Finite Group Databasebeta

← #28486551

Subgroups — #28486551

Order 15104.

454 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 111 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#21433181C211yesmaximal, characteristic
2#21429458C211yesmaximal, characteristic
3#21422524C212yesmaximal
4#21421956C212yesmaximal
5C59×#1801C211yesmaximal, characteristic
6C2×C4×Dic118C2211yescharacteristic
7#16837533—24no—
8#16837459—22no—
9#16837455C2212yes—
10#16837341C412yes—
11#16837322C412yes—
12#16837309—21no—
13#16837298—21no—
14#16837286—22no—
15#16837268C2211yescharacteristic
16#16837132—24no—
17C2×C4×Q8×C59C2211yescharacteristic
18C59×#423C2212yes—
19C2×#101×C59—21no—
20C59×#343—21no—
21C22×Dic118C2×C411yescharacteristic
22C4×Dic118—22no—
23C4×Dic118—22no—
24C2×C4⋊Dic59—21no—
25C2×C4⋊Dic59C2×C411yescharacteristic
26C2×Dic59⋊C4—22no—
27C2×Dic59⋊C4—21no—
28C2×C4×Dic59—21no—
29C4⋊(C59⋊C8)—24no—
30C4⋊(C59⋊C8)—22no—
31C22⋊(C59⋊C8)—22no—
32#14273638—24no—
33C4×C59⋊C8D414yes—
34C59⋊#95—22no—
35C59⋊#95—22no—
36C59⋊#95—22no—
37C59⋊#95—21no—
38C59⋊#95C2×C412yes—
39C22×Q8×C59D411yescharacteristic
40C2×M16×C59—22no—
41C4×Q8×C59—22no—
42C4×Q8×C59—22no—
43C2×C4⋊C4×C59C2×C412yes—
44C2×C4⋊C4×C59D411yescharacteristic
45C2×C42×C59D412yes—
46C2×C42×C59C2311yescharacteristic
47C4⋊C8×C59—22no—
48C4⋊C8×C59—22no—
49#101×C59—42no—
50C2×Q8×C59—22no—
51C2×Q8×C59—21no—
52C2×Q8×C59—21no—
53C2×Q8×C59—22no—
54C22×C4×C59C22⋊C412yes—
55C22×C4×C59C22⋊C412yes—
56C22×C4×C59C22×C411yescharacteristic
57C22×C4×C59C2×D411yescharacteristic
58C22×C4×C59C2×D411yescharacteristic
59C22×Dic59—21no—
60C22×Dic59—22no—
61C22×Dic59—21no—
62C2×Dic118—22no—
63C2×Dic118—42no—
64M16×C59—42no—
65M16×C59—42no—
66C2×C8×C59—42no—
67C4⋊C4×C59—24no—
68C4⋊C4×C59—22no—
69C42×C59—22no—
70C42×C59—22no—
71C42×C59C2×D412yes—
72C42×C59#4112yes—
73C4⋊Dic59—22no—
74C4⋊Dic59—41no—
75C4⋊Dic59—41no—
76Dic59⋊C4—44no—
77Dic59⋊C4—41no—
78Dic59⋊C4—41no—
79C4×Dic59—42no—
80C2×C59⋊C8—24no—
81C23×C59C2×C22⋊C411yescharacteristic
82Q8×C59—42no—
83Q8×C59—42no—
84C2×C4×C59—22no—
85C2×C4×C59—22no—
86C2×C4×C59—22no—
87C2×C4×C59—22no—
88C2×C4×C59—41no—
89C2×C4×C59—41no—
90C2×C4×C59—42no—
91C2×C4×C59C4×D412yes—
92C2×C4×C59#12011yescharacteristic
93C2×C4×C59#12112yes—
94C2×C4×C59#12311yescharacteristic
95C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C59#13712yes—
97C2×C4×C59#13711yescharacteristic
98C2×Dic59—22no—
99C2×Dic59—41no—
100C2×Dic59—41no—
101C2×Dic59—42no—
102C2×Dic59—42no—
103C2×Dic59—42no—
104C2×Dic59—41no—
105C2×Dic59—42no—
106Dic118—24no—
107Dic118—42no—
108C8×C59—82no—
109C59⋊C8—44no—
110#9488—591nomaximal, Sylow
111C22×C59—21no—
112C22×C59—21no—
113C22×C59#38611yescharacteristic
114C22×C59#41711yescharacteristic
115C22×C59#41911yescharacteristic
116C4×C59—22no—
117C4×C59—21no—
118C4×C59—21no—
119C4×C59—42no—
120C4×C59—42no—
121C4×C59#46812yes—
122C4×C59#47012yes—
123Dic59—42no—
124Dic59—82no—
125Dic59—81no—
126#2983—591no—
127#2581—591no—
128#1801D5911yescharacteristic
129#1487—592no—
130#1286—592no—
