Finite Group Databasebeta

← C59×#10257 #28422366

Subgroups — C59×#10257 #28422366

Order 15104.

528 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C59×#3246C211yesmaximal, characteristic
2C59×#3078C211yesmaximal, characteristic, complemented
3C59×#3037C211yesmaximal, characteristic
4C59×#1984C211yesmaximal, characteristic, complemented
5C59×#1493C211yesmaximal, characteristic, complemented
6C59×#1492C211yesmaximal, characteristic
7C59×#1292C211yesmaximal, characteristic, complemented
8C2×#137×C59—21no—
9C59×#533C2211yescharacteristic
10C59×#532C2211yescharacteristic
11C2×C4×Q8×C59C2211yescharacteristic
12C59×#479—22no—
13C59×#477—21no—
14C59×#476—21no—
15C59×#475—21no—
16C59×#475—21no—
17C59×#465—22no—
18C59×#464—22no—
19C59×#462—21no—
20C59×#461—21no—
21C59×#460—21no—
22C59×#460—21no—
23C59×#428—21no—
24C59×#423C2211yescharacteristic
25C59×#423C2211yescharacteristic
26C2×C4⋊C8×C59C2211yescharacteristic
27C59×#419—21no—
28C59×#417—21no—
29C2×C4×C8×C59C2211yescharacteristic
30C59×#336—22no—
31C59×Q8⋊C8—22no—
32C59×D4⋊C8—22no—
33C2×#41×C59—21no—
34C22×Q8×C59D411yescharacteristic
35#137×C59—42no—
36#137×C59—42no—
37C2×Q16×C59—22no—
38C2×SD16×C59—22no—
39C2×SD16×C59—22no—
40C2×M16×C59—21no—
41C2×M16×C59—21no—
42C22×C8×C59—21no—
43C22×C8×C59D412yes—
44C4⋊Q8×C59D411yescharacteristic
45C4⋊Q8×C59D411yescharacteristic
46C4⋊Q8×C59D411yescharacteristic
47#127×C59D411yescharacteristic
48#125×C59—21no—
49#124×C59—21no—
50C22⋊Q8×C59—21no—
51C22⋊Q8×C59—21no—
52#121×C59—21no—
53#121×C59—21no—
54C4×Q8×C59—21no—
55C4×Q8×C59—21no—
56C4×Q8×C59—21no—
57C4×Q8×C59—21no—
58C4×Q8×C59—21no—
59C4×Q8×C59—21no—
60C4×D4×C59—21no—
61C4×D4×C59—21no—
62#117×C59—21no—
63C2×C4⋊C4×C59—21no—
64C2×C4⋊C4×C59D411yescharacteristic
65C2×C42×C59—21no—
66C2×C42×C59C2311yescharacteristic
67#107×C59—22no—
68#106×C59—22no—
69#106×C59—22no—
70C4⋊C8×C59—21no—
71C4⋊C8×C59—21no—
72C4⋊C8×C59—21no—
73C4⋊C8×C59—21no—
74C4⋊C8×C59D412yes—
75C4⋊C8×C59D412yes—
76#103×C59—22no—
77#103×C59—22no—
78#103×C59—22no—
79#103×C59—22no—
80#102×C59—22no—
81#102×C59—22no—
82C4×C8×C59—21no—
83C4×C8×C59—21no—
84#41×C59—41no—
85#41×C59—41no—
86#41×C59—41no—
87#41×C59—41no—
88C2×Q8×C59—21no—
89C2×Q8×C59—21no—
90C2×Q8×C59—21no—
91C2×Q8×C59—21no—
92C2×Q8×C59—21no—
93C2×Q8×C59—21no—
94C2×Q8×C59—21no—
95C2×Q8×C59—21no—
96C2×Q8×C59SD1612yescomplemented
97C2×D4×C59—21no—
98C2×D4×C59—21no—
99C2×D4×C59—21no—
100C2×D4×C59—41no—
101C22×C4×C59—21no—
102C22×C4×C59—21no—
103C22×C4×C59—41no—
104C22×C4×C59C2×D411yescharacteristic
105C22×C4×C59C2×D411yescharacteristic
106C22×C4×C59#4111yescharacteristic
107Q16×C59—44no—
108SD16×C59—44no—
109SD16×C59—82no—
110M16×C59—41no—
111M16×C59—41no—
112M16×C59—41no—
113M16×C59—41no—
114C2×C8×C59—22no—
115C2×C8×C59—22no—
116C2×C8×C59—22no—
117C2×C8×C59—22no—
118C2×C8×C59—41no—
119C2×C8×C59—41no—
120C2×C8×C59—42no—
121C4⋊C4×C59—21no—
122C4⋊C4×C59—21no—
123C4⋊C4×C59—21no—
124C4⋊C4×C59—21no—
125C4⋊C4×C59—21no—
126C4⋊C4×C59—21no—
127C4⋊C4×C59—21no—
128C4⋊C4×C59—21no—
129C4⋊C4×C59—41no—
130C4⋊C4×C59—41no—
131C4⋊C4×C59—41no—
132C4⋊C4×C59—41no—
133C22⋊C4×C59—41no—
134C22⋊C4×C59—41no—
135C22⋊C4×C59—41no—
136C42×C59—41no—
137C42×C59—41no—
138C42×C59—41no—
139C42×C59C2×D411yescharacteristic
