Finite Group Databasebeta

← #28488170

Subgroups — #28488170

Order 15104.

406 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 127 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#21424268C211yesmaximal, characteristic
2#21422781C211yesmaximal, characteristic
3#21422777C211yesmaximal, characteristic
4#21422768C211yesmaximal, characteristic
5#21422768C211yesmaximal, characteristic
6#21422388C211yesmaximal, characteristic
7C2×C59×#487C211yesmaximal, characteristic
8#16837455C2211yescharacteristic
9#16837455C2211yescharacteristic
10C2×#14273663—22no—
11#16837426C2211yescharacteristic
12#16837426C2211yescharacteristic
13#16837151—22no—
14#16837148—22no—
15#16837148—22no—
16#16837148—22no—
17#16837124—22no—
18C2×C4⋊Q8×C59C2211yescharacteristic
19C2×C4⋊Q8×C59C2211yescharacteristic
20C59×#487—22no—
21C59×#487C414yes—
22C2×#103×C59—22no—
23C2×C4×C8×C59C2211yescharacteristic
24C2×#11745664—21no—
25C2×#11745664—21no—
26#14273663—44no—
27C4⋊(C59⋊C8)—21no—
28C4⋊(C59⋊C8)—21no—
29C4⋊(C59⋊C8)—21no—
30C4⋊(C59⋊C8)—21no—
31C4⋊(C59⋊C8)D412yes—
32C4⋊(C59⋊C8)D412yes—
33C22⋊(C59⋊C8)—21no—
34C22⋊(C59⋊C8)—21no—
35C4×C59⋊C8—21no—
36C4×C59⋊C8—21no—
37C22×Q8×C59—21no—
38C22×Q8×C59—21no—
39C22×C8×C59D412yes—
40C4⋊Q8×C59—21no—
41C4⋊Q8×C59—21no—
42C4⋊Q8×C59—21no—
43C4⋊Q8×C59—21no—
44C4⋊Q8×C59—21no—
45C4⋊Q8×C59—21no—
46C4⋊Q8×C59C2×C412yes—
47C4⋊Q8×C59C2×C412yes—
48C2×C4⋊C4×C59—21no—
49C2×C4⋊C4×C59—21no—
50C2×C4⋊C4×C59—21no—
51C2×C4⋊C4×C59—21no—
52C2×C42×C59C2311yescharacteristic
53#103×C59—28no—
54C4×C8×C59C2×C412yes—
55C4×C8×C59D412yes—
56C2×Q8×C59—22no—
57C2×Q8×C59—22no—
58C2×Q8×C59—41no—
59C2×Q8×C59—41no—
60C2×Q8×C59—42no—
61C2×Q8×C59—42no—
62C2×Q8×C59—82no—
63C22×C4×C59—41no—
64C22×C4×C59—41no—
65C22×C4×C59C2×D411yescharacteristic
66C22×C4×C59#4111yescharacteristic
67C22×C4×C59#4111yescharacteristic
68C2×C8×C59—22no—
69C2×C8×C59—22no—
70C2×C8×C59C22⋊C414yes—
71C4⋊C4×C59—22no—
72C4⋊C4×C59—22no—
73C4⋊C4×C59—22no—
74C4⋊C4×C59—22no—
75C4⋊C4×C59—41no—
76C4⋊C4×C59—41no—
77C4⋊C4×C59—41no—
78C4⋊C4×C59—41no—
79C42×C59C22×C411yescharacteristic
80C42×C59C2×D411yescharacteristic
81C42×C59C2×D411yescharacteristic
82C42×C59C2×D411yescharacteristic
83#11745664—42no—
84#11745664—42no—
85C2×C59⋊C8—22no—
86C2×C59⋊C8—22no—
87C2×C59⋊C8—41no—
88C2×C59⋊C8—41no—
89C23×C59#12411yescharacteristic
90Q8×C59—44no—
91Q8×C59—82no—
92Q8×C59—82no—
93Q8×C59—81no—
94Q8×C59—81no—
95C2×C4×C59—22no—
96C2×C4×C59—22no—
97C2×C4×C59—42no—
98C2×C4×C59—42no—
99C2×C4×C59—82no—
100C2×C4×C59—82no—
101C2×C4×C59C2×C22⋊C411yescharacteristic
102C2×C4×C59C4×D411yescharacteristic
103C2×C4×C59C4×D411yescharacteristic
104C2×C4×C59#12111yescharacteristic
105C2×C4×C59#12111yescharacteristic
106C2×C4×C59#12711yescharacteristic
107C2×C4×C59AGL(1,ℤ/8)12yes—
108C2×C4×C59#13712yes—
109C8×C59—24no—
110C59⋊C8—42no—
111C59⋊C8—42no—
112C59⋊C8—81no—
113C59⋊C8—81no—
114#9775—591nomaximal, Sylow
115C22×C59—21no—
116C22×C59—21no—
117C22×C59#39011yescharacteristic
118C22×C59#48912yes—
119C4×C59—21no—
120C4×C59—21no—
121C4×C59—82no—
122C4×C59—82no—
123C4×C59#41712yes—
124C4×C59#47112yes—
125C4×C59#47312yes—
126C4×C59#49611yescharacteristic
127C4×C59#49711yescharacteristic
128C2×#487D5911yescharacteristic
