Finite Group Databasebeta

← #28492761

Subgroups — #28492761

Order 15104.

388 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 81 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#21424215C211yesmaximal, characteristic, complemented
2#21422791C212yesmaximal, complemented
3#21422604C212yesmaximal, complemented
4#21421966C211yesmaximal, characteristic, complemented
5C59×#3245C211yesmaximal, characteristic
6#16837459C2212yes—
7C2×C4⋊(C59⋊C8)C2211yescharacteristic
8#16837248C2211yescharacteristic
9#16837156—22no—
10#16837151—22no—
11#16837146—22no—
12#16837136—24no—
13C59×#532C2212yes—
14C59×#474—22no—
15C59×#466—22no—
16C59×#459—22no—
17C2×C4⋊C8×C59C2211yescharacteristic
18C22×C59⋊C8—21no—
19C2×C4×Dic59—21no—
20C4⋊(C59⋊C8)—22no—
21C4⋊(C59⋊C8)—22no—
22C22⋊(C59⋊C8)—22no—
23#14273638—22no—
24C59⋊#95—21no—
25C59⋊#95D412yes—
26C2×#41×C59—22no—
27C2×D8×C59—22no—
28C22×C8×C59D412yes—
29C4⋊Q8×C59D412yes—
30#127×C59D412yes—
31#124×C59—22no—
32#121×C59—22no—
33#121×C59—22no—
34C4×D4×C59—22no—
35C4×D4×C59—22no—
36C2×C42×C59C2311yescharacteristic
37#107×C59—22no—
38C4⋊C8×C59—22no—
39#102×C59—24no—
40#41×C59—84no—
41C2×Q8×C59—22no—
42C2×D4×C59—22no—
43C2×D4×C59—22no—
44C2×D4×C59—22no—
45C2×D4×C59—42no—
46C22×C4×C59—42no—
47C22×C4×C59C2×D411yescharacteristic
48C22×C4×C59C2×D411yescharacteristic
49C22×C4×C59#4111yescharacteristic
50C22×Dic59—22no—
51D8×C59—82no—
52C2×C8×C59—22no—
53C2×C8×C59—42no—
54C4⋊C4×C59—22no—
55C4⋊C4×C59—22no—
56C22⋊C4×C59—42no—
57C22⋊C4×C59—42no—
58C42×C59C2×D412yes—
59C42×C59#4112yes—
60C4×Dic59—41no—
61C4×Dic59—41no—
62C2×C59⋊C8—42no—
63C2×C59⋊C8—41no—
64C2×C59⋊C8—42no—
65C23×C59—42no—
66C23×C59#12111yescharacteristic
67Q8×C59—82no—
68D4×C59—42no—
69D4×C59—42no—
70D4×C59—82no—
71D4×C59—82no—
72D4×C59—84no—
73C2×C4×C59—22no—
74C2×C4×C59—21no—
75C2×C4×C59—21no—
76C2×C4×C59—41no—
77C2×C4×C59—42no—
78C2×C4×C59—84no—
79C2×C4×C59#12011yescharacteristic
80C2×C4×C59#12112yes—
81C2×C4×C59#12312yes—
82C2×C4×C59#12411yescharacteristic
83C2×Dic59—42no—
84C2×Dic59—42no—
85C2×Dic59—42no—
86C8×C59—42no—
87C59⋊C8—82no—
88C59⋊C8—81no—
89#10513—591nomaximal, Sylow
90C22×C59—21no—
91C22×C59—21no—
92C22×C59—82no—
93C22×C59—82no—
94C22×C59—82no—
95C22×C59#39411yescharacteristic
96C22×C59#47212yes—
97C4×C59—22no—
98C4×C59—21no—
99C4×C59—22no—
100C4×C59—82no—
101C4×C59#45411yescharacteristic
102C4×C59#45511yescharacteristic
103Dic59—82no—
104#3245D5911yescharacteristic
105#1832—591no—
106#1551—592no—
107#1505—592no—
108#1290—591no—
109C2×C59—22no—
110C2×C59—162no—
111C2×C59#204011yescharacteristic
112C2×C59#204511yescharacteristic
113C2×C59#205311yescharacteristic
114#532C2×D5912yes—
115#474—22no—
116#466—22no—
117#459—22no—
118#432—592no—
119C2×C4⋊C8—591no—
120C2×C4⋊C8C2×D5911yescharacteristic
121#376—591no—
122#336—1182no—
123#333—1182no—
124#331—1182no—
125#327—1184no—
126C59#1051311yescharacteristic, complemented, Sylow
127C2×#41—22no—
128C2×D8—22no—
129C22×C8—1181no—
130C22×C8C22⋊D5912yes—
131C4⋊Q8C22⋊D5912yes—
132#127C22⋊D5912yes—
133#124—22no—
134#121—22no—
135#121—22no—
136C4×D4—22no—
137C4×D4—22no—
138C2×C42—1181no—
139C2×C42C22×D5911yescharacteristic
140#107—22no—
141C4⋊C8—22no—
142C4⋊C8—1182no—
143C4⋊C8—1182no—
144#102—24no—
145C22⋊C8—1182no—
146#97—1182no—
147#95—592no—
148#95—1181no—
149#41—84no—
150C2×Q8—22no—
151C2×D4—22no—
152C2×D4—22no—
153C2×D4—22no—
154C2×D4—42no—
155C22×C4—42no—
156C22×C4—1182no—
157C22×C4D4×D5911yescharacteristic
158C22×C4#1174568711yescharacteristic
159C22×C4C2×C22⋊D5911yescharacteristic
160D8—82no—
161C2×C8—22no—
162C2×C8—42no—
163C2×C8—2362no—
164C2×C8—2361no—
165C2×C8—2362no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C42—2361no—
171C42—2361no—
172C42#1174568712yes—
173C42C2×C22⋊D5912yes—
174C23—42no—
175C23#1427369811yescharacteristic
176Q8—82no—
177D4—42no—
178D4—42no—
179D4—82no—
180D4—82no—
181D4—84no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—84no—
188C2×C4—2362no—
189C2×C4—2362no—
190C2×C4—2362no—
191C2×C4#1427369611yescharacteristic
192C2×C4#1427369912yes—
193C2×C4#1427370812yes—
194C2×C4#1427371411yescharacteristic
195C8—42no—
196C8—4722no—
197C8—4721no—
198C22—21no—
199C22—21no—
200C22—82no—
201C22—82no—
202C22—82no—
203C22#1683731611yescharacteristic
204C22#1683766912yes—
205C4—21no—
206C4—22no—
207C4—22no—
208C4—82no—
209C4—4722no—
210C4#1683757511yescharacteristic
211C4#1683758011yescharacteristic
212C2—22no—
213C2—162no—
214C2#2142532211yescharacteristic
215C2#2142534911yescharacteristic
216C2#2142539911yescharacteristic
subgroupG/H
centerC22#16837316
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#3245C2
socleC22×C59#394
hypercenter#3245D59
hypocenterC59#10513
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. #21424215C2semidirect
2. #21422791C2semidirect
3. #21422604C2semidirect
4. #21421966C2semidirect
126. C59#10513semidirect

Maximal subgroups

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

subgroupindex
#214242152
#214227912
#214226042
#214219662
C59×#32452
#1051359