Finite Group Databasebeta

← #28498000

Subgroups — #28498000

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#21432811C211yesmaximal, characteristic
2#21425715C212yesmaximal, complemented
3#21422046C212yesmaximal, complemented
4#21422034C211yesmaximal, characteristic, complemented
5C59×#1321C211yesmaximal, characteristic, complemented
6C2×#14273735C2211yescharacteristic
7C2×#14273734C411yescharacteristic
8#16837948C411yescharacteristic
9M32×D59—22no—
10#16837476—22no—
11#16837470—22no—
12#16837425C2211yescharacteristic
13#16837368C411yescharacteristic
14C2×#14273637C411yescharacteristic
15#16837184C2212yes—
16#16837183—22no—
17C59×#408C2211yescharacteristic
18C59×#349C2212yes—
19C2×#11745685C2×C411yescharacteristic
20#14273735—42no—
21#14273734—42no—
22C2×#11745664—21no—
23C2×#11745664C2×C411yescharacteristic
24C2×#11745659D412yes—
25C22×C59⋊C8—21no—
26C22×C59⋊C8C2×C411yescharacteristic
27C2×C8×D59D412yes—
28#14273725C811yescharacteristic
29#14273723C811yescharacteristic
30#14273716C811yescharacteristic
31#14273714C811yescharacteristic
32#14273699—22no—
33C4×C22⋊D59—22no—
34C2×C4×Dic59C2×C411yescharacteristic
35#14273667—22no—
36#14273666—42no—
37C16×D59—42no—
38Dic59⋊C8—22no—
39(C2×D59)⋊C8—22no—
40#14273637—21no—
41#14273637—21no—
42C2×#41×C59C2×C411yescharacteristic
43C2×M16×C59C2311yescharacteristic
44C22×C8×C59C2×C411yescharacteristic
45M32×C59—22no—
46C22⋊C8×C59—22no—
47#41×C59—41no—
48#41×C59—41no—
49C2×Q8×C59C2×C811yescharacteristic
50C2×D4×C59—21no—
51C2×D4×C59C2×C811yescharacteristic
52C22×C4×C59—21no—
53C22×C4×C59C22×C411yescharacteristic
54C2×C22⋊D59—22no—
55C2×C22⋊D59M1612yes—
56C22×Dic59—21no—
57#11745685—41no—
58#11745685—41no—
59C2×D236C2×C811yescharacteristic
60C2×C4×D59C22⋊C412yes—
61C2×Dic118C2×C811yescharacteristic
62M16×C59—41no—
63C2×C8×C59—21no—
64C2×C8×C59—41no—
65C2×C8×C59C2×D412yes—
66C22⋊Dic59—22no—
67(C2×D59)⋊C4—22no—
68Dic59⋊C4—22no—
69C4×Dic59—21no—
70C4×Dic59C2×C811yescharacteristic
71C4×Dic59C2×C811yescharacteristic
72#11745664—42no—
73#11745664—41no—
74C2×C59⋊C8—22no—
75C2×C59⋊C8—21no—
76C2×C59⋊C8—42no—
77C2×C59⋊C8C22⋊C411yescharacteristic
78C2×C59⋊C8C22⋊C411yescharacteristic
79#11745659—42no—
80C8×D59—22no—
81C8×D59—22no—
82C16×C59—42no—
83C59⋊C16—42no—
84C23×C59—21no—
85C23×C59C2×M1611yescharacteristic
86C22×D59C22⋊C812yes—
87Q8×C59—41no—
88D4×C59—41no—
89D4×C59—42no—
90C2×C4×C59—21no—
91C2×C4×C59—21no—
92C2×C4×C59—21no—
93C2×C4×C59—41no—
94C2×C4×C59—41no—
95C2×C4×C59C2×C22⋊C411yescharacteristic
96C2×C4×C59C22×C811yescharacteristic
97C22⋊D59—42no—
98C22⋊D59—82no—
99C2×Dic59—21no—
100C2×Dic59—21no—
101C2×Dic59—81no—
102C2×Dic59C22⋊C812yes—
103D236—41no—
104C4×D59—22no—
105C4×D59—22no—
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×C59—42no—
116C22×C59—41no—
117C22×C59C2×C22⋊C811yescharacteristic
118C2×D59—22no—
119C2×D59—42no—
120C4×C59—21no—
121C4×C59—41no—
122C4×C59C2×#9911yescharacteristic
123C4×C59#41311yescharacteristic
124Dic59—22no—
125Dic59—41no—
126Dic59—41no—
127#2932—591no—
128#2115—592no—
129#1327—592no—
