Finite Group Databasebeta

← #28500041

Subgroups — #28500041

Order 15104.

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

normal subgroups only

This group has 296 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#21434680C211yesmaximal, characteristic
2#21425898C211yesmaximal, characteristic
3#21422843C211yesmaximal, characteristic
4#21422155C211yesmaximal, characteristic, complemented
5#21422153C211yesmaximal, characteristic, complemented
6#21422145C211yesmaximal, characteristic, complemented
7C59×#1362C211yesmaximal, characteristic, complemented
8#16837957—21no—
9#16837807C2211yescharacteristic
10#16837752—21no—
11#16837640—21no—
12#16837640C412yes—
13Q16×Dic59—21no—
14Q16×Dic59C412yes—
15#16837454—21no—
16C4×C8×D59C2211yescharacteristic
17C4⋊(C59⋊C16)C2211yescharacteristic
18(C2×D59)⋊C16—21no—
19#16837177—21no—
20#16837155C2211yescharacteristic
21#16837152—21no—
22C59×#494C2211yescharacteristic
23C59×C4⋊C16C2211yescharacteristic
24C59×#335C2211yescharacteristic
25C2×C8×D59C2×C411yescharacteristic
26C2×C8×D59D411yescharacteristic
27#14273727—21no—
28#14273718—41no—
29#14273707—41no—
30#14273703—22no—
31Q8×Dic59—22no—
32#14273692—41no—
33#14273685—41no—
34C42×D59D411yescharacteristic
35C2×C59⋊C16—21no—
36#14273661—21no—
37#14273661C2×C412yes—
38C4⋊(C59⋊C8)—21no—
39#14273651—22no—
40#14273651—22no—
41(C2×D59)⋊C8—41no—
42C4×C59⋊C8D411yescharacteristic
43C8×Dic59—41no—
44C8×Dic59C2×C411yescharacteristic
45C8×Dic59D411yescharacteristic
46C2×Q16×C59—21no—
47C2×Q16×C59C2×C412yes—
48C4⋊Q8×C59—21no—
49C2×C16×C59—21no—
50C4⋊C8×C59—21no—
51C4×C8×C59C2311yescharacteristic
52C2×Q8×C59—22no—
53C2×C4×D59C22⋊C411yescharacteristic
54C2×C4×D59D811yescharacteristic
55C2×C4×D59SD1611yescharacteristic
56C2×Dic118—41no—
57Q16×C59—42no—
58Q16×C59—41no—
59Q16×C59—41no—
60C2×C8×C59—41no—
61C2×C8×C59C22×C411yescharacteristic
62C2×C8×C59C2×D411yescharacteristic
63C4⋊C4×C59—41no—
64C4⋊C4×C59—41no—
65C42×C59C2×D411yescharacteristic
66(C2×D59)⋊C4—42no—
67C4⋊Dic59—22no—
68C4⋊Dic59—41no—
69C4⋊Dic59—42no—
70Dic59⋊C4—42no—
71C4×Dic59—42no—
72C4×Dic59C22⋊C411yescharacteristic
73C4×Dic59D811yescharacteristic
74C4×Dic59SD1611yescharacteristic
75C2×C59⋊C8—41no—
76C2×C59⋊C8C22⋊C411yescharacteristic
77C2×C59⋊C8C22⋊C411yescharacteristic
78C8×D59—21no—
79C8×D59—21no—
80C8×D59—41no—
81C16×C59—41no—
82C59⋊C16—41no—
83C22×D59#10211yescharacteristic
84Q8×C59—42no—
85Q8×C59—42no—
86C2×C4×C59—42no—
87C2×C4×C59C2×C22⋊C411yescharacteristic
88C2×C4×C59C2×D811yescharacteristic
89C2×C4×C59C2×SD1611yescharacteristic
90C2×Dic59—42no—
91C2×Dic59#10211yescharacteristic
92C2×Dic59#10211yescharacteristic
93C2×Dic59#10211yescharacteristic
94C4×D59—21no—
95C4×D59—21no—
96C4×D59—21no—
97C4×D59—21no—
98C4×D59—41no—
99Dic118—82no—
100C8×C59—21no—
101C8×C59—81no—
102C8×C59#13511yescharacteristic
103C8×C59#13511yescharacteristic
104C59⋊C8—21no—
105C59⋊C8—21no—
106C59⋊C8—81no—
107#11953—591nomaximal, Sylow
108C22×C59C2×#10211yescharacteristic
109C2×D59—21no—
110C2×D59—21no—
111C2×D59—21no—
112C4×C59—21no—
113C4×C59—82no—
114C4×C59#41311yescharacteristic
115C4×C59#41611yescharacteristic
116C4×C59#51011yescharacteristic
117C4×C59#51011yescharacteristic
118Dic59—21no—
