Finite Group Databasebeta

← #28505751

Subgroups — #28505751

Order 15104.

526 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14273670C211yesmaximal, characteristic
2C2×Q16×Dic59C211yesmaximal, characteristic, complemented
3C2×#16837204C211yesmaximal, characteristic
4#21424960C211yesmaximal, characteristic
5#21422187C211yesmaximal, characteristic, complemented
6#21422173C211yesmaximal, characteristic, complemented
7C2×C59×#358C211yesmaximal, characteristic, complemented
8C22×D472C2211yescharacteristic
9C2×Q8×Dic59—21no—
10C2×#14273670—22no—
11C2×#14273670—22no—
12C2×#14273670C414yescomplemented
13C22×C59⋊C16C2211yescharacteristic
14Q16×Dic59—22no—
15Q16×Dic59—22no—
16C2×#14273661C2211yescharacteristic
17C2×#14273651—21no—
18C2×#14273647—21no—
19C2×C8×Dic59C2211yescharacteristic
20#16837303—21no—
21#16837204—22no—
22C22×Q16×C59C2211yescharacteristic
23C22×C16×C59C2211yescharacteristic
24C2×#107×C59C2211yescharacteristic
25C59×#358—22no—
26C22×D236—21no—
27C2×D472—21no—
28C2×D472—21no—
29C2×D472—21no—
30C2×D472—21no—
31C2×D472—21no—
32C2×D472C2×C412yes—
33C22×C59⋊C8D411yescharacteristic
34Q8×Dic59—24no—
35Q8×Dic59—42no—
36C2×C4⋊Dic59—21no—
37C2×C4⋊Dic59—41no—
38C2×(C2×D59)⋊C4—41no—
39C2×(C2×D59)⋊C4—41no—
40C2×C4×Dic59—41no—
41C2×C4×Dic59D411yescharacteristic
42#14273670—28no—
43C2×C59⋊C16—21no—
44C2×C59⋊C16—21no—
45C2×C59⋊C16C2×C412yes—
46#14273661—21no—
47#14273661D412yes—
48#14273651—24no—
49#14273647—42no—
50C8×Dic59—21no—
51C8×Dic59—21no—
52C22×Q8×C59—21no—
53C2×Q16×C59—21no—
54C2×Q16×C59—21no—
55C2×Q16×C59—21no—
56C2×Q16×C59C2×C412yes—
57C2×Q16×C59D412yes—
58C2×Q16×C59D412yes—
59C22×C8×C59C2311yescharacteristic
60C2×C4⋊C4×C59—21no—
61C2×C16×C59—21no—
62C2×C16×C59—21no—
63C2×C16×C59—21no—
64#107×C59—21no—
65#107×C59—21no—
66C23×D59—41no—
67C2×Q8×C59—22no—
68C2×Q8×C59—22no—
69C2×Q8×C59—22no—
70C2×Q8×C59—41no—
71C2×Q8×C59—41no—
72C2×Q8×C59—41no—
73C22×C4×C59—41no—
74C22×C4×C59—41no—
75C22×C4×C59C2×D411yescharacteristic
76C22×Dic59—41no—
77C22×Dic59D811yescharacteristic
78C22×Dic59D811yescharacteristic
79C2×D236—22no—
80C2×D236—41no—
81C2×D236—41no—
82C2×D236—41no—
83C2×D236—41no—
84C2×D236—41no—
85Q16×C59—22no—
86Q16×C59—22no—
87Q16×C59—22no—
88Q16×C59C22⋊C414yescomplemented
89C2×C8×C59C22×C411yescharacteristic
90C2×C8×C59C2×D411yescharacteristic
91C2×C8×C59C2×D411yescharacteristic
92C2×C8×C59C2×D411yescharacteristic
93C2×C8×C59#4111yescharacteristic
94C2×C8×C59#4111yescharacteristic
95C4⋊C4×C59—41no—
96C4⋊C4×C59—41no—
97(C2×D59)⋊C4—82no—
98(C2×D59)⋊C4—82no—
99C4⋊Dic59—22no—
100C4⋊Dic59—44no—
101C4⋊Dic59—41no—
102C4×Dic59—21no—
103C4×Dic59—21no—
104C4×Dic59—44no—
105C2×C59⋊C8—21no—
106C2×C59⋊C8—21no—
107C2×C59⋊C8—21no—
108D472—22no—
109D472—22no—
110D472—41no—
111D472—41no—
112C16×C59—41no—
113C59⋊C16—22no—
114C23×C59C2×D811yescharacteristic
115C22×D59—42no—
116C22×D59—81no—
117C22×D59—81no—
118C22×D59—81no—
119C22×D59—81no—
120C22×D59—81no—
121C22×D59—81no—
122Q8×C59—24no—
123Q8×C59—42no—
124Q8×C59—42no—
125Q8×C59—42no—
126C2×C4×C59—42no—
127C2×C4×C59—42no—
128C2×C4×C59—42no—