140C42×C59C2×D411yescharacteristic
141C42×C59C2×D411yescharacteristic
142C42×C59C2×D411yescharacteristic
143C23×C59—41no—
144C23×C59#12111yescharacteristic
145Q8×C59—22no—
146Q8×C59—22no—
147Q8×C59—42no—
148Q8×C59—41no—
149Q8×C59—41no—
150Q8×C59—41no—
151Q8×C59—41no—
152Q8×C59—41no—
153Q8×C59—41no—
154D4×C59—42no—
155D4×C59—41no—
156D4×C59—41no—
157D4×C59—41no—
158D4×C59—41no—
159D4×C59—81no—
160D4×C59—81no—
161C2×C4×C59—21no—
162C2×C4×C59—21no—
163C2×C4×C59—22no—
164C2×C4×C59—41no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59—42no—
168C2×C4×C59—41no—
169C2×C4×C59—41no—
170C2×C4×C59—41no—
171C2×C4×C59—41no—
172C2×C4×C59—81no—
173C2×C4×C59—81no—
174C2×C4×C59#12011yescharacteristic
175C2×C4×C59#12011yescharacteristic
176C2×C4×C59#12011yescharacteristic
177C2×C4×C59#12111yescharacteristic
178C2×C4×C59#12111yescharacteristic
179C2×C4×C59#12711yescharacteristic
180C2×C4×C59C2×SD1611yescharacteristic
181C2×C4×C59#13511yescharacteristic
182C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
183C2×C4×C59#13711yescharacteristic
184C8×C59—42no—
185C8×C59—42no—
186C8×C59—42no—
187C8×C59—42no—
188#10257C5911yesmaximal, characteristic, direct product factor, Sylow
189C22×C59—21no—
190C22×C59—21no—
191C22×C59—81no—
192C22×C59—81no—
193C22×C59—81no—
194C22×C59#39211yescharacteristic
195C22×C59#46511yescharacteristic
196C22×C59#46811yescharacteristic
197C4×C59—21no—
198C4×C59—21no—
199C4×C59—21no—
200C4×C59—42no—
201C4×C59—81no—
202C4×C59—81no—
203C4×C59—81no—
204C4×C59—81no—
205C4×C59#44811yescharacteristic
206C4×C59#44911yescharacteristic
207C4×C59#45011yescharacteristic
208C4×C59#45211yescharacteristic
209C4×C59#45511yescharacteristic
210C4×C59#45611yescharacteristic
211#3246C2×C5911yescharacteristic
212#3078C2×C5911yescharacteristic, complemented
213#3037C2×C5911yescharacteristic
214#1984C2×C5911yescharacteristic, complemented
215#1493C2×C5911yescharacteristic, complemented
216#1492C2×C5911yescharacteristic
217#1292C2×C5911yescharacteristic, complemented
218C2×C59—21no—
219C2×C59—21no—
220C2×C59—161no—
221C2×C59#200011yescharacteristic
222C2×C59#200711yescharacteristic
223C2×C59#201711yescharacteristic
224C2×#137—21no—
225#533C22×C5911yescharacteristic
226#532C22×C5911yescharacteristic
227C2×C4×Q8C22×C5911yescharacteristic
228#479—22no—
229#477—21no—
230#476—21no—
231#475—21no—
232#475—21no—
233#465—22no—
234#464—22no—
235#462—21no—
236#461—21no—
237#460—21no—
238#460—21no—
239#428—21no—
240#423C22×C5911yescharacteristic
241#423C22×C5911yescharacteristic
242C2×C4⋊C8C22×C5911yescharacteristic
243#419—21no—
244#417—21no—
245C2×C4×C8C22×C5911yescharacteristic
246#336—22no—
247Q8⋊C8—22no—
248D4⋊C8—22no—
249C59#1025711yescharacteristic, direct product factor, Sylow
250C2×#41—21no—
251C22×Q8D4×C5911yescharacteristic
252#137—42no—
253#137—42no—
254C2×Q16—22no—
255C2×SD16—22no—
256C2×SD16—22no—
257C2×M16—21no—
258C2×M16—21no—
259C22×C8—21no—
260C22×C8D4×C5912yes—
261C4⋊Q8D4×C5911yescharacteristic
262C4⋊Q8D4×C5911yescharacteristic
263C4⋊Q8D4×C5911yescharacteristic
264#127D4×C5911yescharacteristic
265#125—21no—
266#124—21no—
267C22⋊Q8—21no—
268C22⋊Q8—21no—
269#121—21no—
270#121—21no—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×Q8—21no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—21no—
