Finite Group Databasebeta

← #28483912

Subgroups — #28483912

Order 15104.

538 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21432840C211yesmaximal, characteristic, complemented
2#21425016C211yesmaximal, characteristic, complemented
3#21422591C211yesmaximal, characteristic, complemented
4#21422495C211yesmaximal, characteristic, complemented
5#21421936C211yesmaximal, characteristic, complemented
6#21421933C211yesmaximal, characteristic, complemented
7C59×#3169C211yesmaximal, characteristic
8C2×#14273734—21no—
9#16837480—21no—
10#16837474—22no—
11D4×C59⋊C8—21no—
12#16837455C2211yescharacteristic
13C2×C4⋊(C59⋊C8)C2211yescharacteristic
14C2×#14273654—21no—
15C2×#14273654—21no—
16#16837369—21no—
17C4×#11745664C2211yescharacteristic
18C4×C4⋊Dic59C2211yescharacteristic
19#16837136—22no—
20D4⋊(C59⋊C8)—21no—
21D4⋊(C59⋊C8)—21no—
22C59×#532C2211yescharacteristic
23C59×#532C2211yescharacteristic
24C59×#500C411yescharacteristic, complemented
25C59×#493C411yescharacteristic, complemented
26C59×#486C411yescharacteristic, complemented
27C59×#486C411yescharacteristic, complemented
28C59×#466—22no—
29C4×D8×C59—22no—
30C2×C4×C8×C59C2211yescharacteristic
31#14273734—42no—
32#14273734—42no—
33C2×#11745664—22no—
34C2×#11745664D411yescharacteristic
35C2×#11745664D411yescharacteristic
36C22×C59⋊C8—22no—
37C22×C59⋊C8—21no—
38C2×C4⋊Dic59—21no—
39C2×C4×Dic59—21no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)—21no—
42C4⋊(C59⋊C8)D411yescharacteristic
43C4⋊(C59⋊C8)D411yescharacteristic
44#14273654—42no—
45#14273654—42no—
46C22⋊(C59⋊C8)—22no—
47C22⋊(C59⋊C8)—22no—
48C22⋊(C59⋊C8)—21no—
49#14273638—21no—
50C4×C59⋊C8—21no—
51C4×C59⋊C8—21no—
52C59⋊#95—21no—
53C2×#41×C59—21no—
54C2×#41×C59—21no—
55C2×D8×C59—22no—
56C22×C8×C59—21no—
57C4⋊Q8×C59C2×C411yescharacteristic
58C4⋊Q8×C59C2×C411yescharacteristic
59#127×C59C2×C411yescharacteristic
60#127×C59C2×C411yescharacteristic
61#124×C59—21no—
62#124×C59—21no—
63#121×C59—21no—
64#121×C59—21no—
65#121×C59—21no—
66#121×C59D412yes—
67C4×D4×C59—21no—
68C4×D4×C59—21no—
69C4×D4×C59—21no—
70C4×D4×C59D412yes—
71C2×C42×C59C2311yescharacteristic
72#107×C59—22no—
73#102×C59—22no—
74#102×C59—22no—
75C4×C8×C59—21no—
76C4×C8×C59C2×C411yescharacteristic
77C4×C8×C59C2×C411yescharacteristic
78#41×C59—41no—
79#41×C59—41no—
80#41×C59—41no—
81#41×C59—41no—
82#41×C59—41no—
83#41×C59—41no—
84#41×C59—41no—
85#41×C59—41no—
86C2×Q8×C59—21no—
87C2×Q8×C59—21no—
88C2×D4×C59—22no—
89C2×D4×C59—21no—
90C2×D4×C59—21no—
91C2×D4×C59—21no—
92C2×D4×C59—21no—
93C2×D4×C59—21no—
94C2×D4×C59—41no—
95C2×D4×C59C22⋊C412yes—
96C22×C4×C59—22no—
97C22×C4×C59—41no—
98C22×C4×C59C2×D411yescharacteristic
99C22×C4×C59C2×D411yescharacteristic
100C22×C4×C59#4111yescharacteristic
101C22×Dic59—41no—
102D8×C59—44no—
103C2×C8×C59—21no—
104C2×C8×C59—21no—
105C2×C8×C59—42no—
106C4⋊C4×C59—21no—
107C4⋊C4×C59—21no—
108C4⋊C4×C59—21no—
109C4⋊C4×C59C22⋊C412yes—
110C22⋊C4×C59—22no—
111C22⋊C4×C59—22no—
112C22⋊C4×C59—41no—
113C22⋊C4×C59—41no—
114C42×C59C22×C411yescharacteristic
115C42×C59C2×D411yescharacteristic
116C42×C59C2×D411yescharacteristic
117C42×C59#4111yescharacteristic
118C4⋊Dic59—22no—
119C4⋊Dic59—41no—
120C4×Dic59—41no—
121C4×Dic59—41no—
122#11745664—21no—
123#11745664—21no—
124#11745664—21no—
125#11745664—44no—
126#11745664D812yescomplemented
127C2×C59⋊C8—21no—
128C2×C59⋊C8—21no—
129C2×C59⋊C8—21no—
130C2×C59⋊C8—21no—
131C2×C59⋊C8—44no—
132C2×C59⋊C8—41no—
133C2×C59⋊C8—41no—
134C2×C59⋊C8—41no—
135C23×C59—22no—
