Finite Group Databasebeta

← #28506894

Subgroups — #28506894

Order 15104.

438 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 115 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#21432792C211yesmaximal, characteristic
2#21425862C211yesmaximal, characteristic, complemented
3#21425723C212yesmaximal, complemented
4#21425706C212yesmaximal, complemented
5C59×#2110C211yesmaximal, characteristic, complemented
6C2×#14273733C2211yescharacteristic
7C22×C8×D59C2211yescharacteristic
8#16837914C411yescharacteristic
9C2×#14273667C2212yes—
10#16837475—22no—
11C8×D236—22no—
12#16837429C411yescharacteristic
13#16837370C411yescharacteristic
14#16837365C411yescharacteristic
15#16837223—22no—
16#16837185—24no—
17(C2×D59)⋊C16—24no—
18C2×M32×C59C2212yes—
19C59×#405C2211yescharacteristic
20C59×#347—22no—
21C2×#11745685C2×C411yescharacteristic
22C22×C4×D59C2×C411yescharacteristic
23#14273733—22no—
24#14273733—22no—
25C2×#11745664C2×C411yescharacteristic
26C2×#11745659—21no—
27C22×C59⋊C8C2×C411yescharacteristic
28C2×C8×D59—22no—
29C2×C8×D59—21no—
30C2×C8×D59—21no—
31C2×C8×D59—21no—
32C2×C8×D59—22no—
33#14273711—21no—
34#14273707—21no—
35#14273702—21no—
36#14273697—21no—
37C4×D236—22no—
38#14273667—22no—
39#14273667—22no—
40C2×C59⋊C16—22no—
41C4⋊(C59⋊C8)—22no—
42(C2×D59)⋊C8—22no—
43(C2×D59)⋊C8—22no—
44C2×M16×C59C2×C411yescharacteristic
45C22×C8×C59C2311yescharacteristic
46#117×C59C2×C411yescharacteristic
47M32×C59—22no—
48M32×C59—22no—
49C2×C16×C59D414yescomplemented
50#97×C59D412yes—
51C4×C8×C59D412yes—
52C23×D59M1611yescharacteristic
53C22×C4×C59C22×C411yescharacteristic
54C2×C22⋊D59—21no—
55C22×Dic59M1611yescharacteristic
56#11745685—22no—
57#11745685—22no—
58C2×D236—22no—
59C2×D236M1611yescharacteristic
60C2×C4×D59—21no—
61C2×C4×D59—21no—
62C2×C4×D59—22no—
63C2×C4×D59—21no—
64C2×C4×D59—22no—
65C2×Dic118M1611yescharacteristic
66M16×C59—22no—
67C2×C8×C59C22⋊C412yes—
68C2×C8×C59C2×D411yescharacteristic
69C2×C8×C59C2×D412yes—
70C2×C8×C59C2×D411yescharacteristic
71C2×C8×C59#4112yes—
72C4⋊C4×C59—21no—
73C22⋊C4×C59—21no—
74C42×C59C22⋊C412yes—
75(C2×D59)⋊C4—22no—
76(C2×D59)⋊C4—22no—
77C4⋊Dic59—22no—
78#11745664—22no—
79C2×C59⋊C8—21no—
80C2×C59⋊C8—21no—
81C2×C59⋊C8—22no—
82#11745659—42no—
83C8×D59—42no—
84C8×D59—41no—
85C8×D59—41no—
86C8×D59—41no—
87C8×D59—41no—
88C16×C59—24no—
89C59⋊C16—42no—
90C23×C59C2×M1611yescharacteristic
91C22×D59—21no—
92C22×D59—22no—
93C22×D59—21no—
94C22×D59—42no—
95C2×C4×C59—22no—
96C2×C4×C59C2×C22⋊C411yescharacteristic
97C2×C4×C59C4×D412yes—
98C2×C4×C59C2×M1611yescharacteristic
99C2×C4×C59#13112yes—
100C22⋊D59—42no—
101C2×Dic59—21no—
102C2×Dic59—21no—
103C2×Dic59—22no—
104D236—22no—
105D236—44no—
106C4×D59—41no—
107C4×D59—41no—
108C4×D59—41no—
109C4×D59—41no—
110C4×D59—42no—
111Dic118—22no—
112C8×C59—22no—
113C8×C59#12112yes—
114C8×C59#12411yescharacteristic
115C8×C59#12711yescharacteristic
116C59⋊C8—41no—
117C59⋊C8—41no—
118#13200—591nomaximal, Sylow
119C22×C59—41no—
120C22×C59#40711yescharacteristic
121C22×C59#43512yes—
122C2×D59—22no—
123C2×D59—24no—
124C2×D59—41no—
125C2×D59—42no—
126C2×D59—41no—
127C2×D59—42no—
128C2×D59—41no—
129C4×C59—42no—
130C4×C59#39011yescharacteristic
131C4×C59#40811yescharacteristic
132C4×C59#43512yes—
133Dic59—41no—
134Dic59—41no—
135#2929—591no—
136#2153—591no—
137#2117—592no—
138#2112—592no—
139#2110D5911yescharacteristic, complemented
140C2×C59—21no—
