Finite Group Databasebeta

← #28486681

Subgroups — #28486681

Order 15104.

478 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 95 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#21432734C211yesmaximal, characteristic
2#21428379C211yesmaximal, characteristic
3#21422425C212yesmaximal
4#21422005C212yesmaximal
5C59×#1822C211yesmaximal, characteristic
6C22×Dic59⋊C4C2211yescharacteristic
7#16837463—24no—
8#16837426—21no—
9#16837366C2212yes—
10C4×#11745664—21no—
11#16837340—21no—
12#16837333—21no—
13#16837286—21no—
14#16837286—21no—
15#16837274C411yescharacteristic
16#16837274C411yescharacteristic
17#16837265C2211yescharacteristic
18#16837263C411yescharacteristic
19#16837263C411yescharacteristic
20#16837127—24no—
21C2×#117×C59C2211yescharacteristic
22C59×#407C2212yes—
23C59×#343—21no—
24C59×#343—21no—
25C23×Dic59C2×C411yescharacteristic
26C23×Dic59C2×C411yescharacteristic
27C2×#11745664—22no—
28C2×Dic59⋊C4—21no—
29C2×Dic59⋊C4—21no—
30C2×Dic59⋊C4—22no—
31C2×Dic59⋊C4—21no—
32C2×Dic59⋊C4—21no—
33C2×Dic59⋊C4—22no—
34C2×C22⋊Dic59C2×C411yescharacteristic
35C2×C22⋊Dic59C2×C411yescharacteristic
36C4⋊(C59⋊C8)—24no—
37C22⋊(C59⋊C8)—22no—
38C22⋊(C59⋊C8)D414yes—
39#14273638—22no—
40C4×C59⋊C8—22no—
41C59⋊#95—21no—
42C59⋊#95—21no—
43C59⋊#95—21no—
44C59⋊#95—21no—
45C59⋊#95—21no—
46C59⋊#95—21no—
47C59⋊#95—21no—
48C59⋊#95—21no—
49C23×C4×C59C2311yescharacteristic
50C2×M16×C59—22no—
51#117×C59—22no—
52#117×C59—22no—
53C2×C4⋊C4×C59D411yescharacteristic
54C2×C4⋊C4×C59D411yescharacteristic
55C2×C22⋊C4×C59C2×C411yescharacteristic
56C2×C22⋊C4×C59C2×C411yescharacteristic
57C2×C42×C59D411yescharacteristic
58C2×C42×C59D411yescharacteristic
59C22⋊C8×C59—22no—
60C22⋊C8×C59—22no—
61C24×C59C22×C411yescharacteristic
62C22×C4×C59—41no—
63C22×C4×C59—41no—
64C22×C4×C59C22⋊C411yescharacteristic
65C22×C4×C59C22⋊C411yescharacteristic
66C22×C4×C59C22⋊C411yescharacteristic
67C22×C4×C59C22⋊C411yescharacteristic
68C22×C4×C59C2×D411yescharacteristic
69C22×C4×C59C2×D411yescharacteristic
70C22×C4×C59C2×D412yes—
71C22×C4×C59#4112yes—
72C22×Dic59—21no—
73C22×Dic59—22no—
74C22×Dic59—21no—
75C22×Dic59—21no—
76C22×Dic59—21no—
77C22×Dic59—22no—
78C22×Dic59—42no—
79C22×Dic59—41no—
80C22×Dic59—41no—
81M16×C59—42no—
82M16×C59—42no—
83C2×C8×C59—42no—
84C4⋊C4×C59—22no—
85C4⋊C4×C59—22no—
86C22⋊C4×C59—22no—
87C22⋊C4×C59—22no—
88C42×C59—21no—
89C42×C59—21no—
90C42×C59—21no—
91C42×C59—21no—
92C22⋊Dic59—41no—
93C22⋊Dic59—41no—
94Dic59⋊C4—42no—
95Dic59⋊C4—42no—
96Dic59⋊C4—41no—
97Dic59⋊C4—42no—
98Dic59⋊C4—41no—
99#11745664—42no—
100#11745664—42no—
101C2×C59⋊C8—24no—
102C23×C59—21no—
103C23×C59—21no—
104C23×C59—41no—
105C23×C59—41no—
106C23×C59C2×C22⋊C411yescharacteristic
107C23×C59C4×D412yes—
108C2×C4×C59—21no—
109C2×C4×C59—21no—
110C2×C4×C59—21no—
111C2×C4×C59—21no—
112C2×C4×C59—21no—
113C2×C4×C59—21no—
114C2×C4×C59—21no—
115C2×C4×C59—21no—
116C2×C4×C59—42no—
117C2×C4×C59—42no—
118C2×C4×C59—41no—
119C2×C4×C59—41no—
120C2×C4×C59—41no—
121C2×C4×C59—41no—
122C2×C4×C59#12011yescharacteristic
123C2×C4×C59#12112yes—
124C2×C4×C59#12311yescharacteristic
125C2×Dic59—22no—
126C2×Dic59—24no—
127C2×Dic59—41no—
128C2×Dic59—42no—
129C2×Dic59—41no—
130C2×Dic59—41no—
131C2×Dic59—41no—
132C2×Dic59—41no—
133C2×Dic59—41no—
134C2×Dic59—41no—
135C2×Dic59—41no—
136C2×Dic59—41no—
137C2×Dic59—41no—
138C2×Dic59—41no—
139C2×Dic59—82no—
140C8×C59—82no—
141C59⋊C8—44no—
142#9514—591nomaximal, Sylow
143C22×C59—21no—
144C22×C59—21no—
145C22×C59—22no—
146C22×C59—22no—
147C22×C59—22no—
148C22×C59—42no—
149C22×C59—42no—
150C22×C59#38611yescharacteristic
151C22×C59#41011yescharacteristic
152C22×C59#41011yescharacteristic
153C4×C59—21no—
154C4×C59—22no—
155C4×C59—21no—
156C4×C59—41no—
157C4×C59—41no—
158C4×C59—41no—
159C4×C59—41no—
