Finite Group Databasebeta

← #28484584

Subgroups — #28484584

Order 15104.

562 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
1#21432664C211yesmaximal, characteristic, complemented
2#21422764C211yesmaximal, characteristic, complemented
3C2×#16837146C211yesmaximal, characteristic, complemented
4#21422483C211yesmaximal, characteristic, complemented
5#21421991C211yesmaximal, characteristic, complemented
6#21421970C211yesmaximal, characteristic, complemented
7C59×#3038C211yesmaximal, characteristic
8C2×#14273734—21no—
9#16837455C2211yescharacteristic
10#16837429—22no—
11#16837429—21no—
12#16837426—21no—
13#16837426C2211yescharacteristic
14C2×C4⋊(C59⋊C8)—21no—
15C2×C4⋊(C59⋊C8)C2211yescharacteristic
16C2×C4⋊(C59⋊C8)C2211yescharacteristic
17#16837146—21no—
18#16837146—21no—
19#16837146—22no—
20Q8⋊(C59⋊C8)—21no—
21D4⋊(C59⋊C8)—21no—
22C2×AGL(1,ℤ/8)×C59—21no—
23C2×#127×C59C2211yescharacteristic
24C4×#41×C59C2211yescharacteristic
25C59×#460—21no—
26C59×#460C412yes—
27C59×#459—21no—
28C59×#459C412yes—
29C59×#423C2211yescharacteristic
30C59×#418—21no—
31#14273734—42no—
32#14273734—42no—
33C2×#11745664—22no—
34C2×#11745664—21no—
35C22×C59⋊C8—22no—
36C22×C59⋊C8—21no—
37C22×C59⋊C8—21no—
38C4⋊(C59⋊C8)—22no—
39C4⋊(C59⋊C8)—22no—
40C4⋊(C59⋊C8)—21no—
41C4⋊(C59⋊C8)—21no—
42C4⋊(C59⋊C8)—21no—
43C4⋊(C59⋊C8)—21no—
44C4⋊(C59⋊C8)—21no—
45C4⋊(C59⋊C8)D411yescharacteristic
46C4⋊(C59⋊C8)D411yescharacteristic
47C4⋊(C59⋊C8)D412yes—
48C22⋊(C59⋊C8)—21no—
49C4×C59⋊C8—21no—
50C2×#41×C59D411yescharacteristic
51C22×D4×C59—21no—
52C22×D4×C59—21no—
53C22×D4×C59—21no—
54AGL(1,ℤ/8)×C59—42no—
55AGL(1,ℤ/8)×C59—42no—
56C2×SD16×C59—22no—
57C2×D8×C59—22no—
58C2×M16×C59—21no—
59#127×C59—21no—
60#127×C59—21no—
61#127×C59—21no—
62#127×C59C2×C412yes—
63C4×Q8×C59C2×C411yescharacteristic
64C4×Q8×C59D411yescharacteristic
65C4×D4×C59—21no—
66C4×D4×C59—21no—
67C4×D4×C59C2×C411yescharacteristic
68C4×D4×C59D411yescharacteristic
69#117×C59—21no—
70#117×C59D411yescharacteristic
71C2×C42×C59—21no—
72C2×C42×C59C2311yescharacteristic
73C4⋊C8×C59—21no—
74C4⋊C8×C59C2×C412yes—
75#102×C59—22no—
76#102×C59—22no—
77C24×C59—41no—
78#41×C59—21no—
79#41×C59—21no—
80#41×C59D812yes—
81#41×C59SD1612yes—
82C2×Q8×C59C22⋊C411yescharacteristic
83C2×D4×C59—21no—
84C2×D4×C59—22no—
85C2×D4×C59—22no—
86C2×D4×C59—22no—
87C2×D4×C59—41no—
88C2×D4×C59—41no—
89C2×D4×C59—41no—
90C2×D4×C59—41no—
91C2×D4×C59—41no—
92C2×D4×C59—41no—
93C2×D4×C59—41no—
94C2×D4×C59—81no—
95C2×D4×C59—81no—
96C2×D4×C59—82no—
97C2×D4×C59C22⋊C411yescharacteristic
98C22×C4×C59—21no—
99C22×C4×C59—21no—
100C22×C4×C59—21no—
101C22×C4×C59C2×D411yescharacteristic
102C22×C4×C59#4111yescharacteristic
103C22×C4×C59#4111yescharacteristic
104SD16×C59—44no—
105D8×C59—44no—
106M16×C59—41no—
107M16×C59—41no—
108C2×C8×C59—22no—
109C4⋊C4×C59—21no—
110C4⋊C4×C59—21no—
111C4⋊C4×C59C22⋊C411yescharacteristic
112C4⋊C4×C59C22⋊C411yescharacteristic
113C22⋊C4×C59—21no—
114C22⋊C4×C59—21no—
115C22⋊C4×C59—21no—
116C42×C59—21no—
117C42×C59—21no—
118C42×C59—21no—
119C42×C59C22×C411yescharacteristic
120C42×C59C2×D411yescharacteristic
121C42×C59C2×D411yescharacteristic
122C42×C59C2×D411yescharacteristic
123#11745664—44no—
124#11745664—41no—
125#11745664—41no—
126C2×C59⋊C8—21no—
