Finite Group Databasebeta

← #28484467

Subgroups — #28484467

Order 15104.

692 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 143 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#21432427C211yesmaximal, characteristic
2C2×#16837268C211yesmaximal, characteristic, complemented
3C2×#16837137C212yesmaximal
4#21421976C212yesmaximal, complemented
5C59×#1817C211yesmaximal, characteristic, complemented
6C2×#14273710—21no—
7C2×#14273701C2211yescharacteristic
8C22×C4⋊Dic59—21no—
9#16837304C412yescomplemented
10#16837300C412yescomplemented
11#16837293—22no—
12#16837281—21no—
13#16837268—24no—
14#16837262C2211yescharacteristic
15C2×C59⋊#95—21no—
16C2×C59⋊#95C2212yes—
17#16837137—24no—
18C22×C42×C59C2211yescharacteristic
19C2×C22⋊C8×C59C2212yes—
20C59×#336—22no—
21C22×D236C2×C411yescharacteristic
22C22×C22⋊D59—21no—
23C23×Dic59—22no—
24#14273710—42no—
25#14273710—42no—
26#14273701—22no—
27#14273701—22no—
28C2×C4⋊Dic59—24no—
29C2×C4⋊Dic59—42no—
30C2×C4⋊Dic59—42no—
31C2×C4⋊Dic59C2×C411yescharacteristic
32C2×C4⋊Dic59C2×C412yes—
33C2×Dic59⋊C4—22no—
34C2×C22⋊Dic59—21no—
35C2×(C2×D59)⋊C4—22no—
36C2×(C2×D59)⋊C4—22no—
37C2×(C2×D59)⋊C4—21no—
38C59⋊#95—24no—
39C59⋊#95—22no—
40C59⋊#95—22no—
41C59⋊#95—21no—
42C59⋊#95D414yes—
43C23×C4×C59D411yescharacteristic
44C23×C4×C59D411yescharacteristic
45C23×C4×C59C2311yescharacteristic
46C22×C8×C59—22no—
47C2×C42×C59—22no—
48C2×C42×C59—22no—
49C2×C42×C59C2×C412yes—
50C2×C42×C59D412yes—
51C22⋊C8×C59—22no—
52C22⋊C8×C59—22no—
53C24×C59C2×D411yescharacteristic
54C23×D59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—21no—
58C22×C4×C59—21no—
59C22×C4×C59—21no—
60C22×C4×C59—21no—
61C22×C4×C59—42no—
62C22×C4×C59—42no—
63C22×C4×C59C22⋊C412yes—
64C22×C4×C59C22⋊C412yes—
65C22×C4×C59D812yes—
66C22×C4×C59SD1612yes—
67C22×C4×C59C22×C411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59C2×D412yes—
70C22×C4×C59#4112yes—
71C2×C22⋊D59—41no—
72C2×C22⋊D59—41no—
73C2×C22⋊D59—41no—
74C2×C22⋊D59—41no—
75C2×C22⋊D59—41no—
76C2×C22⋊D59—41no—
77C22×Dic59—22no—
78C22×Dic59—21no—
79C22×Dic59—24no—
80C22×Dic59—44no—
81C22×Dic59—44no—
82C22×Dic59—42no—
83C22×Dic59—42no—
84C2×D236—22no—
85C2×D236—42no—
86C2×C8×C59—44no—
87C2×C8×C59—42no—
88C42×C59—24no—
89C42×C59—24no—
90C22⋊Dic59—41no—
91C22⋊Dic59—41no—
92(C2×D59)⋊C4—44no—
93(C2×D59)⋊C4—44no—
94(C2×D59)⋊C4—42no—
95C4⋊Dic59—22no—
96C4⋊Dic59—48no—
97C4⋊Dic59—42no—
98Dic59⋊C4—44no—
99C23×C59—22no—
100C23×C59—22no—
101C23×C59C2×C22⋊C411yescharacteristic
102C23×C59#12011yescharacteristic
103C23×C59#12311yescharacteristic
104C23×C59C2×D811yescharacteristic
105C23×C59C2×SD1611yescharacteristic
106C23×C59AGL(1,ℤ/8)11yescharacteristic
107C23×C59#13711yescharacteristic
108C22×D59—41no—
109C22×D59—42no—
110C22×D59—41no—
