Finite Group Databasebeta

← #28492765

Subgroups — #28492765

Order 15104.

388 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 77 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#21435406C211yesmaximal, characteristic
2#21424215C211yesmaximal, characteristic, complemented
3#21422792C212yesmaximal, complemented
4#21422605C212yesmaximal, complemented
5C59×#1290C211yesmaximal, characteristic, complemented
6#16837952C2212yes—
7#16837683—22no—
8#16837638—22no—
9#16837598—22no—
10C2×C4⋊(C59⋊C8)C2211yescharacteristic
11#16837248C2211yescharacteristic
12#16837156—22no—
13#16837150—22no—
14#16837145—22no—
15#16837137—24no—
16C59×#432C2212yes—
17C2×C4⋊C8×C59C2211yescharacteristic
18C2×#11745685—22no—
19C2×D472—22no—
20C22×C59⋊C8—21no—
21#14273727D412yes—
22#14273724D412yes—
23#14273715—22no—
24#14273701—22no—
25#14273701—22no—
26C4×D236—22no—
27C4×D236—22no—
28C2×C4×Dic59—21no—
29#14273661—22no—
30C4⋊(C59⋊C8)—22no—
31#14273649—24no—
32C59⋊#95—21no—
33C59⋊#95D412yes—
34C22×C8×C59D412yes—
35C2×C42×C59C2311yescharacteristic
36C4⋊C8×C59—22no—
37C4⋊C8×C59—22no—
38C22⋊C8×C59—22no—
39#97×C59—22no—
40C22×C4×C59C2×D411yescharacteristic
41C22×C4×C59C2×D411yescharacteristic
42C22×C4×C59#4111yescharacteristic
43C2×C22⋊D59—42no—
44C22×Dic59—22no—
45#11745685—84no—
46C2×D236—22no—
47C2×D236—22no—
48C2×D236—22no—
49C2×C4×D59—42no—
50C2×Dic118—22no—
51C2×C8×C59—22no—
52C2×C8×C59—42no—
53C2×C8×C59—42no—
54C42×C59C2×D412yes—
55C42×C59#4112yes—
56(C2×D59)⋊C4—42no—
57(C2×D59)⋊C4—42no—
58C4⋊Dic59—22no—
59C4⋊Dic59—22no—
60C4×Dic59—41no—
61C4×Dic59—41no—
62C2×C59⋊C8—42no—
63C2×C59⋊C8—41no—
64D472—82no—
65C23×C59#12111yescharacteristic
66C22×D59—42no—
67C2×C4×C59—22no—
68C2×C4×C59—21no—
69C2×C4×C59—21no—
70C2×C4×C59—41no—
71C2×C4×C59#12011yescharacteristic
72C2×C4×C59#12112yes—
73C2×C4×C59#12312yes—
74C2×C4×C59#12411yescharacteristic
75C22⋊D59—84no—
76C2×Dic59—42no—
77C2×Dic59—42no—
78C2×Dic59—42no—
79C2×Dic59—42no—
80D236—42no—
81D236—42no—
82D236—82no—
83D236—82no—
84C4×D59—84no—
85Dic118—82no—
86C8×C59—42no—
87C8×C59—82no—
88C59⋊C8—81no—
89#10513—591nomaximal, Sylow
90C22×C59—21no—
91C22×C59—21no—
92C22×C59#39411yescharacteristic
93C22×C59#47212yes—
94C2×D59—82no—
95C2×D59—82no—
96C2×D59—82no—
97C4×C59—22no—
98C4×C59—21no—
99C4×C59—22no—
100C4×C59#45411yescharacteristic
101C4×C59#45511yescharacteristic
102Dic59—82no—
103Dic59—82no—
104#3245—591no—
105#1832—591no—
106#1551—592no—
107#1505—592no—
108#1290D5911yescharacteristic, complemented
109C2×C59—22no—
110C2×C59#204011yescharacteristic
111C2×C59#204511yescharacteristic
112C2×C59#205311yescharacteristic
113D59—162no—
114#532—592no—
115#474—1182no—
116#466—1182no—
117#459—1182no—
118#432C2×D5912yes—
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—1182no—
128C2×D8—1182no—
129C22×C8—1181no—
130C22×C8D23612yes—
131C4⋊Q8—592no—
132#127—592no—
133#124—1182no—
134#121—1182no—
135#121—1182no—
136C4×D4—1182no—
137C4×D4—1182no—
138C2×C42—1181no—
139C2×C42C22×D5911yescharacteristic
140#107—1182no—
141C4⋊C8—22no—
142C4⋊C8—22no—
143C4⋊C8—1182no—
144#102—1184no—
145C22⋊C8—22no—
146#97—22no—
147#95—592no—
148#95—1181no—
149#41—4724no—
150C2×Q8—1182no—
151C2×D4—1182no—
152C2×D4—1182no—
153C2×D4—1182no—
154C2×D4—2362no—
155C22×C4—1182no—
156C22×C4—2362no—
157C22×C4C2×D23611yescharacteristic
158C22×C4D4×D5911yescharacteristic
159C22×C4#1174568911yescharacteristic
160D8—4722no—
161C2×C8—22no—
162C2×C8—42no—
163C2×C8—42no—
164C2×C8—2362no—
165C2×C8—2361no—
166C4⋊C4—1182no—
167C4⋊C4—1182no—
168C22⋊C4—2362no—
169C22⋊C4—2362no—
170C42—2361no—
171C42—2361no—
172C42#1174568512yes—
173C42D4×D5912yes—
174C23—2362no—
175C23#1427370211yescharacteristic
176Q8—4722no—
177D4—4722no—
178D4—4722no—
179D4—4724no—
180D4—2362no—
181D4—2362no—
182C2×C4—22no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—41no—
186C2×C4—4724no—
187C2×C4—2362no—
188C2×C4—2362no—
189C2×C4—2362no—
190C2×C4—2362no—
191C2×C4#1427369711yescharacteristic
192C2×C4#1427370012yes—
193C2×C4#1427371212yes—
194C2×C4#1427371511yescharacteristic
195C8—42no—
196C8—82no—
197C8—4721no—
198C22—21no—
199C22—21no—
200C22—4722no—
201C22—4722no—
202C22—4722no—
203C22#1683732011yescharacteristic
204C22#1683767312yes—
205C4—21no—
206C4—22no—
207C4—22no—
208C4—4722no—
209C4—4722no—
210C4#1683757711yescharacteristic
211C4#1683758211yescharacteristic
212C2—22no—
213C2—9442no—
214C2#2142532611yescharacteristic
215C2#2142535311yescharacteristic
216C2#2142540311yescharacteristic
subgroupG/H
centerC22#16837320
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1290C2
socleC22×C59#394
prime radical#21435406C2
hypercenter#1290D59
hypocenterC59#10513
LCSG ▷ C2×C42×C59 ▷ C22×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1290

Complements

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

NH ≅ G/Ntype
2. #21424215C2semidirect
3. #21422792C2semidirect
4. #21422605C2semidirect
5. C59×#1290C2semidirect
108. #1290D59semidirect
126. C59#10513semidirect

Maximal subgroups

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

subgroupindex
#214354062
#214242152
#214227922
#214226052
C59×#12902
#1051359