Finite Group Databasebeta

← #28493089

Subgroups — #28493089

Order 15104.

604 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 121 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#21424283C212yesmaximal
2#21422360C211yesmaximal, characteristic
3#21421974C212yesmaximal
4#21421939C211yesmaximal, characteristic
5C59×#2659C211yesmaximal, characteristic
6C22×C4⋊Dic59C2211yescharacteristic
7C22×Dic59⋊C4—21no—
8C2×#14273660—22no—
9C2×C22⋊(C59⋊C8)C2212yes—
10C2×C59⋊#95—21no—
11C2×C59⋊#95C2211yescharacteristic
12#16837164—22no—
13C2×C22⋊Q8×C59—21no—
14C59×#398—22no—
15C59×#395—22no—
16C59×#380—21no—
17C59×#380C2212yes—
18C2×#95×C59C2211yescharacteristic
19C23×Dic59—21no—
20C23×Dic59—21no—
21C22×C59⋊C8—22no—
22C2×C4⋊Dic59—22no—
23C2×C4⋊Dic59—42no—
24C2×Dic59⋊C4—42no—
25C2×Dic59⋊C4—42no—
26C2×Dic59⋊C4—42no—
27#14273660—44no—
28C22⋊(C59⋊C8)—22no—
29C22⋊(C59⋊C8)Q814yescomplemented
30C59⋊#95—21no—
31C59⋊#95—42no—
32C59⋊#95Q812yescomplemented
33C22×Q8×C59—21no—
34C23×C4×C59D411yescharacteristic
35C23×C4×C59D411yescharacteristic
36C23×C4×C59C2311yescharacteristic
37C22⋊Q8×C59—42no—
38C22⋊Q8×C59—42no—
39C2×C4⋊C4×C59—22no—
40C2×C4⋊C4×C59—21no—
41C2×C4⋊C4×C59—21no—
42C2×C4⋊C4×C59—21no—
43C2×C4⋊C4×C59D412yes—
44C2×C4⋊C4×C59D412yes—
45C2×C22⋊C4×C59—22no—
46#95×C59—22no—
47#95×C59—22no—
48#95×C59—22no—
49C24×C59C2×D411yescharacteristic
50C2×Q8×C59—42no—
51C2×Q8×C59—81no—
52C2×Q8×C59—81no—
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—21no—
57C22×C4×C59—22no—
58C22×C4×C59—22no—
59C22×C4×C59—41no—
60C22×C4×C59—41no—
61C22×C4×C59—41no—
62C22×C4×C59—41no—
63C22×C4×C59—41no—
64C22×C4×C59—41no—
65C22×C4×C59D812yes—
66C22×C4×C59Q1612yes—
67C22×C4×C59C2×D412yes—
68C22×C4×C59C2×Q812yes—
69C22×C4×C59C2×Q811yescharacteristic
70C22×C4×C59#4111yescharacteristic
71C22×Dic59—22no—
72C22×Dic59—22no—
73C22×Dic59—22no—
74C22×Dic59—22no—
75C22×Dic59—41no—
76C22×Dic59—41no—
77C22×Dic59—41no—
78C22×Dic59—41no—
79C22×Dic59—41no—
80C22×Dic59—41no—
81C22×Dic59—41no—
82C22×Dic59—41no—
83C22×Dic59—41no—
84C22×Dic59—41no—
85C4⋊C4×C59—42no—
86C4⋊C4×C59—44no—
87C4⋊C4×C59—42no—
88C4⋊C4×C59—42no—
89C4⋊C4×C59—81no—
90C4⋊C4×C59—81no—
91C22⋊C4×C59—42no—
92C22⋊C4×C59—42no—
93C4⋊Dic59—24no—
94C4⋊Dic59—42no—
95Dic59⋊C4—48no—
96C2×C59⋊C8—24no—
97C2×C59⋊C8—42no—
98C2×C59⋊C8—42no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—21no—
102C23×C59—21no—
103C23×C59#12011yescharacteristic
104C23×C59C22⋊Q811yescharacteristic
105C23×C59C4⋊Q811yescharacteristic
106C23×C59C2×D811yescharacteristic
107C23×C59C2×Q1611yescharacteristic
108C23×C59AGL(1,ℤ/8)11yescharacteristic
109C23×C59#13711yescharacteristic
110Q8×C59—84no—
111C2×C4×C59—22no—
112C2×C4×C59—22no—
113C2×C4×C59—22no—
