Finite Group Databasebeta

← #28492756

Subgroups — #28492756

Order 15104.

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

normal subgroups only

This group has 408 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#21424249C211yesmaximal, characteristic, complemented
2#21422791C212yesmaximal, complemented
3#21422604C212yesmaximal, complemented
4#21421966C211yesmaximal, characteristic, complemented
5C59×#3245C211yesmaximal, characteristic
6#16837459C2212yes—
7C2×C4⋊(C59⋊C8)C2211yescharacteristic
8C2×Dic59⋊C8—22no—
9#16837248C2211yescharacteristic
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—
22Dic59⋊C8—44no—
23C22⋊(C59⋊C8)—22no—
24#14273638—22no—
25C59⋊#95—21no—
26C59⋊#95D412yes—
27C2×#41×C59—22no—
28C2×D8×C59—22no—
29C22×C8×C59D412yes—
30C4⋊Q8×C59D412yes—
31#127×C59D412yes—
32#124×C59—22no—
33#121×C59—22no—
34#121×C59—22no—
35C4×D4×C59—22no—
36C4×D4×C59—22no—
37C2×C42×C59C2311yescharacteristic
38#107×C59—22no—
39C4⋊C8×C59—22no—
40#102×C59—24no—
41#41×C59—84no—
42C2×Q8×C59—22no—
43C2×D4×C59—22no—
44C2×D4×C59—22no—
45C2×D4×C59—22no—
46C2×D4×C59—42no—
47C22×C4×C59—42no—
48C22×C4×C59C2×D411yescharacteristic
49C22×C4×C59C2×D411yescharacteristic
50C22×C4×C59#4111yescharacteristic
51C22×Dic59—22no—
52D8×C59—82no—
53C2×C8×C59—22no—
54C2×C8×C59—42no—
55C4⋊C4×C59—22no—
56C4⋊C4×C59—22no—
57C22⋊C4×C59—42no—
58C22⋊C4×C59—42no—
59C42×C59C2×D412yes—
60C42×C59#4112yes—
61C4×Dic59—22no—
62C4×Dic59—22no—
63C2×C59⋊C8—42no—
64C2×C59⋊C8—41no—
65C2×C59⋊C8—42no—
66C23×C59—42no—
67C23×C59#12111yescharacteristic
68Q8×C59—82no—
69D4×C59—42no—
70D4×C59—42no—
71D4×C59—82no—
72D4×C59—82no—
73D4×C59—84no—
74C2×C4×C59—22no—
75C2×C4×C59—21no—
76C2×C4×C59—21no—
77C2×C4×C59—41no—
78C2×C4×C59—42no—
79C2×C4×C59—84no—
80C2×C4×C59#12011yescharacteristic
81C2×C4×C59#12112yes—
82C2×C4×C59#12312yes—
83C2×C4×C59#12411yescharacteristic
84C2×Dic59—24no—
85C2×Dic59—42no—
86C2×Dic59—42no—
87C8×C59—42no—
88C59⋊C8—82no—
89C59⋊C8—81no—
90#10512—591nomaximal, Sylow
91C22×C59—21no—
92C22×C59—21no—
93C22×C59—82no—
94C22×C59—82no—
95C22×C59—82no—
96C22×C59#39411yescharacteristic
97C22×C59#47212yes—
98C4×C59—22no—
99C4×C59—21no—
100C4×C59—22no—
101C4×C59—82no—
102C4×C59#45311yescharacteristic
103C4×C59#45611yescharacteristic
104Dic59—44no—
105#3245D5911yescharacteristic
106#1843—591no—
107#1551—592no—
108#1505—592no—
109#1290—591no—
110C2×C59—22no—
111C2×C59—162no—
112C2×C59#203911yescharacteristic
113C2×C59#204511yescharacteristic
114C2×C59#205311yescharacteristic
115#532C2×D5912yes—
116#474—22no—
117#466—22no—
118#459—22no—
119#432—592no—
120C2×C4⋊C8—591no—
121C2×C4⋊C8—1182no—
122C2×C4⋊C8C2×D5911yescharacteristic
123#376—591no—
124#333—1182no—
125#331—1182no—
126#327—1184no—
127C59#1051211yescharacteristic, complemented, Sylow
128C2×#41—22no—
129C2×D8—22no—
130C22×C8—1181no—
131C22×C8C22⋊D5912yes—
132C4⋊Q8C22⋊D5912yes—
133#127C22⋊D5912yes—
134#124—22no—
135#121—22no—
136#121—22no—
137C4×D4—22no—
138C4×D4—22no—
139C2×C42—1181no—
140C2×C42C22×D5911yescharacteristic
141#107—22no—
142C4⋊C8—22no—
143C4⋊C8—1182no—
144C4⋊C8—1182no—
145C4⋊C8—2364no—
146#102—24no—
147C22⋊C8—1182no—
148#97—1182no—
149#95—592no—
150#95—1181no—
151#41—84no—
152C2×Q8—22no—
153C2×D4—22no—
154C2×D4—22no—
155C2×D4—22no—
156C2×D4—42no—
157C22×C4—42no—
158C22×C4—1182no—
159C22×C4D4×D5911yescharacteristic
160C22×C4#1174568711yescharacteristic
161C22×C4C2×C22⋊D5911yescharacteristic
162D8—82no—
163C2×C8—22no—
164C2×C8—42no—
165C2×C8—2362no—
166C2×C8—2361no—
167C2×C8—2362no—
168C4⋊C4—22no—
169C4⋊C4—22no—
170C22⋊C4—42no—
171C22⋊C4—42no—
172C42—1182no—
173C42—1182no—
174C42#1174568712yes—
175C42C2×C22⋊D5912yes—
176C23—42no—
177C23#1427369811yescharacteristic
178Q8—82no—
179D4—42no—
180D4—42no—
181D4—82no—
182D4—82no—
183D4—84no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—42no—
189C2×C4—84no—
190C2×C4—1184no—
191C2×C4—2362no—
192C2×C4—2362no—
193C2×C4#1427369611yescharacteristic
194C2×C4#1427369912yes—
195C2×C4#1427370812yes—
196C2×C4#1427371411yescharacteristic
197C8—42no—
198C8—4722no—
199C8—4721no—
200C22—21no—
201C22—21no—
202C22—82no—
203C22—82no—
204C22—82no—
205C22#1683731611yescharacteristic
206C22#1683766912yes—
207C4—21no—
208C4—22no—
209C4—22no—
210C4—82no—
211C4—2364no—
212C4#1683757011yescharacteristic
213C4#1683758511yescharacteristic
214C2—22no—
215C2—162no—
216C2#2142531711yescharacteristic
217C2#2142534911yescharacteristic
218C2#2142539911yescharacteristic
subgroupG/H
centerC22#16837316
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
Fitting, prime radicalC59×#3245C2
socleC22×C59#394
hypercenter#3245D59
hypocenterC59#10512
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. #21424249C2semidirect
2. #21422791C2semidirect
3. #21422604C2semidirect
4. #21421966C2semidirect
127. C59#10512semidirect

Maximal subgroups

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

subgroupindex
#214242492
#214227912
#214226042
#214219662
C59×#32452
#1051259