Finite Group Databasebeta

← #28477704

Subgroups — #28477704

Order 15104.

454 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 79 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#21432076C211yesmaximal, characteristic
2#21425426C211yesmaximal, characteristic, complemented
3#21425196C211yesmaximal, characteristic, complemented
4#21421921C211yesmaximal, characteristic, complemented
5#21421920C211yesmaximal, characteristic, complemented
6#21421917C211yesmaximal, characteristic
7C59×#2023C211yesmaximal, characteristic
8C22×#11745672—21no—
9C2×#14273658—21no—
10C2×#14273651—21no—
11#16837327—21no—
12#16837316—21no—
13#16837314—21no—
14#16837308C2211yescharacteristic
15#16837300C2211yescharacteristic
16#16837295—21no—
17#16837295—21no—
18#16837283—21no—
19#16837164—21no—
20#16837157C2211yescharacteristic
21#16837157C2211yescharacteristic
22C2×C4⋊Q8×C59C2211yescharacteristic
23C2×#102×C59—21no—
24C59×#390C2211yescharacteristic
25C59×#340—21no—
26C59×#336C2211yescharacteristic
27C22×Dic118D411yescharacteristic
28C2×#11745672—42no—
29C2×#11745672—42no—
30C2×#11745672—42no—
31C22×C59⋊C8—21no—
32C22×C59⋊C8—21no—
33C2×C4⋊Dic59D411yescharacteristic
34C2×C4⋊Dic59D411yescharacteristic
35C2×C4⋊Dic59D411yescharacteristic
36C2×Dic59⋊C4—21no—
37C2×C22⋊Dic59—21no—
38C2×C22⋊Dic59—21no—
39C2×C22⋊Dic59—21no—
40C2×C22⋊Dic59—21no—
41C2×C4×Dic59—21no—
42#14273658—42no—
43#14273651—42no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C59⋊#95—21no—
50C22×Q8×C59—21no—
51C22×D4×C59D411yescharacteristic
52C22×C8×C59—21no—
53C4⋊Q8×C59—42no—
54C2×C4⋊C4×C59—21no—
55C2×C4⋊C4×C59—21no—
56C2×C4⋊C4×C59D411yescharacteristic
57C2×C22⋊C4×C59—21no—
58C2×C22⋊C4×C59—21no—
59C2×C42×C59C2311yescharacteristic
60#102×C59—42no—
61C24×C59—21no—
62C2×Q8×C59—22no—
63C2×Q8×C59—41no—
64C2×Q8×C59—41no—
65C2×Q8×C59—41no—
66C2×Q8×C59—41no—
67C2×Q8×C59—41no—
68C2×D4×C59—21no—
69C2×D4×C59—21no—
70C2×D4×C59—41no—
71C2×D4×C59—41no—
72C22×C4×C59—21no—
73C22×C4×C59—41no—
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C22×C4×C59C2×D411yescharacteristic
77C22×Dic59—21no—
78C22×Dic59—21no—
79C22×Dic59—21no—
80C22×Dic59—21no—
81C2×Dic118—21no—
82C2×Dic118—21no—
83C2×Dic118—41no—
84C2×Dic118—41no—
85C2×C8×C59—41no—
86C2×C8×C59—41no—
87C2×C8×C59—41no—
88C4⋊C4×C59—22no—
89C4⋊C4×C59—22no—
90C4⋊C4×C59—41no—
91C4⋊C4×C59—41no—
92C4⋊C4×C59—41no—
93C22⋊C4×C59—41no—
94C22⋊C4×C59—41no—
95C22⋊C4×C59—41no—
96C22⋊C4×C59—41no—
97C42×C59—41no—
98C22⋊Dic59—81no—
99C22⋊Dic59—81no—
100C22⋊Dic59—81no—
101C22⋊Dic59—81no—
102#11745672—48no—
103C4⋊Dic59—21no—
104C4⋊Dic59—21no—
105C4⋊Dic59—41no—
106C4⋊Dic59—41no—
107Dic59⋊C4—41no—
108Dic59⋊C4—41no—
109C4×Dic59—81no—
110C2×C59⋊C8—22no—
111C2×C59⋊C8—22no—
112C2×C59⋊C8—41no—
113C2×C59⋊C8—41no—
114C2×C59⋊C8—41no—
115C2×C59⋊C8—41no—
116C23×C59—41no—
117C23×C59—41no—
118C23×C59—41no—
119C23×C59—41no—
120C23×C59—41no—
