Finite Group Databasebeta

← #28515949

Subgroups — #28515949

Order 15104.

2070 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
1C2×#16837262C211yesmaximal, characteristic
2C2×#16837260C214yesmaximal
3C22×C59⋊#95C211yesmaximal, characteristic
4#21422360C214yesmaximal, complemented
5#21422359C214yesmaximal, complemented
6C2×C59×#380C211yesmaximal, characteristic
7C24×Dic59C2212yes—
8C22×C4⋊Dic59C2212yes—
9C22×Dic59⋊C4C2218yes—
10C22×C22⋊Dic59C2212yes—
11#16837262—24no—
12#16837260—216no—
13C2×C59⋊#95C418yescomplemented
14C2×C59⋊#95C2212yes—
15C2×C59⋊#95C2218yes—
16C2×C59⋊#95C2212yes—
17C2×C59⋊#95C2212yes—
18C24×C4×C59C2211yescharacteristic
19C22×C4⋊C4×C59C2212yes—
20C22×C22⋊C4×C59C2212yes—
21C59×#380—24no—
22C2×#95×C59C2212yes—
23C23×Dic59—44no—
24C23×Dic59—44no—
25C23×Dic59C2×C418yes—
26C23×Dic59D418yes—
27C23×Dic59Q818yes—
28C23×Dic59C2314yes—
29C23×Dic59C2314yes—
30C2×C4⋊Dic59—24no—
31C2×C4⋊Dic59—44no—
32C2×Dic59⋊C4—216no—
33C2×Dic59⋊C4—48no—
34C2×Dic59⋊C4—48no—
35C2×C22⋊Dic59—28no—
36C2×C22⋊Dic59—24no—
37C59⋊#95—24no—
38C59⋊#95—216no—
39C59⋊#95C2×C418yes—
40C59⋊#95C2×C418yes—
41C25×C59C2311yescharacteristic
42C23×C4×C59—44no—
43C23×C4×C59C2×C414yes—
44C23×C4×C59D414yes—
45C23×C4×C59Q814yes—
46C23×C4×C59C2314yes—
47C23×C4×C59C2311yescharacteristic
48C23×C4×C59C2311yescharacteristic
49C2×C4⋊C4×C59—24no—
50C2×C4⋊C4×C59—44no—
51C2×C22⋊C4×C59—24no—
52C2×C22⋊C4×C59—28no—
53#95×C59—24no—
54C24×C59—24no—
55C24×C59—22no—
56C24×C59—22no—
57C24×C59C22×C412yes—
58C24×C59C2×D414yes—
59C24×C59C2×D412yes—
60C24×C59C2×Q812yes—
61C24×C59C2×Q814yes—
62C24×C59C2411yescharacteristic
63C22×C4×C59—24no—
64C22×C4×C59—24no—
65C22×C4×C59—24no—
66C22×C4×C59—28no—
67C22×C4×C59—28no—
68C22×C4×C59—24no—
69C22×C4×C59—24no—
70C22×C4×C59—48no—
71C22×C4×C59—48no—
72C22×C4×C59—48no—
73C22×C4×C59C4⋊C4116yescomplemented
74C22×C4×C59C22×C412yes—
75C22×C4×C59C22×C412yes—
76C22×C4×C59C2×D412yes—
77C22×C4×C59C2×D412yes—
78C22×C4×C59C2×Q812yes—
79C22×C4×C59C2×Q812yes—
80C22×Dic59—28no—
81C22×Dic59—28no—
82C22×Dic59—28no—
83C22×Dic59—24no—
84C22×Dic59—24no—
85C22×Dic59—28no—
86C22×Dic59—24no—
87C22×Dic59—48no—
88C22×Dic59—48no—
89C22×Dic59—416no—
90C22×Dic59—416no—
91C22×Dic59C4⋊C4132yescomplemented
92C22×Dic59C22×C418yes—
93C22×Dic59C2×D418yes—
94C22×Dic59C2×Q818yes—
95C4⋊C4×C59—48no—
96C22⋊C4×C59—48no—
97C22⋊Dic59—48no—
98C4⋊Dic59—48no—
99Dic59⋊C4—432no—
100C23×C59—22no—
101C23×C59—24no—
102C23×C59—22no—
103C23×C59—28no—
104C23×C59—24no—
105C23×C59—28no—
106C23×C59—24no—
107C23×C59—28no—
108C23×C59—24no—
109C23×C59—24no—
110C23×C59—28no—
111C23×C59C2×C4⋊C418yes—
112C23×C59C2×C4⋊C414yes—
113C23×C59#12714yes—
114C23×C59C4⋊Q814yes—
115C23×C59C4⋊Q818yes—
116C23×C59C23×C411yescharacteristic
117C23×C59C22×D412yes—
118C23×C59C22×D411yescharacteristic
119C23×C59C22×Q812yes—
120C23×C59C22×Q811yescharacteristic
121C23×C59H(5,𝔽2)14yes—
