Finite Group Databasebeta

← #28482280

Subgroups — #28482280

Order 15104.

378 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 115 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#21424092C211yesmaximal, characteristic
2#21422420C211yesmaximal, characteristic
3#21421999C212yesmaximal
4#21421954C212yesmaximal
5C59×#2356C211yesmaximal, characteristic
6#16837455C2212yes—
7#16837426C2212yes—
8C2×#14273646—21no—
9#16837171—21no—
10#16837128—22no—
11C2×C4×Q8×C59C2211yescharacteristic
12C59×#391—21no—
13C59×#387—22no—
14C59×#383C412yes—
15C59×#382C2211yescharacteristic
16C4×C4⋊C4×C59—21no—
17C4×C4⋊C4×C59C2211yescharacteristic
18C59×#376C412yes—
19C2×#11745664—22no—
20C4⋊(C59⋊C8)—22no—
21C4⋊(C59⋊C8)—22no—
22C4⋊(C59⋊C8)—22no—
23#14273646—42no—
24C22⋊(C59⋊C8)—22no—
25C4×C59⋊C8—22no—
26C22×Q8×C59D411yescharacteristic
27C4×Q8×C59—22no—
28C4×Q8×C59—22no—
29C2×C4⋊C4×C59—21no—
30C2×C4⋊C4×C59—21no—
31C2×C4⋊C4×C59D412yes—
32C2×C4⋊C4×C59D411yescharacteristic
33C2×C42×C59—22no—
34C2×C42×C59—21no—
35C2×C42×C59C2×C412yes—
36C2×C42×C59C2311yescharacteristic
37#95×C59—22no—
38#95×C59—22no—
39#95×C59—22no—
40#95×C59—21no—
41#95×C59C2×C411yescharacteristic
42#95×C59C2×C412yes—
43#95×C59C2×C411yescharacteristic
44C2×Q8×C59—22no—
45C2×Q8×C59—22no—
46C2×Q8×C59—21no—
47C2×Q8×C59—21no—
48C22×C4×C59—21no—
49C22×C4×C59—22no—
50C22×C4×C59—21no—
51C22×C4×C59C22⋊C412yes—
52C22×C4×C59C22⋊C412yes—
53C22×C4×C59C22×C411yescharacteristic
54C22×C4×C59C2×D411yescharacteristic
55C22×C4×C59C2×D411yescharacteristic
56C4⋊C4×C59—22no—
57C4⋊C4×C59—24no—
58C4⋊C4×C59—42no—
59C4⋊C4×C59—41no—
60C4⋊C4×C59—41no—
61C42×C59—22no—
62C42×C59—22no—
63C42×C59—44no—
64C42×C59—42no—
65C42×C59#4112yes—
66C42×C59#4112yes—
67#11745664—42no—
68#11745664—42no—
69C2×C59⋊C8—42no—
70C2×C59⋊C8—42no—
71C23×C59C2×C22⋊C411yescharacteristic
72Q8×C59—42no—
73Q8×C59—42no—
74C2×C4×C59—22no—
75C2×C4×C59—22no—
76C2×C4×C59—22no—
77C2×C4×C59—22no—
78C2×C4×C59—22no—
79C2×C4×C59—41no—
80C2×C4×C59—41no—
81C2×C4×C59—41no—
82C2×C4×C59—41no—
83C2×C4×C59—42no—
84C2×C4×C59—42no—
85C2×C4×C59—41no—
86C2×C4×C59—41no—
87C2×C4×C59—42no—
88C2×C4×C59—41no—
89C2×C4×C59—42no—
90C2×C4×C59#11711yescharacteristic
91C2×C4×C59#11711yescharacteristic
92C2×C4×C59#12312yes—
93C2×C4×C59#12312yes—
94C2×C4×C59#13512yes—
95C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
96C2×C4×C59#13711yescharacteristic
97C59⋊C8—82no—
98C59⋊C8—82no—
99#8769—591nomaximal, Sylow
100C22×C59—21no—
101C22×C59—21no—
102C22×C59#38111yescharacteristic
103C22×C59#41611yescharacteristic
104C22×C59#41711yescharacteristic
105C4×C59—22no—
106C4×C59—21no—
107C4×C59—21no—
108C4×C59—42no—
109C4×C59—42no—
110C4×C59—42no—
111C4×C59—81no—
112C4×C59—82no—
113C4×C59#48212yes—
114C4×C59#48512yes—
115#2356D5911yescharacteristic
116#1801—591no—
117#1461—591no—
118#1303—592no—
119#1286—592no—