121C23×C59—41no—
122C23×C59—41no—
123C23×C59#12011yescharacteristic
124Q8×C59—42no—
125Q8×C59—42no—
126Q8×C59—81no—
127Q8×C59—81no—
128D4×C59—42no—
129D4×C59—81no—
130C2×C4×C59—21no—
131C2×C4×C59—21no—
132C2×C4×C59—21no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—21no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—41no—
139C2×C4×C59—42no—
140C2×C4×C59—81no—
141C2×C4×C59—81no—
142C2×C4×C59#13511yescharacteristic
143C2×C4×C59#13511yescharacteristic
144C2×C4×C59#13511yescharacteristic
145C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C59#13711yescharacteristic
147C2×C4×C59#13711yescharacteristic
148C2×Dic59—41no—
149C2×Dic59—41no—
150C2×Dic59—41no—
151C2×Dic59—41no—
152C2×Dic59—41no—
153C2×Dic59—41no—
154C2×Dic59—41no—
155C2×Dic59—41no—
156C2×Dic59—41no—
157C2×Dic59—41no—
158C2×Dic59—41no—
159C2×Dic59—41no—
160Dic118—42no—
161Dic118—81no—
162C8×C59—81no—
163C59⋊C8—42no—
164C59⋊C8—42no—
165#7785—591nomaximal, Sylow
166C22×C59—81no—
167C22×C59—81no—
168C22×C59—81no—
169C22×C59—81no—
170C22×C59—81no—
171C22×C59—81no—
172C22×C59—81no—
173C22×C59#44911yescharacteristic
174C22×C59#45211yescharacteristic
175C22×C59#45211yescharacteristic
176C22×C59#45311yescharacteristic
177C22×C59#45511yescharacteristic
178C22×C59#45611yescharacteristic
179C22×C59B(4,𝔽2)11yescharacteristic
180C4×C59—21no—
181C4×C59—21no—
182C4×C59—21no—
183C4×C59—21no—
184C4×C59—21no—
185C4×C59—21no—
186C4×C59—82no—
187C4×C59—81no—
188Dic59—81no—
189Dic59—81no—
190Dic59—81no—
191Dic59—81no—
192#2808—591no—
193#2058—591no—
194#2023D5911yescharacteristic
195#2016—591no—
196#1272—591no—
197#1271—591no—
198#1270—591no—
199C2×C59—161no—
200C2×C59#160011yescharacteristic
201C2×C59#160311yescharacteristic
202C2×C59#160611yescharacteristic
203C2×C59#163411yescharacteristic
204C2×C59#164011yescharacteristic
205C2×C59#165211yescharacteristic
206C2×C59#166111yescharacteristic
207C22×Q16—1181no—
208C2×C4⋊Q8C2×D5911yescharacteristic
209C2×#106—1181no—
210C2×#103—1181no—
211C2×#102—21no—
212#397—1181no—
213#394—1181no—
214#393—1181no—
215#391—591no—
216#390C2×D5911yescharacteristic
217#389—591no—
218#388—1181no—
219#388—1181no—
220#387—1181no—
221#340—21no—
222#340—1181no—
223#336—591no—
224#336—591no—
225#336C2×D5911yescharacteristic
226C59#778511yescharacteristic, complemented, Sylow
227C22×Q8—21no—
228C22×Q8—591no—
229C22×D4C22⋊D5911yescharacteristic
230C2×Q16—2362no—
231C2×Q16—2362no—
232C2×Q16—2362no—
233C22×C8—21no—
234C22×C8—1181no—
235C22×C8—1181no—
236C4⋊Q8—42no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—591no—
240C2×C4⋊C4—591no—
241C2×C4⋊C4—591no—
242C2×C4⋊C4—1181no—
243C2×C4⋊C4C22⋊D5911yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—1181no—
247C2×C22⋊C4—1181no—
248C2×C22⋊C4—1181no—
249C2×C22⋊C4—1181no—
250C2×C42—1181no—
251C2×C42C22×D5911yescharacteristic
252#106—2362no—
253#103—2362no—
254#102—42no—
255#95—1181no—
