Finite Group Databasebeta

← #28500747

Subgroups — #28500747

Order 15104.

458 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 63 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#21425227C211yesmaximal, characteristic
2#21423081C212yesmaximal
3#21422096C212yesmaximal, complemented
4#21422026C211yesmaximal, characteristic, complemented
5C59×#2817C211yesmaximal, characteristic, complemented
6#16837964C2211yescharacteristic
7C2×#14273710—21no—
8#16837715—22no—
9C2×#14273641—21no—
10#16837169C2211yescharacteristic
11#16837135—22no—
12#16837126C2212yes—
13C2×C22⋊Q8×C59—21no—
14C59×#400—22no—
15C59×#397—21no—
16C59×#397C2212yes—
17C59×#395—22no—
18C59×#381C2211yescharacteristic
19C22×C22⋊D59—21no—
20#14273711—41no—
21#14273710—42no—
22#14273710—42no—
23#14273701—22no—
24#14273697—41no—
25C2×Dic59⋊C4—41no—
26C2×C22⋊Dic59—21no—
27C2×(C2×D59)⋊C4—41no—
28#14273660—42no—
29#14273647—42no—
30#14273641—42no—
31C22⋊(C59⋊C8)—22no—
32C22×Q8×C59—21no—
33C23×C4×C59D411yescharacteristic
34C22⋊Q8×C59—42no—
35C22⋊Q8×C59—42no—
36C2×C4⋊C4×C59—22no—
37C2×C4⋊C4×C59—21no—
38C2×C4⋊C4×C59—21no—
39C2×C4⋊C4×C59D412yes—
40C2×C22⋊C4×C59—22no—
41C2×C22⋊C4×C59D411yescharacteristic
42C2×C22⋊C4×C59C2311yescharacteristic
43#95×C59—22no—
44#95×C59—21no—
45#95×C59—22no—
46#95×C59—22no—
47#95×C59D412yes—
48C24×C59C2×D411yescharacteristic
49C23×D59—21no—
50C2×Q8×C59—42no—
51C2×Q8×C59—81no—
52C2×Q8×C59—81no—
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—21no—
56C22×C4×C59—22no—
57C22×C4×C59—22no—
58C22×C4×C59—41no—
59C22×C4×C59—41no—
60C22×C4×C59D812yes—
61C22×C4×C59C2×D412yes—
62C2×C22⋊D59—41no—
63C2×C22⋊D59—41no—
64C2×C22⋊D59—41no—
65C2×C22⋊D59—41no—
66C2×C22⋊D59—41no—
67C2×C22⋊D59—41no—
68C22×Dic59—21no—
69C22×Dic59—41no—
70C2×D236—22no—
71C4⋊C4×C59—42no—
72C4⋊C4×C59—44no—
73C4⋊C4×C59—42no—
74C4⋊C4×C59—81no—
75C22⋊C4×C59—41no—
76C22⋊C4×C59—42no—
77C22⋊C4×C59—42no—
78C22⋊C4×C59—41no—
79C22⋊Dic59—22no—
80C22⋊Dic59—41no—
81(C2×D59)⋊C4—42no—
82(C2×D59)⋊C4—44no—
83C4⋊Dic59—22no—
84Dic59⋊C4—44no—
85C2×C59⋊C8—42no—
86C23×C59—21no—
87C23×C59—21no—
88C23×C59—42no—
89C23×C59#12011yescharacteristic
90C23×C59C2×D811yescharacteristic
91C23×C59AGL(1,ℤ/8)11yescharacteristic
92C22×D59—41no—
93C22×D59—41no—
94C22×D59—41no—
95C22×D59—82no—
96Q8×C59—84no—
97C2×C4×C59—22no—
98C2×C4×C59—22no—
99C2×C4×C59—41no—
100C2×C4×C59—41no—
101C2×C4×C59—42no—
102C2×C4×C59—42no—
103C2×C4×C59—42no—
104C2×C4×C59—44no—
105C2×C4×C59—42no—
106C2×C4×C59—42no—
107C2×C4×C59—41no—
108C2×C4×C59—42no—
109C2×C4×C59—42no—
110C2×C4×C59—81no—
111C22⋊D59—81no—
112C22⋊D59—81no—
113C22⋊D59—81no—
114C22⋊D59—81no—
115C2×Dic59—41no—
116C2×Dic59—42no—
117C2×Dic59—41no—
118C2×Dic59—41no—
119C2×Dic59—82no—
120D236—82no—
121C59⋊C8—82no—
122#12095—591nomaximal, Sylow
123C22×C59—22no—
124C22×C59—21no—
125C22×C59—21no—
126C22×C59—41no—
127C22×C59—41no—
128C22×C59—42no—
129C22×C59—81no—
130C22×C59#44711yescharacteristic
131C22×C59#45311yescharacteristic
132C22×C59B(4,𝔽2)11yescharacteristic
133C2×D59—82no—
134C2×D59—81no—
135C2×D59—82no—
136C2×D59—161no—
137C4×C59—42no—
138C4×C59—42no—
139C4×C59—82no—
140C4×C59—81no—
141C4×C59—82no—
142Dic59—82no—
143Dic59—81no—
144#2817D5911yescharacteristic, complemented
145#2022—591no—
146#1607—592no—
147#1344—592no—
