Finite Group Databasebeta

← #28485180

Subgroups — #28485180

Order 15104.

438 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21433092C211yesmaximal, characteristic, complemented
2#21422716C211yesmaximal, characteristic, complemented
3C2×#16837142C211yesmaximal, characteristic, complemented
4#21422448C211yesmaximal, characteristic, complemented
5#21421996C211yesmaximal, characteristic, complemented
6#21421940C211yesmaximal, characteristic, complemented
7C59×#3211C211yesmaximal, characteristic
8#16837533—21no—
9#16837474—21no—
10#16837474—21no—
11D4×C59⋊C8—21no—
12#16837463—21no—
13#16837459C2211yescharacteristic
14#16837455C2211yescharacteristic
15#16837358—21no—
16C2×#14273638C2211yescharacteristic
17C2×#14273638C2211yescharacteristic
18#16837144—21no—
19#16837142—21no—
20#16837142—21no—
21#16837142—21no—
22#16837126—21no—
23#16837126—21no—
24C59×#529C2211yescharacteristic
25C2×#124×C59C2211yescharacteristic
26C59×#495—21no—
27C59×#488—21no—
28C59×#464—21no—
29C59×#463—21no—
30C59×#451C411yescharacteristic
31C59×#450C411yescharacteristic
32C59×#448C411yescharacteristic
33C59×#447C411yescharacteristic
34C59×#431C2211yescharacteristic
35C2×#11745664—21no—
36C22×C59⋊C8—21no—
37C22×C59⋊C8—21no—
38C22×C59⋊C8—21no—
39C4⋊(C59⋊C8)—21no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)—21no—
42C4⋊(C59⋊C8)—21no—
43C22⋊(C59⋊C8)—21no—
44C22⋊(C59⋊C8)—21no—
45C22⋊(C59⋊C8)—21no—
46C22⋊(C59⋊C8)D411yescharacteristic
47C22⋊(C59⋊C8)D411yescharacteristic
48#14273638—21no—
49#14273638—21no—
50#14273638—21no—
51#14273638—21no—
52#14273638—21no—
53#14273638D411yescharacteristic
54#14273638D411yescharacteristic
55C4×C59⋊C8—21no—
56C4×C59⋊C8—21no—
57C22×Q8×C59C2×C411yescharacteristic
58C22×D4×C59C2×C411yescharacteristic
59C2×Q16×C59—21no—
60C2×SD16×C59—21no—
61C2×SD16×C59—21no—
62C2×D8×C59—21no—
63#126×C59—21no—
64#125×C59D411yescharacteristic
65#124×C59—21no—
66#124×C59—21no—
67#124×C59—21no—
68#124×C59—21no—
69#124×C59D411yescharacteristic
70#123×C59—21no—
71C22⋊Q8×C59C2×C411yescharacteristic
72#121×C59C2×C411yescharacteristic
73C4×Q8×C59D411yescharacteristic
74C4×D4×C59—21no—
75C4×D4×C59D411yescharacteristic
76#117×C59—21no—
77C2×C22⋊C4×C59—21no—
78C2×C22⋊C4×C59—21no—
79C2×C42×C59C2311yescharacteristic
80C4⋊C8×C59—21no—
81#103×C59—21no—
82#103×C59—21no—
83#102×C59—21no—
84#102×C59—21no—
85C22⋊C8×C59C2×C411yescharacteristic
86C22⋊C8×C59C2×C411yescharacteristic
87C4×C8×C59—21no—
88C24×C59—21no—
89C2×Q8×C59—21no—
90C2×Q8×C59—21no—
91C2×Q8×C59—81no—
92C2×Q8×C59C22⋊C411yescharacteristic
93C2×D4×C59—21no—
94C2×D4×C59—21no—
95C2×D4×C59—21no—
96C2×D4×C59—81no—
97C2×D4×C59C22⋊C411yescharacteristic
98C22×C4×C59—21no—
99C22×C4×C59—21no—
100C22×C4×C59C22×C411yescharacteristic
101C22×C4×C59C2×D411yescharacteristic
102C22×C4×C59#4111yescharacteristic
103Q16×C59—81no—
104SD16×C59—81no—
105SD16×C59—81no—
106D8×C59—81no—
107C2×C8×C59—21no—
108C2×C8×C59—21no—
109C4⋊C4×C59—21no—
110C4⋊C4×C59—21no—
111C4⋊C4×C59—21no—
112C4⋊C4×C59—21no—
113C4⋊C4×C59C22⋊C411yescharacteristic
114C4⋊C4×C59C22⋊C411yescharacteristic
115C22⋊C4×C59—21no—
116C22⋊C4×C59—21no—
117C22⋊C4×C59—21no—
118C22⋊C4×C59—21no—
119C22⋊C4×C59—21no—
120C22⋊C4×C59—41no—
