Finite Group Databasebeta

← #28489261

Subgroups — #28489261

Order 15104.

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

normal subgroups only

This group has 658 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
1C2×#16837737C211yesmaximal, characteristic, complemented
2C2×#16837720C211yesmaximal, characteristic
3#21430024C211yesmaximal, characteristic
4C2×#16837138C211yesmaximal, characteristic, complemented
5C2×#16837135C211yesmaximal, characteristic
6#21421939C211yesmaximal, characteristic, complemented
7C59×#1445C211yesmaximal, characteristic, complemented
8C2×#14273706C2211yescharacteristic
9C2×#14273701C2211yescharacteristic
10#16837737—22no—
11#16837737—22no—
12#16837720—22no—
13#16837720—22no—
14C2×#14273660—21no—
15C2×#14273658—21no—
16C2×#14273650—21no—
17C2×#14273647—21no—
18C2×C22⋊(C59⋊C8)C2211yescharacteristic
19C2×C22⋊(C59⋊C8)C2211yescharacteristic
20#16837342—21no—
21#16837329—21no—
22#16837320—21no—
23#16837315—21no—
24#16837281—22no—
25#16837138—22no—
26#16837135—22no—
27C22×C4⋊C4×C59C2211yescharacteristic
28C2×#95×C59—22no—
29C2×#95×C59C2211yescharacteristic
30C2×#95×C59C2211yescharacteristic
31C22×Dic118D411yescharacteristic
32C22×D236D411yescharacteristic
33C22×C22⋊D59—21no—
34C22×C59⋊C8—21no—
35C22×C59⋊C8—21no—
36#14273706—21no—
37#14273706—21no—
38#14273706—21no—
39#14273706—21no—
40#14273701—21no—
41#14273701—21no—
42#14273701—21no—
43#14273701—21no—
44C2×C4⋊Dic59D411yescharacteristic
45C2×C4⋊Dic59D411yescharacteristic
46C2×Dic59⋊C4—21no—
47C2×C22⋊Dic59—21no—
48C2×(C2×D59)⋊C4—21no—
49C2×(C2×D59)⋊C4—22no—
50C2×(C2×D59)⋊C4—22no—
51#14273660—24no—
52#14273658—24no—
53#14273650—24no—
54#14273647—24no—
55C22⋊(C59⋊C8)—21no—
56C22⋊(C59⋊C8)—21no—
57C22⋊(C59⋊C8)D412yes—
58C22⋊(C59⋊C8)D412yes—
59C59⋊#95—22no—
60C59⋊#95—22no—
61C23×C4×C59—21no—
62C23×C4×C59—21no—
63C23×C4×C59D412yes—
64C23×C4×C59C2311yescharacteristic
65C2×C4⋊C4×C59—21no—
66C2×C4⋊C4×C59—21no—
67C2×C4⋊C4×C59—21no—
68C2×C4⋊C4×C59—21no—
69C2×C4⋊C4×C59—21no—
70C2×C4⋊C4×C59D412yes—
71#95×C59—21no—
72#95×C59—21no—
73#95×C59—21no—
74#95×C59—21no—
75#95×C59—44no—
76C24×C59C2×D411yescharacteristic
77C23×D59—21no—
78C22×C4×C59—22no—
79C22×C4×C59—22no—
80C22×C4×C59—22no—
81C22×C4×C59—22no—
82C22×C4×C59—22no—
83C22×C4×C59—42no—
84C22×C4×C59—42no—
85C22×C4×C59—42no—
86C22×C4×C59—42no—
87C22×C4×C59—41no—
88C22×C4×C59—42no—
89C22×C4×C59—42no—
90C22×C4×C59—41no—
91C22×C4×C59—41no—
92C22×C4×C59—41no—
93C22×C4×C59—42no—
94C22×C4×C59C2×D411yescharacteristic
95C22×C4×C59C2×D411yescharacteristic
96C22×C4×C59C2×D411yescharacteristic
97C22×C4×C59C2×D411yescharacteristic
98C22×C4×C59C2×D411yescharacteristic
99C22×C4×C59#4111yescharacteristic
100C2×C22⋊D59—41no—
101C2×C22⋊D59—41no—
102C2×C22⋊D59—42no—
103C2×C22⋊D59—42no—
104C22×Dic59—21no—
105C22×Dic59—21no—
106C22×Dic59—21no—
107C2×D236—21no—
108C2×D236—21no—
109C2×D236—42no—
110C2×Dic118—21no—
111C2×Dic118—21no—
112C2×Dic118—42no—
113C4⋊C4×C59—22no—
114C4⋊C4×C59—22no—
115C4⋊C4×C59—41no—
116C4⋊C4×C59—41no—
117C22⋊Dic59—41no—
118C22⋊Dic59—41no—
119(C2×D59)⋊C4—41no—
120(C2×D59)⋊C4—41no—
121(C2×D59)⋊C4—82no—
122(C2×D59)⋊C4—82no—
123C4⋊Dic59—21no—
124C4⋊Dic59—21no—
125C4⋊Dic59—21no—
126C4⋊Dic59—21no—
