Finite Group Databasebeta

← #28509902

Subgroups — #28509902

Order 15104.

2606 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
1C23×C4×Dic59C211yesmaximal, characteristic
2C2×#16837263C212yesmaximal
3C2×C22⋊C4×Dic59C212yesmaximal
4#21422322C214yesmaximal, complemented
5#21422319C211yesmaximal, characteristic
6#95×Dic59C214yesmaximal, complemented
7C59×C22⋊#95C211yesmaximal, characteristic
8C24×Dic59C2211yescharacteristic
9C24×Dic59C2211yescharacteristic
10C22×C22⋊Dic59C2212yes—
11C22×C4×Dic59—28no—
12C22×C4×Dic59C418yescomplemented
13C22×C4×Dic59C2214yes—
14C22×C4×Dic59C2212yes—
15C22×C4×Dic59C2212yes—
16#16837263C4116yescomplemented
17C22⋊C4×Dic59—28no—
18C2×C59⋊#95C2214yes—
19C2×C59⋊#95C2214yes—
20C2×C59⋊#95C2218yes—
21C24×C4×C59C2211yescharacteristic
22C22×C22⋊C4×C59C2212yes—
23C2×#95×C59C2214yes—
24C23×Dic59—24no—
25C23×Dic59—22no—
26C23×Dic59—22no—
27C23×Dic59—28no—
28C23×Dic59C2×C418yes—
29C23×Dic59C2×C414yes—
30C23×Dic59C2×C414yes—
31C23×Dic59D418yes—
32C23×Dic59C2314yes—
33C23×Dic59C2312yes—
34C23×Dic59C2311yescharacteristic
35C23×Dic59C2311yescharacteristic
36C2×C22⋊Dic59—24no—
37C2×C22⋊Dic59—24no—
38C2×C22⋊Dic59C2×C418yes—
39C2×C4×Dic59—232no—
40C2×C4×Dic59—28no—
41C2×C4×Dic59—28no—
42C2×C4×Dic59—28no—
43C2×C4×Dic59—216no—
44C2×C4×Dic59C2×C418yes—
45C2×C4×Dic59C2×C418yes—
46C59⋊#95—28no—
47C59⋊#95—28no—
48C59⋊#95C2×C4132yescomplemented
49C25×C59C2311yescharacteristic
50C23×C4×C59—24no—
51C23×C4×C59—28no—
52C23×C4×C59C2×C414yes—
53C23×C4×C59C2314yes—
54C23×C4×C59C2312yes—
55C2×C22⋊C4×C59—24no—
56C2×C22⋊C4×C59—24no—
57C2×C22⋊C4×C59C2×C418yes—
58#95×C59—28no—
59C24×C59—24no—
60C24×C59—24no—
61C24×C59C22×C412yes—
62C24×C59C22×C414yes—
63C24×C59C2×D414yes—
64C24×C59#4112yes—
65C24×C59#4112yes—
66C24×C59C2411yescharacteristic
67C22×C4×C59—232no—
68C22×C4×C59—28no—
69C22×C4×C59—24no—
70C22×C4×C59—24no—
71C22×C4×C59—216no—
72C22×C4×C59—24no—
73C22×C4×C59—28no—
74C22×C4×C59—24no—
75C22×C4×C59—28no—
76C22×C4×C59C22×C418yes—
77C22×C4×C59C22×C414yes—
78C22×C4×C59#4118yes—
79C22×Dic59—28no—
80C22×Dic59—28no—
81C22×Dic59—216no—
82C22×Dic59—216no—
83C22×Dic59—24no—
84C22×Dic59—22no—
85C22×Dic59—22no—
86C22×Dic59—24no—
87C22×Dic59—216no—
88C22×Dic59—232no—
89C22×Dic59—24no—
90C22×Dic59—28no—
91C22×Dic59—28no—
92C22×Dic59—24no—
93C22×Dic59C42116yescomplemented
94C22×Dic59C22⋊C4116yes—
95C22×Dic59C22×C418yes—
96C22×Dic59C22×C418yes—
97C22×Dic59C22×C414yes—
98C22×Dic59C22×C412yes—
99C22×Dic59C22×C412yes—
100C22×Dic59C2×D414yes—
101C22×Dic59C2×D414yes—
102C22×Dic59#4118yes—
103C22⋊C4×C59—216no—
104C22⋊Dic59—216no—
105C4×Dic59—216no—
106C4×Dic59—216no—
107C4×Dic59—416no—
108C23×C59—24no—
109C23×C59—24no—
110C23×C59—216no—
111C23×C59—28no—
112C23×C59—28no—
113C23×C59—28no—
114C23×C59—28no—
115C23×C59C2×C4214yes—
116C23×C59C2×C22⋊C414yes—
117C23×C59#11714yes—
118C23×C59C4×D418yes—
119C23×C59C4×D418yes—
120C23×C59C23×C411yescharacteristic
121C23×C59C23×C412yes—
122C23×C59C22×D412yes—