141C2×C59—22no—
142C2×C59#195511yescharacteristic
143D59—42no—
144D59—81no—
145C2×#131—591no—
146C23×C8—591no—
147#525—591no—
148C2×M32—592no—
149C2×M32C2×D5912yes—
150#435—1182no—
151C8×D4—1182no—
152#424—591no—
153#408—591no—
154#407—591no—
155#405C2×D5911yescharacteristic
156#364—1182no—
157#350—1184no—
158C22⋊C16—1184no—
159#347—22no—
160C59#1320011yescharacteristic, complemented, Sylow
161C2×#41—591no—
162C23×C4—591no—
163#131—1182no—
164#131—1182no—
165C2×M16—591no—
166C2×M16—1181no—
167C2×M16C4×D5911yescharacteristic
168C22×C8—591no—
169C22×C8—1182no—
170C22×C8—1181no—
171C22×C8—1181no—
172C22×C8—1182no—
173C22×C8—1181no—
174C22×C8C22×D5911yescharacteristic
175#123—1181no—
176C22⋊Q8—1181no—
177#121—1181no—
178#120—1181no—
179C4×D4—1182no—
180#117C4×D5911yescharacteristic
181M32—22no—
182M32—22no—
183M32—1182no—
184M32—1182no—
185C2×C16—1182no—
186C2×C16D23614yescomplemented
187C4⋊C8—1182no—
188C22⋊C8—1182no—
189C22⋊C8—1182no—
190#97C22⋊D5912yes—
191C4×C8D23612yes—
192C24—591no—
193#41—1182no—
194#41—1182no—
195C2×Q8—591no—
196C2×D4—591no—
197C2×D4—1182no—
198C2×D4—1181no—
199C22×C4—591no—
200C22×C4—1181no—
201C22×C4—1181no—
202C22×C4—1182no—
203C22×C4—1181no—
204C22×C4—1182no—
205C22×C4C2×C4×D5911yescharacteristic
206M16—22no—
207M16—1182no—
208M16—2362no—
209C2×C8—1181no—
210C2×C8—1181no—
211C2×C8—1182no—
212C2×C8—2362no—
213C2×C8—2361no—
214C2×C8—2361no—
215C2×C8—2361no—
216C2×C8—2361no—
217C2×C8(C2×D59)⋊C412yes—
218C2×C8C2×D23612yes—
219C2×C8C2×D23611yescharacteristic
220C2×C8#1174568512yes—
221C2×C8C2×C22⋊D5911yescharacteristic
222C4⋊C4—21no—
223C4⋊C4—1182no—
224C22⋊C4—21no—
225C22⋊C4—1182no—
226C22⋊C4—1182no—
227C42(C2×D59)⋊C412yes—
228C16—24no—
229C16—2362no—
230C23—1182no—
231C23—1181no—
232C23—1181no—
233C23—2362no—
234C23M16×D5911yescharacteristic
235Q8—1182no—
236D4—1182no—
237D4—2364no—
238D4—2362no—
239C2×C4—22no—
240C2×C4—1182no—
241C2×C4—1181no—
242C2×C4—1181no—
243C2×C4—2362no—
244C2×C4—2361no—
245C2×C4—2361no—
246C2×C4—2361no—
247C2×C4—2361no—
248C2×C4C2×(C2×D59)⋊C411yescharacteristic
249C2×C4C4×D23612yes—
250C2×C4M16×D5911yescharacteristic
251C2×C4#1427373512yes—
252C8—22no—
253C8—2361no—
254C8—2361no—
255C8#1427370112yes—
256C8#1427371511yescharacteristic
257C8#1427372411yescharacteristic
258C22—41no—
259C22—1182no—
260C22—1184no—
261C22—2362no—
262C22—2361no—
263C22—2362no—
264C22—2361no—
265C22—2361no—
266C22#1683736511yescharacteristic
267C22#1683747512yes—
268C4—42no—
269C4—2361no—
270C4—2361no—
271C4#1683730411yescharacteristic
272C4#1683737011yescharacteristic
273C4#1683747512yes—
274C2—21no—
275C2—22no—
276C2—4721no—
277C2—2362no—
278C2#2142489411yescharacteristic
subgroupG/H
centerC8#14273724
commutatorC22×C4×C59C22×C4
FrattiniC22×C8C22×D59
FittingC59×#2110C2
socleC2×C59#1955
prime radical#16837914C4
cosocleC22×C8×C59C23
hypercenter#2110D59
hypocenterC59#13200
LCSG ▷ C22×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2110

Complements

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

NH ≅ G/Ntype
2. #21425862C2semidirect
3. #21425723C2semidirect
4. #21425706C2semidirect
5. C59×#2110C2semidirect
49. C2×C16×C59D4semidirect
139. #2110D59semidirect
160. C59#13200semidirect
186. C2×C16D236semidirect

Maximal subgroups

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

subgroupindex
#214327922
#214258622
#214257232
#214257062
C59×#21102
#1320059