Finite Group Databasebeta

← #28486109

Subgroups — #28486109

Order 15104.

450 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 119 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#21429308C211yesmaximal, characteristic
2#21424092C211yesmaximal, characteristic
3#21422780C212yesmaximal
4#21421998C212yesmaximal
5C59×#2983C211yesmaximal, characteristic
6C2×#14273727C2211yescharacteristic
7#16837426C2212yes—
8C2×#14273646—21no—
9#16837341C414yes—
10#16837309—21no—
11#16837308—22no—
12#16837267—21no—
13C4×C4⋊Dic59C2211yescharacteristic
14#16837171—21no—
15#16837149—24no—
16C2×C4×Q8×C59C2211yescharacteristic
17C59×#446—24no—
18C59×#432—22no—
19C59×#431C2212yes—
20C22×Dic118—21no—
21C2×#11745664—22no—
22#14273727—22no—
23#14273727—22no—
24C2×C4⋊Dic59—21no—
25C2×C4⋊Dic59—21no—
26C2×C4⋊Dic59C2×C412yes—
27C2×C4×Dic59—21no—
28C4⋊(C59⋊C8)—22no—
29C4⋊(C59⋊C8)—22no—
30#14273646—42no—
31C59⋊#95—24no—
32C59⋊#95—24no—
33C59⋊#95C2×C412yes—
34C22×Q8×C59D411yescharacteristic
35C4×Q8×C59—22no—
36C4×Q8×C59—22no—
37C2×C4⋊C4×C59C2×C412yes—
38C2×C4⋊C4×C59D411yescharacteristic
39C2×C42×C59D412yes—
40C2×C42×C59C2311yescharacteristic
41C4⋊C8×C59—24no—
42C4⋊C8×C59D414yes—
43C22⋊C8×C59—22no—
44#97×C59—24no—
45C4×C8×C59—22no—
46C2×Q8×C59—22no—
47C2×Q8×C59—21no—
48C2×Q8×C59—21no—
49C2×Q8×C59—22no—
50C22×C4×C59C22⋊C412yes—
51C22×C4×C59C22⋊C412yes—
52C22×C4×C59C22×C411yescharacteristic
53C22×C4×C59C2×D411yescharacteristic
54C22×C4×C59C2×D411yescharacteristic
55C22×Dic59—22no—
56C22×Dic59—22no—
57C2×Dic118—44no—
58C2×Dic118—41no—
59C2×Dic118—41no—
60C2×C8×C59—24no—
61C4⋊C4×C59—24no—
62C4⋊C4×C59—22no—
63C42×C59—22no—
64C42×C59—22no—
65C42×C59C2×D412yes—
66C42×C59#4112yes—
67C4⋊Dic59—24no—
68C4⋊Dic59—42no—
69C4⋊Dic59—42no—
70C4×Dic59—41no—
71C4×Dic59—41no—
72#11745664—42no—
73#11745664—42no—
74C2×C59⋊C8—42no—
75C23×C59C2×C22⋊C411yescharacteristic
76Q8×C59—42no—
77Q8×C59—42no—
78C2×C4×C59—22no—
79C2×C4×C59—22no—
80C2×C4×C59—22no—
81C2×C4×C59—22no—
82C2×C4×C59—22no—
83C2×C4×C59—41no—
84C2×C4×C59—42no—
85C2×C4×C59C4×D412yes—
86C2×C4×C59#12011yescharacteristic
87C2×C4×C59#12112yes—
88C2×C4×C59#12311yescharacteristic
89C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
90C2×C4×C59#13712yes—
91C2×C4×C59#13711yescharacteristic
92C2×Dic59—42no—
93C2×Dic59—44no—
94C2×Dic59—42no—
95C2×Dic59—42no—
96C2×Dic59—42no—
97Dic118—82no—
98Dic118—82no—
99C8×C59—44no—
100C59⋊C8—82no—
101#9404—591nomaximal, Sylow
102C22×C59—21no—
103C22×C59—21no—
104C22×C59#38611yescharacteristic
105C22×C59#41711yescharacteristic
106C22×C59#41911yescharacteristic
107C4×C59—21no—
108C4×C59—22no—
109C4×C59—42no—
110C4×C59—42no—
111C4×C59#45412yes—
112C4×C59#46812yes—
113C4×C59#47012yes—
114Dic59—82no—
115Dic59—82no—
116#2983D5911yescharacteristic
117#2568—591no—
118#1801—591no—
119#1549—592no—
120#1303—592no—
