Finite Group Databasebeta

← #28504495

Subgroups — #28504495

Order 15104.

876 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 131 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
1C2×#16837195C211yesmaximal, characteristic
2C2×#16837192C212yesmaximal, complemented
3#21424510C212yesmaximal, complemented
4#21424495C211yesmaximal, characteristic
5C2×C59×#559C211yesmaximal, characteristic
6#16837877C2211yescharacteristic
7#16837407—21no—
8C2×#14273644C2211yescharacteristic
9C2×#14273642C2212yes—
10#16837265—22no—
11#16837195—22no—
12#16837192—24no—
13#16837168—22no—
14#16837162—21no—
15C59×#559—22no—
16C59×#559—24no—
17C59×#559C414yescomplemented
18C2×#127×C59C2211yescharacteristic
19C2×#124×C59—22no—
20C2×#123×C59—21no—
21C2×#123×C59C2212yes—
22C2×C4×Q8×C59—21no—
23C2×C4×D4×C59—22no—
24C2×#11745664—21no—
25D4×Dic59—42no—
26#14273686—21no—
27C2×C22⋊Dic59—22no—
28C2×C22⋊Dic59—22no—
29#14273648—42no—
30#14273644—22no—
31#14273642—22no—
32#14273642—22no—
33C59⋊#95—44no—
34C22×Q8×C59—21no—
35C22×D4×C59—22no—
36C22×D4×C59D411yescharacteristic
37C22×D4×C59C2311yescharacteristic
38C23×C4×C59—21no—
39C23×C4×C59D412yes—
40#127×C59—22no—
41#127×C59C2×C412yes—
42#127×C59D412yes—
43#124×C59—28no—
44#124×C59—28no—
45#123×C59—24no—
46#123×C59—24no—
47#123×C59—22no—
48#123×C59—22no—
49#123×C59C2×C414yes—
50#123×C59D414yes—
51C4×Q8×C59—24no—
52C4×Q8×C59—24no—
53C4×D4×C59—28no—
54C4×D4×C59—28no—
55C2×C4⋊C4×C59—21no—
56C2×C4⋊C4×C59—22no—
57C2×C22⋊C4×C59—22no—
58C2×C22⋊C4×C59—21no—
59C2×C22⋊C4×C59—22no—
60C2×C22⋊C4×C59D412yes—
61C2×C42×C59—22no—
62C2×C42×C59D411yescharacteristic
63C24×C59—21no—
64C24×C59C2×D412yes—
65C2×Q8×C59—24no—
66C2×Q8×C59—44no—
67C2×D4×C59—24no—
68C2×D4×C59—24no—
69C2×D4×C59—21no—
70C2×D4×C59—41no—
71C2×D4×C59—41no—
72C2×D4×C59—48no—
73C2×D4×C59—41no—
74C2×D4×C59—41no—
75C2×D4×C59C22⋊C412yes—
76C2×D4×C59C22×C411yescharacteristic
77C2×D4×C59C2×D411yescharacteristic
78C2×D4×C59C2×D412yes—
79C22×C4×C59—21no—
80C22×C4×C59—22no—
81C22×C4×C59—22no—
82C22×C4×C59—24no—
83C22×C4×C59—22no—
84C22×C4×C59—42no—
85C22×C4×C59—42no—
86C22×C4×C59—81no—
87C22×C4×C59—81no—
88C22×C4×C59C22⋊C414yes—
89C22×C4×C59C2×D411yescharacteristic
90C22×Dic59—24no—
91C22×Dic59—42no—
92C4⋊C4×C59—24no—
93C4⋊C4×C59—24no—
94C4⋊C4×C59—24no—
95C22⋊C4×C59—28no—
96C22⋊C4×C59—24no—
97C22⋊C4×C59—24no—
98C22⋊C4×C59—24no—
99C22⋊C4×C59—22no—
100C22⋊C4×C59C22⋊C414yes—
101C42×C59—28no—
102C42×C59—21no—
103C42×C59C22⋊C412yes—
104C22⋊Dic59—42no—
105C22⋊Dic59—41no—
106C22⋊Dic59—42no—
107C22⋊Dic59—42no—
108C4⋊Dic59—22no—
109C4×Dic59—41no—
110#11745664—41no—
111#11745664—41no—
112C2×C59⋊C8—41no—
113C23×C59—22no—
114C23×C59—22no—
115C23×C59—22no—
116C23×C59—44no—
117C23×C59—44no—
118C23×C59—41no—
119C23×C59—41no—
120C23×C59C2×C22⋊C412yes—
121C23×C59#12012yes—
122C23×C59#12011yescharacteristic
123Q8×C59—48no—
124D4×C59—41no—
125D4×C59—42no—
126D4×C59—41no—
127D4×C59—42no—
128D4×C59—48no—
129D4×C59—48no—
130D4×C59—81no—
131C2×C4×C59—24no—
132C2×C4×C59—22no—
133C2×C4×C59—21no—
134C2×C4×C59—21no—
135C2×C4×C59—24no—
136C2×C4×C59—24no—
137C2×C4×C59—44no—
138C2×C4×C59—42no—
139C2×C4×C59—42no—
140C2×C4×C59—44no—
141C2×C4×C59—82no—
142C2×C4×C59—84no—
143C2×C4×C59C2×C22⋊C411yescharacteristic
144C2×C4×C59#12011yescharacteristic
145C2×Dic59—42no—
146C2×Dic59—84no—
147C2×Dic59—82no—
148C2×Dic59—84no—
149C59⋊C8—81no—
150#12760—591nomaximal, Sylow
151C22×C59—22no—
152C22×C59—22no—
153C22×C59—22no—
154C22×C59—42no—
155C22×C59—42no—
156C22×C59—44no—
157C22×C59—44no—
158C22×C59—42no—
159C22×C59—82no—
160C22×C59—82no—
161C22×C59—82no—
162C22×C59#37911yescharacteristic
163C22×C59B(4,𝔽2)11yescharacteristic
164C22×C59#45811yescharacteristic
165C4×C59—24no—
166C4×C59—21no—
167C4×C59—21no—
168C4×C59—44no—
169C4×C59—44no—
170Dic59—82no—
171Dic59—162no—
172C2×#559D5911yescharacteristic
