Finite Group Databasebeta

← #28497994

Subgroups — #28497994

Order 15104.

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

normal subgroups only

This group has 368 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#21432798C211yesmaximal, characteristic
2#21425715C212yesmaximal, complemented
3#21422049C211yesmaximal, characteristic, complemented
4#21422031C212yesmaximal, complemented
5C59×#1328C211yesmaximal, characteristic, complemented
6C2×#14273735C2211yescharacteristic
7C22×#11745659C411yescharacteristic
8#16837914C411yescharacteristic
9M32×D59—22no—
10#16837479—22no—
11#16837472—22no—
12#16837368C411yescharacteristic
13#16837365C2211yescharacteristic
14#16837358C411yescharacteristic
15#16837184C2212yes—
16#16837183—22no—
17C59×#424C2211yescharacteristic
18C59×#349C2212yes—
19C2×#11745685C2×C411yescharacteristic
20C22×C4×D59C2×C411yescharacteristic
21#14273735—42no—
22C2×#11745664C2×C411yescharacteristic
23C2×#11745659—21no—
24C2×#11745659—21no—
25C2×#11745659—42no—
26C2×#11745659D412yes—
27C22×C59⋊C8C2×C411yescharacteristic
28C2×C8×D59D412yes—
29#14273711C812yes—
30#14273707—21no—
31#14273702—21no—
32#14273697C812yes—
33C4×D236—22no—
34#14273667—22no—
35#14273666—42no—
36C16×D59—42no—
37(C2×D59)⋊C8—22no—
38(C2×D59)⋊C8—22no—
39#14273638—21no—
40C4×C59⋊C8—21no—
41C2×M16×C59C2311yescharacteristic
42C22×C8×C59C2×C411yescharacteristic
43#117×C59C2×C411yescharacteristic
44M32×C59—22no—
45C4⋊C8×C59—22no—
46C23×D59C2×C811yescharacteristic
47C22×C4×C59C22×C411yescharacteristic
48C2×C22⋊D59C2×C812yes—
49C22×Dic59C2×C811yescharacteristic
50#11745685—41no—
51#11745685—41no—
52C2×D236—22no—
53C2×D236M1611yescharacteristic
54C2×C4×D59—21no—
55C2×C4×D59—21no—
56C2×C4×D59—41no—
57C2×C4×D59—41no—
58C2×C4×D59C22⋊C412yes—
59C2×Dic118M1611yescharacteristic
60M16×C59—41no—
61C2×C8×C59—21no—
62C2×C8×C59—41no—
63C2×C8×C59C2×D412yes—
64C4⋊C4×C59—21no—
65C22⋊C4×C59C2×C812yes—
66C42×C59—21no—
67(C2×D59)⋊C4—22no—
68(C2×D59)⋊C4—22no—
69C4⋊Dic59—22no—
70#11745664—41no—
71C2×C59⋊C8—22no—
72C2×C59⋊C8—21no—
73C2×C59⋊C8C22⋊C411yescharacteristic
74C2×C59⋊C8C22⋊C411yescharacteristic
75#11745659—42no—
76#11745659—42no—
77#11745659—42no—
78C8×D59—22no—
79C8×D59—22no—
80C16×C59—42no—
81C59⋊C16—42no—
82C23×C59C22×C811yescharacteristic
83C22×D59—21no—
84C22×D59—41no—
85C22×D59—81no—
86C22×D59C22⋊C812yes—
87C2×C4×C59—21no—
88C2×C4×C59—21no—
89C2×C4×C59—22no—
90C2×C4×C59C2×C22⋊C411yescharacteristic
91C2×C4×C59C2×M1611yescharacteristic
92C22⋊D59—42no—
93C2×Dic59—21no—
94C2×Dic59—41no—
95C2×Dic59C22⋊C812yes—
96D236—41no—
97D236—82no—
98C4×D59—22no—
99C4×D59—22no—
100C4×D59—42no—
101C4×D59—41no—
102C4×D59—41no—
103Dic118—41no—
104C8×C59—22no—
105C8×C59—41no—
106C59⋊C8—21no—
107C59⋊C8—41no—
108C59⋊C8#9912yes—
109#11484—591nomaximal, Sylow
110C22×C59—21no—
111C22×C59—41no—
112C22×C59C2×C22⋊C811yescharacteristic
113C2×D59—22no—
114C2×D59—42no—
115C2×D59—41no—
116C2×D59—41no—
117C2×D59—41no—
118C2×D59—82no—
119C4×C59—21no—
120C4×C59—42no—
121C4×C59C2×#9911yescharacteristic
122C4×C59#41311yescharacteristic
123Dic59—22no—
124Dic59—41no—
