Finite Group Databasebeta

← #28489205

Subgroups — #28489205

Order 15104.

650 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 650 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C2×#16837730C211yesmaximal, characteristic, complemented
2#21428568C211yesmaximal, characteristic
3#21425082C211yesmaximal, characteristic, complemented
4#21424183C211yesmaximal, characteristic, complemented
5#21422664C211yesmaximal, characteristic
6#21421943C211yesmaximal, characteristic, complemented
7C59×#3187C211yesmaximal, characteristic
8C2×#14273708—22no—
9C22×C4⋊Dic59C2211yescharacteristic
10#16837730—22no—
11#16837730—22no—
12C2×#14273658—21no—
13C2×#14273651—21no—
14C2×C22⋊(C59⋊C8)C2211yescharacteristic
15#16837338—22no—
16#16837305—21no—
17#16837271—21no—
18#16837262C2211yescharacteristic
19C4×C22⋊Dic59C2211yescharacteristic
20#16837169—22no—
21#16837165—22no—
22#16837136—22no—
23C2×C22⋊Q8×C59C2211yescharacteristic
24C2×#121×C59C2211yescharacteristic
25C59×#483—21no—
26C59×#481—21no—
27C59×#465—22no—
28C59×#450—21no—
29C59×#448—21no—
30C2×C22⋊C8×C59C2211yescharacteristic
31C23×Dic59—21no—
32C22×C59⋊C8—21no—
33#14273708—44no—
34#14273708—44no—
35C2×C4⋊Dic59—21no—
36C2×C4⋊Dic59—21no—
37C2×C4⋊Dic59—21no—
38C2×C4⋊Dic59—21no—
39C2×C4⋊Dic59—21no—
40C2×C4⋊Dic59—21no—
41C2×C4⋊Dic59D412yes—
42C2×Dic59⋊C4—24no—
43C2×C22⋊Dic59—24no—
44C2×C22⋊Dic59—21no—
45C2×C22⋊Dic59D412yes—
46C2×C4×Dic59—21no—
47#14273658—24no—
48#14273651—24no—
49C22⋊(C59⋊C8)—21no—
50C22⋊(C59⋊C8)D412yes—
51C59⋊#95—21no—
52C59⋊#95—21no—
53C22×Q8×C59D411yescharacteristic
54C22×D4×C59—21no—
55C22×D4×C59D411yescharacteristic
56C23×C4×C59C2311yescharacteristic
57C2×SD16×C59—22no—
58C22×C8×C59D412yes—
59C22⋊Q8×C59—21no—
60C22⋊Q8×C59—21no—
61C22⋊Q8×C59—21no—
62C22⋊Q8×C59—21no—
63#121×C59—21no—
64#121×C59—21no—
65#121×C59—21no—
66#121×C59—21no—
67C2×C4⋊C4×C59—21no—
68C2×C4⋊C4×C59D411yescharacteristic
69C2×C4⋊C4×C59D411yescharacteristic
70C2×C22⋊C4×C59—21no—
71C2×C22⋊C4×C59—21no—
72#106×C59—22no—
73#103×C59—22no—
74#102×C59—22no—
75C22⋊C8×C59—21no—
76C22⋊C8×C59—21no—
77C24×C59—21no—
78C24×C59C2×D411yescharacteristic
79C2×Q8×C59—21no—
80C2×Q8×C59—21no—
81C2×Q8×C59—42no—
82C2×D4×C59—21no—
83C2×D4×C59—21no—
84C2×D4×C59—42no—
85C2×D4×C59—41no—
86C2×D4×C59—41no—
87C2×D4×C59—42no—
88C2×D4×C59—42no—
89C22×C4×C59—21no—
90C22×C4×C59—21no—
91C22×C4×C59—21no—
92C22×C4×C59—42no—
93C22×C4×C59C2×D411yescharacteristic
94C22×C4×C59C2×D411yescharacteristic
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59C2×D411yescharacteristic
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59#4111yescharacteristic
99C22×Dic59—22no—
100C22×Dic59—22no—
101C22×Dic59—41no—
102C22×Dic59—41no—
103C22×Dic59—42no—
104C22×Dic59—42no—
105C22×Dic59—41no—
106SD16×C59—82no—
107C2×C8×C59—22no—
108C2×C8×C59—42no—
109C4⋊C4×C59—21no—
110C4⋊C4×C59—21no—
111C4⋊C4×C59—21no—
