Finite Group Databasebeta

← #28486480

Subgroups — #28486480

Order 15104.

576 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 109 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#21433122C211yesmaximal, characteristic
2#21428398C211yesmaximal, characteristic
3#21424182C211yesmaximal, characteristic, complemented
4#21424081C211yesmaximal, characteristic
5#21421964C211yesmaximal, characteristic, complemented
6#21421963C211yesmaximal, characteristic, complemented
7C59×#1823C211yesmaximal, characteristic, complemented
8C2×#14273710—21no—
9C2×C4×C22⋊D59C2211yescharacteristic
10#16837473—22no—
11#16837470—22no—
12#16837456—21no—
13C2×#14273641—21no—
14#16837364—21no—
15C2×C22⋊(C59⋊C8)C2211yescharacteristic
16#16837333C412yescomplemented
17#16837328C412yescomplemented
18#16837298—21no—
19#16837265—21no—
20#16837262C2211yescharacteristic
21#16837262C2211yescharacteristic
22#16837259—21no—
23#16837168—21no—
24#16837164—22no—
25C22×C4⋊C4×C59C2211yescharacteristic
26C2×C22⋊C8×C59C2211yescharacteristic
27C59×#380C2211yescharacteristic
28C59×#340—22no—
29C22×C22⋊D59D411yescharacteristic
30C22×C4×D59D411yescharacteristic
31C2×#11745659—22no—
32C22×C59⋊C8D412yes—
33#14273710—42no—
34#14273710—42no—
35C4×C22⋊D59—21no—
36C4×C22⋊D59—21no—
37C4×C22⋊D59—21no—
38C4×C22⋊D59—21no—
39C2×C4⋊Dic59—21no—
40C2×C4⋊Dic59—21no—
41C2×Dic59⋊C4—22no—
42C2×Dic59⋊C4C2×C411yescharacteristic
43C2×C22⋊Dic59—21no—
44C2×C22⋊Dic59—21no—
45C2×C22⋊Dic59D411yescharacteristic
46C2×(C2×D59)⋊C4—22no—
47C2×(C2×D59)⋊C4C2×C411yescharacteristic
48C2×C4×Dic59D411yescharacteristic
49Dic59⋊C8—22no—
50#14273641—42no—
51C22⋊(C59⋊C8)—21no—
52C22⋊(C59⋊C8)—21no—
53(C2×D59)⋊C8—22no—
54#14273637—22no—
55C59⋊#95—22no—
56C59⋊#95—22no—
57C59⋊#95—21no—
58C59⋊#95C2×C412yes—
59C23×C4×C59—21no—
60C23×C4×C59C2311yescharacteristic
61C22×C8×C59D412yes—
62C2×C4⋊C4×C59—21no—
63C2×C4⋊C4×C59—21no—
64C2×C4⋊C4×C59—21no—
65C2×C4⋊C4×C59—21no—
66C2×C4⋊C4×C59—21no—
67C2×C4⋊C4×C59—21no—
68C2×C4⋊C4×C59C2×C412yes—
69C2×C22⋊C4×C59—21no—
70C22⋊C8×C59—21no—
71C22⋊C8×C59—21no—
72#95×C59—21no—
73C24×C59C2×D411yescharacteristic
74C23×D59C22⋊C411yescharacteristic
75C22×C4×C59—22no—
76C22×C4×C59—42no—
77C22×C4×C59—41no—
78C22×C4×C59—41no—
79C22×C4×C59—41no—
80C22×C4×C59—41no—
81C22×C4×C59—41no—
82C22×C4×C59—41no—
83C22×C4×C59—41no—
84C22×C4×C59C22×C411yescharacteristic
85C22×C4×C59C2×D411yescharacteristic
86C22×C4×C59C2×D411yescharacteristic
87C22×C4×C59C2×D411yescharacteristic
88C22×C4×C59#4111yescharacteristic
89C22×C4×C59#4111yescharacteristic
90C2×C22⋊D59—21no—
91C2×C22⋊D59—21no—
92C2×C22⋊D59—21no—
93C2×C22⋊D59—21no—
94C2×C22⋊D59—21no—
95C2×C22⋊D59—21no—
96C22×Dic59—22no—
97C22×Dic59—41no—
98C22×Dic59C22⋊C411yescharacteristic
99C22×Dic59C22⋊C411yescharacteristic
100C22×Dic59C22⋊C411yescharacteristic
101C2×C4×D59—21no—
102C2×C4×D59—21no—
103C2×C4×D59—42no—
104C2×C8×C59—22no—
105C2×C8×C59—42no—
106C4⋊C4×C59—22no—
107C4⋊C4×C59—22no—
108C4⋊C4×C59—42no—
109C4⋊C4×C59—41no—
110C4⋊C4×C59—41no—
111C22⋊C4×C59—41no—
112C22⋊C4×C59—41no—
113C22⋊Dic59—21no—
114C22⋊Dic59—21no—
115C22⋊Dic59—41no—
116C22⋊Dic59—41no—
117C22⋊Dic59—41no—
118C22⋊Dic59—41no—
119(C2×D59)⋊C4—21no—
120(C2×D59)⋊C4—21no—
121(C2×D59)⋊C4—44no—
122C4⋊Dic59—42no—
123C4⋊Dic59—42no—
124Dic59⋊C4—21no—
125Dic59⋊C4—21no—
126Dic59⋊C4—44no—
127C4×Dic59—21no—
128C4×Dic59—21no—
129C2×C59⋊C8—22no—
130C2×C59⋊C8—42no—
