Finite Group Databasebeta

← #28505763

Subgroups — #28505763

Order 15104.

554 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
1C22×#14273670C211yesmaximal, characteristic
2C2×C4×D472C211yesmaximal, characteristic
3C2×#16837206C211yesmaximal, characteristic, complemented
4C2×#16837202C211yesmaximal, characteristic
5#21422187C211yesmaximal, characteristic, complemented
6#21422174C211yesmaximal, characteristic, complemented
7C59×#1969C211yesmaximal, characteristic, complemented
8C22×D472C2211yescharacteristic
9C2×C4×D236—21no—
10C2×#14273670—22no—
11C2×#14273670—22no—
12C2×#14273670C414yescomplemented
13C22×C59⋊C16C2211yescharacteristic
14C22×C59⋊C16C2211yescharacteristic
15C4×D472—22no—
16C4×D472—22no—
17C2×#14273661C2211yescharacteristic
18C2×#14273649—21no—
19#16837206—22no—
20#16837202—22no—
21C22×Q16×C59C2211yescharacteristic
22C2×#107×C59C2211yescharacteristic
23C2×#103×C59—21no—
24C2×C4×C8×C59C2211yescharacteristic
25C59×#391—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#14273670—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×Q8×C59—21no—
49C2×Q16×C59—21no—
50C2×Q16×C59—21no—
51C2×Q16×C59—21no—
52C2×Q16×C59—21no—
53C2×Q16×C59—21no—
54C2×Q16×C59C2×C412yescomplemented
55C22×C8×C59D411yescharacteristic
56C22×C8×C59C2311yescharacteristic
57C2×C4⋊C4×C59—21no—
58C2×C42×C59D411yescharacteristic
59#107×C59—21no—
60#107×C59—21no—
61#103×C59—42no—
62C4×C8×C59—21no—
63C4×C8×C59—21no—
64#95×C59—41no—
65#95×C59—41no—
66C23×D59—41no—
67C2×Q8×C59—22no—
68C2×Q8×C59—41no—
69C2×Q8×C59—41no—
70C2×Q8×C59—41no—
71C2×Q8×C59—41no—
72C2×Q8×C59—41no—
73C22×C4×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—
88Q16×C59—22no—
89Q16×C59—22no—
90Q16×C59—41no—
91Q16×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—
103C42×C59—21no—
104C42×C59—21no—
105(C2×D59)⋊C4—44no—
106C4⋊Dic59—22no—
107C4⋊Dic59—41no—
108D472—22no—
109D472—22no—
110D472—22no—
111D472C22⋊C414yes—
112C59⋊C16—22no—
113C59⋊C16—41no—
114C23×C59C2×D811yescharacteristic
115C22×D59—42no—
116C22×D59—42no—
117C22×D59—42no—
118C22×D59—81no—
119C22×D59—81no—
120C22×D59—81no—
121C22×D59—81no—
122Q8×C59—42no—
123Q8×C59—42no—
124Q8×C59—81no—
125Q8×C59—81no—
126C2×C4×C59—21no—
127C2×C4×C59—21no—
128C2×C4×C59—21no—
129C2×C4×C59—21no—
130C2×C4×C59—21no—
131C2×C4×C59—42no—
132C2×C4×C59—81no—
133C2×C4×C59—81no—
134C2×C4×C59—81no—
135C2×C4×C59—81no—
136C2×C4×C59—81no—
137C2×C4×C59SD3212yescomplemented
138C2×C4×C59C4×D411yescharacteristic
139C2×C4×C59#12011yescharacteristic
140C2×C4×C59#12111yescharacteristic
141C2×C4×C59#13511yescharacteristic
142C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
144C2×Dic59—42no—
145C2×Dic59—81no—
146C2×Dic59—81no—
147D236—24no—
148D236—42no—
149D236—42no—
150D236—42no—
151C4×D59—88no—
152C8×C59—41no—
153C8×C59C2×C22⋊C411yescharacteristic
154C8×C59C4×D411yescharacteristic
155C8×C59#12111yescharacteristic
156C8×C59#12311yescharacteristic
157#13014—591nomaximal, Sylow
158C22×C59C4×D811yescharacteristic
159C22×C59#44711yescharacteristic
160C22×C59#45911yescharacteristic
161C22×C59C2×SD3211yescharacteristic
162C22×C59#50811yescharacteristic
163C22×C59#50911yescharacteristic
164C22×C59#51011yescharacteristic
165C2×D59—44no—
166C2×D59—82no—
167C2×D59—82no—
168C2×D59—82no—
169C2×D59—82no—
170C2×D59—82no—
171C2×D59—82no—
172C4×C59—22no—
173C4×C59—41no—
174C4×C59—82no—
175C4×C59—161no—
176C4×C59#38611yescharacteristic
177C4×C59#44211yescharacteristic
178C4×C59#44911yescharacteristic
179C4×C59#46311yescharacteristic
180Dic59—82no—
181C22×SD32—591no—
182C2×C4×D8—591no—
183C2×#358—591no—
184C2×#357—591no—
185#1969D5911yescharacteristic, complemented
186#1380—591no—
187#1374—591no—
188C2×C59#187911yescharacteristic
189C2×C59C4×SD3211yescharacteristic
190C2×C59#217511yescharacteristic
191C2×C59#218711yescharacteristic
192C2×C59#218911yescharacteristic
193C2×C59#220811yescharacteristic
194C2×C59#221011yescharacteristic
195D59—84no—
196C22×Q16C2×D5911yescharacteristic
197C22×D8—591no—
198C2×C4×D4—1181no—
199C2×SD32—594no—
200C2×SD32—1182no—
201C2×SD32—1182no—
202C22×C16—591no—
203C22×C16—591no—
204C4×D8—1182no—
205C4×D8—1182no—
206C2×#107—591no—
207C2×#107C2×D5911yescharacteristic
208C2×#103—21no—
209C2×#102—1181no—
210C2×C4×C8C2×D5911yescharacteristic
