Finite Group Databasebeta

← #28509035

Subgroups — #28509035

Order 15104.

326 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 59 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#21434440C211yesmaximal, characteristic
2#21425758C212yesmaximal
3#21422267C211yesmaximal, characteristic, complemented
4#21422266C212yesmaximal, complemented
5C59×#1412C211yesmaximal, characteristic, complemented
6#16837958C411yescharacteristic, complemented
7#16837950C2211yescharacteristic
8#16837772C411yescharacteristic, complemented
9#16837771C411yescharacteristic, complemented
10#16837766C2211yescharacteristic
11#16837760—22no—
12#16837395—22no—
13#16837374—22no—
14#97×D59C411yescharacteristic, complemented
15#16837196—22no—
16#16837195C2212yes—
17C59×#490C2211yescharacteristic
18C59×#353C2212yes—
19C2×#11745687D412yes—
20C2×C8×D59—21no—
21#14273727C2×C411yescharacteristic
22#14273726—21no—
23#14273719C2×C411yescharacteristic
24#14273714D412yes—
25#14273705—41no—
26#14273698—41no—
27#14273692—41no—
28D4×Dic59—41no—
29#14273685—21no—
30#14273683—42no—
31C42×D59C2×C411yescharacteristic
32#14273652—22no—
33#14273650—22no—
34#14273648—22no—
35#14273642—22no—
36#14273641—42no—
37#14273638C2×C411yescharacteristic
38#14273637—21no—
39#127×C59C2311yescharacteristic
40#125×C59C2×C411yescharacteristic
41#102×C59—22no—
42#99×C59—22no—
43#97×C59C2×C411yescharacteristic
44C2×D4×C59—41no—
45C2×D4×C59C2×D412yes—
46C2×C22⋊D59—22no—
47C22×Dic59—22no—
48#11745687—42no—
49#11745687—42no—
50C2×C4×D59—21no—
51C2×C4×D59C22⋊C411yescharacteristic
52C2×Dic118—41no—
53C2×Dic118C22⋊C412yes—
54C2×C8×C59—21no—
55C4⋊C4×C59—21no—
56C4⋊C4×C59—41no—
57C22⋊C4×C59—42no—
58C42×C59C22×C411yescharacteristic
59C22⋊Dic59—42no—
60C22⋊Dic59—42no—
61(C2×D59)⋊C4—41no—
62C4⋊Dic59—21no—
63C4⋊Dic59—41no—
64Dic59⋊C4—41no—
65Dic59⋊C4—41no—
66Dic59⋊C4—82no—
67C4×Dic59—21no—
68C4×Dic59—41no—
69C4×Dic59—82no—
70C4×Dic59C22⋊C411yescharacteristic
71C2×C59⋊C8—21no—
72C8×D59—41no—
73C8×D59—41no—
74C23×C59—22no—
75C22×D59#9911yescharacteristic
76D4×C59—42no—
77D4×C59—82no—
78C2×C4×C59—21no—
79C2×C4×C59—41no—
80C2×C4×C59—82no—
81C2×C4×C59C2×C22⋊C411yescharacteristic
82C22⋊D59—82no—
83C2×Dic59—21no—
84C2×Dic59—22no—
85C2×Dic59—42no—
86C2×Dic59—42no—
87C2×Dic59—41no—
88C2×Dic59—82no—
89C2×Dic59#9911yescharacteristic
90C4×D59—21no—
91C4×D59—21no—
92C4×D59—81no—
93Dic118—42no—
94Dic118—82no—
95C8×C59—41no—
96C59⋊C8—41no—
97#13594—591nomaximal, Sylow
98C22×C59—42no—
99C22×C59—82no—
100C22×C59C2×#9911yescharacteristic
101C2×D59—21no—
102C2×D59—41no—
103C4×C59—21no—
104C4×C59—41no—
105C4×C59—81no—
106C4×C59—162no—
107Dic59—41no—
108Dic59—42no—
109Dic59—81no—
110Dic59—162no—
111Dic59#35411yescharacteristic
112Dic59#35411yescharacteristic
113#3135—591no—
114#2126—592no—
115#1412—591no—
116#1412—592no—