131#11745659—82no—
132C23×C59—41no—
133C23×C59—41no—
134C23×C59C2×C22⋊C411yescharacteristic
135C23×C59#12111yescharacteristic
136C23×C59#12111yescharacteristic
137C23×C59AGL(1,ℤ/8)11yescharacteristic
138C23×C59AGL(1,ℤ/8)11yescharacteristic
139C23×C59AGL(1,ℤ/8)11yescharacteristic
140C23×C59#13711yescharacteristic
141C22×D59—21no—
142C22×D59—21no—
143C22×D59—21no—
144C22×D59—41no—
145C22×D59—41no—
146C2×C4×C59—22no—
147C2×C4×C59—22no—
148C2×C4×C59—22no—
149C2×C4×C59—41no—
150C2×C4×C59—41no—
151C2×C4×C59—41no—
152C2×C4×C59—42no—
153C2×C4×C59—42no—
154C2×C4×C59—42no—
155C2×C4×C59—81no—
156C2×C4×C59—81no—
157C2×C4×C59—81no—
158C2×C4×C59—81no—
159C2×C4×C59—81no—
160C2×C4×C59—81no—
161C2×C4×C59—81no—
162C2×C4×C59C4×D411yescharacteristic
163C2×C4×C59C4×D411yescharacteristic
164C2×C4×C59#12011yescharacteristic
165C2×C4×C59#12311yescharacteristic
166C22⋊D59—41no—
167C22⋊D59—41no—
168C22⋊D59—41no—
169C22⋊D59—41no—
170C2×Dic59—21no—
171C2×Dic59—21no—
172C2×Dic59—21no—
173C2×Dic59—21no—
174C2×Dic59—21no—
175C2×Dic59—41no—
176C2×Dic59—41no—
177C2×Dic59—42no—
178C2×Dic59—42no—
179C2×Dic59—42no—
180C2×Dic59—81no—
181C2×Dic59—81no—
182C2×Dic59—81no—
183C4×D59—42no—
184C4×D59—81no—
185C8×C59—42no—
186C59⋊C8—42no—
187#9474—591nomaximal, Sylow
188C22×C59—21no—
189C22×C59—21no—
190C22×C59—21no—
191C22×C59—21no—
192C22×C59—21no—
193C22×C59—21no—
194C22×C59—41no—
195C22×C59—41no—
196C22×C59—41no—
197C22×C59—41no—
198C22×C59#38611yescharacteristic
199C22×C59#41711yescharacteristic
200C22×C59#41811yescharacteristic
201C22×C59#46811yescharacteristic
202C22×C59#46811yescharacteristic
203C22×C59#46911yescharacteristic
204C22×C59#46911yescharacteristic
205C2×D59—42no—
206C2×D59—41no—
207C2×D59—41no—
208C2×D59—41no—
209C2×D59—81no—
210C4×C59—21no—
211C4×C59—21no—
212C4×C59—21no—
213C4×C59—82no—
214C4×C59—161no—
215C4×C59#45812yes—
216Dic59—41no—
217Dic59—41no—
218Dic59—41no—
219Dic59—82no—
220Dic59—161no—
221#2973—591no—
222#2486—591no—
223#1823—591no—
224#1823D5911yescharacteristic, complemented
225#1799—591no—
226#1289—591no—
227#1289—591no—
228C2×C59—21no—
229C2×C59—21no—
230C2×C59—21no—
231C2×C59—21no—
232C2×C59—21no—
233C2×C59—21no—
234C2×C59#189011yescharacteristic
235C2×C59#189511yescharacteristic
236C2×C59#190911yescharacteristic
237D59—81no—
238C2×#123—1181no—
239C2×C4×D4—591no—
240C22×C4⋊C4C2×D5911yescharacteristic
241#435—1182no—
242#435—1182no—
243#432—1181no—
244C2×#99—1181no—
245#407—1181no—
246C2×C22⋊C8—591no—
247C2×C22⋊C8C2×D5911yescharacteristic
248#398—592no—
249#397—592no—
250#389—1181no—
251#381—1181no—
252#380—591no—
253#380—591no—
254#380—1181no—
255#380C2×D5911yescharacteristic
256#342—1181no—
257#340—22no—
258#340—1182no—
259C59#947411yescharacteristic, complemented, Sylow
260C22×D4—591no—
261C23×C4—21no—
262C23×C4—591no—
263C23×C4C22×D5911yescharacteristic
264C2×M16—1182no—
265C22×C8—592no—
266C22×C8C22⋊D5912yes—
267#123—2362no—
268#123—2362no—
269C4×D4—1181no—
270C4×D4—1181no—
271C4×D4—1181no—
272C4×D4—1181no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—591no—
280C2×C4⋊C4—1182no—
281C2×C4⋊C4—1181no—
282C2×C4⋊C4—1181no—
283C2×C4⋊C4C4×D5912yes—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—591no—
286C2×C22⋊C4—591no—
287C2×C22⋊C4—1182no—
288C2×C22⋊C4—1181no—
289C2×C22⋊C4—1181no—
290C2×C42—591no—
291C4⋊C8—1182no—
292#99—2362no—
293C22⋊C8—21no—
294C22⋊C8—21no—
295C22⋊C8—1182no—