148#1317—591no—
149C2×C59—21no—
150C2×C59—21no—
151C2×C59—41no—
152C2×C59—41no—
153C2×C59#159511yescharacteristic
154C2×C59#220112yes—
155D59—161no—
156#534—591no—
157C2×#123—1181no—
158C2×C22⋊Q8—21no—
159#480—1182no—
160C2×#99—1181no—
161#400—22no—
162#397—21no—
163#397C2×D5912yes—
164#395—22no—
165#381C2×D5911yescharacteristic
166#342—591no—
167#327—1182no—
168#323—592no—
169C59#1209511yescharacteristic, complemented, Sylow
170C22×Q8—21no—
171C22×D4—1181no—
172C23×C4C22⋊D5911yescharacteristic
173#123—2361no—
174#123—2362no—
175#123—2362no—
176C22⋊Q8—42no—
177C22⋊Q8—42no—
178#121—1182no—
179#120—2361no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—2361no—
184C2×C4⋊C4C22⋊D5912yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—1181no—
187C2×C22⋊C4—2361no—
188C2×C22⋊C4C22⋊D5911yescharacteristic
189C2×C22⋊C4C22×D5911yescharacteristic
190#107—2362no—
191#102—2362no—
192#99—2362no—
193C22⋊C8—1182no—
194#95—22no—
195#95—21no—
196#95—22no—
197#95—22no—
198#95C22⋊D5912yes—
199C24—1181no—
200C24C2×C22⋊D5911yescharacteristic
201C2×Q8—42no—
202C2×Q8—81no—
203C2×Q8—81no—
204C2×D4—1182no—
205C2×D4—2361no—
206C2×D4—2361no—
207C2×D4—2361no—
208C2×D4—2361no—
209C2×D4—2361no—
210C2×D4—2361no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—1181no—
219C22×C4—2361no—
220C22×C4#1174566912yes—
221C22×C4C2×C22⋊D5912yes—
222C2×C8—2362no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—81no—
227C4⋊C4—1182no—
228C4⋊C4—2364no—
229C22⋊C4—41no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—41no—
233C22⋊C4—1182no—
234C22⋊C4—2362no—
235C22⋊C4—2364no—
236C22⋊C4—2361no—
237C23—21no—
238C23—21no—
239C23—42no—
240C23—4722no—
241C23—2361no—
242C23—2361no—
243C23—2361no—
244C23#1427369611yescharacteristic
245C23C2×#1174566911yescharacteristic
246C23#1427375311yescharacteristic
247Q8—84no—
248D4—4722no—
249D4—4721no—
250D4—4721no—
251D4—4721no—
252D4—4721no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—44no—
261C2×C4—42no—
262C2×C4—42no—
263C2×C4—41no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—81no—
267C2×C4—4722no—
268C2×C4—2362no—
269C2×C4—2361no—
270C2×C4—2361no—
271C2×C4—2361no—
272C8—4722no—
273C22—22no—
274C22—21no—
275C22—21no—
276C22—41no—
277C22—41no—
278C22—42no—
279C22—81no—
280C22—9441no—
281C22—4722no—
282C22—4721no—
283C22—4722no—
284C22#1683753611yescharacteristic
285C22#1683757011yescharacteristic
286C22#1683758911yescharacteristic
287C4—42no—
288C4—42no—
289C4—82no—
290C4—81no—
291C4—82no—
292C4—4722no—
293C4—4721no—
294C2—21no—
295C2—21no—
296C2—41no—
297C2—41no—
298C2—9441no—
299C2#2142301411yescharacteristic
300C2#2142610912yes—
subgroupG/H
centerC22#16837589
commutator, cosocleC2×C22⋊C4×C59C23
FrattiniC2×C22⋊C4C22×D59
FittingC59×#2817C2
socleC22×C59B(4,𝔽2)
prime radical#16837964C22
hypercenter#2817D59
hypocenterC59#12095
LCSG ▷ C2×C22⋊C4×C59 ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2817

Complements

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

NH ≅ G/Ntype
3. #21422096C2semidirect
4. #21422026C2semidirect
5. C59×#2817C2semidirect
144. #2817D59semidirect
169. C59#12095semidirect

Maximal subgroups

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

subgroupindex
#214252272
#214230812
#214220962
#214220262
C59×#28172
#1209559