Finite Group Databasebeta

← #28506995

Subgroups — #28506995

Order 15104.

628 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 107 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×#16837192C212yesmaximal, complemented
2C2×#16837189C212yesmaximal
3#21424088C211yesmaximal, characteristic, complemented
4C4×#14273642C211yesmaximal, characteristic, complemented
5C59×#2534C211yesmaximal, characteristic
6#16837426—21no—
7C2×#14273644C2212yes—
8C2×#14273642C2212yes—
9C4×C22⋊Dic59—21no—
10#16837192—24no—
11#16837189—24no—
12#16837171—21no—
13#16837169—21no—
14C2×#123×C59C2211yescharacteristic
15C2×#120×C59C2211yescharacteristic
16C2×C4×D4×C59C2211yescharacteristic
17C22×C22⋊C4×C59C411yescharacteristic, complemented
18C59×#396—21no—
19C59×#394—21no—
20C59×#388—21no—
21C59×#386—21no—
22C59×#381C411yescharacteristic, complemented
23C59×#380C411yescharacteristic, complemented
24C59×#379C411yescharacteristic, complemented
25C2×#11745664—22no—
26C2×C22⋊Dic59—22no—
27C2×C4×Dic59—41no—
28C4⋊(C59⋊C8)—41no—
29C4⋊(C59⋊C8)—41no—
30#14273644—22no—
31#14273644—22no—
32#14273642—22no—
33#14273642D414yes—
34C59⋊#95—41no—
35C25×C59C2×C411yescharacteristic
36C22×D4×C59—21no—
37C22×D4×C59C2311yescharacteristic
38C23×C4×C59C2×C411yescharacteristic
39C23×C4×C59C2×C411yescharacteristic
40C23×C4×C59D411yescharacteristic
41#123×C59—22no—
42#123×C59—22no—
43#120×C59—22no—
44#120×C59—22no—
45C4×D4×C59—41no—
46C4×D4×C59—41no—
47C2×C4⋊C4×C59—21no—
48C2×C4⋊C4×C59—21no—
49C2×C4⋊C4×C59D411yescharacteristic
50C2×C22⋊C4×C59—21no—
51C2×C22⋊C4×C59—21no—
52C2×C22⋊C4×C59—21no—
53C2×C22⋊C4×C59—21no—
54C2×C22⋊C4×C59—21no—
55C2×C22⋊C4×C59—21no—
56C2×C22⋊C4×C59—41no—
57C2×C22⋊C4×C59—41no—
58C2×C22⋊C4×C59C2×C411yescharacteristic
59C2×C22⋊C4×C59C2×C411yescharacteristic
60C2×C22⋊C4×C59C2×C411yescharacteristic
61C2×C22⋊C4×C59D411yescharacteristic
62C2×C42×C59D411yescharacteristic
63#95×C59—21no—
64#95×C59—21no—
65#95×C59—21no—
66C24×C59—21no—
67C24×C59—22no—
68C24×C59—41no—
69C24×C59—42no—
70C24×C59—41no—
71C24×C59—81no—
72C24×C59C22×C411yescharacteristic
73C24×C59C2×D411yescharacteristic
74C2×D4×C59—41no—
75C2×D4×C59—41no—
76C2×D4×C59—42no—
77C2×D4×C59—42no—
78C2×D4×C59—81no—
79C2×D4×C59C2×D412yes—
80C2×D4×C59#4112yes—
81C22×C4×C59—21no—
82C22×C4×C59—21no—
83C22×C4×C59—21no—
84C22×C4×C59—21no—
85C22×C4×C59—21no—
86C22×C4×C59—21no—
87C22×C4×C59—21no—
88C22×C4×C59—22no—
89C22×C4×C59—41no—
90C22×C4×C59—41no—
91C22×C4×C59—81no—
92C22×C4×C59—81no—
93C22×C4×C59C22⋊C411yescharacteristic
94C22×C4×C59C22⋊C411yescharacteristic
95C22×C4×C59C22⋊C411yescharacteristic
96C22×C4×C59C22⋊C411yescharacteristic
97C22×C4×C59C2×D411yescharacteristic
98C22×Dic59—42no—
99C4⋊C4×C59—41no—
100C4⋊C4×C59—41no—
101C4⋊C4×C59—41no—
102C4⋊C4×C59—81no—
103C22⋊C4×C59—22no—
104C22⋊C4×C59—22no—
105C22⋊C4×C59—41no—
106C22⋊C4×C59—41no—
107C22⋊C4×C59—42no—
108C22⋊C4×C59—42no—
109C22⋊C4×C59—42no—
110C22⋊C4×C59—41no—
111C22⋊C4×C59—41no—
112C22⋊C4×C59—41no—
113C22⋊C4×C59—41no—
114C22⋊C4×C59—81no—
115C42×C59—21no—
116C42×C59—21no—
117C22⋊Dic59—24no—
118C22⋊Dic59—42no—
119C4×Dic59—82no—
120#11745664—42no—
121#11745664—42no—