120C2×C59—21no—
121C2×C59—21no—
122C2×C59#182511yescharacteristic
123C2×C59#182711yescharacteristic
124C2×C59#183011yescharacteristic
125C2×C4×Q8C2×D5911yescharacteristic
126#431—592no—
127#423—592no—
128C2×#101—1181no—
129#391—21no—
130#387—22no—
131#383Dic5912yes—
132#382C2×D5911yescharacteristic
133C4×C4⋊C4—21no—
134C4×C4⋊C4C2×D5911yescharacteristic
135#376Dic5912yes—
136#343—1181no—
137#324—1182no—
138C59#876911yescharacteristic, complemented, Sylow
139C22×Q8C22⋊D5911yescharacteristic
140C2×M16—1182no—
141C4×Q8—22no—
142C4×Q8—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4C22⋊D5912yes—
146C2×C4⋊C4C22⋊D5911yescharacteristic
147C2×C42—22no—
148C2×C42—21no—
149C2×C42C2×Dic5912yes—
150C2×C42C22×D5911yescharacteristic
151C4⋊C8—1182no—
152C4⋊C8—1182no—
153C4⋊C8—1182no—
154#101—2362no—
155C22⋊C8—1182no—
156C4×C8—1182no—
157#95—22no—
158#95—22no—
159#95—22no—
160#95—21no—
161#95C2×Dic5911yescharacteristic
162#95C2×Dic5912yes—
163#95C2×Dic5911yescharacteristic
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8—21no—
167C2×Q8—21no—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4—21no—
171C22×C4C22⋊Dic5912yes—
172C22×C4C22⋊Dic5912yes—
173C22×C4C22×Dic5911yescharacteristic
174C22×C4C2×C22⋊D5911yescharacteristic
175C22×C4C2×C22⋊D5911yescharacteristic
176M16—2362no—
177M16—2362no—
178C2×C8—2362no—
179C2×C8—2362no—
180C4⋊C4—22no—
181C4⋊C4—24no—
182C4⋊C4—42no—
183C4⋊C4—41no—
184C4⋊C4—41no—
185C42—22no—
186C42—22no—
187C42—44no—
188C42—42no—
189C42#1174568512yes—
190C42#1174568512yes—
191C23C2×C22⋊Dic5911yescharacteristic
192Q8—42no—
193Q8—42no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—41no—
209C2×C4—42no—
210C2×C4#1427368611yescharacteristic
211C2×C4#1427368611yescharacteristic
212C2×C4#1427371012yes—
213C2×C4#1427371012yes—
214C2×C4#1427374712yes—
215C2×C4#1427375211yescharacteristic
216C2×C4#1427375711yescharacteristic
217C8—4722no—
218C8—4722no—
219C22—21no—
220C22—21no—
221C22#1683726511yescharacteristic
222C22#1683739811yescharacteristic
223C22#1683740311yescharacteristic
224C4—22no—
225C4—21no—
226C4—21no—
227C4—42no—
228C4—42no—
229C4—42no—
230C4—81no—
231C4—82no—
232C4#1683772612yes—
233C4#1683774312yes—
234C2—21no—
235C2—21no—
236C2#2142419111yescharacteristic
237C2#2142420011yescharacteristic
238C2#2142421011yescharacteristic
subgroupG/H
centerC22#16837265
commutatorC22×C4×C59C22×C4
FrattiniC2×C42C22×D59
FittingC59×#2356C2
socleC22×C59#381
prime radicalC23×C59C2×C22⋊C4
cosocleC2×C42×C59C23
hypercenter#2356D59
hypocenterC59#8769
LCSG ▷ C22×C4×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2356

Complements

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

NH ≅ G/Ntype
138. C59#8769semidirect

Maximal subgroups

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

subgroupindex
#214240922
#214224202
#214219992
#214219542
C59×#23562
#876959