Finite Group Databasebeta

← #28502488

Subgroups — #28502488

Order 15104.

356 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 69 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#21435818C211yesmaximal, characteristic
2#21426325C211yesmaximal, characteristic, complemented
3#21426230C211yesmaximal, characteristic, complemented
4#21422144C211yesmaximal, characteristic, complemented
5#21422056C211yesmaximal, characteristic, complemented
6D4⋊(C59⋊C16)C211yesmaximal, characteristic, complemented
7C59×#1673C211yesmaximal, characteristic, complemented
8C2×#14273749—21no—
9#16838057—21no—
10#16837782C2211yescharacteristic
11#16837643—21no—
12#16837608—21no—
13#16837566—22no—
14C4×D472C2211yescharacteristic
15#16837227—21no—
16C4⋊(C59⋊C16)C2211yescharacteristic
17#16837205—22no—
18#16837202—21no—
19C4×C59⋊C16C2211yescharacteristic
20C59×#500C2211yescharacteristic
21C59×#461—21no—
22C8×D4×C59C2211yescharacteristic
23C59×#335C2211yescharacteristic
24C59×#332—21no—
25C2×#11745685—22no—
26#14273749—42no—
27#14273749—42no—
28C2×Dic236—21no—
29C2×Dic236D412yes—
30C2×#11745660—22no—
31C2×D472D411yescharacteristic
32#14273727—21no—
33#14273709—42no—
34#14273706—22no—
35#14273698—41no—
36C4×D236—21no—
37C4⋊C4×D59—41no—
38C2×C59⋊C16—21no—
39C2×C59⋊C16—21no—
40#14273661D411yescharacteristic
41#14273652—22no—
42#14273649—21no—
43C2×SD16×C59—41no—
44C22×C8×C59—21no—
45C4⋊Q8×C59—21no—
46C4×D4×C59D411yescharacteristic
47#107×C59—21no—
48#107×C59—21no—
49C4⋊C8×C59—21no—
50C4⋊C8×C59D411yescharacteristic
51#103×C59—41no—
52C22⋊C8×C59—21no—
53C4×C8×C59C2311yescharacteristic
54C2×Q8×C59—41no—
55C2×D4×C59D811yescharacteristic
56C22×C4×C59—21no—
57C2×C22⋊D59—42no—
58#11745685—44no—
59#11745685—82no—
60C2×D236—21no—
61C2×C4×D59—42no—
62C2×C4×D59—41no—
63C2×Dic118—22no—
64SD16×C59—82no—
65C2×C8×C59—21no—
66C2×C8×C59—41no—
67C2×C8×C59—41no—
68C2×C8×C59—41no—
69C2×C8×C59C2×D411yescharacteristic
70C2×C8×C59C2×D411yescharacteristic
71C4⋊C4×C59—41no—
72C4⋊C4×C59—41no—
73C4⋊C4×C59D811yescharacteristic
74C22⋊C4×C59—21no—
75C42×C59C2×D411yescharacteristic
76C22⋊Dic59—42no—
77(C2×D59)⋊C4—41no—
78C4⋊Dic59—21no—
79C4⋊Dic59—41no—
80C4⋊Dic59—41no—
81Dic59⋊C4—42no—
82Dic236—42no—
83Dic236—41no—
84Dic236—41no—
85D472—21no—
86D472SD1612yescomplemented
87#11745660—44no—
88C59⋊C16—41no—
89C59⋊C16—41no—
90C23×C59—21no—
91C22×D59—41no—
92Q8×C59—81no—
93Q8×C59—81no—
94D4×C59—21no—
95D4×C59—21no—
96C2×C4×C59—21no—
97C2×C4×C59—41no—
98C2×C4×C59—41no—
99C2×C4×C59—81no—
100C2×C4×C59#12011yescharacteristic
101C2×C4×C59C2×D811yescharacteristic
102C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
103C22⋊D59—84no—
104C2×Dic59—41no—
105C2×Dic59—42no—
106D236—22no—
107D236—41no—
108C4×D59—84no—
109C4×D59—81no—
110C4×D59—81no—
111Dic118—42no—
112Dic118—42no—
113C8×C59—21no—
114C8×C59—41no—
115C8×C59—81no—
116C8×C59C2×SD1611yescharacteristic
117C8×C59AGL(1,ℤ/8)11yescharacteristic
118#12371—591nomaximal, Sylow
119C22×C59—41no—
120C22×C59—41no—
121C22×C59—41no—
122C22×C59#44711yescharacteristic
123C2×D59—42no—
124C2×D59—81no—
125C2×D59—81no—
126C4×C59—21no—
127C4×C59—41no—
128C4×C59—161no—
129C4×C59#45011yescharacteristic
130C4×C59#45511yescharacteristic
131C4×C59#50811yescharacteristic
132C4×C59#50911yescharacteristic
133Dic59—82no—
134Dic59—81no—
135#3283—591no—
136#2241—591no—
137#2224—591no—
138#1673D5911yescharacteristic, complemented
139#1362—591no—
140#1331—591no—
141D4⋊C16—591no—
