Finite Group Databasebeta

← #28498002

Subgroups — #28498002

Order 15104.

364 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 83 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#21432813C211yesmaximal, characteristic
2#21425715C212yesmaximal, complemented
3#21422047C211yesmaximal, characteristic, complemented
4#21422033C212yesmaximal, complemented
5C59×#1327C211yesmaximal, characteristic, complemented
6C2×#14273735C411yescharacteristic
7C2×#14273735C2211yescharacteristic
8#16837952C411yescharacteristic
9M32×D59—22no—
10#16837472—22no—
11#16837472—22no—
12#16837370C411yescharacteristic
13#16837370C2211yescharacteristic
14C2×#14273638C411yescharacteristic
15#16837184C2212yes—
16#16837183—22no—
17C59×#423C2211yescharacteristic
18C59×#349C2212yes—
19C2×#11745685C2×C411yescharacteristic
20C2×#11745685C2×C411yescharacteristic
21#14273735—42no—
22#14273735—42no—
23C2×#11745659—21no—
24C2×#11745659D412yes—
25C22×C59⋊C8C2×C411yescharacteristic
26C22×C59⋊C8C2×C411yescharacteristic
27C2×C8×D59—21no—
28C2×C8×D59D412yes—
29#14273727C811yescharacteristic
30#14273724C811yescharacteristic
31#14273715C811yescharacteristic
32#14273715C811yescharacteristic
33#14273701—22no—
34C4×D236—22no—
35#14273667—22no—
36#14273666—42no—
37C16×D59—42no—
38(C2×D59)⋊C8—22no—
39(C2×D59)⋊C8—22no—
40#14273638—21no—
41#14273638—21no—
42C2×M16×C59C2×C411yescharacteristic
43C2×M16×C59C2311yescharacteristic
44C2×C42×C59C2×C411yescharacteristic
45M32×C59—22no—
46C4⋊C8×C59—22no—
47C22×C4×C59—21no—
48C22×C4×C59C22×C411yescharacteristic
49C2×C22⋊D59—21no—
50C2×C22⋊D59M1612yes—
51#11745685—41no—
52#11745685—41no—
53#11745685—41no—
54#11745685—41no—
55C2×D236—22no—
56C2×D236C2×C811yescharacteristic
57C2×D236C2×C811yescharacteristic
58C2×C4×D59—21no—
59C2×C4×D59C22⋊C412yes—
60C2×Dic118C2×C811yescharacteristic
61C2×Dic118C2×C811yescharacteristic
62M16×C59—41no—
63M16×C59—41no—
64C2×C8×C59—21no—
65C2×C8×C59C2×D412yes—
66C42×C59—21no—
67C42×C59C2×C811yescharacteristic
68C42×C59C2×C811yescharacteristic
69(C2×D59)⋊C4—22no—
70(C2×D59)⋊C4—22no—
71C4⋊Dic59—22no—
72C2×C59⋊C8—22no—
73C2×C59⋊C8—21no—
74C2×C59⋊C8—41no—
75C2×C59⋊C8C22⋊C411yescharacteristic
76C2×C59⋊C8C22⋊C411yescharacteristic
77#11745659—42no—
78#11745659—42no—
79C8×D59—22no—
80C8×D59—22no—
81C8×D59—42no—
82C16×C59—42no—
83C59⋊C16—42no—
84C23×C59C2×M1611yescharacteristic
85C22×D59—21no—
86C22×D59C22⋊C812yes—
87C2×C4×C59—21no—
88C2×C4×C59—21no—
89C2×C4×C59—21no—
90C2×C4×C59—21no—
91C2×C4×C59—81no—
92C2×C4×C59C2×C22⋊C411yescharacteristic
93C2×C4×C59C22×C811yescharacteristic
94C22⋊D59—42no—
95C22⋊D59—42no—
96C2×Dic59—21no—
97C2×Dic59C22⋊C812yes—
98D236—41no—
99D236—41no—
100D236—82no—
101C4×D59—22no—
102C4×D59—22no—
103C4×D59—41no—
104C4×D59—41no—
105Dic118—41no—
106Dic118—41no—
107C8×C59—22no—
108C8×C59—41no—
109C59⋊C8—21no—
110C59⋊C8—41no—
111C59⋊C8#9912yes—
112#11486—591nomaximal, Sylow
113C22×C59—21no—
114C22×C59—41no—
115C22×C59C2×C22⋊C811yescharacteristic
116C2×D59—22no—
117C2×D59—42no—
118C2×D59—42no—
119C2×D59—41no—
120C4×C59—21no—
121C4×C59—41no—
122C4×C59—41no—
123C4×C59C2×#9911yescharacteristic
124C4×C59#41311yescharacteristic
