Finite Group Databasebeta

← #28502210

Subgroups — #28502210

Order 15104.

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

normal subgroups only

This group has 322 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#21437101C211yesmaximal, characteristic, complemented
2#21426351C211yesmaximal, characteristic
3#21423453C211yesmaximal, characteristic
4#21422146C211yesmaximal, characteristic, complemented
5#21422074C211yesmaximal, characteristic, complemented
6#21422060C211yesmaximal, characteristic, complemented
7C59×#2207C211yesmaximal, characteristic
8#16838130—21no—
9C2×#14273671—22no—
10#16837782C2211yescharacteristic
11#16837780—21no—
12#16837612—21no—
13#16837611—22no—
14#16837599—21no—
15C8×Dic118C2211yescharacteristic
16#16837496—21no—
17C4×C59⋊C16C2211yescharacteristic
18#16837155C2211yescharacteristic
19#16837146—21no—
20C2×SD32×C59—21no—
21C59×#493C2211yescharacteristic
22C4×Q16×C59C2211yescharacteristic
23C59×C4⋊C16C2211yescharacteristic
24C59×#357—21no—
25C2×Dic236—21no—
26C2×Dic236D412yes—
27C2×#11745672—22no—
28C2×#11745670—41no—
29#14273727—21no—
30#14273725—22no—
31#14273719—41no—
32#14273717—42no—
33C4×Dic118—41no—
34C4×Dic118D411yescharacteristic
35#14273692—22no—
36#14273671—44no—
37C2×C59⋊C16—21no—
38C4⋊(C59⋊C8)—21no—
39C4⋊(C59⋊C8)D411yescharacteristic
40Dic59⋊C8—21no—
41#14273650—22no—
42#14273648—41no—
43C8×Dic59—21no—
44C2×Q16×C59D411yescharacteristic
45C2×D8×C59—21no—
46C2×D8×C59—21no—
47#127×C59—21no—
48C4×Q8×C59—21no—
49SD32×C59—42no—
50C2×C16×C59—21no—
51#107×C59D411yescharacteristic
52#103×C59—21no—
53C4×C8×C59C2311yescharacteristic
54C2×Q8×C59—21no—
55C2×D4×C59—41no—
56C2×D4×C59—41no—
57C2×D4×C59—41no—
58C2×Dic118—22no—
59C2×Dic118D811yescharacteristic
60Q16×C59—21no—
61Q16×C59—21no—
62D8×C59—42no—
63D8×C59—41no—
64D8×C59—41no—
65C2×C8×C59C2×D411yescharacteristic
66C2×C8×C59C2×D411yescharacteristic
67C4⋊C4×C59—21no—
68C4⋊C4×C59—41no—
69C42×C59—41no—
70C42×C59C2×D411yescharacteristic
71#11745672—82no—
72#11745670—82no—
73C4⋊Dic59—41no—
74C4⋊Dic59—41no—
75C4⋊Dic59D811yescharacteristic
76Dic59⋊C4—21no—
77Dic59⋊C4—42no—
78Dic59⋊C4—42no—
79Dic59⋊C4—42no—
80C4×Dic59—21no—
81C4×Dic59—42no—
82C2×C59⋊C8—21no—
83C2×C59⋊C8—41no—
84Dic236—42no—
85Dic236—41no—
86Dic236—41no—
87C16×C59—41no—
88C59⋊C16—22no—
89C23×C59—81no—
90Q8×C59—41no—
91Q8×C59—41no—
92D4×C59—81no—
93D4×C59—81no—
94D4×C59—81no—
95D4×C59—81no—
96D4×C59—161no—
97C2×C4×C59—41no—
98C2×C4×C59—41no—
99C2×C4×C59#12011yescharacteristic
100C2×C4×C59C2×D811yescharacteristic
101C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
102C2×Dic59—21no—
103C2×Dic59—21no—
104C2×Dic59—42no—
105Dic118—21no—
106Dic118—42no—
107Dic118—42no—
108Dic118SD3212yescomplemented
109C8×C59—21no—
110C8×C59#13511yescharacteristic
111C8×C59AGL(1,ℤ/8)11yescharacteristic
112C59⋊C8—41no—
113C59⋊C8—81no—
114#12330—591nomaximal, Sylow
115C22×C59—161no—
116C22×C59—161no—
117C22×C59—161no—
118C22×C59#44711yescharacteristic
119C4×C59—21no—
120C4×C59—81no—
121C4×C59—81no—
122C4×C59#44911yescharacteristic
123C4×C59#45311yescharacteristic
124C4×C59C2×SD3211yescharacteristic
125C4×C59#51011yescharacteristic
126Dic59—22no—
127Dic59—41no—
128Dic59—82no—
129#3395—591no—
130#2248—591no—
131#2207D5911yescharacteristic