296C22⋊C8—1181no—
297C22⋊C8—1181no—
298#97—1182no—
299#95—21no—
300#95—592no—
301#95—1182no—
302#95—1182no—
303#95—1181no—
304C24—591no—
305C24D4×D5911yescharacteristic
306C2×D4—1181no—
307C2×D4—1181no—
308C2×D4—1181no—
309C2×D4—1181no—
310C2×D4—1181no—
311C2×D4—1181no—
312C22×C4—22no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—591no—
322C22×C4—591no—
323C22×C4—591no—
324C22×C4—1181no—
325C22×C4—1181no—
326C22×C4—1182no—
327C22×C4—2362no—
328C22×C4—2361no—
329C22×C4C2×C4×D5911yescharacteristic
330C22×C4#1174568511yescharacteristic
331C22×C4D4×D5911yescharacteristic
332C22×C4D4×D5911yescharacteristic
333C22×C4#1174568711yescharacteristic
334C22×C4C2×C22⋊D5911yescharacteristic
335M16—4722no—
336C2×C8—22no—
337C2×C8—42no—
338C2×C8—1182no—
339C2×C8—2362no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—1181no—
346C4⋊C4—1181no—
347C4⋊C4—2362no—
348C4⋊C4—2362no—
349C4⋊C4—2364no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—1181no—
353C22⋊C4—1181no—
354C22⋊C4—1181no—
355C22⋊C4—1181no—
356C22⋊C4—2364no—
357C22⋊C4—2361no—
358C22⋊C4—2361no—
359C22⋊C4—2361no—
360C22⋊C4—2361no—
361C42—1181no—
362C42—1181no—
363C23—41no—
364C23—41no—
365C23—1181no—
366C23—1181no—
367C23—1181no—
368C23—2361no—
369C23—2361no—
370C23C22⋊C4×D5911yescharacteristic
371C23#1427369811yescharacteristic
372C23#1427370011yescharacteristic
373C23#1427375111yescharacteristic
374C23#1427375111yescharacteristic
375C23#1427375411yescharacteristic
376C23#1427375611yescharacteristic
377D4—2361no—
378D4—2361no—
379D4—2361no—
380D4—2361no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—1181no—
398C2×C4—1181no—
399C2×C4—1181no—
400C2×C4—1181no—
401C2×C4—1181no—
402C2×C4—4721no—
403C2×C4—4721no—
404C2×C4—4721no—
405C2×C4—4721no—
406C2×C4—2362no—
407C2×C4—2362no—
408C2×C4—2362no—
409C2×C4—2362no—
410C2×C4—2361no—
411C2×C4—2361no—
412C2×C4(C22⋊D59)⋊C411yescharacteristic
413C2×C4C4×C22⋊D5911yescharacteristic
414C2×C4#1427369511yescharacteristic
415C2×C4#1427370911yescharacteristic
416C8—42no—
417C8—2362no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—4721no—
429C22—2362no—
430C22—2361no—
431C22—2361no—
432C22—2361no—
433C22#1683727811yescharacteristic
434C22#1683740011yescharacteristic
435C22#1683740511yescharacteristic
436C22#1683764511yescharacteristic
437C22#1683764711yescharacteristic
438C22#1683765211yescharacteristic
439C22#1683765411yescharacteristic
440C4—21no—
441C4—21no—
442C4—21no—
443C4—82no—
444C4—161no—
445C4—9441no—
446C4—4722no—
447C4—2361no—
448C4—2361no—
449C4—2361no—
450C4#1683759112yes—
451C2—21no—
452C2—21no—
453C2—21no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—4721no—
458C2#2142450611yescharacteristic
459C2#2142453111yescharacteristic
460C2#2142461511yescharacteristic
subgroupG/H
centerC22#16837278
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1823C2
socleC22×C59#386
prime radicalC22×C22⋊D59D4
cosocleC23×C4×C59C23
hypercenter#1823D59
hypocenterC59#9474
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
3. #21424182C2semidirect
5. #21421964C2semidirect
6. #21421963C2semidirect
7. C59×#1823C2semidirect
16. #16837333C4semidirect
17. #16837328C4semidirect
224. #1823D59semidirect
259. C59#9474semidirect

Maximal subgroups

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

subgroupindex
#214331222
#214283982
#214241822
#214240812
#214219642
#214219632
C59×#18232
#947459