256#95—1181no—
257#95—1181no—
258#95—1181no—
259#95—1181no—
260#95—1181no—
261C24—21no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—1181no—
269C2×Q8—1181no—
270C2×Q8—2361no—
271C2×Q8—2361no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C2×D4—41no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—1181no—
279C22×C4—1181no—
280C22×C4—1181no—
281C22×C4—1181no—
282C22×C4D4×D5911yescharacteristic
283C22×C4D4×D5911yescharacteristic
284C22×C4C2×C22⋊D5911yescharacteristic
285Q16—2368no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—1182no—
290C2×C8—1182no—
291C2×C8—2361no—
292C2×C8—2361no—
293C2×C8—2361no—
294C2×C8—2361no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—1181no—
301C4⋊C4—1181no—
302C4⋊C4—2361no—
303C4⋊C4—2361no—
304C4⋊C4—2361no—
305C4⋊C4—2361no—
306C22⋊C4—41no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—4721no—
311C22⋊C4—4721no—
312C22⋊C4—4721no—
313C22⋊C4—4721no—
314C42—41no—
315C42—4721no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23#1427369511yescharacteristic
324Q8—42no—
325Q8—42no—
326Q8—81no—
327Q8—81no—
328Q8—4721no—
329Q8—2362no—
330D4—42no—
331D4—81no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—2361no—
345C2×C4—2361no—
346C2×C4—2361no—
347C2×C4—2361no—
348C2×C4—2361no—
349C2×C4—2361no—
350C2×C4—2361no—
351C2×C4—2361no—
352C2×C4—2361no—
353C2×C4—2361no—
354C2×C4—2361no—
355C2×C4—2361no—
356C2×C4#1427374611yescharacteristic
357C2×C4#1427374711yescharacteristic
358C2×C4#1427374811yescharacteristic
359C2×C4#1427375111yescharacteristic
360C2×C4#1427375611yescharacteristic
361C2×C4#1427375811yescharacteristic
362C8—81no—
363C8—2362no—
364C8—2362no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22#1683754411yescharacteristic
373C22#1683756111yescharacteristic
374C22#1683756211yescharacteristic
375C22#1683756811yescharacteristic
376C22#1683757911yescharacteristic
377C22#1683758311yescharacteristic
378C22#1683758811yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—81no—
386C4—82no—
387C4—4721no—
388C4—4721no—
389C4—4721no—
390C4—4721no—
391C2—161no—
392C2#2142303911yescharacteristic
393C2#2142305611yescharacteristic
394C2#2142307411yescharacteristic
395C2#2142324011yescharacteristic
396C2#2142327911yescharacteristic
397C2#2142336511yescharacteristic
398C2#2142341211yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2023C2
socleC23×C59#120
prime radicalC22×D4×C59D4
hypercenter#2023D59
hypocenterC59#7785
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2023

Complements

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

NH ≅ G/Ntype
2. #21425426C2semidirect
3. #21425196C2semidirect
4. #21421921C2semidirect
5. #21421920C2semidirect
226. C59#7785semidirect

Maximal subgroups

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

subgroupindex
#214320762
#214254262
#214251962
#214219212
#214219202
#214219172
C59×#20232
#778559