Finite Group Databasebeta

← #28509343

Subgroups — #28509343

Order 15104.

324 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 61 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
1C2×D16×D59C211yesmaximal, characteristic
2C2×#16837239C211yesmaximal, characteristic, complemented
3#21426184C211yesmaximal, characteristic, complemented
4#21422275C211yesmaximal, characteristic, complemented
5#21422274C211yesmaximal, characteristic, complemented
6(C2×D59)⋊C32C211yesmaximal, characteristic, complemented
7C2×C59×SD64C211yesmaximal, characteristic, complemented
8C2×D8×D59—21no—
9C2×D944C2211yescharacteristic
10C2×#14273668—21no—
11D16×D59—22no—
12D16×D59—22no—
13C2×C16×D59C2211yescharacteristic
14#16837640—21no—
15#16837239—22no—
16C2×C59⋊C32C2211yescharacteristic, complemented
17#16837228C2211yescharacteristic, complemented
18#16837206—21no—
19C2×Q32×C59C2211yescharacteristic, complemented
20C2×D16×C59C2211yescharacteristic
21C59×SD64—22no—
22C2×C32×C59C2211yescharacteristic, complemented
23C2×D4×D59—41no—
24C2×D472—21no—
25C2×#11745669—41no—
26D8×D59—24no—
27D8×D59—42no—
28C2×C8×D59D411yescharacteristic
29#14273703—41no—
30D944—21no—
31D944D412yes—
32#14273668—24no—
33C2×C59⋊C16D411yescharacteristic
34C16×D59—21no—
35C16×D59—21no—
36#14273661—21no—
37#14273651—41no—
38C59⋊C32—21no—
39C2×Q16×C59—21no—
40C2×D8×C59—21no—
41Q32×C59—21no—
42Q32×C59—21no—
43D16×C59—21no—
44D16×C59D412yes—
45C2×C16×C59C2311yescharacteristic
46C32×C59—21no—
47C23×D59—81no—
48C2×Q8×C59—41no—
49C2×D4×C59—41no—
50C2×C22⋊D59—81no—
51D4×D59—44no—
52D4×D59—82no—
53C2×D236—41no—
54C2×C4×D59D811yescharacteristic
55Q16×C59—41no—
56Q16×C59—41no—
57D8×C59—22no—
58D8×C59—41no—
59C2×C8×C59C2×D411yescharacteristic
60#11745669—44no—
61(C2×D59)⋊C4—81no—
62C4⋊Dic59—41no—
63Dic59⋊C4—81no—
64C2×C59⋊C8D811yescharacteristic
65D472—22no—
66D472—41no—
67C8×D59—21no—
68C8×D59—21no—
69C16×C59C2×D411yescharacteristic
70C16×C59C2×D411yescharacteristic
71C59⋊C16—21no—
72C23×C59—81no—
73C22×D59—84no—
74C22×D59—81no—
75C22×D59—162no—
76C22×D59—162no—
77C22×D59D1611yescharacteristic
78Q8×C59—81no—
79Q8×C59—81no—
80D4×C59—42no—
81D4×C59—81no—
82C2×C4×C59—81no—
83C2×C4×C59C2×D811yescharacteristic
84C22⋊D59—84no—
85C2×Dic59—81no—
86C2×Dic59D1611yescharacteristic
87D236—42no—
88D236—81no—
89C4×D59—21no—
90C4×D59—21no—
91C8×C59#12011yescharacteristic
92C8×C59AGL(1,ℤ/8)11yescharacteristic
93C59⋊C8—21no—
94#13664—591nomaximal, Sylow
95C22×C59—82no—
96C22×C59—161no—
97C22×C59—161no—
98C22×C59C2×D1611yescharacteristic
99C2×D59—21no—
100C2×D59—21no—
101C2×D59—82no—
102C2×D59—168no—
103C2×D59—161no—
104C2×D59—161no—
105C2×D59SD6412yescomplemented
106C4×C59—161no—
107C4×C59#44711yescharacteristic
108C4×C59#50911yescharacteristic
109Dic59—21no—
110Dic59—161no—
111C22×D16—591no—
112C2×SD64—591no—
113C2×SD64D5911yescharacteristic, complemented
114#2216—591no—
115#1415—591no—
116#1415—591no—
117C22⋊C32—591no—
118C2×C59—162no—
119C2×C59#218411yescharacteristic
120C2×C59C2×SD6411yescharacteristic
121C2×C59#226311yescharacteristic
