Finite Group Databasebeta

← #28502521

Subgroups — #28502521

Order 15104.

350 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 69 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#21435817C211yesmaximal, characteristic
2#21426324C211yesmaximal, characteristic, complemented
3#21426243C211yesmaximal, characteristic, complemented
4#21423496C211yesmaximal, characteristic, complemented
5#21422144C211yesmaximal, characteristic, complemented
6#21422052C211yesmaximal, characteristic, complemented
7C59×#1331C211yesmaximal, characteristic, complemented
8C2×#14273746—21no—
9#16838056—21no—
10#16837807C2211yescharacteristic
11#16837782C2211yescharacteristic
12#16837693—21no—
13#16837644—21no—
14#16837607—21no—
15#16837565—22no—
16C8×D236C2211yescharacteristic
17#16837229—21no—
18C4⋊(C59⋊C16)C2211yescharacteristic
19#16837207—22no—
20#16837203—21no—
21#16837150—21no—
22C4×D8×C59C2211yescharacteristic
23C4×C16×C59C2211yescharacteristic
24C59×#335C2211yescharacteristic
25C2×#11745687—22no—
26#14273746—42no—
27#14273746—42no—
28C2×Dic236—21no—
29C2×Dic236D412yes—
30C2×#11745670—22no—
31C2×C8×D59—21no—
32#14273727—21no—
33#14273727—21no—
34#14273711—42no—
35#14273707—22no—
36#14273701—41no—
37C4×D236D411yescharacteristic
38C2×C4⋊Dic59—41no—
39C2×C59⋊C16—21no—
40#14273661—21no—
41#14273661—21no—
42#14273659—41no—
43C4⋊(C59⋊C8)D411yescharacteristic
44#14273650—22no—
45#14273649—41no—
46(C2×D59)⋊C8—21no—
47C2×D8×C59D411yescharacteristic
48C4×D4×C59—21no—
49C2×C16×C59—21no—
50#107×C59D411yescharacteristic
51C4⋊C8×C59—21no—
52#102×C59—21no—
53C4×C8×C59C2311yescharacteristic
54C2×D4×C59—21no—
55C22×C4×C59—41no—
56C2×C22⋊D59—42no—
57C22×Dic59—42no—
58#11745687—44no—
59#11745687—82no—
60C2×D236D811yescharacteristic
61C2×C4×D59—21no—
62C2×Dic118—22no—
63C2×Dic118—41no—
64D8×C59—21no—
65D8×C59SD1612yes—
66C2×C8×C59—41no—
67C2×C8×C59C2×D411yescharacteristic
68C2×C8×C59C2×D411yescharacteristic
69C4⋊C4×C59—21no—
70C22⋊C4×C59—41no—
71C42×C59C2×D411yescharacteristic
72#11745670—44no—
73(C2×D59)⋊C4—21no—
74(C2×D59)⋊C4—42no—
75C4⋊Dic59—42no—
76C4⋊Dic59—41no—
77C4⋊Dic59—41no—
78C4⋊Dic59—41no—
79C4⋊Dic59—41no—
80C4⋊Dic59D811yescharacteristic
81C2×C59⋊C8—21no—
82Dic236—42no—
83Dic236—41no—
84Dic236—41no—
85C8×D59—41no—
86C8×D59—41no—
87C16×C59—41no—
88C59⋊C16—41no—
89C23×C59—41no—
90C22×D59—21no—
91D4×C59—22no—
92D4×C59—41no—
93C2×C4×C59—41no—
94C2×C4×C59—81no—
95C2×C4×C59—81no—
96C2×C4×C59#12011yescharacteristic
97C2×C4×C59C2×D811yescharacteristic
98C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
99C22⋊D59—84no—
100C2×Dic59—21no—
101C2×Dic59—42no—
102C2×Dic59—84no—
103C2×Dic59—81no—
104D236—21no—
105D236—21no—
106C4×D59—41no—
107C4×D59—41no—
108Dic118—42no—
109Dic118—42no—
110Dic118—161no—
111C8×C59—21no—
112C8×C59—81no—
113C8×C59C2×SD1611yescharacteristic
114C8×C59AGL(1,ℤ/8)11yescharacteristic
115C59⋊C8—41no—
116#12375—591nomaximal, Sylow
117C22×C59—42no—
118C22×C59—81no—
119C22×C59—81no—
120C22×C59#44711yescharacteristic
121C2×D59—41no—
122C2×D59—41no—
123C2×D59—41no—
124C4×C59—21no—
125C4×C59—81no—
126C4×C59#45011yescharacteristic
127C4×C59#45511yescharacteristic
128C4×C59#50811yescharacteristic
129C4×C59#51011yescharacteristic
130Dic59—41no—
131Dic59—82no—
132Dic59—161no—
133#3283—591no—
134#2241—591no—
135#2226—591no—
136#1675—591no—
137#1362—591no—
138#1331D5911yescharacteristic, complemented