136C23×C59—41no—
137C23×C59#12111yescharacteristic
138Q8×C59—41no—
139Q8×C59—41no—
140Q8×C59—41no—
141Q8×C59—41no—
142D4×C59—42no—
143D4×C59—42no—
144D4×C59—42no—
145D4×C59—42no—
146D4×C59—41no—
147D4×C59—41no—
148D4×C59—41no—
149D4×C59—41no—
150D4×C59—41no—
151D4×C59—41no—
152D4×C59—41no—
153D4×C59—41no—
154D4×C59—81no—
155D4×C59—81no—
156C2×C4×C59—21no—
157C2×C4×C59—21no—
158C2×C4×C59—22no—
159C2×C4×C59—41no—
160C2×C4×C59—42no—
161C2×C4×C59—42no—
162C2×C4×C59—41no—
163C2×C4×C59—81no—
164C2×C4×C59—81no—
165C2×C4×C59C2×C22⋊C411yescharacteristic
166C2×C4×C59C4×D411yescharacteristic
167C2×C4×C59C4×D411yescharacteristic
168C2×C4×C59#12011yescharacteristic
169C2×C4×C59#12111yescharacteristic
170C2×C4×C59#12311yescharacteristic
171C2×C4×C59C2×D811yescharacteristic
172C2×C4×C59#13511yescharacteristic
173C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
174C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
175C2×Dic59—42no—
176C2×Dic59—81no—
177C2×Dic59—81no—
178C8×C59—22no—
179C8×C59—41no—
180C59⋊C8—22no—
181C59⋊C8—41no—
182C59⋊C8—41no—
183C59⋊C8—41no—
184C59⋊C8—81no—
185#9053—591nomaximal, Sylow
186C22×C59—21no—
187C22×C59—21no—
188C22×C59—42no—
189C22×C59—42no—
190C22×C59—42no—
191C22×C59—81no—
192C22×C59—81no—
193C22×C59—81no—
194C22×C59#38611yescharacteristic
195C22×C59#45911yescharacteristic
196C22×C59#46311yescharacteristic
197C4×C59—21no—
198C4×C59—21no—
199C4×C59—21no—
200C4×C59—42no—
201C4×C59—81no—
202C4×C59C4×D811yescharacteristic
203C4×C59#44211yescharacteristic
204C4×C59#44311yescharacteristic
205C4×C59#44411yescharacteristic
206C4×C59#44711yescharacteristic
207C4×C59#44911yescharacteristic
208Dic59—82no—
209#3169D5911yescharacteristic
210#2935—591no—
211#1980—591no—
212#1502—591no—
213#1480—591no—
214#1279—591no—
215#1278—591no—
216C2×C59—21no—
217C2×C59—21no—
218C2×C59—82no—
219C2×C59—161no—
220C2×C59#187611yescharacteristic
221C2×C59#187911yescharacteristic
222C2×C59#188411yescharacteristic
223C2×#131—1181no—
224#532C2×D5911yescharacteristic
225#532C2×D5911yescharacteristic
226#500Dic5911yescharacteristic, complemented
227#493Dic5911yescharacteristic, complemented
228#486Dic5911yescharacteristic, complemented
229#486Dic5911yescharacteristic, complemented
230#466—22no—
231C4×D8—22no—
232#436—1181no—
233#435—1182no—
234C8×D4—1181no—
235#431—591no—
236C2×C4⋊C8—591no—
237C2×#104—1181no—
238C2×#104—1181no—
239#408—1181no—
240C4×M16—591no—
241C2×C4×C8C2×D5911yescharacteristic
242C4×C4⋊C4—591no—
243#327—1182no—
244D4⋊C8—1181no—
245D4⋊C8—1181no—
246C59#905311yescharacteristic, complemented, Sylow
247C2×#41—21no—
248C2×#41—21no—
249C2×D8—22no—
250#131—2362no—
251#131—2362no—
252C2×M16—591no—
253C2×M16—591no—
254C2×M16—1182no—
255C22×C8—21no—
256C22×C8—1182no—
257C22×C8—1181no—
258C4⋊Q8C2×Dic5911yescharacteristic
259C4⋊Q8C2×Dic5911yescharacteristic
260#127C2×Dic5911yescharacteristic
261#127C2×Dic5911yescharacteristic
262#124—21no—
263#124—21no—
264#121—21no—
265#121—21no—
266#121—21no—
267#121C22⋊D5912yes—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—21no—
271C4×D4C22⋊D5912yes—
272C2×C4⋊C4—1181no—
273C2×C42—1181no—
274C2×C42C22×D5911yescharacteristic
275#107—22no—
276C4⋊C8—591no—
277C4⋊C8—591no—
278C4⋊C8—1181no—
279C4⋊C8—1181no—
280#104—2362no—
281#104—2362no—
282#102—22no—
283#102—22no—
284C22⋊C8—1182no—
285C22⋊C8—1182no—
286C22⋊C8—1181no—
287#97—1181no—
288C4×C8—21no—
289C4×C8—1181no—
290C4×C8—1181no—
291C4×C8C2×Dic5911yescharacteristic
