Finite Group Databasebeta

← #28483769

Subgroups — #28483769

Order 15104.

670 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 111 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#21433125C211yesmaximal, characteristic, complemented
2C4×#14273644C211yesmaximal, characteristic, complemented
3#21422539C212yesmaximal, complemented
4#21421989C212yesmaximal, complemented
5C59×#2397C211yesmaximal, characteristic
6#16837474—24no—
7#16837459C2212yes—
8C2×#14273644—21no—
9#16837366—21no—
10C2×C22⋊(C59⋊C8)—21no—
11C4×#11745664C2212yes—
12#16837171—21no—
13D4⋊(C59⋊C8)—24no—
14C2×C4×D4×C59—21no—
15C2×C4×D4×C59—21no—
16C2×C4×D4×C59C2211yescharacteristic
17C2×C4×D4×C59C2211yescharacteristic
18C59×#399—21no—
19C59×#399C411yescharacteristic
20C59×#399C411yescharacteristic
21C59×#392—21no—
22C59×#392C411yescharacteristic
23C59×#392C411yescharacteristic
24C59×#376C2211yescharacteristic
25C2×#11745664—22no—
26C2×#11745664—22no—
27C22×C59⋊C8—22no—
28C4⋊(C59⋊C8)D414yescomplemented
29#14273644—24no—
30C22⋊(C59⋊C8)—22no—
31C22⋊(C59⋊C8)—22no—
32C22⋊(C59⋊C8)—22no—
33#14273638—22no—
34#14273638—22no—
35C4×C59⋊C8—22no—
36C22×D4×C59—21no—
37C22×D4×C59—21no—
38C22×D4×C59C2×C411yescharacteristic
39C22×D4×C59D411yescharacteristic
40C23×C4×C59—21no—
41C23×C4×C59D411yescharacteristic
42C23×C4×C59D411yescharacteristic
43C4×D4×C59—22no—
44C4×D4×C59—22no—
45C4×D4×C59—22no—
46C4×D4×C59—22no—
47C4×D4×C59—44no—
48C4×D4×C59—44no—
49C2×C4⋊C4×C59—21no—
50C2×C4⋊C4×C59—21no—
51C2×C4⋊C4×C59C2×C411yescharacteristic
52C2×C4⋊C4×C59D411yescharacteristic
53C2×C22⋊C4×C59—21no—
54C2×C22⋊C4×C59—21no—
55C2×C22⋊C4×C59—21no—
56C2×C22⋊C4×C59—21no—
57C2×C22⋊C4×C59—21no—
58C2×C22⋊C4×C59C2×C411yescharacteristic
59C2×C22⋊C4×C59C2×C411yescharacteristic
60C2×C42×C59—21no—
61C2×C42×C59C2311yescharacteristic
62#95×C59—21no—
63#95×C59C2×C411yescharacteristic
64#95×C59C2×C411yescharacteristic
65C24×C59—21no—
66C24×C59C22⋊C411yescharacteristic
67C24×C59C22⋊C411yescharacteristic
68C2×D4×C59—22no—
69C2×D4×C59—22no—
70C2×D4×C59—22no—
71C2×D4×C59—41no—
72C2×D4×C59—42no—
73C2×D4×C59—42no—
74C2×D4×C59—42no—
75C2×D4×C59—42no—
76C2×D4×C59—42no—
77C2×D4×C59—42no—
78C2×D4×C59—42no—
79C22×C4×C59—21no—
80C22×C4×C59—21no—
81C22×C4×C59—21no—
82C22×C4×C59—21no—
83C22×C4×C59—21no—
84C22×C4×C59—21no—
85C22×C4×C59—21no—
86C22×C4×C59—42no—
87C22×C4×C59—42no—
88C22×C4×C59—42no—
89C22×C4×C59—42no—
90C22×C4×C59—41no—
91C22×C4×C59—41no—
92C22×C4×C59C22⋊C411yescharacteristic
93C22×C4×C59C22⋊C411yescharacteristic
94C22×C4×C59C22×C411yescharacteristic
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59C2×D411yescharacteristic
97C4⋊C4×C59—22no—
98C4⋊C4×C59—22no—
99C4⋊C4×C59—42no—
100C4⋊C4×C59—42no—
101C22⋊C4×C59—22no—
102C22⋊C4×C59—22no—
103C22⋊C4×C59—42no—