139D4⋊C16—591no—
140C2×C59—82no—
141C2×C59#163911yescharacteristic
142C2×C59#218911yescharacteristic
143C2×C59#219311yescharacteristic
144D59—81no—
145C2×#135—1181no—
146#547—1181no—
147#500—591no—
148#494—591no—
149#476—1181no—
150#467—1181no—
151#461—1181no—
152#452—1182no—
153C4×D8C2×D5911yescharacteristic
154C8×D4—591no—
155#367—1181no—
156C4⋊C16—591no—
157#358—1182no—
158#357—1181no—
159C4×C16C2×D5911yescharacteristic
160#335C2×D5911yescharacteristic
161#333—1181no—
162C59#1237511yescharacteristic, complemented, Sylow
163C2×#41—1182no—
164#135—2362no—
165#135—2362no—
166C2×Q16—592no—
167C2×Q16—1181no—
168C2×SD16—1182no—
169C2×D8D23611yescharacteristic
170C22×C8—1181no—
171C4⋊Q8—1181no—
172C4⋊Q8—1181no—
173#123—2362no—
174C22⋊Q8—1182no—
175#121—2361no—
176C4×D4—21no—
177C4×D4—591no—
178C2×C4⋊C4—2361no—
179C2×C16—21no—
180C2×C16—1181no—
181#107—1181no—
182#107—1181no—
183#107D23611yescharacteristic
184#106—2361no—
185C4⋊C8—21no—
186C4⋊C8—591no—
187#103—1182no—
188#102—21no—
189#102—2361no—
190C22⋊C8—1181no—
191C4×C8C22×D5911yescharacteristic
192#41—4722no—
193#41—2364no—
194C2×Q8—1182no—
195C2×Q8—2361no—
196C2×D4—21no—
197C2×D4—591no—
198C2×D4—2362no—
199C22×C4—41no—
200C22×C4—1181no—
201C22×C4—2362no—
202Q16—2362no—
203Q16—2361no—
204Q16—2361no—
205SD16—2364no—
206D8—21no—
207D8#1174566012yes—
208C2×C8—41no—
209C2×C8—1181no—
210C2×C8—2361no—
211C2×C8—2361no—
212C2×C8C2×D23611yescharacteristic
213C2×C8D4×D5911yescharacteristic
214C4⋊C4—21no—
215C4⋊C4—591no—
216C4⋊C4—2362no—
217C4⋊C4—2361no—
218C4⋊C4—2361no—
219C4⋊C4—2361no—
220C4⋊C4—2361no—
221C22⋊C4—41no—
222C22⋊C4—1181no—
223C22⋊C4—2362no—
224C42D4×D5911yescharacteristic
225C16—41no—
226C16—2361no—
227C23—41no—
228C23—1181no—
229Q8—9441no—
230Q8—2362no—
231Q8—2362no—
232D4—22no—
233D4—41no—
234D4—1181no—
235D4—1181no—
236D4—4724no—
237C2×C4—41no—
238C2×C4—81no—
239C2×C4—81no—
240C2×C4—1181no—
241C2×C4—4721no—
242C2×C4—4724no—
243C2×C4—2362no—
244C2×C4—2361no—
245C2×C4—2361no—
246C2×C4#1427369711yescharacteristic
247C2×C4D8×D5911yescharacteristic
248C2×C4#1427375411yescharacteristic
249C8—21no—
250C8—81no—
251C8—2361no—
252C8C2×#1174566011yescharacteristic
253C8#1427375511yescharacteristic
254C22—42no—
255C22—81no—
256C22—81no—
257C22—2361no—
258C22—2361no—
259C22—2361no—
260C22#1683753711yescharacteristic
261C4—21no—
262C4—81no—
263C4—9441no—
264C4—4722no—
265C4—2361no—
266C4#1683755511yescharacteristic
267C4#1683758111yescharacteristic
268C4#1683783111yescharacteristic
269C4#1683784111yescharacteristic
270C2—82no—
271C2—4721no—
272C2#2142327811yescharacteristic
273C2#2142603711yescharacteristic
274C2#2142606111yescharacteristic
subgroupG/H
centerC22#16837537
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#1331C2
socleC22×C59#447
prime radical#21435817C2
hypercenter#1331D59
hypocenterC59#12375
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1331

Complements

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

NH ≅ G/Ntype
2. #21426324C2semidirect
3. #21426243C2semidirect
4. #21423496C2semidirect
5. #21422144C2semidirect
6. #21422052C2semidirect
7. C59×#1331C2semidirect
138. #1331D59semidirect
162. C59#12375semidirect

Maximal subgroups

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

subgroupindex
#214358172
#214263242
#214262432
#214234962
#214221442
#214220522
C59×#13312
#1237559