Finite Group Databasebeta

← #28480347

Subgroups — #28480347

Order 15104.

712 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#14273646C211yesmaximal, characteristic
2#21422698C212yesmaximal
3#21422543C212yesmaximal
4#21422440C211yesmaximal, characteristic
5C2×C59×#557C211yesmaximal, characteristic
6#16837459C2212yes—
7C2×#14273646—21no—
8C4×#11745664C2212yes—
9#16837171—21no—
10#16837144—24no—
11Q8⋊(C59⋊C8)—24no—
12#16837128—21no—
13#16837128—21no—
14C59×#557—24no—
15C59×#557—24no—
16C2×C4⋊Q8×C59—21no—
17C2×C4⋊Q8×C59C411yescharacteristic
18C2×#125×C59—21no—
19C2×#125×C59—21no—
20C2×#125×C59C411yescharacteristic
21C2×#125×C59C2211yescharacteristic
22C2×C4×Q8×C59—21no—
23C2×C4×Q8×C59C411yescharacteristic
24C2×C4×Q8×C59C411yescharacteristic
25C2×C4×Q8×C59C2211yescharacteristic
26C2×C4×Q8×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×C59—21no—
35C22×Q8×C59C2×C411yescharacteristic
36C22×Q8×C59D411yescharacteristic
37C4⋊Q8×C59—28no—
38C4⋊Q8×C59—22no—
39C4⋊Q8×C59—22no—
40#125×C59—28no—
41#125×C59—28no—
42#125×C59—22no—
43#125×C59—22no—
44#125×C59—22no—
45#125×C59D414yes—
46C4×Q8×C59—28no—
47C4×Q8×C59—22no—
48C4×Q8×C59—22no—
49C4×Q8×C59—22no—
50C4×Q8×C59—22no—
51C4×Q8×C59—22no—
52C4×Q8×C59—22no—
53C4×Q8×C59—22no—
54C4×Q8×C59D414yes—
55C2×C4⋊C4×C59—21no—
56C2×C4⋊C4×C59—21no—
57C2×C4⋊C4×C59—21no—
58C2×C4⋊C4×C59—21no—
59C2×C4⋊C4×C59—21no—
60C2×C4⋊C4×C59—21no—
61C2×C4⋊C4×C59—21no—
62C2×C4⋊C4×C59—21no—
63C2×C4⋊C4×C59C2×C411yescharacteristic
64C2×C4⋊C4×C59C2×C411yescharacteristic
65C2×C4⋊C4×C59C2×C411yescharacteristic
66C2×C4⋊C4×C59D411yescharacteristic
67C2×C4⋊C4×C59D411yescharacteristic
68C2×C4⋊C4×C59D411yescharacteristic
69C2×C42×C59—21no—
70C2×C42×C59—21no—
71C2×C42×C59—21no—
72C2×C42×C59C2×C411yescharacteristic
73C2×C42×C59C2×C411yescharacteristic
74C2×C42×C59C2311yescharacteristic
75C2×Q8×C59—24no—
76C2×Q8×C59—24no—
77C2×Q8×C59—22no—
78C2×Q8×C59—21no—
79C2×Q8×C59—22no—
80C2×Q8×C59—41no—
81C2×Q8×C59—42no—
82C2×Q8×C59—42no—
83C2×Q8×C59C22⋊C412yes—
84C22×C4×C59—21no—
85C22×C4×C59—21no—
86C22×C4×C59—21no—
87C22×C4×C59—21no—
88C22×C4×C59C22⋊C411yescharacteristic
89C22×C4×C59C22⋊C411yescharacteristic
90C22×C4×C59C22⋊C411yescharacteristic
91C22×C4×C59C22⋊C411yescharacteristic
92C22×C4×C59C22×C411yescharacteristic
93C22×C4×C59C2×D411yescharacteristic
94C22×C4×C59C2×D411yescharacteristic
95C4⋊C4×C59—24no—
96C4⋊C4×C59—24no—
97C4⋊C4×C59—24no—
98C4⋊C4×C59—24no—
99C4⋊C4×C59—24no—
100C4⋊C4×C59—21no—
101C4⋊C4×C59—21no—
102C4⋊C4×C59—21no—
103C4⋊C4×C59—24no—
104C4⋊C4×C59—24no—
105C4⋊C4×C59—24no—
106C4⋊C4×C59—22no—
107C4⋊C4×C59—22no—
108C4⋊C4×C59—22no—
109C4⋊C4×C59C22⋊C412yes—
110C4⋊C4×C59C22⋊C412yes—
111C4⋊C4×C59C22⋊C412yes—
112C42×C59—24no—
113C42×C59—24no—
114C42×C59—21no—
115C42×C59—21no—
116C42×C59—21no—
117C42×C59—21no—
118C42×C59—24no—
119C42×C59C2×D412yes—
120C42×C59C2×D412yes—
121#11745664—24no—
122#11745664—42no—
123C2×C59⋊C8—42no—
124C2×C59⋊C8—42no—
125C23×C59C2×C22⋊C411yescharacteristic
126Q8×C59—24no—
127Q8×C59—22no—
128Q8×C59—44no—
129Q8×C59—48no—
130Q8×C59#10114yescomplemented
131C2×C4×C59—24no—
132C2×C4×C59—22no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—21no—
137C2×C4×C59—22no—
138C2×C4×C59—21no—
139C2×C4×C59—21no—
140C2×C4×C59—22no—
141C2×C4×C59—21no—
142C2×C4×C59—22no—
143C2×C4×C59—22no—
144C2×C4×C59—21no—
145C2×C4×C59—21no—
146C2×C4×C59—41no—
147C2×C4×C59—42no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—41no—
153C2×C4×C59—41no—
154C2×C4×C59#10412yes—
155C2×C4×C59C2×C22⋊C411yescharacteristic
156C2×C4×C59C2×C22⋊C411yescharacteristic
157C2×C4×C59#12012yes—
158C2×C4×C59#12012yes—
159C59⋊C8—44no—
160C59⋊C8—82no—
161#8374—591nomaximal, Sylow
162C22×C59—21no—
163C22×C59—21no—
164C22×C59#37911yescharacteristic
165C22×C59C2×#10411yescharacteristic
166C22×C59#42111yescharacteristic
167C4×C59—22no—
168C4×C59—24no—
169C4×C59—21no—
170C4×C59—22no—
171C4×C59—22no—
