Finite Group Databasebeta

← #28502398

Subgroups — #28502398

Order 15104.

282 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 69 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#21436988C211yesmaximal, characteristic
2#21426323C211yesmaximal, characteristic
3#21426143C211yesmaximal, characteristic
4#21422152C211yesmaximal, characteristic
5Q8⋊(C59⋊C16)C211yesmaximal, characteristic
6#21422059C211yesmaximal, characteristic
7C59×#1670C211yesmaximal, characteristic
8#16838130—21no—
9C2×#14273673—21no—
10#16837807C2211yescharacteristic
11#16837804—21no—
12#16837710—22no—
13#16837688—21no—
14C4×Dic236—21no—
15C4×Dic236C2211yescharacteristic
16C4⋊(C59⋊C16)C2211yescharacteristic
17#16837206—22no—
18#16837206—21no—
19C4×C59⋊C16C2211yescharacteristic
20C59×#494C2211yescharacteristic
21C59×#475—21no—
22C8×Q8×C59C2211yescharacteristic
23C59×#335C2211yescharacteristic
24C59×#332—21no—
25C2×Dic236D411yescharacteristic
26#14273727—21no—
27#14273725—41no—
28#14273719—22no—
29#14273717—42no—
30C4×Dic118—22no—
31C4×Dic118—21no—
32#14273692—41no—
33#14273673—42no—
34C2×C59⋊C16—21no—
35C2×C59⋊C16—21no—
36#14273661—21no—
37#14273661D412yes—
38#14273661D411yescharacteristic
39#14273659—22no—
40#14273652—22no—
41#14273652—21no—
42C2×Q16×C59—21no—
43C2×Q16×C59—21no—
44C4⋊Q8×C59—21no—
45C4×Q8×C59D411yescharacteristic
46#106×C59—41no—
47C4⋊C8×C59—21no—
48C4⋊C8×C59—21no—
49C4⋊C8×C59D411yescharacteristic
50#103×C59—41no—
51C4×C8×C59—21no—
52C4×C8×C59C2311yescharacteristic
53C2×Q8×C59—41no—
54C2×Q8×C59D811yescharacteristic
55C2×Dic118—21no—
56C2×Dic118—41no—
57Q16×C59—42no—
58Q16×C59—41no—
59Q16×C59—41no—
60C2×C8×C59—21no—
61C2×C8×C59—41no—
62C2×C8×C59C2×D411yescharacteristic
63C2×C8×C59C2×D411yescharacteristic
64C4⋊C4×C59—21no—
65C4⋊C4×C59—41no—
66C4⋊C4×C59—41no—
67C4⋊C4×C59D811yescharacteristic
68C42×C59—21no—
69C42×C59C2×D411yescharacteristic
70C4⋊Dic59—21no—
71C4⋊Dic59—22no—
72C4⋊Dic59—41no—
73Dic59⋊C4—42no—
74Dic59⋊C4—42no—
75Dic59⋊C4—42no—
76Dic59⋊C4—41no—
77C4×Dic59—42no—
78C4×Dic59—41no—
79Dic236—21no—
80Dic236SD1612yes—
81C59⋊C16—41no—
82C59⋊C16—41no—
83Q8×C59—21no—
84Q8×C59—21no—
85Q8×C59—81no—
86Q8×C59—81no—
87C2×C4×C59—21no—
88C2×C4×C59—21no—
89C2×C4×C59—81no—
90C2×C4×C59#12011yescharacteristic
91C2×C4×C59C2×D811yescharacteristic
92C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
93C2×Dic59—41no—
94C2×Dic59—41no—
95C2×Dic59—42no—
96Dic118—22no—
97Dic118—41no—
98Dic118—81no—
99Dic118—81no—
100C8×C59—21no—
101C8×C59—41no—
102C8×C59—81no—
103C8×C59C2×SD1611yescharacteristic
104C8×C59#13711yescharacteristic
105#12357—591nomaximal, Sylow
106C22×C59#44711yescharacteristic
107C4×C59—21no—
108C4×C59—41no—
109C4×C59—41no—
110C4×C59—161no—
111C4×C59#44811yescharacteristic
112C4×C59#45411yescharacteristic
113C4×C59#50811yescharacteristic
114C4×C59#51011yescharacteristic
115Dic59—42no—
116Dic59—82no—
117Dic59—81no—
118#3383—591no—
119#2241—591no—
120#2209—591no—
121#1670D5911yescharacteristic
