Finite Group Databasebeta

← #28502232

Subgroups — #28502232

Order 15104.

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

normal subgroups only

This group has 324 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#21426351C211yesmaximal, characteristic
2#21426146C211yesmaximal, characteristic, complemented
3#21423466C211yesmaximal, characteristic, complemented
4#21422142C211yesmaximal, characteristic
5Q8⋊(C59⋊C16)C211yesmaximal, characteristic, complemented
6#21422060C211yesmaximal, characteristic, complemented
7C59×#3396C211yesmaximal, characteristic
8C2×#14273673—21no—
9C2×#14273671—22no—
10#16837782C2211yescharacteristic
11#16837687—21no—
12C4⋊(C59⋊C16)C2211yescharacteristic
13#16837205—21no—
14C4×C59⋊C16C2211yescharacteristic
15#16837155C2211yescharacteristic
16#16837149—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×Dic236—21no—
26C2×Dic236—21no—
27#14273727—21no—
28#14273673—42no—
29#14273671—44no—
30C2×C59⋊C16—21no—
31C2×C59⋊C16—21no—
32#14273658—41no—
33C4⋊(C59⋊C8)—21no—
34#14273651—41no—
35C2×Q16×C59D411yescharacteristic
36C2×SD16×C59—22no—
37C2×D8×C59—21no—
38C2×D8×C59D412yes—
39#127×C59—21no—
40#124×C59—22no—
41#124×C59—41no—
42#123×C59—42no—
43C4×Q8×C59—21no—
44C4×Q8×C59D411yescharacteristic
45C4×D4×C59—22no—
46C4×D4×C59—41no—
47#107×C59D411yescharacteristic
48C4⋊C8×C59—21no—
49C4⋊C8×C59D411yescharacteristic
50#103×C59—21no—
51#102×C59—22no—
52C4×C8×C59—21no—
53C4×C8×C59C2311yescharacteristic
54C2×Q8×C59—21no—
55C2×Q8×C59D811yescharacteristic
56C2×D4×C59—22no—
57C2×D4×C59—41no—
58C2×D4×C59—41no—
59C22×C4×C59—42no—
60C2×Dic118—41no—
61Q16×C59—21no—
62Q16×C59—21no—
63SD16×C59—82no—
64D8×C59—42no—
65D8×C59—41no—
66D8×C59—41no—
67C2×C8×C59—21no—
68C2×C8×C59C2×D411yescharacteristic
69C2×C8×C59C2×D411yescharacteristic
70C4⋊C4×C59—21no—
71C4⋊C4×C59—21no—
72C4⋊C4×C59—41no—
73C4⋊C4×C59D811yescharacteristic
74C22⋊C4×C59—42no—
75C22⋊C4×C59—42no—
76C22⋊C4×C59—42no—
77C42×C59—21no—
78C42×C59—41no—
79C42×C59C2×D411yescharacteristic
80C4⋊Dic59—41no—
81C4⋊Dic59—41no—
82C2×C59⋊C8—41no—
83Dic236—42no—
84Dic236—41no—
85Dic236—41no—
86C59⋊C16—22no—
87C59⋊C16—41no—
88C23×C59—42no—
89Q8×C59—21no—
90Q8×C59—21no—
91Q8×C59—41no—
92Q8×C59—41no—
93D4×C59—42no—
94D4×C59—42no—
95D4×C59—81no—
96D4×C59—81no—
97D4×C59—161no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—41no—
101C2×C4×C59—41no—
102C2×C4×C59—162no—
103C2×C4×C59#12011yescharacteristic
104C2×C4×C59C2×D811yescharacteristic
105C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
106C2×Dic59—81no—
107Dic118—81no—
108Dic118—81no—
109C8×C59—21no—
110C8×C59—41no—
111C8×C59#13511yescharacteristic
112C8×C59AGL(1,ℤ/8)11yescharacteristic
113C59⋊C8—81no—
114#12333—591nomaximal, Sylow
115C22×C59—82no—
116C22×C59—82no—
117C22×C59—82no—
118C22×C59#44711yescharacteristic
119C4×C59—21no—
120C4×C59—41no—
121C4×C59—41no—
122C4×C59—81no—
123C4×C59—81no—
124C4×C59#44911yescharacteristic
125C4×C59#45311yescharacteristic
126C4×C59#50811yescharacteristic
127C4×C59#51011yescharacteristic
128Dic59—161no—
129#3396D5911yescharacteristic
130#2248—591no—
131#2209—591no—
132#1670—591no—