125#2930—591no—
126#2115—592no—
127#1328—591no—
128#1328D5911yescharacteristic, complemented
129#1320—592no—
130C2×C59—21no—
131C2×C59—41no—
132C2×C59#145911yescharacteristic
133D59—42no—
134D59—81no—
135C2×#131—591no—
136C22×M16—591no—
137#525—591no—
138C2×M32—1182no—
139#436—1182no—
140#435—1182no—
141#424C2×D5911yescharacteristic
142#408—591no—
143#407—591no—
144#405—591no—
145#349—592no—
146#349—1182no—
147#349C2×D5912yes—
148C59#1148411yescharacteristic, complemented, Sylow
149C2×#41—591no—
150C23×C4—591no—
151#131—2362no—
152C2×M16—592no—
153C2×M16—591no—
154C2×M16—1181no—
155C2×M16—1181no—
156C2×M16—2362no—
157C2×M16C22×D5911yescharacteristic
158C22×C8—591no—
159C22×C8—592no—
160C22×C8C4×D5911yescharacteristic
161#123—592no—
162C22⋊Q8—1181no—
163#121—1181no—
164#120—592no—
165C4×D4—1182no—
166#117C4×D5911yescharacteristic
167M32—22no—
168M32—1182no—
169M32—2362no—
170C2×C16—2362no—
171C4⋊C8—22no—
172C22⋊C8—1182no—
173C22⋊C8—1182no—
174#97—1181no—
175C4×C8—1181no—
176C24—591no—
177#41—2361no—
178#41—2361no—
179C2×Q8—591no—
180C2×D4—592no—
181C2×D4—591no—
182C2×D4—1182no—
183C22×C4—592no—
184C22×C4—591no—
185C22×C4—1181no—
186C22×C4—1181no—
187C22×C4—2361no—
188C22×C4—2361no—
189C22×C4C2×C4×D5911yescharacteristic
190M16—41no—
191M16—2362no—
192M16—2362no—
193M16—2362no—
194M16—2361no—
195C2×C8—21no—
196C2×C8—41no—
197C2×C8—591no—
198C2×C8—591no—
199C2×C8—1182no—
200C2×C8—1182no—
201C2×C8—1182no—
202C2×C8—1181no—
203C2×C8D4×D5912yes—
204C4⋊C4—21no—
205C4⋊C4—1182no—
206C22⋊C4—1182no—
207C22⋊C4—1182no—
208C22⋊C4C8×D5912yes—
209C42—21no—
210C16—42no—
211C16—2362no—
212C23—592no—
213C23—1181no—
214C23—4721no—
215C23—2361no—
216C23C2×C8×D5911yescharacteristic
217Q8—2361no—
218D4—4722no—
219D4—2361no—
220D4—2362no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—592no—
225C2×C4—1182no—
226C2×C4—1182no—
227C2×C4—1181no—
228C2×C4—2362no—
229C2×C4—2361no—
230C2×C4—2361no—
231C2×C4—2361no—
232C2×C4C22⋊C4×D5911yescharacteristic
233C2×C4M16×D5911yescharacteristic
234C8—22no—
235C8—41no—
236C8—592no—
237C8—1181no—
238C8—2361no—
239C22—21no—
240C22—41no—
241C22—1182no—
242C22—4722no—
243C22—2361no—
244C22—2361no—
245C22—2362no—
246C22—2361no—
247C22C22⋊C8×D5911yescharacteristic
248C4—21no—
249C4—42no—
250C4—1182no—
251C4—2361no—
252C4#99×D5911yescharacteristic
253C4#1683738511yescharacteristic
254C2—21no—
255C2—41no—
256C2—4721no—
257C2—2362no—
258C2#2142241311yescharacteristic
subgroupG/H
centerC4#99×D59
commutatorC23×C59C22×C8
FrattiniC2×M16C22×D59
FittingC59×#1328C2
socleC2×C59#1459
prime radical#16837914C4
cosocleC2×M16×C59C23
hypercenter#1328D59
hypocenterC59#11484
LCSG ▷ C23×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1328

Complements

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

NH ≅ G/Ntype
2. #21425715C2semidirect
3. #21422049C2semidirect
4. #21422031C2semidirect
5. C59×#1328C2semidirect
128. #1328D59semidirect
148. C59#11484semidirect

Maximal subgroups

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

subgroupindex
#214327982
#214257152
#214220492
#214220312
C59×#13282
#1148459