Finite Group Databasebeta

← #28483557

Subgroups — #28483557

Order 15104.

500 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11745664C211yesmaximal, characteristic, complemented
2#21425016C211yesmaximal, characteristic
3C4×#14273651C211yesmaximal, characteristic, complemented
4#21422547C211yesmaximal, characteristic
5#21421998C211yesmaximal, characteristic, complemented
6#21421932C211yesmaximal, characteristic, complemented
7C59×#3103C211yesmaximal, characteristic
8#16837533—22no—
9#16837533—21no—
10Q8×C59⋊C8—21no—
11#16837426—21no—
12#16837426C2211yescharacteristic
13C2×#14273654—21no—
14C2×#14273654—21no—
15C2×#14273651—21no—
16C4×#11745664—21no—
17C4×#11745664C2211yescharacteristic
18C2×C4×C59⋊C8C2211yescharacteristic
19C4×C4⋊Dic59C2211yescharacteristic
20#16837164—21no—
21D4⋊(C59⋊C8)—22no—
22C59×#532C2211yescharacteristic
23C2×C4×Q8×C59C2211yescharacteristic
24C59×#481—21no—
25C59×#475C411yescharacteristic, complemented
26C59×#475C411yescharacteristic, complemented
27C59×#460C411yescharacteristic, complemented
28C59×#460C411yescharacteristic, complemented
29C59×#448—21no—
30C4×SD16×C59—21no—
31C4×SD16×C59—21no—
32C59×#431C2211yescharacteristic
33C2×#11745664—21no—
34C2×#11745664D411yescharacteristic
35C2×#11745664D411yescharacteristic
36C22×C59⋊C8—21no—
37C2×C4⋊Dic59—21no—
38C2×C4×Dic59—21no—
39C4⋊(C59⋊C8)—22no—
40C4⋊(C59⋊C8)—22no—
41C4⋊(C59⋊C8)—21no—
42C4⋊(C59⋊C8)D412yes—
43#14273654—42no—
44#14273654—42no—
45#14273651—24no—
46#14273638—22no—
47#14273638—21no—
48C4×C59⋊C8—22no—
49C4×C59⋊C8—21no—
50C4×C59⋊C8—21no—
51C4×C59⋊C8—21no—
52C59⋊#95—21no—
53C2×#41×C59—21no—
54C22×Q8×C59D411yescharacteristic
55C2×SD16×C59—21no—
56C2×SD16×C59—21no—
57C4⋊Q8×C59C2×C411yescharacteristic
58#127×C59C2×C411yescharacteristic
59#124×C59—21no—
60#121×C59—21no—
61#121×C59—21no—
62C4×Q8×C59—21no—
63C4×Q8×C59—21no—
64C4×Q8×C59C2×C411yescharacteristic
65C4×Q8×C59C2×C411yescharacteristic
66C4×Q8×C59D411yescharacteristic
67C4×Q8×C59D411yescharacteristic
68C4×D4×C59—21no—
69C4×D4×C59—21no—
70C2×C4⋊C4×C59—21no—
71C2×C4⋊C4×C59D411yescharacteristic
72C2×C42×C59—21no—
73C2×C42×C59C2311yescharacteristic
74#106×C59—21no—
75#106×C59—21no—
76C4⋊C8×C59C2×C411yescharacteristic
77C4⋊C8×C59C2×C411yescharacteristic
78#103×C59—21no—
79#103×C59—21no—
80#102×C59—21no—
81#102×C59—21no—
82C22⋊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—22no—
90C2×Q8×C59—21no—
91C2×Q8×C59—41no—
92C2×Q8×C59C22⋊C411yescharacteristic
93C2×Q8×C59C22⋊C411yescharacteristic
94C2×D4×C59—21no—
95C2×D4×C59—21no—
96C2×D4×C59—21no—
97C2×D4×C59—41no—
98C22×C4×C59—21no—
99C22×C4×C59—21no—
100C22×C4×C59—41no—
101C22×C4×C59C2×D411yescharacteristic
102C22×C4×C59C2×D411yescharacteristic
103C22×C4×C59#4111yescharacteristic
104C22×Dic59—41no—
105SD16×C59—42no—
106SD16×C59—42no—
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×C59—21no—
114C4⋊C4×C59—21no—
115C4⋊C4×C59—21no—
116C4⋊C4×C59C22⋊C411yescharacteristic
117C4⋊C4×C59C22⋊C411yescharacteristic
118C22⋊C4×C59—41no—
119C22⋊C4×C59—41no—
120C42×C59—21no—
121C42×C59—21no—
122C42×C59—21no—
123C42×C59—21no—
124C42×C59C22×C411yescharacteristic
125C42×C59C2×D411yescharacteristic
126C42×C59C2×D411yescharacteristic
127C42×C59#4111yescharacteristic
128C4⋊Dic59—22no—
129C4⋊Dic59—41no—
130C4×Dic59—41no—
131C4×Dic59—41no—
132#11745664—21no—
133#11745664—21no—
134#11745664—21no—
135#11745664—41no—
136#11745664—41no—
137#11745664SD1612yescomplemented
138C2×C59⋊C8—21no—
139C2×C59⋊C8—22no—
140C2×C59⋊C8—21no—
141C2×C59⋊C8—22no—
142C2×C59⋊C8—41no—
