Finite Group Databasebeta

← #28502383

Subgroups — #28502383

Order 15104.

312 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 67 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#21436965C211yesmaximal, characteristic, complemented
2#21426128C211yesmaximal, characteristic
3#21423453C211yesmaximal, characteristic, complemented
4#21422154C211yesmaximal, characteristic, complemented
5#21422076C211yesmaximal, characteristic, complemented
6#21422058C211yesmaximal, characteristic, complemented
7C59×#2240C211yesmaximal, characteristic, complemented
8#16838113—21no—
9C2×#14273668—21no—
10#16837803—21no—
11#16837705—22no—
12#16837689—21no—
13#16837599—21no—
14C8×Dic118C2211yescharacteristic
15C4×D472C2211yescharacteristic
16#16837484—21no—
17C4⋊(C59⋊C16)C2211yescharacteristic
18#16837203—21no—
19#16837155C2211yescharacteristic
20#16837146—21no—
21C59×#500C2211yescharacteristic
22C59×#493C2211yescharacteristic
23C59×#357—22no—
24C4×C16×C59C2211yescharacteristic
25C2×#11745670—41no—
26C2×D472D411yescharacteristic
27#14273718—22no—
28#14273707—41no—
29#14273703—42no—
30C4×Dic118D411yescharacteristic
31C4×D236—21no—
32#14273689—22no—
33C4⋊C4×D59—41no—
34#14273668—42no—
35C2×C59⋊C16—21no—
36#14273661D411yescharacteristic
37#14273658—22no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)D411yescharacteristic
40Dic59⋊C8—21no—
41#14273649—21no—
42#14273648—41no—
43#14273647—22no—
44C8×Dic59—21no—
45C2×D8×C59—21no—
46C2×D8×C59—21no—
47C4⋊Q8×C59—21no—
48#127×C59—21no—
49C2×C16×C59—21no—
50#107×C59—21no—
51#107×C59D412yes—
52C4×C8×C59C2311yescharacteristic
53C2×Q8×C59—41no—
54C2×D4×C59—41no—
55C2×D4×C59—41no—
56C2×D4×C59—41no—
57C2×D236—21no—
58C2×C4×D59—41no—
59C2×C4×D59—42no—
60C2×Dic118D811yescharacteristic
61D8×C59—42no—
62D8×C59—41no—
63D8×C59—41no—
64C2×C8×C59C2×D411yescharacteristic
65C2×C8×C59C2×D411yescharacteristic
66C4⋊C4×C59—22no—
67C4⋊C4×C59—41no—
68C42×C59C2×D411yescharacteristic
69#11745670—82no—
70(C2×D59)⋊C4—42no—
71(C2×D59)⋊C4—41no—
72C4⋊Dic59—21no—
73C4⋊Dic59—42no—
74C4⋊Dic59D811yescharacteristic
75Dic59⋊C4—21no—
76Dic59⋊C4—42no—
77C4×Dic59—21no—
78C2×C59⋊C8—21no—
79C2×C59⋊C8—41no—
80D472—21no—
81D472SD1612yescomplemented
82C16×C59—41no—
83C59⋊C16—41no—
84C23×C59—81no—
85C22×D59—41no—
86Q8×C59—161no—
87D4×C59—81no—
88D4×C59—81no—
89D4×C59—81no—
90D4×C59—81no—
91D4×C59—161no—
92C2×C4×C59—42no—
93C2×C4×C59#12011yescharacteristic
94C2×C4×C59C2×D811yescharacteristic
95C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
96C2×Dic59—21no—
97C2×Dic59—21no—
98C2×Dic59—41no—
99D236—22no—
100D236—41no—
101C4×D59—81no—
102C4×D59—81no—
103C4×D59—162no—
104Dic118—21no—
105Dic118—21no—
106C8×C59—21no—
107C8×C59C2×SD1611yescharacteristic
108C8×C59#13711yescharacteristic
109C59⋊C8—41no—
110C59⋊C8—81no—
111#12355—591nomaximal, Sylow
112C22×C59—161no—
113C22×C59—161no—
114C22×C59—161no—
115C22×C59#44711yescharacteristic
116C2×D59—42no—
117C2×D59—81no—
118C2×D59—81no—
119C4×C59—21no—
120C4×C59—82no—
121C4×C59#44811yescharacteristic
122C4×C59#45411yescharacteristic
123C4×C59#50811yescharacteristic
124C4×C59#50911yescharacteristic
125Dic59—41no—
126Dic59—41no—
127Dic59—81no—
128#3380—591no—
129#2240D5911yescharacteristic, complemented
130#2206—591no—
131#1668—591no—
132#1365—591no—
