Finite Group Databasebeta

← #28505749

Subgroups — #28505749

Order 15104.

590 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273668C211yesmaximal, characteristic
2C2×C4×D472C211yesmaximal, characteristic, complemented
3C2×#16837203C211yesmaximal, characteristic, complemented
4C2×#16837202C211yesmaximal, characteristic, complemented
5#21422187C211yesmaximal, characteristic, complemented
6#21422174C211yesmaximal, characteristic, complemented
7C59×#1970C211yesmaximal, characteristic, complemented
8C22×D472C2211yescharacteristic
9C2×C4×D236—21no—
10C2×#14273668—22no—
11C2×#14273668—22no—
12C2×#14273668C414yescomplemented
13C22×C59⋊C16C2211yescharacteristic
14C22×C59⋊C16C2211yescharacteristic
15C4×D472—22no—
16C4×D472—22no—
17C2×#14273661C2211yescharacteristic
18C2×#14273649—21no—
19#16837203—22no—
20#16837202—22no—
21C22×D8×C59C2211yescharacteristic
22C2×#107×C59C2211yescharacteristic
23C2×#102×C59—21no—
24C2×C4×C8×C59C2211yescharacteristic
25C59×#390—21no—
26C22×D236—21no—
27C22×C4×D59—41no—
28C2×D472—21no—
29C2×D472—21no—
30C2×D472—21no—
31C2×D472C2×C412yes—
32C2×D472D412yes—
33C2×D472D412yes—
34C4×D236—24no—
35C4×D236—42no—
36C2×C4⋊Dic59—21no—
37C2×(C2×D59)⋊C4—41no—
38#14273668—28no—
39C2×C59⋊C16—21no—
40C2×C59⋊C16—21no—
41C2×C59⋊C16—21no—
42C2×C59⋊C16—21no—
43C2×C59⋊C16—21no—
44C2×C59⋊C16C2×C412yescomplemented
45#14273661—21no—
46#14273661D412yes—
47#14273649—24no—
48C22×D4×C59—21no—
49C2×D8×C59—21no—
50C2×D8×C59—21no—
51C2×D8×C59—21no—
52C2×D8×C59—21no—
53C2×D8×C59—21no—
54C2×D8×C59C2×C412yescomplemented
55C22×C8×C59D411yescharacteristic
56C22×C8×C59C2311yescharacteristic
57C2×C4⋊C4×C59—21no—
58C2×C22⋊C4×C59—41no—
59C2×C22⋊C4×C59—41no—
60C2×C42×C59D411yescharacteristic
61#107×C59—21no—
62#107×C59—21no—
63#102×C59—42no—
64C4×C8×C59—21no—
65C4×C8×C59—21no—
66C24×C59—41no—
67C23×D59—41no—
68C2×D4×C59—22no—
69C2×D4×C59—41no—
70C2×D4×C59—41no—
71C2×D4×C59—41no—
72C2×D4×C59—41no—
73C2×D4×C59—41no—
74C22×C4×C59—41no—
75C22×C4×C59D811yescharacteristic
76C22×C4×C59D811yescharacteristic
77C22×C4×C59C2×D411yescharacteristic
78C22×Dic59—41no—
79C2×D236—22no—
80C2×D236—22no—
81C2×D236—22no—
82C2×D236—41no—
83C2×D236—41no—
84C2×D236—41no—
85C2×C4×D59—44no—
86C2×C4×D59—82no—
87C2×C4×D59—82no—
88D8×C59—22no—
89D8×C59—22no—
90D8×C59—41no—
91D8×C59—41no—
92C2×C8×C59—21no—
93C2×C8×C59—21no—
94C2×C8×C59—21no—
95C2×C8×C59C22×C411yescharacteristic
96C2×C8×C59C2×D411yescharacteristic
97C2×C8×C59C2×D411yescharacteristic
98C2×C8×C59C2×D411yescharacteristic
99C2×C8×C59#4111yescharacteristic
100C2×C8×C59#4111yescharacteristic
101C4⋊C4×C59—41no—
102C4⋊C4×C59—41no—
103C22⋊C4×C59—82no—
104C22⋊C4×C59—82no—
105C42×C59—21no—
106C42×C59—21no—
107(C2×D59)⋊C4—44no—
108C4⋊Dic59—22no—
109C4⋊Dic59—41no—
110D472—22no—
111D472—22no—
112D472—22no—
113D472C22⋊C414yescomplemented
114C59⋊C16—22no—
115C59⋊C16—41no—
116C23×C59—42no—
117C23×C59—81no—
118C23×C59—81no—
119C23×C59—81no—
120C23×C59—81no—
121C23×C59—81no—
122C23×C59—81no—
123C23×C59C2×D811yescharacteristic
124C22×D59—42no—
125C22×D59—42no—
126C22×D59—42no—
127C22×D59—81no—
128C22×D59—81no—
129C22×D59—81no—
130C22×D59—81no—
131D4×C59—42no—
132D4×C59—42no—
133D4×C59—81no—
134D4×C59—81no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—21no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—81no—