122C2×C59⋊C8—42no—
123C23×C59—21no—
124C23×C59—21no—
125C23×C59—21no—
126C23×C59—21no—
127C23×C59—22no—
128C23×C59—41no—
129C23×C59—41no—
130C23×C59—41no—
131C23×C59—41no—
132C23×C59—41no—
133C23×C59—41no—
134C23×C59—41no—
135C23×C59—42no—
136C23×C59—42no—
137C23×C59—41no—
138C23×C59—41no—
139C23×C59—41no—
140C23×C59—82no—
141C23×C59—84no—
142C23×C59—84no—
143C23×C59—82no—
144C23×C59C2×C22⋊C411yescharacteristic
145C23×C59C4×D412yes—
146C23×C59#12112yes—
147D4×C59—42no—
148D4×C59—42no—
149D4×C59—84no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—21no—
154C2×C4×C59—21no—
155C2×C4×C59—41no—
156C2×C4×C59—42no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59—41no—
160C2×C4×C59—41no—
161C2×C4×C59—41no—
162C2×C4×C59—41no—
163C2×C4×C59—41no—
164C2×C4×C59—41no—
165C2×C4×C59—41no—
166C2×C4×C59—41no—
167C2×C4×C59—41no—
168C2×C4×C59—41no—
169C2×C4×C59—81no—
170C2×C4×C59—81no—
171C2×C4×C59—81no—
172C2×C4×C59—82no—
173C2×C4×C59#9911yescharacteristic
174C2×C4×C59#9911yescharacteristic
175C2×C4×C59#12411yescharacteristic
176C2×C4×C59#12711yescharacteristic
177C2×Dic59—44no—
178C2×Dic59—82no—
179C2×Dic59—82no—
180C59⋊C8—82no—
181#13222—591nomaximal, Sylow
182C22×C59—21no—
183C22×C59—21no—
184C22×C59—22no—
185C22×C59—22no—
186C22×C59—42no—
187C22×C59—42no—
188C22×C59—42no—
189C22×C59—42no—
190C22×C59—42no—
191C22×C59—41no—
192C22×C59—41no—
193C22×C59—41no—
194C22×C59—41no—
195C22×C59—42no—
196C22×C59—81no—
197C22×C59—81no—
198C22×C59—81no—
199C22×C59—84no—
200C22×C59—84no—
201C22×C59—81no—
202C22×C59—81no—
203C22×C59#39011yescharacteristic
204C22×C59C2×#9911yescharacteristic
205C22×C59#41011yescharacteristic
206C4×C59—21no—
207C4×C59—21no—
208C4×C59—21no—
209C4×C59—21no—
210C4×C59—41no—
211C4×C59—41no—
212C4×C59—41no—
213C4×C59—81no—
214C4×C59—81no—
215C4×C59—81no—
216Dic59—84no—
217#2534D5911yescharacteristic
218C2×#352—592no—
219C2×AΣL(1,𝔽16)—592no—
220#1800—591no—
221C4×#99—591no—
222C2×C59—21no—
223C2×C59—21no—
224C2×C59—42no—
225C2×C59—42no—
226C2×C59—82no—
227C2×C59#196411yescharacteristic
228C2×C59#212011yescharacteristic
229C2×C59#212111yescharacteristic
230C2×#123C2×D5911yescharacteristic
231C2×#120C2×D5911yescharacteristic
232C2×C4×D4C2×D5911yescharacteristic
233C22×C22⋊C4Dic5911yescharacteristic, complemented
234#423—1181no—
235C2×#100—592no—
236C2×#99—592no—
237#396—21no—
238#394—21no—
239#388—21no—
240#386—21no—
241#381Dic5911yescharacteristic, complemented
242#380Dic5911yescharacteristic, complemented
243#379Dic5911yescharacteristic, complemented
244C4×C22⋊C4—1181no—
245#352—1184no—
246AΣL(1,𝔽16)—1184no—
247#343—1181no—
248#342—1181no—
249C59#1322211yescharacteristic, complemented, Sylow
250C25C2×Dic5911yescharacteristic
251C22×D4—21no—
252C22×D4C22×D5911yescharacteristic
253C23×C4C2×Dic5911yescharacteristic
254C23×C4C2×Dic5911yescharacteristic
255C23×C4C22⋊D5911yescharacteristic
256C2×M16—1182no—
257#123—22no—
258#123—22no—
259#120—22no—
260#120—22no—
261C4×D4—41no—
262C4×D4—41no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C22⋊D5911yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—1182no—
275C2×C22⋊C4C2×Dic5911yescharacteristic