114C2×C4×C59—22no—
115C2×C4×C59—42no—
116C2×C4×C59—41no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59—42no—
123C2×C4×C59—42no—
124C2×C4×C59—42no—
125C2×C4×C59—42no—
126C2×C4×C59—42no—
127C2×C4×C59—42no—
128C2×C4×C59—42no—
129C2×C4×C59—42no—
130C2×C4×C59—81no—
131C2×C4×C59—81no—
132C2×C4×C59C22⋊Q812yes—
133C2×C4×C59C4⋊Q812yes—
134C2×Dic59—24no—
135C2×Dic59—42no—
136C2×Dic59—42no—
137C2×Dic59—42no—
138C2×Dic59—42no—
139C2×Dic59—42no—
140C2×Dic59—42no—
141C2×Dic59—42no—
142C2×Dic59—42no—
143C2×Dic59—42no—
144C2×Dic59—81no—
145C2×Dic59—81no—
146C2×Dic59—81no—
147C2×Dic59—81no—
148C59⋊C8—44no—
149#10567—591nomaximal, Sylow
150C22×C59—21no—
151C22×C59—21no—
152C22×C59—21no—
153C22×C59—21no—
154C22×C59—21no—
155C22×C59—21no—
156C22×C59—21no—
157C22×C59—21no—
158C22×C59—21no—
159C22×C59—21no—
160C22×C59—21no—
161C22×C59—21no—
162C22×C59#39511yescharacteristic
163C22×C59#44711yescharacteristic
164C22×C59#45111yescharacteristic
165C22×C59#45311yescharacteristic
166C22×C59#45611yescharacteristic
167C22×C59B(4,𝔽2)11yescharacteristic
168C22×C59#45811yescharacteristic
169C22×C59#47411yescharacteristic
170C22×C59#47711yescharacteristic
171C22×C59#50011yescharacteristic
172C22×C59#50111yescharacteristic
173C4×C59—22no—
174C4×C59—22no—
175C4×C59—42no—
176C4×C59—42no—
177C4×C59—82no—
178C4×C59—82no—
179Dic59—44no—
180Dic59—82no—
181#2659D5911yescharacteristic
182#1852—592no—
183#1444—591no—
184#1294—592no—
185#1280—591no—
186C2×C59—21no—
187C2×C59—21no—
188C2×C59—21no—
189C2×C59—21no—
190C2×C59#159511yescharacteristic
191C2×C59#160511yescharacteristic
192C2×C59#161011yescharacteristic
193C2×C59#161411yescharacteristic
194C2×C59#205711yescharacteristic
195C2×C59#206011yescharacteristic
196C2×C59#206211yescharacteristic
197C2×C22⋊Q8—21no—
198C22×C4⋊C4—591no—
199C22×C4⋊C4—1181no—
200C2×#107—1182no—
201C2×C22⋊C8—592no—
202#398—22no—
203#395—22no—
204#380—21no—
205#380C2×D5912yes—
206C2×#95—591no—
207C2×#95—1181no—
208C2×#95C2×D5911yescharacteristic
209#340—1182no—
210C59#1056711yescharacteristic, complemented, Sylow
211C22×Q8—21no—
212C23×C4—1181no—
213C23×C4—1181no—
214C23×C4C22⋊D5911yescharacteristic
215C23×C4C22⋊D5911yescharacteristic
216C23×C4C22×D5911yescharacteristic
217C22×C8—1182no—
218C22⋊Q8—42no—
219C22⋊Q8—42no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—1182no—
225C2×C4⋊C4—2362no—
226C2×C4⋊C4—2362no—
227C2×C4⋊C4—2362no—
228C2×C4⋊C4—2362no—
229C2×C4⋊C4C22⋊D5912yes—
230C2×C4⋊C4C22⋊D5912yes—
231C2×C22⋊C4—22no—
232#107—2364no—
233C22⋊C8—594no—
234C22⋊C8—1182no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—592no—
239#95—1181no—
240#95—2362no—
241C24C2×C22⋊D5911yescharacteristic
242C2×Q8—42no—
243C2×Q8—81no—