111C22×D59—41no—
112C22×D59—42no—
113C2×C4×C59—22no—
114C2×C4×C59—22no—
115C2×C4×C59—22no—
116C2×C4×C59—22no—
117C2×C4×C59—22no—
118C2×C4×C59—22no—
119C2×C4×C59—22no—
120C2×C4×C59—22no—
121C2×C4×C59—22no—
122C2×C4×C59—22no—
123C2×C4×C59—22no—
124C2×C4×C59—42no—
125C2×C4×C59—42no—
126C2×C4×C59—42no—
127C2×C4×C59—42no—
128C2×C4×C59#10214yes—
129C2×C4×C59C4×D412yes—
130C2×C4×C59#12112yes—
131C22⋊D59—81no—
132C22⋊D59—81no—
133C22⋊D59—81no—
134C22⋊D59—81no—
135C2×Dic59—44no—
136C2×Dic59—44no—
137C2×Dic59—41no—
138C2×Dic59—42no—
139C2×Dic59—41no—
140C2×Dic59—42no—
141C2×Dic59—41no—
142C2×Dic59—42no—
143C2×Dic59—44no—
144C2×Dic59—84no—
145C2×Dic59—84no—
146D236—82no—
147C8×C59—82no—
148#9156—591nomaximal, Sylow
149C22×C59—21no—
150C22×C59—21no—
151C22×C59—21no—
152C22×C59—21no—
153C22×C59—22no—
154C22×C59—22no—
155C22×C59—22no—
156C22×C59—22no—
157C22×C59#38611yescharacteristic
158C22×C59#41012yes—
159C22×C59C2×#10211yescharacteristic
160C22×C59#41711yescharacteristic
161C22×C59#42112yes—
162C22×C59#44711yescharacteristic
163C22×C59#44811yescharacteristic
164C22×C59#48011yescharacteristic
165C22×C59#48111yescharacteristic
166C2×D59—81no—
167C2×D59—81no—
168C2×D59—81no—
169C2×D59—82no—
170C2×D59—82no—
171C4×C59—24no—
172C4×C59—22no—
173C4×C59—22no—
174C4×C59—42no—
175Dic59—84no—
176Dic59—81no—
177Dic59—82no—
178#2854—591no—
179C2×#382—591no—
180#1817D5911yescharacteristic, complemented
181C2×#327—592no—
182#1294—592no—
183C2×C59—21no—
184C2×C59—21no—
185C2×C59—21no—
186C2×C59—21no—
187C2×C59#151212yes—
188C2×C59#151712yes—
189C2×C59#187411yescharacteristic
190C2×C59#189711yescharacteristic
191C2×C59#189911yescharacteristic
192D59—161no—
193C2×#123—1181no—
194C2×#121—591no—
195C22×C4⋊C4—1181no—
196C22×C42C2×D5911yescharacteristic
197C2×C22⋊C8C2×D5912yes—
198#390—592no—
199#389—592no—
200#388—1182no—
201#386—1181no—
202#382—1184no—
203#380—591no—
204C2×#95—592no—
205C2×#95—1181no—
206#336—22no—
207#327—1184no—
208C59#915611yescharacteristic, complemented, Sylow
209C22×D4—591no—
210C22×D4—1181no—
211C23×C4—1182no—
212C23×C4D23611yescharacteristic
213C23×C4D23611yescharacteristic
214C23×C4C22×D5911yescharacteristic
215C22×C8—22no—
216#123—2362no—
217#123—2362no—
218#121—1182no—
219#121—1182no—
220C2×C4⋊C4—591no—
221C2×C4⋊C4—592no—
222C2×C4⋊C4—1184no—
223C2×C4⋊C4—1182no—
224C2×C4⋊C4—2362no—
225C2×C4⋊C4—2362no—
226C2×C22⋊C4—1181no—
227C2×C22⋊C4—1182no—
228C2×C22⋊C4—1182no—
229C2×C22⋊C4—1181no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42C4×D5912yes—
233C2×C42C22⋊D5912yes—
234C22⋊C8—22no—
235C22⋊C8—22no—
236#95—594no—
237#95—1184no—
238#95—1182no—
239#95—1182no—
240#95—1181no—
241C24—1181no—
242C24C2×D23611yescharacteristic
243C2×D4—1182no—
244C2×D4—2362no—
245C2×D4—2361no—
246C2×D4—2361no—
