Finite Group Databasebeta

← #28498192

Subgroups — #28498192

Order 15104.

334 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 71 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#21432802C211yesmaximal, characteristic
2#21425909C212yesmaximal, complemented
3#21422034C211yesmaximal, characteristic, complemented
4#21422031C212yesmaximal, complemented
5C59×#1321C211yesmaximal, characteristic, complemented
6C2×#14273734C411yescharacteristic
7C22×#11745659C411yescharacteristic
8#16837909C411yescharacteristic
9#16837477—22no—
10#16837470—22no—
11#16837428C411yescharacteristic
12#16837365C2211yescharacteristic
13#16837360C2211yescharacteristic
14#16837222—22no—
15#16837184C2212yes—
16#16837179—22no—
17C59×#408C2211yescharacteristic
18C59×#349C2212yes—
19C22×C4×D59C2×C411yescharacteristic
20#14273734—42no—
21C2×#11745664—21no—
22C2×#11745664C2×C411yescharacteristic
23C2×#11745659—21no—
24C2×#11745659—21no—
25C2×#11745659—42no—
26C22×C59⋊C8—21no—
27C22×C59⋊C8C2×C411yescharacteristic
28#14273708—21no—
29#14273703—21no—
30#14273698—21no—
31#14273695—21no—
32C4×C22⋊D59—22no—
33#14273686C2×C411yescharacteristic
34#14273667—22no—
35Dic59⋊C8—22no—
36(C2×D59)⋊C8—22no—
37#14273637D412yes—
38C8×Dic59D412yes—
39C2×#41×C59C2×C411yescharacteristic
40C2×M16×C59C2311yescharacteristic
41C22×C8×C59C2×C411yescharacteristic
42M32×C59—22no—
43C22⋊C8×C59—22no—
44C23×D59M1611yescharacteristic
45#41×C59—41no—
46#41×C59—41no—
47C2×Q8×C59M1611yescharacteristic
48C2×D4×C59—21no—
49C2×D4×C59M1611yescharacteristic
50C22×C4×C59—21no—
51C22×C4×C59C22×C411yescharacteristic
52C2×C22⋊D59—22no—
53C22×Dic59M1611yescharacteristic
54C2×C4×D59—21no—
55C2×C4×D59—21no—
56C2×C4×D59—41no—
57C2×C4×D59—41no—
58M16×C59—41no—
59C2×C8×C59—21no—
60C2×C8×C59—41no—
61C2×C8×C59C2×D412yes—
62C22⋊Dic59—22no—
63C22⋊Dic59—21no—
64(C2×D59)⋊C4—22no—
65C4⋊Dic59—21no—
66Dic59⋊C4—22no—
67C4×Dic59C22⋊C412yes—
68#11745664—42no—
69#11745664—41no—
70C2×C59⋊C8—22no—
71C2×C59⋊C8—21no—
72C2×C59⋊C8—42no—
73C2×C59⋊C8C22⋊C411yescharacteristic
74C2×C59⋊C8C22⋊C411yescharacteristic
75#11745659—42no—
76#11745659—42no—
77C16×C59—42no—
78C59⋊C16—42no—
79C23×C59—21no—
80C23×C59C2×M1611yescharacteristic
81C22×D59—21no—
82C22×D59—41no—
83C22×D59—81no—
84Q8×C59—41no—
85D4×C59—41no—
86D4×C59—42no—
87C2×C4×C59—21no—
88C2×C4×C59—21no—
89C2×C4×C59—21no—
90C2×C4×C59—41no—
91C2×C4×C59—41no—
92C2×C4×C59C2×C22⋊C411yescharacteristic
93C2×C4×C59C2×M1611yescharacteristic
94C22⋊D59—82no—
95C2×Dic59—22no—
96C2×Dic59—21no—
97C2×Dic59—41no—
98C4×D59—42no—
99C4×D59—41no—
100C4×D59—41no—
101C8×C59—22no—
102C8×C59—41no—
103C59⋊C8—21no—
104C59⋊C8—41no—
105C59⋊C8#10012yes—
106#11538—591nomaximal, Sylow
107C22×C59—21no—
108C22×C59—41no—
109C22×C59—42no—
110C22×C59—41no—
111C22×C59#40711yescharacteristic
112C2×D59—41no—
113C2×D59—41no—
114C2×D59—41no—
115C2×D59—82no—
116C4×C59—21no—
117C4×C59—41no—
118C4×C59#41011yescharacteristic
119C4×C59C2×#10011yescharacteristic
120Dic59—42no—
121Dic59—41no—
122#2930—591no—
123#2166—592no—