160Dic59—42no—
161Dic59—81no—
162Dic59—81no—
163Dic59—81no—
164#2921—591no—
165#2484—591no—
166#1822D5911yescharacteristic
167#1463—592no—
168#1306—592no—
169C2×C59—21no—
170C2×C59—21no—
171C2×C59—42no—
172C2×C59#189111yescharacteristic
173C2×C59#189911yescharacteristic
174C2×C59#191511yescharacteristic
175C2×#117C2×D5911yescharacteristic
176C22×C4⋊C4—591no—
177#433—1184no—
178#423—1181no—
179#407—592no—
180#407C2×D5912yes—
181C4×M16—1181no—
182#400—1181no—
183#398—1181no—
184#387—1181no—
185#387—1181no—
186#385—591no—
187#385—591no—
188#381—591no—
189#381—591no—
190#381—591no—
191#343—21no—
192#343—21no—
193#324—1184no—
194C59#951411yescharacteristic, complemented, Sylow
195C23×C4—591no—
196C23×C4—591no—
197C23×C4C22×D5911yescharacteristic
198C2×M16—22no—
199C2×M16—1182no—
200#117—22no—
201#117—22no—
202C2×C4⋊C4—1181no—
203C2×C4⋊C4—1181no—
204C2×C4⋊C4—1181no—
205C2×C4⋊C4—1182no—
206C2×C4⋊C4—1182no—
207C2×C4⋊C4—1181no—
208C2×C4⋊C4C22⋊D5911yescharacteristic
209C2×C4⋊C4C22⋊D5911yescharacteristic
210C2×C22⋊C4—591no—
211C2×C22⋊C4—591no—
212C2×C22⋊C4C4×D5911yescharacteristic
213C2×C22⋊C4C4×D5911yescharacteristic
214C2×C42D23611yescharacteristic
215C2×C42D23611yescharacteristic
216C4⋊C8—1184no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8—594no—
220C22⋊C8—1182no—
221#97—1182no—
222C4×C8—1182no—
223#95—1181no—
224#95—1181no—
225#95—1181no—
226#95—1181no—
227#95—1181no—
228#95—1181no—
229#95—1181no—
230#95—1181no—
231C24C2×C4×D5911yescharacteristic
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—1181no—
235C22×C4—1182no—
236C22×C4—1182no—
237C22×C4—1181no—
238C22×C4—1181no—
239C22×C4—1181no—
240C22×C4—2362no—
241C22×C4—2361no—
242C22×C4—2361no—
243C22×C4(C2×D59)⋊C411yescharacteristic
244C22×C4(C2×D59)⋊C411yescharacteristic
245C22×C4(C2×D59)⋊C411yescharacteristic
246C22×C4(C2×D59)⋊C411yescharacteristic
247C22×C4C2×D23611yescharacteristic
248C22×C4D4×D5912yes—
249C22×C4#1174568712yes—
250C22×C4C2×C22⋊D5911yescharacteristic
251M16—42no—
252M16—42no—
253M16—2362no—
254M16—2362no—
255C2×C8—42no—
256C2×C8—1184no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—2362no—
260C4⋊C4—2362no—
261C4⋊C4—2361no—
262C4⋊C4—2362no—
263C4⋊C4—2361no—
264C22⋊C4—22no—
265C22⋊C4—22no—
266C22⋊C4—2361no—
267C22⋊C4—2361no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23C2×(C2×D59)⋊C411yescharacteristic
277C23(C22⋊D59)⋊C412yes—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—1182no—
293C2×C4—1184no—
294C2×C4—4722no—
295C2×C4—2361no—
296C2×C4—2361no—
297C2×C4—2361no—
298C2×C4—2361no—
299C2×C4—2361no—
300C2×C4—2361no—
301C2×C4—2361no—
302C2×C4—2361no—
303C2×C4—2361no—
304C2×C4—2361no—
305C2×C4—2362no—
306C2×C4—2361no—
307C2×C4#1427369711yescharacteristic
308C2×C4#1427369912yes—
309C2×C4#1427371111yescharacteristic
310C8—82no—
311C8—2364no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—42no—
318C22—42no—
319C22#1683727911yescharacteristic
320C22#1683737511yescharacteristic
321C22#1683737511yescharacteristic
322C4—22no—
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—4721no—
330C4—4721no—
331C4—4721no—
332C4—2362no—
333C2—21no—
334C2—21no—
335C2—42no—
336C2#2142451211yescharacteristic
337C2#2142455211yescharacteristic
338C2#2142464611yescharacteristic
subgroupG/H
centerC22#16837279
commutator, prime radicalC24×C59C22×C4
FrattiniC23×C4C22×D59
FittingC59×#1822C2
socleC22×C59#386
cosocleC23×C4×C59C23
hypercenter#1822D59
hypocenterC59#9514
LCSG ▷ C24×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1822

Complements

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

NH ≅ G/Ntype
194. C59#9514semidirect

Maximal subgroups

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

subgroupindex
#214327342
#214283792
#214224252
#214220052
C59×#18222
#951459