127C2×C59⋊C8—22no—
128C2×C59⋊C8—21no—
129C2×C59⋊C8—44no—
130C2×C59⋊C8—41no—
131C2×C59⋊C8—41no—
132C2×C59⋊C8—41no—
133C2×C59⋊C8—41no—
134C2×C59⋊C8—81no—
135C23×C59—21no—
136C23×C59—42no—
137C23×C59—82no—
138C23×C59—82no—
139C23×C59—81no—
140C23×C59—81no—
141C23×C59#12311yescharacteristic
142Q8×C59—21no—
143Q8×C59#10212yes—
144D4×C59—22no—
145D4×C59—22no—
146D4×C59—21no—
147D4×C59—42no—
148D4×C59—42no—
149D4×C59—82no—
150D4×C59—82no—
151D4×C59—82no—
152D4×C59—81no—
153D4×C59—81no—
154D4×C59—81no—
155D4×C59—81no—
156D4×C59#10212yes—
157C2×C4×C59—21no—
158C2×C4×C59—22no—
159C2×C4×C59—22no—
160C2×C4×C59—21no—
161C2×C4×C59—21no—
162C2×C4×C59—21no—
163C2×C4×C59—21no—
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×C59C4×D411yescharacteristic
171C2×C4×C59#12011yescharacteristic
172C2×C4×C59#12111yescharacteristic
173C2×C4×C59#12111yescharacteristic
174C2×C4×C59C2×D811yescharacteristic
175C2×C4×C59C2×SD1611yescharacteristic
176C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C59#13711yescharacteristic
178C8×C59—42no—
179C59⋊C8—41no—
180C59⋊C8—42no—
181C59⋊C8—41no—
182C59⋊C8—81no—
183C59⋊C8—81no—
184#9173—591nomaximal, Sylow
185C22×C59—21no—
186C22×C59—21no—
187C22×C59—22no—
188C22×C59—41no—
189C22×C59—41no—
190C22×C59—82no—
191C22×C59—82no—
192C22×C59—82no—
193C22×C59—162no—
194C22×C59—162no—
195C22×C59#38611yescharacteristic
196C22×C59#48011yescharacteristic
197C22×C59#48111yescharacteristic
198C4×C59—22no—
199C4×C59—21no—
200C4×C59—21no—
201C4×C59—21no—
202C4×C59—41no—
203C4×C59—41no—
204C4×C59C2×#10211yescharacteristic
205C4×C59#41711yescharacteristic
206C4×C59#44711yescharacteristic
207C4×C59#44811yescharacteristic
208C4×C59#47111yescharacteristic
209C4×C59#47211yescharacteristic
210#3038D5911yescharacteristic
211#2906—591no—
212#1545—591no—
213C2×#331—591no—
214#1477—591no—
215#1301—591no—
216#1292—591no—
217C2×C59—21no—
218C2×C59—21no—
219C2×C59—42no—
220C2×C59—162no—
221C2×C59#187411yescharacteristic
222C2×C59#190311yescharacteristic
223C2×C59#190711yescharacteristic
224C2×AGL(1,ℤ/8)—21no—
225C2×#131—1181no—
226C2×#127C2×D5911yescharacteristic
227C4×#41C2×D5911yescharacteristic
228#460—21no—
229#460Dic5912yes—
230#459—21no—
231#459Dic5912yes—
232#431—591no—
233#424—1182no—
234#424—1181no—
235#423—591no—
236#423—1181no—
237#423C2×D5911yescharacteristic
238C2×C4⋊C8—591no—
239C2×C4⋊C8—591no—
240C2×C4⋊C8—1181no—
241#418—21no—
242#331—1181no—
243#331—1181no—
244#331—1182no—
245Q8⋊C8—1181no—
246D4⋊C8—1181no—
247C59#917311yescharacteristic, complemented, Sylow
248C2×#41C22⋊D5911yescharacteristic
249C22×D4—21no—
250C22×D4—21no—
251C22×D4—21no—
252AGL(1,ℤ/8)—42no—
253AGL(1,ℤ/8)—42no—
254C2×SD16—22no—
255C2×D8—22no—
256#131—2362no—
257#131—2362no—
258C2×M16—21no—
259C2×M16—1182no—
260C2×M16—1181no—
261C22×C8—1182no—
262C22×C8—1181no—
263C22×C8—1181no—
264#127—21no—
265#127—21no—
266#127—21no—
267#127C2×Dic5912yes—
268C4×Q8C2×Dic5911yescharacteristic
269C4×Q8C22⋊D5911yescharacteristic
270C4×D4—21no—
271C4×D4—21no—
272C4×D4C2×Dic5911yescharacteristic
273C4×D4C22⋊D5911yescharacteristic
274#117—21no—
275#117C22⋊D5911yescharacteristic
276C2×C42—21no—
277C2×C42C22×D5911yescharacteristic
278C4⋊C8—21no—
279C4⋊C8—591no—
280C4⋊C8—591no—
281C4⋊C8—592no—
282C4⋊C8—1182no—