133#1361—591no—
134Q8⋊C16—591no—
135#1332—591no—
136C2×C59—162no—
137C2×C59#162111yescharacteristic
138C2×C59#218911yescharacteristic
139C2×C59#219411yescharacteristic
140#559—21no—
141C2×Q32—1181no—
142C2×SD32—1182no—
143#495—21no—
144#494—591no—
145#493C2×D5911yescharacteristic
146#475—1181no—
147#463—22no—
148#460—21no—
149C8×Q8C2×D5911yescharacteristic
150C4×Q16C2×D5911yescharacteristic
151C4×D8—21no—
152C4⋊C16—591no—
153#358—1181no—
154C4×C16—591no—
155#335—591no—
156#332—1181no—
157C59#1233311yescharacteristic, complemented, Sylow
158C2×Q16—1181no—
159C2×Q16—1181no—
160C2×Q16C22⋊D5911yescharacteristic
161C2×SD16—22no—
162C2×D8—21no—
163C2×D8C22⋊D5912yes—
164C4⋊Q8—1181no—
165#127—21no—
166#124—22no—
167#124—41no—
168#123—42no—
169C4×Q8—21no—
170C4×Q8C22⋊D5911yescharacteristic
171C4×D4—22no—
172C4×D4—41no—
173Q32—2362no—
174SD32—2364no—
175C2×C16—1181no—
176C2×C16—1181no—
177#107C22⋊D5911yescharacteristic
178#106—2361no—
179C4⋊C8—21no—
180C4⋊C8—1181no—
181C4⋊C8C22⋊D5911yescharacteristic
182#103—21no—
183#103—2361no—
184#102—22no—
185C4×C8—21no—
186C4×C8C22×D5911yescharacteristic
187C2×Q8—21no—
188C2×Q8—2361no—
189C2×Q8#1174566911yescharacteristic
190C2×D4—22no—
191C2×D4—41no—
192C2×D4—41no—
193C22×C4—42no—
194Q16—21no—
195Q16—21no—
196Q16—2362no—
197Q16—2361no—
198Q16—2361no—
199SD16—82no—
200D8—42no—
201D8—41no—
202D8—41no—
203C2×C8—21no—
204C2×C8—2361no—
205C2×C8C2×C22⋊D5911yescharacteristic
206C2×C8C2×C22⋊D5911yescharacteristic
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—41no—
210C4⋊C4—2361no—
211C4⋊C4—2361no—
212C4⋊C4#1174566911yescharacteristic
213C22⋊C4—42no—
214C22⋊C4—42no—
215C22⋊C4—42no—
216C42—21no—
217C42—41no—
218C42C2×C22⋊D5911yescharacteristic
219C16—1182no—
220C16—2361no—
221C23—42no—
222Q8—21no—
223Q8—21no—
224Q8—41no—
225Q8—41no—
226Q8—4721no—
227Q8—4721no—
228D4—42no—
229D4—42no—
230D4—81no—
231D4—81no—
232D4—161no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—162no—
238C2×C4—4721no—
239C2×C4#1427369611yescharacteristic
240C2×C4C2×#1174566911yescharacteristic
241C2×C4#1427375311yescharacteristic
242C8—21no—
243C8—41no—
244C8—4721no—
245C8#1427374711yescharacteristic
246C8#1427375211yescharacteristic
247C22—82no—
248C22—82no—
249C22—82no—
250C22#1683753611yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—81no—
255C4—81no—
256C4—9441no—
257C4#1683754711yescharacteristic
258C4#1683757011yescharacteristic
259C4#1683783211yescharacteristic
260C4#1683784211yescharacteristic
261C2—162no—
262C2#2142316111yescharacteristic
263C2#2142603611yescharacteristic
264C2#2142606711yescharacteristic
subgroupG/H
centerC22#16837536
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#3396C2
socleC22×C59#447
prime radicalC59×#493C22
hypercenter#3396D59
hypocenterC59#12333
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. #21426146C2semidirect
3. #21423466C2semidirect
5. Q8⋊(C59⋊C16)C2semidirect
6. #21422060C2semidirect
157. C59#12333semidirect

Maximal subgroups

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

subgroupindex
#214263512
#214261462
#214234662
#214221422
Q8⋊(C59⋊C16)2
#214220602
C59×#33962
#1233359