122#1364—591no—
123Q8⋊C16—591no—
124#1332—591no—
125C2×C59#162711yescharacteristic
126C2×C59#218911yescharacteristic
127C2×C59#219511yescharacteristic
128#557—1181no—
129C2×Q32—1181no—
130#500—591no—
131#499—1181no—
132#494C2×D5911yescharacteristic
133#479—1182no—
134#475—21no—
135#475—1181no—
136C8×Q8C2×D5911yescharacteristic
137C4×Q16—591no—
138C4×Q16—1181no—
139C4⋊C16—591no—
140#358—1181no—
141#358—1182no—
142C4×C16—591no—
143#335C2×D5911yescharacteristic
144#332—21no—
145C59#1235711yescharacteristic, complemented, Sylow
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—591no—
149C4⋊Q8—21no—
150C4⋊Q8—1181no—
151C4⋊Q8—2361no—
152#125—1182no—
153#125—2362no—
154C4×Q8—1182no—
155C4×Q8—1181no—
156C4×Q8—2361no—
157C4×Q8C22⋊D5911yescharacteristic
158Q32—2362no—
159C2×C16—1181no—
160C2×C16—1181no—
161#107—591no—
162#107—592no—
163#107—1181no—
164#106—41no—
165#106—1182no—
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8C22⋊D5911yescharacteristic
169#103—41no—
170#103—1182no—
171#103—1181no—
172C4×C8—21no—
173C4×C8C22×D5911yescharacteristic
174C2×Q8—41no—
175C2×Q8—1181no—
176C2×Q8—2361no—
177C2×Q8#1174566911yescharacteristic
178Q16—42no—
179Q16—41no—
180Q16—41no—
181Q16—592no—
182Q16—1181no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8D4×D5911yescharacteristic
186C2×C8D4×D5911yescharacteristic
187C4⋊C4—21no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—1181no—
191C4⋊C4—1182no—
192C4⋊C4—2362no—
193C4⋊C4—2362no—
194C4⋊C4—2362no—
195C4⋊C4—2361no—
196C4⋊C4—2361no—
197C4⋊C4#1174566911yescharacteristic
198C42—21no—
199C42—2362no—
200C42—2361no—
201C42C2×C22⋊D5911yescharacteristic
202C16—2361no—
203C16—2361no—
204Q8—21no—
205Q8—21no—
206Q8—81no—
207Q8—81no—
208Q8—1182no—
209Q8—4721no—
210Q8—4721no—
211Q8—2361no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—81no—
215C2×C4—2361no—
216C2×C4—2362no—
217C2×C4—2361no—
218C2×C4#1427369511yescharacteristic
219C2×C4C2×#1174566911yescharacteristic
220C2×C4#1427375211yescharacteristic
221C8—21no—
222C8—41no—
223C8—81no—
224C8SD16×D5911yescharacteristic
225C8#1427375611yescharacteristic
226C22#1683753511yescharacteristic
227C4—21no—
228C4—41no—
229C4—41no—
230C4—161no—
231C4—4722no—
232C4—4721no—
233C4—2362no—
234C4#1683753911yescharacteristic
235C4#1683757311yescharacteristic
236C4#1683783211yescharacteristic
237C4#1683784311yescharacteristic
238C2#2142319711yescharacteristic
239C2#2142603411yescharacteristic
240C2#2142607211yescharacteristic
subgroupG/H
centerC22#16837535
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1670C2
socle, prime radicalC22×C59#447
hypercenter#1670D59
hypocenterC59#12357
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1670

Complements

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

NH ≅ G/Ntype
145. C59#12357semidirect

Maximal subgroups

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

subgroupindex
#214369882
#214263232
#214261432
#214221522
Q8⋊(C59⋊C16)2
#214220592
C59×#16702
#1235759