129C2×C4×C59—81no—
130C2×C4×C59—81no—
131C2×C4×C59—81no—
132C2×C4×C59C4×D411yescharacteristic
133C2×C4×C59#12011yescharacteristic
134C2×C4×C59#12111yescharacteristic
135C2×C4×C59#13511yescharacteristic
136C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
138C2×Dic59—21no—
139C2×Dic59—21no—
140C2×Dic59—21no—
141C2×Dic59—21no—
142C2×Dic59—21no—
143C2×Dic59—42no—
144C2×Dic59—81no—
145C2×Dic59—81no—
146C2×Dic59SD3212yescomplemented
147D236—42no—
148D236—42no—
149D236—81no—
150D236—81no—
151C8×C59C2×C22⋊C411yescharacteristic
152C8×C59C4×D411yescharacteristic
153C8×C59#12111yescharacteristic
154C8×C59#12311yescharacteristic
155C59⋊C8—41no—
156#13013—591nomaximal, Sylow
157C22×C59C4×D811yescharacteristic
158C22×C59#44711yescharacteristic
159C22×C59#45911yescharacteristic
160C22×C59C2×SD3211yescharacteristic
161C22×C59#50811yescharacteristic
162C22×C59#50911yescharacteristic
163C22×C59#51011yescharacteristic
164C2×D59—82no—
165C2×D59—82no—
166C2×D59—82no—
167C2×D59—161no—
168C2×D59—161no—
169C2×D59—161no—
170C2×D59—161no—
171C4×C59—44no—
172C4×C59—161no—
173C4×C59#38611yescharacteristic
174C4×C59#44211yescharacteristic
175C4×C59#44911yescharacteristic
176C4×C59#46311yescharacteristic
177Dic59—22no—
178Dic59—41no—
179Dic59—82no—
180C22×SD32—591no—
181C2×C4×Q16—591no—
182C2×#358D5911yescharacteristic, complemented
183C2×#357—591no—
184#1970—591no—
185#1380—591no—
186#1374—591no—
187C2×C59#187911yescharacteristic
188C2×C59C4×SD3211yescharacteristic
189C2×C59#217511yescharacteristic
190C2×C59#218511yescharacteristic
191C2×C59#218611yescharacteristic
192C2×C59#220711yescharacteristic
193C2×C59#221111yescharacteristic
194D59—162no—
195C22×Q16C2×D5911yescharacteristic
196C22×D8—591no—
197C2×C4×Q8—1181no—
198C2×SD32—594no—
199C2×SD32—1182no—
200C2×SD32—1182no—
201C22×C16—591no—
202C22×C16C2×D5911yescharacteristic
203C4×Q16—1182no—
204C4×Q16—1182no—
205C2×#107—591no—
206C2×#107C2×D5911yescharacteristic
207C2×#103—1181no—
208C2×#102—1181no—
209C2×C4×C8—591no—
210#390—1181no—
211#358—22no—
212#357—1182no—
213C59#1301311yescharacteristic, complemented, Sylow
214C22×Q8—21no—
215C22×D4—1181no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16—21no—
219C2×Q16C4×D5912yes—
220C2×Q16D23612yes—
221C2×Q16C22⋊D5912yes—
222C2×D8—592no—
223C2×D8—1181no—
224C2×D8—1181no—
225C2×D8—1181no—
226C2×D8—1181no—
227C2×D8—1181no—
228C22×C8—591no—
229C22×C8C22×D5911yescharacteristic
230C4×Q8—1184no—
231C4×Q8—2362no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—1181no—
234C2×C4⋊C4—2361no—
235C2×C22⋊C4—2361no—
236C2×C22⋊C4—2361no—
237C2×C42—591no—
238C2×C42—2361no—
239SD32—1188no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—592no—
244C2×C16—1181no—
245C2×C16—1181no—
246#107—21no—
247#107—21no—
248#107—592no—
249#107—1181no—
250#103—1184no—
251#102—2362no—
252C4×C8—1181no—
253C4×C8—1181no—
254C24—2361no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—1182no—
262C2×D4—2361no—
263C2×D4—2361no—
264C2×D4—2361no—
265C2×D4—2361no—
266C2×D4—2361no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—591no—
270C22×C4—591no—