121C22⋊C4×C59—41no—
122C22⋊C4×C59—41no—
123C22⋊C4×C59—41no—
124C42×C59—21no—
125C42×C59C2×D411yescharacteristic
126C42×C59C2×D411yescharacteristic
127C42×C59C2×D411yescharacteristic
128C42×C59#4111yescharacteristic
129#11745664—42no—
130C2×C59⋊C8—21no—
131C2×C59⋊C8—22no—
132C2×C59⋊C8—21no—
133C2×C59⋊C8—21no—
134C2×C59⋊C8—21no—
135C2×C59⋊C8—42no—
136C2×C59⋊C8—42no—
137C2×C59⋊C8—41no—
138C2×C59⋊C8—41no—
139C2×C59⋊C8—41no—
140C23×C59—21no—
141C23×C59—41no—
142C23×C59—41no—
143C23×C59—41no—
144C23×C59—81no—
145C23×C59—81no—
146C23×C59C4×D411yescharacteristic
147Q8×C59—41no—
148Q8×C59—41no—
149Q8×C59—41no—
150Q8×C59—81no—
151D4×C59—41no—
152D4×C59—41no—
153D4×C59—41no—
154D4×C59—41no—
155D4×C59—41no—
156D4×C59—81no—
157C2×C4×C59—21no—
158C2×C4×C59—21no—
159C2×C4×C59—21no—
160C2×C4×C59—21no—
161C2×C4×C59—21no—
162C2×C4×C59—41no—
163C2×C4×C59—41no—
164C2×C4×C59—41no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59—81no—
168C2×C4×C59C2×C22⋊C411yescharacteristic
169C2×C4×C59C4×D411yescharacteristic
170C2×C4×C59#12011yescharacteristic
171C2×C4×C59#12111yescharacteristic
172C2×C4×C59#12111yescharacteristic
173C2×C4×C59#12311yescharacteristic
174C8×C59—41no—
175C8×C59—41no—
176C59⋊C8—22no—
177C59⋊C8—42no—
178C59⋊C8—41no—
179C59⋊C8—41no—
180C59⋊C8—41no—
181C59⋊C8—81no—
182#9265—591nomaximal, Sylow
183C22×C59—21no—
184C22×C59—21no—
185C22×C59—41no—
186C22×C59—41no—
187C22×C59—41no—
188C22×C59—81no—
189C22×C59—81no—
190C22×C59—81no—
191C22×C59—161no—
192C22×C59—161no—
193C22×C59#38611yescharacteristic
194C22×C59#44311yescharacteristic
195C22×C59#44411yescharacteristic
196C4×C59—21no—
197C4×C59—21no—
198C4×C59—21no—
199C4×C59—21no—
200C4×C59—21no—
201C4×C59—21no—
202C4×C59—41no—
203C4×C59—41no—
204C4×C59—41no—
205C4×C59—81no—
206#3211D5911yescharacteristic
207#2968—591no—
208#1532—591no—
209C2×#329—591no—
210#1469—591no—
211#1302—591no—
212#1281—591no—
213C2×C59—21no—
214C2×C59—21no—
215C2×C59—81no—
216C2×C59—161no—
217C2×C59#187611yescharacteristic
218C2×C59#190611yescharacteristic
219C2×C59#191411yescharacteristic
220#529C2×D5911yescharacteristic
221C2×#124C2×D5911yescharacteristic
222#495—21no—
223#488—21no—
224#464—21no—
225#463—21no—
226#451Dic5911yescharacteristic
227#450Dic5911yescharacteristic
228#448Dic5911yescharacteristic
229#447Dic5911yescharacteristic
230#446—1181no—
231#435—1181no—
232#435—1181no—
233C8×D4—1181no—
234#433—1181no—
235#432—591no—
236#431—591no—
237#431C2×D5911yescharacteristic
238#405—1181no—
239C2×#97—591no—
240C2×#97—591no—
241#330—1181no—
242#329—1181no—
243#329—1181no—
244#329—1181no—
245#323—1181no—
246#323—1181no—
247C59#926511yescharacteristic, complemented, Sylow
248C22×Q8C2×Dic5911yescharacteristic
249C22×D4C2×Dic5911yescharacteristic
250C2×Q16—21no—
251C2×SD16—21no—
252C2×SD16—21no—
253C2×D8—21no—
254C2×M16—1181no—
255C22×C8—1181no—
256C22×C8—1181no—
257C22×C8—1181no—
258#126—21no—
259#125C22⋊D5911yescharacteristic
260#124—21no—
261#124—21no—
262#124—21no—
263#124—21no—
264#124C22⋊D5911yescharacteristic
265#123—21no—
266C22⋊Q8C2×Dic5911yescharacteristic
267#121C2×Dic5911yescharacteristic
268C4×Q8C22⋊D5911yescharacteristic
269C4×D4—21no—
270C4×D4C22⋊D5911yescharacteristic
