Finite Group Databasebeta

← #28502218

Subgroups — #28502218

Order 15104.

328 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 71 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#21426353C211yesmaximal, characteristic, complemented
2#21426131C211yesmaximal, characteristic
3#21423456C211yesmaximal, characteristic, complemented
4#21422145C211yesmaximal, characteristic, complemented
5Q8⋊(C59⋊C16)C211yesmaximal, characteristic, complemented
6#21422057C211yesmaximal, characteristic, complemented
7C59×#3393C211yesmaximal, characteristic, complemented
8C2×#14273670—22no—
9C2×#14273668—21no—
10#16837779C2211yescharacteristic
11#16837602—21no—
12C4⋊(C59⋊C16)C2211yescharacteristic
13#16837202—21no—
14C4×C59⋊C16C2211yescharacteristic
15#16837155C2211yescharacteristic
16#16837145—21no—
17C59×#556—21no—
18C59×#495—21no—
19C59×#494C2211yescharacteristic
20C59×#464—22no—
21C59×#461—21no—
22C8×Q8×C59C2211yescharacteristic
23C4×Q16×C59—21no—
24C4×D8×C59C2211yescharacteristic
25C2×#11745671—41no—
26C2×D472—21no—
27C2×D472—21no—
28#14273724—21no—
29#14273670—44no—
30#14273668—42no—
31C2×C59⋊C16—21no—
32C2×C59⋊C16—21no—
33C4⋊(C59⋊C8)—21no—
34#14273647—41no—
35C2×Q16×C59—21no—
36C2×Q16×C59D412yes—
37C2×SD16×C59—22no—
38C2×D8×C59D411yescharacteristic
39C4⋊Q8×C59—21no—
40#126×C59—42no—
41#124×C59—22no—
42C22⋊Q8×C59—41no—
43C4×Q8×C59—22no—
44C4×Q8×C59D411yescharacteristic
45C4×D4×C59—21no—
46#117×C59—41no—
47#107×C59D411yescharacteristic
48C4⋊C8×C59—21no—
49C4⋊C8×C59D411yescharacteristic
50#103×C59—22no—
51#102×C59—21no—
52C4×C8×C59—21no—
53C4×C8×C59C2311yescharacteristic
54C2×Q8×C59—22no—
55C2×Q8×C59D811yescharacteristic
56C2×D4×C59—21no—
57C22×C4×C59—41no—
58C2×D236—41no—
59C2×D236—41no—
60C2×D236—41no—
61Q16×C59—42no—
62Q16×C59—41no—
63Q16×C59—41no—
64SD16×C59—82no—
65D8×C59—21no—
66D8×C59—21no—
67C2×C8×C59—21no—
68C2×C8×C59C2×D411yescharacteristic
69C2×C8×C59C2×D411yescharacteristic
70C4⋊C4×C59—21no—
71C4⋊C4×C59—21no—
72C4⋊C4×C59—42no—
73C4⋊C4×C59—41no—
74C4⋊C4×C59—41no—
75C4⋊C4×C59D811yescharacteristic
76C22⋊C4×C59—42no—
77C22⋊C4×C59—42no—
78C22⋊C4×C59—41no—
79C42×C59—21no—
80C42×C59—42no—
81C42×C59C2×D411yescharacteristic
82#11745671—82no—
83C2×C59⋊C8—41no—
84D472—42no—
85D472—41no—
86D472—41no—
87C59⋊C16—22no—
88C59⋊C16—41no—
89C23×C59—41no—
90C22×D59—81no—
91Q8×C59—21no—
92Q8×C59—21no—
93Q8×C59—42no—
94Q8×C59—42no—
95D4×C59—41no—
96D4×C59—41no—
97C2×C4×C59—21no—
98C2×C4×C59—21no—
99C2×C4×C59—41no—
100C2×C4×C59—42no—
101C2×C4×C59—81no—
102C2×C4×C59—81no—
103C2×C4×C59#12011yescharacteristic
104C2×C4×C59C2×D811yescharacteristic
105C2×C4×C59AGL(1,ℤ/8)11yescharacteristic
106D236—81no—
107D236—81no—
108D236—81no—
109D236—81no—
110D236—161no—
111C8×C59—21no—
112C8×C59—41no—
113C8×C59#13511yescharacteristic
114C8×C59AGL(1,ℤ/8)11yescharacteristic
115C59⋊C8—81no—
116#12331—591nomaximal, Sylow
117C22×C59—81no—
118C22×C59—81no—
119C22×C59—81no—
120C22×C59#44711yescharacteristic
121C2×D59—161no—
122C2×D59—161no—
123C2×D59—161no—
124C4×C59—21no—
125C4×C59—41no—
126C4×C59—41no—
127C4×C59—82no—
128C4×C59—81no—
129C4×C59#44911yescharacteristic
130C4×C59#45311yescharacteristic
131C4×C59#50811yescharacteristic
132C4×C59#50911yescharacteristic
133#3393D5911yescharacteristic, complemented
134#2248—591no—
135#2206—591no—
136#1668—591no—
137#1362—591no—
138Q8⋊C16—591no—
139#1331—591no—