123C23×C59C2×#4114yes—
124C23×C59C2×#4111yescharacteristic
125C23×C59C2×#4114yes—
126C23×C59C2×#4111yescharacteristic
127C2×C4×C59—232no—
128C2×C4×C59—24no—
129C2×C4×C59—24no—
130C2×C4×C59—216no—
131C2×C4×C59—28no—
132C2×C4×C59—216no—
133C2×C4×C59—44no—
134C2×C4×C59—44no—
135C2×C4×C59—48no—
136C2×C4×C59#11718yes—
137C2×Dic59—24no—
138C2×Dic59—24no—
139C2×Dic59—24no—
140C2×Dic59—28no—
141C2×Dic59—28no—
142C2×Dic59—28no—
143C2×Dic59—232no—
144C2×Dic59—232no—
145C2×Dic59—28no—
146C2×Dic59—216no—
147C2×Dic59—44no—
148C2×Dic59—44no—
149C2×Dic59—48no—
150C2×Dic59C2×C4218yes—
151C2×Dic59C2×C22⋊C414yes—
152C2×Dic59C2×C22⋊C414yes—
153C2×Dic59#11718yes—
154C2×Dic59#123116yes—
155C4×#381—591nomaximal, Sylow
156C22×C59—28no—
157C22×C59—24no—
158C22×C59—24no—
159C22×C59—28no—
160C22×C59—216no—
161C22×C59—216no—
162C22×C59C4×C22⋊C418yes—
163C22×C59C22×C4211yescharacteristic
164C22×C59C22×C22⋊C411yescharacteristic
165C22×C59C2×#11712yes—
166C22×C59C2×#11712yes—
167C22×C59C2×#11711yescharacteristic
168C22×C59C2×C4×D412yes—
169C22×C59C2×C4×D412yes—
170C22×C59C4×#4118yes—
171C22×C59C2×#12314yes—
172C22×C59#52514yes—
173C22×C59#52914yes—
174C22×C59#52914yes—
175C4×C59—28no—
176C4×C59—48no—
177Dic59—28no—
178Dic59—24no—
179Dic59—48no—
180Dic59#38118yescomplemented
181C23×C42—591no—
182C2×#381—592no—
183C2×C4×C22⋊C4—592no—
184#1434—594no—
185C22⋊#95—591no—
186C22⋊#95D5911yescharacteristic
187C4×#95—594no—
188C2×C59—28no—
189C2×C59C2×C4×C22⋊C411yescharacteristic
190C2×C59C4×#11712yes—
191C2×C59C2×#38111yescharacteristic
192C2×C59#228011yescharacteristic
193C2×C59#228312yes—
194C2×C59C4×#12314yes—
195C2×C59C4×#12314yes—
196C24×C4—591no—
197C24×C4—591no—
198C24×C4C2×D5911yescharacteristic
199C22×C22⋊C4—592no—
200C22×C22⋊C4C2×D5912yes—
201C22×C42—598no—
202C22×C42—592no—
203C22×C42—592no—
204C22×C42—594no—
205C22×C42—1188no—
206#381—5916no—
207C4×C22⋊C4—1188no—
208C2×#95—598no—
209C2×#95—594no—
210C2×#95—594no—
211C2×#95C2×D5914yes—
212C59C4×#38111yescharacteristic, complemented, Sylow
213C25C22×D5911yescharacteristic
214C23×C4—24no—
215C23×C4—28no—
216C23×C4—591no—
217C23×C4—598no—
218C23×C4—594no—
219C23×C4—594no—
220C23×C4—598no—
221C23×C4—592no—
222C23×C4—591no—
223C23×C4—594no—
224C23×C4—1188no—
225C23×C4—1184no—
226C23×C4—1182no—
227C23×C4—1182no—
228C23×C4C4×D5914yes—
229C23×C4C22×D5914yes—
230C23×C4C22×D5912yes—
231C2×C22⋊C4—24no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—598no—
234C2×C22⋊C4—1184no—
235C2×C22⋊C4—1184no—
236C2×C22⋊C4C4×D5918yes—
237C2×C42—598no—
238C2×C42—598no—
239C2×C42—11832no—
240C2×C42—1188no—
241C2×C42—1188no—
242C2×C42—1188no—
243C2×C42—11816no—
244#95—28no—
245#95—5932no—
246#95—1188no—
247#95—1188no—
248C24—24no—
249C24—24no—
250C24C2×C4×D5912yes—
251C24C2×C4×D5914yes—
252C24D4×D5914yes—
253C24#1174568712yes—
254C24#1174568912yes—
255C24C23×D5911yescharacteristic
256C22×C4—232no—
257C22×C4—28no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—216no—