112C4⋊C4×C59—21no—
113C4⋊C4×C59—41no—
114C4⋊C4×C59—41no—
115C22⋊C4×C59—41no—
116C22⋊C4×C59—41no—
117C22⋊C4×C59—41no—
118C22⋊C4×C59—41no—
119C22⋊Dic59—22no—
120C22⋊Dic59—48no—
121C22⋊Dic59—42no—
122C22⋊Dic59SD1614yescomplemented
123C4⋊Dic59—22no—
124C4⋊Dic59—22no—
125C4⋊Dic59—41no—
126C4⋊Dic59—41no—
127C4⋊Dic59—42no—
128Dic59⋊C4—48no—
129C4×Dic59—42no—
130C2×C59⋊C8—22no—
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—
143Q8×C59—42no—
144Q8×C59—42no—
145D4×C59—42no—
146D4×C59—81no—
147D4×C59—82no—
148D4×C59—81no—
149D4×C59—81no—
150C2×C4×C59—22no—
151C2×C4×C59—22no—
152C2×C4×C59—22no—
153C2×C4×C59—41no—
154C2×C4×C59—41no—
155C2×C4×C59—41no—
156C2×C4×C59—41no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59—42no—
160C2×C4×C59—81no—
161C2×C4×C59—81no—
162C2×C4×C59#12011yescharacteristic
163C2×C4×C59#12011yescharacteristic
164C2×C4×C59#12111yescharacteristic
165C2×C4×C59#12111yescharacteristic
166C2×Dic59—24no—
167C2×Dic59—42no—
168C2×Dic59—42no—
169C2×Dic59—42no—
170C2×Dic59—44no—
171C2×Dic59—82no—
172C2×Dic59—81no—
173C2×Dic59—81no—
174C2×Dic59—81no—
175C2×Dic59—81no—
176C2×Dic59—81no—
177C8×C59—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—81no—
201C4×C59—81no—
202C4×C59—81no—
203C4×C59#45812yes—
204Dic59—44no—
205Dic59—82no—
206Dic59—161no—
207#3187D5911yescharacteristic
208C2×#483—591no—
209#2501—591no—
210#1996—591no—
211#1823—591no—
212#1519—591no—
213#1282—591no—
214C2×C59—22no—
215C2×C59—21no—
216C2×C59—22no—
217C2×C59—21no—
218C2×C59—161no—
219C2×C59#200911yescharacteristic
220C2×C59#201411yescharacteristic
221C2×C59#202211yescharacteristic
222C2×#123—1182no—
223C2×C22⋊Q8C2×D5911yescharacteristic
224C2×#121C2×D5911yescharacteristic
225C22×C4⋊C4—591no—
226#483—21no—
227#483—1182no—
228#483—1182no—
229#481—21no—
230#465—22no—
231#450—21no—
232#448—21no—
233C2×#106—1181no—
234C2×#103—1181no—
235C2×C22⋊C8—591no—
236C2×C22⋊C8C2×D5911yescharacteristic
237#399—1182no—
238#390—1181no—
239#384—1181no—
240#380—591no—
241C4×C22⋊C4—591no—
242#342—1182no—
243#340—1182no—
244#327—1182no—
245C59#996011yescharacteristic, complemented, Sylow
246C22×Q8C22⋊D5911yescharacteristic
247C22×D4—21no—
248C22×D4C22⋊D5911yescharacteristic
249C23×C4—1181no—
250C23×C4C22×D5911yescharacteristic
251C2×SD16—22no—
252C22×C8—1181no—
253C22×C8C22⋊D5912yes—
254#123—2364no—
255#123—2364no—
256C22⋊Q8—21no—
257C22⋊Q8—21no—
258C22⋊Q8—21no—
259C22⋊Q8—21no—
260#121—21no—
261#121—21no—
262#121—21no—
263#121—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—592no—
266C2×C4⋊C4—1181no—
267C2×C4⋊C4—1181no—
268C2×C4⋊C4—1181no—
269C2×C4⋊C4—1181no—
270C2×C4⋊C4—1184no—
271C2×C4⋊C4—1181no—
272C2×C4⋊C4—1181no—
273C2×C4⋊C4C22⋊D5911yescharacteristic
274C2×C4⋊C4C22⋊D5911yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—592no—
278C2×C22⋊C4—1184no—
279C2×C22⋊C4—1181no—
280C2×C42—1181no—
281#106—22no—
282#106—1184no—
