Finite Group Databasebeta

← #28491664

Subgroups — #28491664

Order 15104.

684 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 115 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#21430210C211yesmaximal, characteristic
2#21424031C212yesmaximal
3#21421975C212yesmaximal
4#21421939C211yesmaximal, characteristic
5C59×#1446C211yesmaximal, characteristic
6C2×#14273706—21no—
7C2×#14273651—22no—
8C2×#14273651—22no—
9C2×C22⋊(C59⋊C8)C2212yes—
10#16837334—22no—
11#16837323—22no—
12#16837262—21no—
13#16837262C2212yes—
14C23×Q8×C59C2211yescharacteristic
15C2×#95×C59—21no—
16C2×#95×C59—21no—
17C2×#95×C59C2211yescharacteristic
18C2×#95×C59C2211yescharacteristic
19C22×Dic118—21no—
20C22×C59⋊C8—22no—
21#14273706—42no—
22#14273706—42no—
23C2×C4⋊Dic59—21no—
24C2×C4⋊Dic59—21no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59D412yes—
27C2×C4⋊Dic59D412yes—
28C2×Dic59⋊C4—22no—
29C2×C22⋊Dic59—22no—
30#14273651—44no—
31#14273651—44no—
32C22⋊(C59⋊C8)—22no—
33C22⋊(C59⋊C8)—22no—
34C59⋊#95—22no—
35C59⋊#95—22no—
36C22×Q8×C59—21no—
37C22×Q8×C59—22no—
38C22×Q8×C59—22no—
39C22×Q8×C59—42no—
40C22×Q8×C59—42no—
41C22×Q8×C59Q812yescomplemented
42C23×C4×C59—21no—
43C23×C4×C59—21no—
44C23×C4×C59D411yescharacteristic
45C23×C4×C59D411yescharacteristic
46C23×C4×C59C2311yescharacteristic
47#95×C59—22no—
48#95×C59—22no—
49#95×C59—42no—
50#95×C59—42no—
51C24×C59C2×D411yescharacteristic
52C2×Q8×C59—24no—
53C2×Q8×C59—24no—
54C2×Q8×C59—24no—
55C2×Q8×C59—44no—
56C2×Q8×C59—42no—
57C2×Q8×C59—42no—
58C2×Q8×C59—44no—
59C2×Q8×C59—48no—
60C2×Q8×C59—42no—
61C22×C4×C59—22no—
62C22×C4×C59—22no—
63C22×C4×C59—21no—
64C22×C4×C59—21no—
65C22×C4×C59—22no—
66C22×C4×C59—21no—
67C22×C4×C59—21no—
68C22×C4×C59—22no—
69C22×C4×C59—41no—
70C22×C4×C59—41no—
71C22×C4×C59—41no—
72C22×C4×C59—41no—
73C22×C4×C59—41no—
74C22×C4×C59—41no—
75C22×C4×C59—41no—
76C22×C4×C59—41no—
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×C59—41no—
85C22×C4×C59SD1612yes—
86C22×C4×C59Q1612yes—
87C22×C4×C59C2×D412yes—
88C22×C4×C59C2×Q811yescharacteristic
89C22×C4×C59#4112yes—
90C22×C4×C59#4111yescharacteristic
91C22×Dic59—22no—
92C22×Dic59—22no—
93C2×Dic118—42no—
94C2×Dic118—81no—
95C2×Dic118—81no—
96C22⋊Dic59—42no—
97C22⋊Dic59—42no—
98C4⋊Dic59—42no—
99C4⋊Dic59—42no—
100C4⋊Dic59—42no—
101C4⋊Dic59—81no—
102C4⋊Dic59—81no—
103Dic59⋊C4—44no—
104C2×C59⋊C8—42no—
105C2×C59⋊C8—42no—
106C2×C59⋊C8—42no—
107C23×C59—21no—
108C23×C59—21no—
109C23×C59—21no—
110C23×C59—21no—
111C23×C59#12011yescharacteristic
112C23×C59C22⋊Q811yescharacteristic
113C23×C59#12411yescharacteristic
114C23×C59C2×SD1611yescharacteristic
115C23×C59C2×Q1611yescharacteristic
116C23×C59AGL(1,ℤ/8)11yescharacteristic
117C23×C59#13711yescharacteristic
118Q8×C59—28no—
119Q8×C59—44no—
120Q8×C59—44no—
