Finite Group Databasebeta

← #28477789

Subgroups — #28477789

Order 15104.

478 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 87 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#21430468C211yesmaximal, characteristic
2#21425624C211yesmaximal, characteristic
3#21425569C212yesmaximal, complemented
4#21421920C212yesmaximal, complemented
5C59×#1270C211yesmaximal, characteristic
6C2×#14273704—22no—
7C2×#14273661—22no—
8C2×#14273660—21no—
9C2×#14273648—21no—
10#16837325—22no—
11#16837322—21no—
12#16837300C2212yes—
13#16837282—21no—
14#16837272C2211yescharacteristic
15#16837165—22no—
16#16837157C2211yescharacteristic
17C59×#390C2211yescharacteristic
18C59×#336C2212yes—
19C22×Dic118—21no—
20C23×Dic59—21no—
21C22×C59⋊C8—21no—
22#14273704—44no—
23#14273704—44no—
24C2×C4⋊Dic59—21no—
25C2×C4⋊Dic59—22no—
26C2×C4⋊Dic59D412yes—
27C2×C4⋊Dic59D412yes—
28C2×Dic59⋊C4—22no—
29C2×Dic59⋊C4—22no—
30C2×C22⋊Dic59—22no—
31C2×C22⋊Dic59—21no—
32#14273661—44no—
33#14273660—42no—
34#14273648—42no—
35C59⋊#95—21no—
36C59⋊#95—22no—
37C22×D4×C59D411yescharacteristic
38C22×C8×C59—22no—
39C2×C4⋊C4×C59D411yescharacteristic
40C2×C22⋊C4×C59—22no—
41C2×C42×C59C2311yescharacteristic
42C24×C59—21no—
43C2×D4×C59—21no—
44C2×D4×C59—41no—
45C2×D4×C59—41no—
46C2×D4×C59Q1612yes—
47C22×C4×C59—21no—
48C22×C4×C59C2×D412yes—
49C22×C4×C59C2×D411yescharacteristic
50C22×Dic59—22no—
51C22×Dic59—22no—
52C22×Dic59—44no—
53C22×Dic59—41no—
54C22×Dic59—81no—
55C2×Dic118—44no—
56C2×Dic118—81no—
57C2×C8×C59—24no—
58C2×C8×C59—42no—
59C2×C8×C59—42no—
60C4⋊C4×C59—41no—
61C22⋊C4×C59—42no—
62C22⋊C4×C59—42no—
63C42×C59—41no—
64C22⋊Dic59—44no—
65C22⋊Dic59—81no—
66C4⋊Dic59—22no—
67C4⋊Dic59—42no—
68C4⋊Dic59—41no—
69C4⋊Dic59—41no—
70C4⋊Dic59—42no—
71C4⋊Dic59—42no—
72C4⋊Dic59D814yes—
73Dic59⋊C4—44no—
74Dic59⋊C4—42no—
75Dic59⋊C4—42no—
76C2×C59⋊C8—22no—
77C2×C59⋊C8—41no—
78C2×C59⋊C8—41no—
79C23×C59—22no—
80C23×C59—42no—
81C23×C59—41no—
82C23×C59—42no—
83C23×C59—41no—
84C23×C59#12011yescharacteristic
85D4×C59—42no—
86D4×C59—81no—
87C2×C4×C59—21no—
88C2×C4×C59—22no—
89C2×C4×C59—21no—
90C2×C4×C59—22no—
91C2×C4×C59—42no—
92C2×C4×C59—41no—
93C2×C4×C59C2×D812yes—
94C2×C4×C59C2×Q1611yescharacteristic
95C2×C4×C59AGL(1,ℤ/8)12yes—
96C2×C4×C59#13711yescharacteristic
97C2×Dic59—24no—
98C2×Dic59—42no—
99C2×Dic59—42no—
100C2×Dic59—42no—
101C2×Dic59—42no—
102C2×Dic59—42no—
103C2×Dic59—84no—
104C2×Dic59—82no—
105C2×Dic59—81no—
106Dic118—84no—
107C8×C59—44no—
108C59⋊C8—42no—
109#7797—591nomaximal, Sylow
110C22×C59—42no—
111C22×C59—44no—
112C22×C59—82no—
113C22×C59—81no—
114C22×C59—81no—
115C22×C59#44712yes—
116C22×C59#45111yescharacteristic
117C22×C59#45311yescharacteristic
118C22×C59#45612yes—
119C22×C59#45811yescharacteristic
120C4×C59—22no—
121C4×C59—22no—
122C4×C59—21no—
123C4×C59—21no—
124C4×C59—81no—
125Dic59—44no—
126Dic59—82no—
127Dic59—161no—
128#2683—591no—
129#2093—591no—
130#2083—592no—
131#1271—592no—
132#1270D5911yescharacteristic
133C2×C59—82no—
134C2×C59#161012yes—
135C2×C59#161411yescharacteristic
