—
173#127—1182no—
174#127D23611yescharacteristic
175#124C4×D5912yes—
176#121—22no—
177C4×D4—22no—
178C2×C42C22×D5911yescharacteristic
179C4⋊C8—1182no—
180C4⋊C8—1182no—
181C4⋊C8C22⋊D5912yes—
182C4⋊C8C22⋊D5912yes—
183C22⋊C8—24no—
184C22⋊C8—22no—
185#97—592no—
186C4×C8—22no—
187C4×C8—1181no—
188C24—2361no—
189#41—42no—
190#41—42no—
191C2×Q8(C2×D59)⋊C412yes—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—1182no—
195C2×D4—1184no—
196C2×D4—4722no—
197C2×D4—4722no—
198C2×D4—2362no—
199C2×D4—2362no—
200C2×D4—2361no—
201C2×D4—2362no—
202C2×D4(C2×D59)⋊C412yes—
203C22×C4—22no—
204C22×C4#1174568512yes—
205C22×C4C2×C22⋊D5911yescharacteristic
206SD16—2368no—
207D8—2368no—
208M16—82no—
209M16—1182no—
210M16—1182no—
211M16—4722no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—82no—
215C2×C8—1182no—
216C2×C8—2362no—
217C4⋊C4—22no—
218C22⋊C4—24no—
219C42C2×C4×D5911yescharacteristic
220C42C2×D23611yescharacteristic
221C42C2×C22⋊D5912yes—
222C23—22no—
223C23—4724no—
224C23—4722no—
225C23—2362no—
226C23#1427371011yescharacteristic
227Q8—42no—
228D4—42no—
229D4—42no—
230D4—42no—
231D4—82no—
232D4—4724no—
233D4—4722no—
234D4—4722no—
235D4—4721no—
236D4—4721no—
237D4—2364no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4C2×(C2×D59)⋊C411yescharacteristic
246C2×C4C4×C22⋊D5912yes—
247C2×C4#1427369611yescharacteristic
248C2×C4#1427370112yes—
249C8—42no—
250C8—42no—
251C8—1184no—
252C8—4722no—
253C22—21no—
254C22—22no—
255C22—42no—
256C22—44no—
257C22—9444no—
258C22—4722no—
259C22—4724no—
260C22#1683728111yescharacteristic
261C4—21no—
262C4—22no—
263C4—42no—
264C4#1683757011yescharacteristic
265C4#1683757011yescharacteristic
266C4#1683766512yes—
267C4#1683767212yes—
268C2—41no—
269C2—82no—
270C2—9442no—
271C2#2142449411yescharacteristic
272C2#2142460412yes—
subgroupG/H
centerC22#16837281
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2979C2
socleC22×C59#386
prime radical#21434549C2
cosocleC2×C42×C59C23
hypercenter#2979D59
hypocenterC59#9437
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2979

Complements

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

NH ≅ G/Ntype
2. #21422831C2semidirect
3. #21422763C2semidirect
4. #21421987C2semidirect
5. C59×#2979C2semidirect
133. #2979D59semidirect
158. C59#9437semidirect

Maximal subgroups

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

subgroupindex
#214345492
#214228312
#214227632
#214219872
C59×#29792
#943759