261C22×C4—24no—
262C22×C4—28no—
263C22×C4—28no—
264C22×C4—24no—
265C22×C4—592no—
266C22×C4—594no—
267C22×C4—598no—
268C22×C4—5916no—
269C22×C4—5916no—
270C22×C4—598no—
271C22×C4—594no—
272C22×C4—592no—
273C22×C4—594no—
274C22×C4—598no—
275C22×C4—11832no—
276C22×C4—11816no—
277C22×C4—1188no—
278C22×C4—1188no—
279C22×C4—11816no—
280C22×C4—11816no—
281C22×C4—1184no—
282C22×C4—1184no—
283C22×C4—1182no—
284C22×C4—1182no—
285C22×C4—1184no—
286C22×C4—1188no—
287C22×C4—1188no—
288C22×C4—1184no—
289C22×C4C2×C4×D5918yes—
290C22×C4C2×C4×D5914yes—
291C22×C4#1174568718yes—
292C22⋊C4—216no—
293C22⋊C4—11816no—
294C42—11816no—
295C42—11816no—
296C42—23616no—
297C23—28no—
298C23—28no—
299C23—28no—
300C23—28no—
301C23—24no—
302C23—24no—
303C23—216no—
304C23C42×D5914yes—
305C23C22⋊C4×D5914yes—
306C23#1427368514yes—
307C23(C22⋊D59)⋊C418yes—
308C23#1427368918yes—
309C23C22×C4×D5911yescharacteristic
310C23C22×C4×D5912yes—
311C23C2×D4×D5912yes—
312C23#41×D5914yes—
313C23C2×#1174568714yes—
314C23C2×#1174568711yescharacteristic
315C23C2×#1174568911yescharacteristic
316C2×C4—232no—
317C2×C4—216no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—28no—
321C2×C4—216no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—48no—
325C2×C4—5916no—
326C2×C4—594no—
327C2×C4—598no—
328C2×C4—598no—
329C2×C4—594no—
330C2×C4—11832no—
331C2×C4—1188no—
332C2×C4—11832no—
333C2×C4—1184no—
334C2×C4—1184no—
335C2×C4—1184no—
336C2×C4—1188no—
337C2×C4—1188no—
338C2×C4—1188no—
339C2×C4—11816no—
340C2×C4—2364no—
341C2×C4—2364no—
342C2×C4—2368no—
343C2×C4#1427368318yes—
344C22—28no—
345C22—24no—
346C22—24no—
347C22—28no—
348C22—216no—
349C22—216no—
350C22#1683724918yes—
351C22C2×C42×D5911yescharacteristic
352C22C2×C22⋊C4×D5911yescharacteristic
353C22#117×D5912yes—
354C22C2×#1427368312yes—
355C22C2×#1427368511yescharacteristic
356C22C2×(C22⋊D59)⋊C412yes—
357C22C2×#1427368912yes—
358C22C4×#1174568718yes—
359C22#123×D5914yes—
360C22#1683790814yes—
361C22#1683793314yes—
362C22#1683793814yes—
363C4—28no—
364C4—48no—
365C4—598no—
366C4—1184no—
367C4—1188no—
368C4—2368no—
369C2—28no—
370C2C2×#1683724911yescharacteristic
371C2C4×#1427368312yes—
372C2#381×D5911yescharacteristic
373C2#2142647411yescharacteristic
374C2#2142649112yes—
375C2#2142661514yes—
376C2#2142662114yes—
subgroupG/H
center, FrattiniC24C23×D59
commutatorC22×C59C22×C42
FittingC59×C22⋊#95C2
socle, cosocleC24×C59C24
prime radicalC25×C59C23
hypercenterC22⋊#95D59
hypocenterC59C4×#381
LCSG ▷ C22×C59 ▷ C59
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
4. #21422322C2semidirect
6. #95×Dic59C2semidirect
12. C22×C4×Dic59C4semidirect
16. #16837263C4semidirect
48. C59⋊#95C2×C4semidirect
93. C22×Dic59C42semidirect
180. Dic59#381semidirect
212. C59C4×#381semidirect

Maximal subgroups

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

subgroupindex
C23×C4×Dic592
C2×#168372632
C2×C22⋊C4×Dic592
#214223222
#214223192
#95×Dic592
C59×C22⋊#952
C4×#38159