Finite Group Databasebeta

← #28500780

Subgroups — #28500780

Order 15104.

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

normal subgroups only

This group has 442 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#21431128C211yesmaximal, characteristic, complemented
2#21423099C212yesmaximal
3#21422101C212yesmaximal, complemented
4#21422030C211yesmaximal, characteristic, complemented
5C59×#2007C211yesmaximal, characteristic
6C2×#14273725—21no—
7#16837713—22no—
8#16837332C2212yes—
9#16837323—22no—
10#16837300—21no—
11#16837287—22no—
12#16837136—22no—
13#16837128C2212yes—
14C2×#137×C59—21no—
15C59×#537C2211yescharacteristic
16C2×#104×C59—21no—
17C59×#391C2211yescharacteristic
18C59×#343C2211yescharacteristic
19C22×Dic118—21no—
20#14273725—42no—
21#14273725—42no—
22C2×C4⋊Dic59—21no—
23C2×C4⋊Dic59—21no—
24C2×C4⋊Dic59D412yes—
25C2×Dic59⋊C4—22no—
26C2×Dic59⋊C4—22no—
27C2×C4×Dic59—21no—
28#14273660—42no—
29#14273648—42no—
30C22⋊(C59⋊C8)—22no—
31C59⋊#95—22no—
32C59⋊#95—22no—
33C59⋊#95D412yes—
34C2×#41×C59—21no—
35C22×Q8×C59D411yescharacteristic
36#137×C59—42no—
37#137×C59—42no—
38C2×Q16×C59—41no—
39C2×SD16×C59—41no—
40C2×M16×C59—21no—
41C22⋊Q8×C59—41no—
42#121×C59—22no—
43#121×C59—41no—
44C2×C4⋊C4×C59C2311yescharacteristic
45C2×C42×C59D411yescharacteristic
46#104×C59—42no—
47#95×C59—22no—
48#41×C59—41no—
49#41×C59—41no—
50#41×C59—41no—
51#41×C59—41no—
52C2×Q8×C59—21no—
53C2×Q8×C59—21no—
54C2×Q8×C59—41no—
55C2×Q8×C59—41no—
56C2×Q8×C59D812yes—
57C2×D4×C59—21no—
58C2×D4×C59—22no—
59C2×D4×C59—41no—
60C22×C4×C59—21no—
61C22×C4×C59—21no—
62C22×C4×C59—41no—
63C22×C4×C59C2×D411yescharacteristic
64C22×C4×C59C2×D412yes—
65C22×Dic59—22no—
66C22×Dic59—22no—
67C2×Dic118—42no—
68C2×Dic118—81no—
69C2×Dic118—81no—
70Q16×C59—44no—
71SD16×C59—44no—
72M16×C59—22no—
73M16×C59—41no—
74C2×C8×C59—41no—
75C4⋊C4×C59—22no—
76C4⋊C4×C59—41no—
77C22⋊C4×C59—42no—
78C42×C59—41no—
79C4⋊Dic59—24no—
80C4⋊Dic59—42no—
81C4⋊Dic59—81no—
82Dic59⋊C4—44no—
83Dic59⋊C4—42no—
84Dic59⋊C4—42no—
85C4×Dic59—42no—
86C2×C59⋊C8—42no—
87C23×C59—41no—
88C23×C59#12011yescharacteristic
89Q8×C59—42no—
90Q8×C59—41no—
91Q8×C59—41no—
92Q8×C59—81no—
93D4×C59—41no—
94D4×C59—41no—
95D4×C59—82no—
96D4×C59—81no—
97D4×C59—81no—
98C2×C4×C59—21no—
99C2×C4×C59—21no—
100C2×C4×C59—22no—
101C2×C4×C59—22no—
102C2×C4×C59—22no—
103C2×C4×C59—41no—
104C2×C4×C59—41no—
105C2×C4×C59—41no—
106C2×C4×C59—41no—
107C2×C4×C59—42no—
108C2×C4×C59—41no—
109C2×C4×C59—81no—
110C2×C4×C59—81no—
111C2×C4×C59C2×D811yescharacteristic
112C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
113C2×Dic59—24no—
114C2×Dic59—42no—
115C2×Dic59—42no—
116C2×Dic59—42no—
117C2×Dic59—42no—
118C2×Dic59—42no—
119Dic118—84no—
120C8×C59—42no—
121C59⋊C8—82no—
122#12101—591nomaximal, Sylow
123C22×C59—21no—
124C22×C59—21no—
125C22×C59—81no—
126C22×C59—81no—
127C22×C59—81no—
128C22×C59#44711yescharacteristic
129C22×C59#45311yescharacteristic
130C22×C59B(4,𝔽2)11yescharacteristic
131C4×C59—21no—
132C4×C59—21no—
133C4×C59—42no—
134C4×C59—42no—
135C4×C59—42no—
136C4×C59—81no—
137Dic59—44no—
138Dic59—82no—
139#2734—591no—
140#2007D5911yescharacteristic
141#1610—592no—
142#1346—592no—
143#1319—591no—
144C2×C59—21no—
