Finite Group Databasebeta

← #28492760

Subgroups — #28492760

Order 15104.

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

normal subgroups only

This group has 408 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#21424249C211yesmaximal, 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
11C2×Dic59⋊C8—22no—
12#16837248C2211yescharacteristic
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—
31Dic59⋊C8—44no—
32#14273649—24no—
33C59⋊#95—21no—
34C59⋊#95D412yes—
35C22×C8×C59D412yes—
36C2×C42×C59C2311yescharacteristic
37C4⋊C8×C59—22no—
38C4⋊C8×C59—22no—
39C22⋊C8×C59—22no—
40#97×C59—22no—
41C22×C4×C59C2×D411yescharacteristic
42C22×C4×C59C2×D411yescharacteristic
43C22×C4×C59#4111yescharacteristic
44C2×C22⋊D59—42no—
45C22×Dic59—22no—
46#11745685—84no—
47C2×D236—22no—
48C2×D236—22no—
49C2×D236—22no—
50C2×C4×D59—42no—
51C2×Dic118—22no—
52C2×C8×C59—22no—
53C2×C8×C59—42no—
54C2×C8×C59—42no—
55C42×C59C2×D412yes—
56C42×C59#4112yes—
57(C2×D59)⋊C4—42no—
58(C2×D59)⋊C4—42no—
59C4⋊Dic59—22no—
60C4⋊Dic59—22no—
61C4×Dic59—22no—
62C4×Dic59—22no—
63C2×C59⋊C8—42no—
64C2×C59⋊C8—41no—
65D472—82no—
66C23×C59#12111yescharacteristic
67C22×D59—42no—
68C2×C4×C59—22no—
69C2×C4×C59—21no—
70C2×C4×C59—21no—
71C2×C4×C59—41no—
72C2×C4×C59#12011yescharacteristic
73C2×C4×C59#12112yes—
74C2×C4×C59#12312yes—
75C2×C4×C59#12411yescharacteristic
76C22⋊D59—84no—
77C2×Dic59—24no—
78C2×Dic59—42no—
79C2×Dic59—42no—
80C2×Dic59—42no—
81D236—42no—
82D236—42no—
83D236—82no—
84D236—82no—
85C4×D59—84no—
86Dic118—82no—
87C8×C59—42no—
88C8×C59—82no—
89C59⋊C8—81no—
90#10512—591nomaximal, Sylow
91C22×C59—21no—
92C22×C59—21no—
93C22×C59#39411yescharacteristic
94C22×C59#47212yes—
95C2×D59—82no—
96C2×D59—82no—
97C2×D59—82no—
98C4×C59—22no—
99C4×C59—21no—
100C4×C59—22no—
101C4×C59#45311yescharacteristic
102C4×C59#45611yescharacteristic
103Dic59—44no—
104Dic59—82no—
105#3245—591no—
106#1843—591no—
107#1551—592no—
108#1505—592no—
109#1290D5911yescharacteristic, complemented
110C2×C59—22no—
111C2×C59#203911yescharacteristic
112C2×C59#204511yescharacteristic
113C2×C59#205311yescharacteristic
114D59—162no—
115#532—592no—
116#474—1182no—
117#466—1182no—
118#459—1182no—
119#432C2×D5912yes—
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—1182no—
129C2×D8—1182no—
130C22×C8—1181no—
131C22×C8D23612yes—
132C4⋊Q8—592no—
133#127—592no—
134#124—1182no—
135#121—1182no—
136#121—1182no—
137C4×D4—1182no—
138C4×D4—1182no—
139C2×C42—1181no—
140C2×C42C22×D5911yescharacteristic
141#107—1182no—
142C4⋊C8—22no—
143C4⋊C8—22no—
144C4⋊C8—1182no—
145C4⋊C8—2364no—
146#102—1184no—
147C22⋊C8—22no—
148#97—22no—
149#95—592no—
150#95—1181no—
151#41—4724no—
152C2×Q8—1182no—
153C2×D4—1182no—
154C2×D4—1182no—
155C2×D4—1182no—
156C2×D4—2362no—
157C22×C4—1182no—
158C22×C4—2362no—
159C22×C4C2×D23611yescharacteristic
160C22×C4D4×D5911yescharacteristic
161C22×C4#1174568911yescharacteristic
162D8—4722no—
163C2×C8—22no—
164C2×C8—42no—
165C2×C8—42no—
166C2×C8—2362no—
167C2×C8—2361no—
168C4⋊C4—1182no—
169C4⋊C4—1182no—
170C22⋊C4—2362no—
171C22⋊C4—2362no—
172C42—1182no—
173C42—1182no—
174C42#1174568512yes—
175C42D4×D5912yes—
176C23—2362no—
177C23#1427370211yescharacteristic
178Q8—4722no—
179D4—4722no—
180D4—4722no—
181D4—4724no—
182D4—2362no—
183D4—2362no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—41no—
188C2×C4—1184no—
189C2×C4—4724no—
190C2×C4—2362no—
191C2×C4—2362no—
192C2×C4—2362no—
193C2×C4#1427369711yescharacteristic
194C2×C4#1427370012yes—
195C2×C4#1427371212yes—
196C2×C4#1427371511yescharacteristic
197C8—42no—
198C8—82no—
199C8—4721no—
200C22—21no—
201C22—21no—
202C22—4722no—
203C22—4722no—
204C22—4722no—
205C22#1683732011yescharacteristic
206C22#1683767312yes—
207C4—21no—
208C4—22no—
209C4—22no—
210C4—4722no—
211C4—2364no—
212C4#1683757211yescharacteristic
213C4#1683758711yescharacteristic
214C2—22no—
215C2—9442no—
216C2#2142532111yescharacteristic
217C2#2142535311yescharacteristic
218C2#2142540311yescharacteristic
subgroupG/H
centerC22#16837320
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1290C2
socleC22×C59#394
prime radical#21435406C2
hypercenter#1290D59
hypocenterC59#10512
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. #21424249C2semidirect
3. #21422792C2semidirect
4. #21422605C2semidirect
5. C59×#1290C2semidirect
109. #1290D59semidirect
127. C59#10512semidirect

Maximal subgroups

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

subgroupindex
#214354062
#214242492
#214227922
#214226052
C59×#12902
#1051259