143C2×C59⋊C8—41no—
144C23×C59—41no—
145C23×C59#12111yescharacteristic
146Q8×C59—21no—
147Q8×C59—21no—
148Q8×C59—21no—
149Q8×C59—42no—
150Q8×C59—41no—
151Q8×C59—41no—
152Q8×C59#10412yescomplemented
153D4×C59—42no—
154D4×C59—41no—
155D4×C59—41no—
156D4×C59—41no—
157D4×C59—41no—
158D4×C59—81no—
159D4×C59—81no—
160C2×C4×C59—21no—
161C2×C4×C59—21no—
162C2×C4×C59—21no—
163C2×C4×C59—21no—
164C2×C4×C59—21no—
165C2×C4×C59—21no—
166C2×C4×C59—41no—
167C2×C4×C59—42no—
168C2×C4×C59—41no—
169C2×C4×C59—81no—
170C2×C4×C59—81no—
171C2×C4×C59—81no—
172C2×C4×C59C2×C22⋊C411yescharacteristic
173C2×C4×C59C4×D411yescharacteristic
174C2×C4×C59C4×D411yescharacteristic
175C2×C4×C59#12011yescharacteristic
176C2×C4×C59#12111yescharacteristic
177C2×C4×C59#12311yescharacteristic
178C2×C4×C59C2×SD1611yescharacteristic
179C2×C4×C59#13511yescharacteristic
180C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
181C2×C4×C59#13711yescharacteristic
182C2×Dic59—42no—
183C2×Dic59—81no—
184C2×Dic59—81no—
185C8×C59—41no—
186C8×C59—41no—
187C59⋊C8—22no—
188C59⋊C8—42no—
189C59⋊C8—41no—
190C59⋊C8—42no—
191#8992—591nomaximal, Sylow
192C22×C59—21no—
193C22×C59—21no—
194C22×C59—81no—
195C22×C59—81no—
196C22×C59—81no—
197C22×C59#38611yescharacteristic
198C22×C59#46011yescharacteristic
199C22×C59#46411yescharacteristic
200C4×C59—21no—
201C4×C59—22no—
202C4×C59—21no—
203C4×C59—21no—
204C4×C59—41no—
205C4×C59—41no—
206C4×C59—41no—
207C4×C59—81no—
208C4×C59C2×#10411yescharacteristic
209C4×C59#42111yescharacteristic
210C4×C59C4×SD1611yescharacteristic
211C4×C59#44011yescharacteristic
212C4×C59#44811yescharacteristic
213C4×C59#44911yescharacteristic
214Dic59—82no—
215#3103D5911yescharacteristic
216Q8×M16—591no—
217#1980—591no—
218C4×#103—591no—
219#1492—591no—
220#1303—591no—
221#1278—591no—
222C2×C59—21no—
223C2×C59—21no—
224C2×C59—161no—
225C2×C59#187111yescharacteristic
226C2×C59#188011yescharacteristic
227C2×C59#188111yescharacteristic
228#532C2×D5911yescharacteristic
229C2×C4×Q8C2×D5911yescharacteristic
230#481—21no—
231#475Dic5911yescharacteristic, complemented
232#475Dic5911yescharacteristic, complemented
233#460Dic5911yescharacteristic, complemented
234#460Dic5911yescharacteristic, complemented
235#448—21no—
236#446—1182no—
237#446—1181no—
238C8×Q8—1181no—
239C4×SD16—21no—
240C4×SD16—21no—
241#431C2×D5911yescharacteristic
242#423—591no—
243#423—1181no—
244C2×#104—1181no—
245C2×#104—1181no—
246C2×#103—1181no—
247C4×M16—591no—
248C4×M16—1181no—
249C2×C4×C8—591no—
250C4×C4⋊C4—591no—
251#340—1181no—
252D4⋊C8—1182no—
253C59#899211yescharacteristic, complemented, Sylow
254C2×#41—21no—
255C22×Q8C22⋊D5911yescharacteristic
256C2×SD16—21no—
257C2×SD16—21no—
258C2×M16—591no—
259C2×M16—591no—
260C2×M16—1181no—
261C22×C8—1181no—
262C4⋊Q8C2×Dic5911yescharacteristic
263#127C2×Dic5911yescharacteristic
264#124—21no—
265#121—21no—
266#121—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8C2×Dic5911yescharacteristic
270C4×Q8C2×Dic5911yescharacteristic
271C4×Q8C22⋊D5911yescharacteristic
272C4×Q8C22⋊D5911yescharacteristic
273C4×D4—21no—
274C4×D4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—1181no—
277C2×C4⋊C4C22⋊D5911yescharacteristic
278C2×C42—21no—
279C2×C42—1181no—
280C2×C42C22×D5911yescharacteristic
281#106—21no—
282#106—21no—
283C4⋊C8—592no—
284C4⋊C8—1182no—
285C4⋊C8—1182no—
286C4⋊C8—1181no—
287C4⋊C8C2×Dic5911yescharacteristic
288C4⋊C8C2×Dic5911yescharacteristic
289#104—2362no—
290#104—2362no—
291#103—21no—
292#103—21no—
293#103—1184no—
294#102—21no—
295#102—21no—
296C22⋊C8—21no—
