Finite Group Databasebeta

← #28735569

Subgroups — #28735569

Order 15104.

628 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C59⋊#2104C212yesmaximal, complemented
2#21425273C212yesmaximal, complemented
3#21425263C211yesmaximal, characteristic, complemented
4#21424639C212yesmaximal
5#21424555C212yesmaximal, complemented
6#21424510C212yesmaximal, complemented
7#21424202C212yesmaximal, complemented
8#21424171C211yesmaximal, characteristic, complemented
9C59×#2022C211yesmaximal, characteristic
10#16838028C2211yescharacteristic
11#16837877C2212yes—
12C2×#14273686C2211yescharacteristic
13C2×#14273642C2212yes—
14C2×#14273642C2212yes—
15C2×#14273642C2212yes—
16#16837265C2211yescharacteristic
17#16837265C2212yes—
18#16837169C2212yes—
19#16837169C2212yes—
20#16837169C2211yescharacteristic
21#16837168C2212yes—
22#16837168C2212yes—
23#16837168C2212yes—
24#16837168C2212yes—
25#16837168C2212yes—
26C59×#534C2211yescharacteristic
27C2×#123×C59C2212yes—
28C2×#99×C59C2212yes—
29C59×#381C2211yescharacteristic
30C59×#342C2211yescharacteristic
31#14273706—41no—
32#14273704—42no—
33D4×Dic59—42no—
34D4×Dic59—42no—
35#14273686—22no—
36#14273686—41no—
37#14273686—41no—
38C2×C4⋊Dic59—21no—
39C2×C22⋊Dic59—22no—
40C2×C22⋊Dic59C2312yes—
41C2×C22⋊Dic59C2312yes—
42C2×C22⋊Dic59C2312yes—
43C2×C22⋊Dic59C2312yes—
44C2×C4×Dic59—21no—
45#14273642—42no—
46#14273642—42no—
47#14273642—42no—
48C59⋊#95—41no—
49C59⋊#95—42no—
50C22×D4×C59C2312yes—
51C23×C4×C59C2311yescharacteristic
52#123×C59—22no—
53#123×C59—22no—
54#123×C59—22no—
55#123×C59—42no—
56#121×C59—41no—
57#120×C59—22no—
58C2×C4⋊C4×C59—22no—
59C2×C22⋊C4×C59—22no—
60C2×C22⋊C4×C59C2311yescharacteristic
61C2×C22⋊C4×C59C2312yes—
62C2×C22⋊C4×C59C2311yescharacteristic
63#99×C59—42no—
64#95×C59—41no—
65C24×C59#4112yes—
66C24×C59C2411yescharacteristic
67C2×D4×C59—22no—
68C2×D4×C59—22no—
69C2×D4×C59—22no—
70C2×D4×C59—22no—
71C2×D4×C59—22no—
72C2×D4×C59—22no—
73C2×D4×C59—41no—
74C22×C4×C59—22no—
75C22×C4×C59—21no—
76C22×C4×C59—22no—
77C22×C4×C59—21no—
78C22×C4×C59—22no—
79C22×C4×C59—21no—
80C22×C4×C59—21no—
81C22×C4×C59#4112yes—
82C22×C4×C59#4111yescharacteristic
83C22×C4×C59#4111yescharacteristic
84C22×Dic59—22no—
85C22×Dic59—22no—
86C22×Dic59—22no—
87C22×Dic59—22no—
88C22×Dic59—42no—
89C2×Dic118—41no—
90C4⋊C4×C59—42no—
91C4⋊C4×C59—42no—
92C4⋊C4×C59—41no—
93C22⋊C4×C59—22no—
94C22⋊C4×C59—22no—
95C22⋊C4×C59—22no—
96C22⋊C4×C59—22no—
97C22⋊C4×C59—42no—
98C22⋊C4×C59—42no—
99C22⋊C4×C59—42no—
100C22⋊Dic59—22no—
101C22⋊Dic59—22no—
102C22⋊Dic59—22no—
103C22⋊Dic59—22no—
104C22⋊Dic59—22no—
105C22⋊Dic59—22no—
106C22⋊Dic59—22no—
107C22⋊Dic59—22no—
108C22⋊Dic59—42no—
109C4⋊Dic59—42no—
