Finite Group Databasebeta

← #28505652

Subgroups — #28505652

Order 15104.

512 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16837672C211yesmaximal, characteristic
2#21432696C211yesmaximal, characteristic, complemented
3#21425813C211yesmaximal, characteristic, complemented
4C2×#16837204C211yesmaximal, characteristic, complemented
5#21422205C211yesmaximal, characteristic, complemented
6#21422192C211yesmaximal, characteristic, complemented
7C2×C59×#350C211yesmaximal, characteristic, complemented
8C2×#14273752—21no—
9C22×D472C2211yescharacteristic
10#16837885—21no—
11#16837672—22no—
12#16837672C414yescomplemented
13C2×#14273663C2211yescharacteristic
14#16837444—22no—
15#16837444—21no—
16#16837440—21no—
17#16837440C2211yescharacteristic
18C2×#14273661—21no—
19C2×#14273661C2211yescharacteristic
20C2×#14273643—21no—
21#16837204—22no—
22#16837204—21no—
23#16837204—21no—
24C2×#131×C59C2211yescharacteristic
25C2×M32×C59C2211yescharacteristic
26C22×C16×C59C2211yescharacteristic
27C59×#350—22no—
28C22×D236—21no—
29#14273752—24no—
30#14273752—42no—
31C2×#11745671—41no—
32C2×D472—21no—
33C2×D472—21no—
34C2×D472—21no—
35C2×D472—21no—
36C2×D472—21no—
37C2×D472C2×C412yescomplemented
38C2×#11745669—41no—
39C2×#11745664—21no—
40Q8×Dic59—41no—
41D4×Dic59—42no—
42D4×Dic59—41no—
43#14273686—22no—
44#14273686—21no—
45C2×C4⋊Dic59—22no—
46C2×C4⋊Dic59—21no—
47#14273663—21no—
48#14273663C2×C412yescomplemented
49#14273661—22no—
50#14273661—21no—
51#14273661D411yescharacteristic
52#14273661D412yes—
53#14273661D411yescharacteristic
54#14273659—22no—
55#14273659—21no—
56#14273643—24no—
57C2×#41×C59D411yescharacteristic
58#131×C59—21no—
59#131×C59—21no—
60#131×C59C2×C412yes—
61#131×C59D412yes—
62C2×M16×C59—21no—
63C2×M16×C59D411yescharacteristic
64C22×C8×C59—21no—
65C22×C8×C59C2311yescharacteristic
66M32×C59—21no—
67M32×C59—21no—
68C2×C16×C59—21no—
69C2×C16×C59—21no—
70C2×C16×C59—21no—
71C2×C16×C59—21no—
72C23×D59—41no—
73#41×C59—21no—
74#41×C59—21no—
75#41×C59C22⋊C412yes—
76#41×C59SD1612yes—
77C2×Q8×C59D811yescharacteristic
78C2×D4×C59—21no—
79C2×D4×C59D811yescharacteristic
80C22×C4×C59—21no—
81C22×C4×C59C2×D411yescharacteristic
82C22×Dic59—42no—
83C22×Dic59—41no—
84C2×D236—22no—
85C2×D236—41no—
86C2×D236—41no—
87C2×D236—41no—
88C2×D236—41no—
89C2×D236—41no—
90M16×C59—21no—
91M16×C59—22no—
92M16×C59—22no—
93M16×C59C22⋊C412yes—
94C2×C8×C59—21no—
95C2×C8×C59—22no—
96C2×C8×C59—22no—
97C2×C8×C59C22×C411yescharacteristic
98C2×C8×C59C2×D411yescharacteristic
99C2×C8×C59C2×D411yescharacteristic
100C2×C8×C59C2×D411yescharacteristic
101C2×C8×C59#4111yescharacteristic
102C2×C8×C59#4111yescharacteristic
103C22⋊Dic59—42no—
104C22⋊Dic59—41no—
105#11745671—44no—
106#11745669—44no—
107C4⋊Dic59—21no—
108C4⋊Dic59—22no—
109C4⋊Dic59—21no—
110C4⋊Dic59—42no—
111C4⋊Dic59—41no—
112C4×Dic59—42no—
113C4×Dic59—41no—
114#11745664—22no—
115#11745664—41no—
116C2×C59⋊C8—41no—
117D472—22no—
118D472—22no—
119D472—41no—
120D472—41no—
121C16×C59—41no—
122C16×C59—41no—
123C23×C59—21no—
