Finite Group Databasebeta

← #28486539

Subgroups — #28486539

Order 15104.

626 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 135 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#21433120C211yesmaximal, characteristic, complemented
2#21424088C211yesmaximal, characteristic, complemented
3#21422521C212yesmaximal, complemented
4#21421954C212yesmaximal, complemented
5C59×#2535C211yesmaximal, characteristic
6D4×C59⋊C8—24no—
7#16837455C2212yes—
8#16837426C2212yes—
9C2×#14273644—21no—
10C2×C22⋊(C59⋊C8)—22no—
11#16837171—21no—
12D4⋊(C59⋊C8)—24no—
13C2×#121×C59—21no—
14C2×C4×D4×C59C2211yescharacteristic
15C2×C4×D4×C59C2211yescharacteristic
16C59×#397C412yes—
17C59×#392C412yes—
18C59×#386—22no—
19C59×#382C2211yescharacteristic
20C59×#380—21no—
21C2×#11745664—22no—
22C22×C59⋊C8—24no—
23C4⋊(C59⋊C8)—22no—
24C4⋊(C59⋊C8)—22no—
25C4⋊(C59⋊C8)—22no—
26#14273644—42no—
27C22⋊(C59⋊C8)—24no—
28C22⋊(C59⋊C8)—22no—
29C4×C59⋊C8D414yescomplemented
30C22×D4×C59—22no—
31C22×D4×C59C2×C411yescharacteristic
32C22×D4×C59D411yescharacteristic
33C23×C4×C59—21no—
34C23×C4×C59D412yes—
35#121×C59—42no—
36#121×C59—42no—
37C4×D4×C59—22no—
38C4×D4×C59—22no—
39C4×D4×C59—22no—
40C4×D4×C59—22no—
41C2×C4⋊C4×C59—21no—
42C2×C4⋊C4×C59C2×C411yescharacteristic
43C2×C4⋊C4×C59D411yescharacteristic
44C2×C22⋊C4×C59—22no—
45C2×C22⋊C4×C59—21no—
46C2×C22⋊C4×C59—22no—
47C2×C22⋊C4×C59C2×C412yes—
48C2×C42×C59C2311yescharacteristic
49#95×C59—22no—
50#95×C59—21no—
51#95×C59C2×C412yes—
52C24×C59—21no—
53C24×C59C22⋊C412yes—
54C2×D4×C59—22no—
55C2×D4×C59—22no—
56C2×D4×C59—22no—
57C2×D4×C59—42no—
58C2×D4×C59—44no—
59C2×D4×C59—44no—
60C2×D4×C59—82no—
61C2×D4×C59D812yes—
62C2×D4×C59SD1612yes—
63C22×C4×C59—22no—
64C22×C4×C59—22no—
65C22×C4×C59—22no—
66C22×C4×C59—21no—
67C22×C4×C59—42no—
68C22×C4×C59—41no—
69C22×C4×C59—41no—
70C22×C4×C59—42no—
71C22×C4×C59—82no—
72C22×C4×C59C22⋊C412yes—
73C22×C4×C59C22×C411yescharacteristic
74C22×C4×C59C2×D411yescharacteristic
75C22×C4×C59C2×D411yescharacteristic
76C4⋊C4×C59—22no—
77C4⋊C4×C59—22no—
78C4⋊C4×C59—41no—
79C4⋊C4×C59—41no—
80C22⋊C4×C59—24no—
81C22⋊C4×C59—44no—
82C22⋊C4×C59—41no—
83C22⋊C4×C59—41no—
84C22⋊C4×C59—82no—
85C42×C59C2×D412yes—
86C42×C59#4112yes—
87#11745664—42no—
88#11745664—42no—
89C2×C59⋊C8—24no—
90C2×C59⋊C8—42no—
91C2×C59⋊C8—84no—
92C23×C59—22no—
93C23×C59—22no—
94C23×C59—22no—
95C23×C59—42no—
96C23×C59—42no—
97C23×C59—41no—
98C23×C59—41no—
99C23×C59—41no—
100C23×C59—41no—
101C23×C59—41no—
102C23×C59—41no—
103C23×C59#10214yes—
104C23×C59C2×C22⋊C411yescharacteristic
105D4×C59—24no—
106D4×C59—24no—
107D4×C59—24no—
108D4×C59—42no—
109D4×C59—88no—
110C2×C4×C59—22no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59—44no—
114C2×C4×C59—42no—
115C2×C4×C59—42no—
116C2×C4×C59—42no—
117C2×C4×C59—41no—
118C2×C4×C59—41no—
119C2×C4×C59—42no—
120C2×C4×C59—41no—
121C2×C4×C59—82no—
122C2×C4×C59—82no—
123C2×C4×C59—82no—
124C2×C4×C59C4×D412yes—
125C2×C4×C59#12011yescharacteristic
126C2×C4×C59#12112yes—
127C2×C4×C59#12311yescharacteristic
128C2×C4×C59C2×D811yescharacteristic
129C2×C4×C59C2×SD1611yescharacteristic
130C2×C4×C59#13512yes—
131C59⋊C8—44no—
132C59⋊C8—82no—
133#9485—591nomaximal, Sylow
134C22×C59—21no—
135C22×C59—21no—
136C22×C59—24no—
137C22×C59—44no—
138C22×C59—42no—
139C22×C59—42no—
140C22×C59—42no—
141C22×C59—42no—
142C22×C59—42no—
143C22×C59—82no—
144C22×C59—81no—
145C22×C59—81no—
146C22×C59—81no—
147C22×C59—81no—
148C22×C59#38611yescharacteristic
149C22×C59C2×#10211yescharacteristic
150C22×C59#41611yescharacteristic
151C4×C59—22no—
152C4×C59—21no—
153C4×C59—21no—
154C4×C59—42no—
155C4×C59—82no—
156C4×C59—81no—
