Finite Group Databasebeta

← #28486481

Subgroups — #28486481

Order 15104.

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

normal subgroups only

This group has 576 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#21433125C211yesmaximal, characteristic
2#21428404C211yesmaximal, characteristic
3#21424180C212yesmaximal, complemented
4#21421963C212yesmaximal, complemented
5C59×#1799C211yesmaximal, characteristic
6C2×#14273708—21no—
7C22×C4⋊Dic59C2211yescharacteristic
8#16837474—24no—
9#16837459—21no—
10#16837366—21no—
11C2×C22⋊(C59⋊C8)C2212yes—
12#16837332C412yescomplemented
13#16837327C412yescomplemented
14#16837300—21no—
15#16837263—21no—
16#16837262—21no—
17#16837262C2211yescharacteristic
18#16837164—24no—
19C2×C4×D4×C59C2211yescharacteristic
20C2×#99×C59—21no—
21C59×#380C2212yes—
22C59×#342—21no—
23C23×Dic59—21no—
24C2×#11745664—22no—
25C22×C59⋊C8D414yes—
26#14273708—44no—
27C2×C4⋊Dic59—22no—
28C2×C4⋊Dic59—22no—
29C2×C4⋊Dic59—21no—
30C2×C4⋊Dic59—21no—
31C2×C4⋊Dic59C2×C412yes—
32C2×Dic59⋊C4—22no—
33C2×C22⋊Dic59—22no—
34C2×C22⋊Dic59—21no—
35C4⋊(C59⋊C8)—22no—
36C22⋊(C59⋊C8)—22no—
37C22⋊(C59⋊C8)—22no—
38C22⋊(C59⋊C8)—22no—
39#14273638—22no—
40C59⋊#95—22no—
41C59⋊#95—22no—
42C59⋊#95C2×C412yes—
43C22×D4×C59D411yescharacteristic
44C23×C4×C59D411yescharacteristic
45C23×C4×C59C2311yescharacteristic
46C4×D4×C59—22no—
47C4×D4×C59—22no—
48C2×C4⋊C4×C59—22no—
49C2×C4⋊C4×C59C2×C411yescharacteristic
50C2×C22⋊C4×C59—22no—
51C2×C22⋊C4×C59C2×C411yescharacteristic
52C2×C22⋊C4×C59D411yescharacteristic
53C2×C42×C59D411yescharacteristic
54#99×C59—42no—
55#95×C59—22no—
56C24×C59C22⋊C411yescharacteristic
57C24×C59C2×D411yescharacteristic
58C2×D4×C59—22no—
59C2×D4×C59—21no—
60C2×D4×C59—22no—
61C2×D4×C59—21no—
62C22×C4×C59—21no—
63C22×C4×C59—21no—
64C22×C4×C59—42no—
65C22×C4×C59—42no—
66C22×C4×C59—42no—
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C22⋊C411yescharacteristic
69C22×C4×C59C22⋊C411yescharacteristic
70C22×C4×C59C22×C411yescharacteristic
71C22×C4×C59C2×D412yes—
72C22×C4×C59C2×D411yescharacteristic
73C22×C4×C59#4112yes—
74C22×Dic59—22no—
75C22×Dic59—22no—
76C22×Dic59—42no—
77C22×Dic59—42no—
78C22×Dic59—41no—
79C22×Dic59—41no—
80C4⋊C4×C59—22no—
81C4⋊C4×C59—44no—
82C22⋊C4×C59—22no—
83C22⋊C4×C59—22no—
84C22⋊C4×C59—42no—
85C22⋊C4×C59—42no—
86C42×C59—21no—
87C42×C59—21no—
88C22⋊Dic59—44no—
89C22⋊Dic59—42no—
90C4⋊Dic59—24no—
91C4⋊Dic59—42no—
92C4⋊Dic59—41no—
93C4⋊Dic59—41no—
94Dic59⋊C4—44no—
95#11745664—82no—
96C2×C59⋊C8—24no—
97C2×C59⋊C8—44no—
98C23×C59—21no—
99C23×C59—21no—
100C23×C59—21no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59—41no—
104C23×C59—41no—
105C23×C59C2×C22⋊C411yescharacteristic
106C23×C59#12112yes—
107C23×C59AGL(1,ℤ/8)12yes—
108C23×C59AGL(1,ℤ/8)11yescharacteristic
109C23×C59#13711yescharacteristic
110D4×C59—42no—
111D4×C59—42no—
112C2×C4×C59—24no—
113C2×C4×C59—22no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—21no—
117C2×C4×C59—21no—
118C2×C4×C59—21no—
119C2×C4×C59—41no—
120C2×C4×C59—42no—
121C2×C4×C59—42no—
122C2×C4×C59—41no—
123C2×C4×C59—41no—
124C2×C4×C59—81no—
125C2×C4×C59—82no—
126C2×C4×C59—82no—
127C2×C4×C59—82no—
128C2×C4×C59C4×D412yes—
129C2×C4×C59#12011yescharacteristic
130C2×C4×C59#12311yescharacteristic
131C2×Dic59—42no—
132C2×Dic59—42no—
133C2×Dic59—42no—
134C2×Dic59—42no—
135C2×Dic59—42no—
136C2×Dic59—42no—
137C2×Dic59—82no—
138C2×Dic59—82no—
139C59⋊C8—44no—
140#9474—591nomaximal, Sylow
141C22×C59—21no—
142C22×C59—22no—
143C22×C59—21no—
144C22×C59—22no—
145C22×C59—42no—
146C22×C59—41no—
147C22×C59—41no—
148C22×C59—42no—
149C22×C59—41no—
150C22×C59—41no—
151C22×C59—41no—
152C22×C59—81no—
153C22×C59#38611yescharacteristic
154C22×C59#41711yescharacteristic
155C22×C59#41811yescharacteristic
156C22×C59#46812yes—
157C22×C59#46912yes—
158C4×C59—21no—
159C4×C59—22no—