104C22⋊C4×C59—42no—
105C22⋊C4×C59—42no—
106C22⋊C4×C59—42no—
107C22⋊C4×C59—42no—
108C42×C59—24no—
109C42×C59C2×D412yes—
110C42×C59#4112yes—
111#11745664—24no—
112#11745664—42no—
113#11745664—82no—
114C2×C59⋊C8—24no—
115C2×C59⋊C8—42no—
116C2×C59⋊C8—82no—
117C23×C59—21no—
118C23×C59—21no—
119C23×C59—22no—
120C23×C59—21no—
121C23×C59—21no—
122C23×C59—21no—
123C23×C59—41no—
124C23×C59—41no—
125C23×C59—41no—
126C23×C59—41no—
127C23×C59—41no—
128C23×C59—41no—
129C23×C59—41no—
130C23×C59—41no—
131C23×C59—41no—
132C23×C59—41no—
133C23×C59#10412yes—
134C23×C59C2×C22⋊C411yescharacteristic
135D4×C59—24no—
136D4×C59—44no—
137D4×C59—42no—
138D4×C59—48no—
139D4×C59—48no—
140C2×C4×C59—21no—
141C2×C4×C59—21no—
142C2×C4×C59—22no—
143C2×C4×C59—22no—
144C2×C4×C59—22no—
145C2×C4×C59—21no—
146C2×C4×C59—21no—
147C2×C4×C59—41no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—42no—
153C2×C4×C59—42no—
154C2×C4×C59—41no—
155C2×C4×C59—41no—
156C2×C4×C59—41no—
157C2×C4×C59—41no—
158C2×C4×C59—41no—
159C2×C4×C59—81no—
160C2×C4×C59—81no—
161C2×C4×C59—82no—
162C2×C4×C59—82no—
163C2×C4×C59C4×D412yes—
164C2×C4×C59#12011yescharacteristic
165C2×C4×C59#12112yes—
166C2×C4×C59#12311yescharacteristic
167C59⋊C8—44no—
168C59⋊C8—44no—
169#9031—591nomaximal, Sylow
170C22×C59—21no—
171C22×C59—21no—
172C22×C59—22no—
173C22×C59—42no—
174C22×C59—42no—
175C22×C59—42no—
176C22×C59—42no—
177C22×C59—41no—
178C22×C59—41no—
179C22×C59—41no—
180C22×C59—41no—
181C22×C59—42no—
182C22×C59—41no—
183C22×C59—81no—
184C22×C59—81no—
185C22×C59—81no—
186C22×C59—81no—
187C22×C59—81no—
188C22×C59—81no—
189C22×C59#38611yescharacteristic
190C22×C59C2×#10411yescharacteristic
191C22×C59#42111yescharacteristic
192C4×C59—22no—
193C4×C59—21no—
194C4×C59—22no—
195C4×C59—42no—
196C4×C59—42no—
197C4×C59—41no—
198C4×C59—41no—
199C4×C59—81no—
200C4×C59#45312yes—
201#2973—591no—
202#2397D5911yescharacteristic
203C4×#100—591no—
204#1490—592no—
205#1300—592no—
206C2×C59—21no—
207C2×C59—21no—
208C2×C59—42no—
209C2×C59—81no—
210C2×C59—82no—
211C2×C59#187111yescharacteristic
212C2×C59#188711yescharacteristic
213C2×C59#189111yescharacteristic
214C2×C4×D4—21no—
215C2×C4×D4—21no—
216C2×C4×D4C2×D5911yescharacteristic
217C2×C4×D4C2×D5911yescharacteristic
218#435—1184no—
219#432—592no—
220C2×#100—1181no—
221#407—1181no—
222C2×C22⋊C8—1181no—
223C4×M16—592no—
224#399—21no—
225#399Dic5911yescharacteristic
226#399Dic5911yescharacteristic
227#392—21no—
228#392Dic5911yescharacteristic
229#392Dic5911yescharacteristic
230#376C2×D5911yescharacteristic
231#343—1181no—
232D4⋊C8—1184no—
233C59#903111yescharacteristic, complemented, Sylow
234C22×D4—21no—
235C22×D4—21no—
236C22×D4C2×Dic5911yescharacteristic
237C22×D4C22⋊D5911yescharacteristic
238C23×C4—21no—