110C4⋊Dic59—42no—
111C4⋊Dic59—41no—
112Dic59⋊C4—42no—
113C4×Dic59—42no—
114C4×Dic59—42no—
115C23×C59—22no—
116C23×C59—22no—
117C23×C59—22no—
118C23×C59—81no—
119C23×C59—82no—
120C23×C59C2×#4112yes—
121C23×C59C2×#4111yescharacteristic
122C23×C59H(5,𝔽2)11yescharacteristic
123C23×C59H(5,𝔽2)12yes—
124C23×C59H(5,𝔽2)11yescharacteristic
125D4×C59—42no—
126D4×C59—42no—
127D4×C59—42no—
128D4×C59—42no—
129D4×C59—161no—
130C2×C4×C59—21no—
131C2×C4×C59—22no—
132C2×C4×C59—21no—
133C2×C4×C59—22no—
134C2×C4×C59—22no—
135C2×C4×C59—22no—
136C2×C4×C59—22no—
137C2×C4×C59—42no—
138C2×C4×C59—41no—
139C2×C4×C59—41no—
140C2×C4×C59—42no—
141C2×C4×C59—41no—
142C2×C4×C59—41no—
143C2×C4×C59—81no—
144C2×C4×C59—82no—
145C2×C4×C59—81no—
146C2×Dic59—42no—
147C2×Dic59—42no—
148C2×Dic59—42no—
149C2×Dic59—42no—
150C2×Dic59—82no—
151C2×Dic59—82no—
152C2×Dic59—82no—
153C2×Dic59—82no—
154C2×Dic59—162no—
155Dic118—161no—
156#32701—591nomaximal, Sylow
157C22×C59—41no—
158C22×C59—42no—
159C22×C59—41no—
160C22×C59—41no—
161C22×C59—42no—
162C22×C59—42no—
163C22×C59—82no—
164C22×C59—82no—
165C22×C59—82no—
166C22×C59#53811yescharacteristic
167C22×C59#53812yes—
168C22×C59#55112yes—
169C22×C59#55111yescharacteristic
170C22×C59#56011yescharacteristic
171C4×C59—41no—
172C4×C59—41no—
173C4×C59—42no—
174C4×C59—81no—
175C4×C59—82no—
176C4×C59—81no—
177Dic59—82no—
178Dic59—82no—
179Dic59—82no—
180Dic59—82no—
181#2104—592no—
182#2032—592no—
183#2030—591no—
184#2022D5911yescharacteristic
185#1913—592no—
186#1899—592no—
187#1890—592no—
188#1828—592no—
189#1820—591no—
190C2×C59—21no—
191C2×C59—22no—
192C2×C59—21no—
193C2×C59—21no—
194C2×C59—22no—
195C2×C59—82no—
196C2×C59#276011yescharacteristic
197#543—591no—
198#534C2×D5911yescharacteristic
199C2×#123C2×D5912yes—
200#518—592no—
201C2×#117—591no—
202C2×#99—592no—
203C2×#99—592no—
204C2×#99—592no—
205C2×#99C2×D5912yes—
206#381—591no—
207#381—592no—
208#381C2×D5911yescharacteristic
209#342—592no—
210#342—592no—
211#342—591no—
212#342—592no—
213#342—592no—
214#342—592no—
215#342—592no—
216#342—592no—
217#342C2×D5911yescharacteristic
218C59#3270111yescharacteristic, complemented, Sylow
219C22×D4C22×D5912yes—
220C23×C4C22×D5911yescharacteristic
221#123—22no—
222#123—22no—
223#123—22no—
224#123—42no—
225C22⋊Q8—2362no—
226C22⋊Q8—2361no—
227#121—41no—
228#120—22no—
229C4×D4—2362no—
230C4×D4—2362no—
231#117—1182no—
232#117—2361no—
233#117—2361no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—1181no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—592no—
238C2×C22⋊C4—592no—
239C2×C22⋊C4—592no—
240C2×C22⋊C4—592no—
241C2×C22⋊C4—1182no—
242C2×C22⋊C4C22×D5912yes—
243C2×C22⋊C4C22×D5911yescharacteristic
