Finite Group Databasebeta

← #28489207

Subgroups — #28489207

Order 15104.

650 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21434853C211yesmaximal, characteristic
2C2×#16837732C211yesmaximal, characteristic, complemented
3#21425081C211yesmaximal, characteristic, complemented
4#21424182C211yesmaximal, characteristic, complemented
5#21422664C211yesmaximal, characteristic
6#21421944C211yesmaximal, characteristic, complemented
7C59×#2501C211yesmaximal, characteristic
8C2×#14273706C2211yescharacteristic
9#16837732—22no—
10#16837732—22no—
11#16837732—21no—
12#16837718—21no—
13#16837629—22no—
14#16837553—21no—
15#16837540—21no—
16C2×#14273658—21no—
17C2×#14273650—21no—
18C2×C22⋊(C59⋊C8)C2211yescharacteristic
19C2×C22⋊(C59⋊C8)C2211yescharacteristic
20#16837262C2211yescharacteristic
21#16837168—22no—
22#16837164—22no—
23#16837136—22no—
24C2×#123×C59—22no—
25C2×#121×C59C2211yescharacteristic
26C22×C4⋊C4×C59C2211yescharacteristic
27C59×#399—22no—
28C59×#390—21no—
29C59×#384—21no—
30C4×C22⋊C4×C59C2211yescharacteristic
31C22×Dic118D411yescharacteristic
32C2×#11745670—22no—
33C22×C59⋊C8—21no—
34C22×C59⋊C8D412yes—
35#14273706—21no—
36#14273706—21no—
37#14273706—21no—
38#14273706—21no—
39C2×C4⋊Dic59—21no—
40C2×C4⋊Dic59D411yescharacteristic
41C2×Dic59⋊C4—21no—
42C2×C22⋊Dic59—21no—
43C2×C22⋊Dic59—21no—
44#14273658—24no—
45#14273658—22no—
46#14273650—24no—
47#14273650—22no—
48#14273648—22no—
49C22⋊(C59⋊C8)—21no—
50C22⋊(C59⋊C8)—21no—
51C22⋊(C59⋊C8)—21no—
52C22⋊(C59⋊C8)D412yes—
53C59⋊#95—21no—
54C22×D4×C59—21no—
55C22×D4×C59D411yescharacteristic
56C23×C4×C59—21no—
57C23×C4×C59C2311yescharacteristic
58#123×C59—44no—
59#123×C59—44no—
60#121×C59—21no—
61#121×C59—21no—
62#121×C59—21no—
63#121×C59—21no—
64C2×C4⋊C4×C59—21no—
65C2×C4⋊C4×C59—21no—
66C2×C4⋊C4×C59—21no—
67C2×C4⋊C4×C59—21no—
68C2×C4⋊C4×C59—24no—
69C2×C4⋊C4×C59—21no—
70C2×C4⋊C4×C59D411yescharacteristic
71C2×C4⋊C4×C59D412yes—
72C2×C22⋊C4×C59—24no—
73C2×C22⋊C4×C59—21no—
74C2×C22⋊C4×C59D412yes—
75C2×C42×C59—21no—
76#95×C59—21no—
77C24×C59—21no—
78C24×C59C2×D411yescharacteristic
79C2×D4×C59—21no—
80C2×D4×C59—21no—
81C2×D4×C59—42no—
82C2×D4×C59—41no—
83C2×D4×C59—41no—
84C2×D4×C59—42no—
85C2×D4×C59—42no—
86C22×C4×C59—22no—
87C22×C4×C59—22no—
88C22×C4×C59—21no—
89C22×C4×C59—42no—
90C22×C4×C59—41no—
91C22×C4×C59—41no—
92C22×C4×C59—42no—
93C22×C4×C59—42no—
94C22×C4×C59C2×D411yescharacteristic
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59C2×D411yescharacteristic
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59C2×D411yescharacteristic
99C22×C4×C59#4111yescharacteristic
100C22×Dic59—21no—
101C22×Dic59—21no—
102C22×Dic59—41no—
103C2×Dic118—21no—
104C2×Dic118—21no—
105C2×Dic118—42no—
106C4⋊C4×C59—22no—
107C4⋊C4×C59—21no—
108C4⋊C4×C59—22no—
109C4⋊C4×C59—21no—
110C4⋊C4×C59—48no—
111C4⋊C4×C59—41no—
112C4⋊C4×C59—41no—
113C22⋊C4×C59—22no—
114C22⋊C4×C59—48no—
115C22⋊C4×C59—41no—
116C22⋊C4×C59—41no—
117C22⋊C4×C59SD1614yescomplemented
118C42×C59—42no—
119C22⋊Dic59—41no—
120C22⋊Dic59—41no—
121C22⋊Dic59—42no—
122#11745670—82no—
123C4⋊Dic59—21no—