244C2×Q8—81no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—1182no—
258C22×C4—1182no—
259C22×C4—1182no—
260C22×C4—1182no—
261C22×C4—2361no—
262C22×C4—2361no—
263C22×C4—2361no—
264C22×C4—2361no—
265C22×C4—2361no—
266C22×C4—2361no—
267C22×C4—2361no—
268C22×C4—2361no—
269C22×C4—2361no—
270C22×C4—2361no—
271C22×C4#1174566912yes—
272C22×C4#1174567212yes—
273C22×C4Q8×D5912yes—
274C22×C4Q8×D5911yescharacteristic
275C22×C4#1174568911yescharacteristic
276C22×C4C2×C22⋊D5912yes—
277C2×C8—1184no—
278C2×C8—2362no—
279C2×C8—2362no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—81no—
285C4⋊C4—81no—
286C4⋊C4—1184no—
287C4⋊C4—2362no—
288C4⋊C4—2368no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23#1427369611yescharacteristic
296C23#1427370311yescharacteristic
297C23#1427372511yescharacteristic
298C23C2×#1174566911yescharacteristic
299C23C2×#1174567211yescharacteristic
300C23#1427375311yescharacteristic
301C23#1427375811yescharacteristic
302Q8—84no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—1184no—
325C2×C4—4721no—
326C2×C4—4721no—
327C2×C4—4721no—
328C2×C4—4721no—
329C2×C4—2362no—
330C2×C4—2362no—
331C2×C4—2362no—
332C2×C4—2362no—
333C2×C4—2362no—
334C2×C4—2362no—
335C2×C4—2362no—
336C2×C4—2362no—
337C2×C4—2362no—
338C2×C4#1427370312yes—
339C2×C4#1427372512yes—
340C8—2364no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22#1683732111yescharacteristic
354C22#1683753611yescharacteristic
355C22#1683755811yescharacteristic
356C22#1683757011yescharacteristic
357C22#1683758511yescharacteristic
358C22#1683758911yescharacteristic
359C22#1683759211yescharacteristic
360C22#1683767911yescharacteristic
361C22#1683769411yescharacteristic
362C22#1683780511yescharacteristic
363C22#1683780811yescharacteristic
364C4—22no—
365C4—22no—
366C4—42no—
367C4—42no—
368C4—82no—
369C4—82no—
370C4—4722no—
371C4—2364no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2#2142301411yescharacteristic
377C2#2142307011yescharacteristic
378C2#2142309911yescharacteristic
379C2#2142312111yescharacteristic
380C2#2142542111yescharacteristic
381C2#2142543811yescharacteristic
382C2#2142544811yescharacteristic
subgroupG/H
centerC23#14273696
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC59×#2659C2
socleC23×C59#120
prime radicalC24×C59C2×D4
hypercenter#2659D59
hypocenterC59#10567
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2659

Complements

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

NH ≅ G/Ntype
29. C22⋊(C59⋊C8)Q8semidirect
32. C59⋊#95Q8semidirect
210. C59#10567semidirect

Maximal subgroups

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

subgroupindex
#214242832
#214223602
#214219742
#214219392
C59×#26592
#1056759