136C2×C59#161911yescharacteristic
137C2×C59#164912yes—
138C2×C59#166011yescharacteristic
139C2×C22⋊Q8—1182no—
140C2×#107—1182no—
141C2×#107—1181no—
142C2×#102—1181no—
143#396—1182no—
144#395—1181no—
145#390C2×D5911yescharacteristic
146#389—592no—
147#386—1181no—
148#384—591no—
149#340—1182no—
150#336—591no—
151#336C2×D5912yes—
152C59#779711yescharacteristic, complemented, Sylow
153C22×Q8—1181no—
154C22×D4D23611yescharacteristic
155C23×C4—1181no—
156C22×C8—22no—
157C22×C8—1181no—
158C22⋊Q8—2364no—
159C22⋊Q8—2364no—
160C2×C4⋊C4—592no—
161C2×C4⋊C4—592no—
162C2×C4⋊C4—1182no—
163C2×C4⋊C4—1182no—
164C2×C4⋊C4—1181no—
165C2×C4⋊C4—1182no—
166C2×C4⋊C4D23611yescharacteristic
167C2×C22⋊C4—22no—
168C2×C22⋊C4—1181no—
169C2×C22⋊C4—1182no—
170C2×C42C22×D5911yescharacteristic
171#107—2362no—
172#107—2364no—
173#102—2362no—
174#95—1181no—
175#95—1182no—
176C24—21no—
177C2×Q8—4721no—
178C2×Q8—2364no—
179C2×D4—21no—
180C2×D4—41no—
181C2×D4—41no—
182C2×D4Dic23612yes—
183C22×C4—21no—
184C22×C4—1182no—
185C22×C4—1182no—
186C22×C4—4721no—
187C22×C4—2364no—
188C22×C4—2361no—
189C22×C4C2×D23611yescharacteristic
190C22×C4D4×D5912yes—
191C2×C8—24no—
192C2×C8—42no—
193C2×C8—42no—
194C2×C8—1182no—
195C2×C8—2361no—
196C2×C8—2361no—
197C4⋊C4—41no—
198C4⋊C4—594no—
199C4⋊C4—1182no—
200C4⋊C4—2361no—
201C4⋊C4—2361no—
202C4⋊C4—2362no—
203C4⋊C4—2364no—
204C4⋊C4—2362no—
205C4⋊C4—2362no—
206C4⋊C4—2362no—
207C4⋊C4—2362no—
208C22⋊C4—42no—
209C22⋊C4—42no—
210C22⋊C4—4721no—
211C22⋊C4—2364no—
212C42—41no—
213C23—22no—
214C23—42no—
215C23—41no—
216C23—42no—
217C23—41no—
218C23#1427369711yescharacteristic
219Q8—4724no—
220D4—42no—
221D4—81no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—21no—
226C2×C4—42no—
227C2×C4—41no—
228C2×C4—1184no—
229C2×C4—4722no—
230C2×C4—4721no—
231C2×C4—4724no—
232C2×C4—2362no—
233C2×C4—2362no—
234C2×C4—2362no—
235C2×C4—2362no—
236C2×C4—2362no—
237C2×C4D8×D5912yes—
238C2×C4C2×Dic23611yescharacteristic
239C2×C4#1427375412yes—
240C2×C4#1427376011yescharacteristic
241C8—44no—
242C8—2362no—
243C22—42no—
244C22—44no—
245C22—81no—
246C22—81no—
247C22—82no—
248C22#1683753712yes—
249C22#1683756011yescharacteristic
250C22#1683757211yescharacteristic
251C22#1683758612yes—
252C22#1683759311yescharacteristic
253C4—21no—
254C4—21no—
255C4—22no—
256C4—22no—
257C4—81no—
258C4—9441no—
259C4—4722no—
260C4—2364no—
261C2—82no—
262C2#2142310012yes—
263C2#2142312311yescharacteristic
264C2#2142315111yescharacteristic
265C2#2142334812yes—
266C2#2142340811yescharacteristic
subgroupG/H
centerC23#14273697
commutator, cosocleC2×C42×C59C23
FrattiniC2×C42C22×D59
FittingC59×#1270C2
socleC23×C59#120
prime radicalC22×D4×C59D4
hypercenter#1270D59
hypocenterC59#7797
LCSG ▷ C2×C42×C59 ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
3. #21425569C2semidirect
4. #21421920C2semidirect
152. C59#7797semidirect

Maximal subgroups

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

subgroupindex
#214304682
#214256242
#214255692
#214219202
C59×#12702
#779759