276C2×C22⋊C4C2×Dic5911yescharacteristic
277C2×C22⋊C4C2×Dic5911yescharacteristic
278C2×C22⋊C4C22⋊D5911yescharacteristic
279C2×C42—2361no—
280C2×C42C22⋊D5911yescharacteristic
281C4⋊C8—2361no—
282C4⋊C8—2361no—
283#100—1182no—
284#100—1182no—
285#99—594no—
286#99—1182no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95—2361no—
291C24—21no—
292C24—22no—
293C24—41no—
294C24—42no—
295C24—41no—
296C24—81no—
297C24C22×Dic5911yescharacteristic
298C24C2×C22⋊D5911yescharacteristic
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—81no—
304C2×D4D4×D5912yes—
305C2×D4#1174568712yes—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—81no—
317C22×C4—81no—
318C22×C4—2362no—
319C22×C4C22⋊Dic5911yescharacteristic
320C22×C4C22⋊Dic5911yescharacteristic
321C22×C4C22⋊Dic5911yescharacteristic
322C22×C4C22⋊Dic5911yescharacteristic
323C22×C4C2×C22⋊D5911yescharacteristic
324M16—2362no—
325M16—2362no—
326C2×C8—2362no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—81no—
331C22⋊C4—22no—
332C22⋊C4—22no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—81no—
343C22⋊C4—1184no—
344C22⋊C4—2362no—
345C42—21no—
346C42—21no—
347C42—4722no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—22no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—42no—
361C23—42no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—82no—
366C23—84no—
367C23—84no—
368C23—82no—
369C23C2×C22⋊Dic5911yescharacteristic
370C23D4×Dic5912yes—
371C23#1427369812yes—
372D4—42no—
373D4—42no—
374D4—84no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—4722no—
399C2×C4—4722no—
400C2×C4—2364no—
401C2×C4#1427364211yescharacteristic
402C2×C4#1427364211yescharacteristic
403C2×C4#1427371411yescharacteristic
404C2×C4#1427372311yescharacteristic
405C8—4722no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—22no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—42no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—84no—
424C22—84no—
425C22—81no—
426C22—81no—
427C22#1683730511yescharacteristic
428C22C2×#1427364211yescharacteristic
429C22#1683737611yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4—4724no—
441C2—21no—
442C2—21no—
443C2—42no—
444C2—42no—
445C2—82no—
446C2#2142493311yescharacteristic
447C2#2142573711yescharacteristic
448C2#2142574211yescharacteristic
subgroupG/H
centerC22C2×#14273642
commutatorC24×C59C22×C4
FrattiniC22×D4C22×D59
FittingC59×#2534C2
socleC22×C59C2×#99
prime radicalC2×#120×C59C22
cosocleC22×D4×C59C23
hypercenter#2534D59
hypocenterC59#13222
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2534

Complements

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

NH ≅ G/Ntype
1. C2×#16837192C2semidirect
3. #21424088C2semidirect
4. C4×#14273642C2semidirect
17. C22×C22⋊C4×C59C4semidirect
22. C59×#381C4semidirect
23. C59×#380C4semidirect
24. C59×#379C4semidirect
233. C22×C22⋊C4Dic59semidirect
241. #381Dic59semidirect
242. #380Dic59semidirect
243. #379Dic59semidirect
249. C59#13222semidirect

Maximal subgroups

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

subgroupindex
C2×#168371922
C2×#168371892
#214240882
C4×#142736422
C59×#25342
#1322259