122D59—22no—
123D59—162no—
124C22×D8—1181no—
125C2×Q32C2×D5911yescharacteristic, complemented
126C2×D16—591no—
127C2×D16—1181no—
128C2×D16—1182no—
129C2×D16—1182no—
130C2×D16C2×D5911yescharacteristic
131C22×C16—591no—
132#467—1181no—
133SD64—22no—
134SD64—1182no—
135C2×C32—591no—
136C2×C32C2×D5911yescharacteristic, complemented
137#366—591no—
138#358—1181no—
139C59#1366411yescharacteristic, complemented, Sylow
140C22×D4—2361no—
141C2×Q16—21no—
142C2×D8—21no—
143C2×D8—1184no—
144C2×D8—1181no—
145C2×D8—2361no—
146C2×D8—2362no—
147C22×C8—591no—
148C22⋊Q8—2361no—
149Q32—21no—
150Q32—21no—
151D16—21no—
152D16—592no—
153D16—1184no—
154D16—1181no—
155D16C22⋊D5912yes—
156C2×C16—591no—
157C2×C16—1181no—
158C2×C16—1181no—
159C2×C16C22×D5911yescharacteristic
160#107—1181no—
161#103—2361no—
162C32—21no—
163C32—1181no—
164C24—4721no—
165C2×Q8—41no—
166C2×D4—41no—
167C2×D4—4721no—
168C2×D4—4722no—
169C2×D4—2364no—
170C2×D4—2361no—
171C22×C4—591no—
172Q16—41no—
173Q16—41no—
174D8—22no—
175D8—41no—
176D8—1182no—
177D8—2364no—
178D8—2361no—
179C2×C8—591no—
180C2×C8—1181no—
181C2×C8—1181no—
182C2×C8D4×D5911yescharacteristic
183C4⋊C4—4721no—
184C4⋊C4—2361no—
185C22⋊C4—4721no—
186C16—1181no—
187C16D4×D5911yescharacteristic
188C16C2×C22⋊D5911yescharacteristic
189C23—81no—
190C23—591no—
191C23—9442no—
192C23—9442no—
193C23—4724no—
194C23—4721no—
195Q8—81no—
196Q8—81no—
197D4—42no—
198D4—81no—
199D4—4724no—
200D4—4721no—
201D4—2362no—
202C2×C4—81no—
203C2×C4—591no—
204C2×C4—1181no—
205C2×C4—1181no—
206C2×C4—4721no—
207C2×C4D8×D5911yescharacteristic
208C8—1181no—
209C8#1427369511yescharacteristic
210C8#1427375111yescharacteristic
211C22—82no—
212C22—161no—
213C22—161no—
214C22—592no—
215C22—1181no—
216C22—1181no—
217C22—9448no—
218C22—9441no—
219C22—9441no—
220C22—4722no—
221C22D16×D5911yescharacteristic
222C4—161no—
223C4—1181no—
224C4—9441no—
225C4#1683753411yescharacteristic
226C4#1683783611yescharacteristic
227C2—162no—
228C2—1182no—
229C2—9442no—
230C2#2142600611yescharacteristic
231C2SD64×D5911yescharacteristic
232C2#2142641011yescharacteristic
subgroupG/H
centerC22D16×D59
commutator, cosocleC2×C16×C59C23
FrattiniC2×C16C22×D59
FittingC2×C59×SD64C2
socleC22×C59C2×D16
prime radicalC2×D16×D59C2
hypercenterC2×SD64D59
hypocenterC59#13664
LCSG ▷ C2×C16×C59 ▷ C8×C59 ▷ C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×SD64

Complements

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

NH ≅ G/Ntype
2. C2×#16837239C2semidirect
3. #21426184C2semidirect
4. #21422275C2semidirect
5. #21422274C2semidirect
6. (C2×D59)⋊C32C2semidirect
7. C2×C59×SD64C2semidirect
16. C2×C59⋊C32C22semidirect
17. #16837228C22semidirect
19. C2×Q32×C59C22semidirect
22. C2×C32×C59C22semidirect
105. C2×D59SD64semidirect
113. C2×SD64D59semidirect
125. C2×Q32C2×D59semidirect
136. C2×C32C2×D59semidirect
139. C59#13664semidirect

Maximal subgroups

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

subgroupindex
C2×D16×D592
C2×#168372392
#214261842
#214222752
#214222742
(C2×D59)⋊C322
C2×C59×SD642
#1366459