Finite Group Databasebeta

← #28477995

Subgroups — #28477995

Order 15104.

494 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 83 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#21431122C211yesmaximal, characteristic
2#21425634C211yesmaximal, characteristic, complemented
3#21425628C212yesmaximal, complemented
4#21421916C212yesmaximal, complemented
5C59×#1272C211yesmaximal, characteristic, complemented
6C2×#14273716—21no—
7C2×#14273660—21no—
8C2×#14273659—22no—
9C2×#14273651—21no—
10C2×#14273649—22no—
11#16837319—21no—
12#16837312—22no—
13#16837304C2212yes—
14#16837294—22no—
15#16837272C2211yescharacteristic
16#16837157C2211yescharacteristic
17C59×#391C2211yescharacteristic
18C59×#336C2212yes—
19C22×D236—21no—
20C22×D236—21no—
21C22×D236D412yes—
22C22×C59⋊C8—21no—
23#14273716—42no—
24#14273716—42no—
25C2×C4⋊Dic59—22no—
26C2×C4⋊Dic59—21no—
27C2×C4⋊Dic59D412yes—
28C2×(C2×D59)⋊C4—22no—
29C2×(C2×D59)⋊C4—22no—
30C2×(C2×D59)⋊C4—22no—
31C2×(C2×D59)⋊C4—22no—
32C2×C4×Dic59—21no—
33#14273660—42no—
34#14273659—44no—
35#14273651—42no—
36#14273649—44no—
37C59⋊#95—21no—
38C22×Q8×C59D411yescharacteristic
39C22×C8×C59—22no—
40C2×C4⋊C4×C59D411yescharacteristic
41C2×C42×C59C2311yescharacteristic
42#95×C59—22no—
43C23×D59—22no—
44C2×Q8×C59—21no—
45C2×Q8×C59—41no—
46C2×Q8×C59—41no—
47C2×Q8×C59SD1612yes—
48C22×C4×C59—21no—
49C22×C4×C59—21no—
50C22×C4×C59C2×D411yescharacteristic
51C22×C4×C59C2×D412yes—
52C22×Dic59—22no—
53C22×Dic59—41no—
54C2×D236—22no—
55C2×D236—22no—
56C2×D236—42no—
57C2×D236—42no—
58C2×D236—42no—
59C2×D236—81no—
60C2×D236—81no—
61C2×D236—82no—
62C2×D236—81no—
63C2×C8×C59—24no—
64C2×C8×C59—42no—
65C2×C8×C59—42no—
66C4⋊C4×C59—41no—
67C42×C59—41no—
68(C2×D59)⋊C4—44no—
69(C2×D59)⋊C4—42no—
70(C2×D59)⋊C4—42no—
71(C2×D59)⋊C4—42no—
72(C2×D59)⋊C4—42no—
73(C2×D59)⋊C4—82no—
74C4⋊Dic59—42no—
75C4⋊Dic59—42no—
76C4⋊Dic59—41no—
77C4⋊Dic59—42no—
78C4⋊Dic59—41no—
79C4×Dic59—42no—
80C2×C59⋊C8—22no—
81C2×C59⋊C8—41no—
82C2×C59⋊C8—41no—
83C23×C59#12011yescharacteristic
84C22×D59—42no—
85C22×D59—42no—
86C22×D59—42no—
87C22×D59—42no—
88C22×D59—42no—
89C22×D59—42no—
90C22×D59—42no—
91Q8×C59—42no—
92Q8×C59—81no—
93C2×C4×C59—21no—
94C2×C4×C59—22no—
95C2×C4×C59—21no—
96C2×C4×C59—22no—
97C2×C4×C59—22no—
98C2×C4×C59—41no—
99C2×C4×C59—41no—
100C2×C4×C59—42no—
101C2×C4×C59—41no—
102C2×C4×C59C2×SD1611yescharacteristic
103C2×C4×C59#13512yes—
104C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
105C2×C4×C59#13712yes—
106C2×Dic59—42no—
107C2×Dic59—42no—
108C2×Dic59—42no—
109C2×Dic59—82no—
110C2×Dic59—81no—
111D236—44no—
112D236—82no—
113D236—84no—
114D236—84no—
115C8×C59—44no—
116C59⋊C8—42no—
117#7830—591nomaximal, Sylow
118C22×C59#45011yescharacteristic
119C22×C59#45212yes—
120C22×C59#45312yes—
121C22×C59#45311yescharacteristic
122C22×C59#45811yescharacteristic
123C2×D59—82no—
124C2×D59—82no—
125C2×D59—82no—
126C2×D59—82no—
127C2×D59—82no—
128C2×D59—82no—
129C2×D59—82no—
130C4×C59—21no—
131C4×C59—21no—
132C4×C59—22no—
133C4×C59—22no—
134C4×C59—42no—
135C4×C59—81no—
136Dic59—82no—
137Dic59—161no—
138#2733—591no—
139#2095—591no—
140#2094—592no—
141#1272D5911yescharacteristic, complemented
142#1270—592no—
143C2×C59#161311yescharacteristic
144C2×C59#161812yes—