125Dic59—22no—
126Dic59—41no—
127#2932—591no—
128#2115—592no—
129#1327—591no—
130#1327D5911yescharacteristic, complemented
131#1321—592no—
132C2×C59—21no—
133C2×C59—41no—
134C2×C59#145911yescharacteristic
135D59—42no—
136D59—81no—
137C2×#131—591no—
138C2×#131—591no—
139#532—591no—
140C2×M32—1182no—
141#435—1182no—
142#435—1182no—
143#423C2×D5911yescharacteristic
144#408—591no—
145#408—591no—
146C2×#97—591no—
147#349—592no—
148#349—1182no—
149#349C2×D5912yes—
150C59#1148611yescharacteristic, complemented, Sylow
151C2×#41—591no—
152C2×#41—591no—
153#131—2362no—
154#131—2362no—
155C2×M16—592no—
156C2×M16—1181no—
157C2×M16C4×D5911yescharacteristic
158C2×M16C22×D5911yescharacteristic
159C22×C8—591no—
160C22×C8—592no—
161C22×C8—591no—
162C22×C8—1181no—
163C4⋊Q8—591no—
164#127—591no—
165#124—591no—
166#124—591no—
167#121—1182no—
168C4×D4—1182no—
169C2×C42C4×D5911yescharacteristic
170M32—22no—
171M32—1182no—
172M32—2362no—
173C2×C16—2362no—
174C4⋊C8—22no—
175C22⋊C8—1182no—
176C22⋊C8—1182no—
177#97—1181no—
178#97—1181no—
179#41—2361no—
180#41—2361no—
181#41—2361no—
182#41—2361no—
183C2×Q8—591no—
184C2×Q8—591no—
185C2×D4—592no—
186C2×D4—591no—
187C2×D4—591no—
188C2×D4—1182no—
189C2×D4—1181no—
190C22×C4—21no—
191C22×C4—592no—
192C22×C4—1181no—
193C22×C4C2×C4×D5911yescharacteristic
194M16—41no—
195M16—41no—
196M16—2362no—
197M16—2362no—
198C2×C8—21no—
199C2×C8—591no—
200C2×C8—591no—
201C2×C8—1182no—
202C2×C8—1182no—
203C2×C8—1182no—
204C2×C8—1181no—
205C2×C8—2362no—
206C2×C8—2361no—
207C2×C8D4×D5912yes—
208C4⋊C4—1182no—
209C22⋊C4—1182no—
210C22⋊C4—1182no—
211C42—21no—
212C42C8×D5911yescharacteristic
213C42C8×D5911yescharacteristic
214C16—42no—
215C16—2362no—
216C23—592no—
217C23—1181no—
218C23M16×D5911yescharacteristic
219Q8—2361no—
220Q8—2361no—
221D4—4722no—
222D4—2361no—
223D4—2361no—
224D4—2362no—
225D4—2362no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—81no—
231C2×C4—592no—
232C2×C4—1182no—
233C2×C4—1182no—
234C2×C4—1181no—
235C2×C4—2361no—
236C2×C4—2361no—
237C2×C4C22⋊C4×D5911yescharacteristic
238C2×C4C2×C8×D5911yescharacteristic
239C8—22no—
240C8—41no—
241C8—592no—
242C8—1181no—
243C8—2361no—
244C22—21no—
245C22—41no—
246C22—1182no—
247C22—2362no—
248C22—2362no—
249C22—2361no—
250C22C22⋊C8×D5911yescharacteristic
251C4—21no—
252C4—41no—
253C4—41no—
254C4—1182no—
255C4—2361no—
256C4#99×D5911yescharacteristic
257C4#1683738511yescharacteristic
258C2—21no—
259C2—41no—
260C2—4721no—
261C2—2362no—
262C2#2142241311yescharacteristic
subgroupG/H
centerC4#99×D59
commutatorC2×C4×C59C22×C8
FrattiniC2×M16C22×D59
FittingC59×#1327C2
socleC2×C59#1459
prime radical#16837952C4
cosocleC2×M16×C59C23
hypercenter#1327D59
hypocenterC59#11486
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1327

Complements

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

NH ≅ G/Ntype
2. #21425715C2semidirect
3. #21422047C2semidirect
4. #21422033C2semidirect
5. C59×#1327C2semidirect
130. #1327D59semidirect
150. C59#11486semidirect

Maximal subgroups

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

subgroupindex
#214328132
#214257152
#214220472
#214220332
C59×#13272
#1148659