142C2×C59—81no—
143C2×C59#163911yescharacteristic
144C2×C59#218611yescharacteristic
145C2×C59#219511yescharacteristic
146D59—82no—
147C2×#135—1181no—
148#547—1181no—
149#500C2×D5911yescharacteristic
150#494—591no—
151#467—1181no—
152#461—21no—
153#461—1181no—
154#452—1182no—
155C4×D8—591no—
156C8×D4C2×D5911yescharacteristic
157#366—1181no—
158C4⋊C16—591no—
159#358—1182no—
160#357—1181no—
161C4×C16—591no—
162#335C2×D5911yescharacteristic
163#332—21no—
164C59#1237111yescharacteristic, complemented, Sylow
165C2×#41—1182no—
166#135—2362no—
167#135—2362no—
168C2×Q16—592no—
169C2×Q16—1181no—
170C2×SD16—41no—
171C2×SD16—1182no—
172C2×D8—591no—
173C22×C8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—1181no—
176#123—2362no—
177C22⋊Q8—1182no—
178#121—2361no—
179C4×D4—1181no—
180C4×D4C22⋊D5911yescharacteristic
181C2×C4⋊C4—2361no—
182C2×C16—1181no—
183C2×C16—1181no—
184#107—21no—
185#107—21no—
186#107—591no—
187C4⋊C8—21no—
188C4⋊C8C22⋊D5911yescharacteristic
189#103—41no—
190#103—1182no—
191#102—1181no—
192C22⋊C8—21no—
193C4×C8C22×D5911yescharacteristic
194#41—4722no—
195#41—2364no—
196C2×Q8—41no—
197C2×Q8—1182no—
198C2×D4—1181no—
199C2×D4—2362no—
200C2×D4#1174566911yescharacteristic
201C22×C4—21no—
202C22×C4—2362no—
203C22×C4—2361no—
204Q16—2362no—
205Q16—2361no—
206Q16—2361no—
207SD16—82no—
208SD16—2364no—
209D8—592no—
210D8—1181no—
211C2×C8—21no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8D4×D5911yescharacteristic
216C2×C8D4×D5911yescharacteristic
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—1181no—
220C4⋊C4—2362no—
221C4⋊C4—2361no—
222C4⋊C4—2361no—
223C4⋊C4#1174566911yescharacteristic
224C22⋊C4—21no—
225C22⋊C4—2362no—
226C22⋊C4—2361no—
227C42C2×C22⋊D5911yescharacteristic
228C16—2361no—
229C16—2361no—
230C23—21no—
231C23—2361no—
232Q8—81no—
233Q8—81no—
234Q8—2362no—
235Q8—2362no—
236D4—21no—
237D4—21no—
238D4—1182no—
239D4—4724no—
240D4—2361no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—81no—
245C2×C4—4724no—
246C2×C4—4721no—
247C2×C4—4721no—
248C2×C4—2361no—
249C2×C4—2362no—
250C2×C4#1427369511yescharacteristic
251C2×C4C2×#1174566911yescharacteristic
252C2×C4#1427375211yescharacteristic
253C8—21no—
254C8—41no—
255C8—81no—
256C8SD16×D5911yescharacteristic
257C8#1427375411yescharacteristic
258C22—41no—
259C22—41no—
260C22—41no—
261C22—4721no—
262C22—4721no—
263C22—2362no—
264C22#1683753511yescharacteristic
265C4—21no—
266C4—41no—
267C4—161no—
268C4—4721no—
269C4—4722no—
270C4#1683755111yescharacteristic
271C4#1683757811yescharacteristic
272C4#1683783211yescharacteristic
273C4#1683783811yescharacteristic
274C2—81no—
275C2—4722no—
276C2#2142327311yescharacteristic
277C2#2142601711yescharacteristic
278C2#2142607211yescharacteristic
subgroupG/H
centerC22#16837535
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1673C2
socleC22×C59#447
prime radical#21435818C2
hypercenter#1673D59
hypocenterC59#12371
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ #1673

Complements

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

NH ≅ G/Ntype
2. #21426325C2semidirect
3. #21426230C2semidirect
4. #21422144C2semidirect
5. #21422056C2semidirect
6. D4⋊(C59⋊C16)C2semidirect
7. C59×#1673C2semidirect
86. D472SD16semidirect
138. #1673D59semidirect
164. C59#12371semidirect

Maximal subgroups

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

subgroupindex
#214358182
#214263252
#214262302
#214221442
#214220562
D4⋊(C59⋊C16)2
C59×#16732
#1237159