Finite Group Databasebeta

← #28480327

Subgroups — #28480327

Order 15104.

808 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14273646C211yesmaximal, characteristic, complemented
2#21422694C212yesmaximal, complemented
3#21422539C212yesmaximal, complemented
4#21422440C211yesmaximal, characteristic, complemented
5C2×C59×#559C211yesmaximal, characteristic
6#16837459C2212yes—
7C2×#14273646—21no—
8C4×#11745664C2212yes—
9#16837171—21no—
10#16837142—24no—
11D4⋊(C59⋊C8)—24no—
12#16837128—21no—
13#16837128—21no—
14C59×#559—24no—
15C59×#559—24no—
16C2×#127×C59C411yescharacteristic
17C2×#124×C59—21no—
18C2×#124×C59C411yescharacteristic
19C2×#124×C59C2211yescharacteristic
20C2×#123×C59—21no—
21C2×#123×C59—21no—
22C2×C4×Q8×C59C411yescharacteristic
23C2×C4×Q8×C59C2211yescharacteristic
24C2×C4×D4×C59—21no—
25C2×C4×D4×C59C411yescharacteristic
26C2×C4×D4×C59C2211yescharacteristic
27C2×#11745664—22no—
28C4⋊(C59⋊C8)—22no—
29#14273646—24no—
30C22⋊(C59⋊C8)—22no—
31#14273638—22no—
32#14273638—22no—
33C4×C59⋊C8—22no—
34C22×Q8×C59C2×C411yescharacteristic
35C22×Q8×C59D411yescharacteristic
36C22×D4×C59—21no—
37C22×D4×C59—21no—
38C22×D4×C59C2×C411yescharacteristic
39C22×D4×C59C2×C411yescharacteristic
40C23×C4×C59—21no—
41C23×C4×C59—21no—
42#127×C59—22no—
43#127×C59—22no—
44#124×C59—28no—
45#124×C59—22no—
46#124×C59—22no—
47#124×C59—22no—
48#124×C59D414yes—
49#123×C59—28no—
50#123×C59—28no—
51C4×Q8×C59—22no—
52C4×Q8×C59—22no—
53C4×Q8×C59—22no—
54C4×Q8×C59—22no—
55C4×D4×C59—28no—
56C4×D4×C59—22no—
57C4×D4×C59—22no—
58C4×D4×C59—22no—
59C4×D4×C59D414yes—
60C2×C4⋊C4×C59—21no—
61C2×C4⋊C4×C59C2×C411yescharacteristic
62C2×C4⋊C4×C59D411yescharacteristic
63C2×C4⋊C4×C59D411yescharacteristic
64C2×C4⋊C4×C59D411yescharacteristic
65C2×C22⋊C4×C59—21no—
66C2×C22⋊C4×C59—21no—
67C2×C22⋊C4×C59—21no—
68C2×C22⋊C4×C59—21no—
69C2×C22⋊C4×C59—21no—
70C2×C22⋊C4×C59—21no—
71C2×C42×C59—21no—
72C2×C42×C59C2×C411yescharacteristic
73C2×C42×C59C2×C411yescharacteristic
74C2×C42×C59C2311yescharacteristic
75C24×C59—21no—
76C24×C59—21no—
77C2×Q8×C59—22no—
78C2×Q8×C59—21no—
79C2×Q8×C59—22no—
80C2×Q8×C59—41no—
81C2×Q8×C59—42no—
82C2×Q8×C59C22⋊C412yes—
83C2×D4×C59—24no—
84C2×D4×C59—24no—
85C2×D4×C59—21no—
86C2×D4×C59—21no—
87C2×D4×C59—42no—
88C2×D4×C59—42no—
89C2×D4×C59—42no—
90C2×D4×C59—42no—
91C2×D4×C59—82no—
92C2×D4×C59C22⋊C412yes—
93C2×D4×C59C22⋊C412yes—
94C22×C4×C59—21no—
95C22×C4×C59—21no—
96C22×C4×C59—24no—
97C22×C4×C59—24no—
98C22×C4×C59—82no—
99C22×C4×C59—82no—
100C22×C4×C59C22⋊C411yescharacteristic
101C22×C4×C59C22⋊C411yescharacteristic
102C22×C4×C59C22⋊C411yescharacteristic
103C22×C4×C59C22⋊C411yescharacteristic
104C22×C4×C59C22×C411yescharacteristic
105C22×C4×C59C2×D411yescharacteristic
106C22×C4×C59C2×D411yescharacteristic
107C4⋊C4×C59—24no—
108C4⋊C4×C59—21no—
109C4⋊C4×C59—22no—
110C4⋊C4×C59—22no—
111C4⋊C4×C59—22no—
112C4⋊C4×C59C22⋊C412yes—
113C22⋊C4×C59—24no—
114C22⋊C4×C59—24no—
115C22⋊C4×C59—24no—
116C22⋊C4×C59—24no—