244C2×C22⋊C4C22×D5911yescharacteristic
245C2×C42—1181no—
246#99—42no—
247#99—2362no—
248#99—2362no—
249#99—2362no—
250#95—41no—
251#95—2362no—
252#95—2361no—
253C24#1174568712yes—
254C24C23×D5911yescharacteristic
255C2×Q8—2361no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—41no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—1182no—
271C22×C4—1182no—
272C22×C4—1182no—
273C22×C4—1182no—
274C22×C4—2362no—
275C22×C4#1174568711yescharacteristic
276C22×C4#1174568712yes—
277C22×C4#1174568711yescharacteristic
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—2362no—
282C4⋊C4—2362no—
283C4⋊C4—2362no—
284C4⋊C4—2361no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—1182no—
293C22⋊C4—1182no—
294C22⋊C4—1182no—
295C22⋊C4—1182no—
296C22⋊C4—1182no—
297C22⋊C4—1182no—
298C22⋊C4—1182no—
299C22⋊C4—1182no—
300C22⋊C4—2362no—
301C42—2362no—
302C42—2362no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—81no—
307C23—82no—
308C23C2×#1174568711yescharacteristic
309C23C2×#1174568712yes—
310C23#1427377212yes—
311C23#1427377211yescharacteristic
312C23#1427377311yescharacteristic
313Q8—9441no—
314D4—42no—
315D4—42no—
316D4—42no—
317D4—42no—
318D4—161no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—82no—
335C2×C4—9442no—
336C2×C4—4722no—
337C2×C4—4722no—
338C2×C4—4722no—
339C2×C4—4722no—
340C2×C4—2362no—
341C2×C4—2362no—
342C2×C4—2362no—
343C2×C4—2362no—
344C22—41no—
345C22—42no—
346C22—41no—
347C22—42no—
348C22—41no—
349C22—42no—
350C22—82no—
351C22—82no—
352C22—82no—
353C22#1683798911yescharacteristic
354C22#1683798912yes—
355C22#1683808312yes—
356C22#1683808311yescharacteristic
357C22#1683813911yescharacteristic
358C4—41no—
359C4—41no—
360C4—42no—
361C4—81no—
362C4—81no—
363C4—82no—
364C4—4722no—
365C4—4722no—
366C4—4722no—
367C4—4722no—
368C2—21no—
369C2—21no—
370C2—22no—
371C2—22no—
372C2—21no—
373C2—82no—
374C2#2143147511yescharacteristic
subgroupG/H
centerC2#21431475
commutator, cosocleC24×C59C24
FrattiniC24C23×D59
FittingC59×#2022C2
socleC2×C59#2760
prime radicalC59×#534C22
hypercenter#2022D59
hypocenterC59#32701
LCSG ▷ C24×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C24 ◁ #2022

Complements

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

NH ≅ G/Ntype
1. C59⋊#2104C2semidirect
2. #21425273C2semidirect
3. #21425263C2semidirect
5. #21424555C2semidirect
6. #21424510C2semidirect
7. #21424202C2semidirect
8. #21424171C2semidirect
218. C59#32701semidirect

Maximal subgroups

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

subgroupindex
C59⋊#21042
#214252732
#214252632
#214246392
#214245552
#214245102
#214242022
#214241712
C59×#20222
#3270159