Finite Group Databasebeta

← #28489530

Subgroups — #28489530

Order 15104.

734 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 105 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#21434096C212yesmaximal, complemented
2#21424928C211yesmaximal, characteristic, complemented
3#21422521C212yesmaximal, complemented
4#21421925C211yesmaximal, characteristic, complemented
5C59×#2397C211yesmaximal, characteristic, complemented
6#16837952C2211yescharacteristic
7#16837715—22no—
8#16837682—22no—
9#16837597—22no—
10#16837536—22no—
11C4×#11745669—24no—
12#16837455C2212yes—
13#16837426C2211yescharacteristic
14C2×#14273655—22no—
15D4⋊(C59⋊C8)—24no—
16C2×C4×D4×C59—22no—
17C2×C4×D4×C59C2212yes—
18C59×#399—22no—
19C59×#392—22no—
20C59×#376C2211yescharacteristic
21C2×#11745685—21no—
22C2×#11745669—24no—
23C2×#11745664—21no—
24#14273727D411yescharacteristic, complemented
25#14273724D411yescharacteristic, complemented
26#14273715—21no—
27#14273701—22no—
28C4×D236—22no—
29#14273660—24no—
30C4⋊(C59⋊C8)—22no—
31C4⋊(C59⋊C8)—22no—
32#14273655—44no—
33#14273648—24no—
34#14273647—24no—
35C22⋊(C59⋊C8)—22no—
36C4×C59⋊C8D414yescomplemented
37C22×D4×C59—22no—
38C22×D4×C59D412yes—
39C23×C4×C59—22no—
40C4×D4×C59—22no—
41C4×D4×C59—22no—
42C4×D4×C59—22no—
43C4×D4×C59—22no—
44C4×D4×C59—44no—
45C4×D4×C59—44no—
46C2×C4⋊C4×C59—22no—
47C2×C4⋊C4×C59D412yes—
48C2×C22⋊C4×C59—24no—
49C2×C22⋊C4×C59—22no—
50C2×C42×C59D412yes—
51C2×C42×C59C2311yescharacteristic
52#95×C59—22no—
53C24×C59—22no—
54C2×D4×C59—22no—
55C2×D4×C59—22no—
56C2×D4×C59—44no—
57C2×D4×C59—44no—
58C2×D4×C59—48no—
59C22×C4×C59—22no—
60C22×C4×C59—22no—
61C22×C4×C59—42no—
62C22×C4×C59—42no—
63C22×C4×C59—44no—
64C22×C4×C59—44no—
65C22×C4×C59C2×D412yes—
66C22×C4×C59C2×D411yescharacteristic
67C2×C22⋊D59—41no—
68#11745685—42no—
69#11745685—42no—
70C2×D236—22no—
71C2×D236—21no—
72C2×C4×D59—41no—
73C2×Dic118—21no—
74C4⋊C4×C59—22no—
75C4⋊C4×C59—22no—
76C4⋊C4×C59—44no—
77C22⋊C4×C59—48no—
78C22⋊C4×C59—42no—
79C22⋊C4×C59—42no—
80C42×C59—22no—
81C42×C59D814yescomplemented
82C42×C59C2×D411yescharacteristic
83C42×C59C2×D412yes—
84C42×C59#4111yescharacteristic
85#11745669—48no—
86(C2×D59)⋊C4—42no—
87C4⋊Dic59—22no—
88#11745664—42no—
89C2×C59⋊C8—24no—
90C2×C59⋊C8—41no—
91C23×C59—24no—
92C23×C59—44no—
93C23×C59—42no—
94C23×C59—42no—
95C23×C59—44no—
96C23×C59#12011yescharacteristic
97C22×D59—41no—
98D4×C59—24no—
99D4×C59—24no—
100D4×C59—42no—
101D4×C59—42no—
102D4×C59—416no—
103C2×C4×C59—24no—
104C2×C4×C59—42no—
105C2×C4×C59—44no—
106C2×C4×C59—42no—
107C2×C4×C59—44no—
108C2×C4×C59—84no—
109C2×C4×C59—82no—
110C2×C4×C59—82no—
111C2×C4×C59#12012yes—
112C2×C4×C59#12111yescharacteristic
113C2×C4×C59#12112yes—
114C2×C4×C59#12711yescharacteristic
115C2×C4×C59C2×D812yes—
116C2×C4×C59AGL(1,ℤ/8)12yes—
117C22⋊D59—82no—
118C2×Dic59—41no—
119D236—44no—
120D236—42no—
121C4×D59—81no—
122C4×D59—81no—
123Dic118—42no—
124C59⋊C8—44no—
125C59⋊C8—81no—
126#10012—591nomaximal, Sylow
127C22×C59—22no—
128C22×C59—48no—
129C22×C59—44no—
130C22×C59—84no—
131C22×C59—82no—
132C22×C59—82no—
133C22×C59#39211yescharacteristic
134C22×C59#44712yes—
135C2×D59—82no—
136C2×D59—81no—
137C4×C59—22no—
138C4×C59—22no—
139C4×C59—44no—
140C4×C59—82no—
141C4×C59#45912yes—
142C4×C59#49311yescharacteristic
143C4×C59#49611yescharacteristic
144Dic59—81no—
145#3100—592no—
146#2397D5911yescharacteristic, complemented
147#1963—591no—
148#1486—592no—
149#1274—591no—
150C2×C59—22no—
151C2×C59—84no—