117C22⋊C4×C59—24no—
118C22⋊C4×C59—24no—
119C42×C59—24no—
120C42×C59—21no—
121C42×C59—21no—
122C42×C59—21no—
123C42×C59—21no—
124C42×C59C2×D412yes—
125C42×C59C2×D412yes—
126#11745664—24no—
127#11745664—42no—
128C2×C59⋊C8—42no—
129C2×C59⋊C8—42no—
130C23×C59—22no—
131C23×C59—22no—
132C23×C59—42no—
133C23×C59—42no—
134C23×C59—42no—
135C23×C59—42no—
136C23×C59—41no—
137C23×C59—41no—
138C23×C59—41no—
139C23×C59—41no—
140C23×C59C2×C22⋊C411yescharacteristic
141Q8×C59—44no—
142Q8×C59—42no—
143Q8×C59—42no—
144D4×C59—24no—
145D4×C59—42no—
146D4×C59—42no—
147D4×C59—48no—
148D4×C59—42no—
149D4×C59—84no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—21no—
154C2×C4×C59—22no—
155C2×C4×C59—22no—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—22no—
159C2×C4×C59—21no—
160C2×C4×C59—21no—
161C2×C4×C59—21no—
162C2×C4×C59—41no—
163C2×C4×C59—42no—
164C2×C4×C59—42no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59—84no—
168C2×C4×C59—84no—
169C2×C4×C59#10412yes—
170C2×C4×C59C2×C22⋊C411yescharacteristic
171C2×C4×C59C2×C22⋊C411yescharacteristic
172C2×C4×C59#12012yes—
173C2×C4×C59#12012yes—
174C59⋊C8—44no—
175C59⋊C8—82no—
176#8370—591nomaximal, Sylow
177C22×C59—21no—
178C22×C59—21no—
179C22×C59—42no—
180C22×C59—42no—
181C22×C59—42no—
182C22×C59—42no—
183C22×C59—42no—
184C22×C59—42no—
185C22×C59—82no—
186C22×C59—82no—
187C22×C59—82no—
188C22×C59—82no—
189C22×C59#37911yescharacteristic
190C22×C59C2×#10411yescharacteristic
191C22×C59#42111yescharacteristic
192C4×C59—22no—
193C4×C59—21no—
194C4×C59—22no—
195C4×C59—22no—
196C4×C59—42no—
197C4×C59—41no—
198C4×C59—41no—
199C4×C59—42no—
200C4×C59—41no—
201C4×C59#41312yes—
202C2×#559D5911yescharacteristic
203C4×#101—591no—
204#1527—592no—
205#1490—592no—
206#1467—591no—
207C2×C59—21no—
208C2×C59—21no—
209C2×C59—82no—
210C2×C59—82no—
211C2×C59#178511yescharacteristic
212C2×C59#178911yescharacteristic
213C2×C59#179511yescharacteristic
214#559—24no—
215#559—24no—
216C2×#127Dic5911yescharacteristic
217C2×#124—21no—
218C2×#124Dic5911yescharacteristic
219C2×#124C2×D5911yescharacteristic
220C2×#123—21no—
221C2×#123—21no—
222C2×C4×Q8Dic5911yescharacteristic
223C2×C4×Q8C2×D5911yescharacteristic
224C2×C4×D4—21no—
225C2×C4×D4Dic5911yescharacteristic
226C2×C4×D4C2×D5911yescharacteristic
227#432—592no—
228C2×#101—1181no—
229C4×M16—592no—
230#343—1181no—
231#329—1184no—
232D4⋊C8—1184no—
233#324—1181no—
234#324—1181no—
235C59#837011yescharacteristic, complemented, Sylow
236C22×Q8C2×Dic5911yescharacteristic
237C22×Q8C22⋊D5911yescharacteristic
238C22×D4—21no—
239C22×D4—21no—
240C22×D4C2×Dic5911yescharacteristic
241C22×D4C2×Dic5911yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C2×M16—1182no—
245#127—22no—
246#127—22no—
247#124—28no—
248#124—22no—
249#124—22no—
250#124—22no—
251#124C22⋊D5914yes—
252#123—28no—
253#123—28no—
254C4×Q8—22no—
255C4×Q8—22no—
256C4×Q8—22no—
257C4×Q8—22no—
258C4×D4—28no—
259C4×D4—22no—
260C4×D4—22no—
261C4×D4—22no—
262C4×D4C22⋊D5914yes—
