Finite Group Databasebeta

← #28502059

Subgroups — #28502059

Order 15104.

274 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 59 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#21426313C211yesmaximal, characteristic, complemented
2#21426308C211yesmaximal, characteristic
3#21425676C211yesmaximal, characteristic, complemented
4#21422090C211yesmaximal, characteristic, complemented
5#21422087C211yesmaximal, characteristic
6#21422049C211yesmaximal, characteristic, complemented
7C59×#3263C211yesmaximal, characteristic
8#16837451C2211yescharacteristic
9#16837224—21no—
10#16837224—21no—
11#16837218—21no—
12#16837215—21no—
13#16837184C2211yescharacteristic
14#16837184C2211yescharacteristic
15#16837172—21no—
16#16837167—21no—
17C59⋊#338C2211yescharacteristic
18C59×#542—21no—
19C59×#476—21no—
20C59×#475—21no—
21C59×#470C2211yescharacteristic
22C59×#469C2211yescharacteristic
23C59×#465—21no—
24C59×#464—21no—
25C59×#463—21no—
26C59×#424C2211yescharacteristic
27C2×#11745664—21no—
28C2×#11745664—21no—
29#14273667—21no—
30#14273667—21no—
31#14273663—21no—
32#14273663—21no—
33C2×Q16×C59D411yescharacteristic
34C2×SD16×C59—21no—
35C2×D8×C59D411yescharacteristic
36C2×M16×C59C2311yescharacteristic
37C22×C8×C59D411yescharacteristic
38C4⋊Q8×C59—21no—
39#126×C59—41no—
40#124×C59—21no—
41#124×C59—41no—
42#123×C59—41no—
43C22⋊Q8×C59—21no—
44C22⋊Q8×C59—41no—
45#121×C59—21no—
46C4×Q8×C59—21no—
47C4×D4×C59—21no—
48#117×C59D411yescharacteristic
49#106×C59—21no—
50#106×C59D411yescharacteristic
51#106×C59D411yescharacteristic
52C4⋊C8×C59—21no—
53C4⋊C8×C59—21no—
54#103×C59—21no—
55#103×C59—21no—
56#102×C59—21no—
57#102×C59—21no—
58C2×Q8×C59—21no—
59C2×Q8×C59—41no—
60C2×D4×C59—21no—
61C2×D4×C59—41no—
62C22×C4×C59—41no—
63C22×C4×C59C2×D411yescharacteristic
64Q16×C59—41no—
65SD16×C59—81no—
66D8×C59—41no—
67M16×C59—41no—
68C2×C8×C59—21no—
69C2×C8×C59—41no—
70C2×C8×C59C2×D411yescharacteristic
71C2×C8×C59C2×D411yescharacteristic
72C4⋊C4×C59—21no—
73C4⋊C4×C59—21no—
74C4⋊C4×C59—21no—
75C4⋊C4×C59—41no—
76C4⋊C4×C59—41no—
77C4⋊C4×C59—41no—
78C22⋊C4×C59—41no—
79C22⋊C4×C59—41no—
80C22⋊C4×C59—41no—
81C22⋊C4×C59—41no—
82C22⋊C4×C59—41no—
83C22⋊C4×C59SD1611yescharacteristic
84C22⋊C4×C59SD1611yescharacteristic
85C42×C59—21no—
86C42×C59—41no—
87#11745664—41no—
88#11745664—41no—
89#11745664—41no—
90#11745664—41no—
91C2×C59⋊C8—41no—
92C2×C59⋊C8—41no—
93C59⋊C16—41no—
94C59⋊C16—41no—
95C23×C59—41no—
96C23×C59C2×SD1611yescharacteristic
97Q8×C59—41no—
98Q8×C59—41no—
99Q8×C59—161no—
100D4×C59—41no—
101D4×C59—41no—
102D4×C59—161no—
103C2×C4×C59—21no—
104C2×C4×C59—21no—
105C2×C4×C59—21no—
106C2×C4×C59—21no—
107C2×C4×C59—41no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—161no—
111C2×C4×C59#12011yescharacteristic
112C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
113C8×C59—21no—
114C8×C59—21no—
115C8×C59—41no—
116C59⋊C8—81no—
117C59⋊C8—81no—
118#12306—591nomaximal, Sylow
119C22×C59—21no—
120C22×C59—41no—
121C22×C59—81no—
122C22×C59—81no—
123C22×C59—81no—
124C22×C59#45011yescharacteristic
125C4×C59—21no—
126C4×C59—41no—
127C4×C59—41no—
128C4×C59—81no—
129C4×C59—81no—
130C4×C59—81no—