157C4×C59#46512yes—
158C4×C59#46612yes—
159#2972—591no—
160#2535D5911yescharacteristic
161#1800—591no—
162#1486—592no—
163#1286—592no—
164C2×C59—21no—
165C2×C59—21no—
166C2×C59—44no—
167C2×C59—82no—
168C2×C59#189111yescharacteristic
169C2×C59#189311yescharacteristic
170C2×C59#191111yescharacteristic
171C2×#121—21no—
172C2×C4×D4C2×D5911yescharacteristic
173C2×C4×D4C2×D5911yescharacteristic
174C8×D4—1184no—
175#431—592no—
176#423—592no—
177C2×#100—1181no—
178C2×C22⋊C8—1182no—
179#397Dic5912yes—
180#392Dic5912yes—
181#386—22no—
182#382C2×D5911yescharacteristic
183#380—21no—
184#343—1181no—
185D4⋊C8—1184no—
186C59#948511yescharacteristic, complemented, Sylow
187C22×D4—22no—
188C22×D4C2×Dic5911yescharacteristic
189C22×D4C22⋊D5911yescharacteristic
190C23×C4—21no—
191C23×C4C22⋊D5912yes—
192C2×M16—1182no—
193C22×C8—1184no—
194#121—42no—
195#121—42no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C2×Dic5911yescharacteristic
202C2×C4⋊C4C22⋊D5911yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4C2×Dic5912yes—
207C2×C42C22×D5911yescharacteristic
208C4⋊C8—1182no—
209C4⋊C8—1182no—
210C4⋊C8—1182no—
211#100—2362no—
212C22⋊C8—1184no—
213C22⋊C8—1182no—
214C4×C8—594no—
215#95—22no—
216#95—21no—
217#95C2×Dic5912yes—
218C24—21no—
219C24C22⋊Dic5912yes—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—42no—
224C2×D4—44no—
225C2×D4—44no—
226C2×D4—82no—
227C2×D4#1174566912yes—
228C2×D4#1174567012yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—82no—
238C22×C4C22⋊Dic5912yes—
239C22×C4C22×Dic5911yescharacteristic
240C22×C4C2×C22⋊D5911yescharacteristic
241C22×C4C2×C22⋊D5911yescharacteristic
242M16—2362no—
243M16—2362no—
244C2×C8—1184no—
245C2×C8—4724no—
246C2×C8—2362no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C22⋊C4—24no—
252C22⋊C4—44no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—82no—
256C42D4×D5912yes—
257C42#1174568712yes—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—42no—
262C23—42no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23—41no—
268C23—41no—
269C23#1427364814yes—
270C23C2×C22⋊Dic5911yescharacteristic
271D4—24no—
272D4—24no—
273D4—24no—
274D4—42no—
275D4—88no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4D4×Dic5912yes—
291C2×C4#1427369511yescharacteristic
292C2×C4#1427369912yes—
293C2×C4#1427370811yescharacteristic
294C2×C4C2×#1174566911yescharacteristic
295C2×C4C2×#1174567011yescharacteristic
296C2×C4#1427374712yes—
297C8—4722no—
298C8—2364no—
299C22—21no—
300C22—21no—
301C22—24no—
302C22—44no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—42no—
308C22—82no—
309C22—81no—
310C22—81no—
311C22—81no—
312C22—81no—
313C22#1683728211yescharacteristic
314C22C2×#1427364811yescharacteristic
315C22#1683739811yescharacteristic
316C4—22no—
317C4—21no—
318C4—21no—
319C4—42no—
320C4—82no—
321C4—81no—
322C4#1683762912yes—
323C4#1683763612yes—
324C2—21no—
325C2—21no—
326C2—44no—
327C2—82no—
328C2#2142451511yescharacteristic
329C2#2142452511yescharacteristic
330C2#2142462911yescharacteristic
subgroupG/H
centerC22#16837282
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#2535C2
socleC22×C59#386
cosocleC2×C42×C59C23
hypercenter#2535D59
hypocenterC59#9485
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2535

Complements

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

NH ≅ G/Ntype
1. #21433120C2semidirect
2. #21424088C2semidirect
3. #21422521C2semidirect
4. #21421954C2semidirect
29. C4×C59⋊C8D4semidirect
186. C59#9485semidirect

Maximal subgroups

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

subgroupindex
#214331202
#214240882
#214225212
#214219542
C59×#25352
#948559