129#1848—591no—
130#1549—591no—
131#1548—591no—
132#1546—591no—
133#1546—591no—
134#1452—591no—
135C2×C59—22no—
136C2×C59#195911yescharacteristic
137C2×C59#197711yescharacteristic
138C2×C59#197911yescharacteristic
139C2×C4⋊Q8C2×D5911yescharacteristic
140C2×C4⋊Q8C2×D5911yescharacteristic
141#487—22no—
142#487Dic5914yes—
143#431—591no—
144#431—591no—
145C2×#108—1182no—
146#423—591no—
147#423—591no—
148C2×#103—22no—
149C2×C4×C8C2×D5911yescharacteristic
150#333—1182no—
151#332—1182no—
152#332—1182no—
153#332—1182no—
154#322—1182no—
155C59#977511yescharacteristic, complemented, Sylow
156C22×Q8—21no—
157C22×Q8—21no—
158C2×M16—1181no—
159C2×M16—1181no—
160C22×C8C22⋊D5912yes—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8C2×Dic5912yes—
168C4⋊Q8C2×Dic5912yes—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C42C22×D5911yescharacteristic
174#108—2364no—
175C4⋊C8—592no—
176C4⋊C8—592no—
177C4⋊C8—1181no—
178C4⋊C8—1181no—
179C4⋊C8—1181no—
180C4⋊C8—1181no—
181#103—28no—
182C22⋊C8—1181no—
183C22⋊C8—1181no—
184C4×C8—1181no—
185C4×C8—1181no—
186C4×C8C2×Dic5912yes—
187C4×C8C22⋊D5912yes—
188C2×Q8—22no—
189C2×Q8—22no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—82no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4#1174568711yescharacteristic
198C22×C4#1174568711yescharacteristic
199C22×C4C2×C22⋊D5911yescharacteristic
200M16—2362no—
201M16—2362no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—1182no—
205C2×C8—1182no—
206C2×C8—2361no—
207C2×C8—2361no—
208C2×C8C22⋊Dic5914yes—
209C4⋊C4—22no—
210C4⋊C4—22no—
211C4⋊C4—22no—
212C4⋊C4—22no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C42D4×D5911yescharacteristic
218C42D4×D5911yescharacteristic
219C42C22×Dic5911yescharacteristic
220C42C2×C22⋊D5911yescharacteristic
221C23#1427371411yescharacteristic
222Q8—44no—
223Q8—82no—
224Q8—82no—
225Q8—81no—
226Q8—81no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—82no—
232C2×C4—82no—
233C2×C4C2×C22⋊Dic5911yescharacteristic
234C2×C4D4×Dic5911yescharacteristic
235C2×C4D4×Dic5911yescharacteristic
236C2×C4#1427369811yescharacteristic
237C2×C4#1427369811yescharacteristic
238C2×C4#1427372311yescharacteristic
239C2×C4#1427375212yes—
240C2×C4#1427375712yes—
241C8—24no—
242C8—4721no—
243C8—4721no—
244C8—2362no—
245C8—2362no—
246C22—21no—
247C22—21no—
248C22#1683730511yescharacteristic
249C22#1683776112yes—
250C4—21no—
251C4—21no—
252C4—82no—
253C4—82no—
254C4#1683740312yes—
255C4#1683766212yes—
256C4#1683767412yes—
257C4#1683779211yescharacteristic
258C4#1683779711yescharacteristic
259C2—22no—
260C2#2142491311yescharacteristic
261C2#2142500111yescharacteristic
262C2#2142501211yescharacteristic
subgroupG/H
centerC22#16837305
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC2×C59×#487C2
socleC22×C59#390
prime radicalC23×C59#124
cosocleC2×C42×C59C23
hypercenterC2×#487D59
hypocenterC59#9775
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#487

Complements

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

NH ≅ G/Ntype
155. C59#9775semidirect

Maximal subgroups

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

subgroupindex
#214242682
#214227812
#214227772
#214227682
#214223882
C2×C59×#4872
#977559