277C4×D4—21no—
278C4×D4—21no—
279#117—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4D4×C5911yescharacteristic
282C2×C42—21no—
283C2×C42C23×C5911yescharacteristic
284#107—22no—
285#106—22no—
286#106—22no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8D4×C5912yes—
292C4⋊C8D4×C5912yes—
293#103—22no—
294#103—22no—
295#103—22no—
296#103—22no—
297#102—22no—
298#102—22no—
299C4×C8—21no—
300C4×C8—21no—
301#41—41no—
302#41—41no—
303#41—41no—
304#41—41no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8SD16×C5912yescomplemented
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—41no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—41no—
321C22×C4C2×D4×C5911yescharacteristic
322C22×C4C2×D4×C5911yescharacteristic
323C22×C4#41×C5911yescharacteristic
324Q16—44no—
325SD16—44no—
326SD16—82no—
327M16—41no—
328M16—41no—
329M16—41no—
330M16—41no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—42no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42C2×D4×C5911yescharacteristic
357C42C2×D4×C5911yescharacteristic
358C42C2×D4×C5911yescharacteristic
359C42C2×D4×C5911yescharacteristic
360C23—41no—
361C23#121×C5911yescharacteristic
362Q8—22no—
363Q8—22no—
364Q8—42no—
365Q8—41no—
366Q8—41no—
367Q8—41no—
368Q8—41no—
369Q8—41no—
370Q8—41no—
371D4—42no—
372D4—41no—
373D4—41no—
374D4—41no—
375D4—41no—
376D4—81no—
377D4—81no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4#120×C5911yescharacteristic
392C2×C4#120×C5911yescharacteristic
393C2×C4#120×C5911yescharacteristic
394C2×C4#121×C5911yescharacteristic
395C2×C4#121×C5911yescharacteristic
396C2×C4#127×C5911yescharacteristic
397C2×C4C2×SD16×C5911yescharacteristic
398C2×C4#135×C5911yescharacteristic
399C2×C4AGL(1,ℤ/8)×C5911yescharacteristic
400C2×C4#137×C5911yescharacteristic
401C8—42no—
402C8—42no—
403C8—42no—
404C8—42no—
405C22—21no—
406C22—21no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22C59×#39211yescharacteristic
411C22C59×#46511yescharacteristic
412C22C59×#46811yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—42no—
417C4—81no—
418C4—81no—
419C4—81no—
420C4—81no—
421C4C59×#44811yescharacteristic
422C4C59×#44911yescharacteristic
423C4C59×#45011yescharacteristic
424C4C59×#45211yescharacteristic
425C4C59×#45511yescharacteristic
426C4C59×#45611yescharacteristic
427C2—21no—
428C2—21no—
429C2—161no—
430C2C59×#200011yescharacteristic
431C2C59×#200711yescharacteristic
432C2C59×#201711yescharacteristic
subgroupG/H
center, socleC22×C59#392
commutator, Frattini, cosocleC2×C42C23×C59
prime radicalC59×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C59 ◁ C2×C42×C59 ◁ G

Complements

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

NH ≅ G/Ntype
2. C59×#3078C2semidirect
4. C59×#1984C2semidirect
5. C59×#1493C2semidirect
7. C59×#1292C2semidirect
96. C2×Q8×C59SD16semidirect
188. #10257C59direct (N × H)
212. #3078C2×C59semidirect
214. #1984C2×C59semidirect
215. #1493C2×C59semidirect
217. #1292C2×C59semidirect
249. C59#10257direct (N × H)
313. C2×Q8SD16×C59semidirect

Maximal subgroups

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

subgroupindex
C59×#32462
C59×#30782
C59×#30372
C59×#19842
C59×#14932
C59×#14922
C59×#12922
#1025759