121C2×C59—21no—
122C2×C59—21no—
123C2×C59#188811yescharacteristic
124C2×C59#189411yescharacteristic
125C2×C59#191511yescharacteristic
126C2×C4⋊Q8—591no—
127C2×C4×Q8C2×D5911yescharacteristic
128#446—24no—
129#432—22no—
130#431C2×D5912yes—
131#423—592no—
132C2×#101—1181no—
133#400—594no—
134#391—1181no—
135#391—1182no—
136#382—1181no—
137C4×C4⋊C4—591no—
138#343—1181no—
139#332—1184no—
140C59#940411yescharacteristic, complemented, Sylow
141C22×Q8—1181no—
142C22×Q8D23611yescharacteristic
143C2×M16—1182no—
144C4⋊Q8—1182no—
145C4⋊Q8—1182no—
146C4×Q8—22no—
147C4×Q8—22no—
148C2×C4⋊C4—592no—
149C2×C4⋊C4—1181no—
150C2×C4⋊C4—1181no—
151C2×C4⋊C4C4×D5912yes—
152C2×C4⋊C4D23611yescharacteristic
153C2×C42—1181no—
154C2×C42C22⋊D5912yes—
155C2×C42C22×D5911yescharacteristic
156C4⋊C8—24no—
157C4⋊C8—1182no—
158C4⋊C8—1182no—
159C4⋊C8C22⋊D5914yes—
160#101—2362no—
161C22⋊C8—22no—
162#97—24no—
163C4×C8—22no—
164#95—592no—
165#95—1184no—
166#95—1184no—
167C2×Q8—22no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—22no—
171C2×Q8—2364no—
172C2×Q8—2361no—
173C2×Q8—2361no—
174C22×C4—1182no—
175C22×C4—1182no—
176C22×C4(C2×D59)⋊C412yes—
177C22×C4(C2×D59)⋊C412yes—
178C22×C4C2×C4×D5911yescharacteristic
179C22×C4C2×D23611yescharacteristic
180C22×C4C2×C22⋊D5911yescharacteristic
181M16—2362no—
182M16—2362no—
183C2×C8—24no—
184C2×C8—2362no—
185C4⋊C4—24no—
186C4⋊C4—22no—
187C4⋊C4—1184no—
188C4⋊C4—2362no—
189C4⋊C4—2362no—
190C42—22no—
191C42—22no—
192C42—2361no—
193C42—2361no—
194C42#1174568512yes—
195C42C2×C22⋊D5912yes—
196C23C2×(C2×D59)⋊C411yescharacteristic
197Q8—42no—
198Q8—42no—
199Q8—4722no—
200Q8—4722no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—2362no—
209C2×C4—2362no—
210C2×C4—2364no—
211C2×C4—2362no—
212C2×C4—2362no—
213C2×C4C4×C22⋊D5912yes—
214C2×C4#1427369611yescharacteristic
215C2×C4#1427370112yes—
216C2×C4#1427371011yescharacteristic
217C2×C4#1427375511yescharacteristic
218C2×C4#1427376012yes—
219C2×C4#1427376011yescharacteristic
220C8—44no—
221C8—4722no—
222C22—21no—
223C22—21no—
224C22#1683728111yescharacteristic
225C22#1683740411yescharacteristic
226C22#1683741211yescharacteristic
227C4—22no—
228C4—21no—
229C4—42no—
230C4—42no—
231C4—4722no—
232C4—4722no—
233C4#1683757512yes—
234C4#1683764812yes—
235C4#1683766012yes—
236C2—21no—
237C2—21no—
238C2#2142449911yescharacteristic
239C2#2142452911yescharacteristic
240C2#2142464811yescharacteristic
subgroupG/H
centerC22#16837281
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2983C2
socleC22×C59#386
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C42×C59C23
hypercenter#2983D59
hypocenterC59#9404
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
140. C59#9404semidirect

Maximal subgroups

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

subgroupindex
#214293082
#214240922
#214227802
#214219982
C59×#29832
#940459