124C23×C59AGL(1,ℤ/8)11yescharacteristic
125C22×D59—42no—
126C22×D59—81no—
127C22×D59—81no—
128C22×D59—161no—
129C22×D59—161no—
130Q8×C59—21no—
131Q8×C59#10212yes—
132D4×C59—22no—
133D4×C59—22no—
134D4×C59—21no—
135D4×C59#10212yes—
136C2×C4×C59—21no—
137C2×C4×C59—22no—
138C2×C4×C59—22no—
139C2×C4×C59C2×C22⋊C411yescharacteristic
140C2×C4×C59#12011yescharacteristic
141C2×C4×C59#12311yescharacteristic
142C2×C4×C59C2×D811yescharacteristic
143C2×C4×C59C2×SD1611yescharacteristic
144C2×C4×C59#13711yescharacteristic
145C2×Dic59—41no—
146C2×Dic59—42no—
147C2×Dic59—41no—
148C2×Dic59—161no—
149C2×Dic59—162no—
150D236—42no—
151D236—42no—
152D236—81no—
153D236—81no—
154C8×C59—22no—
155C8×C59C4×D411yescharacteristic
156C8×C59C4×D411yescharacteristic
157C8×C59#12111yescharacteristic
158C8×C59#12111yescharacteristic
159C59⋊C8—42no—
160#12996—591nomaximal, Sylow
161C22×C59—21no—
162C22×C59—21no—
163C22×C59—22no—
164C22×C59—41no—
165C22×C59—41no—
166C22×C59#41711yescharacteristic
167C22×C59#44711yescharacteristic
168C22×C59#48111yescharacteristic
169C2×D59—82no—
170C2×D59—162no—
171C2×D59—161no—
172C2×D59—161no—
173C2×D59—162no—
174C4×C59—22no—
175C4×C59#38611yescharacteristic
176C4×C59C2×#10211yescharacteristic
177C4×C59#44811yescharacteristic
178C4×C59#48011yescharacteristic
179Dic59—82no—
180Dic59—81no—
181Dic59—81no—
182C2×#472—591no—
183#2913—591no—
184#2140—591no—
185C2×#357—591no—
186C2×#350D5911yescharacteristic, complemented
187#1387—591no—
188#1382—591no—
189C2×C59—21no—
190C2×C59—21no—
191C2×C59—42no—
192C2×C59#187411yescharacteristic
193C2×C59#219211yescharacteristic
194C2×C59#219311yescharacteristic
195D59—162no—
196C2×AGL(1,ℤ/8)—1181no—
197C22×D8—591no—
198C2×#131C2×D5911yescharacteristic
199#520—1181no—
200C2×M32C2×D5911yescharacteristic
201C22×C16C2×D5911yescharacteristic
202#472—594no—
203#472—1182no—
204C2×#108—591no—
205#428—1182no—
206#428—1181no—
207#427—591no—
208#427—1181no—
209C2×#107—591no—
210C2×#107—1181no—
211C2×#100—1181no—
212#357—1182no—
213#357—1181no—
214#357—1181no—
215#350—22no—
216C59#1299611yescharacteristic, complemented, Sylow
217C2×#41D23611yescharacteristic
218C22×D4—1181no—
219AGL(1,ℤ/8)—1184no—
220AGL(1,ℤ/8)—2362no—
221C2×SD16—2361no—
222C2×D8—592no—
223C2×D8—1181no—
224C2×D8—1181no—
225C2×D8—1181no—
226C2×D8—1181no—
227C2×D8—1181no—
228C2×D8—2361no—
229#131—21no—
230#131—21no—
231#131C4×D5912yes—
232#131C22⋊D5912yes—
233C2×M16—21no—
234C2×M16—1181no—
235C2×M16D23611yescharacteristic
236C22×C8—21no—
237C22×C8C22×D5911yescharacteristic
238C4×Q8—2361no—
239C4×D4—2362no—
240C4×D4—2361no—
241#117—1182no—
242#117—1181no—
243C2×C4⋊C4—1182no—
244C2×C4⋊C4—1181no—
245M32—21no—
246M32—21no—
247C2×C16—21no—
248C2×C16—21no—
249C2×C16—21no—
250C2×C16—21no—
251#108—592no—
252#108—1181no—
253#107—592no—
254#107—591no—
255#107—591no—
256#107—1182no—
257#107—1181no—
258#106—1182no—
259#106—1181no—
260#100—1184no—
261C24—2361no—
262#41—21no—
263#41—21no—
264#41#1174566012yes—
265#41(C2×D59)⋊C412yes—
266C2×Q8D47211yescharacteristic
267C2×D4—21no—