140C2×C59—161no—
141C2×C59#162111yescharacteristic
142C2×C59#218611yescharacteristic
143C2×C59#219311yescharacteristic
144D59—321no—
145#556—21no—
146C2×SD32—1182no—
147C2×D16—1181no—
148#495—21no—
149#494C2×D5911yescharacteristic
150#493—591no—
151#464—22no—
152#461—21no—
153#460—1181no—
154C8×Q8C2×D5911yescharacteristic
155C4×Q16—21no—
156C4×D8C2×D5911yescharacteristic
157C4⋊C16—591no—
158#357—1181no—
159C4×C16—591no—
160#335—591no—
161#331—1181no—
162C59#1233111yescharacteristic, complemented, Sylow
163C2×Q16—21no—
164C2×Q16C22⋊D5912yes—
165C2×SD16—22no—
166C2×SD16—2361no—
167C2×D8—1181no—
168C2×D8—1181no—
169C2×D8C22⋊D5911yescharacteristic
170C4⋊Q8—21no—
171#127—1181no—
172#126—42no—
173#124—22no—
174C22⋊Q8—41no—
175C4×Q8—22no—
176C4×Q8C22⋊D5911yescharacteristic
177C4×D4—21no—
178#117—41no—
179SD32—2364no—
180D16—2362no—
181C2×C16—1181no—
182C2×C16—1181no—
183#107C22⋊D5911yescharacteristic
184C4⋊C8—21no—
185C4⋊C8—1181no—
186C4⋊C8C22⋊D5911yescharacteristic
187#103—22no—
188#102—21no—
189#102—2361no—
190C4×C8—21no—
191C4×C8C22×D5911yescharacteristic
192C2×Q8—22no—
193C2×Q8#1174566911yescharacteristic
194C2×D4—21no—
195C2×D4—2361no—
196C2×D4—2361no—
197C2×D4—2361no—
198C22×C4—41no—
199Q16—42no—
200Q16—41no—
201Q16—41no—
202SD16—82no—
203SD16—4722no—
204D8—21no—
205D8—21no—
206D8—2362no—
207D8—2361no—
208D8—2361no—
209C2×C8—21no—
210C2×C8—2361no—
211C2×C8C2×C22⋊D5911yescharacteristic
212C2×C8C2×C22⋊D5911yescharacteristic
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4#1174566911yescharacteristic
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—41no—
222C42—21no—
223C42—42no—
224C42C2×C22⋊D5911yescharacteristic
225C16—1182no—
226C16—2361no—
227C23—41no—
228C23—4721no—
229Q8—21no—
230Q8—21no—
231Q8—42no—
232Q8—42no—
233D4—41no—
234D4—41no—
235D4—9441no—
236D4—4721no—
237D4—4721no—
238D4—4721no—
239D4—4721no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—81no—
245C2×C4—81no—
246C2×C4#1427369611yescharacteristic
247C2×C4C2×#1174566911yescharacteristic
248C2×C4#1427375311yescharacteristic
249C8—21no—
250C8—41no—
251C8—4721no—
252C8#1427374711yescharacteristic
253C8#1427375211yescharacteristic
254C22—81no—
255C22—81no—
256C22—81no—
257C22—9441no—
258C22—9441no—
259C22—9441no—
260C22#1683753611yescharacteristic
261C4—21no—
262C4—41no—
263C4—41no—
264C4—82no—
265C4—81no—
266C4#1683754711yescharacteristic
267C4#1683757011yescharacteristic
268C4#1683783211yescharacteristic
269C4#1683783711yescharacteristic
270C2—161no—
271C2—18881no—
272C2#2142316111yescharacteristic
273C2#2142601911yescharacteristic
274C2#2142606011yescharacteristic
subgroupG/H
centerC22#16837536
commutator, cosocleC4×C8×C59C23
FrattiniC4×C8C22×D59
FittingC59×#3393C2
socleC22×C59#447
prime radical#21426131C2
hypercenter#3393D59
hypocenterC59#12331
LCSG ▷ C4×C8×C59 ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3393

Complements

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

NH ≅ G/Ntype
1. #21426353C2semidirect
3. #21423456C2semidirect
4. #21422145C2semidirect
5. Q8⋊(C59⋊C16)C2semidirect
6. #21422057C2semidirect
7. C59×#3393C2semidirect
133. #3393D59semidirect
162. C59#12331semidirect

Maximal subgroups

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

subgroupindex
#214263532
#214261312
#214234562
#214221452
Q8⋊(C59⋊C16)2
#214220572
C59×#33932
#1233159