Finite Group Databasebeta

← #28476241

Subgroups — #28476241

Order 15104.

228 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 43 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#21422042C211yesmaximal, characteristic, complemented
2#21421979C211yesmaximal, characteristic, complemented
3C59×#2039C211yesmaximal, characteristic
4C2×C4⋊(C59⋊C8)—21no—
5#16837128—21no—
6C59×#537C411yescharacteristic
7C2×#104×C59—21no—
8C59×#417—21no—
9C59×#384C2211yescharacteristic
10C59×#343C411yescharacteristic
11C22×C59⋊C8—41no—
12C4⋊(C59⋊C8)—22no—
13C4⋊(C59⋊C8)—41no—
14C22⋊(C59⋊C8)—41no—
15C2×#41×C59—21no—
16C2×M16×C59—21no—
17C22⋊Q8×C59—41no—
18#121×C59—41no—
19#121×C59—41no—
20C2×C4⋊C4×C59—41no—
21C2×C4⋊C4×C59C2×C411yescharacteristic
22C2×C4⋊C4×C59D411yescharacteristic
23C2×C42×C59D411yescharacteristic
24#104×C59—41no—
25#104×C59—41no—
26#103×C59—41no—
27#102×C59—41no—
28#41×C59—41no—
29#41×C59—41no—
30#41×C59—41no—
31#41×C59—41no—
32C2×Q8×C59—21no—
33C2×D4×C59—21no—
34C2×D4×C59—41no—
35C2×D4×C59—41no—
36C22×C4×C59—21no—
37C22×C4×C59—21no—
38C22×C4×C59—21no—
39C22×C4×C59—41no—
40C22×C4×C59C22⋊C411yescharacteristic
41M16×C59—41no—
42M16×C59—41no—
43C2×C8×C59—41no—
44C4⋊C4×C59—41no—
45C4⋊C4×C59—41no—
46C4⋊C4×C59—41no—
47C4⋊C4×C59—81no—
48C4⋊C4×C59—81no—
49C22⋊C4×C59—42no—
50C42×C59—21no—
51C42×C59D811yescharacteristic
52C42×C59SD1611yescharacteristic
53C2×C59⋊C8—42no—
54C2×C59⋊C8—81no—
55C2×C59⋊C8—81no—
56C2×C59⋊C8—81no—
57C23×C59—41no—
58C23×C59#9911yescharacteristic
59Q8×C59—41no—
60Q8×C59—41no—
61D4×C59—41no—
62D4×C59—41no—
63D4×C59—81no—
64D4×C59—81no—
65D4×C59—161no—
66C2×C4×C59—21no—
67C2×C4×C59—21no—
68C2×C4×C59—21no—
69C2×C4×C59—21no—
70C2×C4×C59—21no—
71C2×C4×C59—21no—
72C2×C4×C59—21no—
73C2×C4×C59—21no—
74C2×C4×C59—41no—
75C2×C4×C59—41no—
76C2×C4×C59—41no—
77C2×C4×C59—81no—
78C2×C4×C59—81no—
79C2×C4×C59—81no—
80C2×C4×C59—81no—
81C2×C4×C59#10211yescharacteristic
82C2×C4×C59#10411yescharacteristic
83C8×C59—81no—
84C59⋊C8—82no—
85C59⋊C8—161no—
86#7417—591nomaximal, Sylow
87C22×C59—21no—
88C22×C59—21no—
89C22×C59—81no—
90C22×C59—81no—
91C22×C59—81no—
92C22×C59#32711yescharacteristic
93C22×C59AΣL(1,𝔽16)11yescharacteristic
94C22×C59#35511yescharacteristic
95C4×C59—21no—
96C4×C59—21no—
97C4×C59—21no—
98C4×C59—21no—
99C4×C59—21no—
100C4×C59—21no—
101C4×C59—81no—
102C4×C59—81no—
103C4×C59—81no—
104#2039D5911yescharacteristic
105#1325—591no—
106#1296—591no—
107C2×C59—21no—
108C2×C59—21no—
109C2×C59—161no—
110C2×C59#134511yescharacteristic
111C2×C59#140211yescharacteristic
112C2×C59#140511yescharacteristic
113#537Dic5911yescharacteristic
114C2×C4⋊C8—1181no—
115C2×#104—21no—
116#417—21no—
117#384C2×D5911yescharacteristic
118#343Dic5911yescharacteristic
119#324—1181no—
120C59#741711yescharacteristic, complemented, Sylow
121C2×#41—21no—
122C2×M16—21no—
123C22×C8—2361no—
124C22⋊Q8—41no—
125#121—41no—
126#121—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4C2×Dic5911yescharacteristic
129C2×C4⋊C4C22⋊D5911yescharacteristic
130C2×C42C22⋊D5911yescharacteristic
131C4⋊C8—1182no—
132C4⋊C8—2361no—
133#104—41no—
134#104—41no—
135#103—41no—
136#102—41no—
137C22⋊C8—2361no—
138#41—41no—
139#41—41no—
140#41—41no—
141#41—41no—
142C2×Q8—21no—
143C2×D4—21no—
144C2×D4—41no—
145C2×D4—41no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4C22⋊Dic5911yescharacteristic
151M16—41no—
152M16—41no—
153C2×C8—41no—
154C2×C8—4721no—
155C2×C8—4721no—
156C2×C8—4721no—
157C2×C8—2362no—
158C4⋊C4—41no—
159C4⋊C4—41no—
160C4⋊C4—41no—
161C4⋊C4—81no—
162C4⋊C4—81no—
163C22⋊C4—42no—
164C42—21no—
165C42#1174566911yescharacteristic
166C42#1174567011yescharacteristic
167C23—41no—
168C23#1427364211yescharacteristic
169Q8—41no—
170Q8—41no—
171D4—41no—
172D4—41no—
173D4—81no—
174D4—81no—
175D4—161no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—81no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—81no—
191C2×C4#1427364811yescharacteristic
192C2×C4#1427365411yescharacteristic
193C8—81no—
194C8—9441no—
195C8—4722no—
196C22—21no—
197C22—21no—
198C22—81no—
199C22—81no—
200C22—81no—
201C22#1683713611yescharacteristic
202C22#1683718911yescharacteristic
203C22#1683719911yescharacteristic
204C4—21no—
205C4—21no—
206C4—21no—
207C4—21no—
208C4—21no—
209C4—21no—
210C4—81no—
211C4—81no—
212C4—81no—
213C2—21no—
214C2—21no—
215C2—161no—
216C2#2142209911yescharacteristic
217C2#2142224111yescharacteristic
218C2#2142225011yescharacteristic
subgroupG/H
centerC22#16837189
commutatorC2×C4⋊C4×C59C2×C4
Frattini#384C2×D59
FittingC59×#2039C2
socleC22×C59AΣL(1,𝔽16)
prime radicalC59×#537C4
cosocleC59×#384C22
hypercenter#2039D59
hypocenterC59#7417
LCSG ▷ C2×C4⋊C4×C59 ▷ C22×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #384 ◁ #2039

Complements

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

NH ≅ G/Ntype
1. #21422042C2semidirect
2. #21421979C2semidirect
120. C59#7417semidirect

Maximal subgroups

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

subgroupindex
#214220422
#214219792
C59×#20392
#741759