121Q8×C59—44no—
122C2×C4×C59—22no—
123C2×C4×C59—22no—
124C2×C4×C59—24no—
125C2×C4×C59—22no—
126C2×C4×C59—22no—
127C2×C4×C59—41no—
128C2×C4×C59—41no—
129C2×C4×C59—42no—
130C2×C4×C59—42no—
131C2×C4×C59—42no—
132C2×C4×C59—42no—
133C2×C4×C59—42no—
134C2×C4×C59—41no—
135C2×C4×C59—41no—
136C2×C4×C59—41no—
137C2×C4×C59—41no—
138C2×C4×C59—42no—
139C2×C4×C59—42no—
140C2×C4×C59—42no—
141C2×C4×C59—42no—
142C2×C4×C59—42no—
143C2×C4×C59—42no—
144C2×C4×C59—42no—
145C2×C4×C59—81no—
146C2×C4×C59—81no—
147C2×C4×C59—81no—
148C2×C4×C59—81no—
149C2×C4×C59—81no—
150C2×C4×C59—81no—
151C2×C4×C59C22⋊Q812yes—
152C2×C4×C59#12412yes—
153C2×Dic59—42no—
154C2×Dic59—42no—
155C2×Dic59—42no—
156C2×Dic59—42no—
157C2×Dic59—42no—
158C2×Dic59—42no—
159Dic118—84no—
160C59⋊C8—82no—
161#10341—591nomaximal, Sylow
162C22×C59—21no—
163C22×C59—21no—
164C22×C59—21no—
165C22×C59—21no—
166C22×C59—21no—
167C22×C59—21no—
168C22×C59—21no—
169C22×C59—21no—
170C22×C59—21no—
171C22×C59—21no—
172C22×C59—21no—
173C22×C59—21no—
174C22×C59#39311yescharacteristic
175C22×C59#45011yescharacteristic
176C22×C59#45111yescharacteristic
177C22×C59#45511yescharacteristic
178C22×C59#45611yescharacteristic
179C22×C59B(4,𝔽2)11yescharacteristic
180C22×C59B(4,𝔽2)11yescharacteristic
181C22×C59#47511yescharacteristic
182C22×C59#47711yescharacteristic
183C22×C59#48711yescharacteristic
184C22×C59#48911yescharacteristic
185C4×C59—22no—
186C4×C59—22no—
187C4×C59—42no—
188C4×C59—42no—
189C4×C59—44no—
190C4×C59—82no—
191Dic59—82no—
192Dic59—82no—
193#2659—591no—
194#1787—592no—
195#1446D5911yescharacteristic
196#1294—592no—
197#1280—591no—
198C2×C59—21no—
199C2×C59—21no—
200C2×C59—21no—
201C2×C59—21no—
202C2×C59#160111yescharacteristic
203C2×C59#160211yescharacteristic
204C2×C59#160411yescharacteristic
205C2×C59#160511yescharacteristic
206C2×C59#202411yescharacteristic
207C2×C59#202811yescharacteristic
208C2×C59#202911yescharacteristic
209C23×Q8C2×D5911yescharacteristic
210C2×C22⋊Q8—1181no—
211C2×#103—1182no—
212C2×#103—1182no—
213C2×C22⋊C8—592no—
214#398—1182no—
215#395—1182no—
216#380—592no—
217#380—1181no—
218C2×#95—21no—
219C2×#95—21no—
220C2×#95C2×D5911yescharacteristic
221C2×#95C2×D5911yescharacteristic
222C59#1034111yescharacteristic, complemented, Sylow
223C22×Q8—21no—
224C22×Q8—22no—
225C22×Q8—22no—
226C22×Q8—42no—
227C22×Q8—42no—
228C22×Q8—1181no—
229C22×Q8Dic11812yescomplemented
230C23×C4—21no—
231C23×C4—21no—
232C23×C4C22⋊D5911yescharacteristic
233C23×C4C22⋊D5911yescharacteristic
234C23×C4C22×D5911yescharacteristic
235C22×C8—1182no—
236C22⋊Q8—2362no—
237C22⋊Q8—2362no—
238C2×C4⋊C4—592no—
239C2×C4⋊C4—592no—
240C2×C4⋊C4—1181no—
241C2×C4⋊C4—1181no—
242C2×C4⋊C4—1182no—
243C2×C4⋊C4—1181no—
244C2×C22⋊C4—1182no—
245#103—2364no—
246#103—2364no—
