Finite Group Databasebeta

← #28506994

Subgroups — #28506994

Order 15104.

628 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 103 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
4#21424088C211yesmaximal, characteristic, complemented
5C4×#99×C59C211yesmaximal, characteristic, complemented
6C2×#14273708C2211yescharacteristic
7C2×#14273695C2211yescharacteristic
8C22×(C2×D59)⋊C4C411yescharacteristic, complemented
9#16837426—21no—
10C2×#14273644C2212yes—
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—
23C4×C22⋊C4×C59—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—
45C4⋊(C59⋊C8)—41no—
46C4⋊(C59⋊C8)—41no—
47#14273644—22no—
48#14273644—22no—
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×C59D411yescharacteristic
58C2×C22⋊C4×C59—22no—
59C2×C22⋊C4×C59C2×C411yescharacteristic
60C2×C22⋊C4×C59D411yescharacteristic
61C2×C42×C59—41no—
62C2×C42×C59D411yescharacteristic
63#99×C59—22no—
64#99×C59D414yes—
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—42no—
92C2×C22⋊D59—81no—
93C22×Dic59—21no—
94C22×Dic59—21no—
95C22×Dic59—21no—
96C22×Dic59—22no—
97C22×Dic59—41no—
98C22×Dic59—41no—
99C4⋊C4×C59—41no—
100C22⋊C4×C59—24no—
101C22⋊C4×C59—41no—
102C22⋊C4×C59—41no—
103C22⋊C4×C59—42no—
104C42×C59—21no—
105C42×C59—21no—
106C42×C59—82no—
107C22⋊Dic59—22no—
108C22⋊Dic59—22no—
109C22⋊Dic59—42no—
110(C2×D59)⋊C4—42no—
111(C2×D59)⋊C4—42no—
112(C2×D59)⋊C4—41no—
113(C2×D59)⋊C4—41no—
114(C2×D59)⋊C4—41no—
115(C2×D59)⋊C4—41no—
116(C2×D59)⋊C4—81no—
117C4⋊Dic59—81no—
118Dic59⋊C4—41no—
119Dic59⋊C4—41no—
120#11745664—42no—
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#12112yes—
134C22×D59—22no—
135C22×D59—41no—
136C22×D59—41no—
137C22×D59—41no—
138C22×D59—42no—
139C22×D59—41no—
140C22×D59—41no—
141C22×D59—41no—
142C22×D59—42no—
143C22×D59—84no—
144C22×D59—84no—
145C22×D59—82no—
146C22×D59—82no—
147D4×C59—42no—
148D4×C59—42no—
149C2×C4×C59—21no—
150C2×C4×C59—21no—
151C2×C4×C59—21no—
152C2×C4×C59—21no—
153C2×C4×C59—21no—
154C2×C4×C59—41no—
155C2×C4×C59—42no—
156C2×C4×C59—41no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59—44no—
160C2×C4×C59—81no—
161C2×C4×C59—81no—
162C2×C4×C59—81no—
163C2×C4×C59—82no—
164C2×C4×C59—82no—
165C2×C4×C59#9911yescharacteristic
166C2×C4×C59#9911yescharacteristic
167C2×C4×C59#12411yescharacteristic
168C2×C4×C59#12711yescharacteristic
169C22⋊D59—84no—
170C2×Dic59—41no—
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—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—41no—
190C22×C59—41no—
191C22×C59—41no—
192C22×C59—41no—
193C22×C59—81no—
194C22×C59#39011yescharacteristic
195C22×C59C2×#9911yescharacteristic
196C22×C59#41011yescharacteristic
197C2×D59—42no—
198C2×D59—42no—
199C2×D59—42no—
200C2×D59—81no—
201C2×D59—81no—
202C2×D59—84no—
203C2×D59—81no—
204C2×D59—81no—
205C2×D59—84no—
206C4×C59—21no—
207C4×C59—21no—
208C4×C59—21no—
209C4×C59—21no—
210C4×C59—41no—
211C4×C59—41no—
212C4×C59—41no—
213C4×C59—84no—
214Dic59—81no—
215Dic59—81no—
216Dic59—81no—
217#2534—591no—
218C2×#352—592no—
219C2×AΣL(1,𝔽16)—592no—
220#1800—591no—
221C4×#99D5911yescharacteristic, complemented
222C2×C59—21no—
223C2×C59—21no—
224C2×C59—42no—
225C2×C59—42no—
226C2×C59#196411yescharacteristic
227C2×C59#212011yescharacteristic
228C2×C59#212111yescharacteristic
229D59—82no—
230C2×#123—591no—
231C2×#120—591no—
232C2×C4×D4C2×D5911yescharacteristic
233C22×C22⋊C4—591no—
234#423—1181no—
235C2×#100—592no—
236C2×#99C2×D5912yes—
237#396—1181no—
238#394—1181no—
239#388—1181no—
240#386—1181no—
241#381—591no—
242#380—591no—
243#379—591no—
244C4×C22⋊C4—21no—
245#352—1184no—
246AΣL(1,𝔽16)—1184no—
247#343—1181no—
248#342—21no—
249C59#1322211yescharacteristic, complemented, Sylow
250C25—591no—
251C22×D4—1181no—
252C22×D4C22×D5911yescharacteristic
253C23×C4—591no—
254C23×C4C4×D5911yescharacteristic
255C23×C4C22⋊D5911yescharacteristic
256C2×M16—1182no—
257#123—1182no—