160C4×C59—41no—
161C4×C59—41no—
162C4×C59—41no—
163C4×C59—162no—
164C4×C59#45812yes—
165Dic59—82no—
166Dic59—82no—
167#2973—591no—
168#2486—591no—
169#1823—592no—
170#1799D5911yescharacteristic
171#1289—592no—
172C2×C59—21no—
173C2×C59—22no—
174C2×C59—21no—
175C2×C59—22no—
176C2×C59—81no—
177C2×C59#189011yescharacteristic
178C2×C59#189511yescharacteristic
179C2×C59#190911yescharacteristic
180C2×#123—1181no—
181C2×C4×D4C2×D5911yescharacteristic
182C22×C4⋊C4—591no—
183#435—1184no—
184#432—1181no—
185C2×#99—21no—
186#407—1181no—
187C2×C22⋊C8—592no—
188#398—592no—
189#397—592no—
190#389—1181no—
191#381—1181no—
192#380—591no—
193#380—1181no—
194#380C2×D5912yes—
195#342—21no—
196#340—1184no—
197C59#947411yescharacteristic, complemented, Sylow
198C22×D4C22⋊D5911yescharacteristic
199C23×C4—1181no—
200C23×C4D23611yescharacteristic
201C23×C4C22×D5911yescharacteristic
202C2×M16—1182no—
203C22×C8—594no—
204#123—2364no—
205C4×D4—22no—
206C4×D4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—592no—
209C2×C4⋊C4—1182no—
210C2×C4⋊C4—1182no—
211C2×C4⋊C4—1182no—
212C2×C4⋊C4—1181no—
213C2×C4⋊C4—1181no—
214C2×C4⋊C4C4×D5911yescharacteristic
215C2×C22⋊C4—22no—
216C2×C22⋊C4—1182no—
217C2×C22⋊C4—1181no—
218C2×C22⋊C4C4×D5911yescharacteristic
219C2×C22⋊C4C22⋊D5911yescharacteristic
220C2×C42D23611yescharacteristic
221C4⋊C8—1182no—
222#99—42no—
223C22⋊C8—1182no—
224C22⋊C8—1182no—
225C22⋊C8—1182no—
226#97—1182no—
227#95—22no—
228#95—592no—
229#95—1182no—
230#95—1182no—
231C24(C2×D59)⋊C411yescharacteristic
232C24C2×C22⋊D5911yescharacteristic
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—21no—
236C2×D4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—1182no—
243C22×C4—1182no—
244C22×C4—2362no—
245C22×C4—2362no—
246C22×C4—2361no—
247C22×C4—2361no—
248C22×C4(C2×D59)⋊C411yescharacteristic
249C22×C4(C2×D59)⋊C411yescharacteristic
250C22×C4(C2×D59)⋊C411yescharacteristic
251C22×C4C2×C4×D5911yescharacteristic
252C22×C4C2×D23611yescharacteristic
253C22×C4D4×D5912yes—
254C22×C4#1174568712yes—
255M16—4722no—
256C2×C8—1184no—
257C2×C8—2364no—
258C4⋊C4—22no—
259C4⋊C4—44no—
260C4⋊C4—1184no—
261C4⋊C4—2364no—
262C4⋊C4—2362no—
263C4⋊C4—2361no—
264C4⋊C4—2361no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—2364no—
270C22⋊C4—2362no—
271C42—21no—
272C42—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23C2×(C2×D59)⋊C411yescharacteristic
281C23#1427369912yes—
282C23#1427375212yes—
283C23#1427375311yescharacteristic
284C23#1427375811yescharacteristic
285D4—42no—
286D4—42no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4—4722no—
304C2×C4—4722no—
305C2×C4—2362no—
306C2×C4—2362no—
307C2×C4—2362no—
308C2×C4—2362no—
309C2×C4—2362no—
310C2×C4—2362no—
311C2×C4(C22⋊D59)⋊C412yes—
312C2×C4#1427369711yescharacteristic
313C2×C4#1427371111yescharacteristic
314C8—2364no—
315C22—21no—
316C22—22no—
317C22—21no—
318C22—22no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22#1683727911yescharacteristic
328C22#1683740111yescharacteristic
329C22#1683740611yescharacteristic
330C22#1683765012yes—
331C22#1683765312yes—
332C4—21no—
333C4—22no—
334C4—41no—
335C4—41no—
336C4—41no—
337C4—162no—
338C4—4722no—
339C4—4722no—
340C4#1683759312yes—
341C2—22no—
342C2—21no—
343C2—21no—
344C2—22no—
345C2—81no—
346C2#2142450711yescharacteristic
347C2#2142453211yescharacteristic
348C2#2142461611yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1799C2
socleC22×C59#386
prime radicalC22×D4×C59D4
cosocleC23×C4×C59C23
hypercenter#1799D59
hypocenterC59#9474
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #21424180C2semidirect
4. #21421963C2semidirect
12. #16837332C4semidirect
13. #16837327C4semidirect
197. C59#9474semidirect

Maximal subgroups

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

subgroupindex
#214331252
#214284042
#214241802
#214219632
C59×#17992
#947459