283C4⋊C8—1182no—
284C4⋊C8—1181no—
285C4⋊C8—1181no—
286C4⋊C8—1181no—
287C4⋊C8—1181no—
288C4⋊C8—1181no—
289C4⋊C8C2×Dic5912yes—
290#102—22no—
291#102—22no—
292C22⋊C8—1181no—
293C4×C8—1181no—
294C24—41no—
295#41—21no—
296#41—21no—
297#41#1174566912yes—
298#41#1174567012yes—
299C2×Q8C22⋊Dic5911yescharacteristic
300C2×D4—21no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—81no—
312C2×D4—81no—
313C2×D4—82no—
314C2×D4C22⋊Dic5911yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4#1174568711yescharacteristic
319C22×C4#1174568711yescharacteristic
320C22×C4C2×C22⋊D5911yescharacteristic
321SD16—44no—
322D8—44no—
323M16—41no—
324M16—41no—
325M16—2364no—
326M16—2361no—
327M16—2361no—
328C2×C8—22no—
329C2×C8—1181no—
330C2×C8—1181no—
331C2×C8—1182no—
332C2×C8—4721no—
333C2×C8—2361no—
334C2×C8—2361no—
335C2×C8—2364no—
336C2×C8—2361no—
337C2×C8—2361no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4C22⋊Dic5911yescharacteristic
341C4⋊C4C22⋊Dic5911yescharacteristic
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42D4×D5911yescharacteristic
349C42D4×D5911yescharacteristic
350C42C22×Dic5911yescharacteristic
351C42C2×C22⋊D5911yescharacteristic
352C23—21no—
353C23—42no—
354C23—82no—
355C23—82no—
356C23—81no—
357C23—81no—
358C23#1427370811yescharacteristic
359Q8—21no—
360Q8#1427364812yes—
361D4—22no—
362D4—22no—
363D4—21no—
364D4—42no—
365D4—42no—
366D4—82no—
367D4—82no—
368D4—82no—
369D4—81no—
370D4—81no—
371D4—81no—
372D4—81no—
373D4#1427364812yes—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—81no—
385C2×C4C2×C22⋊Dic5911yescharacteristic
386C2×C4D4×Dic5911yescharacteristic
387C2×C4D4×Dic5911yescharacteristic
388C2×C4#1427369511yescharacteristic
389C2×C4#1427369911yescharacteristic
390C2×C4#1427369911yescharacteristic
391C2×C4C2×#1174566911yescharacteristic
392C2×C4C2×#1174567011yescharacteristic
393C2×C4#1427375211yescharacteristic
394C2×C4#1427375711yescharacteristic
395C8—42no—
396C8—4721no—
397C8—4721no—
398C8—2361no—
399C8—2362no—
400C8—2361no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—162no—
410C22—162no—
411C22#1683728211yescharacteristic
412C22#1683771311yescharacteristic
413C22#1683771811yescharacteristic
414C4—22no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—41no—
419C4—41no—
420C4C2×#1427364811yescharacteristic
421C4#1683740311yescharacteristic
422C4#1683753511yescharacteristic
423C4#1683754011yescharacteristic
424C4#1683766311yescharacteristic
425C4#1683767011yescharacteristic
426C2—21no—
427C2—21no—
428C2—42no—
429C2—162no—
430C2#2142441411yescharacteristic
431C2#2142457811yescharacteristic
432C2#2142460611yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC42×C59C22×C4
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#3038C2
socleC22×C59#386
cosocleC2×C42×C59C23
hypercenter#3038D59
hypocenterC59#9173
LCSG ▷ C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #21432664C2semidirect
2. #21422764C2semidirect
3. C2×#16837146C2semidirect
4. #21422483C2semidirect
5. #21421991C2semidirect
6. #21421970C2semidirect
247. C59#9173semidirect

Maximal subgroups

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

subgroupindex
#214326642
#214227642
C2×#168371462
#214224832
#214219912
#214219702
C59×#30382
#917359