Finite Group Databasebeta

← #28506979

Subgroups — #28506979

Order 15104.

624 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 99 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#21428982C211yesmaximal, characteristic
2C2×#16837191C212yesmaximal, complemented
3C2×#16837188C212yesmaximal
4C4×#14273644C211yesmaximal, characteristic, complemented
5C59×#1799C211yesmaximal, characteristic, complemented
6C2×#14273708C2211yescharacteristic
7C2×#14273695C2211yescharacteristic
8C22×(C2×D59)⋊C4C411yescharacteristic, complemented
9C2×#14273644C2212yes—
10C4×#11745664—21no—
11#16837324—21no—
12#16837318—21no—
13#16837293—21no—
14#16837281—21no—
15#16837263C411yescharacteristic, complemented
16#16837262C411yescharacteristic, complemented
17#16837257C411yescharacteristic, complemented
18#16837191—24no—
19#16837188—24no—
20#16837171—21no—
21C2×C4×D4×C59C2211yescharacteristic
22C2×#99×C59C2212yes—
23C59×#380—21no—
24C59×#342—21no—
25C24×D59C2×C411yescharacteristic
26C22×C22⋊D59—21no—
27C23×Dic59C2×C411yescharacteristic
28C2×#11745664—22no—
29#14273708—22no—
30#14273708—22no—
31#14273695—22no—
32#14273695—22no—
33C2×C4⋊Dic59—21no—
34C2×Dic59⋊C4—21no—
35C2×C22⋊Dic59—21no—
36C2×C22⋊Dic59C2×C411yescharacteristic
37C2×C22⋊Dic59C2×C411yescharacteristic
38C2×(C2×D59)⋊C4—21no—
39C2×(C2×D59)⋊C4—21no—
40C2×(C2×D59)⋊C4—21no—
41C2×(C2×D59)⋊C4—21no—
42C2×(C2×D59)⋊C4—21no—
43C2×(C2×D59)⋊C4—41no—
44C2×(C2×D59)⋊C4—41no—
45#14273644—22no—
46#14273644D414yes—
47#14273638—41no—
48C4×C59⋊C8—41no—
49C59⋊#95—21no—
50C59⋊#95—21no—
51C59⋊#95—21no—
52C22×D4×C59C2311yescharacteristic
53C23×C4×C59C2×C411yescharacteristic
54C23×C4×C59D411yescharacteristic
55C4×D4×C59—41no—
56C4×D4×C59—41no—
57C2×C4⋊C4×C59—41no—
58C2×C4⋊C4×C59D411yescharacteristic
59C2×C22⋊C4×C59—22no—
60C2×C22⋊C4×C59C2×C411yescharacteristic
61C2×C22⋊C4×C59D411yescharacteristic
62C2×C42×C59D411yescharacteristic
63#99×C59—22no—
64#99×C59—22no—
65#95×C59—41no—
66C24×C59C22×C411yescharacteristic
67C24×C59C2×D411yescharacteristic
68C23×D59—21no—
69C23×D59—22no—
70C23×D59—42no—
71C23×D59—41no—
72C23×D59—41no—
73C23×D59—81no—
74C2×D4×C59—41no—
75C2×D4×C59—41no—
76C2×D4×C59C2×D412yes—
77C2×D4×C59#4112yes—
78C22×C4×C59—21no—
79C22×C4×C59—21no—
80C22×C4×C59—21no—
81C22×C4×C59—21no—
82C22×C4×C59—42no—
83C22×C4×C59—81no—
84C22×C4×C59—81no—
85C22×C4×C59C22⋊C411yescharacteristic
86C22×C4×C59C22⋊C411yescharacteristic
87C22×C4×C59C22⋊C411yescharacteristic
88C22×C4×C59C22⋊C411yescharacteristic
89C22×C4×C59C2×D411yescharacteristic
90C2×C22⋊D59—42no—
91C2×C22⋊D59—41no—
92C2×C22⋊D59—41no—
93C2×C22⋊D59—81no—
94C22×Dic59—21no—
95C22×Dic59—21no—
96C22×Dic59—21no—
97C22×Dic59—22no—
98C22×Dic59—41no—
99C22×Dic59—41no—
100C4⋊C4×C59—41no—
101C4⋊C4×C59—82no—
102C22⋊C4×C59—41no—
103C22⋊C4×C59—41no—
104C22⋊C4×C59—42no—
105C22⋊C4×C59—42no—
106C42×C59—21no—
107C42×C59—21no—
108C22⋊Dic59—22no—
109C22⋊Dic59—22no—
110C22⋊Dic59—41no—
111C22⋊Dic59—41no—
112(C2×D59)⋊C4—42no—
113(C2×D59)⋊C4—42no—
114(C2×D59)⋊C4—41no—
115(C2×D59)⋊C4—41no—
116(C2×D59)⋊C4—42no—
117(C2×D59)⋊C4—81no—
118C4⋊Dic59—81no—
119Dic59⋊C4—42no—
120#11745664—24no—
121#11745664—42no—
122C2×C59⋊C8—42no—
123C23×C59—21no—
124C23×C59—21no—
125C23×C59—21no—
126C23×C59—21no—