271#117—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C42C22×D5911yescharacteristic
275C4⋊C8—21no—
276C4⋊C8—1181no—
277C4⋊C8—1181no—
278C4⋊C8—1181no—
279C4⋊C8—1181no—
280#103—21no—
281#103—21no—
282#102—21no—
283#102—21no—
284C22⋊C8—591no—
285C22⋊C8—591no—
286C22⋊C8—1181no—
287C22⋊C8—1181no—
288C22⋊C8—1181no—
289C22⋊C8C2×Dic5911yescharacteristic
290C22⋊C8C2×Dic5911yescharacteristic
291#97—591no—
292#97—591no—
293#97—1181no—
294#97—1181no—
295#97—1181no—
296#97—1181no—
297#97—1181no—
298C4×C8—21no—
299C4×C8—1181no—
300C4×C8—1181no—
301C24—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—81no—
305C2×Q8C22⋊Dic5911yescharacteristic
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—81no—
310C2×D4C22⋊Dic5911yescharacteristic
311C22×C4—21no—
312C22×C4—21no—
313C22×C4D4×D5911yescharacteristic
314C22×C4#1174568711yescharacteristic
315C22×C4C22×Dic5911yescharacteristic
316Q16—81no—
317SD16—81no—
318SD16—81no—
319D8—81no—
320M16—2362no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—1181no—
324C2×C8—1182no—
325C2×C8—1181no—
326C2×C8—1181no—
327C2×C8—1181no—
328C2×C8—2362no—
329C2×C8—2362no—
330C2×C8—2361no—
331C2×C8—2361no—
332C2×C8—2361no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4C22⋊Dic5911yescharacteristic
338C4⋊C4C22⋊Dic5911yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C42—21no—
349C42D4×D5911yescharacteristic
350C42#1174568711yescharacteristic
351C42C2×C22⋊D5911yescharacteristic
352C42C2×C22⋊D5911yescharacteristic
353C23—21no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—81no—
358C23—81no—
359C23D4×Dic5911yescharacteristic
360Q8—41no—
361Q8—41no—
362Q8—41no—
363Q8—81no—
364D4—41no—
365D4—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—81no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—81no—
381C2×C4C2×C22⋊Dic5911yescharacteristic
382C2×C4D4×Dic5911yescharacteristic
383C2×C4#1427369511yescharacteristic
384C2×C4#1427369911yescharacteristic
385C2×C4#1427369911yescharacteristic
386C2×C4#1427370811yescharacteristic
387C8—41no—
388C8—41no—
389C8—1182no—
390C8—4721no—
391C8—2362no—
392C8—2361no—
393C8—2361no—
394C8—2361no—
395C22—21no—
396C22—21no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—161no—
404C22—161no—
405C22#1683728211yescharacteristic
406C22#1683751911yescharacteristic
407C22#1683752411yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—81no—
418C2—21no—
419C2—21no—
420C2—81no—
421C2—161no—
422C2#2142442511yescharacteristic
423C2#2142459911yescharacteristic
424C2#2142464411yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3211C2
socleC22×C59#386
prime radicalC59×#447C4
cosocleC2×C42×C59C23
hypercenter#3211D59
hypocenterC59#9265
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3211

Complements

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

NH ≅ G/Ntype
1. #21433092C2semidirect
2. #21422716C2semidirect
3. C2×#16837142C2semidirect
4. #21422448C2semidirect
5. #21421996C2semidirect
6. #21421940C2semidirect
247. C59#9265semidirect

Maximal subgroups

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

subgroupindex
#214330922
#214227162
C2×#168371422
#214224482
#214219962
#214219402
C59×#32112
#926559