131C4×C59#45411yescharacteristic
132C4×C59#45811yescharacteristic
133#3263D5911yescharacteristic
134#2239—591no—
135#2238—591no—
136#2103—591no—
137#1342—591no—
138#1341—591no—
139#1328—591no—
140C2×C59—21no—
141C2×C59—41no—
142C2×C59—161no—
143C2×C59#161611yescharacteristic
144#542—21no—
145#476—21no—
146#475—21no—
147#470C2×D5911yescharacteristic
148#469C2×D5911yescharacteristic
149#465—21no—
150#464—21no—
151#463—21no—
152#430—591no—
153#424C2×D5911yescharacteristic
154#365—1181no—
155#365—1181no—
156#362—1181no—
157#361—1181no—
158#349—591no—
159#349—591no—
160#344—1181no—
161#341—1181no—
162#338—591no—
163C59#1230611yescharacteristic, complemented, Sylow
164C2×Q16C22⋊D5911yescharacteristic
165C2×SD16—21no—
166C2×D8C22⋊D5911yescharacteristic
167C2×M16—1181no—
168C2×M16—1181no—
169C2×M16C22×D5911yescharacteristic
170C22×C8C22⋊D5911yescharacteristic
171C4⋊Q8—21no—
172#126—41no—
173#124—21no—
174#124—41no—
175#123—41no—
176C22⋊Q8—21no—
177C22⋊Q8—41no—
178#121—21no—
179C4×Q8—21no—
180C4×D4—21no—
181#117C22⋊D5911yescharacteristic
182M32—1181no—
183M32—1181no—
184#108—1181no—
185#108—1181no—
186#106—21no—
187#106C22⋊D5911yescharacteristic
188#106C22⋊D5911yescharacteristic
189C4⋊C8—21no—
190C4⋊C8—21no—
191#103—21no—
192#103—21no—
193#102—21no—
194#102—21no—
195C2×Q8—21no—
196C2×Q8—41no—
197C2×D4—21no—
198C2×D4—41no—
199C22×C4—41no—
200C22×C4C2×C22⋊D5911yescharacteristic
201Q16—41no—
202SD16—81no—
203D8—41no—
204M16—41no—
205M16—2361no—
206M16—2361no—
207M16—2361no—
208M16—2361no—
209C2×C8—21no—
210C2×C8—41no—
211C2×C8—2361no—
212C2×C8—2361no—
213C2×C8C2×C22⋊D5911yescharacteristic
214C2×C8C2×C22⋊D5911yescharacteristic
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4#1174567011yescharacteristic
227C22⋊C4#1174567011yescharacteristic
228C42—21no—
229C42—41no—
230C16—2361no—
231C16—2361no—
232C23—41no—
233C23C2×#1174567011yescharacteristic
234Q8—41no—
235Q8—41no—
236Q8—161no—
237D4—41no—
238D4—41no—
239D4—161no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—161no—
248C2×C4#1427369611yescharacteristic
249C2×C4#1427375311yescharacteristic
250C8—21no—
251C8—21no—
252C8—41no—
253C8—4721no—
254C8—4721no—
255C22—21no—
256C22—41no—
257C22—81no—
258C22—81no—
259C22—81no—
260C22#1683755311yescharacteristic
261C4—21no—
262C4—41no—
263C4—41no—
264C4—81no—
265C4—81no—
266C4—81no—
267C4#1683757511yescharacteristic
268C4#1683759211yescharacteristic
269C2—21no—
270C2—41no—
271C2—161no—
272C2#2142313311yescharacteristic
subgroupG/H
centerC2#21423133
commutator, cosocleC2×M16×C59C23
FrattiniC2×M16C22×D59
FittingC59×#3263C2
socleC2×C59#1616
prime radicalC59×#469C22
hypercenter#3263D59
hypocenterC59#12306
LCSG ▷ C2×M16×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #3263

Complements

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

NH ≅ G/Ntype
1. #21426313C2semidirect
3. #21425676C2semidirect
4. #21422090C2semidirect
6. #21422049C2semidirect
163. C59#12306semidirect

Maximal subgroups

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

subgroupindex
#214263132
#214263082
#214256762
#214220902
#214220872
#214220492
C59×#32632
#1230659