263C2×C4⋊C4—21no—
264C2×C4⋊C4C2×Dic5911yescharacteristic
265C2×C4⋊C4C22⋊D5911yescharacteristic
266C2×C4⋊C4C22⋊D5911yescharacteristic
267C2×C4⋊C4C22⋊D5911yescharacteristic
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42—21no—
275C2×C42C2×Dic5911yescharacteristic
276C2×C42C2×Dic5911yescharacteristic
277C2×C42C22×D5911yescharacteristic
278C4⋊C8—1182no—
279#101—1184no—
280C22⋊C8—1182no—
281#97—1182no—
282#97—1182no—
283C4×C8—1182no—
284C24—21no—
285C24—21no—
286C2×Q8—22no—
287C2×Q8—21no—
288C2×Q8—22no—
289C2×Q8—41no—
290C2×Q8—42no—
291C2×Q8C22⋊Dic5912yes—
292C2×D4—24no—
293C2×D4—24no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—82no—
301C2×D4C22⋊Dic5912yes—
302C2×D4C22⋊Dic5912yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—24no—
306C22×C4—24no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4C22⋊Dic5911yescharacteristic
310C22×C4C22⋊Dic5911yescharacteristic
311C22×C4C22⋊Dic5911yescharacteristic
312C22×C4C22⋊Dic5911yescharacteristic
313C22×C4C22×Dic5911yescharacteristic
314C22×C4C2×C22⋊D5911yescharacteristic
315C22×C4C2×C22⋊D5911yescharacteristic
316M16—1184no—
317M16—2362no—
318C2×C8—2362no—
319C2×C8—2362no—
320C4⋊C4—24no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4C22⋊Dic5912yes—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C42—24no—
333C42—21no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42C2×C22⋊D5912yes—
338C42C2×C22⋊D5912yes—
339C23—22no—
340C23—22no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23C2×C22⋊Dic5911yescharacteristic
350Q8—44no—
351Q8—42no—
352Q8—42no—
353D4—24no—
354D4—42no—
355D4—42no—
356D4—48no—
357D4—42no—
358D4—84no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4#1427365412yes—
379C2×C4C2×C22⋊Dic5911yescharacteristic
380C2×C4C2×C22⋊Dic5911yescharacteristic
381C2×C4#1427369612yes—
382C2×C4#1427369612yes—
383C8—4722no—
384C8—2364no—
385C22—21no—
386C22—21no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22—42no—
391C22—42no—
392C22—42no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22#1683725811yescharacteristic
398C22C2×#1427365411yescharacteristic
399C22#1683741911yescharacteristic
400C4—21no—
401C4—22no—
402C4—22no—
403C4—22no—
404C4—42no—
405C4—41no—
406C4—42no—
407C4—41no—
408C4—41no—
409C4#1683738612yes—
410C2—21no—
411C2—21no—
412C2—82no—
413C2—82no—
414C2#2142402411yescharacteristic
415C2#2142404011yescharacteristic
416C2#2142406611yescharacteristic
subgroupG/H
centerC22#16837258
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC2×C59×#559C2
socleC22×C59#379
prime radicalC2×#127×C59C4
cosocleC2×C42×C59C23
hypercenterC2×#559D59
hypocenterC59#8370
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
1. C4×#14273646C2semidirect
2. #21422694C2semidirect
3. #21422539C2semidirect
4. #21422440C2semidirect
235. C59#8370semidirect

Maximal subgroups

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

subgroupindex
C4×#142736462
#214226942
#214225392
#214224402
C2×C59×#5592
#837059