130#1321—591no—
131#1321D5911yescharacteristic, complemented
132C2×C59—21no—
133C2×C59—41no—
134C2×C59—81no—
135C2×C59#145911yescharacteristic
136D59—42no—
137C2×#131—591no—
138C2×#131—591no—
139#532—591no—
140C2×M32—1182no—
141#435—1182no—
142#435—1182no—
143#423—591no—
144#408—591no—
145#408C2×D5911yescharacteristic
146C2×#97—591no—
147#349—592no—
148#349—1182no—
149#349C2×D5912yes—
150C59#1148611yescharacteristic, complemented, Sylow
151C2×#41—591no—
152C2×#41C4×D5911yescharacteristic
153#131—2362no—
154#131—2362no—
155C2×M16—592no—
156C2×M16—591no—
157C2×M16—1181no—
158C2×M16C22×D5911yescharacteristic
159C22×C8—591no—
160C22×C8—592no—
161C22×C8—1181no—
162C22×C8C4×D5911yescharacteristic
163C4⋊Q8—591no—
164#127—591no—
165#124—591no—
166#124—591no—
167#121—1182no—
168C4×D4—1182no—
169C2×C42—591no—
170M32—22no—
171M32—1182no—
172M32—2362no—
173C2×C16—2362no—
174C4⋊C8—1182no—
175C22⋊C8—22no—
176C22⋊C8—1182no—
177#97—1181no—
178#97—1181no—
179#41—41no—
180#41—41no—
181#41—2361no—
182#41—2361no—
183C2×Q8—591no—
184C2×Q8C8×D5911yescharacteristic
185C2×D4—21no—
186C2×D4—592no—
187C2×D4—591no—
188C2×D4—1182no—
189C2×D4C8×D5911yescharacteristic
190C22×C4—21no—
191C22×C4—592no—
192C22×C4—1181no—
193C22×C4C2×C4×D5911yescharacteristic
194M16—41no—
195M16—2362no—
196M16—2362no—
197M16—2361no—
198C2×C8—21no—
199C2×C8—41no—
200C2×C8—591no—
201C2×C8—591no—
202C2×C8—1182no—
203C2×C8—1182no—
204C2×C8—1182no—
205C2×C8—1181no—
206C2×C8—2362no—
207C2×C8D4×D5912yes—
208C4⋊C4—1182no—
209C22⋊C4—1182no—
210C22⋊C4—1182no—
211C42—591no—
212C42—591no—
213C42—1181no—
214C16—42no—
215C16—2362no—
216C23—21no—
217C23—592no—
218C23M16×D5911yescharacteristic
219Q8—41no—
220Q8—2361no—
221D4—41no—
222D4—42no—
223D4—4722no—
224D4—2361no—
225D4—2362no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—592no—
232C2×C4—1182no—
233C2×C4—1182no—
234C2×C4—1181no—
235C2×C4—1181no—
236C2×C4—4721no—
237C2×C4C22⋊C4×D5911yescharacteristic
238C2×C4C2×C8×D5911yescharacteristic
239C8—22no—
240C8—41no—
241C8—592no—
242C8—1181no—
243C8—2361no—
244C22—21no—
245C22—41no—
246C22—42no—
247C22—41no—
248C22—1182no—
249C22—2362no—
250C22C22⋊C8×D5911yescharacteristic
251C4—21no—
252C4—41no—
253C4—1182no—
254C4—2361no—
255C4—2361no—
256C4#99×D5911yescharacteristic
257C4#1683738311yescharacteristic
258C2—21no—
259C2—41no—
260C2—81no—
261C2—2362no—
262C2#2142241111yescharacteristic
subgroupG/H
centerC4#99×D59
commutatorC2×C4×C59C22×C8
FrattiniC2×M16C22×D59
FittingC59×#1321C2
socleC2×C59#1459
prime radical#16837948C4
cosocleC2×M16×C59C23
hypercenter#1321D59
hypocenterC59#11486
LCSG ▷ C2×C4×C59 ▷ C22×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1321

Complements

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

NH ≅ G/Ntype
2. #21425715C2semidirect
3. #21422046C2semidirect
4. #21422034C2semidirect
5. C59×#1321C2semidirect
131. #1321D59semidirect
150. C59#11486semidirect

Maximal subgroups

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

subgroupindex
#214328112
#214257152
#214220462
#214220342
C59×#13212
#1148659