Finite Group Databasebeta

← #28502560

Subgroups — #28502560

Order 15104.

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

normal subgroups only

This group has 306 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#21426319C211yesmaximal, characteristic
2#21426238C211yesmaximal, characteristic, complemented
3#21423490C211yesmaximal, characteristic, complemented
4#21422142C211yesmaximal, characteristic
5Q8⋊(C59⋊C16)C211yesmaximal, characteristic, complemented
6#21422060C211yesmaximal, characteristic, complemented
7C59×#3396C211yesmaximal, characteristic
8#16837807C2211yescharacteristic
9#16837612—21no—
10#16837229—21no—
11C4⋊(C59⋊C16)C2211yescharacteristic
12#16837206—21no—
13#16837203—22no—
14C4×C59⋊C16C2211yescharacteristic
15#16837155C2211yescharacteristic
16#16837147—21no—
17C59×#559—21no—
18C59×#495—21no—
19C59×#493C2211yescharacteristic
20C59×#463—22no—
21C59×#460—21no—
22C8×Q8×C59C2211yescharacteristic
23C4×Q16×C59C2211yescharacteristic
24C4×D8×C59—21no—
25C2×#11745672—41no—
26#14273727—21no—
27C2×C59⋊C16—21no—
28C2×C59⋊C16—21no—
29#14273661—21no—
30#14273661—21no—
31C4⋊(C59⋊C8)—21no—
32#14273650—41no—
33C2×Q16×C59D411yescharacteristic
34C2×SD16×C59—22no—
35C2×D8×C59—21no—
36C2×D8×C59D412yes—
37#127×C59—21no—
38#124×C59—22no—
39#124×C59—41no—
40#123×C59—42no—
41C4×Q8×C59—21no—
42C4×Q8×C59D411yescharacteristic
43C4×D4×C59—22no—
44C4×D4×C59—41no—
45#107×C59D411yescharacteristic
46C4⋊C8×C59—21no—
47C4⋊C8×C59D411yescharacteristic
48#103×C59—21no—
49#102×C59—22no—
50C4×C8×C59—21no—
51C4×C8×C59C2311yescharacteristic
52C2×Q8×C59—21no—
53C2×Q8×C59D811yescharacteristic
54C2×D4×C59—22no—
55C2×D4×C59—41no—
56C2×D4×C59—41no—
57C22×C4×C59—42no—
58C2×Dic118—41no—
59Q16×C59—21no—
60Q16×C59—21no—
61SD16×C59—82no—
62D8×C59—42no—
63D8×C59—41no—
64D8×C59—41no—
65C2×C8×C59—21no—
66C2×C8×C59C2×D411yescharacteristic
67C2×C8×C59C2×D411yescharacteristic
68C4⋊C4×C59—21no—
69C4⋊C4×C59—21no—
70C4⋊C4×C59—41no—
71C4⋊C4×C59D811yescharacteristic
72C22⋊C4×C59—42no—
73C22⋊C4×C59—42no—
74C22⋊C4×C59—42no—
75C42×C59—21no—
76C42×C59—41no—
77C42×C59C2×D411yescharacteristic
78#11745672—82no—
79C4⋊Dic59—41no—
80C4⋊Dic59—41no—
81C2×C59⋊C8—41no—
82C59⋊C16—41no—
83C59⋊C16—41no—
84C23×C59—42no—
85Q8×C59—21no—
86Q8×C59—21no—
87Q8×C59—41no—
88Q8×C59—41no—
89D4×C59—42no—
90D4×C59—42no—
91D4×C59—81no—
92D4×C59—81no—
93D4×C59—161no—
94C2×C4×C59—21no—
95C2×C4×C59—21no—
96C2×C4×C59—41no—
97C2×C4×C59—41no—
98C2×C4×C59—162no—
99C2×C4×C59#12011yescharacteristic
100C2×C4×C59C2×D811yescharacteristic
101C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
102C2×Dic59—81no—
103Dic118—81no—
104Dic118—81no—
105C8×C59—21no—
106C8×C59—41no—
107C8×C59#13511yescharacteristic
108C8×C59#13711yescharacteristic
109C59⋊C8—81no—
110#12381—591nomaximal, Sylow
111C22×C59—82no—
112C22×C59—82no—
113C22×C59—82no—
114C22×C59#44711yescharacteristic
115C4×C59—21no—
116C4×C59—41no—
117C4×C59—41no—
118C4×C59—81no—
119C4×C59—81no—
120C4×C59#45211yescharacteristic
121C4×C59#45511yescharacteristic
122C4×C59#50811yescharacteristic
123C4×C59#50911yescharacteristic
124Dic59—161no—
125#3396D5911yescharacteristic
126#2240—591no—
127#2225—591no—
128#1674—591no—