297#97—1182no—
298#97—1181no—
299C4×C8—21no—
300C4×C8—1182no—
301C4×C8—1181no—
302C4×C8—1181no—
303C4×C8—1181no—
304#95—1181no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309C2×Q8—21no—
310C2×Q8—22no—
311C2×Q8—21no—
312C2×Q8—41no—
313C2×Q8C22⋊Dic5911yescharacteristic
314C2×Q8C22⋊Dic5911yescharacteristic
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—41no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—41no—
322C22×C4—2361no—
323C22×C4D4×D5911yescharacteristic
324C22×C4#1174568711yescharacteristic
325C22×C4C2×C22⋊D5911yescharacteristic
326SD16—42no—
327SD16—42no—
328M16—592no—
329M16—1181no—
330M16—1181no—
331M16—1181no—
332M16—2361no—
333M16—2361no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—1181no—
337C2×C8—1182no—
338C2×C8—1182no—
339C2×C8—1181no—
340C2×C8—2361no—
341C2×C8—2361no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—1182no—
350C4⋊C4—2361no—
351C4⋊C4C22⋊Dic5911yescharacteristic
352C4⋊C4C22⋊Dic5911yescharacteristic
353C22⋊C4—41no—
354C22⋊C4—41no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—2361no—
360C42—2361no—
361C42D4×D5911yescharacteristic
362C42#1174568711yescharacteristic
363C42C22×Dic5911yescharacteristic
364C42C2×C22⋊D5911yescharacteristic
365C23—41no—
366C23#1427369911yescharacteristic
367Q8—21no—
368Q8—21no—
369Q8—21no—
370Q8—42no—
371Q8—41no—
372Q8—41no—
373Q8#1427365412yescomplemented
374D4—42no—
375D4—41no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—81no—
380D4—81no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—4721no—
394C2×C4—4721no—
395C2×C4—2362no—
396C2×C4C2×C22⋊Dic5911yescharacteristic
397C2×C4D4×Dic5911yescharacteristic
398C2×C4D4×Dic5911yescharacteristic
399C2×C4#1427369511yescharacteristic
400C2×C4#1427369911yescharacteristic
401C2×C4#1427370811yescharacteristic
402C2×C4SD16×D5911yescharacteristic
403C2×C4#1427374811yescharacteristic
404C2×C4#1427375411yescharacteristic
405C2×C4#1427375611yescharacteristic
406C8—41no—
407C8—41no—
408C8—1182no—
409C8—2362no—
410C8—2361no—
411C8—2362no—
412C22—21no—
413C22—21no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22#1683728211yescharacteristic
418C22#1683760011yescharacteristic
419C22#1683762211yescharacteristic
420C4—21no—
421C4—22no—
422C4—21no—
423C4—21no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—81no—
428C4—4722no—
429C4C2×#1427365411yescharacteristic
430C4#1683741911yescharacteristic
431C4SD16×Dic5911yescharacteristic
432C4#1683750311yescharacteristic
433C4#1683753911yescharacteristic
434C4#1683754611yescharacteristic
435C2—21no—
436C2—21no—
437C2—161no—
438C2#2142439911yescharacteristic
439C2#2142445311yescharacteristic
440C2#2142446011yescharacteristic
subgroupG/H
centerC2×C4#14273695
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#3103C2
socleC22×C59#386
prime radicalC59×#532C22
cosocleC2×C42×C59C23
hypercenter#3103D59
hypocenterC59#8992
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
1. Q8×#11745664C2semidirect
3. C4×#14273651C2semidirect
5. #21421998C2semidirect
6. #21421932C2semidirect
25. C59×#475C4semidirect
26. C59×#475C4semidirect
27. C59×#460C4semidirect
28. C59×#460C4semidirect
137. #11745664SD16semidirect
152. Q8×C59#104semidirect
231. #475Dic59semidirect
232. #475Dic59semidirect
233. #460Dic59semidirect
234. #460Dic59semidirect
253. C59#8992semidirect
373. Q8#14273654semidirect

Maximal subgroups

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

subgroupindex
Q8×#117456642
#214250162
C4×#142736512
#214225472
#214219982
#214219322
C59×#31032
#899259