Finite Group Databasebeta

← #28486112

Subgroups — #28486112

Order 15104.

706 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 115 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#21433137C211yesmaximal, characteristic, complemented
2#21429042C211yesmaximal, characteristic
3#21422765C212yesmaximal, complemented
4#21421997C212yesmaximal, complemented
5C59×#1800C211yesmaximal, characteristic, complemented
6C2×#14273724C2211yescharacteristic
7C2×#14273723—21no—
8C2×#14273701—22no—
9#16837480—24no—
10#16837455C2212yes—
11#16837366—22no—
12#16837311C414yes—
13#16837303—21no—
14C4×C4⋊Dic59C2211yescharacteristic
15#16837145—24no—
16C2×C4×D4×C59C2211yescharacteristic
17C59×#423C2212yes—
18C2×#100×C59—21no—
19C59×#343—21no—
20C22×D236—21no—
21C22×D236—21no—
22C22×D236C2×C412yes—
23C22×C22⋊D59—24no—
24C2×#11745664—24no—
25#14273724—22no—
26#14273724—22no—
27#14273723—44no—
28#14273701—48no—
29C2×C4⋊Dic59—21no—
30C2×(C2×D59)⋊C4—24no—
31C2×C4×Dic59—21no—
32C4⋊(C59⋊C8)D414yescomplemented
33C22⋊(C59⋊C8)—24no—
34C22⋊(C59⋊C8)—22no—
35C4×C59⋊C8—22no—
36C59⋊#95C2×C412yes—
37C22×D4×C59D411yescharacteristic
38C23×C4×C59D412yes—
39C2×M16×C59—22no—
40C4×D4×C59—22no—
41C4×D4×C59—22no—
42C2×C4⋊C4×C59D411yescharacteristic
43C2×C22⋊C4×C59C2×C412yes—
44C2×C42×C59C2311yescharacteristic
45C4⋊C8×C59—22no—
46C4⋊C8×C59—22no—
47#100×C59—42no—
48C24×C59C22⋊C412yes—
49C23×D59—22no—
50C2×D4×C59—22no—
51C2×D4×C59—21no—
52C2×D4×C59—22no—
53C2×D4×C59—21no—
54C22×C4×C59—22no—
55C22×C4×C59—22no—
56C22×C4×C59—44no—
57C22×C4×C59C22⋊C412yes—
58C22×C4×C59C22×C411yescharacteristic
59C22×C4×C59C2×D411yescharacteristic
60C22×C4×C59C2×D411yescharacteristic
61C2×C22⋊D59—48no—
62C2×C22⋊D59—48no—
63C2×C22⋊D59—84no—
64C22×Dic59—22no—
65C2×D236—24no—
66C2×D236—44no—
67C2×D236—41no—
68C2×D236—42no—
69C2×D236—41no—
70C2×D236—42no—
71C2×D236—41no—
72C2×D236—41no—
73C2×D236—82no—
74M16×C59—42no—
75M16×C59—42no—
76C2×C8×C59—42no—
77C4⋊C4×C59—22no—
78C22⋊C4×C59—24no—
79C42×C59C2×D412yes—
80C42×C59#4112yes—
81(C2×D59)⋊C4—48no—
82C4⋊Dic59—24no—
83C4×Dic59—42no—
84#11745664—84no—
85C2×C59⋊C8—24no—
86C23×C59—22no—
87C23×C59—22no—
88C23×C59—22no—
89C23×C59—42no—
90C23×C59—42no—
91C23×C59C2×C22⋊C411yescharacteristic
92C22×D59—44no—
93C22×D59—44no—
94C22×D59—42no—
95C22×D59—42no—
96C22×D59—42no—
97D4×C59—42no—
98D4×C59—42no—
99C2×C4×C59—22no—
100C2×C4×C59—22no—
101C2×C4×C59—41no—
102C2×C4×C59—44no—
103C2×C4×C59—42no—
104C2×C4×C59—82no—
105C2×C4×C59C4×D412yes—
106C2×C4×C59#12011yescharacteristic
107C2×C4×C59#12112yes—
108C2×C4×C59#12311yescharacteristic
109C2×C4×C59AGL(1,ℤ/8)12yes—
110C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
111C2×C4×C59#13711yescharacteristic
112C22⋊D59—816no—
113C2×Dic59—24no—
114C2×Dic59—42no—
115C2×Dic59—42no—
116D236—84no—
117D236—82no—
118D236—82no—
119D236—82no—
120D236—82no—
121C8×C59—82no—
122C59⋊C8—44no—
123#9405—591nomaximal, Sylow
124C22×C59—21no—
125C22×C59—21no—
126C22×C59—44no—
127C22×C59—42no—
128C22×C59—42no—
129C22×C59—42no—
130C22×C59—82no—
131C22×C59#38611yescharacteristic
132C22×C59#41711yescharacteristic
133C22×C59#41811yescharacteristic
134C2×D59—84no—
135C2×D59—84no—
136C2×D59—82no—
137C2×D59—82no—
138C2×D59—82no—
139C4×C59—21no—
140C4×C59—22no—
141C4×C59—42no—
142C4×C59#45312yes—
143C4×C59#46812yes—
144C4×C59#46912yes—
145Dic59—44no—
146#2975—591no—
147#2541—591no—
148#1800D5911yescharacteristic, complemented
149#1545—592no—
150#1303—592no—
151C2×C59—21no—
152C2×C59—21no—
153C2×C59—82no—
154C2×C59#188711yescharacteristic