141C2×C4×C59—81no—
142C2×C4×C59—81no—
143C2×C4×C59D1612yescomplemented
144C2×C4×C59C4×D411yescharacteristic
145C2×C4×C59#12011yescharacteristic
146C2×C4×C59#12111yescharacteristic
147C2×C4×C59#13511yescharacteristic
148C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
150C2×Dic59—42no—
151C2×Dic59—81no—
152C2×Dic59—81no—
153D236—24no—
154D236—42no—
155D236—42no—
156D236—42no—
157C4×D59—88no—
158C8×C59—41no—
159C8×C59C2×C22⋊C411yescharacteristic
160C8×C59C4×D411yescharacteristic
161C8×C59#12111yescharacteristic
162C8×C59#12311yescharacteristic
163#13012—591nomaximal, Sylow
164C22×C59—82no—
165C22×C59—82no—
166C22×C59—82no—
167C22×C59—161no—
168C22×C59—161no—
169C22×C59—161no—
170C22×C59—161no—
171C22×C59C4×D811yescharacteristic
172C22×C59#44711yescharacteristic
173C22×C59#45911yescharacteristic
174C22×C59C2×D1611yescharacteristic
175C22×C59#50811yescharacteristic
176C22×C59#50911yescharacteristic
177C22×C59#50911yescharacteristic
178C2×D59—44no—
179C2×D59—82no—
180C2×D59—82no—
181C2×D59—82no—
182C2×D59—82no—
183C2×D59—82no—
184C2×D59—82no—
185C4×C59—22no—
186C4×C59—41no—
187C4×C59—161no—
188C4×C59#38611yescharacteristic
189C4×C59#44211yescharacteristic
190C4×C59#44911yescharacteristic
191C4×C59#46311yescharacteristic
192Dic59—82no—
193C22×D16—591no—
194C2×C4×D8—591no—
195C2×#357—591no—
196C2×#357—591no—
197#1970D5911yescharacteristic, complemented
198#1380—591no—
199#1374—591no—
200C2×C59—162no—
201C2×C59#187911yescharacteristic
202C2×C59C4×D1611yescharacteristic
203C2×C59#217711yescharacteristic
204C2×C59#218411yescharacteristic
205C2×C59#218611yescharacteristic
206C2×C59#220611yescharacteristic
207C2×C59#221011yescharacteristic
208D59—84no—
209C22×D8—591no—
210C22×D8C2×D5911yescharacteristic
211C2×C4×D4—1181no—
212C2×D16—594no—
213C2×D16—1182no—
214C2×D16—1182no—
215C22×C16—591no—
216C22×C16—591no—
217C4×D8—1182no—
218C4×D8—1182no—
219C2×#107—591no—
220C2×#107C2×D5911yescharacteristic
221C2×#102—21no—
222C2×#102—1181no—
223C2×C4×C8C2×D5911yescharacteristic
224#390—21no—
225#357—1182no—
226#357—1182no—
227C59#1301211yescharacteristic, complemented, Sylow
228C22×D4—21no—
229C22×D4—1181no—
230C23×C4—2361no—
231C2×D8—21no—
232C2×D8—21no—
233C2×D8—21no—
234C2×D8—21no—
235C2×D8—21no—
236C2×D8—592no—
237C2×D8—592no—
238C2×D8—592no—
239C2×D8—1181no—
240C2×D8—1181no—
241C2×D8—1181no—
242C2×D8C4×D5912yescomplemented
243C22×C8C22⋊D5911yescharacteristic
244C22×C8C22×D5911yescharacteristic
245C4×D4—1184no—
246C4×D4—2362no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—1181no—
249C2×C22⋊C4—41no—
250C2×C22⋊C4—41no—
251C2×C22⋊C4—2361no—
252C2×C42C22⋊D5911yescharacteristic
253D16—1188no—
254C2×C16—592no—
255C2×C16—1181no—
256C2×C16—1181no—
257C2×C16—1181no—
258C2×C16—1181no—
259C2×C16—1181no—
260#107—21no—
261#107—21no—
262#107—592no—
263#107—1181no—
264#102—42no—
265#102—1184no—
266C4×C8—21no—
267C4×C8—21no—
268C24—41no—
269C24—2361no—
270C2×D4—22no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—1182no—
277C2×D4—1182no—
278C2×D4—1182no—
279C2×D4—2361no—
280C2×D4—2361no—
281C2×D4—2361no—
282C22×C4—41no—
283C22×C4—4722no—
284C22×C4—4722no—
285C22×C4—2364no—
286C22×C4—2361no—
287C22×C4#1174566911yescharacteristic
288C22×C4#1174566911yescharacteristic