271C22×C4—2361no—
272C22×C4D4×D5911yescharacteristic
273Q16—22no—
274Q16—22no—
275Q16—22no—
276Q16(C2×D59)⋊C414yescomplemented
277D8—1182no—
278D8—1182no—
279D8—2361no—
280D8—2361no—
281C2×C8—1181no—
282C2×C8—1181no—
283C2×C8—1181no—
284C2×C8C2×C4×D5911yescharacteristic
285C2×C8C2×D23611yescharacteristic
286C2×C8D4×D5911yescharacteristic
287C2×C8#1174568711yescharacteristic
288C2×C8#1174568711yescharacteristic
289C2×C8C2×C22⋊D5911yescharacteristic
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—1182no—
293C4⋊C4—2364no—
294C4⋊C4—2361no—
295C22⋊C4—4722no—
296C22⋊C4—4722no—
297C42—1181no—
298C42—1181no—
299C42—2364no—
300C16—41no—
301C16—1182no—
302C23—4721no—
303C23—4721no—
304C23—4721no—
305C23—4721no—
306C23—4721no—
307C23—4721no—
308C23—2362no—
309C23D8×D5911yescharacteristic
310Q8—24no—
311Q8—42no—
312Q8—42no—
313Q8—42no—
314D4—4721no—
315D4—4721no—
316D4—2362no—
317D4—2362no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—592no—
325C2×C4—1181no—
326C2×C4—1181no—
327C2×C4—1181no—
328C2×C4—1181no—
329C2×C4—1181no—
330C2×C4—4721no—
331C2×C4—4721no—
332C2×C4—2362no—
333C2×C4(C22⋊D59)⋊C411yescharacteristic
334C2×C4#1427369711yescharacteristic
335C2×C4#1427369911yescharacteristic
336C2×C4#1427374811yescharacteristic
337C2×C4#1427375111yescharacteristic
338C2×C4#1427375411yescharacteristic
339C8—2361no—
340C8C2×(C2×D59)⋊C411yescharacteristic
341C8(C22⋊D59)⋊C411yescharacteristic
342C8#1427369911yescharacteristic
343C8#1427371111yescharacteristic
344C22—9441no—
345C22—9441no—
346C22—9441no—
347C22—9441no—
348C22—4722no—
349C22—4722no—
350C22—4722no—
351C22#1683748211yescharacteristic
352C22#1683753711yescharacteristic
353C22#1683759511yescharacteristic
354C22SD32×D5911yescharacteristic
355C22#1683783511yescharacteristic
356C22#1683783911yescharacteristic
357C22#1683784411yescharacteristic
358C4—44no—
359C4—161no—
360C4—1182no—
361C4—4722no—
362C4—2361no—
363C4#1683727911yescharacteristic
364C4#1683751111yescharacteristic
365C4#1683754811yescharacteristic
366C4#1683761911yescharacteristic
367C2—9442no—
368C2#2142444211yescharacteristic
369C2#2142594511yescharacteristic
370C2#2142596311yescharacteristic
371C2#2142601411yescharacteristic
372C2#2142602211yescharacteristic
373C2#2142613411yescharacteristic
374C2#2142616011yescharacteristic
subgroupG/H
centerC23D8×D59
commutatorC2×C8×C59C22×C4
FrattiniC22×C8C22×D59
FittingC2×C59×#358C2
socleC23×C59C2×D8
prime radicalC22×D472C22
cosocleC22×C8×C59C23
hypercenterC2×#358D59
hypocenterC59#13013
LCSG ▷ C2×C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×C8 ◁ C2×#358

Complements

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

NH ≅ G/Ntype
2. C2×Q16×Dic59C2semidirect
5. #21422187C2semidirect
6. #21422173C2semidirect
7. C2×C59×#358C2semidirect
12. C2×#14273670C4semidirect
88. Q16×C59C22⋊C4semidirect
146. C2×Dic59SD32semidirect
182. C2×#358D59semidirect
213. C59#13013semidirect
276. Q16(C2×D59)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#142736702
C2×Q16×Dic592
C2×#168372042
#214249602
#214221872
#214221732
C2×C59×#3582
#1301359