155C2×C59#189511yescharacteristic
156C2×C59#191411yescharacteristic
157D59—162no—
158C2×#127—591no—
159C2×#127—1181no—
160C2×#121—1182no—
161C2×C4×D4C2×D5911yescharacteristic
162#436—1184no—
163#431—592no—
164#423C2×D5912yes—
165C2×#100—21no—
166#407—1182no—
167#392—594no—
168#390—1181no—
169C4×C4⋊C4—591no—
170#343—21no—
171#331—1184no—
172C59#940511yescharacteristic, complemented, Sylow
173C22×D4—592no—
174C22×D4—1184no—
175C22×D4—1181no—
176C22×D4—1181no—
177C22×D4C22⋊D5911yescharacteristic
178C23×C4D23612yes—
179C2×M16—22no—
180C2×M16—1184no—
181#127—1182no—
182#127—1182no—
183#127—2364no—
184#121—2368no—
185C4×D4—22no—
186C4×D4—22no—
187C2×C4⋊C4—1181no—
188C2×C4⋊C4C22⋊D5911yescharacteristic
189C2×C22⋊C4—1184no—
190C2×C22⋊C4C4×D5912yes—
191C2×C42—1181no—
192C2×C42C22×D5911yescharacteristic
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—594no—
196#100—42no—
197C22⋊C8—1184no—
198C22⋊C8—1182no—
199C4×C8—1182no—
200#95—592no—
201C24—1182no—
202C24(C2×D59)⋊C412yes—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—21no—
207C2×D4—1184no—
208C2×D4—4722no—
209C2×D4—4724no—
210C2×D4—2364no—
211C2×D4—2368no—
212C2×D4—2368no—
213C2×D4—2361no—
214C2×D4—2362no—
215C2×D4—2361no—
216C2×D4—2362no—
217C2×D4—2361no—
218C2×D4—2361no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—44no—
222C22×C4—1182no—
223C22×C4(C2×D59)⋊C412yes—
224C22×C4C2×C4×D5911yescharacteristic
225C22×C4C2×D23611yescharacteristic
226C22×C4C2×C22⋊D5911yescharacteristic
227M16—42no—
228M16—42no—
229M16—4724no—
230C2×C8—42no—
231C2×C8—1184no—
232C4⋊C4—22no—
233C4⋊C4—1184no—
234C22⋊C4—24no—
235C22⋊C4—2368no—
236C42—2362no—
237C42D4×D5912yes—
238C42#1174568712yes—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—2364no—
245C23—2364no—
246C23—2362no—
247C23—2362no—
248C23—2362no—
249C23C2×(C2×D59)⋊C411yescharacteristic
250D4—42no—
251D4—42no—
252D4—4724no—
253D4—4722no—
254D4—47216no—
255D4—4722no—
256D4—4722no—
257D4—4722no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—41no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—82no—
264C2×C4—1184no—
265C2×C4—2362no—
266C2×C4—2362no—
267C2×C4(C22⋊D59)⋊C412yes—
268C2×C4#1427369711yescharacteristic
269C2×C4#1427369912yes—
270C2×C4#1427371111yescharacteristic
271C2×C4#1427375212yes—
272C2×C4#1427375311yescharacteristic
273C2×C4#1427375811yescharacteristic
274C8—82no—
275C8—2364no—
276C22—21no—
277C22—21no—
278C22—44no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—82no—
283C22—4724no—
284C22—4724no—
285C22—4722no—
286C22—4722no—
287C22—4722no—
288C22#1683727911yescharacteristic
289C22#1683740111yescharacteristic
290C22#1683740611yescharacteristic
291C4—22no—
292C4—21no—
293C4—42no—
294C4—2364no—
295C4#1683757212yes—
296C4#1683765012yes—
297C4#1683765312yes—
298C2—21no—
299C2—21no—
300C2—82no—
301C2—9442no—
302C2#2142449211yescharacteristic
303C2#2142453211yescharacteristic
304C2#2142464111yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1800C2
socleC22×C59#386
prime radical#21429042C2
cosocleC2×C42×C59C23
hypercenter#1800D59
hypocenterC59#9405
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1800

Complements

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

NH ≅ G/Ntype
1. #21433137C2semidirect
3. #21422765C2semidirect
4. #21421997C2semidirect
5. C59×#1800C2semidirect
32. C4⋊(C59⋊C8)D4semidirect
148. #1800D59semidirect
172. C59#9405semidirect

Maximal subgroups

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

subgroupindex
#214331372
#214290422
#214227652
#214219972
C59×#18002
#940559