239C23×C4C22⋊D5911yescharacteristic
240C23×C4C22⋊D5911yescharacteristic
241C2×M16—1182no—
242C2×M16—1182no—
243C22×C8—1182no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—44no—
249C4×D4—44no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C2×Dic5911yescharacteristic
253C2×C4⋊C4C22⋊D5911yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4C2×Dic5911yescharacteristic
260C2×C22⋊C4C2×Dic5911yescharacteristic
261C2×C42—21no—
262C2×C42C22×D5911yescharacteristic
263C4⋊C8—594no—
264#100—1184no—
265C22⋊C8—1182no—
266C22⋊C8—1182no—
267C22⋊C8—1182no—
268#97—1182no—
269#97—1182no—
270C4×C8—1182no—
271#95—21no—
272#95C2×Dic5911yescharacteristic
273#95C2×Dic5911yescharacteristic
274C24—21no—
275C24C22⋊Dic5911yescharacteristic
276C24C22⋊Dic5911yescharacteristic
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4C22⋊Dic5911yescharacteristic
302C22×C4C22⋊Dic5911yescharacteristic
303C22×C4C22×Dic5911yescharacteristic
304C22×C4C2×C22⋊D5911yescharacteristic
305C22×C4C2×C22⋊D5911yescharacteristic
306M16—1184no—
307M16—4722no—
308M16—2362no—
309C2×C8—1184no—
310C2×C8—4722no—
311C2×C8—2362no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C42—24no—
324C42D4×D5912yes—
325C42#1174568712yes—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23#1427365412yes—
343C23C2×C22⋊Dic5911yescharacteristic
344D4—24no—
345D4—44no—
346D4—42no—
347D4—48no—
348D4—48no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4D4×Dic5912yes—
373C2×C4#1427369511yescharacteristic
374C2×C4#1427369912yes—
375C2×C4#1427370811yescharacteristic
376C8—2364no—
377C8—2364no—
378C22—21no—
379C22—21no—
380C22—22no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—42no—
390C22—41no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22#1683728211yescharacteristic
398C22C2×#1427365411yescharacteristic
399C22#1683741911yescharacteristic
400C4—22no—
401C4—21no—
402C4—22no—
403C4—42no—
404C4—42no—
405C4—41no—
406C4—41no—
407C4—81no—
408C4#1683756912yes—
409C2—21no—
410C2—21no—
411C2—42no—
412C2—81no—
413C2—82no—
414C2#2142439911yescharacteristic
415C2#2142449511yescharacteristic
416C2#2142451511yescharacteristic
subgroupG/H
centerC2×C4#14273695
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#2397C2
socleC22×C59#386
cosocleC2×C42×C59C23
hypercenter#2397D59
hypocenterC59#9031
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2397

Complements

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

NH ≅ G/Ntype
1. #21433125C2semidirect
2. C4×#14273644C2semidirect
3. #21422539C2semidirect
4. #21421989C2semidirect
28. C4⋊(C59⋊C8)D4semidirect
233. C59#9031semidirect

Maximal subgroups

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

subgroupindex
#214331252
C4×#142736442
#214225392
#214219892
C59×#23972
#903159