289C22×C4C2×C22⋊D5911yescharacteristic
290D8—22no—
291D8—22no—
292D8—41no—
293D8—41no—
294D8—594no—
295D8—1182no—
296D8—1182no—
297D8—1182no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8C2×C4×D5911yescharacteristic
302C2×C8#1174568511yescharacteristic
303C2×C8D4×D5911yescharacteristic
304C2×C8D4×D5911yescharacteristic
305C2×C8D4×D5911yescharacteristic
306C2×C8#1174568711yescharacteristic
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—1182no—
310C4⋊C4—2361no—
311C22⋊C4—82no—
312C22⋊C4—82no—
313C22⋊C4—2364no—
314C42—21no—
315C42—21no—
316C16—1182no—
317C16—2361no—
318C23—42no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—4721no—
326C23—4721no—
327C23—4721no—
328C23—4721no—
329C23—2362no—
330C23—2362no—
331C23—2362no—
332C23C2×#1174566911yescharacteristic
333D4—42no—
334D4—42no—
335D4—81no—
336D4—81no—
337D4—1184no—
338D4—2362no—
339D4—2362no—
340D4—2362no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—4728no—
350C2×C4—4721no—
351C2×C4—4721no—
352C2×C4—2362no—
353C2×C4#1427366812yescomplemented
354C2×C4C4×C22⋊D5911yescharacteristic
355C2×C4#1427369511yescharacteristic
356C2×C4#1427369811yescharacteristic
357C2×C4#1427374711yescharacteristic
358C2×C4#1427375211yescharacteristic
359C2×C4#1427375311yescharacteristic
360C8—41no—
361C8C22⋊C4×D5911yescharacteristic
362C8(C22⋊D59)⋊C411yescharacteristic
363C8#1427370011yescharacteristic
364C8#1427370911yescharacteristic
365C22—82no—
366C22—82no—
367C22—82no—
368C22—161no—
369C22—161no—
370C22—161no—
371C22—161no—
372C22—4722no—
373C22—4722no—
374C22—4722no—
375C22—4722no—
376C22—4722no—
377C22—4722no—
378C22—2364no—
379C22C4×#1174566911yescharacteristic
380C22#1683753511yescharacteristic
381C22#1683759411yescharacteristic
382C22C2×#1427366811yescharacteristic
383C22#1683783211yescharacteristic
384C22#1683783711yescharacteristic
385C22#1683783811yescharacteristic
386C4—22no—
387C4—41no—
388C4—161no—
389C4—4722no—
390C4#1683727811yescharacteristic
391C4#1683751211yescharacteristic
392C4#1683754511yescharacteristic
393C4#1683761411yescharacteristic
394C2—162no—
395C2—4724no—
396C2#2142444711yescharacteristic
397C2C4×#1427366811yescharacteristic
398C2#2142597011yescharacteristic
399C2#2142600711yescharacteristic
400C2#2142601711yescharacteristic
401C2#2142612811yescharacteristic
402C2#2142614811yescharacteristic
subgroupG/H
centerC23C2×#11745669
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#1970C2
socleC23×C59C2×D8
prime radicalC22×#14273668C2
cosocleC22×C8×C59C23
hypercenter#1970D59
hypocenterC59#13012
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ #1970

Complements

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

NH ≅ G/Ntype
2. C2×C4×D472C2semidirect
3. C2×#16837203C2semidirect
4. C2×#16837202C2semidirect
5. #21422187C2semidirect
6. #21422174C2semidirect
7. C59×#1970C2semidirect
12. C2×#14273668C4semidirect
44. C2×C59⋊C16C2×C4semidirect
54. C2×D8×C59C2×C4semidirect
113. D472C22⋊C4semidirect
143. C2×C4×C59D16semidirect
197. #1970D59semidirect
227. C59#13012semidirect
242. C2×D8C4×D59semidirect
353. C2×C4#14273668semidirect

Maximal subgroups

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

subgroupindex
C22×#142736682
C2×C4×D4722
C2×#168372032
C2×#168372022
#214221872
#214221742
C59×#19702
#1301259