247C2×D4—2361no—
248C2×D4—2361no—
249C2×D4—2361no—
250C2×D4—2361no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—1182no—
260C22×C4—1181no—
261C22×C4—1184no—
262C22×C4—2364no—
263C22×C4—2364no—
264C22×C4—2362no—
265C22×C4—2362no—
266C22×C4#1174566012yes—
267C22×C4D47212yes—
268C22×C4(C2×D59)⋊C412yes—
269C22×C4(C2×D59)⋊C412yes—
270C22×C4C2×C4×D5911yescharacteristic
271C22×C4D4×D5912yes—
272C22×C4#1174568712yes—
273C22×C4C2×C22⋊D5911yescharacteristic
274C2×C8—44no—
275C2×C8—42no—
276C4⋊C4—1182no—
277C4⋊C4—2362no—
278C4⋊C4—2368no—
279C4⋊C4—2364no—
280C22⋊C4—2364no—
281C22⋊C4—2361no—
282C22⋊C4—2361no—
283C22⋊C4—2364no—
284C22⋊C4—2362no—
285C42—24no—
286C42—24no—
287C23—22no—
288C23—22no—
289C23—2361no—
290C23—2361no—
291C23—2362no—
292C23—2361no—
293C23—2362no—
294C23C2×(C2×D59)⋊C411yescharacteristic
295C23#1427369711yescharacteristic
296C23#1427371111yescharacteristic
297C23C2×D47211yescharacteristic
298C23C2×#1174566011yescharacteristic
299C23#1427375511yescharacteristic
300C23#1427376011yescharacteristic
301D4—4722no—
302D4—4721no—
303D4—4721no—
304D4—4721no—
305D4—4721no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—4724no—
322C2×C4—4724no—
323C2×C4—2364no—
324C2×C4—2364no—
325C2×C4—2362no—
326C2×C4—2361no—
327C2×C4—2361no—
328C2×C4—2362no—
329C2×C4—2361no—
330C2×C4—2364no—
331C2×C4—2362no—
332C2×C4#1427364914yes—
333C2×C4(C22⋊D59)⋊C412yes—
334C2×C4#1427369912yes—
335C8—82no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—4721no—
345C22—4722no—
346C22—4722no—
347C22—4721no—
348C22—4721no—
349C22#1683727911yescharacteristic
350C22#1683737512yes—
351C22C2×#1427364911yescharacteristic
352C22#1683740411yescharacteristic
353C22#1683741812yes—
354C22#1683753811yescharacteristic
355C22#1683754311yescharacteristic
356C22#1683771611yescharacteristic
357C22#1683772111yescharacteristic
358C4—24no—
359C4—22no—
360C4—22no—
361C4—42no—
362C4—4724no—
363C4—4721no—
364C4—4722no—
365C2—21no—
366C2—21no—
367C2—21no—
368C2—21no—
369C2—9441no—
370C2#2142263512yes—
371C2#2142265712yes—
372C2#2142441111yescharacteristic
373C2#2142454211yescharacteristic
374C2#2142455211yescharacteristic
subgroupG/H
centerC23C2×(C2×D59)⋊C4
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1817C2
socleC23×C59C2×C22⋊C4
prime radicalC2×#14273701C22
cosocleC23×C4×C59C23
hypercenter#1817D59
hypocenterC59#9156
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
2. C2×#16837268C2semidirect
4. #21421976C2semidirect
5. C59×#1817C2semidirect
9. #16837304C4semidirect
10. #16837300C4semidirect
180. #1817D59semidirect
208. C59#9156semidirect

Maximal subgroups

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

subgroupindex
#214324272
C2×#168372682
C2×#168371372
#214219762
C59×#18172
#915659