131C2×C59—21no—
132C2×C59—21no—
133C2×C59#189111yescharacteristic
134C2×C59#189411yescharacteristic
135C2×C59#191211yescharacteristic
136C2×C4×Q8—591no—
137C2×C4×Q8C2×D5911yescharacteristic
138#446—1184no—
139#432—1182no—
140#431—592no—
141#423C2×D5912yes—
142C2×#101—21no—
143#400—592no—
144#395—592no—
145#391—1181no—
146#389—1181no—
147#387—1182no—
148#382—591no—
149#343—21no—
150Q8⋊C8—1184no—
151C59#948811yescharacteristic, complemented, Sylow
152C22×Q8—591no—
153C22×Q8C22⋊D5911yescharacteristic
154C2×M16—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—1182no—
158C4×Q8—1182no—
159C2×C4⋊C4—591no—
160C2×C4⋊C4—1182no—
161C2×C4⋊C4—1181no—
162C2×C4⋊C4—1181no—
163C2×C4⋊C4C4×D5912yes—
164C2×C4⋊C4C22⋊D5911yescharacteristic
165C2×C42—1181no—
166C2×C42D23612yes—
167C2×C42C22×D5911yescharacteristic
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—1184no—
171C4⋊C8—1182no—
172#101—42no—
173C22⋊C8—1182no—
174#97—1184no—
175C4×C8—594no—
176#95—592no—
177#95—1182no—
178#95—1182no—
179#95—1182no—
180#95—1181no—
181C2×Q8—22no—
182C2×Q8—21no—
183C2×Q8—22no—
184C2×Q8—21no—
185C2×Q8—1182no—
186C2×Q8—2362no—
187C22×C4—1181no—
188C22×C4—1182no—
189C22×C4—1181no—
190C22×C4(C2×D59)⋊C412yes—
191C22×C4(C2×D59)⋊C412yes—
192C22×C4C2×C4×D5911yescharacteristic
193C22×C4C2×D23611yescharacteristic
194C22×C4C2×C22⋊D5911yescharacteristic
195M16—42no—
196M16—42no—
197C2×C8—42no—
198C2×C8—1184no—
199C4⋊C4—24no—
200C4⋊C4—22no—
201C4⋊C4—1182no—
202C4⋊C4—2364no—
203C4⋊C4—2361no—
204C4⋊C4—2361no—
205C4⋊C4—2361no—
206C4⋊C4—2361no—
207C42—22no—
208C42—22no—
209C42—2362no—
210C42D4×D5912yes—
211C42#1174568712yes—
212C23C2×(C2×D59)⋊C411yescharacteristic
213Q8—42no—
214Q8—42no—
215Q8—1184no—
216Q8—2362no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—42no—
224C2×C4—1182no—
225C2×C4—2362no—
226C2×C4—2362no—
227C2×C4—2362no—
228C2×C4—2362no—
229C2×C4—2361no—
230C2×C4—2361no—
231C2×C4—2361no—
232C2×C4(C22⋊D59)⋊C412yes—
233C2×C4#1427369711yescharacteristic
234C2×C4#1427369912yes—
235C2×C4#1427371111yescharacteristic
236C2×C4#1427375311yescharacteristic
237C2×C4#1427375712yes—
238C2×C4#1427375811yescharacteristic
239C8—82no—
240C8—2364no—
241C22—21no—
242C22—21no—
243C22#1683727911yescharacteristic
244C22#1683740111yescharacteristic
245C22#1683741011yescharacteristic
246C4—22no—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—4722no—
252C4—4721no—
253C4—2362no—
254C4#1683764612yes—
255C4#1683765812yes—
256C2—21no—
257C2—21no—
258C2#2142451211yescharacteristic
259C2#2142452711yescharacteristic
260C2#2142463111yescharacteristic
subgroupG/H
centerC22#16837279
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#1801C2
socleC22×C59#386
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C42×C59C23
hypercenter#1801D59
hypocenterC59#9488
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
151. C59#9488semidirect

Maximal subgroups

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

subgroupindex
#214331812
#214294582
#214225242
#214219562
C59×#18012
#948859