172C4×C59—41no—
173C4×C59—41no—
174C4×C59—42no—
175C4×C59—41no—
176C4×C59—42no—
177C4×C59—42no—
178C4×C59C2×#10112yes—
179C2×#557D5911yescharacteristic
180C4×#101—591no—
181#1528—592no—
182#1491—592no—
183#1467—591no—
184C2×C59—21no—
185C2×C59—21no—
186C2×C59#178411yescharacteristic
187C2×C59#178911yescharacteristic
188C2×C59#179611yescharacteristic
189#557—24no—
190#557—24no—
191C2×C4⋊Q8—21no—
192C2×C4⋊Q8Dic5911yescharacteristic
193C2×#125—21no—
194C2×#125—21no—
195C2×#125Dic5911yescharacteristic
196C2×#125C2×D5911yescharacteristic
197C2×C4×Q8—21no—
198C2×C4×Q8Dic5911yescharacteristic
199C2×C4×Q8Dic5911yescharacteristic
200C2×C4×Q8C2×D5911yescharacteristic
201C2×C4×Q8C2×D5911yescharacteristic
202#432—592no—
203C2×#101—1181no—
204C4×M16—592no—
205#343—1181no—
206#330—1184no—
207Q8⋊C8—1184no—
208#324—1181no—
209#324—1181no—
210C59#837411yescharacteristic, complemented, Sylow
211C22×Q8—21no—
212C22×Q8C2×Dic5911yescharacteristic
213C22×Q8C22⋊D5911yescharacteristic
214C2×M16—1182no—
215C4⋊Q8—28no—
216C4⋊Q8—22no—
217C4⋊Q8—22no—
218#125—28no—
219#125—28no—
220#125—22no—
221#125—22no—
222#125—22no—
223#125C22⋊D5914yes—
224C4×Q8—28no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8C22⋊D5914yes—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4C2×Dic5911yescharacteristic
242C2×C4⋊C4C2×Dic5911yescharacteristic
243C2×C4⋊C4C2×Dic5911yescharacteristic
244C2×C4⋊C4C22⋊D5911yescharacteristic
245C2×C4⋊C4C22⋊D5911yescharacteristic
246C2×C4⋊C4C22⋊D5911yescharacteristic
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42C2×Dic5911yescharacteristic
251C2×C42C2×Dic5911yescharacteristic
252C2×C42C22×D5911yescharacteristic
253C4⋊C8—1182no—
254#101—1184no—
255C22⋊C8—1182no—
256#97—1182no—
257#97—1182no—
258C4×C8—1182no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—21no—
263C2×Q8—22no—
264C2×Q8—41no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8C22⋊Dic5912yes—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4C22⋊Dic5911yescharacteristic
273C22×C4C22⋊Dic5911yescharacteristic
274C22×C4C22⋊Dic5911yescharacteristic
275C22×C4C22⋊Dic5911yescharacteristic
276C22×C4C22×Dic5911yescharacteristic
277C22×C4C2×C22⋊D5911yescharacteristic
278C22×C4C2×C22⋊D5911yescharacteristic
279M16—1184no—
280M16—2362no—
281C2×C8—2362no—
282C2×C8—2362no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—21no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4C22⋊Dic5912yes—
298C4⋊C4C22⋊Dic5912yes—
299C4⋊C4C22⋊Dic5912yes—
300C42—24no—
301C42—24no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—24no—
307C42C2×C22⋊D5912yes—
308C42C2×C22⋊D5912yes—
309C23C2×C22⋊Dic5911yescharacteristic
310Q8—24no—
311Q8—22no—
312Q8—44no—
313Q8—48no—
314Q8#1427364614yescomplemented
315C2×C4—24no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4#1427365412yes—
339C2×C4C2×C22⋊Dic5911yescharacteristic
340C2×C4C2×C22⋊Dic5911yescharacteristic
341C2×C4#1427369612yes—
342C2×C4#1427369612yes—
343C8—4722no—
344C8—2364no—
345C22—21no—
346C22—21no—
347C22#1683725811yescharacteristic
348C22C2×#1427365411yescharacteristic
349C22#1683741911yescharacteristic
350C4—24no—
351C4—21no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—42no—
357C4—42no—
358C4—42no—
359C4—41no—
360C4—41no—
361C4C2×#1427364612yes—
362C2—21no—
363C2—21no—
364C2#2142402011yescharacteristic
365C2#2142404011yescharacteristic
366C2#2142407111yescharacteristic
subgroupG/H
centerC22#16837258
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC2×C59×#557C2
socleC22×C59#379
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C42×C59C23
hypercenterC2×#557D59
hypocenterC59#8374
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
130. Q8×C59#101semidirect
210. C59#8374semidirect
314. Q8#14273646semidirect

Maximal subgroups

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

subgroupindex
C4×#142736462
#214226982
#214225432
#214224402
C2×C59×#5572
#837459