268C2×D4—1182no—
269C2×D4—2361no—
270C2×D4—2361no—
271C2×D4—2361no—
272C2×D4—2361no—
273C2×D4—2361no—
274C2×D4D47211yescharacteristic
275C22×C4—21no—
276C22×C4—2362no—
277C22×C4—2361no—
278C22×C4C2×D23611yescharacteristic
279SD16—2364no—
280D8—1182no—
281D8—1182no—
282D8—2364no—
283D8—2361no—
284D8—2361no—
285M16—21no—
286M16—22no—
287M16—22no—
288M16—1182no—
289M16—2361no—
290M16(C2×D59)⋊C412yes—
291C2×C8—21no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—2361no—
295C2×C8C2×C4×D5911yescharacteristic
296C2×C8D4×D5911yescharacteristic
297C2×C8D4×D5911yescharacteristic
298C2×C8#1174568711yescharacteristic
299C2×C8#1174568711yescharacteristic
300C2×C8C2×C22⋊D5911yescharacteristic
301C4⋊C4—1181no—
302C4⋊C4—1182no—
303C4⋊C4—1181no—
304C4⋊C4—2362no—
305C4⋊C4—2361no—
306C22⋊C4—2362no—
307C22⋊C4—2361no—
308C42—2362no—
309C42—2361no—
310C16—41no—
311C16—41no—
312C23—21no—
313C23—9441no—
314C23—9441no—
315C23—4721no—
316C23—4721no—
317C23—2362no—
318C23#1427375511yescharacteristic
319Q8—21no—
320Q8#1427364912yes—
321D4—22no—
322D4—22no—
323D4—21no—
324D4—4721no—
325D4—4721no—
326D4—2362no—
327D4—2362no—
328D4#1427364912yes—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—9441no—
333C2×C4—9442no—
334C2×C4—2361no—
335C2×C4—2362no—
336C2×C4—2361no—
337C2×C4C2×(C2×D59)⋊C411yescharacteristic
338C2×C4#1427369711yescharacteristic
339C2×C4#1427371111yescharacteristic
340C2×C4C2×D47211yescharacteristic
341C2×C4C2×#1174566011yescharacteristic
342C2×C4#1427376011yescharacteristic
343C8—22no—
344C8—2362no—
345C8(C22⋊D59)⋊C411yescharacteristic
346C8(C22⋊D59)⋊C411yescharacteristic
347C8#1427369911yescharacteristic
348C8#1427369911yescharacteristic
349C22—21no—
350C22—21no—
351C22—22no—
352C22—41no—
353C22—41no—
354C22—9442no—
355C22—9442no—
356C22—9441no—
357C22—9441no—
358C22—4722no—
359C22#1683740411yescharacteristic
360C22#1683753811yescharacteristic
361C22#1683772111yescharacteristic
362C4—22no—
363C4—4722no—
364C4—4721no—
365C4—4721no—
366C4#1683727911yescharacteristic
367C4C2×#1427364911yescharacteristic
368C4#1683754311yescharacteristic
369C4#1683771611yescharacteristic
370C2—21no—
371C2—21no—
372C2—42no—
373C2—9442no—
374C2#2142441111yescharacteristic
375C2#2142605511yescharacteristic
376C2#2142606211yescharacteristic
subgroupG/H
centerC22#16837538
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC2×C59×#350C2
socleC22×C59#447
prime radicalC2×#16837672C2
cosocleC22×C8×C59C23
hypercenterC2×#350D59
hypocenterC59#12996
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
2. #21432696C2semidirect
3. #21425813C2semidirect
4. C2×#16837204C2semidirect
5. #21422205C2semidirect
6. #21422192C2semidirect
7. C2×C59×#350C2semidirect
12. #16837672C4semidirect
37. C2×D472C2×C4semidirect
48. #14273663C2×C4semidirect
186. C2×#350D59semidirect
216. C59#12996semidirect

Maximal subgroups

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

subgroupindex
C2×#168376722
#214326962
#214258132
C2×#168372042
#214222052
#214221922
C2×C59×#3502
#1299659