124C4⋊Dic59—21no—
125C4⋊Dic59—42no—
126Dic59⋊C4—41no—
127Dic59⋊C4—41no—
128C2×C59⋊C8—22no—
129C2×C59⋊C8—22no—
130C2×C59⋊C8—42no—
131C2×C59⋊C8—42no—
132C23×C59—41no—
133C23×C59—41no—
134C23×C59—42no—
135C23×C59—42no—
136C23×C59—41no—
137C23×C59—42no—
138C23×C59#12011yescharacteristic
139C23×C59#12111yescharacteristic
140C23×C59#12711yescharacteristic
141C23×C59C2×SD1612yes—
142C23×C59AGL(1,ℤ/8)12yes—
143D4×C59—42no—
144D4×C59—81no—
145D4×C59—82no—
146D4×C59—81no—
147D4×C59—81no—
148C2×C4×C59—22no—
149C2×C4×C59—22no—
150C2×C4×C59—22no—
151C2×C4×C59—24no—
152C2×C4×C59—41no—
153C2×C4×C59—41no—
154C2×C4×C59—41no—
155C2×C4×C59—42no—
156C2×C4×C59—42no—
157C2×C4×C59—41no—
158C2×C4×C59—42no—
159C2×C4×C59—44no—
160C2×C4×C59—42no—
161C2×C4×C59—82no—
162C2×C4×C59—81no—
163C2×C4×C59—81no—
164C2×C4×C59#12011yescharacteristic
165C2×C4×C59#12011yescharacteristic
166C2×C4×C59#12111yescharacteristic
167C2×C4×C59#12111yescharacteristic
168C2×Dic59—41no—
169C2×Dic59—41no—
170C2×Dic59—81no—
171C2×Dic59—81no—
172C2×Dic59—81no—
173C2×Dic59—81no—
174C2×Dic59—81no—
175Dic118—42no—
176Dic118—42no—
177C59⋊C8—42no—
178C59⋊C8—42no—
179#9960—591nomaximal, Sylow
180C22×C59—21no—
181C22×C59—21no—
182C22×C59—21no—
183C22×C59—21no—
184C22×C59—21no—
185C22×C59—21no—
186C22×C59—42no—
187C22×C59—42no—
188C22×C59—84no—
189C22×C59—81no—
190C22×C59—81no—
191C22×C59—81no—
192C22×C59#39211yescharacteristic
193C22×C59#45012yes—
194C22×C59#46012yes—
195C22×C59#49211yescharacteristic
196C22×C59#49611yescharacteristic
197C4×C59—21no—
198C4×C59—21no—
199C4×C59—21no—
200C4×C59—44no—
201C4×C59—82no—
202C4×C59—81no—
203C4×C59#45812yes—
204Dic59—81no—
205Dic59—81no—
206Dic59—161no—
207#3187—591no—
208C2×#483—591no—
209#2501D5911yescharacteristic
210#1996—591no—
211#1823—591no—
212#1519—591no—
213#1282—591no—
214C2×C59—21no—
215C2×C59—22no—
216C2×C59—22no—
217C2×C59—21no—
218C2×C59—161no—
219C2×C59#200911yescharacteristic
220C2×C59#201411yescharacteristic
221C2×C59#202211yescharacteristic
222C2×#123—22no—
223C2×C22⋊Q8—591no—
224C2×#121C2×D5911yescharacteristic
225C22×C4⋊C4C2×D5911yescharacteristic
226#483—1182no—
227#483—1182no—
228#483—1181no—
229#481—1181no—
230#465—1182no—
231#450—1181no—
232#448—1181no—
233C2×#106—1181no—
234C2×#103—1181no—
235C2×C22⋊C8—591no—
236C2×C22⋊C8—591no—
237#399—22no—
238#390—21no—
239#384—21no—
240#380—591no—
241C4×C22⋊C4C2×D5911yescharacteristic
242#342—1182no—
243#340—1182no—
244#327—1182no—
245C59#996011yescharacteristic, complemented, Sylow
246C22×Q8—591no—
247C22×D4—21no—
248C22×D4C22⋊D5911yescharacteristic
249C23×C4—21no—
250C23×C4C22×D5911yescharacteristic
251C2×SD16—1182no—
252C22×C8—592no—
253C22×C8—1181no—
254#123—44no—
255#123—44no—
256C22⋊Q8—1181no—
257C22⋊Q8—1181no—
258C22⋊Q8—1181no—
259C22⋊Q8—1181no—
260#121—21no—
261#121—21no—
262#121—21no—
263#121—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—591no—
271C2×C4⋊C4—1181no—
272C2×C4⋊C4—1181no—
273C2×C4⋊C4C22⋊D5912yes—
274C2×C4⋊C4C22⋊D5911yescharacteristic
275C2×C22⋊C4—24no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—1181no—
278C2×C22⋊C4—1181no—
