Finite Group Databasebeta

← #28493030

Subgroups — #28493030

Order 15104.

500 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 79 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#21434847C211yesmaximal, characteristic
2#21424559C212yesmaximal, complemented
3#21422611C212yesmaximal, complemented
4#21421973C211yesmaximal, characteristic, complemented
5C59×#1851C211yesmaximal, characteristic, complemented
6C2×#14273701C2212yes—
7#16837716—22no—
8#16837638—21no—
9#16837638—21no—
10#16837538—22no—
11C2×#14273643—22no—
12#16837366C2212yes—
13#16837170—22no—
14#16837135—22no—
15#16837135—22no—
16C22×M16×C59C2211yescharacteristic
17C2×C4⋊C8×C59—21no—
18C2×C22⋊C8×C59C2211yescharacteristic
19C4×C22⋊C4×C59C2211yescharacteristic
20C59×#336—21no—
21C22×D236D412yes—
22C22×C22⋊D59—22no—
23C2×D472—21no—
24C2×D472—21no—
25C2×#11745664—22no—
26#14273701—22no—
27#14273701—22no—
28#14273701—22no—
29#14273701—22no—
30C2×C4⋊Dic59D412yes—
31C2×(C2×D59)⋊C4—22no—
32#14273661—21no—
33#14273661—21no—
34#14273649—22no—
35#14273649—22no—
36#14273643—44no—
37C22⋊(C59⋊C8)—22no—
38C22⋊(C59⋊C8)—22no—
39C23×C4×C59C2311yescharacteristic
40C2×M16×C59—22no—
41C2×M16×C59—42no—
42C22×C8×C59—21no—
43C22×C8×C59D411yescharacteristic
44C22×C8×C59D411yescharacteristic
45C2×C22⋊C4×C59—21no—
46C2×C42×C59—21no—
47C4⋊C8×C59—42no—
48C22⋊C8×C59—22no—
49#95×C59D411yescharacteristic
50#95×C59D411yescharacteristic
51C24×C59#4111yescharacteristic
52C23×D59—22no—
53C22×C4×C59—21no—
54C22×C4×C59—21no—
55C22×C4×C59—41no—
56C22×C4×C59—41no—
57C22×C4×C59C2×D412yes—
58C22×C4×C59C2×D411yescharacteristic
59C22×C4×C59C2×D411yescharacteristic
60C22×C4×C59#4112yes—
61C2×C22⋊D59—42no—
62C2×C22⋊D59—42no—
63C2×C22⋊D59—82no—
64C2×C22⋊D59—82no—
65C22×Dic59—22no—
66C2×D236—22no—
67C2×D236—22no—
68C2×D236—42no—
69C2×D236—42no—
70M16×C59—24no—
71M16×C59—42no—
72C2×C8×C59—22no—
73C2×C8×C59—21no—
74C2×C8×C59—21no—
75C2×C8×C59—41no—
76C2×C8×C59—41no—
77C2×C8×C59—41no—
78C2×C8×C59—41no—
79C22⋊C4×C59—42no—
80C42×C59—41no—
81C42×C59—41no—
82(C2×D59)⋊C4—42no—
83(C2×D59)⋊C4—42no—
84C4⋊Dic59—22no—
85C4⋊Dic59—22no—
86#11745664—42no—
87#11745664—42no—
88C2×C59⋊C8—42no—
89D472—81no—
90D472—81no—
91C23×C59—21no—
92C23×C59—21no—
93C23×C59—41no—
94C23×C59—41no—
95C23×C59#12111yescharacteristic
96C23×C59#12312yes—
97C22×D59—42no—
98C22×D59—44no—
99C22×D59—82no—
100C22×D59—82no—
101C2×C4×C59—22no—
102C2×C4×C59—21no—
103C2×C4×C59—22no—
104C2×C4×C59—21no—
105C2×C4×C59—22no—
106C2×C4×C59—41no—
107C2×C4×C59—42no—
108C2×C4×C59—41no—
109C2×C4×C59—41no—
110C2×C4×C59—41no—
111C2×C4×C59—41no—
112C2×C4×C59—41no—
113C2×C4×C59#12011yescharacteristic
114C2×C4×C59#12112yes—
115C2×C4×C59#12411yescharacteristic
116C22⋊D59—84no—
117C22⋊D59—82no—
118C22⋊D59—82no—
119C2×Dic59—42no—
120C2×Dic59—82no—
121D236—42no—
122D236—42no—
123D236—82no—
124C8×C59—41no—
125C8×C59—42no—
126C8×C59—41no—
127C59⋊C8—82no—
128#10557—591nomaximal, Sylow
129C22×C59—21no—
130C22×C59—21no—
131C22×C59—22no—
132C22×C59—22no—
133C22×C59—22no—
134C22×C59—81no—
135C22×C59—81no—
136C22×C59#39411yescharacteristic
137C22×C59#47211yescharacteristic
138C22×C59#47211yescharacteristic
139C2×D59—82no—
140C2×D59—82no—
141C2×D59—82no—
142C2×D59—164no—
143C4×C59—22no—
144C4×C59—21no—
145C4×C59—42no—
146C4×C59—81no—
147C4×C59B(4,𝔽2)11yescharacteristic
148C4×C59#45811yescharacteristic
149Dic59—82no—
150#3186—591no—
151#1900—592no—
152#1851D5911yescharacteristic, complemented