127C23×C59—41no—
128C23×C59—41no—
129C23×C59—41no—
130C23×C59—41no—
131C23×C59C2×C22⋊C411yescharacteristic
132C23×C59C4×D412yes—
133C23×C59#12411yescharacteristic
134C23×C59#12711yescharacteristic
135C22×D59—22no—
136C22×D59—41no—
137C22×D59—41no—
138C22×D59—41no—
139C22×D59—42no—
140C22×D59—41no—
141C22×D59—41no—
142C22×D59—41no—
143C22×D59—42no—
144C22×D59—84no—
145C22×D59—84no—
146C22×D59—82no—
147C22×D59—82no—
148D4×C59—42no—
149D4×C59—42no—
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—81no—
161C2×C4×C59—81no—
162C2×C4×C59—81no—
163C2×C4×C59—82no—
164C2×C4×C59—82no—
165C2×C4×C59—82no—
166C2×C4×C59#9911yescharacteristic
167C2×C4×C59#9911yescharacteristic
168C2×C4×C59#12112yes—
169C22⋊D59—82no—
170C22⋊D59—82no—
171C2×Dic59—41no—
172C2×Dic59—41no—
173C2×Dic59—41no—
174C2×Dic59—41no—
175C2×Dic59—41no—
176C2×Dic59—41no—
177C2×Dic59—41no—
178C2×Dic59—41no—
179C2×Dic59—41no—
180C2×Dic59—82no—
181C59⋊C8—44no—
182#13219—591nomaximal, Sylow
183C22×C59—21no—
184C22×C59—21no—
185C22×C59—22no—
186C22×C59—21no—
187C22×C59—21no—
188C22×C59—41no—
189C22×C59—41no—
190C22×C59—42no—
191C22×C59—42no—
192C22×C59—41no—
193C22×C59—41no—
194C22×C59—41no—
195C22×C59—41no—
196C22×C59—81no—
197C22×C59#39011yescharacteristic
198C22×C59C2×#9911yescharacteristic
199C22×C59#41011yescharacteristic
200C2×D59—42no—
201C2×D59—42no—
202C2×D59—42no—
203C2×D59—81no—
204C2×D59—81no—
205C2×D59—84no—
206C2×D59—81no—
207C2×D59—81no—
208C2×D59—84no—
209C4×C59—21no—
210C4×C59—21no—
211C4×C59—22no—
212C4×C59—41no—
213C4×C59—41no—
214C4×C59—41no—
215C4×C59—162no—
216Dic59—81no—
217Dic59—81no—
218Dic59—81no—
219#2534—591no—
220C2×#352—592no—
221C2×AΣL(1,𝔽16)—592no—
222#1799D5911yescharacteristic, complemented
223C4×#100—591no—
224C2×C59—21no—
225C2×C59—21no—
226C2×C59—41no—
227C2×C59—42no—
228C2×C59—41no—
229C2×C59#196411yescharacteristic
230C2×C59#212011yescharacteristic
231C2×C59#212111yescharacteristic
232D59—82no—
233C2×#123—591no—
234C2×#120—591no—
235C2×C4×D4C2×D5911yescharacteristic
236C22×C22⋊C4—591no—
237C2×#100—592no—
238C2×#99C2×D5912yes—
239C4×M16—1181no—
240#396—1181no—
241#394—1181no—
242#388—1181no—
243#386—1181no—
244#381—591no—
245#380—21no—
246#380—591no—
247#379—591no—
248#352—1184no—
249AΣL(1,𝔽16)—1184no—
250#343—1181no—
251#342—21no—
252C59#1321911yescharacteristic, complemented, Sylow
253C25—591no—
254C22×D4—1181no—
255C22×D4C22×D5911yescharacteristic
256C23×C4—591no—
257C23×C4C4×D5911yescharacteristic
258C23×C4C22⋊D5911yescharacteristic
259C2×M16—1182no—
260#123—1182no—
261#123—1182no—
262#120—1182no—
263#120—1182no—
264C4×D4—41no—
265C4×D4—41no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—1181no—
268C2×C4⋊C4—1181no—
269C2×C4⋊C4D23611yescharacteristic
270C2×C22⋊C4—22no—
271C2×C22⋊C4—591no—
272C2×C22⋊C4—591no—
273C2×C22⋊C4—1181no—
274C2×C22⋊C4—1181no—
275C2×C22⋊C4—1181no—
276C2×C22⋊C4—1181no—
277C2×C22⋊C4—1181no—
278C2×C22⋊C4—1181no—
279C2×C22⋊C4—2361no—
280C2×C22⋊C4—2361no—
281C2×C22⋊C4C4×D5911yescharacteristic
282C2×C22⋊C4C22⋊D5911yescharacteristic
283C2×C42D23611yescharacteristic
284#100—594no—
285#100—1182no—
286#99—22no—
287#99—22no—
288#97—2361no—
289C4×C8—2361no—
290#95—41no—
291#95—1181no—
292#95—1181no—
293#95—1181no—