122C23×C59H(5,𝔽2)12yes—
123C23×C592−1+412yes—
124C2×C4×C59—24no—
125C2×C4×C59—24no—
126C2×C4×C59—24no—
127C2×C4×C59—24no—
128C2×C4×C59—24no—
129C2×C4×C59—24no—
130C2×C4×C59—416no—
131C2×C4×C59—44no—
132C2×C4×C59—48no—
133C2×C4×C59—44no—
134C2×C4×C59—48no—
135C2×C4×C59—44no—
136C2×C4×C59C2×C4⋊C414yes—
137C2×C4×C59C2×C4⋊C414yes—
138C2×Dic59—28no—
139C2×Dic59—28no—
140C2×Dic59—216no—
141C2×Dic59—216no—
142C2×Dic59—432no—
143C2×Dic59—44no—
144C2×Dic59—48no—
145C2×Dic59—44no—
146C2×Dic59—48no—
147C2×Dic59—44no—
148C2×Dic59C2×C4⋊C4116yes—
149#14353—591nomaximal, Sylow
150C22×C59—28no—
151C22×C59—24no—
152C22×C59—24no—
153C22×C59—28no—
154C22×C59—28no—
155C22×C59—28no—
156C22×C59—216no—
157C22×C59#38418yes—
158C22×C59C22×C4⋊C412yes—
159C22×C59C22×C4⋊C411yescharacteristic
160C22×C59#51812yes—
161C22×C59#51811yescharacteristic
162C22×C59#51911yescharacteristic
163C22×C59C2×#12711yescharacteristic
164C22×C59C2×C4⋊Q811yescharacteristic
165C22×C59C2×C4⋊Q812yes—
166C22×C59#53414yes—
167C22×C59#53412yes—
168C22×C59#53714yes—
169C22×C59#53712yes—
170C22×C59#54314yes—
171C22×C59#54312yes—
172C22×C59#54412yes—
173C22×C59#54414yes—
174C4×C59—24no—
175C4×C59—24no—
176C4×C59—84no—
177Dic59—216no—
178Dic59—84no—
179C2×#380—591no—
180C2×#380—594no—
181C2×#380D5911yescharacteristic
182C22×#95—591no—
183#1444—594no—
184#1444—594no—
185C2×C59—28no—
186C2×C59C2×#38411yescharacteristic
187C2×C59#231511yescharacteristic
188C2×C59#231512yes—
189C2×C59#231611yescharacteristic
190C2×C59#231612yes—
191C2×C59#265612yes—
192C2×C59#265912yes—
193C2×C59#265914yes—
194C24×C4—592no—
195C24×C4C2×D5911yescharacteristic
196C22×C4⋊C4—598no—
197C22×C4⋊C4—592no—
198C22×C4⋊C4C2×D5912yes—
199C22×C22⋊C4—592no—
200C22×C22⋊C4C2×D5912yes—
201#380—24no—
202#380—11816no—
203#380—1184no—
204C2×#95—598no—
205C2×#95—592no—
206C2×#95—592no—
207C2×#95—598no—
208C2×#95—592no—
209C2×#95C2×D5912yes—
210C59#1435311yescharacteristic, complemented, Sylow
211C25C22×D5911yescharacteristic
212C23×C4—44no—
213C23×C4—594no—
214C23×C4—598no—
215C23×C4—598no—
216C23×C4—598no—
217C23×C4—594no—
218C23×C4—2364no—
219C23×C4—2364no—
220C23×C4Dic11814yes—
221C23×C4C4×D5914yes—
222C23×C4C22⋊D5914yes—
223C23×C4C22×D5914yes—
224C23×C4C22×D5911yescharacteristic
225C23×C4C22×D5911yescharacteristic
226C2×C4⋊C4—24no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—11816no—
229C2×C4⋊C4—1184no—
230C2×C4⋊C4—2368no—
231C2×C4⋊C4—2368no—
232C2×C4⋊C4—2364no—
233C2×C22⋊C4—28no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—1188no—
236C2×C22⋊C4—1184no—
237#95—24no—
238#95—598no—
239#95—598no—
240#95—11816no—
241#95—1184no—
242C24—24no—
243C24—22no—
244C24—22no—
245C24C2×Dic11812yes—
246C24C2×C4×D5912yes—
247C24D4×D5914yes—
248C24Q8×D5914yes—
249C24C2×C22⋊D5912yes—
250C24C23×D5911yescharacteristic
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—28no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—48no—