145C2×C59—21no—
146C2×C59—161no—
147C2×C59#159511yescharacteristic
148C2×C59#220111yescharacteristic
149C2×C59#220511yescharacteristic
150C2×#137—21no—
151#537C2×D5911yescharacteristic
152C2×C4⋊Q8—1181no—
153#480—1182no—
154C2×#104—21no—
155#398—592no—
156#395—1182no—
157#391C2×D5911yescharacteristic
158#389—1181no—
159#387—1182no—
160#343C2×D5911yescharacteristic
161#327—1182no—
162#324—592no—
163C59#1210111yescharacteristic, complemented, Sylow
164C2×#41—21no—
165C22×Q8—1181no—
166C22×Q8C22⋊D5911yescharacteristic
167#137—42no—
168#137—42no—
169C2×Q16—41no—
170C2×SD16—41no—
171C2×M16—21no—
172C4⋊Q8—2362no—
173C4⋊Q8—2362no—
174C22⋊Q8—41no—
175#121—22no—
176#121—41no—
177C2×C4⋊C4—592no—
178C2×C4⋊C4—1181no—
179C2×C4⋊C4—1181no—
180C2×C4⋊C4—1182no—
181C2×C4⋊C4—1182no—
182C2×C4⋊C4C22×D5911yescharacteristic
183C2×C42—1181no—
184C2×C42C22⋊D5911yescharacteristic
185#107—2362no—
186#104—42no—
187#102—2362no—
188C22⋊C8—1182no—
189#95—22no—
190#95—592no—
191#95—1182no—
192#95—1182no—
193#41—41no—
194#41—41no—
195#41—41no—
196#41—41no—
197C2×Q8—21no—
198C2×Q8—21no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—4721no—
202C2×Q8—4721no—
203C2×Q8—2362no—
204C2×Q8#1174566912yes—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—41no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—41no—
211C22×C4—1182no—
212C22×C4—1182no—
213C22×C4D4×D5912yes—
214C22×C4C2×C22⋊D5911yescharacteristic
215Q16—44no—
216SD16—44no—
217M16—22no—
218M16—41no—
219C2×C8—41no—
220C2×C8—2362no—
221C4⋊C4—22no—
222C4⋊C4—41no—
223C4⋊C4—1184no—
224C4⋊C4—4721no—
225C4⋊C4—2362no—
226C4⋊C4—2364no—
227C4⋊C4—2362no—
228C4⋊C4—2362no—
229C22⋊C4—42no—
230C42—41no—
231C42—2362no—
232C23—41no—
233C23#1427369511yescharacteristic
234Q8—42no—
235Q8—41no—
236Q8—41no—
237Q8—81no—
238Q8—4724no—
239D4—41no—
240D4—41no—
241D4—82no—
242D4—81no—
243D4—81no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—41no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—1184no—
258C2×C4—2362no—
259C2×C4—2362no—
260C2×C4—2362no—
261C2×C4—2362no—
262C2×C4—2362no—
263C2×C4C2×#1174566911yescharacteristic
264C2×C4#1427375211yescharacteristic
265C8—42no—
266C8—4722no—
267C22—21no—
268C22—21no—
269C22—81no—
270C22—81no—
271C22—81no—
272C22#1683753511yescharacteristic
273C22#1683756911yescharacteristic
274C22#1683758811yescharacteristic
275C4—21no—
276C4—21no—
277C4—42no—
278C4—42no—
279C4—42no—
280C4—81no—
281C4—4722no—
282C4—2364no—
283C2—21no—
284C2—21no—
285C2—161no—
286C2#2142301311yescharacteristic
287C2#2142610811yescharacteristic
288C2#2142612511yescharacteristic
subgroupG/H
centerC22#16837588
commutator, cosocleC2×C4⋊C4×C59C23
FrattiniC2×C4⋊C4C22×D59
FittingC59×#2007C2
socleC22×C59B(4,𝔽2)
prime radicalC59×#537C22
hypercenter#2007D59
hypocenterC59#12101
LCSG ▷ C2×C4⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2007

Complements

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

NH ≅ G/Ntype
1. #21431128C2semidirect
3. #21422101C2semidirect
4. #21422030C2semidirect
163. C59#12101semidirect

Maximal subgroups

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

subgroupindex
#214311282
#214230992
#214221012
#214220302
C59×#20072
#1210159