117#1412D5911yescharacteristic, complemented
118C2×C59—21no—
119C2×C59—82no—
120C2×C59C2×#35411yescharacteristic
121D59—41no—
122#533—591no—
123#532—591no—
124#491—591no—
125#491—591no—
126#490—591no—
127#490C2×D5911yescharacteristic
128#489—1182no—
129#416—1182no—
130#410—1182no—
131C2×#97—591no—
132#354—1182no—
133#353—592no—
134#353C2×D5912yes—
135C59#1359411yescharacteristic, complemented, Sylow
136C2×#41—592no—
137C22×C8—1181no—
138C4⋊Q8—591no—
139C4⋊Q8—1181no—
140#127C22×D5911yescharacteristic
141#125—591no—
142#125C4×D5911yescharacteristic
143#124—592no—
144C22⋊Q8—2361no—
145#121—2361no—
146C4×Q8—2361no—
147C4×D4—2361no—
148#117—1181no—
149#117—2362no—
150C2×C42—591no—
151#103—1182no—
152#103—1182no—
153#102—22no—
154#102—1182no—
155#99—22no—
156#99—1182no—
157#99—2362no—
158#97—591no—
159#97—1181no—
160#97C4×D5911yescharacteristic
161#41—2362no—
162#41—2362no—
163C2×Q8—592no—
164C2×Q8—2361no—
165C2×D4—41no—
166C2×D4—1182no—
167C2×D4D4×D5912yes—
168C22×C4—591no—
169C22×C4—1181no—
170C22×C4—1182no—
171C2×C8—21no—
172C2×C8—1181no—
173C2×C8—2361no—
174C2×C8—2361no—
175C4⋊C4—21no—
176C4⋊C4—41no—
177C4⋊C4—1181no—
178C4⋊C4—4722no—
179C4⋊C4—2361no—
180C4⋊C4—2361no—
181C4⋊C4—2361no—
182C22⋊C4—42no—
183C22⋊C4—2361no—
184C22⋊C4—2362no—
185C22⋊C4—2362no—
186C42—591no—
187C42—1181no—
188C42—4722no—
189C42—2361no—
190C42C2×C4×D5911yescharacteristic
191C23—22no—
192C23—591no—
193Q8—4722no—
194Q8—2362no—
195D4—42no—
196D4—82no—
197D4—4722no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—82no—
201C2×C4—591no—
202C2×C4—1181no—
203C2×C4—1181no—
204C2×C4—1181no—
205C2×C4—1182no—
206C2×C4—4721no—
207C2×C4—4722no—
208C2×C4—2362no—
209C2×C4—2362no—
210C2×C4—2361no—
211C2×C4C22⋊C4×D5911yescharacteristic
212C8—41no—
213C8—2361no—
214C22—42no—
215C22—82no—
216C22—1181no—
217C22—2361no—
218C22#99×D5911yescharacteristic
219C4—21no—
220C4—41no—
221C4—81no—
222C4—162no—
223C4—591no—
224C4—591no—
225C4—9442no—
226C4—4721no—
227C4—2361no—
228C4—2362no—
229C2—21no—
230C2—82no—
231C2—2361no—
232C2#354×D5911yescharacteristic
subgroupG/H
centerC2#354×D59
commutatorC42×C59C22×C4
Frattini#127C22×D59
FittingC59×#1412C2
socleC2×C59C2×#354
prime radical#16837950C22
cosocle#127×C59C23
hypercenter#1412D59
hypocenterC59#13594
LCSG ▷ C42×C59 ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #1412

Complements

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

NH ≅ G/Ntype
3. #21422267C2semidirect
4. #21422266C2semidirect
5. C59×#1412C2semidirect
6. #16837958C4semidirect
8. #16837772C4semidirect
9. #16837771C4semidirect
14. #97×D59C4semidirect
117. #1412D59semidirect
135. C59#13594semidirect

Maximal subgroups

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

subgroupindex
#214344402
#214257582
#214222672
#214222662
C59×#14122
#1359459