173C2×#353—591no—
174C2×#352—592no—
175#1890—592no—
176#1887—591no—
177C2×C59—21no—
178C2×C59—21no—
179C2×C59—42no—
180C2×C59—42no—
181C2×C59—82no—
182C2×C59#179011yescharacteristic
183C2×C59#219812yes—
184#559—22no—
185#559—24no—
186#559Dic5914yescomplemented
187C2×#127C2×D5911yescharacteristic
188C2×#124—22no—
189C2×#123—21no—
190C2×#123C2×D5912yes—
191#518—591no—
192C2×C4×Q8—21no—
193C2×C4×D4—22no—
194#418—1181no—
195C2×#100—591no—
196C2×#99—592no—
197#381—1182no—
198#353—1182no—
199#352—1184no—
200#342—1182no—
201#339—1181no—
202C59#1276011yescharacteristic, complemented, Sylow
203C22×Q8—21no—
204C22×D4—22no—
205C22×D4C22⋊D5911yescharacteristic
206C22×D4C22×D5911yescharacteristic
207C23×C4—21no—
208C23×C4C22⋊D5912yes—
209C2×M16—1181no—
210#127—22no—
211#127C2×Dic5912yes—
212#127C22⋊D5912yes—
213#124—28no—
214#124—28no—
215#123—24no—
216#123—24no—
217#123—22no—
218#123—22no—
219#123C2×Dic5914yes—
220#123C22⋊D5914yes—
221C4×Q8—24no—
222C4×Q8—24no—
223C4×D4—28no—
224C4×D4—28no—
225C4×D4—2362no—
226#117—1181no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—1182no—
233C2×C22⋊C4—1182no—
234C2×C22⋊C4C22⋊D5912yes—
235C2×C42—22no—
236C2×C42C22⋊D5911yescharacteristic
237#102—2362no—
238#100—1182no—
239#99—1182no—
240#99—1182no—
241#95—2364no—
242C24—21no—
243C24C2×C22⋊D5912yes—
244C2×Q8—24no—
245C2×Q8—44no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—48no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4C22⋊Dic5912yes—
255C2×D4C22×Dic5911yescharacteristic
256C2×D4C2×C22⋊D5911yescharacteristic
257C2×D4C2×C22⋊D5912yes—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—81no—
266C22×C4—81no—
267C22×C4—1184no—
268C22×C4—2362no—
269C22×C4C22⋊Dic5914yes—
270C22×C4C2×C22⋊D5911yescharacteristic
271M16—2361no—
272M16—2361no—
273C2×C8—2361no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—1182no—
278C22⋊C4—28no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—24no—
282C22⋊C4—22no—
283C22⋊C4—2362no—
284C22⋊C4—2362no—
285C22⋊C4—2362no—
286C22⋊C4—2361no—
287C22⋊C4C22⋊Dic5914yes—
288C42—28no—
289C42—21no—
290C42—2361no—
291C42C22⋊Dic5912yes—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—44no—
296C23—44no—
297C23—41no—
298C23—41no—
299C23C2×C22⋊Dic5912yes—
300C23#1427369612yes—
301C23#1427369611yescharacteristic
302Q8—48no—
303D4—41no—
304D4—42no—
305D4—41no—
306D4—42no—
307D4—48no—
308D4—48no—
309D4—81no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—44no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—82no—
321C2×C4—84no—
322C2×C4—4722no—
323C2×C4—4724no—
324C2×C4—4724no—
325C2×C4—2362no—
326C2×C4C2×C22⋊Dic5911yescharacteristic
327C2×C4#1427369611yescharacteristic
328C8—4721no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—42no—
333C22—42no—
334C22—44no—
335C22—44no—
336C22—42no—
337C22—82no—
338C22—82no—
339C22—82no—
340C22#1683725811yescharacteristic
341C22#1683758911yescharacteristic
342C22#1683759211yescharacteristic
343C4—24no—
344C4—21no—
345C4—21no—
346C4—44no—
347C4—44no—
348C4—9442no—
349C4—4722no—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—42no—
354C2—82no—
355C2#2142404411yescharacteristic
356C2#2142609212yes—
subgroupG/H
centerC22#16837589
commutatorC2×D4×C59C22×C4
FrattiniC22×D4C22×D59
FittingC2×C59×#559C2
socleC22×C59B(4,𝔽2)
prime radicalC2×#127×C59C22
cosocleC22×D4×C59C23
hypercenterC2×#559D59
hypocenterC59#12760
LCSG ▷ C2×D4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#559

Complements

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

NH ≅ G/Ntype
2. C2×#16837192C2semidirect
3. #21424510C2semidirect
17. C59×#559C4semidirect
186. #559Dic59semidirect
202. C59#12760semidirect

Maximal subgroups

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

subgroupindex
C2×#168371952
C2×#168371922
#214245102
#214244952
C2×C59×#5592
#1276059