127Dic59⋊C4—41no—
128Dic59⋊C4—41no—
129C2×C59⋊C8—22no—
130C2×C59⋊C8—22no—
131C2×C59⋊C8—42no—
132C2×C59⋊C8—42no—
133C23×C59—21no—
134C23×C59—21no—
135C23×C59—22no—
136C23×C59#12011yescharacteristic
137C23×C59#12111yescharacteristic
138C23×C59#12711yescharacteristic
139C23×C59#13512yes—
140C23×C59#13712yes—
141C22×D59—41no—
142C22×D59—42no—
143C22×D59—41no—
144C22×D59—41no—
145C22×D59—42no—
146C2×C4×C59—22no—
147C2×C4×C59—22no—
148C2×C4×C59—41no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—44no—
153C2×C4×C59—42no—
154C2×C4×C59—44no—
155C2×C4×C59—44no—
156C2×C4×C59—42no—
157C2×C4×C59—42no—
158C2×C4×C59—42no—
159C2×C4×C59—42no—
160C2×C4×C59—82no—
161C2×C4×C59—81no—
162C2×C4×C59—82no—
163C2×C4×C59—81no—
164C2×C4×C59—81no—
165C2×C4×C59—81no—
166C2×C4×C59—82no—
167C2×C4×C59#12011yescharacteristic
168C2×C4×C59#12011yescharacteristic
169C2×C4×C59#12111yescharacteristic
170C2×C4×C59#12111yescharacteristic
171C22⋊D59—82no—
172C22⋊D59—81no—
173C22⋊D59—81no—
174C2×Dic59—41no—
175C2×Dic59—41no—
176C2×Dic59—41no—
177C2×Dic59—42no—
178C2×Dic59—41no—
179C2×Dic59—41no—
180C2×Dic59—42no—
181D236—42no—
182D236—81no—
183Dic118—42no—
184Dic118—81no—
185C59⋊C8—42no—
186C59⋊C8—42no—
187#9969—591nomaximal, Sylow
188C22×C59—22no—
189C22×C59—22no—
190C22×C59—22no—
191C22×C59—22no—
192C22×C59—21no—
193C22×C59—21no—
194C22×C59—21no—
195C22×C59—21no—
196C22×C59#39211yescharacteristic
197C22×C59#45212yes—
198C22×C59#45311yescharacteristic
199C22×C59#45411yescharacteristic
200C22×C59#45812yes—
201C22×C59#46412yes—
202C22×C59#49511yescharacteristic
203C22×C59#49711yescharacteristic
204C2×D59—82no—
205C2×D59—82no—
206C2×D59—81no—
207C2×D59—81no—
208C2×D59—81no—
209C4×C59—21no—
210C4×C59—21no—
211C4×C59—21no—
212C4×C59—21no—
213C4×C59—82no—
214C4×C59—82no—
215C4×C59—82no—
216Dic59—81no—
217Dic59—81no—
218Dic59—81no—
219C2×#484—591no—
220C2×#481—591no—
221#2639—591no—
222C2×#328—591no—
223C2×#327—591no—
224#1445D5911yescharacteristic, complemented
225#1280—591no—
226C2×C59—21no—
227C2×C59—22no—
228C2×C59—21no—
229C2×C59#161512yes—
230C2×C59#161812yes—
231C2×C59#200611yescharacteristic
232C2×C59#201511yescharacteristic
233C2×C59#202211yescharacteristic
234D59—161no—
235C2×C22⋊Q8—591no—
236C2×#121—591no—
237C22×C4⋊C4C2×D5911yescharacteristic
238#484—1182no—
239#484—1182no—
240#481—1182no—
241#481—1182no—
242C2×#107—1181no—
243C2×#106—1181no—
244C2×#103—1181no—
245C2×#102—1181no—
246C2×C22⋊C8—591no—
247C2×C22⋊C8—591no—
248#400—1181no—
249#397—1181no—
250#394—1181no—
251#393—1181no—
252#386—1182no—
253C2×#95—22no—
254C2×#95C2×D5911yescharacteristic
255C2×#95C2×D5911yescharacteristic
256#328—1182no—
257#327—1182no—
258C59#996911yescharacteristic, complemented, Sylow
259C22×Q8—591no—
260C22×D4—591no—
261C22×D4—1181no—
262C23×C4—21no—
263C23×C4—21no—
264C23×C4C22⋊D5912yes—
265C23×C4C22×D5911yescharacteristic
266C22×C8—1181no—
267C22×C8—1181no—
268C22⋊Q8—1181no—
269C22⋊Q8—1181no—
270C22⋊Q8—1181no—
271C22⋊Q8—1181no—
272#121—1181no—
273#121—1181no—
274#121—1181no—
275#121—1181no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—591no—
282C2×C4⋊C4—591no—
283C2×C4⋊C4—1181no—
284C2×C4⋊C4D23612yes—
285C2×C22⋊C4—1182no—
286C2×C22⋊C4—1182no—
287C2×C22⋊C4—1181no—