129#1361—591no—
130Q8⋊C16—591no—
131#1332—591no—
132C2×C59—162no—
133C2×C59#164111yescharacteristic
134C2×C59#218611yescharacteristic
135C2×C59#219411yescharacteristic
136#559—21no—
137#500—591no—
138#495—21no—
139#493C2×D5911yescharacteristic
140#463—22no—
141#462—1181no—
142#460—21no—
143C8×Q8C2×D5911yescharacteristic
144C4×Q16C2×D5911yescharacteristic
145C4×D8—21no—
146#367—1181no—
147C4⋊C16—591no—
148#358—1181no—
149#357—1182no—
150C4×C16—591no—
151#335—591no—
152#332—1181no—
153C59#1238111yescharacteristic, complemented, Sylow
154C2×Q16—2361no—
155C2×Q16C22⋊D5911yescharacteristic
156C2×SD16—22no—
157C2×D8—21no—
158C2×D8C22⋊D5912yes—
159C4⋊Q8—1181no—
160#127—21no—
161#124—22no—
162#124—41no—
163#123—42no—
164C4×Q8—21no—
165C4×Q8C22⋊D5911yescharacteristic
166C4×D4—22no—
167C4×D4—41no—
168C2×C16—1181no—
169C2×C16—1181no—
170#107—1181no—
171#107—1181no—
172#107C22⋊D5911yescharacteristic
173C4⋊C8—21no—
174C4⋊C8—1181no—
175C4⋊C8C22⋊D5911yescharacteristic
176#103—21no—
177#103—2361no—
178#102—22no—
179C4×C8—21no—
180C4×C8C22×D5911yescharacteristic
181C2×Q8—21no—
182C2×Q8—2361no—
183C2×Q8#1174566911yescharacteristic
184C2×D4—22no—
185C2×D4—41no—
186C2×D4—41no—
187C22×C4—42no—
188Q16—21no—
189Q16—21no—
190Q16—4722no—
191SD16—82no—
192D8—42no—
193D8—41no—
194D8—41no—
195C2×C8—21no—
196C2×C8—2361no—
197C2×C8C2×C22⋊D5911yescharacteristic
198C2×C8C2×C22⋊D5911yescharacteristic
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—41no—
202C4⋊C4—2361no—
203C4⋊C4—2361no—
204C4⋊C4#1174566911yescharacteristic
205C22⋊C4—42no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C42—21no—
209C42—41no—
210C42C2×C22⋊D5911yescharacteristic
211C16—2361no—
212C16—2361no—
213C23—42no—
214Q8—21no—
215Q8—21no—
216Q8—41no—
217Q8—41no—
218Q8—4721no—
219Q8—4721no—
220D4—42no—
221D4—42no—
222D4—81no—
223D4—81no—
224D4—161no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—162no—
230C2×C4—4721no—
231C2×C4#1427369611yescharacteristic
232C2×C4C2×#1174566911yescharacteristic
233C2×C4#1427375311yescharacteristic
234C8—21no—
235C8—41no—
236C8—4721no—
237C8#1427374711yescharacteristic
238C8#1427375711yescharacteristic
239C22—82no—
240C22—82no—
241C22—82no—
242C22#1683753611yescharacteristic
243C4—21no—
244C4—41no—
245C4—41no—
246C4—81no—
247C4—81no—
248C4—9441no—
249C4#1683756411yescharacteristic
250C4#1683758011yescharacteristic
251C4#1683783211yescharacteristic
252C4#1683783711yescharacteristic
253C2—162no—
254C2#2142328911yescharacteristic
255C2#2142601911yescharacteristic
256C2#2142606711yescharacteristic
subgroupG/H
centerC22#16837536
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#3396C2
socleC22×C59#447
prime radicalC59×#493C22
hypercenter#3396D59
hypocenterC59#12381
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3396

Complements

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

NH ≅ G/Ntype
2. #21426238C2semidirect
3. #21423490C2semidirect
5. Q8⋊(C59⋊C16)C2semidirect
6. #21422060C2semidirect
153. C59#12381semidirect

Maximal subgroups

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

subgroupindex
#214263192
#214262382
#214234902
#214221422
Q8⋊(C59⋊C16)2
#214220602
C59×#33962
#1238159