211#391—21no—
212#358—1182no—
213#357—1182no—
214C59#1301411yescharacteristic, complemented, Sylow
215C22×Q8—21no—
216C22×D4—1181no—
217C23×C4—2361no—
218C2×Q16—21no—
219C2×Q16—21no—
220C2×Q16—21no—
221C2×Q16—21no—
222C2×Q16—21no—
223C2×Q16C4×D5912yescomplemented
224C2×D8—592no—
225C2×D8—592no—
226C2×D8—592no—
227C2×D8—1181no—
228C2×D8—1181no—
229C2×D8—1181no—
230C22×C8C22⋊D5911yescharacteristic
231C22×C8C22×D5911yescharacteristic
232C4×D4—1184no—
233C4×D4—2362no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—1181no—
236C2×C22⋊C4—2361no—
237C2×C42C22⋊D5911yescharacteristic
238SD32—1188no—
239C2×C16—592no—
240C2×C16—1181no—
241C2×C16—1181no—
242C2×C16—1181no—
243C2×C16—1181no—
244C2×C16—1181no—
245#107—21no—
246#107—21no—
247#107—592no—
248#107—1181no—
249#103—42no—
250#102—1184no—
251C4×C8—21no—
252C4×C8—21no—
253#95—41no—
254#95—41no—
255C24—2361no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×D4—1182no—
263C2×D4—1182no—
264C2×D4—1182no—
265C2×D4—2361no—
266C2×D4—2361no—
267C2×D4—2361no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—4722no—
271C22×C4—4722no—
272C22×C4—2364no—
273C22×C4—2361no—
274C22×C4#1174566911yescharacteristic
275C22×C4#1174566911yescharacteristic
276C22×C4C2×C22⋊D5911yescharacteristic
277Q16—22no—
278Q16—22no—
279Q16—41no—
280Q16—41no—
281D8—594no—
282D8—1182no—
283D8—1182no—
284D8—1182no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8C2×C4×D5911yescharacteristic
289C2×C8#1174568511yescharacteristic
290C2×C8D4×D5911yescharacteristic
291C2×C8D4×D5911yescharacteristic
292C2×C8D4×D5911yescharacteristic
293C2×C8#1174568711yescharacteristic
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—1182no—
297C4⋊C4—2361no—
298C22⋊C4—2364no—
299C42—21no—
300C42—21no—
301C16—1182no—
302C16—2361no—
303C23—4721no—
304C23—4721no—
305C23—4721no—
306C23—4721no—
307C23—2362no—
308C23—2362no—
309C23—2362no—
310C23C2×#1174566911yescharacteristic
311Q8—42no—
312Q8—42no—
313Q8—81no—
314Q8—81no—
315D4—1184no—
316D4—2362no—
317D4—2362no—
318D4—2362no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—4728no—
331C2×C4—4721no—
332C2×C4—4721no—
333C2×C4—2362no—
334C2×C4#1427367012yescomplemented
335C2×C4C4×C22⋊D5911yescharacteristic
336C2×C4#1427369511yescharacteristic
337C2×C4#1427369811yescharacteristic
338C2×C4#1427374711yescharacteristic
339C2×C4#1427375211yescharacteristic
340C2×C4#1427375311yescharacteristic
341C8—41no—
342C8C22⋊C4×D5911yescharacteristic
343C8(C22⋊D59)⋊C411yescharacteristic
344C8#1427370011yescharacteristic
345C8#1427370911yescharacteristic
346C22—4722no—
347C22—4722no—
348C22—4722no—
349C22—4722no—
350C22—4722no—
351C22—4722no—
352C22—2364no—
353C22C4×#1174566911yescharacteristic
354C22#1683753511yescharacteristic
355C22#1683759411yescharacteristic
356C22C2×#1427367011yescharacteristic
357C22#1683783211yescharacteristic
358C22#1683783711yescharacteristic
359C22#1683784311yescharacteristic
360C4—22no—
361C4—41no—
362C4—82no—
363C4—161no—
364C4—4722no—
365C4#1683727811yescharacteristic
366C4#1683751211yescharacteristic
367C4#1683754511yescharacteristic
368C4#1683761411yescharacteristic
369C2—4724no—
370C2#2142444711yescharacteristic
371C2C4×#1427367011yescharacteristic
372C2#2142595811yescharacteristic
373C2#2142602411yescharacteristic
374C2#2142603411yescharacteristic
375C2#2142613811yescharacteristic
376C2#2142614811yescharacteristic
subgroupG/H
centerC23C2×#11745669
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#1969C2
socleC23×C59C2×D8
prime radicalC22×D472C22
cosocleC22×C8×C59C23
hypercenter#1969D59
hypocenterC59#13014
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ #1969

Complements

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

NH ≅ G/Ntype
3. C2×#16837206C2semidirect
5. #21422187C2semidirect
6. #21422174C2semidirect
7. C59×#1969C2semidirect
12. C2×#14273670C4semidirect
44. C2×C59⋊C16C2×C4semidirect
54. C2×Q16×C59C2×C4semidirect
137. C2×C4×C59SD32semidirect
185. #1969D59semidirect
214. C59#13014semidirect
223. C2×Q16C4×D59semidirect
334. C2×C4#14273670semidirect

Maximal subgroups

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

subgroupindex
C22×#142736702
C2×C4×D4722
C2×#168372062
C2×#168372022
#214221872
#214221742
C59×#19692
#1301459