133Q8⋊C16—591no—
134#1331—591no—
135C2×C59—321no—
136C2×C59#162711yescharacteristic
137C2×C59#218611yescharacteristic
138C2×C59#219211yescharacteristic
139D59—82no—
140#554—1181no—
141C2×D16—1181no—
142#500C2×D5911yescharacteristic
143#499—1181no—
144#493C2×D5911yescharacteristic
145#478—1182no—
146#476—1181no—
147#460—1181no—
148C8×Q8—591no—
149C4×D8—591no—
150C4×D8—1181no—
151C4⋊C16—591no—
152#357—22no—
153#357—1181no—
154C4×C16C2×D5911yescharacteristic
155#335—591no—
156#331—1181no—
157C59#1235511yescharacteristic, complemented, Sylow
158C2×SD16—2361no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—591no—
162C4⋊Q8—21no—
163#127—21no—
164#125—1182no—
165C22⋊Q8—2361no—
166C22⋊Q8—2362no—
167C4×Q8—591no—
168C4×D4—1182no—
169C4×D4—1181no—
170C2×C4⋊C4—2361no—
171D16—2362no—
172C2×C16—21no—
173C2×C16—1181no—
174#107—21no—
175#107—591no—
176#107C22⋊D5912yes—
177#106—1182no—
178C4⋊C8—591no—
179C4⋊C8—1181no—
180C4⋊C8—1181no—
181#102—1182no—
182#102—1181no—
183#102—2361no—
184C4×C8—1181no—
185C4×C8C22×D5911yescharacteristic
186C2×Q8—41no—
187C2×Q8—591no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—1181no—
192C22×C4—2362no—
193C22×C4—2361no—
194SD16—4722no—
195D8—42no—
196D8—41no—
197D8—41no—
198D8—592no—
199D8—1181no—
200C2×C8—1181no—
201C2×C8—2361no—
202C2×C8D4×D5911yescharacteristic
203C2×C8C2×C22⋊D5911yescharacteristic
204C4⋊C4—22no—
205C4⋊C4—41no—
206C4⋊C4—591no—
207C4⋊C4—1181no—
208C4⋊C4—1181no—
209C4⋊C4—2362no—
210C4⋊C4—2362no—
211C22⋊C4—2362no—
212C22⋊C4—2361no—
213C42—1181no—
214C42D4×D5911yescharacteristic
215C16—41no—
216C16—2361no—
217C23—81no—
218C23—2361no—
219Q8—161no—
220Q8—1181no—
221Q8—1181no—
222D4—81no—
223D4—81no—
224D4—81no—
225D4—81no—
226D4—161no—
227D4—1182no—
228D4—2361no—
229C2×C4—42no—
230C2×C4—1181no—
231C2×C4—1181no—
232C2×C4—9442no—
233C2×C4—4721no—
234C2×C4—4721no—
235C2×C4—2361no—
236C2×C4#1427369511yescharacteristic
237C2×C4D8×D5911yescharacteristic
238C2×C4#1427375111yescharacteristic
239C8—21no—
240C8—4721no—
241C8—2361no—
242C8SD16×D5911yescharacteristic
243C8#1427375611yescharacteristic
244C22—161no—
245C22—161no—
246C22—161no—
247C22—4721no—
248C22—4721no—
249C22—2362no—
250C22#1683753411yescharacteristic
251C4—21no—
252C4—82no—
253C4—4721no—
254C4—2361no—
255C4—2361no—
256C4#1683753911yescharacteristic
257C4#1683757411yescharacteristic
258C4#1683783311yescharacteristic
259C4#1683783911yescharacteristic
260C2—321no—
261C2—4722no—
262C2#2142319611yescharacteristic
263C2#2142601811yescharacteristic
264C2#2142605211yescharacteristic
subgroupG/H
centerC22#16837534
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#2240C2
socleC22×C59#447
prime radical#21426128C2
hypercenter#2240D59
hypocenterC59#12355
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2240

Complements

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

NH ≅ G/Ntype
1. #21436965C2semidirect
3. #21423453C2semidirect
4. #21422154C2semidirect
5. #21422076C2semidirect
6. #21422058C2semidirect
7. C59×#2240C2semidirect
81. D472SD16semidirect
129. #2240D59semidirect
157. C59#12355semidirect

Maximal subgroups

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

subgroupindex
#214369652
#214261282
#214234532
#214221542
#214220762
#214220582
C59×#22402
#1235559