124#1321—591no—
125#1321D5911yescharacteristic, complemented
126#1320—592no—
127C2×C59—21no—
128C2×C59—41no—
129C2×C59—81no—
130C2×C59#146611yescharacteristic
131D59—81no—
132C2×#131—591no—
133C22×M16—591no—
134#525—591no—
135#436—1182no—
136#435—1182no—
137#424—591no—
138#408C2×D5911yescharacteristic
139#407—591no—
140#405—591no—
141#364—1182no—
142#349—592no—
143#349C2×D5912yes—
144#347—1182no—
145C59#1153811yescharacteristic, complemented, Sylow
146C2×#41C4×D5911yescharacteristic
147C23×C4—591no—
148#131—2362no—
149C2×M16—591no—
150C2×M16—1181no—
151C2×M16—1181no—
152C2×M16—1181no—
153C2×M16—2362no—
154C2×M16C22×D5911yescharacteristic
155C22×C8—591no—
156C22×C8—1181no—
157C22×C8C4×D5911yescharacteristic
158#123—1181no—
159C22⋊Q8—1181no—
160#121—1181no—
161#120—1181no—
162C4×D4—1182no—
163#117—591no—
164M32—22no—
165M32—1182no—
166C4⋊C8—1182no—
167C22⋊C8—22no—
168C22⋊C8—1182no—
169#97—592no—
170C4×C8—592no—
171C24—591no—
172#41—41no—
173#41—41no—
174C2×Q8#1174565911yescharacteristic
175C2×D4—21no—
176C2×D4—1182no—
177C2×D4#1174565911yescharacteristic
178C22×C4—21no—
179C22×C4—591no—
180C22×C4—1181no—
181C22×C4—1181no—
182C22×C4—2361no—
183C22×C4—2361no—
184C22×C4C2×C4×D5911yescharacteristic
185M16—41no—
186M16—2362no—
187M16—2362no—
188M16—2362no—
189M16—2361no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8—591no—
193C2×C8—591no—
194C2×C8—1182no—
195C2×C8—1181no—
196C2×C8—2362no—
197C2×C8D4×D5912yes—
198C4⋊C4—1182no—
199C4⋊C4—1181no—
200C22⋊C4—1182no—
201C22⋊C4—1182no—
202C22⋊C4—1181no—
203C42—592no—
204C16—42no—
205C16—2362no—
206C23—21no—
207C23—1181no—
208C23—4721no—
209C23—2361no—
210C23M16×D5911yescharacteristic
211Q8—41no—
212D4—41no—
213D4—42no—
214D4—4722no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—1182no—
221C2×C4—1181no—
222C2×C4—2362no—
223C2×C4—2361no—
224C2×C4—2361no—
225C2×C4—2361no—
226C2×C4C22⋊C4×D5911yescharacteristic
227C2×C4C2×#1174565911yescharacteristic
228C8—22no—
229C8—41no—
230C8—592no—
231C8—1181no—
232C8—2361no—
233C22—21no—
234C22—41no—
235C22—42no—
236C22—41no—
237C22—4722no—
238C22—2361no—
239C22—2361no—
240C22—2361no—
241C22#1683736411yescharacteristic
242C4—21no—
243C4—41no—
244C4—2362no—
245C4—2361no—
246C4#1683737411yescharacteristic
247C4#100×D5911yescharacteristic
248C2—21no—
249C2—41no—
250C2—81no—
251C2—4721no—
252C2#2142243511yescharacteristic
subgroupG/H
centerC4#16837374
commutatorC22×C4×C59C22×C4
FrattiniC2×M16C22×D59
FittingC59×#1321C2
socleC2×C59#1466
prime radical#16837909C4
cosocleC2×M16×C59C23
hypercenter#1321D59
hypocenterC59#11538
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1321

Complements

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

NH ≅ G/Ntype
2. #21425909C2semidirect
3. #21422034C2semidirect
4. #21422031C2semidirect
5. C59×#1321C2semidirect
125. #1321D59semidirect
145. C59#11538semidirect

Maximal subgroups

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

subgroupindex
#214328022
#214259092
#214220342
#214220312
C59×#13212
#1153859