288C2×C22⋊C4—1181no—
289#107—1184no—
290#106—1184no—
291#103—1184no—
292#102—1184no—
293C22⋊C8—592no—
294C22⋊C8—592no—
295C22⋊C8—1181no—
296C22⋊C8—1181no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—44no—
302#95—1182no—
303#95—1182no—
304C24—1181no—
305C24C2×C22⋊D5911yescharacteristic
306C2×Q8—1181no—
307C2×Q8—1181no—
308C2×Q8—2362no—
309C2×D4—1181no—
310C2×D4—1181no—
311C2×D4—2362no—
312C2×D4—2361no—
313C2×D4—2361no—
314C2×D4—2362no—
315C2×D4—2362no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—1181no—
333C22×C4—1181no—
334C22×C4—1181no—
335C22×C4C2×D23611yescharacteristic
336C22×C4#1174568511yescharacteristic
337C22×C4D4×D5911yescharacteristic
338C22×C4D4×D5911yescharacteristic
339C22×C4D4×D5911yescharacteristic
340C22×C4D4×D5911yescharacteristic
341C2×C8—1182no—
342C2×C8—1182no—
343C2×C8—2362no—
344C2×C8—2362no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—1181no—
350C4⋊C4—1181no—
351C4⋊C4—1181no—
352C4⋊C4—1181no—
353C4⋊C4—2361no—
354C4⋊C4—2361no—
355C22⋊C4—4722no—
356C22⋊C4—4722no—
357C22⋊C4—2361no—
358C22⋊C4—2361no—
359C22⋊C4—2361no—
360C22⋊C4—2361no—
361C23—21no—
362C23—21no—
363C23—22no—
364C23—2362no—
365C23—2361no—
366C23—2361no—
367C23—2361no—
368C23—2362no—
369C23#1427369511yescharacteristic
370C23#1427370111yescharacteristic
371C23#1427372311yescharacteristic
372C23#1427374712yes—
373C23#1427375812yes—
374Q8—4721no—
375Q8—2362no—
376D4—4721no—
377D4—4721no—
378D4—4721no—
379D4—4722no—
380D4—2362no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—82no—
396C2×C4—81no—
397C2×C4—82no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—82no—
402C2×C4—2362no—
403C2×C4—2361no—
404C2×C4—2361no—
405C2×C4—2361no—
406C2×C4—2361no—
407C2×C4—2361no—
408C2×C4—2362no—
409C2×C4#1427369711yescharacteristic
410C2×C4#1427369711yescharacteristic
411C2×C4#1427370011yescharacteristic
412C2×C4#1427370011yescharacteristic
413C8—2362no—
414C8—2362no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—4722no—
424C22—4721no—
425C22—4722no—
426C22—4721no—
427C22—4721no—
428C22#1683731111yescharacteristic
429C22#1683756212yes—
430C22#1683756911yescharacteristic
431C22#1683757411yescharacteristic
432C22#1683759112yes—
433C22#1683762412yes—
434C22#1683778411yescharacteristic
435C22#1683779411yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—82no—
441C4—82no—
442C4—82no—
443C4—4721no—
444C4—4721no—
445C4—4721no—
446C2—21no—
447C2—21no—
448C2—22no—
449C2—9441no—
450C2#2142312512yes—
451C2#2142314212yes—
452C2#2142513711yescharacteristic
453C2#2142518911yescharacteristic
454C2#2142522711yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC59×#1445C2
socleC23×C59#120
prime radicalC2×#14273701C22
hypercenter#1445D59
hypocenterC59#9969
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1445

Complements

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

NH ≅ G/Ntype
1. C2×#16837737C2semidirect
4. C2×#16837138C2semidirect
6. #21421939C2semidirect
7. C59×#1445C2semidirect
224. #1445D59semidirect
258. C59#9969semidirect

Maximal subgroups

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

subgroupindex
C2×#168377372
C2×#168377202
#214300242
C2×#168371382
C2×#168371352
#214219392
C59×#14452
#996959