259C22×C4—48no—
260C22×C4—48no—
261C22×C4—598no—
262C22×C4—598no—
263C22×C4—598no—
264C22×C4—5932no—
265C22×C4—1188no—
266C22×C4—1188no—
267C22×C4—1188no—
268C22×C4—1184no—
269C22×C4—1188no—
270C22×C4—1184no—
271C22×C4—1184no—
272C22×C4—23616no—
273C22×C4—23616no—
274C22×C4—2368no—
275C22×C4—2368no—
276C22×C4Dic59⋊C4116yescomplemented
277C22×C4C2×Dic11812yes—
278C22×C4C2×Dic11812yes—
279C22×C4C2×C4×D5912yes—
280C22×C4C2×C4×D5912yes—
281C22×C4C2×C22⋊D5912yes—
282C22×C4C2×C22⋊D5912yes—
283C4⋊C4—48no—
284C4⋊C4—2368no—
285C4⋊C4—23632no—
286C22⋊C4—48no—
287C22⋊C4—2368no—
288C23—28no—
289C23—24no—
290C23—28no—
291C23—24no—
292C23—28no—
293C23—24no—
294C23—24no—
295C23—28no—
296C23—24no—
297C23—22no—
298C23—22no—
299C23C4⋊C4×D5918yes—
300C23C2×Dic59⋊C414yes—
301C23#1427372314yes—
302C23#1427372514yes—
303C23#1427372618yes—
304C23C22×C4×D5911yescharacteristic
305C23C2×D4×D5912yes—
306C23C22×C22⋊D5911yescharacteristic
307C23C2×Q8×D5912yes—
308C23C22×Dic11811yescharacteristic
309C23#1427377214yes—
310C23#1427377312yes—
311C23#1427377412yes—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—416no—
319C2×C4—44no—
320C2×C4—48no—
321C2×C4—44no—
322C2×C4—48no—
323C2×C4—44no—
324C2×C4—5916no—
325C2×C4—11816no—
326C2×C4—11816no—
327C2×C4—1188no—
328C2×C4—1188no—
329C2×C4—23632no—
330C2×C4—2364no—
331C2×C4—2368no—
332C2×C4—2364no—
333C2×C4—2368no—
334C2×C4—2364no—
335C2×C4C2×Dic59⋊C414yes—
336C2×C4C2×Dic59⋊C414yes—
337C22—24no—
338C22—28no—
339C22—24no—
340C22—28no—
341C22—28no—
342C22—28no—
343C22—216no—
344C22#1683727118yes—
345C22C2×C4⋊C4×D5912yes—
346C22C22×Dic59⋊C411yescharacteristic
347C22#1683787512yes—
348C22#1683787611yescharacteristic
349C22#1683787911yescharacteristic
350C22C2×#1427372311yescharacteristic
351C22C2×#1427372511yescharacteristic
352C22C2×#1427372612yes—
353C22#1683796014yes—
354C22#1683796112yes—
355C22#1683797914yes—
356C22#1683798012yes—
357C22#1683802714yes—
358C22#1683802812yes—
359C22#1683803012yes—
360C22#1683803114yes—
361C4—24no—
362C4—24no—
363C4—84no—
364C4—11816no—
365C4—4724no—
366C2—28no—
367C2C2×#1683727111yescharacteristic
368C2#2142670811yescharacteristic
369C2#2142670912yes—
370C2#2142671211yescharacteristic
371C2#2142671312yes—
372C2#2143020012yes—
373C2#2143021212yes—
374C2#2143021314yes—
subgroupG/H
center, FrattiniC24C23×D59
commutatorC23×C59C23×C4
FittingC2×C59×#380C2
socle, cosocleC24×C59C24
prime radicalC25×C59C23
hypercenterC2×#380D59
hypocenterC59#14353
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
4. #21422360C2semidirect
5. #21422359C2semidirect
13. C2×C59⋊#95C4semidirect
73. C22×C4×C59C4⋊C4semidirect
91. C22×Dic59C4⋊C4semidirect
210. C59#14353semidirect
276. C22×C4Dic59⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#168372622
C2×#168372602
C22×C59⋊#952
#214223602
#214223592
C2×C59×#3802
#1435359