132#1668—591no—
133#1362—591no—
134Q8⋊C16—591no—
135#1332—591no—
136C2×C59—321no—
137C2×C59#162111yescharacteristic
138C2×C59#218511yescharacteristic
139C2×C59#219311yescharacteristic
140#557—1181no—
141C2×SD32—21no—
142C2×SD32—1182no—
143#494—591no—
144#494—1181no—
145#493C2×D5911yescharacteristic
146#462—1182no—
147#462—1181no—
148#460—1181no—
149C8×Q8—591no—
150C4×Q16—1181no—
151C4×Q16C2×D5911yescharacteristic
152C4⋊C16C2×D5911yescharacteristic
153#357—21no—
154C4×C16—591no—
155#335—591no—
156#331—1181no—
157C59#1233011yescharacteristic, complemented, Sylow
158C2×Q16—592no—
159C2×Q16—1182no—
160C2×Q16—1181no—
161C2×Q16C22⋊D5911yescharacteristic
162C2×SD16—2361no—
163C2×D8—21no—
164C2×D8—21no—
165C4⋊Q8—1182no—
166C4⋊Q8—1181no—
167#127—21no—
168#125—2361no—
169#125—2362no—
170C4×Q8—21no—
171C4×Q8—591no—
172C4×Q8—1182no—
173C4×Q8—2361no—
174SD32—42no—
175SD32—2364no—
176C2×C16—21no—
177C2×C16—1181no—
178#107C22⋊D5911yescharacteristic
179C4⋊C8—591no—
180C4⋊C8—1181no—
181C4⋊C8—1181no—
182#103—21no—
183#103—1182no—
184#102—2361no—
185C4×C8—1181no—
186C4×C8C22×D5911yescharacteristic
187C2×Q8—21no—
188C2×Q8—591no—
189C2×Q8—1182no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193Q16—21no—
194Q16—21no—
195Q16—4722no—
196Q16—2362no—
197Q16—2361no—
198Q16—2361no—
199SD16—4722no—
200D8—42no—
201D8—41no—
202D8—41no—
203C2×C8—1181no—
204C2×C8—2361no—
205C2×C8D4×D5911yescharacteristic
206C2×C8C2×C22⋊D5911yescharacteristic
207C4⋊C4—21no—
208C4⋊C4—41no—
209C4⋊C4—591no—
210C4⋊C4—1181no—
211C4⋊C4—2362no—
212C4⋊C4—2362no—
213C4⋊C4—2362no—
214C4⋊C4—2361no—
215C4⋊C4—2361no—
216C42—41no—
217C42—1181no—
218C42—2362no—
219C42D4×D5911yescharacteristic
220C16—41no—
221C16—1182no—
222C23—81no—
223Q8—41no—
224Q8—41no—
225Q8—592no—
226Q8—1181no—
227Q8—2362no—
228Q8—2362no—
229D4—81no—
230D4—81no—
231D4—81no—
232D4—81no—
233D4—161no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—1181no—
237C2×C4—1181no—
238C2×C4—2362no—
239C2×C4#1427369511yescharacteristic
240C2×C4D8×D5911yescharacteristic
241C2×C4#1427375111yescharacteristic
242C8—21no—
243C8—4721no—
244C8—2361no—
245C8#1427374711yescharacteristic
246C8#1427375311yescharacteristic
247C22—161no—
248C22—161no—
249C22—161no—
250C22#1683753411yescharacteristic
251C4—21no—
252C4—81no—
253C4—81no—
254C4—1182no—
255C4—4722no—
256C4—2361no—
257C4#1683754511yescharacteristic
258C4#1683756811yescharacteristic
259C4SD32×D5911yescharacteristic
260C4#1683784111yescharacteristic
261C2—321no—
262C2#2142316011yescharacteristic
263C2#2142601111yescharacteristic
264C2#2142605711yescharacteristic
subgroupG/H
centerC22#16837534
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#2207C2
socleC22×C59#447
prime radicalC59×#493C22
hypercenter#2207D59
hypocenterC59#12330
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2207

Complements

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

NH ≅ G/Ntype
1. #21437101C2semidirect
4. #21422146C2semidirect
5. #21422074C2semidirect
6. #21422060C2semidirect
108. Dic118SD32semidirect
157. C59#12330semidirect

Maximal subgroups

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

subgroupindex
#214371012
#214263512
#214234532
#214221462
#214220742
#214220602
C59×#22072
#1233059