Finite Group Databasebeta

← #28499881

Subgroups — #28499881

Order 15104.

326 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 61 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#21434469C211yesmaximal, characteristic
2#21425906C212yesmaximal, complemented
3#21422129C211yesmaximal, characteristic, complemented
4#21422128C212yesmaximal, complemented
5C59×#1355C211yesmaximal, characteristic, complemented
6#16837958C411yescharacteristic
7#127×D59C411yescharacteristic
8#16837768C411yescharacteristic
9#16837767C2211yescharacteristic
10#16837766C411yescharacteristic
11#16837747—22no—
12#16837405—22no—
13#16837354C2211yescharacteristic
14#16837222—22no—
15#16837183—22no—
16#16837180C2212yescomplemented
17C59×#490C2211yescharacteristic
18C59×#347C2212yescomplemented
19C2×D4×D59—21no—
20C2×D4×D59—41no—
21C2×#11745659D412yes—
22#14273726—21no—
23#14273724C2×C411yescharacteristic
24#14273723—21no—
25#14273723—41no—
26#14273719C2×C411yescharacteristic
27#14273705—41no—
28#14273692—41no—
29#14273685—21no—
30C42×D59C2×C411yescharacteristic
31#14273667—22no—
32#14273649—22no—
33#14273648—22no—
34#14273647—22no—
35#14273638C2×C411yescharacteristic
36C8×Dic59D412yes—
37#127×C59C2×C411yescharacteristic
38#125×C59C2×C411yescharacteristic
39M32×C59—22no—
40#102×C59—22no—
41#97×C59C2311yescharacteristic
42C23×D59—41no—
43C2×D4×C59—21no—
44C2×D4×C59—41no—
45C2×C22⋊D59—42no—
46C2×C22⋊D59—41no—
47D4×D59—44no—
48D4×D59—84no—
49C2×D236—21no—
50C2×D236—41no—
51C2×C4×D59—21no—
52C2×C4×D59C22⋊C411yescharacteristic
53C2×Dic118—41no—
54C2×C8×C59C2×D412yes—
55C4⋊C4×C59—21no—
56C4⋊C4×C59—41no—
57C42×C59C22×C411yescharacteristic
58(C2×D59)⋊C4—41no—
59C4⋊Dic59—21no—
60Dic59⋊C4—41no—
61Dic59⋊C4—41no—
62C4×Dic59—21no—
63C4×Dic59—41no—
64C4×Dic59C22⋊C411yescharacteristic
65C2×C59⋊C8C22⋊C412yes—
66#11745659—42no—
67C16×C59—42no—
68C59⋊C16—42no—
69C23×C59—41no—
70C22×D59—41no—
71C22×D59—82no—
72C22×D59—84no—
73C22×D59#10011yescharacteristic
74D4×C59—42no—
75D4×C59—82no—
76C2×C4×C59—21no—
77C2×C4×C59—41no—
78C2×C4×C59C2×C22⋊C411yescharacteristic
79C22⋊D59—82no—
80C22⋊D59—84no—
81C2×Dic59—21no—
82C2×Dic59—41no—
83C2×Dic59#10011yescharacteristic
84D236—42no—
85D236—82no—
86C4×D59—22no—
87C4×D59—42no—
88Dic118—82no—
89C8×C59—22no—
90C59⋊C8—22no—
91#11915—591nomaximal, Sylow
92C22×C59—82no—
93C22×C59—81no—
94C22×C59C2×#10011yescharacteristic
95C2×D59—21no—
96C2×D59—41no—
97C2×D59—82no—
98C2×D59—81no—
99C2×D59—164no—
100C4×C59—41no—
101C4×C59—81no—
102C4×C59#42112yes—
103Dic59—22no—
104Dic59—21no—
105Dic59—81no—
106#3138—591no—
107#2165—592no—
108#1355—591no—
109#1355—592no—
110#1355D5911yescharacteristic, complemented
111C2×C59—21no—
112C2×C59—161no—
113C2×C59#153311yescharacteristic
114D59—41no—
115D59—161no—
116#533—591no—
117C2×#127—591no—
118#490—591no—
119#490—591no—
120#490—591no—
121#490C2×D5911yescharacteristic
122#486—1182no—
123#418—1182no—
124C4×M16—591no—
125#364—1182no—
126#349—1182no—
127#347—592no—
128#347C2×D5912yescomplemented
129C59#1191511yescharacteristic, complemented, Sylow
130C22×D4—1181no—
131C22×D4—2361no—
132C2×M16—592no—
133C4⋊Q8—1181no—
134#127—591no—
135#127—1181no—
136#127—2361no—
137#127C4×D5911yescharacteristic
138#125—591no—
139#125C4×D5911yescharacteristic
140C22⋊Q8—2361no—
141C4×Q8—2361no—
142#117—1181no—
143C2×C42—591no—
144M32—22no—
145M32—1182no—
146#102—22no—
147#102—1182no—
148#102—1182no—
149#102—1182no—
150#97—591no—
151#97C22×D5911yescharacteristic
152C4×C8—592no—
153C24—2361no—
154C2×Q8—2361no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—1181no—
158C2×D4—4724no—
159C2×D4—2361no—
160C2×D4—2361no—
161C2×D4—2364no—
162C2×D4—2362no—
163C22×C4—591no—
164C22×C4—1181no—
165M16—2362no—
166C2×C8—592no—
167C2×C8D4×D5912yes—
168C4⋊C4—21no—
169C4⋊C4—41no—
170C4⋊C4—1181no—
171C4⋊C4—2361no—
172C4⋊C4—2361no—
173C22⋊C4—2361no—
174C42—591no—
175C42—1181no—
176C42—2361no—
177C42C2×C4×D5911yescharacteristic
178C16—42no—
179C16—2362no—
180C23—41no—
181C23—591no—
182C23—4722no—
183C23—4724no—
184C23—2361no—
185Q8—4722no—
186D4—42no—
187D4—82no—
188D4—4722no—
189D4—4724no—
190D4—4722no—
191D4—2362no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—591no—
195C2×C4—1181no—
196C2×C4—1182no—
197C2×C4—2362no—
198C2×C4—2361no—
199C2×C4C22⋊C4×D5911yescharacteristic
200C8—22no—
201C8—1182no—
202C22—82no—
203C22—81no—
204C22—1181no—
205C22—9444no—
206C22—4722no—
207C22—4721no—
208C22—2361no—
209C22#100×D5911yescharacteristic
210C4—41no—
211C4—81no—
212C4—1182no—
213C4—1181no—
214C4—4721no—
215C4#1683741712yes—
216C2—21no—
217C2—161no—
218C2—9441no—
219C2—2361no—
220C2#2142271811yescharacteristic
subgroupG/H
centerC2#21422718
commutatorC42×C59C22×C4
Frattini#97C22×D59
FittingC59×#1355C2
socleC2×C59#1533
prime radical#127×D59C4
cosocle#97×C59C23
hypercenter#1355D59
hypocenterC59#11915
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C2×C4 ◁ #97 ◁ #1355

Complements

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

NH ≅ G/Ntype
2. #21425906C2semidirect
3. #21422129C2semidirect
4. #21422128C2semidirect
5. C59×#1355C2semidirect
16. #16837180C22semidirect
18. C59×#347C22semidirect
110. #1355D59semidirect
128. #347C2×D59semidirect
129. C59#11915semidirect

Maximal subgroups

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

subgroupindex
#214344692
#214259062
#214221292
#214221282
C59×#13552
#1191559