294C24—1181no—
295C24—1182no—
296C24—4721no—
297C24—2361no—
298C24—2361no—
299C24—2362no—
300C24C2×C4×D5911yescharacteristic
301C24C2×C22⋊D5911yescharacteristic
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—4721no—
305C2×D4—2362no—
306C2×D4—2361no—
307C2×D4—2361no—
308C2×D4D4×D5912yes—
309C2×D4#1174568912yes—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—42no—
315C22×C4—81no—
316C22×C4—81no—
317C22×C4—1182no—
318C22×C4—1181no—
319C22×C4—1181no—
320C22×C4—1181no—
321C22×C4—2361no—
322C22×C4—2361no—
323C22×C4(C2×D59)⋊C411yescharacteristic
324C22×C4(C2×D59)⋊C411yescharacteristic
325C22×C4(C2×D59)⋊C411yescharacteristic
326C22×C4(C2×D59)⋊C411yescharacteristic
327C22×C4C2×D23611yescharacteristic
328M16—1184no—
329M16—2362no—
330C2×C8—2362no—
331C4⋊C4—41no—
332C4⋊C4—82no—
333C4⋊C4—4721no—
334C4⋊C4—2362no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—1182no—
340C22⋊C4—1182no—
341C22⋊C4—4721no—
342C22⋊C4—2362no—
343C22⋊C4—2362no—
344C22⋊C4—2361no—
345C22⋊C4—2361no—
346C22⋊C4—2361no—
347C22⋊C4—2361no—
348C22⋊C4—2362no—
349C42—21no—
350C42—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—1182no—
360C23—4722no—
361C23—4722no—
362C23—4724no—
363C23—4724no—
364C23—2361no—
365C23—2361no—
366C23—2361no—
367C23—2362no—
368C23—2362no—
369C23—2361no—
370C23—2361no—
371C23—2361no—
372C23C2×(C2×D59)⋊C411yescharacteristic
373C23#1427368912yes—
374C23#1427371611yescharacteristic
375C23#1427372311yescharacteristic
376D4—42no—
377D4—42no—
378D4—4722no—
379D4—4722no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—82no—
396C2×C4—4722no—
397C2×C4—2361no—
398C2×C4—2361no—
399C2×C4—2361no—
400C2×C4—2361no—
401C2×C4—2361no—
402C2×C4—2361no—
403C2×C4—2361no—
404C2×C4—2361no—
405C2×C4—2361no—
406C2×C4#1427364111yescharacteristic
407C2×C4#1427364111yescharacteristic
408C2×C4#1427370212yes—
409C8—2364no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—21no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—42no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—81no—
424C22—4721no—
425C22—4721no—
426C22—4724no—
427C22—4724no—
428C22—4721no—
429C22—4721no—
430C22—2362no—
431C22—2362no—
432C22—2362no—
433C22#1683730311yescharacteristic
434C22C2×#1427364111yescharacteristic
435C22#1683737511yescharacteristic
436C4—22no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4—162no—
443C4—4721no—
444C4—4721no—
445C4—4721no—
446C2—21no—
447C2—21no—
448C2—41no—
449C2—42no—
450C2—41no—
451C2—4722no—
452C2#2142493111yescharacteristic
453C2#2142573511yescharacteristic
454C2#2142574011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC24×C59C22×C4
FrattiniC22×D4C22×D59
FittingC59×#1799C2
socleC22×C59C2×#99
prime radicalC2×#14273695C22
cosocleC22×D4×C59C23
hypercenter#1799D59
hypocenterC59#13219
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
2. C2×#16837191C2semidirect
4. C4×#14273644C2semidirect
5. C59×#1799C2semidirect
8. C22×(C2×D59)⋊C4C4semidirect
15. #16837263C4semidirect
16. #16837262C4semidirect
17. #16837257C4semidirect
222. #1799D59semidirect
252. C59#13219semidirect

Maximal subgroups

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

subgroupindex
#214289822
C2×#168371912
C2×#168371882
C4×#142736442
C59×#17992
#1321959