119Dic59—21no—
120Dic59—21no—
121Dic59—82no—
122#3170—591no—
123#2163—591no—
124#1564—591no—
125#1365—591no—
126#1365—591no—
127#1362—591no—
128#1362D5911yescharacteristic, complemented
129C2×C59#154011yescharacteristic
130C2×C59#214111yescharacteristic
131C2×C59#214811yescharacteristic
132D59—41no—
133#533—1181no—
134#500—591no—
135#494C2×D5911yescharacteristic
136#487—1181no—
137#467—592no—
138#467—1181no—
139C4×Q16—592no—
140C4×Q16—1181no—
141#431—1181no—
142C2×C4×C8—591no—
143C4⋊C16—591no—
144C4⋊C16C2×D5911yescharacteristic
145C22⋊C16—1181no—
146#346—1181no—
147#335—591no—
148#335C2×D5911yescharacteristic
149#334—1181no—
150C59#1195311yescharacteristic, complemented, Sylow
151C2×Q16—21no—
152C2×Q16C4×D5912yes—
153C22×C8—591no—
154C22×C8—591no—
155C4⋊Q8—21no—
156C4⋊Q8—1181no—
157#125—2361no—
158C22⋊Q8—1182no—
159C22⋊Q8—2361no—
160C4×Q8—1182no—
161C4×Q8—2361no—
162#117—2361no—
163C2×C42—591no—
164C2×C16—21no—
165C2×C16—1181no—
166#107—592no—
167#107—1181no—
168C4⋊C8—21no—
169C4⋊C8—1181no—
170#103—1182no—
171#103—1182no—
172C22⋊C8—2361no—
173C4×C8—591no—
174C4×C8—591no—
175C4×C8—591no—
176C4×C8—2361no—
177C4×C8C22×D5911yescharacteristic
178C2×Q8—22no—
179C2×Q8—2361no—
180C22×C4—591no—
181C22×C4—591no—
182C22×C4—591no—
183Q16—42no—
184Q16—41no—
185Q16—41no—
186C2×C8—41no—
187C2×C8—591no—
188C2×C8—591no—
189C2×C8—1181no—
190C2×C8—1181no—
191C2×C8—2361no—
192C2×C8—2361no—
193C2×C8C2×C4×D5911yescharacteristic
194C2×C8D4×D5911yescharacteristic
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—1182no—
198C4⋊C4—2362no—
199C4⋊C4—2362no—
200C4⋊C4—2361no—
201C22⋊C4—2362no—
202C42—591no—
203C42—591no—
204C42—591no—
205C42—2362no—
206C42D4×D5911yescharacteristic
207C16—41no—
208C16—2361no—
209C23—591no—
210Q8—42no—
211Q8—42no—
212Q8—4722no—
213C2×C4—42no—
214C2×C4—591no—
215C2×C4—591no—
216C2×C4—591no—
217C2×C4—1181no—
218C2×C4—1181no—
219C2×C4—1181no—
220C2×C4—1181no—
221C2×C4—2361no—
222C2×C4—2362no—
223C2×C4C22⋊C4×D5911yescharacteristic
224C2×C4D8×D5911yescharacteristic
225C2×C4SD16×D5911yescharacteristic
226C8—21no—
227C8—81no—
228C8—1181no—
229C8—1181no—
230C8—4721no—
231C8#1427374611yescharacteristic
232C8#1427374811yescharacteristic
233C22—1181no—
234C22—1181no—
235C22—1181no—
236C22#102×D5911yescharacteristic
237C4—21no—
238C4—82no—
239C4—1181no—
240C4—1181no—
241C4—1181no—
242C4—4722no—
243C4#1683738311yescharacteristic
244C4#1683739511yescharacteristic
245C4#1683784111yescharacteristic
246C4#1683784411yescharacteristic
247C2—2361no—
248C2#2142274211yescharacteristic
249C2#2142581411yescharacteristic
250C2#2142584411yescharacteristic
subgroupG/H
centerC22#102×D59
commutatorC2×C8×C59C22×C4
FrattiniC4×C8C22×D59
FittingC59×#1362C2
socleC22×C59C2×#102
prime radicalC22×D59#102
cosocleC4×C8×C59C23
hypercenter#1362D59
hypocenterC59#11953
LCSG ▷ C2×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
4. #21422155C2semidirect
5. #21422153C2semidirect
6. #21422145C2semidirect
7. C59×#1362C2semidirect
128. #1362D59semidirect
150. C59#11953semidirect

Maximal subgroups

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

subgroupindex
#214346802
#214258982
#214228432
#214221552
#214221532
#214221452
C59×#13622
#1195359