292C4×C8C2×Dic5911yescharacteristic
293#95—1181no—
294#41—41no—
295#41—41no—
296#41—41no—
297#41—41no—
298#41—41no—
299#41—41no—
300#41—41no—
301#41—41no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×D4—22no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—41no—
311C2×D4C22⋊Dic5912yes—
312C22×C4—22no—
313C22×C4—41no—
314C22×C4—2361no—
315C22×C4D4×D5911yescharacteristic
316C22×C4#1174568711yescharacteristic
317C22×C4C2×C22⋊D5911yescharacteristic
318D8—44no—
319M16—592no—
320M16—1181no—
321M16—1181no—
322M16—1181no—
323M16—2364no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—42no—
327C2×C8—1181no—
328C2×C8—1181no—
329C2×C8—1181no—
330C2×C8—1181no—
331C2×C8—2364no—
332C2×C8—2361no—
333C2×C8—2361no—
334C2×C8—2361no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—1182no—
339C4⋊C4—2361no—
340C4⋊C4C22⋊Dic5912yes—
341C22⋊C4—22no—
342C22⋊C4—22no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C42—2361no—
346C42—2361no—
347C42D4×D5911yescharacteristic
348C42#1174568711yescharacteristic
349C42C22×Dic5911yescharacteristic
350C42C2×C22⋊D5911yescharacteristic
351C23—22no—
352C23—41no—
353C23#1427369911yescharacteristic
354Q8—41no—
355Q8—41no—
356Q8—41no—
357Q8—41no—
358D4—42no—
359D4—42no—
360D4—42no—
361D4—42no—
362D4—41no—
363D4—41no—
364D4—41no—
365D4—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—41no—
370D4—81no—
371D4—81no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—4721no—
382C2×C4—4721no—
383C2×C4—2362no—
384C2×C4C2×C22⋊Dic5911yescharacteristic
385C2×C4D4×Dic5911yescharacteristic
386C2×C4D4×Dic5911yescharacteristic
387C2×C4#1427369511yescharacteristic
388C2×C4#1427369911yescharacteristic
389C2×C4#1427370811yescharacteristic
390C2×C4D8×D5911yescharacteristic
391C2×C4#1427374611yescharacteristic
392C2×C4#1427375111yescharacteristic
393C2×C4#1427375111yescharacteristic
394C8—22no—
395C8—41no—
396C8—1182no—
397C8—4721no—
398C8—2361no—
399C8—2361no—
400C8—2361no—
401C22—21no—
402C22—21no—
403C22—42no—
404C22—42no—
405C22—42no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22#1683728211yescharacteristic
410C22#1683759511yescharacteristic
411C22#1683761511yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—42no—
416C4—81no—
417C4—4722no—
418C4D8×Dic5911yescharacteristic
419C4#1683751411yescharacteristic
420C4#1683751911yescharacteristic
421C4#1683752411yescharacteristic
422C4#1683753411yescharacteristic
423C4#1683754411yescharacteristic
424C2—21no—
425C2—21no—
426C2—82no—
427C2—161no—
428C2#2142442511yescharacteristic
429C2#2142444611yescharacteristic
430C2#2142448011yescharacteristic
subgroupG/H
centerC2×C4#14273695
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#3169C2
socleC22×C59#386
cosocleC2×C42×C59C23
hypercenter#3169D59
hypocenterC59#9053
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3169

Complements

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

NH ≅ G/Ntype
1. #21432840C2semidirect
2. #21425016C2semidirect
3. #21422591C2semidirect
4. #21422495C2semidirect
5. #21421936C2semidirect
6. #21421933C2semidirect
24. C59×#500C4semidirect
25. C59×#493C4semidirect
26. C59×#486C4semidirect
27. C59×#486C4semidirect
126. #11745664D8semidirect
226. #500Dic59semidirect
227. #493Dic59semidirect
228. #486Dic59semidirect
229. #486Dic59semidirect
246. C59#9053semidirect

Maximal subgroups

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

subgroupindex
#214328402
#214250162
#214225912
#214224952
#214219362
#214219332
C59×#31692
#905359