258#123—1182no—
259#120—1182no—
260#120—1182no—
261C4×D4—41no—
262C4×D4—41no—
263C2×C4⋊C4—1181no—
264C2×C4⋊C4—1181no—
265C2×C4⋊C4D23611yescharacteristic
266C2×C22⋊C4—22no—
267C2×C22⋊C4—591no—
268C2×C22⋊C4—591no—
269C2×C22⋊C4—1181no—
270C2×C22⋊C4—1181no—
271C2×C22⋊C4—1181no—
272C2×C22⋊C4—1181no—
273C2×C22⋊C4—1181no—
274C2×C22⋊C4—1181no—
275C2×C22⋊C4—2361no—
276C2×C22⋊C4—2361no—
277C2×C22⋊C4C4×D5911yescharacteristic
278C2×C22⋊C4C22⋊D5911yescharacteristic
279C2×C42—41no—
280C2×C42D23611yescharacteristic
281C4⋊C8—2361no—
282C4⋊C8—2361no—
283#100—1182no—
284#100—1182no—
285#99—22no—
286#99D23614yes—
287#95—41no—
288#95—1181no—
289#95—1181no—
290#95—1181no—
291C24—1181no—
292C24—1182no—
293C24—4721no—
294C24—2361no—
295C24—2361no—
296C24—2362no—
297C24C2×C4×D5911yescharacteristic
298C24C2×C22⋊D5911yescharacteristic
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—4721no—
302C2×D4—2362no—
303C2×D4—2362no—
304C2×D4C2×D23612yes—
305C2×D4#1174568512yes—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—42no—
311C22×C4—81no—
312C22×C4—81no—
313C22×C4—1182no—
314C22×C4—1181no—
315C22×C4—1181no—
316C22×C4—1181no—
317C22×C4—2361no—
318C22×C4—2361no—
319C22×C4(C2×D59)⋊C411yescharacteristic
320C22×C4(C2×D59)⋊C411yescharacteristic
321C22×C4(C2×D59)⋊C411yescharacteristic
322C22×C4(C2×D59)⋊C411yescharacteristic
323C22×C4C2×D23611yescharacteristic
324M16—2362no—
325M16—2362no—
326C2×C8—2362no—
327C4⋊C4—41no—
328C4⋊C4—4721no—
329C4⋊C4—2361no—
330C4⋊C4—2361no—
331C22⋊C4—24no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—42no—
335C22⋊C4—1182no—
336C22⋊C4—1182no—
337C22⋊C4—4721no—
338C22⋊C4—2362no—
339C22⋊C4—2362no—
340C22⋊C4—2362no—
341C22⋊C4—2361no—
342C22⋊C4—2361no—
343C22⋊C4—2361no—
344C22⋊C4—2361no—
345C42—21no—
346C42—21no—
347C42—82no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—1182no—
357C23—4722no—
358C23—4722no—
359C23—4724no—
360C23—4724no—
361C23—2361no—
362C23—2361no—
363C23—2361no—
364C23—2362no—
365C23—2362no—
366C23—2361no—
367C23—2361no—
368C23—2361no—
369C23C2×(C2×D59)⋊C411yescharacteristic
370C23C4×D23612yes—
371C23#1427370112yes—
372D4—42no—
373D4—42no—
374D4—4724no—
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—44no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—4722no—
392C2×C4—2361no—
393C2×C4—2361no—
394C2×C4—2361no—
395C2×C4—2361no—
396C2×C4—2361no—
397C2×C4—2361no—
398C2×C4—2361no—
399C2×C4—2361no—
400C2×C4—2361no—
401C2×C4#1427364111yescharacteristic
402C2×C4#1427364111yescharacteristic
403C2×C4#1427371511yescharacteristic
404C2×C4#1427372411yescharacteristic
405C8—4722no—
406C22—21no—
407C22—21no—
408C22—22no—
409C22—22no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—81no—
418C22—4721no—
419C22—4721no—
420C22—4724no—
421C22—4724no—
422C22—4721no—
423C22—4721no—
424C22—2362no—
425C22—2362no—
426C22—2362no—
427C22#1683730411yescharacteristic
428C22C2×#1427364111yescharacteristic
429C22#1683737511yescharacteristic
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—84no—
438C4—4721no—
439C4—4721no—
440C4—4721no—
441C2—21no—
442C2—21no—
443C2—42no—
444C2—42no—
445C2—4722no—
446C2#2142493211yescharacteristic
447C2#2142573511yescharacteristic
448C2#2142574011yescharacteristic
subgroupG/H
centerC22C2×#14273641
commutatorC24×C59C22×C4
FrattiniC22×D4C22×D59
FittingC4×#99×C59C2
socleC22×C59C2×#99
prime radicalC2×#14273695C22
cosocleC22×D4×C59C23
hypercenterC4×#99D59
hypocenterC59#13222
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ C4×#99

Complements

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

NH ≅ G/Ntype
2. C2×#16837191C2semidirect
4. #21424088C2semidirect
5. C4×#99×C59C2semidirect
8. C22×(C2×D59)⋊C4C4semidirect
15. #16837263C4semidirect
16. #16837262C4semidirect
17. #16837257C4semidirect
221. C4×#99D59semidirect
249. C59#13222semidirect

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
#214240882
C4×#99×C592
#1322259