153#1507—592no—
154#1293—591no—
155C2×C59—21no—
156C2×C59—21no—
157C2×C59—42no—
158C2×C59#204411yescharacteristic
159C2×C59#204511yescharacteristic
160C2×C59#205011yescharacteristic
161D59—162no—
162C22×M16C2×D5911yescharacteristic
163C2×#121—592no—
164#480—1182no—
165#466—1181no—
166#466—1181no—
167#447—1182no—
168C2×C4⋊C8—21no—
169C2×#100—1182no—
170#407—592no—
171C2×C22⋊C8C2×D5911yescharacteristic
172C4×C22⋊C4C2×D5911yescharacteristic
173#343—1182no—
174#336—21no—
175#327—1182no—
176#327—1182no—
177C59#1055711yescharacteristic, complemented, Sylow
178C22×D4—592no—
179C22×D4—1182no—
180C23×C4C22×D5911yescharacteristic
181C2×D8—1181no—
182C2×D8—1181no—
183C2×M16—22no—
184C2×M16—42no—
185C2×M16—1182no—
186C22×C8—21no—
187C22×C8C22⋊D5911yescharacteristic
188C22×C8C22⋊D5911yescharacteristic
189#121—1182no—
190#121—1182no—
191#121—1182no—
192#121—1182no—
193C2×C4⋊C4—592no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—1182no—
196C2×C42—21no—
197#107—1181no—
198#107—1181no—
199C4⋊C8—42no—
200#102—1182no—
201#102—1182no—
202#100—2364no—
203C22⋊C8—22no—
204C22⋊C8—1182no—
205C22⋊C8—1182no—
206#95D23611yescharacteristic
207#95D23611yescharacteristic
208C24—1182no—
209C24#1174568511yescharacteristic
210C2×D4—1182no—
211C2×D4—1182no—
212C2×D4—4722no—
213C2×D4—4722no—
214C2×D4—2362no—
215C2×D4—2362no—
216C2×D4—2362no—
217C2×D4—2362no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—1182no—
223C22×C4C2×D23611yescharacteristic
224C22×C4D4×D5912yes—
225C22×C4#1174568912yes—
226C22×C4C2×C22⋊D5911yescharacteristic
227D8—4721no—
228D8—4721no—
229M16—24no—
230M16—42no—
231M16—2362no—
232M16—2362no—
233C2×C8—22no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—2362no—
241C4⋊C4—1182no—
242C4⋊C4—1182no—
243C22⋊C4—42no—
244C22⋊C4—2362no—
245C22⋊C4—2362no—
246C42—41no—
247C42—41no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—4722no—
253C23—4722no—
254C23—2364no—
255C23—2362no—
256C23#1427370111yescharacteristic
257C23#1427371212yes—
258D4—4722no—
259D4—4724no—
260D4—4722no—
261D4—4722no—
262D4—2362no—
263D4—2362no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—41no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—4722no—
277C2×C4—2362no—
278C2×C4#1427369511yescharacteristic
279C2×C4#1427370212yes—
280C2×C4#1427371611yescharacteristic
281C8—42no—
282C8—41no—
283C8—41no—
284C8—4722no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—22no—
290C22—81no—
291C22—81no—
292C22—9444no—
293C22—4722no—
294C22—4722no—
295C22—4722no—
296C22#1683731911yescharacteristic
297C22#1683767211yescharacteristic
298C22#1683767211yescharacteristic
299C4—22no—
300C4—21no—
301C4—42no—
302C4—81no—
303C4—4722no—
304C4#1683758811yescharacteristic
305C4#1683759111yescharacteristic
306C2—21no—
307C2—21no—
308C2—42no—
309C2—9442no—
310C2#2142534711yescharacteristic
311C2#2142535211yescharacteristic
312C2#2142538311yescharacteristic
subgroupG/H
centerC22#16837319
commutator, cosocleC23×C4×C59C23
FrattiniC23×C4C22×D59
FittingC59×#1851C2
socleC22×C59#394
prime radical#21434847C2
hypercenter#1851D59
hypocenterC59#10557
LCSG ▷ C23×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
2. #21424559C2semidirect
3. #21422611C2semidirect
4. #21421973C2semidirect
5. C59×#1851C2semidirect
152. #1851D59semidirect
177. C59#10557semidirect

Maximal subgroups

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

subgroupindex
#214348472
#214245592
#214226112
#214219732
C59×#18512
#1055759