279C2×C22⋊C4C22⋊D5912yes—
280C2×C42—21no—
281#106—1184no—
282#106—1182no—
283#103—1184no—
284#103—1182no—
285#102—1182no—
286C22⋊C8—592no—
287C22⋊C8—1181no—
288C22⋊C8—1181no—
289C22⋊C8—1181no—
290#95—21no—
291#95—1181no—
292C24—21no—
293C24C2×C22⋊D5911yescharacteristic
294C2×Q8—1181no—
295C2×Q8—1181no—
296C2×Q8—2362no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—42no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—1181no—
313C22×C4—1181no—
314C22×C4—2361no—
315C22×C4D4×D5911yescharacteristic
316C22×C4D4×D5911yescharacteristic
317C22×C4D4×D5911yescharacteristic
318C22×C4#1174568711yescharacteristic
319C22×C4C2×C22⋊D5911yescharacteristic
320C22×C4C2×C22⋊D5911yescharacteristic
321SD16—4722no—
322C2×C8—1182no—
323C2×C8—1182no—
324C2×C8—2362no—
325C2×C8—2362no—
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—48no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—1181no—
334C4⋊C4—1181no—
335C4⋊C4—2362no—
336C4⋊C4—2361no—
337C4⋊C4—2361no—
338C22⋊C4—22no—
339C22⋊C4—48no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—2362no—
343C22⋊C4—2361no—
344C22⋊C4—2361no—
345C22⋊C4#1174567014yescomplemented
346C42—42no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23#1427369611yescharacteristic
354C23#1427369811yescharacteristic
355C23#1427372311yescharacteristic
356C23C2×#1174567012yes—
357C23#1427375312yes—
358Q8—2362no—
359Q8—2362no—
360D4—42no—
361D4—81no—
362D4—81no—
363D4—81no—
364D4—82no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—82no—
381C2×C4—4721no—
382C2×C4—4721no—
383C2×C4—4721no—
384C2×C4—4721no—
385C2×C4—4721no—
386C2×C4—2361no—
387C2×C4—2361no—
388C2×C4#1427369511yescharacteristic
389C2×C4#1427369511yescharacteristic
390C2×C4#1427369911yescharacteristic
391C2×C4#1427369911yescharacteristic
392C8—2362no—
393C8—2362no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—42no—
401C22—42no—
402C22—84no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22#1683731011yescharacteristic
407C22#1683755312yes—
408C22#1683759912yes—
409C22#1683777411yescharacteristic
410C22#1683778911yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—44no—
415C4—82no—
416C4—81no—
417C4—9441no—
418C4—4721no—
419C4—4721no—
420C4#1683759112yes—
421C2—22no—
422C2—21no—
423C2—21no—
424C2—22no—
425C2—161no—
426C2#2142515611yescharacteristic
427C2#2142518411yescharacteristic
428C2#2142522411yescharacteristic
subgroupG/H
centerC22#16837310
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC59×#2501C2
socleC22×C59#392
prime radicalC2×#121×C59C22
hypercenter#2501D59
hypocenterC59#9960
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2501

Complements

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

NH ≅ G/Ntype
2. C2×#16837732C2semidirect
3. #21425081C2semidirect
4. #21424182C2semidirect
6. #21421944C2semidirect
117. C22⋊C4×C59SD16semidirect
245. C59#9960semidirect
345. C22⋊C4#11745670semidirect

Maximal subgroups

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

subgroupindex
#214348532
C2×#168377322
#214250812
#214241822
#214226642
#214219442
C59×#25012
#996059