283#103—22no—
284#103—1184no—
285#102—22no—
286C22⋊C8—21no—
287C22⋊C8—21no—
288C22⋊C8—592no—
289C22⋊C8—1181no—
290#95—1181no—
291#95—1181no—
292C24—21no—
293C24D4×D5911yescharacteristic
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—42no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—42no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—42no—
308C22×C4—1182no—
309C22×C4—1182no—
310C22×C4—2361no—
311C22×C4—2361no—
312C22×C4—2361no—
313C22×C4—2362no—
314C22×C4—2362no—
315C22×C4D4×D5911yescharacteristic
316C22×C4D4×D5911yescharacteristic
317C22×C4#1174568711yescharacteristic
318C22×C4C2×C22⋊D5911yescharacteristic
319C22×C4C2×C22⋊D5911yescharacteristic
320C22×C4C2×C22⋊D5911yescharacteristic
321SD16—82no—
322C2×C8—22no—
323C2×C8—42no—
324C2×C8—1182no—
325C2×C8—2362no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—1182no—
333C4⋊C4—1182no—
334C4⋊C4—2362no—
335C4⋊C4—2368no—
336C4⋊C4—2361no—
337C4⋊C4—2361no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—594no—
343C22⋊C4—1182no—
344C22⋊C4—2368no—
345C22⋊C4—2362no—
346C42—2362no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23#1427369511yescharacteristic
354C23#1427369911yescharacteristic
355C23#1427372311yescharacteristic
356C23SD16×D5912yes—
357C23#1427375412yes—
358Q8—42no—
359Q8—42no—
360D4—42no—
361D4—81no—
362D4—81no—
363D4—81no—
364D4—82no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—1184no—
378C2×C4—4721no—
379C2×C4—4721no—
380C2×C4—4721no—
381C2×C4—4721no—
382C2×C4—4721no—
383C2×C4—4722no—
384C2×C4—2364no—
385C2×C4—2362no—
386C2×C4—2362no—
387C2×C4—2362no—
388C2×C4#1427369511yescharacteristic
389C2×C4#1427369611yescharacteristic
390C2×C4#1427369811yescharacteristic
391C2×C4#1427369911yescharacteristic
392C8—42no—
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#1683755112yes—
408C22#1683760012yes—
409C22#1683777311yescharacteristic
410C22#1683778811yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—81no—
415C4—81no—
416C4—81no—
417C4—9441no—
418C4—4722no—
419C4—2364no—
420C4#1683759212yes—
421C2—21no—
422C2—22no—
423C2—21no—
424C2—22no—
425C2—161no—
426C2#2142515311yescharacteristic
427C2#2142518211yescharacteristic
428C2#2142522511yescharacteristic
subgroupG/H
centerC22#16837310
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC59×#3187C2
socleC22×C59#392
prime radicalC2×#121×C59C22
hypercenter#3187D59
hypocenterC59#9960
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3187

Complements

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

NH ≅ G/Ntype
1. C2×#16837730C2semidirect
3. #21425082C2semidirect
4. #21424183C2semidirect
6. #21421943C2semidirect
122. C22⋊Dic59SD16semidirect
245. C59#9960semidirect

Maximal subgroups

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

subgroupindex
C2×#168377302
#214285682
#214250822
#214241832
#214226642
#214219432
C59×#31872
#996059