152C2×C59#201012yes—
153C2×C59#201111yescharacteristic
154D59—161no—
155#532—591no—
156C2×C4×D4—22no—
157C2×C4×D4C2×D5912yes—
158#480—1182no—
159#474—1182no—
160#459—1182no—
161#447—1182no—
162C4×D8—1184no—
163#431—592no—
164#423—591no—
165C2×#104—1182no—
166#399—22no—
167#392—22no—
168#376C2×D5911yescharacteristic
169D4⋊C8—1184no—
170C59#1001211yescharacteristic, complemented, Sylow
171C2×#41—1181no—
172C22×D4—22no—
173C22×D4C22⋊D5912yes—
174C23×C4—22no—
175C2×D8—1184no—
176C2×M16—1181no—
177C4⋊Q8—591no—
178#127—591no—
179#124—1181no—
180#121—1182no—
181C4×D4—22no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—44no—
186C4×D4—44no—
187C4×D4—1182no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C22⋊D5912yes—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—22no—
192C2×C42C22⋊D5912yes—
193C2×C42C22×D5911yescharacteristic
194#107—1184no—
195C4⋊C8—1182no—
196C4⋊C8—1182no—
197#104—2364no—
198#102—1184no—
199#102—1184no—
200C22⋊C8—1182no—
201C4×C8—594no—
202#95—22no—
203C24—22no—
204#41—2362no—
205#41—2362no—
206C2×Q8—1181no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—44no—
210C2×D4—44no—
211C2×D4—48no—
212C2×D4—1182no—
213C2×D4—1181no—
214C2×D4—2361no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—2361no—
222C22×C4C2×C22⋊D5912yes—
223C22×C4C2×C22⋊D5911yescharacteristic
224D8—2368no—
225M16—2362no—
226C2×C8—1184no—
227C2×C8—2361no—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—44no—
231C4⋊C4—1182no—
232C22⋊C4—48no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—2362no—
236C42—22no—
237C42#1174566914yescomplemented
238C42D4×D5911yescharacteristic
239C42D4×D5912yes—
240C42#1174568711yescharacteristic
241C23—24no—
242C23—44no—
243C23—42no—
244C23—44no—
245C23—42no—
246C23—2361no—
247C23#1427369611yescharacteristic
248Q8—2362no—
249D4—24no—
250D4—24no—
251D4—42no—
252D4—42no—
253D4—416no—
254D4—4722no—
255D4—2364no—
256D4—2362no—
257C2×C4—24no—
258C2×C4—42no—
259C2×C4—44no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—82no—
263C2×C4—82no—
264C2×C4—84no—
265C2×C4—4721no—
266C2×C4—4721no—
267C2×C4—2361no—
268C2×C4#1427369512yes—
269C2×C4#1427369811yescharacteristic
270C2×C4#1427369912yes—
271C2×C4#1427372311yescharacteristic
272C2×C4C2×#1174566912yes—
273C2×C4#1427375312yes—
274C8—4721no—
275C8—2364no—
276C22—22no—
277C22—48no—
278C22—44no—
279C22—84no—
280C22—82no—
281C22—82no—
282C22—4722no—
283C22—4721no—
284C22#1683731011yescharacteristic
285C22#1683753612yes—
286C4—22no—
287C4—22no—
288C4—44no—
289C4—82no—
290C4—4721no—
291C4#1683759412yes—
292C4#1683777811yescharacteristic
293C4#1683778911yescharacteristic
294C2—22no—
295C2—84no—
296C2—9441no—
297C2#2142516112yes—
298C2#2142516711yescharacteristic
subgroupG/H
centerC2×C4#14273723
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#2397C2
socleC22×C59#392
hypercenter#2397D59
hypocenterC59#10012
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2397

Complements

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

NH ≅ G/Ntype
1. #21434096C2semidirect
2. #21424928C2semidirect
3. #21422521C2semidirect
4. #21421925C2semidirect
5. C59×#2397C2semidirect
24. #14273727D4semidirect
25. #14273724D4semidirect
36. C4×C59⋊C8D4semidirect
81. C42×C59D8semidirect
146. #2397D59semidirect
170. C59#10012semidirect
237. C42#11745669semidirect

Maximal subgroups

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

subgroupindex
#214340962
#214249282
#214225212
#214219252
C59×#23972
#1001259