247C22⋊C8—1182no—
248C22⋊C8—1182no—
249#95—22no—
250#95—22no—
251#95—42no—
252#95—42no—
253#95—1182no—
254#95—1182no—
255C24C2×C22⋊D5911yescharacteristic
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—44no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—44no—
263C2×Q8—48no—
264C2×Q8—42no—
265C2×Q8—4721no—
266C2×Q8—4721no—
267C2×Q8—2362no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—1182no—
293C22×C4—1182no—
294C22×C4#1174567112yes—
295C22×C4#1174567212yes—
296C22×C4C2×Dic11811yescharacteristic
297C22×C4#1174568511yescharacteristic
298C22×C4D4×D5912yes—
299C22×C4#1174568712yes—
300C2×C8—2362no—
301C2×C8—2362no—
302C2×C8—2362no—
303C4⋊C4—4721no—
304C4⋊C4—4721no—
305C4⋊C4—2362no—
306C4⋊C4—2362no—
307C4⋊C4—2364no—
308C4⋊C4—2362no—
309C22⋊C4—2362no—
310C22⋊C4—2362no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23#1427369511yescharacteristic
316C23#1427370611yescharacteristic
317C23#1427371411yescharacteristic
318C23C2×#1174567111yescharacteristic
319C23C2×#1174567211yescharacteristic
320C23#1427375211yescharacteristic
321C23#1427375711yescharacteristic
322Q8—28no—
323Q8—44no—
324Q8—44no—
325Q8—44no—
326Q8—4724no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—2362no—
357C2×C4—2362no—
358C2×C4—2362no—
359C2×C4—2362no—
360C2×C4—2362no—
361C2×C4—2362no—
362C2×C4#1427370412yes—
363C2×C4#1427371312yes—
364C8—4722no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22#1683731411yescharacteristic
378C22#1683755211yescharacteristic
379C22#1683755711yescharacteristic
380C22#1683757911yescharacteristic
381C22#1683758411yescharacteristic
382C22#1683758811yescharacteristic
383C22#1683758811yescharacteristic
384C22#1683768711yescharacteristic
385C22#1683769711yescharacteristic
386C22#1683775211yescharacteristic
387C22#1683776111yescharacteristic
388C4—22no—
389C4—22no—
390C4—44no—
391C4—42no—
392C4—42no—
393C4—82no—
394C4—4722no—
395C4—4722no—
396C2—21no—
397C2—21no—
398C2—21no—
399C2—21no—
400C2#2142304711yescharacteristic
401C2#2142305211yescharacteristic
402C2#2142306411yescharacteristic
403C2#2142306911yescharacteristic
404C2#2142523711yescharacteristic
405C2#2142525711yescharacteristic
406C2#2142526011yescharacteristic
subgroupG/H
centerC23#14273695
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC59×#1446C2
socleC23×C59#120
prime radicalC24×C59C2×D4
hypercenter#1446D59
hypocenterC59#10341
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1446

Complements

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

NH ≅ G/Ntype
41. C22×Q8×C59Q8semidirect
222. C59#10341semidirect
229. C22×Q8Dic118semidirect

Maximal subgroups

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

subgroupindex
#214302102
#214240312
#214219752
#214219392
C59×#14462
#1034159