145C2×C59#163112yes—
146C2×C59#163411yescharacteristic
147C2×C59#165511yescharacteristic
148D59—162no—
149C2×#124—1181no—
150C2×#107—1181no—
151C2×#106—1182no—
152C2×#103—1181no—
153C2×#102—1182no—
154#394—1181no—
155#392—1182no—
156#391C2×D5911yescharacteristic
157#390—592no—
158#388—1182no—
159#384—591no—
160#336—591no—
161#336C2×D5912yes—
162C59#783011yescharacteristic, complemented, Sylow
163C22×Q8D23611yescharacteristic
164C22×D4—592no—
165C22×D4—1181no—
166C22×D4—1181no—
167C22×C8—22no—
168C22×C8—1181no—
169#124—2362no—
170#124—2362no—
171C2×C4⋊C4—592no—
172C2×C4⋊C4—1181no—
173C2×C4⋊C4—1182no—
174C2×C4⋊C4D23611yescharacteristic
175C2×C22⋊C4—1182no—
176C2×C22⋊C4—1182no—
177C2×C22⋊C4—1182no—
178C2×C22⋊C4—1182no—
179C2×C42—1181no—
180C2×C42C22×D5911yescharacteristic
181#107—2362no—
182#106—2364no—
183#103—2362no—
184#102—2364no—
185#95—22no—
186#95—1181no—
187C24—1182no—
188C2×Q8—21no—
189C2×Q8—41no—
190C2×Q8—41no—
191C2×Q8#1174566012yes—
192C2×D4—1182no—
193C2×D4—1182no—
194C2×D4—4721no—
195C2×D4—4721no—
196C2×D4—4722no—
197C2×D4—4721no—
198C2×D4—2362no—
199C2×D4—2362no—
200C2×D4—2362no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—1182no—
204C22×C4—2361no—
205C22×C4C2×D23611yescharacteristic
206C22×C4D4×D5912yes—
207C2×C8—24no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—1182no—
211C2×C8—2361no—
212C2×C8—2361no—
213C4⋊C4—41no—
214C4⋊C4—2362no—
215C4⋊C4—2361no—
216C4⋊C4—2362no—
217C4⋊C4—2361no—
218C4⋊C4—2362no—
219C22⋊C4—4722no—
220C22⋊C4—2364no—
221C22⋊C4—2362no—
222C22⋊C4—2362no—
223C22⋊C4—2362no—
224C22⋊C4—2362no—
225C42—41no—
226C42—2362no—
227C23—2362no—
228C23—2362no—
229C23—2362no—
230C23—2362no—
231C23—2362no—
232C23—2362no—
233C23—2362no—
234C23#1427369711yescharacteristic
235Q8—42no—
236Q8—81no—
237D4—4722no—
238D4—4724no—
239D4—4724no—
240D4—2364no—
241C2×C4—22no—
242C2×C4—21no—
243C2×C4—22no—
244C2×C4—21no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—4722no—
251C2×C4—4721no—
252C2×C4—2362no—
253C2×C4—2362no—
254C2×C4—2362no—
255C2×C4C2×#1174566011yescharacteristic
256C2×C4#1427374812yes—
257C2×C4#1427375511yescharacteristic
258C2×C4#1427375912yes—
259C8—44no—
260C8—2362no—
261C22—4722no—
262C22—4722no—
263C22—4722no—
264C22—4722no—
265C22—4722no—
266C22—4722no—
267C22—4722no—
268C22#1683755511yescharacteristic
269C22#1683756712yes—
270C22#1683757112yes—
271C22#1683757211yescharacteristic
272C22#1683759311yescharacteristic
273C4—21no—
274C4—21no—
275C4—22no—
276C4—22no—
277C4—42no—
278C4—81no—
279C4—9441no—
280C4—4722no—
281C2—9442no—
282C2#2142311811yescharacteristic
283C2#2142314712yes—
284C2#2142322512yes—
285C2#2142324211yescharacteristic
286C2#2142338611yescharacteristic
subgroupG/H
centerC23#14273697
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1272C2
socleC23×C59#120
prime radical#21431122C2
hypercenter#1272D59
hypocenterC59#7830
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #21425634C2semidirect
3. #21425628C2semidirect
4. #21421916C2semidirect
5. C59×#1272C2semidirect
141. #1272D59semidirect
162. C59#7830semidirect

Maximal subgroups

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

subgroupindex
#214311222
#214256342
#214256282
#214219162
C59×#12722
#783059